microsoft word reviewer acknowledgement 2021.docx * if you were a reviewer for a manuscript during 2021 and your name and affiliation is not listed, please contact robert m. capraro at rcapraro@tamu.edu. it is our sincere intent to recognize all those who have generously given of their expertise and time to the success of jume. journal of urban mathematics education december 2021, vol. 14, no. 2, p. 117 ©jume. https://journals.tdl.org/jume reviewer acknowledgment 2021* reda abuelwan, sultan qaboos university tufan adiguzel, ozyegin university joel amidon, university of mississippi tonya bartell, michigan state university dan battey, rutgers university mary margaret capraro, texas a&m university matthew g. caputo, harmony school of science–high marta civil, university of arizona sarah coppersmith, university of missouri–st. louis sencer corlu, bahçeşehir university jahneille cunningham, university of california, los angeles elizabeth de freitas, manchester metropolitan university gregory downing, north carolina central university melva r. grant, old dominion university brian r. lawler, kennesaw state university danny bernard martin, university of illinois at chicago alesia mickle moldavan, fordham university terrell morton, university of missouri vern nelson, university of minnesota kristie jones newton, temple university francis m. nzuki, stockton university nickolaus ortiz, georgia state university ayanna perry, knowles teacher initiative diana piccolo, missouri state university judtih quander, university of houston–downtown mary candace raygoza, saint mary’s college of california madeline ortiz rodriguez, university of florida ksenija simic-muller, pacific lutheran university nathalie sinclair, simon fraser university carmen georgina thomas-browne, university of pittsburgh brian tweed, massey university jamaal rashad young, texas a&m university journal of urban mathematics education december 2017, vol. 10, no. 2, p. 146 ©jume. http://education.gsu.edu/jume journal of urban mathematics education vol. 10, no. 2 reviewer acknowledgment january 2016–december 2017* glenda anthony, massey university lorraine baron tonya bartell, michigan state university dan battey, rutgers university nermin bayazit, fitchburg state university john bragelman, university of illinois at chicago susan cannon, georgia state university robert capraro, texas a&m university theodore chao, the ohio state university ervin china, georgia state university marta civil, university of arizona stephanie cross, georgia state university corey drake, michigan state university anthony fernandes, university of north carolina charlotte mary foote, queens college, cuny toya jones frank, george mason university maisie gholson, university of michigan susan gregson, university of cincinnati eric gutstein, university of illinois at chicago jessica hale, georgia state university victoria hand, university of colorado shandy hauk, wested crystal hill, indian university-purdue university indianapolis keith howard, chapman university signe kastberg, purdue university rick kitchen, university of wyoming gregory larnell, university of illinois at chicago brian lawler, kennesaw state university jacqueline leonard, university of wyoming danny martin, university of illinois at chicago jasmine mathis, georgia state university percival matthews, university of wisconsinmadison maxine mckinney de royston, university of wisconsin-madison alesia mickle moldavan, georgia state university eduardo mosqueda, university of california, santa cruz sarah oppland-cordell, northern illinois university alexandre pais, manchester metropolitan university angela lopez pedrana, university of houston downtown elijah porter, georgia state university arthur powell, rutgers university-newark mary raygoza, saint mary’s college laurie rubel, brooklyn college, cuny tesha sengupta-irving, vanderbilt university james telese, university of texas at brownsville luz valoyes-chávez, university of chile, santiago anita wager, vanderbilt university erica walker, teachers college columbia university craig willey, indiana university-purdue university indianapolis morgin jones williams, university of south carolina beaufort maria zavala, san francisco state university * if you were a reviewer for a manuscript during january 2016–december 2017 and your name and affiliation is not listed, please contact david stinson at dstinson@gsu.edu. it is our sincere intent to recognize all those who generously give of their time and expertise to the continued success of jume. http://education.gsu.edu/jume mailto:dstinson@gsu.edu microsoft word reviewer acknowledgement 2020.docx journal of urban mathematics education december 2020, vol. 13, no. 2, p. 87 ©jume. https://journals.tdl.org/jume reviewer acknowledgment 2020* nathan alexander, university of san francisco joel amidon, university of mississippi david barnes, washington university in st. louis tonya bartell, michigan state university dan battey, rutgers university joanne becker, san jose state university clare bell, university of missouri – kansas city alexandra bella, university of british columbia robert berry iii, university of virginia viveka borum, spelman college jonathan bostic, bowling green state university angela brown, piedmont college theodore chao, ohio state university marta civil, university of arizona david cook, texas tech university gregory downing, north carolina central university gheorghita faitar, d’youville college imani goffney, university of maryland susan gregson, university of cincinnati barbro grevholm, universitetet i agder shandy hauk, san francisco state university jodie hunter, massey university kara jackson, university of washington jennifer jones, farleigh dickinson university nicole joseph, vanderbilt university joanne kantner, joliet junior college richard kitchen, university of wyoming matt larson, national council of teachers of mathematics brian lawler, kennesaw state university luis leyva, vanderbilt university danny martin, university of illinois – chicago oren mcclain, loyola university maryland ebony mcgee, vanderbilt university hazel mckenna, utah valley university alexander moore, virginia polytechnic institute and state university eduardo mosqueda, university of california – santa cruz charles munter, university of missouri vern nelson, university of minnesota – twin cities kristie newton, temple university sheila orr, michigan state university ayanna perry, knowles teacher initiative diana piccolo, missouri state university elijah porter, georgia state university judith quander, university of houston – downtown mary raygoza, st. mary’s college warren roane, humble independent school district madeline ortiz-rodriguez, university of florida walter secada, university of miami james sheldon, university of arizona ksenija simic-muller, pacific lutheran university megan staples, university of connecticut dorian stoilescu, western sydney university marilyn strutchens, auburn university dalene swanson, university of stirling brian tweed, massey university luz valoyes-chávez, universidad de chile katherine vela, utah state university trevor warburton, utah valley university craig willey, indiana university – perdue university indiana john williams iii, texas a&m university emily yanisko, urban teachers baltimore * if you were a reviewer for a manuscript during 2020 and your name and affiliation is not listed, please contact robert m. capraro at rcapraro@tamu.edu. it is our sincere intent to recognize all those who have generously given of their expertise and time to the success of jume. microsoft word 376-article text no abstract-1774-1-18-20191217 jc.docx journal of urban mathematics education december 2019, vol. 12, no. 1, pp. 15–18 ©jume. https://journals.tdl.org/jume chance w. lewis is the carol grotnes belk distinguished professor of urban education in the cato college of education at the university of north carolina at charlotte, 320 e. 9th center, unc charlotte center city campus, charlotte, nc 28202; e-mail: chance.lewis@uncc.edu. his research interests are centered around the improvement of academic achievement for students of color, particularly african american students. editorial the decision: do we really want urban students to achieve in mathematics? chance w. lewis university of north carolina at charlotte he date of july 7, 2010 will forever be etched into the memory of national basketball association (nba) fans. this date was the day that millions of sports fans around the world watched the highly anticipated 1-hour television special entitled the decision, focused on which team lebron james would select for the next phase of his career. when he uttered the words, “i’m going to take my talents to south beach,” the landscape of the nba changed. reflecting on this monumental moment almost 10 years later, i was proud of his ability to make a decision to join his colleagues/friends and pursue the ultimate prize of an nba championship (of course, he has won 3 championships after making this decision). while this scholarly venue is not the best space to debate whether he made the right decision, the important item to consider is the fact that he made a decision. as we, the new editorial team of the journal of urban mathematics education (jume), enter a new era with this scholarly journal, i’m honored to write this editorial to set a vision of the urgency for the work that needs to be addressed specifically in urban mathematics education. this editorial will challenge each of us to consider if and how we are going to maximize our positions within the academy to improve the mathematical identity and agency of urban students. i want to be crystal clear that anytime i note the terms urban students, urban education, or urban environments, i am focusing on the broader population of urban students that are the aim of the mission of this journal. because urban spaces have evolved through gentrification and other social changes, contributors to this journal have exciting opportunities to influence the mathematical identity and agency that focuses not only on race but also gender, english language learners, and disability within the urban context of schooling. based on my years of scholarly contributions, i do provide specific reference to black and brown students to raise our sense of urgency, because these are the students that have been the most underserved in key areas (landsman & lewis, 2011; national center for education statistics, 2017; toldson & lewis, 2012) such as, but not limited to, the following: (1) a lack of financial resources to fully support their schooling experience; (2) increased likelihood to have either unqualified and/or t lewis editorial journal of urban mathematics education vol. 12, no. 1 16 underqualified mathematics teachers; (3) limited access to higher-level mathematics classes (i.e., calculus,); and (4) a digital divide in technological and internet access in the school, home, and community that has ultimately led to underachievement and decreased access to college and career opportunities (toldson & lewis, 2012). for those of us who have found entry into the “ivory tower” better known as the academy (with all the rights and privileges of our positions), we must also make a decision about whether our work will lead to positive effects in mathematics learning in urban settings or if we are seeking just to add to our curriculum vitae for promotion within the academy––that is a question about your legacy! when we ultimately decided to take our talents to the academy to pursue scholarly investigations that would make a positive difference in mathematics education, did we have urban students in mind? for those that have scholarly research agendas focused on black and brown students, did we make a conscious decision about addressing their mathematical identity and agency in these educational environments? remember, our answers to these questions have direct implications on the lives of these students, their career opportunities, and even national security. given that we have made this decision to influence research, policy, and practice, we must collectively make sure we are not just focused on being comfortable in the academy. although higher education is known for bestowing many lofty titles and positions, we, the editors, are looking for contributions to jume that truly “move the needle” for all urban students. a reminder – urban education has reached your neighborhood as you consider the decision you have made and/or will make regarding your research, i want to challenge you to look into your own neighborhood, at its schools and school districts, to see why your work is needed now more than ever before. milner (2012) has eloquently noted the evolving typology of urban education (i.e., urban intensive, urban emergent, and urban characteristics). within this context, urban mathematics education is not serving students well, as can be inferred from the mathematics achievement of black and brown students across major urban centers in the united states (see table 1). table 1 illustrates the mathematics academic proficiency at grade 4 and grade 8 on the national assessment of educational progress (national center of education statistics, 2017). at each of these grade levels, we see the four major categories of below basic, at basic, at proficient, and at advanced, which highlight mathematics achievement in urban mathematics education at these grade levels. instead of “gapgazing” (gutiérrez, 2008; young, young, & capraro, 2018), we must take a deeper look at what is occurring in urban (tate, 2008) mathematics education in classrooms across the united states (martin & larnell, 2013; stinson, 2014). table 1 clearly documents the urgency in urban mathematics education in these major urban centers, lewis editorial journal of urban mathematics education vol. 12, no. 1 17 with detroit leading the way with 97% of black students in grade 4 and 95% of hispanic students in grade 8 not reaching proficiency in mathematics––this documents the urgency of the work that is needed to increase the mathematical identity and agency of not only black and brown students in urban contexts but also all urban students. although these may be the data on urban centers, the same trends hold true for the neighborhood near you. taken holistically, i have a question to pose: what type of decision do we need to make to maximize our positions in order to make a positive difference for urban students in their mathematics identity and agency? table 1 percentage of black and hispanic students at each achievement level on naep assessments at grade 4 and grade 8 mathematics in 2017 for selected public urban school districts grade 4 grade 8 urban district race below basic at basic at proficient at advanced below basic at basic at proficient at advanced atlanta black 42 44 13 1 60 30 9 1 hispanic 35 46 18 * 53 28 15 5 baltimore black 52 37 10 * 65 28 7 1 hispanic 38 44 17 1 56 27 13 3 boston black 33 47 19 1 52 32 13 2 hispanic 30 48 20 3 45 35 16 4 charlotte black 27 45 26 3 43 32 20 5 hispanic 20 44 31 4 38 37 19 5 chicago black 38 43 17 2 53 36 9 2 hispanic 27 46 24 3 36 38 21 5 dallas black 35 46 18 57 31 12 1 hispanic 20 47 29 4 45 36 16 3 detroit black 73 23 3 * 53 40 7 hispanic 64 30 6 * 74 21 4 1 washington, dc black 43 38 16 2 64 27 8 1 hispanic 30 42 23 5 50 32 13 5 los angeles black 47 36 16 1 62 26 10 2 hispanic 45 42 12 1 53 34 11 1 miami black 17 53 27 4 57 34 8 1 hispanic 11 44 39 6 36 38 20 5 source: u.s. department of education, institute of education sciences, national center for education statistics, national assessment of educational progress, 2017 mathematics assessment. = not enough students to equal 1 percent * = did not meet naep sample requirements ** = urban center is defined as 1,000 or more residents per square mile maximizing the research potential of jume one critical area we must pursue if we are going to make positive change is to maximize the research potential of jume. because jume is a scholarly refereed journal with a “mission to foster a transformative global academic space in mathematics that embraces critical research, emancipatory pedagogy, and scholarship of lewis editorial journal of urban mathematics education vol. 12, no. 1 18 engagement in urban communities,” we must submit our highest quality empirical research for consideration. if we send less than our best, we demonstrate to the scholarly community that we, the producers and consumers of research in jume, are not fulfilling the mission of this journal. we must view jume as a clearinghouse of highquality research pertaining to improving mathematical identity and agency for urban students. we know the issues in urban environments have been well documented over the last 50 years; however, we are seeking contributions that fulfill our mission. conclusion in closing, i must say that i am excited about the possibilities that jume can provide via research. i must remind us that this scholarly journal will only be as great as we allow it to be. now is the time for us to carry the torch to make sure we adequately address the urban context in our work in mathematics education. references gutiérrez, r. (2008). a “gap-gazing” fetish in mathematics education? problematizing research on the achievement gap. journal for research in mathematics education, 39(4), 357–364. landsman, j. a., & lewis, c. w. (eds.). (2011). white teachers/diverse classrooms: creating inclusive schools, building on students’ diversity and providing true educational equity (2nd ed.). sterling, va: stylus. martin, d. b., & larnell, g. v. (2013). urban mathematics education. in h. r. milner & k. lomotey (eds.), handbook of urban education (pp. 373–393). new york, ny: routledge. milner, h. r. (2012). but what is urban education? urban education, 47(3), 556–561. national center for education statistics. (2017). the nation’s report card: 2017 mathematics assessment. washington, dc: u.s. government publishing office. stinson, d. w. (2014). urban mathematics education. in s. lerman (ed.), encyclopedia of mathematics education (pp. 631–632). dordrecht, the netherlands: springer. tate, w. f. (2008). putting the “urban” in mathematics education scholarship. journal of urban mathematics education, 1(1), 5–9. retrieved from https://jume-ojs-tamu.tdl.org/jume/index.php/jume/article/view/19 toldson, i., & lewis, c. (2012). challenging the status quo: academic success among school-age african american males. washington, dc: congressional black caucus foundation. young, j. l., young, y. r., & capraro, r. m. (2018). gazing past the gaps: a growth-based assessment of the mathematics achievement of black girls. the urban review, 50(1), 156–176. copyright: © 2019 lewis. this is an open access article distributed under the terms of a creative commons attribution-noncommercial-sharealike 4.0 international license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. journal of urban mathematics education december 2018, vol. 11, no. 1&2, pp. 7–10 ©jume. http://education.gsu.edu/jume lou edward matthews lou edward matthews is an assistant professor of mathematics education in the college of education at georgia state university, 30 pryor street, atlanta, ga, 30303; e-mail: lmatthews@gsu.edu. his research focuses on how teachers incorporate visions of culturally relevant teaching into practice, as well as with the critical examination of mathematics reform ideology, black masculinity and schooling, and teacher experiences in the context of reform. dr. matthews is currently serving as the 2007– 2009 president of the benjamin banneker association, a national non-profit organization advocating excellence in mathematics for black students. dr. matthews is also co-founder and currently inaugural editor-inchief of the journal of urban mathematics education. editorial illuminating urban excellence: a movement of change within mathematics education1 lou edward matthews georgia state university ver a year ago my colleagues and i embarked on an unchartered quest to “open up” within the mathematics education community a scholarly space that could honor—not marginalize—the professional work in the domain we characterized as urban. we sought to open up a space in mathematics education that would honor and enrich the work in this domain which had become central to our endeavors as reformers. admittedly, with only one tenured professor in our group of six, the catalyst for this risky endeavor laid in our own frustrations within the academy to gain access to, and collectively synthesize, the complexities of mathematics reform taking place in urban schools. our initial “conversational” surveys of the landscape of mathematics education discourse in the fall of 2007 (e.g., “top-tier” mathematics education journals and research conference offerings) revealed a suspicious absence of urban scholarship. in addition, access to existing voices outside of the community has been significantly restricted with limited access to the eric database, the proliferation of pay-to-read scholarship, and narrowly defined notions of what counts as “scientific” research. after months of painstaking deliberation, our efforts culminated in the launching of this journal, the journal of urban mathematics education (jume), on january 15, 2008 with a national call for manuscripts and our home webpage: http://education.gsu.edu/jume. the following mission statement heralds this initiative: to foster a transformative global academic space in mathematics that embraces critical research, emancipatory pedagogy, and scholarship of engagement in urban communities. as we met to frame the components of this statement, several tensions surfaced, centering around three crucial questions that i wish to discuss: (1) how should we 1 originally published in the inaugural december 2008 issue of the journal of urban mathematics education (jume); see http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/20/9. o http://education.gsu.edu/jume mailto:lmatthews@gsu.edu http://education.gsu.edu/jume http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/20/9 matthews editorial journal of urban mathematics education vol. 11, no. 1&2 8 define “urban” in mathematics education? (2) how should we orient ourselves toward work in the urban domain? and (3) how should we give “voice” to the complex dynamics of change within the urban domain? when we say urban while engaging in our discussion around what happens in the urban domain, we often found ourselves at odds as to how to define the domain. in one sense, it was obvious that urban is used to define a particular geographical space, for example “metro atlanta.” yet, geography alone was not enough to draw attention to the various complexities regarding the racial and ethnic makeup of schools and communities, degree of economic hardships, and neighborhood and community traditions. by default, many educators forgo delineating these complexities, focusing only on specific groups within urban schools and communities. in this negligent manner, the term urban is often relegated to an umbrella term used indiscriminately to denote african american, hispanic, immigrant, or low-income students. furthermore, given that mathematics “achievement gaps” are popularly depicted in terms of race, ethnicity, and/or income, the term urban is often utilized as an all encompassing deficit term. not wanting to continue the status quo exercised in practice, we settled on the following definition of the urban domain, which will no doubt undergo extensive, ongoing reflection and refinement: here, the view of the urban domain extends beyond the geographical context, into the lives of people within the multitude of cultural, social, and political spaces in which mathematics teaching and learning takes place. with this definition, we set a standard that all scholarship, which has the urban domain as its primary focus, should give a thorough accounting of the complexities of cultural, social, and political elements through which mathematics teaching and learning and mathematics education reform is experienced. excellence as a frame of reference the focus of jume is aligned with a counter-trend of equity-focused organizations (e.g., benjamin banneker association’s national leadership summit on the mathematics education excellence of black children) to replace the standard practice of “gap-gazing” as a catalyst for action with what we call “illuminating excellence.” why do this? certainly, this is not to ignore the strained efforts of educators working to meet yearly nclb progress goals, counter budgets cutbacks, and develop and implement new mathematics standards for teaching and content quality. matthews editorial journal of urban mathematics education vol. 11, no. 1&2 9 these concerns are not to be ignored. we are aware that within georgia there is growing concern that the implementation of new standards in mathematics has contributed to an alarming number of students who have not passed recent standardized tests in mathematics at the upper elementary and middle grades (georgia department of education, 2008; strepp, 2008a, 2008b). we were equally concerned about the invisibility of exemplary teachers and administrative practice facilitating mathematics success like the kind we encounter in our everyday duties. an example of this kind of scholarship that positions excellence as a starting point from which to examine urban mathematics reform is gutiérrez’s (2000) “urban youth in mathematics: unpacking the success of one math department.” we have all borne witness to the underutilized, hidden wisdom found in some of our partner schools and classrooms. while this wisdom of teaching, curriculum management, culturally relevant pedagogy, collaboration, and local action has provided powerful counter narratives to the diatribe on urban achievement in mathematics, exclusion of such narratives in mainstream mathematics education journals relegates them to largely “asystemic” to the community of mathematics education reformers. the reporting of excellence within the urban domain has been suspiciously underreported in top-tier mathematics education journals. the existence of this work outside of the espoused canons of mathematics education literature is cause for significant concern (see, e.g., gutiérrez, 2000). for one, it calls into question whether mathematics educators consider the urban domain as relevant engagement (i.e., beyond a sample size comparison), or, even more ominous, whether our “major” scholarship is relevant for truly reforming urban practice. the latter question of relevance for practice, interestingly enough, was the primary agenda topic at a research agenda conference hosted by the national council of teachers of mathematics in hyacinth, maryland. the conference gathered 60 to70 national representatives from the research community for the primary purpose of developing an agenda for research that could be used to inform practice to a greater extent than is currently seen. in lay terms, researchers pondered the relevance of current research to practice as we asked: do practitioners not use our work? the notion of relevance is an important standard for the professional lives of mathematics educators. without such a notion, there is little real community to speak of beyond the ivory towers of academia. urban change as a movement of people we choose solidarity as an important focus for the work of mathematics education reformers. in that, another decision was made to honor the way in which urban groups move amidst the aforementioned complexities of urban mathematics reform. a commitment to excellence meant broadening a definition of mathematics matthews editorial journal of urban mathematics education vol. 11, no. 1&2 10 reform to include the social movement of people. the prevailing view of reform in mathematics posits that true change vis-a-vis improvement in learning experiences is accomplished by increasing the content and pedagogical-content knowledge of practicing teachers. less attention is given to the racialized, cultural, and political experiences within the urban domain that influence people to move urban practice. the prevailing view essentially sidesteps any critical analysis of race, culture, and/or policy constraints that have been documented by a substantial number of equity researchers in mathematics education (albeit, seldom in mainstream mathematics education journals). social aspects of human development have been largely ignored in mathematics education research. although we are taught that the discipline lies at the nexus of social change (along with changes in psychology and mathematics), this change is most often articulated in terms of shifts in government ideologies (e.g., sputnik) and economic trends (e.g., “mathematically literate workers”). mathematics reform has scarcely been defined in terms of “ground up” movements of people. little is documented about the local actions of school and community families to “right” the inequities espoused in mathematics reform. social movements of equity such as the civil rights movements for gender and racial equality have been scarcely emphasized as critical to mathematics reform success. capturing the excellence of local groups as they author change has the potential to connect mathematics education scholarship to the very communities it intends to serve. what this look likes as a base of scholarship remains to be seen, but in jume we open this space. the articles in this inaugural issue range in nature in the ways in which the scholars tackle the questions we, ourselves, have struggled through; in that, they extend the very definition and context of urban, challenge racist conventions of urban schooling, and offer insights for finding excellent practice. we trust that you will be challenged as you join us on this journey in which old questions are explored (differently) and new questions formulated. this space is open and freely accessible to all who have as their primary interest the illumination of urban excellence. references georgia department of education. (2008). scores rise, gap closes on new crct. retrieved from http://www.gadoe.org/external-affairs-andpolicy/communications/pages/pressreleasedetails.aspx?pressview=archive&pid=210 gutiérrez, r. (2000). advancing african-american, urban youth in mathematics: unpacking the success of one math department. american journal of education, 109(1), 63–111. strepp, d. (2008a). failed math tests = swollen summer classrooms. the atlanta journal-constitution. retrieved from http://www.ajc.com/metro/content/metro/stories/2008/05/23/summerskl_0525.html strepp, d. (2008b). unhappy students: classes start right away for those failing crct. the atlanta journalconstitution. retrieved from http://www.ajc.com/search/content/metro/stories/2008/06/01/summer_school_crct.html http://www.ajc.com/metro/content/metro/stories/2008/05/23/summerskl_0525.html http://www.ajc.com/search/content/metro/stories/2008/06/01/summer_school_crct.html 404 not found microsoft word 404-article text no abstract-1988-1-6-20200608 (proof 1).docx journal of urban mathematics education july 2020, vol. 13, no. 1b (special issue), pp. 5–11 ©jume. https://journals.tdl.org/jume jacqueline leonard is professor of mathematics education in the school of teacher education, university of wyoming, 1000 e. university avenue, department 3374, laramie, wy 82071; email: jleona12@uwyo.edu. her research interests include computational thinking, self-efficacy in stem education, culturally specific pedagogy, and teaching mathematics for social justice. black lives matter in teaching mathematics for social justice1 jacqueline leonard university of wyoming prior to becoming a mathematics educator, i was a teacher in prince george’s county, maryland. because of the success i experienced with culturally relevant pedagogy (crp), it became part of my research agenda along with teaching mathematics for social justice (tmfsj). databases like 23andme and ancestry can be used as a context for crp and tmfsj. data related to education, occupation, military service, and voting records can be accessed online. our ancestors’ experiences can shape our identity and serve as powerful tools for contextualizing mathematics. the stories and counterstories of african americans can be problematized to show black lives matter. keywords: #blacklivesmatter, culturally relevant pedagogy, identity, teaching mathematics for social justice 1this article was first published by the urban education collaborative of the university of north carolina at charlotte: leonard, j. (2018). black lives matter in teaching mathematics for social justice. in s. richardson, a. davis, & c. w. lewis (eds.), proceedings of the third international conference on urban education (pp.129–137). urban education collaborative. https://1e1dd9d7-3209-48d8-90a27a4cef33595c.filesusr.com/ugd/a4a250_e4a5a6994b914a68975203db95e688ae.pdf permission to reprint was given to the journal of urban mathematics education by dr. chance lewis on june 8, 2020. leonard black lives matter in teaching mathematics journal of urban mathematics education vol. 13, no. 1b (special issue) 6 hen alex haley (1976) wrote his poignant novel, roots, it was not only ground-breaking for black america but an opportunity for america to come to grips with its historical past. the subject of slavery tends to draw angst from broad sectors of the u.s. population, both black and white alike. yet, haley’s story, while painful, was a powerful story of survival and resilience. at the time of the production of the mini-series, i wondered how haley conducted the research to find his ancestors. sheltered in a close-knit family that welcomed few relatives, i had no idea how to find my own roots. then in 1996, as a mathematics teacher in prince george’s county, maryland, i chose to use genetics and genealogy as a way to engage my students in culturally relevant mathematics. i used the base two pattern to help students understand exponentiation. beginning with themselves (20), their parents (21), and grandparents (22), students could easily see how they had 32 (25) third great-grandparents. after being given a simple family tree, some students went to the national archives in washington, d.c., to learn more about their family history. four students’ projects exceeded my expectations. one african american female student reported on six generations in her family, ending with a woman who was born into slavery and simply known as lizzy. a white female student engraved a tree into a wooden countertop. her father brought the artifact to class, where she reported on her ancestry. another white female student brought a dot matrix printout to class that revealed relatives born in the 15th century. she also discovered that she was related to benjamin franklin, one of the nation’s founders. the fourth student, who was white and male, learned that his grandparents married at the age of fourteen. he was twelve at the time and clearly rejected the idea. their enthusiasm encouraged me to go to the national archives as well. there, i found census records showing my grandmother at the age of two in 1920 and twelve in 1930. her grandfather, who was 70 years old, was living in the same household according to the 1930 census. it was surreal to imagine what their lives as sharecroppers were like in oktibbeha county, mississippi. at the archives, i was able to trace the maternal side of the family back to my fourth great-grandfather, who was born in south carolina about 1801. purpose some teacher educators and k–12 teachers realize the importance of the sociopolitical context in schooling but often ponder how to engage students in discourse that is meaningfully connected to mathematics content. the purpose of the paper is to show mathematics teacher educators and teachers of mathematics how to problematize issues of significance to the community, teach mathematics for social justice, and advocate for equality in education and society more generally. one issue that has emerged in the black community in recent years is racial profiling and state violence. after the death of trayvon martin in florida, w leonard black lives matter in teaching mathematics journal of urban mathematics education vol. 13, no. 1b (special issue) 7 #blacklivesmatter (#blm) became a national cry when three women—patrisse cullors, opal tometi, and alicia garza—created the hashtag (taylor, 2016). #blm uses decentralized leadership and local chapter organizing methods similar to the occupy movement of 2011 to speak to all issues related to human dignity while asserting the value of black life (taylor, 2016). quality education (ladson-billings, 2017), equal protection under the law (hill, 2016), equal housing (rothstein, 2017), and health and wellness (akom, 2011) are topics that can be mathematicized to teach mathematics for cultural relevance and social justice. some justice-oriented lessons have already been produced and can be used to link mathematics to #blm. himmelstein (2013) used stop-and-frisk as the basis for learning about central tendency. similarly, gustein (2013) investigated driving while black or brown with his students in chicago, illinois. in these lessons, students used probability to compare the actual number and percentage of traffic stops by race. using data as evidence, students may engage in letter writing, public service announcements, and other forms of civic engagement to advocate for justice and equality. yet, even more powerful, as my experience teaching in prince george’s county revealed, is using such lessons to help students to develop personal and social identity. the case for student identity the advent of the internet and dna testing has changed genealogy research. while my research at the national archives more than 20 years ago ended with discoveries on the maternal side of the family tree, dna testing provided matches on the paternal side of my family that were historical and eye-opening. moreover, these discoveries changed how i viewed myself as an american citizen (i.e., personal identity) and my relationship to others (i.e., social identity). furthermore, the intersection of race, nationality, gender, and class (i.e., intersectionality) had an impact on my self-efficacy and career goals. the results of my dna tests as reported by 23andme and ancestry are shown in table 1 below. leonard black lives matter in teaching mathematics journal of urban mathematics education vol. 13, no. 1b (special issue) 8 table 1 ancestry reports countries of origin 23andme ancestry west african 75.6% 76% ghana/ivory coast 31% cameroon 27% nigeria 7% benin/togo 4% senegal 4% bantu (cultural group) 3% european 21.8% 22% great britain, ireland, scotland, & wales 8.6% 10.5% scandinavian (i.e., norway, sweden, & denmark) 1.3% 6% other european (i.e., france, finland) 11.9% 5.5% south & southeast asian 1.6% 2% data like these can be used to create pie charts for students to study and compare their ancestry. a pie chart of my ancestry based on ancestry.com is shown in figure 1 below. figure 1. percentage of dna from different countries of origin through dna matches, evaluation of family trees, and census records, i learned that i am related to notable figures in american history. some of these figures dna analysis: countries of origin benin/togo cameroon/congo ivory coast/ghana germany mali ireland/scotland england/wales philippines leonard black lives matter in teaching mathematics journal of urban mathematics education vol. 13, no. 1b (special issue) 9 are pre-civil war presidents of the united states, famous statesmen, and patriots who fought in the revolutionary war. furthermore, u.s. census records revealed powerful information about my black ancestors. on the 1870 census (the first census listing former slaves and newly freed blacks), the racial category for some of my ancestors was mulatto, which is a derogatory term used to indicate mixed african and european ancestry. my second great-grandfather was a harrison slave who was born in sumter, alabama. yet, once freed, the former harrison slave registered to vote in sumter, alabama, in 1867 (see figure 2, data obtained from ancestry.com). figure 2. 1867 voting record while black men had the right to vote after 1865, exercising the right to vote in the south was usurped through poll taxes and literacy tests. the poll tax in birmingham, alabama (see figure 3), which was $2.50 in 1895, can be used as a mathematics problem (leonard, 2019). wages for farm labor in alabama were approximately $11.76 per month in 1895. thus, the poll tax was roughly equivalent to an entire week’s work. few sharecroppers could afford to pay the tax, and, consequently, the number of black voters in alabama decreased substantially. engaging in this type of problematizing connects issues of voting rights to the current sociopolitical context where voter suppression laws are in place (leonard, 2019). figure 3. 1985 poll tax receipt (publication granted by the smithsonian institute) leonard black lives matter in teaching mathematics journal of urban mathematics education vol. 13, no. 1b (special issue) 10 the knowledge that this ancestor voted so early after the civil war is aweinspiring considering the hardships and the violence that he must have endured. as a result, the stories of my ancestors became deeply personal. roots was no longer a story about someone else’s family. it signified my story and the stories of my slave and slaveholding ancestors, some of whom fought in the revolutionary war, signed the constitution, served as president of the united states, and served in political office. shortly after the civil war, one the presidents in my family tree proposed federal education funding and voting rights enforcement for african americans but was unsuccessful. how does one reconcile the right to life, liberty, and the pursuit of happiness while also refusing to grant those same rights to others? this is the american dilemma. nevertheless, knowing my background has reshaped my identity and encouraged me to continue breaking down racial and gender barriers as a professor, researcher, and scholar. as the first african american to receive the fulbright canada research chair award in stem education at the university of calgary in alberta in 2018, i engage in culturally relevant pedagogy by encouraging indigenous, african american, and latinx students in north america to tell their stories. solutions in this era of anti-blackness and white nationalism, it is more important than ever to discover one’s roots and to learn how people of every race and background are interdependent and interconnected. teachers of mathematics should use students’ culture and history to mathematize problems to show that black lives matter. perhaps the common ancestry shared among descendants of former slaves and slaveholders will help us to recognize our humanity. lessons related to #blacklivesmatter have already been developed on racial profiling and equal housing. additional lessons may be developed around voting rights and wages as illustrated above. the stories and counterstories of generations who lived before us provide the backdrop for culturally relevant and social justice-oriented mathematics lessons. the data for these lessons are only a click away. references akom, a. (2011). eco-apartheid: linking environmental health to educational outcomes. teachers college record, 113(4), 831–859. gutstein, e. (2013). understanding the mathematics of neighborhood replacement. in e. gutstein & b. peterson (eds.), rethinking mathematics: teaching mathematics by the numbers (2nd ed., pp. 101–109). rethinking schools. haley, a. (1976). roots: the saga of an american family. doubleday. hill, m. l. (2016). nobody: casualties of america’s war on the vulnerable from ferguson to flint and beyond. atria books. leonard black lives matter in teaching mathematics journal of urban mathematics education vol. 13, no. 1b (special issue) 11 himmelstein, k. (2013). racism and stop-and-frisk. in e. gutstein & b. peterson (eds.), rethinking mathematics: teaching mathematics by the numbers (2nd ed., pp. 122–128). rethinking schools. ladson-billings, g. (2017). “makes me wanna holler”: refuting the “culture of poverty” discourse in urban schooling. the annals of the american academy of political and social science, 673(1), 80–90. https://doi.org/10.1177%2f0002716217718793 leonard, j. (2019). culturally specific pedagogy in the mathematics classroom: strategies for teachers and students (2nd ed.). routledge. https://doi.org/10.4324/9781351255837 rothstein, r. (2017). no blacks allowed. the crisis magazine, 124(3), 12–17. taylor, k-y. (2016). from #blacklivesmatter to black liberation. haymarket books. copyright: © 2020 leonard. this is an open access article distributed under the terms of a creative commons attribution-noncommercial-sharealike 4.0 international license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. microsoft word 444-article text no abstract-2383-1-4-20210714-1 (proof 1).docx journal of urban mathematics education november 2021, vol. 14, no. 1b (special issue), pp. 1–5 ©jume. https://journals.tdl.org/jume gloria ladson-billings is the former kellner family distinguished professor of urban education in the department of curriculum and instruction at the university of wisconsin–madison and currently serves as the president of the national academy of education. her research on culturally relevant pedagogy and critical race theory has been widely influential in the field of mathematics education, and many others. guest editorial does that count? how mathematics education can support justice-focused anti-racist teaching and learning gloria ladson-billings university of wisconsin–madison “numbers is hardly real and they never have feelings…” – mos def, 1999, 3:30 n 1999 the hip hop artist mos def (a.k.a. yasiin bey, born dante terrell smith) released an album that contained the song “mathematics.” the song speaks to the disparities between black urban and white lives in the united states by quoting a litany of statistics, including the differences in defense spending and spending on social services along with statistics on incarceration rates, minimum wage, unemployment rates, and the way that mathematics reduces human beings to numbers— social security numbers, phone numbers, zip codes. mos def’s (1999) declaration that “it’s a number game, but sh%t don’t add up somehow” (1:09) helps to dispel the notion that mathematics does not enter into the question of equity and justice in our society. for more than 30 years i have been working with teachers, teacher candidates, graduate students, and community members about the role of teaching and learning in promoting and supporting equity, justice, and democracy in our schools and broader society. those who teach history/social science, english/language arts, and fine arts often recognize the connections among questions of culture, race, and inequality and their disciplines. however, colleagues in what we deem the stem fields (i.e., science, technology, engineering, and mathematics) often reject the notion that their subjects are amenable to what we consider racial, ethnic, or cultural issues. it is not unusual for me to hear someone remark, “i teach physics (or mathematics); this stuff doesn’t apply to me!” they make claims of objectivity or neutrality that should shield them from having to engage in conversations of diversity, equity, and inclusion. other colleagues who do not reject the notion that there is a relationship between stem fields and issues of diversity, equity, and justice often connect with superficial aspects of “multiculturalism” and believe sharing information about diverse scientists i ladson-billings does that count? journal of urban mathematics education vol. 14, no. 1b (special issue) 2 and mathematicians is sufficient. thus, students hear about benjamin banneker, garrett morgan, katherine johnson, or the ancient egyptian pyramids and the mayan concept of zero as exemplars that “different” people participated in stem fields. however, these isolated bits of information are rarely, if ever, connected to big ideas that aid students in the understanding and application of the science. mathematics educator eric “rico” gutstein from the university of illinois– chicago has demonstrated a more seamless integration of justice issues when teaching students at social justice high school in chicago (see gutstein, 2013). gutstein taught the students how much people actually paid when they purchased a home by showing them what happens with compound interest and the need to get private mortgage insurance for homeowners who purchase a home with a conventional mortgage and less than 20 percent down payment. gutstein helped students see that poorer buyers of color were more likely to be in this position, and once the students began to calculate what those buyers would pay overall versus what a buyer with a 20 percent or greater down payment would pay, they were shocked to see the large difference in the amount paid for houses of the exact same costs. the justice applications of mathematics both surprised and angered the students while making them determined to understand mathematics at a level that would prevent their being cheated in the future. in another instance, a student complained to his teacher that the school’s “no hat” rule was applied in a discriminatory way against black students. according to the student, black male students were singled out by teachers and administrators for violating the hat policy far more than their white peers. the teacher asked the student for evidence of his claim. when the student could only provide anecdotal evidence from his own encounters and those of a few of his friends, the teacher worked with the class to design a study. the teacher helped the students write a hypothesis about the likelihood of race being a predictor of whether or not a student would be stopped and sanctioned for wearing a hat inside the building. the class was divided into four groups with each group charged with sampling at least 25% of each of the classes (i.e., freshmen, sophomores, juniors, and seniors). then the students designed a brief survey with student demographics (race, gender) and two questions—"have you ever been stopped for wearing a hat in the building?” and “if yes, what happened after you were stopped?” when the students brought their data back, the teacher used their results to teach mean, mode, and median and how to construct graphs. their results proved the students’ hypothesis. the students asked to look at gender as a predictor and found that, taken together, race and gender (i.e., being black and male) were the strongest indicators of whether one would receive a sanction for wearing a hat indoors. the students wrote up their findings and presented them to the school principal. to his credit, the principal shared the findings with the faculty and declared that the results were unacceptable. he required the teachers and staff to modify their ladson-billings does that count? journal of urban mathematics education vol. 14, no. 1b (special issue) 3 behavior in the monitoring of the hat rule. the students were able to use their mathematics knowledge to solve a real personal social problem. mathematics, like most science disciplines, is about solving problems and understanding patterns. the patterns that emerge in mathematics can be understood through the prism of race, class, gender, and other forms of difference. on almost any dimension of human well-being, we can see a pattern of disparity. the question for mathematics is how to explain the pattern followed by what problem can we solve that emerges from the pattern. for example, students might examine the pattern of the relationships among educational attainment, employment, and income. if higher educational attainment is a predictor of better employment and higher income, what explains the fact that the unemployment rate for black college graduates is higher than that for white high school graduates (morrison, 2020)? within the mathematics education community, scholars have taken a position to align mathematics education with social justice. for instance, in mathematics education through the lens of social justice: acknowledgment, actions, and accountability, a joint position statement from the national council of supervisors of mathematics and todos: mathematics for all, the organizations committed to four broad goals: eliminating deficit views of mathematics learning; eradicating mathematics as a gatekeeper; engaging the sociopolitical turn of mathematics education; and elevating the professional learning of mathematics teachers and leaders, with a dual focus on mathematics and social justice. tate, ladson-billings, and grant (1993) discussed how one of the nation’s most noted supreme court cases, brown v. board of education (1954), allowed school districts to provide a mathematical remedy to the problem of school segregation. schools could have just as easily redrawn attendance boundaries or paired schools so that part of the education students received was in each of the two schools. instead, most school districts determined that a certain number of students was needed to desegregate a school. because mathematics is so embedded, indeed reified in u.s. society, there are numerous mathematical examples throughout the nation’s history that can be used to explain patterns of inequity, injustice, and the denial of democratic citizenship rights. for example, when students learn of the 3/5ths compromise, are they ever asked to determine exactly how many representatives that added to slave holding states? or students can use u.s. census data to determine the various ways groups have been counted over time (e.g., mexican americans were once counted as white). presidential elections make excellent data sources for understanding polling. on a more personal level, students can look at housing assessments in their communities versus assessments in other neighborhoods in their cities or towns. they can draw inferences about what the differences in assessments may mean. an interesting exercise for students is to look up the assessed value of their homes when they were initially sold and determine the difference between then and the current value. ladson-billings does that count? journal of urban mathematics education vol. 14, no. 1b (special issue) 4 this exercise can give students an understanding of how people who can afford to stay in a home can build equity and accumulate wealth, whereas those who are renters are not building equity or wealth through real estate. the challenging headlines provide mathematics teachers with ample data for constructing mathematics problems. for instance, the horrific stories of police shootings of unarmed citizens can be placed in a larger context. how many unarmed people were actually shot in a particular municipality? what are the characteristics of those people? what is the likelihood that certain people will become victims of a police shooting? the very topic of disparity is an excellent one for exploring mathematical problems—issues of differentials in life spans (exploring actuarial tables), incarceration rates, suspension rates, high school graduation rates, or dropout rates are good sources of problems. if we return to mos def’s (1999) “mathematics,” we find a line that says, “like i got, sixteen to thirty-two bars to rock it but only 15% of profits, ever see my pockets…” (1:11). for students, this can mean tracking album sales and calculating how much the artist actually nets. students with part-time jobs should be able to calculate how much of their income goes to federal, state, and local taxes, social security contributions, and other deductions. the point is mathematics surrounds our lives and should never be relegated to only one group of students. perhaps the most significant reason i think mathematics educators must be engaged in work dealing with diversity, equity, and justice has to do with its civic imperative. in the 1950s and 1960s when african americans were fighting for their constitutional right to the franchise, many civil rights activists realized how important it was for african americans to be literate. from the earliest days of arrival on american shores, people of african descent who were enslaved were prohibited from learning to read. becoming literate meant they could read documents that pertained to their futures (e.g., proposed sales and mortgages involving enslaved people as collateral, etc.). being able to write meant enslaved people could produce manumission papers and make their way to free states and territories. this placed literacy at the center of the movement for liberation. as we moved into the 20th century, it became clear that success in u.s. society would require citizens to have more advanced education in a variety of areas. one area that was a key to career success in this modern world was mathematics. although the “average” american was thought to only need a rudimentary knowledge of mathematics (i.e., arithmetic), those who would be leaders and innovators would need knowledge and facility in advanced mathematics. they would have to understand algebra, geometry, trigonometry, calculus, and statistics. civil rights leader and mathematician robert moses recognized the connection between mathematics and civil rights. his book radical equations: civil rights from mississippi to the algebra project (moses & cobb, 2002) described how he saw black students being excluded from the thinking and reasoning that undergirds mathematics. moses saw ladson-billings does that count? journal of urban mathematics education vol. 14, no. 1b (special issue) 5 economic freedom as the key to full participation in u.s. society and mathematical literacy as the key to economic advancement. by insisting that all students can and should have access to algebra, moses was breaking a longstanding paradigm that suggested only certain students should have access to algebra. moses’s notion that mathematical knowledge was a civil right brought the discipline out of the narrow confines of academic learning and into the notion of basic skills citizens need to function in a democracy. in 2009, the commission on mathematics and science education declared, our nation needs an educated young citizenry with the capacity to contribute to and gain from the country’s future productivity understand policy choices and participate in building a sustainable future. knowledge and skills from science, technology, engineering, and mathematics—the so-called stem fields—are crucial to virtually every endeavor of individual and community life. all young americans should be educated to be “stem-capable” no matter where they live, what educational path they pursue, or in which field they choose to work. (p. vii) so, if we see mathematics as a key aspect of our citizenship and we know that large segments of our students are not able to access high-quality mathematics teaching and learning, what does that say about us as a nation? references brown v. board of education, 347 u.s. 483 (1954). commission on mathematics and science education. (2009). the opportunity equation: transforming mathematics and science education for citizenship and the global economy. carnegie corporation of new york & institute for advanced study. gutstein, e. (2013). home buying while brown or black: teaching mathematics for racial justice. in e. gutstein & b. peterson (eds.), rethinking mathematics: teaching social justice by the numbers (2nd ed., pp. 61–66). rethinking schools. morrison, n. (2020, june 18). black graduates twice as likely to be unemployed. forbes. https://www.forbes.com/sites/nickmorrison/2020/06/18/black-graduates-twice-as-likely-to-beunemployed/?sh=5157d47377eb mos def. (1999). mathematics [song]. on black on both sides. rawkus; priority. moses, r., & cobb, c. e., jr. (2002). radical equations: civil rights from mississippi to the algebra project. beacon press. national council of supervisors of mathematics & todos: mathematics for all. (2016). mathematics education through the lens of social justice: acknowledgement, actions, and accountability. https://www.todos-math.org/assets/docs2016/2016enews/3.pospaper16_wtodos_8pp.pdf tate, w. f., ladson-billings, g., & grant, c. a. (1993). the brown decision revisited: mathematizing social problems. educational policy, 7(3), 255–275. copyright: © 2021 ladson-billings. this is an open access article distributed under the terms of a creative commons attribution-noncommercial-sharealike 4.0 international license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. journal of urban mathematics education december 2009, vol. 2, no. 2, pp. 66–71 ©jume. http://education.gsu.edu/jume christopher c. jett is a clinical assistant professor of mathematics education in the department of middle-secondary education and instructional technology, in the college of education at georgia state university, p.o. box 3978, atlanta, ga, 30303-3978; e-mail: mstccjx@langate.gsu.edu. his research interests are centered on employing a critical race theoretical perspective to mathematics education research and investigating the experiences of successful african american students in mathematics. book review mathematics, an empowering tool of liberation?: a review of mathematics teaching, learning, and liberation in the lives of black children 1 christopher c. jett georgia state university ver the years, the phrase mathematics teaching and learning has become synonymous with mathematics education. there have been some scholars who have extended the phrase by specifically intertwining mathematics teaching and learning with black 2 children. but what about enhancing the phrase—both in concept and in practice—by coupling mathematics teaching and learning with liberation in the name of black children? that is exactly what danny bernard martin‘s edited book mathematics teaching, learning, and liberation in the lives of black children (routledge, 2009) does so poignantly. organization of the book the book is organized into four sections: (a) mapping a liberatory research and policy agenda, (b) pedagogy, standards, and assessment, (c) socialization, learning, and identity, and (d) collaboration and reform. all 14 chapters are thought provoking, with figures and tables interspersed throughout to validate the authors‘ arguments. in addition, the real-world examples of ―mathematics in context‖ in mathematics classrooms offer a direct application of research presented throughout the volume to the mathematics experiences of black children. the first section, mapping a liberatory research and policy agenda, includes a single chapter from martin (2009) who provides a critical race analysis on the knowledge production of black children and mathematics. that is to say, 1 martin, d. b. (ed.). (2009). mathematics teaching, learning, and liberation in the lives of black children. new york: routledge. 376 pp., $54.95 (paper), isbn 0-8058-6464-4 http://www.routledgeeducation.com/books/mathematics-teaching-learning-and-liberation-inthe-lives-of-black-children-isbn9780805864649 2 i use the terms black and african american interchangeably throughout this review. o http://www.routledgeeducation.com/books/mathematics-teaching-learning-and-liberation-in-the-lives-of-black-children-isbn9780805864649 http://www.routledgeeducation.com/books/mathematics-teaching-learning-and-liberation-in-the-lives-of-black-children-isbn9780805864649 jett book review journal of urban mathematics education vol. 2, no. 2 67 martin‘s analysis is an examination of much of the existing mathematics education research and policy regarding black children from a critical race theoretical perspective—a perspective that highlights the often negative effects of existing research and policy on the lives of black children. to counteract these negative effects, martin convincingly argues for culturally sensitive approaches to research and policy that are liberatory in nature. as a result, his chapter frames the book, underscoring the overall theme of mathematics liberation for black children. in the second section pedagogy, standards, and assessment, theories such as culturally relevant and culturally responsive pedagogy refreshingly run rampant throughout the chapters. these culturally centered discussions, however, are not limited to african american students. examples of the mathematics experiences of children and mathematics educators from bermuda and south africa are also included in the discussions. drawing on teachers‘ and students‘ voices, these chapters offer the reader the opportunity to explore culturally specific pedagogical frameworks related to teaching mathematics to black students. the third section socialization, learning, and identity, which resonated with me the most, examines how african american students are socially constructed, how they are framed as mathematics learners, and how their identities are shaped as african americans and students of mathematics. the authors in this section explore socialization and identity issues as it relates to african american students‘ mathematics performance. the student discussions concerning various racial stereotypes are robust and provide a frame to investigate socialization, learning, and identity as it pertains to african american children and mathematics. the last section, collaboration and reform, includes one chapter. in this final chapter, the maryland institute for minority achievement and urban education is presented as a successful model that serves as a liberatory initiative for black children and adults alike. highlighting engaging events such as family math nights, an elementary school based african american male mentoring program, and university colloquia, to name a few, johnson and brown (2009) express that these efforts have been liberatory in nature and have influenced the (mathematics) academic achievement rates of african american students. johnson and brown also outline some specific suggestions to consider when forming liberating school–university partnerships that seek to improve the mathematics achievement rates of black children. nuances in black mathematics education as i began to write this review, reflecting on the book both as a whole and as individual chapters, the chapter that i returned to often was leonard‘s (2009) chapter: ―‗still not saved‘‖: the power of mathematics to liberate the op jett book review journal of urban mathematics education vol. 2, no. 2 68 pressed.‖ although she wrote about her experiences in the methodist church, i am reminded of my own experiences in an african american ―sanctified‖ church. the church has faithfully served as a liberating place for me from the liberating preacher to the spirit-filled soloists leading the choir, with the tambourines, drums, and organs accompanying these liberating sprits. when we have been ―moved‖ in the sanctified church, we are compelled to think and act differently, hopefully, seeking to become liberated individuals. i translate those ideas to this area of inquiry challenging those vested in the mathematics teaching and learning of black children to be moved by some of the nuances explicated in this volume. in this context, acting and thinking differently might imply conducting research that highlights the strengths of black children in mathematics classrooms as well as deconstructing inferiority discourses concerning the mathematics abilities of black children. i pay particular attention to a couple of the nuances suggested by some of the authors below. first, several of the authors intersect issues of race and/or racism to their mathematics education research regarding black children. in this volume, critical race theory (crt) seems to be the leading theoretical approach to this end. there is also, however, a charge for researchers to employ black liberation theology to mathematics education research and pedagogy. black liberation theology is an underutilized theory in mathematics education scholarship. furthermore, black liberation theology might be another avenue that leads to better understandings of how african american students are engaged in and connected to mathematics. in addition, many of the authors express their disdain for how (mathematics education) research and policy categorize african american mathematics learners as unintelligible. more important, the authors critically examine policy efforts supposedly designed to serve the educational (mathematics) needs of black students. these authors argue that many of the policy efforts have worked to perpetuate existing notions in mainstream discourse concerning the perceived mathematical deficiencies of black children. policymakers and other stakeholders must reconceptualize how they bring policy into fruition, especially when they start with the premise that african american students are lacking the mathematical aptitude to succeed in their mathematical endeavors. along with this charge is the need for more research that highlights the mathematics successes and strengths of african american learners to be the primary sources to inform research and policy. furthermore, this volume causes researchers to think differently about mathematics education research and its implications for black children. king (2005), in her work with the american educational research association‘s commission on research in black education, explored the question: ―how can research become one of the forms of struggle for black education?‖ (p. 6). in like manner, how can mathematics education research (as well as mathematics teaching and jett book review journal of urban mathematics education vol. 2, no. 2 69 learning) serve as an emancipatory and liberating force in the ―struggle‖ for mathematics achievement among black students? critiques of the book in the second section pedagogy, standards, and assessment there are examples presented of teachers‘ instructional strategies. it seems as if there is an overwhelming emphasis on the pedagogical aspects of teachers of black children in this section. for this reason, the second section might have been entitled pedagogy. while references are made to the national council of teachers of mathematics‘ standards as well as various assessments, it appears that issues regarding the pedagogical practices of teachers of black students are the primary foci of the discussions. while there is some discussion of successful mathematics educators, it might have been useful to read interview data from black mathematics teacher educators who have a legacy of liberating black children as well as liberating preservice and inservice teachers of black children. a case study devoted to highlighting the success of black mathematics teacher educators might serve as a springboard to facilitate discussions surrounding issues concerning being a liberated black mathematics educator and producing liberating mathematics teachers. hilliard (2003) argues that it takes gap-closing teacher educators to teach and produce gap-closing teachers and school leaders. similarly, only those mathematics teacher educators who are liberated themselves might be able to produce mathematics teachers who seek liberation in the lives of black children. what better way to learn than from the voices and experiences of successful black mathematics teacher educators. with the emphasis on the shortage of black mathematics majors, a chapter focusing specifically on black college students‘ mathematics experiences might have been insightful. although the book‘s title emphasizes black children, it might be far-fetched to expect a chapter regarding the experiences of undergraduate students. given the application of various theoretical frameworks such as crt, black feminism, and black liberation ideology in this volume, however, examining the experiences of undergraduate students employing these theoretical approaches would have aligned closely with the theme of the book and offered additional insights for researchers and policymakers in this area of inquiry. furthermore, institutions of higher learning are an exceptional space to explore issues of race and/or racism given the racial diversity of various higher education institutions (e.g., historically black colleges and universities, predominantly white institutions, hispanic-serving institutions, etc.). because this volume, i believe, is such important work, it would have been beneficial to see an additional chapter or two added to the final section (collabo jett book review journal of urban mathematics education vol. 2, no. 2 70 ration and reform) to further emphasize the seriousness of putting these liberation ideas into action. another example of a school–university partnership could have been a starting point to this end. after reading such liberating work throughout the first three sections of the book, it would seem plausible that more collaborative and reform initiatives would be presented to expose readers to the great collaborative models that are being constructed on many fronts. the driving force of such powerful work, i argue, should be the ―call to action‖ from university, k– 12, and community advocates to collaboratively undertake some of the charges presented in this volume. concluding thoughts i must admit i am a biased reader when it comes to this area of inquiry. first, i am an african american male scholar and mathematics educator who seeks to establish liberatory mathematics practices in both my scholarship and pedagogy. second, i am a huge ―fan‖ of danny martin‘s scholarship. i identify with his research and, too, like perry, steele, and hilliard (2003), conceptualize african american learners as young, gifted, and black. and last, i seek to problematize issues of race and/or racism in my own research. with that being said, i have a deep appreciation for this volume and understand its value for the educational enterprise in general and black children in particular. even those who do not share my same sentiments will find this book to be insightful, informative, and thought provoking concerning the mathematics teaching and learning of black children. the chapters offer different perspectives on an important, relevant topic in mathematics education research and practice. with the limited exposure of african american mathematics education scholars‘ research, this work exposes readers to the many voices in mathematics education that differ from mainstream discourses in mathematics education research. while there are a couple of chapters written by white scholars, these white scholars‘ language is extremely different from the dominant discourse that often frames black children as unintelligible and doomed to mathematics failure. notwithstanding, this book is a significant contribution to the education community in general and the mathematics education community in particular because of the dominance of voices of black mathematics education scholars and teacher educators and the powerful narratives produced by black children themselves. as we move forward with the challenge of liberation in the lives of black children, i believe it is worth drawing attention to some crucial questions posed in the book. these questions include: ―why should african american children learn mathematics?‖ (martin, 2009, p. 25); ―who is a highly qualified mathematics teacher relative to the needs of african american children?‖ (martin, 2009, p. 27); and ―in what ways might african american teachers influence their african jett book review journal of urban mathematics education vol. 2, no. 2 71 american students‘ perceptions of themselves as ‗doers of mathematics‘?‖ (clark, johnson, & chazan, 2009, p. 47). 3 these questions, i believe, are the hallmarks of future research in this area of inquiry. in sum, this book is worth reading because it gives ―voice‖ to both black mathematics educators and black children. additionally, it is a must read, i believe, for those seeking to develop deeper understandings of the dynamics concerning the mathematics teaching and learning of black children. i cannot glean all of the unique perspectives of the book, but i hope this book review motivates potential readers to engage in a reflective, critical read. moreover, i hope and trust that those who read this book will seek to (re)establish mathematics teaching and learning practices that utilize mathematics as an empowering tool for liberation, especially in the lives of black children. references clark, l. m., johnson, w., & chazan, d. (2009). researching african american mathematics teachers of african american students: conceptual and methodological considerations. in d. b. martin (ed.), mathematics teaching, learning, and liberation in the lives of black children (pp. 39–62). new york: routledge. hilliard, a. g., iii. (2003). no mystery: closing the achievement gap between africans and excellence. in t. perry, c. steele, & a. g. hilliard, iii, young, gifted, and black: promoting high achievement among african-american students (pp. 131–165). boston: beacon press. hooks, b. (1994). teaching to transgress: education as the practice of freedom. new york: routledge. johnson, m. l., & brown, s. t. (2009). university/k–12 partnerships: a collaborative approach to school reform. in d. b. martin (ed.), mathematics teaching, learning, and liberation in the lives of black children (pp. 333–350). new york: routledge. king, j. e. (2005). a transformative vision of black education for human freedom. in j. e. king (ed.), black education: a transformative research and action agenda for the new century (pp. 3–17). mahwah, nj: erlbaum. leonard, j. (2009). ―still not saved‖: the power of mathematics to liberate the oppressed. in d. b. martin (ed.), mathematics teaching, learning, and liberation in the lives of black children (pp. 304–330). new york: routledge. martin, d. b. (2009). liberating the production of knowledge about african american children mathematics. in d. b. martin (ed.), mathematics teaching, learning, and liberation in the lives of black children (pp. 3–36). new york: routledge. perry, t., steele, c., & hilliard, a. g, iii (2003). young, gifted, and black: promoting high achievement among african-american students. boston: beacon press. 3 hooks (1994) argues that whites can come to know blacks‘ realities (i.e., be successful at teaching black children), but that they know them differently. therefore, i am not suggesting that only african american teachers should teach african american students. in this instance, i only wish to highlight the unique influences that black teachers might have on black children. journal of urban mathematics education december 2009, vol. 2, no. 2, pp. 12–17 ©jume. http://education.gsu.edu/jume ira david dawson is a mathematics teacher at the walker school, marietta, ga 30062, and part-time specialist student of mathematics education in the department of middle-secondary education and instructional technology, in the college of education at georgia state university, atlanta, ga 30303; email: idawson2@student.gsu.edu. public stories of mathematics educators  how did i get this way? how bad is the damage? and how do i fix it? ira david dawson the walker school here is an old saying, ―ignorance is bliss.‖ i never really knew what the elders meant by this statement until the last few years of teaching mathematics to high-ses students at a private school. you see it has been 3 years since i decided to take the red pill (like neo in the matrix) and this decision has led to numerous days of frustration, confusion, and unfulfilling results. before doing so, i was in a state of ―ignorant bliss.‖ i stood in front of my students, poked my chest out, and indoctrinated them with knowledge relevant or irrelevant to their lives. in return, they wrote down word-for-word everything i said, never questioned my mathematics, and passed the assessments. dialogue 1 was not present, numeracy was not present, and mathemacy 2 was nowhere in sight. then the decision—the red pill—and i became conflicted and began to reflect differently on my teaching practices. i started to see the discourses that surround my students (and yes the wealthy too have discourses), discourses created by their teachers and their teachers‘ practices, and the discourses created by lofty and unsatisfied expectations of their parents. these practices and discourses work  lou matthews, in his editorial in jume, 2(1), argued that one of the greatest challenges for mathematics educators has been in defining a people-centric mathematics education, claiming that to do so would require that we begin to tell our stories in the face of perplexing times in urban education. the ―public stories of mathematics educators‖ section of jume is a newly created section to provide an intellectual space for k–16 urban mathematics teachers and teacher educators to tell their stories as they reflect on and transform their pedagogical philosophies and practices and, in turn, the opportunities to learn for the students they serve. 1 stinson (2009) summarizes freirian dialogue as ―a loving, humble, hopeful, trusting, critical, and horizontal relationship between persons, a ‗relation of ‗empathy‘ between two ‗poles‘ who are engaged in a joint search‘‖ (p. 516). 2 skovsmose (2005) writes: ―mathemacy must contain mathematical as well as reflective elements. …as an idealized notion, mathemacy must also include reflections on (mathematical) knowledge in action. …mathemacy includes the hope of critical mathematics education that…address[es] the paradox of reason and…develop[s] a critical citizenship‖ (p. 188). t dawson public stories journal of urban mathematics education vol. 2, no. 2 13 in concert, indoctrinating my students to receive, to memorize, to repeat. freire (1970/2000) argues: education thus becomes an act of depositing, in which the students are the depositories and the teacher is the depositor. instead of communicating, the teacher issues communiqués and makes deposits which the students patiently receive, memorize, and repeat. this is the ―banking‖ concept of education. (p. 72) this banking concept embodied my teaching practices. this embodiment created a sense of panic and frustration upon revelation. how did i get this way? how bad is the damage? and how do i fix it? these are the major questions that i sought out to answer in order to begin my quest for liberation and redemption—not only for me but also, and most important, for my students. how did i get this way? in october 1997, secretary of education richard riley released a white paper that was intended to help students, parents, and educators understand the significance of a solid foundation in mathematics as a key to college and career success. the preparers of the report wrote: in the united states today, mastering mathematics has become more important than ever. students with a strong grasp of mathematics have an advantage in academics and in the job market. the 8th grade is a critical point in mathematics education. achievement at that stage clears the way for students to take rigorous high school mathematics and science courses—keys to college entrance and success in the labor force. students who take rigorous mathematics and science courses are much more likely to go to college than those who do not. algebra is the ―gateway‖ to advanced mathematics and science in high school, yet most students do not take it in middle school. (u.s. department of education, as cited in stinson, 2004, p. 11) i was 16 years of age at the time this report was released, and though i knew nothing of it, i knew of its cause. mathematics education was a huge question mark at the time and educators and policymakers were searching for answers. mathematics began to be referred to as a ―gatekeeper,‖ an authoritative quandary. at the time, i knew of this gatekeeper status very well and effectively used its authority to empower myself and gain access to privilege. at home, my mother indoctrinated my older brother and me with the belief that achieving mathematical success could be our key to success in life, and like many, i took mother‘s advice seriously. in the 3rd grade, i used my mathematics privilege to gain less supervision during independent study time. in 5th grade, i used it to become a member of an elite program called tag (talented and gifted). in this program, i was able to leave school and travel to an educational dawson public stories journal of urban mathematics education vol. 2, no. 2 14 facility where tag students would be isolated from all the ―others‖ so that we might continue in peace to do the ―school thing.‖ in 8th grade, i continued to utilize my mathematics privilege and spent most of my school hours in small classes, surrounded by fellow students that abused their mathematics privilege as well. my high school years were no different; mathematics allowed me to again receive benefits such as small classes, effective teachers, and minimal behavioral interruptions. with all this being said, at an early age, i understood the importance of a quality mathematics education. i recognized the power of mathematics to open doors that other disciplines just could not. i was fortunate because mathemacy was present only because my privilege allowed it to be. the gate was open— mathematics had provided the key. my athletic privilege had limitations. my male privilege was hindered often by my skin color. but my mathematics privilege was unlimited and ever so effective. to this day, i use my mathematics privilege to open doors and mathemacy to read the world. this understanding of the gate-keeping status of mathematics is why i see my role as a mathematics teacher as important as any person‘s role in shaping the leaders of tomorrow. but had my consistent state of mathematics privilege made me ignorant to the oppression of others? had i become an oppressor? how bad is the damage? i still remember the first day of teaching mathematics in an intercity, public school in charlotte, north carolina. after useless days of in-service training where the administration prepared teachers on how not to fail instead of preparing us on how to succeed, i had little time to plan my first few days, but i managed to do so quickly. i had it all mapped out. i would hit the students quick, hard, and smooth. they would not know what hit them. i was in for a rude awakening, however. they were unprepared, unmotivated, and mathematically malnourished, so it seemed. did they not know that mathematics was the gatekeeper? the students appeared to be ignorant to the privileges mathematical success brought, and thus felt as if mathematics was just another thing they had to do. that whole year, i pushed and they pulled; i fussed and they fought. it was me versus them. ―to simply think about the people, as the dominators do, without any self-giving in that thought, to fail to think with the people, is a sure way to cease being revolutionary leaders‖ (freire, 1970/2000, p. 132, italics in original)—i was no revolutionary leader and there was no liberation present. a year later, i found myself in front of a different group of students. a classroom filled with the sons and daughters of lawyers and policymakers. their parents knew how to do the ―school thing well,‖ so as children these students had been indoctrinated with the secrets. these students were prepared, enthusiastic, and recognizing of their mathematics privilege. to them, mathematics opened dawson public stories journal of urban mathematics education vol. 2, no. 2 15 doors to better opportunities, and thus, they recognized it as a gatekeeper. as a mathematics teacher, i saw early on my role as only a depositor; i held the knowledge and dispensed it evenly and consistently using the same methods i had used before. there was no dialogue present, and mathemacy was still nowhere near. it was not until i involuntarily bumped into the writings of john dewey (see, e.g., 1938/1998) and paulo freire (see, e.g., 1970/2000) that i began to realize the dehumanizing acts that i had performed. i had become an oppressor. both dewey and freire highlighted areas of my teaching practices that actually aided in the mathematical illiteracy of my students. i was the master of manipulation, and i enjoyed the role. freire claims, ―one of the methods of manipulation is to inoculate individuals with the bourgeois appetite for personal success‖ (p. 149). manipulation of their bourgeois appetite was my means of motivating my students and they either responded or withered. those that responded were motivated primarily by the accolade of a variable representing their success in doing the school thing, but those that withered just moved on to hate mathematics, seeing the world as math-less. how do i fix it? ―is it too late to begin?‖ is the resonating question within my mind and conscience. freire (1997) notes, ―i have…encountered many teachers…who while being oppressed by the political system in which they operate, were in turn oppressors of their students‖ (p. 311). here, freire exemplifies my past (and somewhat my current) role in the oppression of my students. while reflecting and critiquing on my past (and current) role, however, i have begun to understand what i want to become and what changes need to be made. these critiques of my teaching practices and me have taken me on a whirlwind of reflections. some of these reflections are about the different discourses that have constructed both my students and me, and many are about the new and old ethical trends that affect our teaching, learning, and living experiences. freire believes, ―it has become necessary for teachers, especially critical teachers, to deconstruct the social construction of this fatalism [of the market] so as to unveil the inherent ideology that informs and shapes and maintains an ethic of greed‖ (p. 313). so i began to deconstruct, realizing that to become a critical teacher, in the freirian sense, was a first step. in doing so, i aim to be an agent of change within the mathematics education community; a critical mathematics teacher creating an environment in which mathemacy is encouraged and nourished. i have taking initial steps toward this becoming by initiating the agonizing process of reflection. not absent of pain, and narrowly close to depression, the process of self-reflection is extensive, ongoing, and crucial to growth. and given that ―reflection and action, [are] in such radical interaction that if one is sacrificed—even in part—the dawson public stories journal of urban mathematics education vol. 2, no. 2 16 other immediately suffers‖ (freire, 19970/2000, p. 87), i am prompted to reflect, critique, and reflect again, in concert with action. in other words, these reflections are public, demanding not only introspections that address personal experience but also interpersonal interactions (skovsmose, 2005). the next step in rehabilitating my teaching practices has been to determine what i believe critical mathematics education might ―look like‖ within my classroom. stinson, bidwell, and powell (2009), drawing on the work of leistyna and woodrum, claim that the role of a critical pedagogue is to ―encourage both teachers and students to develop an understanding of the interconnecting relationship among ideology, power, and culture, rejecting any claim to universal foundations for truth and culture, as well as any claim to objectivity.‖ this role resonates with my emerging philosophy of teaching and learning. critical mathematics pedagogues therefore should create lifelong learners who strive to learn and use the full potential of the growing power and privilege of mathemacy. these learners then begin to read and write the world with mathematics (gutstein, 2006), ―understand[ing] relations of power, resource inequities, disparate opportunities and explicit discrimination among different social groups based on race, gender, class, language, and other differences‖ (e. gutstein, as cited in stinson et al., 2009). as for my becoming a critical mathematics pedagogue, it will be everchanging, controversial, and conflicted. i will continue to work at engaging my students in dialogue and refrain from depositing: dialogue is thus an existential necessity. and since dialogue is the encounter in which the united reflection and action of the dialoguers are addressed to the world which is to be transformed and humanized, this dialogue cannot be reduced to the act of one person‘s ―depositing‖ ideas in another, nor can it become a simple exchange of ideas to be ―consumed‖ by the discussants. (freire, 1970/2000, pp. 88–89) that is to say, i will not only be the teacher but also the student, as i listen to and learn from (and with) my students in mutually humanizing dialogue. in the past, i have refused ―the dialogical character of education as the practice of freedom‖ (freire, 1970/2000, p. 93), slighting the possibilities of education as a means for social change (dewey, 1937/1987). in speaking about educators who adopt such a position, dewey argues: but i am surprised when educators adopt this position, for it shows a profound lack of faith in their own calling. it assumes that education as education has nothing or next to nothing to contribute; that formation of understanding and disposition counts for nothing; that only immediate overt action counts and that it can count equally whether or not it has been modified by education. (p. 412) i was one of those educators—slighting the possibilities of social change via education. i conducted my teaching practices as if the primary goal was to only dawson public stories journal of urban mathematics education vol. 2, no. 2 17 spread the gospel of mathematics. but now, i am deconstructing, i am reflecting, i am transforming and becoming an agent of change. my classroom is a think tank, a greenhouse for mathemacy, and though the old is not removed, it is being entrenched in reflection, critique, and action. and for the guilt that i feel about the indoctrination that occurred in my first 8 years of teaching, i soothe it with the thoughts of creating mathematicians that read and re-write the world and, in turn, create a society where mathemacy (and justice) is indeed for all. references dewey, j. (1987). education and social change. in j. a. boydston (ed.), john dewey: the later works, 1925–1953 (vol. 11, pp. 408–415). carbondale, il: southern illinois university press. (original work published 1937) dewey, j. (1998). experience and education: the 60th anniversary edition (60th ann. ed.). west lafayette, in: kappa delta pi. (original work published 1938) freire, p. (1997). a response. in p. freire (ed.), mentoring the mentor: a critical dialogue with paulo freire (pp. 303–330). new york: peter lang. freire, p. (2000). pedagogy of the oppressed (m. b. ramos, trans., 30th ann. ed.). new york: continuum. (original work published 1970) gutstein, e. (2006). reading and writing the world with mathematics: toward a pedagogy for social justice. new york: routledge. skovsmose, o. (2005). travelling through education: uncertainty, mathematics, responsibility. rotterdam, nl: sense. stinson, d. w. (2004). mathematics as ―gate-keeper‖ (?): three theoretical perspectives that aim toward empowering all children with a key to the gate. the mathematics educator, 14(1), 8–18. stinson, d., w. (2009). the proliferation of theoretical paradigms quandary: how one novice researcher used eclecticism as a solution. the qualitative report, 14(3), 498–523. retrieved from http://www.nova.edu/ssss/qr/qr14-3/stinson.pdf. stinson, d. w., bidwell, c. r., & powell, g. c (2009). critical pedagogy and teaching mathematics for social justice. unpublished manuscript, atlanta: georgia state university. http://www.nova.edu/ssss/qr/qr14-3/stinson.pdf journal of urban mathematics education december 2009, vol. 2, no. 2, pp. 1–5 ©jume. http://education.gsu.edu/jume david w. stinson is an assistant professor of mathematics education in the department of middle-secondary education and instructional technology, in the college of education at georgia state university, p.o. box 3978, atlanta, ga, 30303-3978; e-mail: dstinson@gsu.edu. his research interests include exploring sociocultural and sociohistorical aspects of mathematics and mathematics teaching and learning from a critical postmodern theoretical (and methodological) perspective. he is co-founder and current editor-in-chief of the journal of urban mathematics education. editorial mathematics teacher educators as cultural workers: a dare to those who dare to teach (urban?) teachers david w. stinson georgia state university ecently, there has been a stormy whirlwind of disparaging rhetoric blowingabout regarding the preparation of teachers, and the work that colleges and schools of education do—work that, i believe, is the most vital work any institution of higher learning might undertake. being relatively new to the profession of teacher education, i thought that the current assault on colleges and schools of education was a new phenomenon, somehow a means of surveillance and discipline (foucault, 1975/1995), similar to the solidifying surveillance and discipline that is occurring in k–12 schools. william ―bill‖ mcdiarmid (2009), dean of the school of education at the university of north carolina at chapel hill, however, informed me: a new phenomenon—it is not. dean mcdiarmid (2009), responding, in part, to secretary of education arne duncan’s (2009a) october 9th speech delivered at the rotunda at the university of virginia and, in part, in anticipation of a forthcoming similar, highly critical speech, claimed that ever since colleges and schools of education were established within universities over a century ago they ―have been the frequent whipping boy of politicians, commentators, liberal arts faculty and others‖ (mcdiarmid, 2009, ¶ 1). to substantiate his claim, mcdiarmid provided a 1928 quote from h. l. mencken, the noted american journalist and essayist: the great majority of american colleges [of education] are so incompetent and vicious that, in any really civilized country, they would be closed by the police. …in the typical american state they are staffed by quacks and hag-ridden by fanatics. … the profession mainly attracts…flabby, feeble fellows who yearn for easy jobs. (as cited in mcdiarmid, 2009, ¶ 1) but even with this enlightened history regarding the ―status‖ of colleges and schools of education, i waited (and i suppose like many other teacher education professionals) with hopeful anticipation for secretary duncan’s (2009b) followr stinson editorial journal of urban mathematics education vol. 2, no. 2 2 up speech that he was to deliver on october 22nd at teachers college, columbia university. the 23 days between the two speeches, i believed, provided secretary duncan ample time to re-reflect, re-think, and re-engage with several teacher educators and education scholars who hold different and divergent perspectives on colleges and schools of education, and the work that they (we) do—or could do. after all, didn’t president obama win the presidency, in part, on a platform that emphasized the crucial need to bring different and divergent voices to the table when discussing grave national concerns? and isn’t developing a more complex—perhaps, even a rhizomatic (deleuze & guattari, 1980/1987)—understanding of how the nation might provide all its children with a humanizing pedagogy (bartolomé, 1994) equal to any other current national concern? therefore, i imagined that secretary duncan, as an agent at the highest level of the obama administration, would bring different and divergent voices to the table in those intervening days. 1 i, however, was sadly disappointed. when secretary duncan (2009b), during his speech at teachers college, referred to e. d. hirsch as the ―father of the acclaimed, content-rich core knowledge program‖ (¶ 29), i knew then that secretary duncan and i have very different understandings of knowledge (cf. foucault, 1969/1972), and therefore, (most likely?) very different visions for ―effective‖ teacher education programs and colleges and schools of education. as he applauded e. d. hirsch’s (1988) book cultural literacy: what every american needs to know, i wondered what he might think in regards to how i use hirsch’s text, and other similar text (see, e.g., schlesinger, 1998), in my mathematics teacher education courses. along with scholars such as joel spring (2008) and others, i do use hirsch’s text as an extraordinary exemplar, an exemplar of the delimiting and debilitating hegemony that structures u.s. public schools and society in general—a hegemony that, on the whole, continues to reify and privilege ―only one universal subject of human history—the white, anglo, heterosexual male of bourgeois privilege‖ (p. mclaren as cited in torres, 1998, p. 178). i also wondered why secretary duncan, in either speech, failed to mention any one of the three books that have become the foundation in which the learningteaching–teaching-learning experience is built upon in my mathematics teacher education courses: john dewey’s (1938/1998) experience and education, paulo freire’s (1970/2000) pedagogy of the oppressed, and gloria ladson-billings’s (1994) dreamkeepers: successful teachers of african american children. my 1 henry giroux (2009) noted, ―just as his economic advisory team and his security council include not one progressive or antiwar advocate, obama’s education team is divorced from liberal and left-liberal perspectives‖ (p. 259), and claimed that president obama’s ―call for education reform in many ways embraces many of the same arguments made by george w. bush‖ (p. 264). stinson editorial journal of urban mathematics education vol. 2, no. 2 3 hope—and yes, i am still hopeful 2 —is that many of the scholars whose work has informed not only my scholarship but also, and most important, my pedagogy will be asked to the table (and soon) to provide their different and divergent perspectives on the national concern of teaching all of the nation’s children and, in turn, the teacher educators who teach (and learn with) the teachers 3 (see harvard educational review, volume 79, number 2: ―education and the obama presidency‖ for perspectives from many of these scholars). although i was disappointed by secretary duncan’s remarks at both the rotunda and teachers college, the whirlwind that his remarks blew up did give me pause to think and reflect, and begin to examine more precisely what i—as an urban mathematics teacher educator—do in my courses, and how and why i do the things i do. in other words, his remarks made me think: what if i was asked to the table? just how might i precisely articulate my vision for an effective urban mathematics teacher education program or urban teacher education program in general? but then again, why do i even use the word ―urban‖ as a descriptor? is an urban mathematics teacher educator, urban mathematics teacher educator program, or, for that matter, an urban college or school of education different from say, a ―non-urban‖ mathematics teacher educator? should they be different? or should they be the same? is an urban mathematics teacher educator one who establishes issues such as diversity, equity, democracy, freedom, and social justice as primary and reoccurring themes in her or his mathematics education courses? but shouldn’t such issues be primary and reoccurring themes in every mathematics teacher educators’ courses? or, for that matter, in every teacher educators’ courses? even in non-urban (mathematics) teacher educators’ courses? i could go on ad infinitum posing questions that secretary duncan’s remarks motivated for me as i began to examine more precisely the what, how, and why i do the things i do in my courses, and how the things i do might or might not be compatible or 2 my hope is derived, in part, by comments made by linda darling-hammond (2009) who headed president obama’s education policy transition team; she stated: what attracted me to [obama’s] campaign...were his early pronouncements of education. i sensed a sincerity and a depth of commitment to education, a genuine concern for improving the quality of teaching and learning, an intolerance of a status quo that promotes inequality, and a drive to move our education system into the twenty-first century—not only in math, science, and technology but also in developing creativity, critical thinking skills, and the capacity to innovate—a much-needed change from the narrow views of the last eight years. (pp. 210–211) 3 for a bit of speculative storytelling on the probable outcomes of secretary duncan’s current ―corporate model‖ strategy, see gloria ladson-billings (2009). and although i do not claim that dr. ladson-billings possesses some mystical powers of predicting the future (nor does she); i have discussed with students that her speculative storytelling or predictions regarding new orleans after katrina have been and continue to become frighteningly accurate (see ladson-billings, 2006), an acknowledgment that she makes note of as well (2009). stinson editorial journal of urban mathematics education vol. 2, no. 2 4 consistent with secretary duncan’s vision for effective teacher education programs and colleges and schools of education in general. nonetheless, so that i might ―scientifically‖ respond to secretary duncan if i am asked to the table, i have begun doing science on the what, how, and why i do the things i do in my mathematics education courses (and yes, the work that a critical postmodernist does is science). as i began this new science project, i turned to the grand master of transformative, empowering pedagogy: paulo freire. it is freire’s (1998b) posthumously published book teachers as cultural workers: letters to those who dare teach that inspired the title of this editorial. (i recommend this book as a starting point for any [mathematics] teacher educator, urban and non-urban alike, who might choose to begin a similar such project.) although freire wrote his 10 letters to brazilin teachers, i thought that i might use his letters to begin organizing—in a more precise, scientific fashion—the what, how, and why i do the things i do as a teacher (and learner) of (with) urban mathematics teachers in the united states. i did not turn to his letters to provide me with prescriptive answers to what i should do; freire, throughout his prolific scholarship (see, e.g., 1970/2000, 1985, 1994, 1997, 1998a, 1998b), was adamant that his work is not to be duplicated but reinvented. rather, i turned to freire’s letters to remind me that, as in all of his scholarship, he asks teachers, teacher educators, and, in turn, colleges and schools of education to dare to think differently, to dare to act differently, to dare to dream differently: we must dare to learn how to dare in order to say no to the bureaucratization of the mind to which we are exposed everyday. we must dare so that we can continue to do so even when it is so much more materially advantageous to stop daring. (1998a, p. 3) as teachers of (urban?) mathematics teachers, i want to challenge—no, i dare—each of us to institute as the primary goal for the community of mathematics educators the cultural transformation of the discipline of mathematics from the psychologically brutalizing discipline of stratification (bourdieu, 1989/1998) into the psychologically humanizing discipline of freedom. we have, i believe, an ethical responsibility to do so. and for me, this dare is how i plan to begin the discussion if (when?) i am asked to the table. how might you begin the discussion? references bartolomé, l. i. (1994). beyond the methods fetish: toward a humanizing pedagogy. harvard educational review, 64(2), 173–194. bourdieu, p. (1998). the new capital, in practical reason: on the theory of action (pp. 19–30). stanford, ca: stanford university press. (lecture delivered 1989) darling-hammond, l. (2009). president obama and education: the possibility for dramatic improvements in teaching and learning. harvard educational review, 79(2), 210–223. stinson editorial journal of urban mathematics education vol. 2, no. 2 5 deleuze, g., & guattari, f. (1987). introduction: rhizome, in a thousand plateaus: capitalism and schizophrenia (b. massumi, trans., pp. 3–25). minneapolis, mn: university of minnesota press. (original work published 1980) dewey, j. (1998). experience and education: the 60th anniversary edition (60th ann. ed.). west lafayette, in: kappa delta pi. (original work published 1938) duncan, a. (2009a). a call to teaching: secretary arne duncan’s remarks at the rotunda at the university of virginia. retrieved from http://www.ed.gov/news/speeches/2009/10/10092009.html duncan, a. (2009b). teacher preparation: reforming the uncertain profession—remarks of secretary arne duncan at teachers college, columbia university. retrieved from http://www.ed.gov/news/speeches/2009/10/10222009.html foucault, m. (1972). the archaeology of knowledge (a. m. sheridan smith, trans.). new york: pantheon books. (original work published 1969) foucault, m. (1995). discipline and punish: the birth of the prison (a.m. sheridan, trans.). new york: vintage books. (original work published 1975) freire, p. (1985). the politics of education: culture, power, and liberation (d. macedo, trans.). westport, ct: bergin & garvey. freire, p. (1994). pedagogy of hope: reliving pedagogy of the oppressed (r. b. barr, trans.). new york: continuum. freire, p. (1998a). pedagogy of freedom: ethics, democracy, and civic courage (p. clarke, trans.). lanham, md: rowman & littlefield. freire, p. (1998b). teachers as cultural workers: letters to those who dare teach (d. macedo, d. koike, & a. oliveira, trans.). boulder, co: westview. freire, p. (1997). pedagogy of the heart (d. macedo & a. oliveira, trans.). new york: continuum. freire, p. (2000). pedagogy of the oppressed (m. b. ramos, trans., 30th ann. ed.). new york: continuum. (original work published 1970) giroux, h. a. (2009). obama’s dilemma: postpartisan politics and the crisis of american education. harvard educational review, 79(2), 250–266. hirsch, e. d. (1988). cultural literacy: what every american needs to know (1st vintage books ed.). new york: vintage books. ladson-billings, g. (1994). the dreamkeepers: successful teachers of african american children. san francisco: jossey-bass. ladson-billings, g. (2006). introduction. in g. ladson-billings & w. f. tate (eds.), education research in the public interest: social justice, action, and policy (pp. 1–13). new york: teachers college press. ladson-billings, g. (2009). education for everyday people: obstacles and opportunities facing the obama administration. harvard educational review, 79(2), 345–359. mcdiarmid, w. (2009). from the dean: are ed schools really the problem? retrieved from http://soe.unc.edu/news_events/news/2009/091020_from_the_dean.php schlesinger, a. m. (1998). the disuniting of america: reflections on a multicultural society (rev. and enl., 2nd ed.). new york: w. w. norton. spring, j. h. (2008).the intersection of cultures: multicultural education in the united states and the global economy (4th ed.). new york: erlbaum. torres, c. a. (1998). multiculturalism, in democracy, education, and multiculturalism: dilemmas of citizenship in a global world (pp. 175–222). lanham, md: rowman & littlefield. http://www.ed.gov/news/speeches/2009/10/10092009.html http://www.ed.gov/news/speeches/2009/10/10222009.html http://soe.unc.edu/news_events/news/2009/091020_from_the_dean.php microsoft word review_ 379-article text no abstract-1697-1-4-20191213.docx journal of urban mathematics education december 2019, vol. 12, no. 1, pp. 1–7 ©jume. https://journals.tdl.org/jume robert m. capraro is professor of mathematics education and co-director of the aggie stem center at texas a&m university, department of teaching, learning, and culture, 4232 tamu, college station, tx 77843-4232; e-mail: rcapraro@tamu.edu. his research interests are centered on stem educational research initiatives, urban mathematics achievement and representational models, and quantitative methods. editorial duty is necessary, passion is sufficient: it takes both robert m. capraro texas a&m university enjoyed reading the journal of urban mathematics education (jume); i was published in jume, and several of my doctoral mentees were published in it. i reviewed for dr. stinson and used his editorial letters to me as a model for my graduate classes to talk about an editor who does “it” right. i liked the scholarly dialog that took place on the journal’s pages and the balance of new scholars who found a home for their work amongst the work of more established scholars. recently, one of my newest doctoral mentees was developing a manuscript dealing with urban mathematics issues, so i directed him to jume, just to be told the journal had gone inactive for nearly a year. i felt a tremendous sense of loss, both personally and for our field. i also felt immense regret that the work of the prior editorial team was going to pass into history and that their sacrifice and contribution to all of us would only survive as a commemorative footnote. i contacted dr. stinson, learned about the application process, and immediately put together a package to present to him and his team. assuming the mantle of leadership with jume became a perceived duty of mine. i assembled a team, and together we have moved the journal to a new, more modern operating system, developed a journal handbook, established an editorial board, and applied for inclusion in scopus. we were able to do this because we inherited an excellent journal that was already listed in the directory of open access journals and had a robust list of potential reviewers and a prestigious list of scholars who have been published within its pages. i am so pleased that i am at this perfect moment in time with the journal of urban mathematics education to be able to put all my thoughts, hopes, dreams, and promises in print so that you, our readers, can hold me and the team accountable. when i completed my application for the editorship of jume, i was asked to respond to several questions from the committee. one was to conceptualize the mission statement, and below is what i wrote on the application: the mission of the journal is to foster discourse among a community of scholars to catalyze and transform the global academic space in mathematics education into one that embraces critical research, emancipatory pedagogy, and scholarship of engagement in urban communities. here, “urban” transcends geographical, socio-economic, gender i capraro editorial journal of urban mathematics education vol. 12, no. 1 2 identity, and political contexts and rejects the use of “urban” as code. specifically, our focus is on the teaching and learning in formal and informal places related to mathematics situated within urban contexts and wherever else teaching and learning takes place. this mission is not completely unrelated to what was already in place. at most, it was my opportunity to read each word, consider meanings, and plan for how i would act and think about research and potential manuscripts. the deep reflection on the mission precipitated an internal dialog about how i would foster and nurture new and more seasoned scholars to work outside the box and encourage jume to be a safe place to try out new ideas, suggest new paradigms, and enact new methodologies. my driving question was about the nature of duty and what my duty is in this situation. i was struck with the idea that a good editorial team pushes limits, but i was haunted by what dr. carl grant, editorial board chair, said to me during an interview for another journal much earlier in my career. dr. grant said, “a great editorial team creates a space where the field pushes limits and takes the heat when it goes wrong.” to me, that meant that the editors do not push limits. instead, they create the environment where scholars challenge the status quo, where junior scholars challenge long-standing beliefs and practices, where new ideas have an opportunity to be heard, and where senior scholars can express new or evolving understandings of their long-standing positions. as such, it is the scholars in the field who must push the limits, take risks, and submit their work. when ideas receive critique, the editorial team must assume its responsibilities of making the decision to publish and providing opportunities for vetted professional dialogs to play out in published literature. in my response to the question on my application, “why are you the best person to be the next editor of jume?” i responded as honestly as i could have managed. i wrote: i am not the best person to do the job. i can think of three or four others who would likely do a better and perhaps more efficient job than i will be able to do. however, what i bring with me is broad experiences in mathematics education research, institutional support, a large committed team, and a history of successful editing practices. there are some amazing scholars in this space, and i am humbled by their accomplishments and the scope and breadth of their research. looking at the list of now-senior scholars published in the journal’s first 10 years alongside the many new scholars who make up the early ranks of assistant professors or new associate professors in mathematics education, i see that there is a great deal of room for the journal to grow alongside this highly motivated and enthusiastic field. i believe that my work ethic and the current team will ensure that jume reaches its potential as a journal that serves the field and its many voices while retaining the amazing quality it has had in the past. therefore, i stand by my short but complex statement, and i feel capraro editorial journal of urban mathematics education vol. 12, no. 1 3 strongly that this team is uniquely situated to nurture and support the journal mission and evolution of jume. my experience with editing and my service on editorial boards provide the background and skills necessary to deal with the complexity of our field and the flexibility that it demands. there may be many people who received a rejection from me in my previous editorial roles; after all, the acceptance rate of the editorial boards i served on ranged from 5–25%. this percentage means that many people received rejection letters from me, and that group far outnumbers those who received acceptance letters. so, there will always be more people with a reason to be angry with me than to be happy. i expect that jume will also have an acceptance rate between 5–25%, and i cannot promise that everyone will be happy with this. i can promise, however, to value the work entrusted to our review, to respect the efforts that our authors have made, and to treat each manuscript as i would expect mine to be treated. my editorial practices have always fostered a diverse conversation without bias toward a paradigm, entitled university group, or legendary professor status in the field. this journal will not favor one research paradigm over another, and we will not honor or harbor any preconceived notions that “good research” comes from a handful of universities or graduates of a small number of professors (past or present). i promise that no one associated with the journal will ever imply that one person’s work is always good or that a newly submitted manuscript will be good by association. each piece, each effort, each hard-fought research battle will stand on its own, judged for its merits and accepted or rejected based on its individual quality. with false hurdles and preconceptions removed from the editorial process, i believe the jume team will be uniquely situated to nurture junior and mid-career faculty and underserved scholars with important messages, and thus help move marginalized research lines to a broader audience. for our vision to move marginalized research lines to a broader audience, the right team must be in place, one that is not hobbled by allegiances that propagate biases toward certain research lines while marginalizing others. the former editorial team of jume functioned well before our tenure. i am very grateful for all the work they put forth! they did the work for free, off the sides of their desks and with little recognition. much of this is still true for our current team, but times change and transparency is paramount during periods of transition. the best journals in academia have an editorial board: scholars in the field whose perspectives frame our work. to ensure that jume’s editorial team remains dedicated to our high standards and lofty goals, we decided to solicit the most forward-thinking, provocative, and committed scholars in mathematics education for our first editorial board. one editorial board member will be a reviewer on each and every manuscript we send out for review. the most prominent and recognizable scholars in our field were solicited, as well as some incredible junior scholars with amazing promise. to choose a scholar to nominate for the editorial board, we read capraro editorial journal of urban mathematics education vol. 12, no. 1 4 the work of each nominee, listened to their published voice, and were persuaded by their logical arguments, reasoned perspectives, and fearless confrontation of the status quo. i am pleased that we had a 72% acceptance rate to serve on the board and that only one person did not respond to our invitation. the outstanding scholars who will serve on the editorial board will function as the rudder to ensure that although the jume ship may tack against the wind, the journal’s course remains true. there were those who declined to serve on the board. the main reason for this decision is best characterized by this quote from one of those who declined: “. . . i am just swamped! however, i am happy to be a reviewer though, so please keep me on as a reviewer.” i am honored that so many capable scholars have agreed to lend their skills and experience to jume. in my application, i was also asked to respond to a question about how i would move beyond criticism or stir controversy. my response–– not sure what this is really attempting to disentangle. i am not sure what the “controversial” refers to, nor what we might call controversial. i believe that the role of the editor is to foster diverse voices and provide opportunities for those who have diverse research interests and perspectives, to have those discussions in the public space where commentary is permitted, but more importantly, the work is welcomed and solicited when appropriate. extreme controversial topics often increase readership, citations, and the overall prestige of the journal. receiving those extreme perspectives is probably the most important job of any editor. as i reflect on what i wrote, i realized there was subtext that reveals my true feelings, particularly in the statement, “receiving those extreme perspectives is probably the most important job of any editor.” what i meant here is that the field is responsible for submitting extreme perspectives, and the editorial team is responsible for fostering diverse voices and welcoming those perspectives when they are submitted. in my response, i also revealed that i believe that most of us do not think our ideas are controversial or proffer an idea just to be controversial. we move forward with ideas because we believe they are right and just. we believe that it would make the world, even when that world is just our neighborhood, a better place. i truly believe that controversial ideas are perceived to be controversial by those who disagree with an idea as a way to label what they do not understand well. another idea that needs to be unpacked from the lack of context in my response about what is thought to be controversial was that i substituted the word “perspectives” toward the end. it is not totally unrelated to my positionality for an editorial team nor my belief that “controversiality” resides in the receiver, listener, or reader. i also do not believe that controversiality is inherently characteristic of an idea that the word is ascribed to. i believe that an extreme or controversial perspective simply comes from a scholar moved by extreme passion to act, one whose belief in change stands in conflict with the status quo. capraro editorial journal of urban mathematics education vol. 12, no. 1 5 at this point, i would like to discuss two different words for a person who disrupts: disrupters and disruptors. during my 15 years as a public-school educator and 20 years as a higher education faculty member, i tended to find that those who were termed a disrupter often expressed their passion about a situation or context, and someone outside that situation or context who found that person’s opinion “controversial” labeled them a disrupter. this was typically intended to be negative. however, thanks to business, we have the term disruptor (see snihur, thomas, & burgelman, 2017; webb & gile, 2001). a disruptor is a person or entity that fosters an idea that creates a new niche or network that eventually revolutionizes an existing niche or network. as a result, a disruptor often displaces established ideas, ways of doing things, and/or alliances. it is important to note that revolutionaries are not all disruptors. the problem with the term having two etymologies, two different uses, and two different interpretations is that it is often difficult to know what a person means when it is used, and these multiple uses create a great deal of ambiguity. the more meanings a term has, the more difficult it is for those from across disciplines to understand intended meanings (see barroso et al., 2017; rugh et al., 2018). because we do not know which sense is intended when disrupter/disruptor is used orally, it is difficult to know if we have been complimented or insulted. for an excellent example of early positive disruptors, read about the impactful and important statements made in the editorial by my colleague entitled “…and a little child shall lead them.” perhaps an answer to internal and external struggles might be to spend less time labeling others and to invest more time understanding the passion underlying their actions. paradigm war free zone i can imagine many authors will want to know how my own personal methodological paradigm will influence the paradigms accepted and published in the journal. when i thought of how to respond to this question during the application process, i was reminded of something dr. richard duran once said to me: “the best research is published research!” he said this statement to me in conversation as a response to my concern that we, as a field, only value what is published. unfortunately, there are many potential obstacles that can keep phenomenal research from ever being published. therefore, editors, reviewers, and everyone else in the publication process must be cognizant that what we do impacts real people doing amazing work. we must not be blinded by our own subjective standards or biases; these must be set aside when we review the work of others. we must avoid prejudice in the review process and avoid pushing our paradigm on others by asking researchers to conduct a study we would design. we must respect the works of our peers and offer fair critique and discussion in our reviews and commentaries. capraro editorial journal of urban mathematics education vol. 12, no. 1 6 my conversation with dr. duran occurred during the time that the american educational research association (aera) was developing its empirical standards (duran et al., 2006). so, to be clear, i did not interpret dr. duran’s comment to mean that we should not hold each other to high standards when reporting research but that those standards should be transparent when possible and always objective and free of our own personal biases. as i reflected on this matter while assembling my application for the jume editor position, i was confronted with some import myths and truths. the greatest of these is one that i heard during my professional training to become a faculty member and have repeatedly heard since: “the question always drives the paradigmatic choice.” funny that in my career, during which i have helped author almost 200 publications, i have only come across nine questions that necessitated qualitative inquiry. i only became aware of this fact when i worked in collaboration with someone just like me, except she on only rare occasion found a question that necessitated quantitative inquiry. i think that the reality is that a person cannot divorce the “self” from the inquiry or the type of questions that, at least subliminally, aligns to one’s own perspectives and beliefs. i have come to learn that my lens is quantitative and that i need a team who is as strong qualitatively as i am quantitatively. unapologetically, i believe the jume editorial team is comprised of strongly positioned senior and junior scholars who excel in both quantitative and qualitative methods. this both–and ensures that all “good work,” regardless of paradigm, gets published in jume. we also believe that the aera reporting practices article provides valuable assistance when conducting quantitative studies. to further assist with this commitment, we have solicited an editorial from a senior scholar and a more junior scholar in qualitative research and a senior scholar in quantitative research to help make “good work” a bit more transparent. our sense of duty has brought the entire team to action, but it is our collective passion that will sustain us and make what could have been an arduous endeavor a work of love through an act of selfless service. warmest thanks to david and his amazing team of scholars, we are immensely grateful that you allowed us to continue your work and keep your dreams alive in the amazing journal you nurtured for 10 years. we stand on your shoulders as we continue your work and build jume’s next chapter upon the foundation your team created. we hope our team makes you proud when you reflect on our progress and accomplishments. we trust you can manage a nod and a kind word or two in the twilight of our leadership and the dawn of our own successors in 2025. to our readership, reviewers, and potential jume authors, we are humbled to lead jume and are in awe of the work undertaken in our field. we hope that if you capraro editorial journal of urban mathematics education vol. 12, no. 1 7 are not currently engaged with jume, that you sign up to be a reviewer. if you are interested in serving on the editorial board, please make personal contact with an editor. look for us at the editor’s roundtable at aera. stop in, pull up a chair, and tell us what you are doing or ask questions. we look forward to engaging in broad dialogs and brainstorming for ways to bring the larger community closer together. isn’t it a pleasure to study and practice what you have learned? isn’t it also great when friends visit from distant places? if one remains not annoyed when he is not understood by people around him, isn’t he a sage? –confucius references barroso, l. r., bicer, a., capraro, m. m., capraro, r. m., foran, a., grant, m. l., lincoln, y. s., nite, s. b., oner, a. t., & rice, d. (abc order). (2017). run! spot. run! – vocabulary development and the evolution of stem disciplinary language for secondary teachers. zdm, 48(2), 187–201. duran, r. p., eisenhart, m. a., erickson, f. d., grant, c. a., green, j. l., hedges, l. v., . . . schneider, b. l. (2006). standards for reporting on empirical social science research in aera publications: american educational research association. educational researcher, 35(6), 33–40. rugh, m. s., calabrese, j. e., madson, m. a., capraro, r. m., barroso, l. r., capraro, m. m., bicer, a. (2018). stem language can be the stem of the problem. in proceedings of the 48th annual ieee frontiers in education conference (pp. 1–8). piscataway, nj: ieee. snihur, y., thomas, l. d., & burgelman, r. a. (2017). the disruptor’s gambit: how business model disruptors use framing for strategic advantage. academy of management proceedings, 2017(1), 1–6. webb, j., & gile, c. (2001). reversing the value chain. journal of business strategy, 22(2), 13–17. copyright: © 2019 capraro. this is an open access article distributed under the terms of a creative commons attribution-noncommercial-sharealike 4.0 international license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. microsoft word young_and_capraro_spring_2023 jume editorial.docx journal of urban mathematics education june 2023, vol. 16, 1, pp. 1–9 ©jume. https://journals.tdl.org/jume 1 journal of urban mathematics education vol. 16, no. 1 editorial equity, excellence, and editorial ethics: you’re an editor, now what? ll the parties involved in the publishing process, that is, the author, the editorial board, reviewers, the editors, and aggie stem, the organization responsible for publishing the contents, must function under the same umbrella of ethical behavior. our editorial ethics statement is based on the committee on publication ethics (cope) guidelines of good publication practices (https://publicationethics.org/files/u7141/1999pdf13.pdf). however, these guidelines do not go as far as we would at the journal for urban mathematics education. the editorial team of the journal for urban mathematics education is responsible for deciding which manuscripts will be published, with guidance from the editorial board and external reviewers. in the sections that follow, we provide readers, authors, and other editors an overview of the key elements of editorial ethics that guide the journal for urban mathematics education: (1) publication decisions, (2) peer review, (3) equal opportunity, (4) neutrality, (5) journal metrics, (6) confidentiality, and (7) editorial independence. using these elements as our foundation, we hope to become the standard for equity and excellence in editorial ethics. in this editorial, we unpack these key elements and how each is handled by the journal, and we conclude with ten considerations for newly appointed editors. publication decisions the publication decision refers to the process by which editors decide whether to accept or reject a manuscript for publication. the importance of ethical considerations in publication decisions is highlighted in the literature (avanzas et al., 2011; carver et al., 2011). however, documented advice for journal editors in the social and behavioral sciences remains elusive. the cope best practice guidelines for journal editors emphasized the need for fairness, impartiality, and transparency in decision-making processes (cope, 2017). additionally, graf et al. (2007) stressed the significance of maintaining integrity and avoiding biases in publication decisions. they stated, “editors should base their decisions solely on the merits of the manuscripts and their relevance to the journal’s content” (p. 2). this implies that editors should prioritize the quality and scientific rigor of the work, rather than a robert m. capraro professor emeritus texas a&m university jamaal young texas a&m university capraro & young editorial 2 journal of urban mathematics education vol. 16, no. 1 personal or subjective preferences. the journal for urban mathematics education was founded on the ideals of independence of thought and action. yet, much of what we have learned has been handed down in conversations and informal discussions. so, the ideas of independence must be unpacked to be understood. they are complex because they deal with a human endeavor that is both messy and dynamic. to help us to explicate independence, we have divided it into two categories: independence of thought and independence of action. 1. independence of thought deals with three aspects. first, it protects the author from undue influence from perversion of their work at the whim of reviewers or from heavy-handed editors. second, it also deals with protecting our dedicated readership from bias and slanted or stilted presentations. when possible, all sides of issues are given equal weight in the journal, free of censorship. finally, independence of thought protects the editors and editorial board from undue thought constraints that might sway the mission of the journal for urban mathematics education. from the start through the finished product, every outcome is the work of a dedicated group of people who have a stake in the outcome, who work diligently to ensure that independence of thought is preserved. 2. independence of action is about two primary goals. first, independence of action ensures that authors are not coerced to follow a trend, pursue a framework that is gaining prominence, or shun work in a declining frame. in no way should editors, editorial board members, or reviewers in any way use their positions of power to promote or demote an author’s and researcher’s choice of what they research or how they research. second, independence of action ensures that the editorial team is free of external pressures to make decisions about a work based on an external entity’s perspectives or expectations. peer review peer review is a critical component of the publication process, involving the evaluation of manuscripts by experts in the field. the importance of peer review in ensuring the quality and validity of published research is emphasized across the literature. specifically, the peer review process is paramount to making a decision. it is fraught with the potential for misuse and abuse. therefore, we use a process that attempts to mitigate the negatives. our assistant editors work independently to assign at least two reviewers and one editorial board member based on reviewers ’ selected interests in the journal of urban mathematics education database. this capraro & young editorial 3 journal of urban mathematics education vol. 16, no. 1 process ensures that no editor can manipulate who is assigned to review any manuscript. in an ideal world, each of the solicited reviewers would complete the review; however, the solicited reviewers who subsequently complete the review is always surprising. though who completes the review is always unpredictable, the most problematic issue of the editorial process is dissonance between the review comments and the recommendations. first, reviewers have a pattern that we believe is completely unacknowledged by the reviewers themselves. some reviewers almost always recommend to reject, whereas others almost always recommend to revise and resubmit or accept. the pattern holds even when the comments they make do not support the decision, with often a chasm between the comments provided by any one reviewer and their recommendation. therefore, as an editorial team, we focus more on the substance of the comments and rely less on the recommendation. for example, we recently received a very detailed review, probably a best-case review. the reviewer pointed out issues of conflict between the theoretical framework and conduct of the study, misalignment between the data collected and how it was analyzed, and that the findings had little to do with the data that were collected. the reviewer made many helpful suggestions on how to revise the work and to make it more rigorous and publishable. the recommendation was to accept with minor revisions. from the editor’s perspective, this study would almost have to be completely reconstructed. moreover, many scholars who identify issues between the data collected and the research would determine that the study as reported was fatally flawed and could not be revised. thus, many reviewers would recommend rejecting the manuscript. it is important to note that fatal flaws are highly contextualized. though one use of the data creates a problem, applying another lens and taking a second look might result in an exemplary study guided by different questions and possibly different analyses. another issue that arises is reviewers recommending rejection but providing little evidence on what led to the decision. for example, identifying minor issues with language or formatting, minor alignment issues, or a lack of detail in a particular section. this review and recommendation alignment issue creates an additional level of concern for editors because peer review serves as a critical filter to assess the validity, reliability, and relevance of scientific work before it reaches publication (benatar, 1998, p. 155). however, when reviews do not provide substantive feedback with explicit examples, it creates a conundrum for editors who have the final say in publication. these are just a few examples of the challenges that arise during the peer review process. in subsequent sections below, we explicate how these and similar challenges are addressed by the editorial team. capraro & young editorial 4 journal of urban mathematics education vol. 16, no. 1 equal opportunity equal opportunity is a fundamental principle that should guide editorial practices, ensuring fairness and inclusivity in the selection and publication of research (reich, 2013). a journal editor evaluates manuscripts solely based on their intellectual content, irrespective of the author’s race, gender, sexual orientation, religious beliefs, ethnic origin, nationality, political philosophy, or geographic location. thus, editors should actively promote inclusivity and diversity in their authorship and editorial boards (islam & greenwood, 2022, p. 1). this is a major objective at the journal that we remain dedicated to as an editorial team. we also work to ensure that all reviews are not only objective but educative as well. during the beginning of the process, once the initial review is complete, articles unlikely to receive a constructive external review are returned to the author with a clear explanation about the fit issues of the article. this process is often defined as a desk review. however, articles judged likely to receive a constructive external review are sent out for full review. the journal of urban mathematics education takes this responsibility so intensely that we even have section editors whose responsibility is to ensure that articles have a fair opportunity for evaluation. our sections are carefully crafted to reflect the key lines of inquiry related to urban mathematics education. additionally, some of these sections are research strands historically underrepresented in urban mathematics education; thus, we created specific sections to help increase the visibility and expertise in these strands (i.e., early college and community college experiences). however, equal opportunity not only pertains to authors and reviewers but also data. good data are the foundation of “good” science. data are hard-won and often come at great expense, both personal and financial. those data can result from hard-won and highly competitive grants. the editorial team believes that the field must make the most of these precious resources. data should be conserved and preserved, it should be shared and compared, reused, and reproduced. this broad acceptance of reanalysis facilitates scrutiny of previously published findings by those who might be diligently pursuing related questions. thus, to support equal opportunity, the data from published work should be easy to find, comprehensively described, and accessibly stored. to accelerate data publication and sharing, the journal of urban mathematics education will pursue and eventually offer its authors secure and permanent storage. we are committed to equal opportunity in all aspects of the term. neutrality neutrality in the context of editorial ethics refers to maintaining an impartial stance and avoiding conflicts of interest that could compromise the integrity of the capraro & young editorial 5 journal of urban mathematics education vol. 16, no. 1 publication. stichler (2018) highlighted the ethical responsibility of editors to remain neutral and unbiased in their decision-making. moreover, ethics in the editorial process are essential to the integrity of the peer-review process, to the credibility of the journal and its content, and to ensuring that authors and readers can trust the fairness and objectivity of the review process (mack, 2017, p. 030101-1).this includes disclosing any potential conflicts of interest and ensuring that reviewers are also free from conflicts that could influence their evaluations. all of the members of the editorial team and board for journal of urban mathematics education are required to disclose conflicts of interest that may influence their evaluations. common conflicts occur when current and former students of a colleague choose to submit manuscripts. when these conflicts arise, an external review panel is convened to help maintain objectivity and rigor through neutrality of the peer-review process and decision-making. in summary, neutrality is independence from external forces, conferences, donors, researchers, and national and private funding agencies. the journal of urban mathematics education is dedicated to publishing highquality research in urban mathematics education based on objective decisions that place the quality of the scientific product at the forefront by reducing the effects of nepotism within our publication decision process. journal metrics journal metrics, such as impact factor, citation counts, and altmetrics, are quantitative measures used to assess the influence and reach of a journal’s publications. while these metrics have gained importance in academia, ethical concerns have emerged regarding their use. huggett (2013) discussed the potential pitfalls and ethical issues associated with the use of journal metrics. the author argued that metrics should not be the sole determinant of a publication’s value and that they should be used cautiously to avoid distorting incentives or creating biases in publication decisions. moreover, metrics should be used in a responsible and ethical manner, with a focus on transparency, fairness, and accuracy in evaluating research and scholarly publishing (islam & greenwood, 2022). as an editorial team, we recognize that given the positionality of journal of urban mathematics education in the field of mathematics education “journal metrics matter.” the editorial team has focused on three major aspects to highlight both immediacy and transparency to improve journal metrics. first, we decided to implement digital object identifiers. as the world becomes more digital, internet crawlers are being utilized more frequently to gather information to direct searchers to resources. we believe that prioritizing digital object identifiers (doi) is in the best interest of our authors and readers. capraro & young editorial 6 journal of urban mathematics education vol. 16, no. 1 open access and dois are just one part of ensuring equitable access to the contents of journal of urban mathematics education. the other consideration, and arguably the most important, is ensuring that the average person can find the articles without needing a privileged level of access. the doi ensures that all crawlers can index the articles and ensures maximizing accessibility. second, we sought scopus indexing. with the transition to open access and the sheer number of openaccess options being able to distinguish between excellent open-access journals and others was a foundational concern. our authors need the confidence to submit their work to the journal of urban mathematics education. thus, our readers need to know we have received the green check (i.e., scopus indexing), the highest level of quality for the director of open access journals (doaj). scopus indexing provides another measure of quality assurance and provides a set of standards for our journal editors, editorial board, and reviewers. finally, equitable access is essential but wholly insufficient if our authors are not receiving maximum exposure of their work. now authors will be able to link their open researcher and contributor id (orcid) to their article, allowing a reader to immediately access more articles by that researcher, to see other projects, and to facilitate connections between the researcher and the reader achieving the ultimate connectivity. researchers often move affiliations but listed contacts in an article is static. linking the orcid allows up-to-date information and an easy way to help find authors regardless of where their career may take them. we are working to improve our metrics because as every journal grows, nurturing the metrics nurtures the authors and feeds the readers. these pathways ensure that the entire editorial team meets the demands for immediacy and transparency. confidentiality confidentiality is a crucial aspect of editorial ethics, ensuring the protection of authors ’rights and the integrity of the peer-review process. editors are expected to maintain the confidentiality of submitted manuscripts and the identities of reviewers. the cope best practice guidelines explicitly state that editors should “keep all information about a submitted manuscript confidential” (cope, 2017). this principle fosters trust between authors, reviewers, and editors and safeguards the integrity of the publication process. the editorial team works diligently to protect confidentiality. the journal of urban mathematics education uses a doubleblind review process, which means that the author and reviewer information is “blinded” on both ends (i.e., for the author and the reviewer). this process extends beyond the initial decision. editors must prioritize the confidentiality of authors and peer reviewers, ensuring that all information related to manuscripts remains confidential and is not disclosed without proper authorization (stichler, 2018, p. 6). capraro & young editorial 7 journal of urban mathematics education vol. 16, no. 1 hence, the editor and the editorial staff must not reveal any information about any manuscript that has been submitted to the journal for revision. the authors, proofreaders, editorial advisors, and members of the editorial and scientific committees are the only people allowed to exchange information, and then only when appropriate. any manuscripts received for revision shall be treated with the utmost confidentiality. they must not be shown, nor must their contents be disclosed to anyone who has not been authorized by the editor. this is one of the simplest yet most important ways that the journal of urban mathematics education strives to maintain rigor and intellectual freedom as an academic outlet. editorial independence the editorial team is completely independent. editorial independence refers to the freedom of editors to make decisions based on their judgment and expertise, free from external influence or pressure. benatar (1998) emphasized the significance of editorial independence in upholding the integrity and credibility of a journal. editors should be able to make decisions based on the scientific merit of the work, without interference from funding sources, institutions, or other external factors. while each editorial member has autonomy over the manuscripts they handle, weight is given to the reviews and perspectives of the reviewers and editorial panel. these reviews can drive the review, but they do not outweigh the expertise of the editorial team member who ultimately makes the final publication decision. a reviewer assists the editorial team when the time comes to make any editorial decisions and via the editorial communications with the author based on the details provided in the submitted review. he/she will be able to help the author improve the content of the paper. however, reviewer recommendations are recommendations, and the action editor is solely responsible for the publication decision of the manuscript. this is an important consideration, as it can create situations where there is disagreement between the reviewer comments, editorial panel expert, and the associate editor. although the associate editor has the final say, the team meets regularly to discuss the manuscripts in the pipeline and any possible concerns or editorial challenges. this process helps to provide a sounding board for editors who may be struggling to reach a decision on a manuscript due to divergence in reviewer comments, intellectual merit, or a number of other related challenges. ten tips for new editors in the preceding sections, we argue that publication decisions should be based on the quality and scientific rigor of the work, prioritizing fairness, impartiality, and transparency. independence of thought and action are crucial in protecting authors, readers, and editors from undue influence and biases. furthermore, capraro & young editorial 8 journal of urban mathematics education vol. 16, no. 1 peer review plays a critical role in ensuring the quality and validity of published research. the editorial team strives to maintain a fair and unbiased review process, assigning reviewers based on their expertise and focusing on the substance of their comments rather than solely relying on recommendations. equal opportunity is fundamental in editorial practices, ensuring inclusivity and fairness irrespective of authors’ backgrounds. the journal actively promotes diversity and inclusivity in its authorship and editorial boards. neutrality is essential in maintaining an impartial stance and avoiding conflicts of interest that could compromise the integrity of the publication. the editorial team discloses and manages conflicts of interest and seeks external review panels when needed. journal metrics should be used responsibly, with a focus on transparency, fairness, and accuracy. the journal for urban mathematics education prioritizes digital object identifiers, scopus indexing, and linking an author’s orcid to improve accessibility, visibility, and connectivity. confidentiality is crucial to protect authors’ rights and maintain trust in the peer-review process. the editorial team follows a double-blind review process and ensures that all information related to manuscripts remains confidential. finally, editorial independence is upheld to make decisions based on scientific merit without external influence. when considering reviewer recommendations, the editorial team has the final say in publication decisions. based on these considerations, we present the following ten tips for newly appointed editors. 1. prioritize independence of thought. protect authors from undue influence and ensure a balanced presentation of all sides of issues without censorship. preserve the mission and integrity of the publication. 2. maintain independence of action. avoid pressuring authors to follow trends or frameworks and make decisions based on external perspectives. uphold the editorial team’s freedom from external pressures. 3. understand the importance of peer review. recognize the critical role of peer review in the publication process and its potential for misuse and abuse. strive to mitigate negative aspects in the process. 4. focus on the substance of reviewer comments. pay more attention to the content of reviewers’ comments rather than relying solely on their recommendations. consider the quality and rigor of the work when making publication decisions. 5. ensure equal opportunity. evaluate manuscripts solely based on their intellectual content, disregarding factors such as author demographics. promote inclusivity and diversity in authorship and editorial boards. 6. preserve and share data. recognize the value of data and promote responsible data publication and sharing. make data easy to find, welldescribed, and accessible to facilitate scrutiny and reproducibility. capraro & young editorial 9 journal of urban mathematics education vol. 16, no. 1 7. maintain neutrality. remain impartial and unbiased in decision-making processes. disclose and manage conflicts of interest for both editors and reviewers. seek external perspectives when conflicts arise. 8. use journal metrics responsibly. be cautious with the use of journal metrics and avoid overreliance on them. consider metrics as one aspect of evaluation, ensuring transparency, fairness, and accuracy. 9. safeguard confidentiality. respect the confidentiality of submitted manuscripts and reviewers’ identities. only exchange information with authorized individuals. protect the integrity of the peer-review process. 10. uphold editorial independence. make decisions based on scientific merit, free from external influence or pressure. consider input from reviewers and panel experts but maintain autonomy over the final publication decision. conclusion in conclusion, this editorial emphasizes the importance of ethical behavior in the publishing process and outlines the key elements of editorial ethics that guide the journal for urban mathematics education. these elements include publication decisions, peer review, equal opportunity, neutrality, journal metrics, confidentiality, and editorial independence. in sum, the journal for urban mathematics education strives to be a standard-bearer for equity and excellence in editorial ethics, fostering a publication process that upholds integrity, fairness, and inclusivity. references avanzas, p., bayes-genis, a., de isla, l. p., sanchis, j., & heras, m. (2011). ethical considerations in the publication of scientific articles. revista española de cardiología (english edition), 64(5), 427–429. benatar, s. r. (1998). editorial ethics. bmj, 316(7125), 155–156. carver, j. d., dellva, b., emmanuel, p. j., & parchure, r. (2011). ethical considerations in scientific writing. indian journal of sexually transmitted diseases and aids, 32(2), 124–128. https://doi.org/10.4103/0253-7184.85425 graf, c., wager, e., bowman, a., fiack, s., scott‐lichter, d., & robinson, a. (2007). best practice guidelines on publication ethics: a publisher’s perspective. international journal of clinical practice, 61, 1–26. https://doi.org/10.1111/j.1742-1241.2006.01230.x islam, g., & greenwood, m. (2022). the metrics of ethics and the ethics of metrics. journal of business ethics, 175, 1–5. stichler, j. f. (2018). journal editors: are we friend of foe? herd: health environments research & design journal, 11(1), 6–10. copyright: © 2023 capraro & young. this is an open access article distributed under the terms of a creative commons attribution-noncommercial-sharealike 4.0 international license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. microsoft word 398 article20200922 (gallley proof 2).docx journal of urban mathematics education december 2020, vol. 13, no. 2, pp. 17–25 ©jume. https://journals.tdl.org/jume alesia mickle moldavan is an assistant professor of mathematics education in the graduate school of education at fordham university, 113 west 60th street, new york, ny 10023-7414; email: amoldavan@fordham.edu. her research interests include culturally responsive pedagogy, ethnomathematics, and teacher preparation that promotes equity and social justice advocacy in mathematics education. editorial a call for critical reads of “trouble” texts that inform urban mathematics education alesia mickle moldavan fordham university each of us has a moral obligation to stand up, speak up and speak out. when you see something that is not right, you must say something. you must do something… ordinary people with extraordinary vision can redeem the soul of america by getting in what i call good trouble, necessary trouble. – john lewis, “together, you can redeem the soul of our nation” choing the decade-long mission of the journal of urban mathematics education (jume) as restated in dr. robert m. capraro’s (2019) editorial, the journal’s purpose is to “foster discourse among a community of scholars to catalyze and transform the global academic space in mathematics education into one that embraces critical research, emancipatory pedagogy, and scholarship of engagement in urban communities” (p. 1). as jume readers and potential contributors, the new critical reads section expands on jume’s previous book review section (see jett, 2015) and provides an extended, scholarly space to highlight critical texts (e.g., books, research articles, short stories, social media posts, poetry, play/film scripts) in order to push conversations in the field that offer an empowering and transformative vision for urban mathematics education. as the section’s editor and an active scholar with jume since my early doctoral days, i am well aware of how the journal has inspired the field of urban mathematics education for over a decade by offering a “revolutionary spirit” of critical mathematics education knowledge dissemination (stinson, 2018, p. 2). i anticipate that the critical reads section will continue the legacy of jume honoring—not marginalizing—the professional work central to urban scholarship. this new section serves as another outlet within jume where scholars can recognize foundational and newly published works that can be described as necessary reads for keeping current in the field of urban mathematics education. whether the texts inspire new understandings, motivate the interrogation and dismantling of systemic inequities, or catalyze social justice advocacy, the critical reads section opens possibilities for readers to learn of (and revisit) new and old texts focused on “illuminating urban excellence” (matthews, 2008, p. 1). e moldavan editorial journal of urban mathematics education vol. 13, no. 2 18 in this section editorial, i invite you to learn about the critical reads section, including its purpose and expectations for those wishing to contribute. although the section is open to both new and seasoned scholars in writing critical reviews, i also pose questions to urge all scholars to get in “good trouble” with the ways they engage in critical reading and reflection in the urban context. these questions serve to not only strengthen professional practice but also raise collective awareness that catalyzes newfound conceptualizations and emancipatory actions focused on improving mathematics education in urban communities. writing for the critical reads section of jume critical readers and writers should “take themselves in hand and become agents of curiosity, become investigators, become subjects in an ongoing process of quest for the revelation of the ‘why’ of things and facts” (freire, 1992/1994, p. 105). the purpose of the critical reads section of jume is to create a scholarly space to reflect on texts that challenge the status quo, reconceptualize beliefs and practices, and/or respond to new problems and new possibilities in urban mathematics education. authors interested in contributing to this section are encouraged to thoughtfully reflect on texts that explore the complexities of mathematics education in urban contexts. these texts should have the potential to introduce new critical perspectives or reflect on foundational pieces that elicit provocative conversations that disrupt societal norms and inspire new ways of thinking. furthermore, the emphasis on the variety of texts from various mediums invites interdisciplinary works that are often overlooked yet beneficial in advancing our knowledge of urban mathematics education, promoting discussion and controversy, and shaping high quality research of the urban domain. submissions for the critical reads section should be approximately 1,500– 2,500 words, inclusive of references, appendices, footnotes, tables, and figures, and create a public dialogue in which the author(s) of the referenced text or others might be invited to respond to your critical review. outside of the length constraints, the structure of your submission is open to interpretation, as it should reflect your objective for writing as well as your writing style. although a suggested template could guide you to focus your attention on targeted areas of “typical” review feedback (e.g., relevance, rigor, quality of writing, implications), i recognize that templates can also be restrictive and discourage creative liberties that reflect how you personally respond to the text. thus, i encourage you to select a well-organized structure that promotes meaning making crucial to your (and others’) ongoing professional growth that accounts for the complexities (e.g., social, cultural, political) of urban mathematics education reform. to guide the process of writing your critical review, i recommend referencing the work of wallace and wray’s (2016) critical synopsis questions: moldavan editorial journal of urban mathematics education vol. 13, no. 2 19 1. why am i reading this? 2. what are the authors trying to achieve in writing this? 3. what are the authors claiming that is relevant to my work? 4. how convincing are these claims, and why? 5. in conclusion, what use can i make of this? (p. 42) these questions elicit critique and challenge you to look beyond the surface of the text and peel back its many layers. asking these questions will not only shift your perspective as a reader but will also offer you more insight into the text, which will help you develop your stance, adopt a reaction, and provide critical feedback to inform others. additionally, it is equally, if not more, important to know how to interweave your response into a critical summary that invites reflection from the author of the text as well as the audience (see cannon & myers, 2016; hamilton, 2015; martin, 2015; meyer, 2016). as you begin to reflect on what drew you into the piece, consider who else may be interested in reading the text and why the text is worth reading. when you introduce the text, you may want to say more about the author and, as an extension of that, describe the context of the text. consider making a statement about the motivation behind the text to not only set a scene but also determine whether the purpose was fulfilled. it may also be advantageous to describe the structure of the text. for instance, how is the book organized and what is the purpose of the book’s structure? to support your claims, you should reference the author’s arguments, finding that balance between restating the shared information and explaining why the text should be of interest to jume readers. it may be helpful to consider how the text can be used as a resource among other works to reframe a vision for empowering and advancing urban mathematics education. in your evaluation of the text, you may want to assess if the author appropriately communicated the key takeaways and if their claims were warranted. you can do this by going beyond the surface and challenging the structural and political boundaries of the text. you may also want to reference your own expertise on the content or highlight how the key takeaways disrupted or aligned to your own experiences, beliefs, and values. thus, do not forget to say more about yourself and your positionality in relation to the work you are reviewing. consider how the text inspired your work moving forward and how it can similarly inspire others. therefore, a critical review, yet flexible in nature, must inform the readers about the text and clearly communicate what you as a reviewer want to convey. connecting a critical review to urban mathematics education another major point of emphasis for authors submitting to the critical reads section is to ensure the critical review is thoroughly connected to urban mathematics moldavan editorial journal of urban mathematics education vol. 13, no. 2 20 education. the text under critique may not classify itself in the urban or mathematics education domain; however, your positioning of the text should clearly be associated. before i begin to offer my recommendation for such positioning, i find it necessary to address what topics and themes may elicit provocative conversations that disrupt societal norms and inspire new ways of thinking (or rethinking) about urban mathematics education. for authors of jume and the field at large to be critical in productive ways that push boundaries and advance new breakthroughs that illuminate urban excellence often overlooked by mainstream discourses, efforts must be made to define (or redefine) the essence of “urban” or “urbanicity” in teaching and learning mathematics (matthews, 2008; tate, 2008; walker, 2012). welsh and swain (2020) address this very idea of problematizing what constitutes “urban” education, which is often viewed as a socially constructed and disputed concept with no common definition (buendía, 2011; milner, 2012; schaffer et al., 2018). in their research to conceptualize the nuances of urban education, welsh and swain (2020) explored how urban can be defined in terms of six categories: geographical location, enrollment size, student demographic composition, school resources (e.g., pupil-teacher ratios, instructional expenditures), discrepancies in educational inequities, and social/economic context (e.g., poverty, unemployment, housing, family structure). these categories not only challenge traditional ways of conceptualizing urban education in terms of location, size, and population (buendía, 2011; schaffer et al., 2018) but also blur the geographical/place boundaries by illuminating the crossroads of other social, oft-neglected, issues (e.g., segregation, systemic institutional failures, racial and environmental inequities, economic and resource gaps) that impact research and policy initiatives in education. when urban education is described as a dynamic, multifaceted concept, we begin to include schools that encompass their own “urban characteristic” in suburban and rural areas that experience challenges (e.g., increases in english language learners, access to technology) shared with those in large cities and metropolitan areas (milner, 2012; tatum & muhammad, 2012). as noted by welsh and swain (2020), “urban can be defined as a continuum of conditions dependent on the characteristics, challenges, and context” (p. 97); thus, schools with such conditions should not be overlooked because they lack the scale and scope of other classified “urban” schools. a further look into the described conditions of urbanicity calls attention to the educational inequities as well as the sociohistorical, sociopolitical, and socioeconomic hardships faced by traditionally underserved communities (leonardo & hunter, 2007). historically, urban communities have been (and continue to be) shaped by discrimination and systems of oppression (kohli et al., 2017). such challenges have led to socioeconomic segregation (e.g., free and reduced price lunch (frl) vs non-frl, latinx/black-white) in school districts, student underperformance in educational outcomes, and deficit perspectives/stereotypes by those in (and seeking) the teaching profession (gadsden & dixon-román, 2017; jacobs, 2015). moldavan editorial journal of urban mathematics education vol. 13, no. 2 21 urban has become common nomenclature to describe race and class (milner, 2012) and is often used to label impoverished and uneducated inner city black and latinx children (buendía, 2011; jacobs, 2015). to challenge the deficit-oriented language and perspectives rooted in such definitions of urban, counternarratives must be shared to disrupt deficit discourses and assumptions (e.g., shortcomings, limitations), debunk common stereotypes, and highlight students’ assets (e.g., social, cultural, linguistic) in urban schools and communities (buendía & ares, 2006; popkewitz, 1998; schaffer et al., 2018). for instance, welsh and swain (2020) described urban schools in terms of the following: “there are safe, academically successful, and desirable urban schools. urban is success as well as failure” (p. 95). likewise, milner (2012) stated: “not all urban districts and the people in them are ‘bad.’ there is a rich array of excellence, intellect, and talent among the people in urban environments” (p. 558). when considering urban mathematics teaching and learning, it is also necessary to recognize the contributing factors of out-of-school settings that impact students’ learning environments, experiences, and educational outcomes in mathematics. for example, infrastructure, housing, poverty, parental education, and access to transportation, health care, and teacher quality are essential factors that play a role in students’ educational experiences and trajectories (carter & welner, 2013). deficiencies in these essential factors coupled with the scarcity of adequate resources (e.g., highquality teachers, rigorous curriculum, appropriate texts and technology) may limit participation in learning mathematics, especially when available resources do not support students’ mathematical, cultural, and linguistic needs. thus, there is a need to unpack the cultural context of where students live and learn, both in and out of the classroom, to consider what constitutes an effective environment for teaching (gadsden & dixon-román, 2017; ladson-billings, 2008). furthermore, consideration must be made for the development (and redevelopment) of communities where students attend school. gentrification projects revitalizing cityscapes are often changing the city landscape of communities and public schools at the cost of displacing families who historically inhabited the area (keels et al., 2013; kennedy & leonard, 2001). as families are financially forced to relocate to suburban and rural areas, new urbanism requires us to expand our research beyond traditional ideologies of what constitutes an urban place (e.g., density, geographical locations). this forces the field of urban mathematics education to engage in “interpreting and reinterpreting the definition and conceptualization of urban education amid a confluence of significant changes in demographics as well as economic and social circumstances” (welsh & swain, 2020, p. 99). a growing population of ethnically and racially diverse students rapidly changing the context of urban education also forces reexamination for what signifies “urbanness” to educators, researchers, and policymakers. additionally, what urban looks like in one context (e.g., the united states) should not be generalized to what (and how) urban is defined across the globe. although welsh and swain (2020) note that moldavan editorial journal of urban mathematics education vol. 13, no. 2 22 “urban education collapses race (people), place, and space” (p. 94), researchers are challenged to not lose the essence of what constitutes urban in the context of their work. urban is not monolithic (jacobs, 2015), so urban needs to be seen as an epicenter of varying racial, linguistic, socioeconomic, and cultural differences that extend both in and out of the classroom. thus, authors considering submission to the critical reads section for publication in jume should connect their critical review to the urban context as follows: § problematize the work in terms of place, space, people, and educational processes within new landscapes of urbanicity (see welsh & swain, 2020). recent works (e.g., jacobs, 2015; schaffer et al., 2018) have reinforced the significance for urban to be explicitly defined to not only foster a common understanding and language about what is meant by urban but also to guide others in exploring the nuances of urban to provide better support and greater equity for all students. be sure to state how the author of the referenced text defines urban and how this aligns (or not) with the multifaceted definition of urban in mathematics education (see martin & larnell, 2013; matthews, 2008; tate, 2008). § position yourself within your work in relation to the scope of the referenced text. use creative liberties to personalize your work by discussing your identity, experiences, and ideologies to strengthen your positionality within the critique. this will assist readers in understanding your perspective; it will also serve as an opportunity to model how others might do the same in reading the referenced text or other texts. § consider the contribution of the text you are reviewing in terms of challenging the complexities of mathematics education in urban contexts, including how the text’s ontological and epistemological constructs engage with the field. furthermore, describe how the text can be used to shape theoretical, conceptual, and empirical work to make sense of urban mathematics education. you may also consider the following questions: what assumptions are made about mathematics in urban schools? what assumptions are made about students doing mathematics in urban schools? and finally, what connections between schools, mathematics, and the community does the text suggest may be supportive or problematic? be sure to communicate how urban mathematics educators can benefit from reading the referenced text and what might be missing or further explored in other texts. moldavan editorial journal of urban mathematics education vol. 13, no. 2 23 a call (and obligation) to get in “good trouble” for the last decade, jume has fostered “revolutionary” works in mathematics that have inspired us to engage in critical research, embrace emancipatory pedagogy, and unite in scholarship and advocacy in our urban communities. as we embark on the next decade, let us use the steppingstones that we have traveled on thus far to continue revolutionizing the future of urban mathematics education. the critical reads section of jume is a scholarly space to engage in critical thinking and creative rethinking that unpacks the concept of urbanicity in ways that may get authors in “trouble” for questioning existing truths concerning learning conditions, students, schools, communities, and mathematics teaching and learning and summarily subjecting these to criticism and discussion. submissions must reframe teaching and learning mathematics in the urban context to not only shift the discourse but also improve educational experiences and outcomes for historically marginalized students by capitalizing on both foundational and new works from the field. such an orientation focuses on the mathematical success and excellence in urban spaces while also drawing attention to issues of access, power, race, and identity (gutiérrez, 2007, 2009; martin et al., 2017). the need for such a practice is summarized in martin and larnell’s (2013) statement: urban mathematics education scholars and practitioners must continue to … generate significations that position urban education not simply as a counternarrative to mainstream ideology but that transform the entire field. … this transformation is not simply about rearranging power relations and hierarchies inside and outside the field to produce new inequities but about realizing the potential of urban mathematics education to aid in the struggle against inequities. (p. 32) thus, it is my hope that the critical reads section serves as a model for what critical reviews of texts and critical engagement in the broader context of the field can offer to reform urban practice and excellence. references buendía, e. (2011). reconsidering the urban in urban education: interdisciplinary conversations. the urban review, 43(1), 1–21. https://doi.org/10.1007/s11256-010-0152-z buendía, e., & ares, n. (2006). geographies of difference: the social production of the east side, west side, and central city school. peter lang. cannon, s. o., & myers, k. d. (2016). radical reconfiguring(s) for equity in urban mathematics classrooms: lines of flight in mathematics and the body: material entanglements in the classroom. journal of urban mathematics education, 9(2), 185–194. https://doi.org/10.21423/jume-v9i2a314 carter, p. l., & welner, k. g. (eds.). (2013). closing the opportunity gap: what america must do to give every child an even chance. oxford university press. capraro, r. m. (2019). duty is necessary, passion is sufficient: it takes both. journal of urban mathematics education, 12(1), 1–7. https://doi.org/10.21423/jume-v12i1a379 moldavan editorial journal of urban mathematics education vol. 13, no. 2 24 freire, p. (1994). pedagogy of hope: reliving pedagogy of the oppressed (r. b. barr, trans.). continuum. (original work published 1992) gadsden, v. l., & dixon-román, e. j. (2017). “urban” schooling and “urban” families: the role of context and place. urban education, 52(4), 431–459. https://doi.org/10.1177%2f0042085916652189 gutiérrez, r. (2007). context matters: equity, success, and the future of mathematics education. in t. lamberg & l. r. wiest (eds.), proceedings of the 29th annual meeting of the north american chapter of the international group for the psychology of mathematics education (pp. 1– 18). university of nevada. gutiérrez, r. (2009). framing equity: helping students “play the game” and “change the game.” teaching for excellence and equity in mathematics, 1(1), 4–8. hamilton, b. (2015). book review: empowering science and mathematics education in urban schools. urban education, 50(3), 377–381. https://doi.org/10.1177%2f0042085913507460 jacobs, k. b. (2015). “i want to see real urban schools”: teacher learners’ discourse and discussion of urban-based field experiences. perspectives on urban education, 12(1), 18–37. jett, c. c. (2015). an urban mathematics education book review?: considerations for jume book review authors. journal of urban mathematics education, 8(1), 14–16. https://doi.org/10.21423/jume-v8i1a271 keels, m., burdick-will, j., & keene, s. (2013). the effects of gentrification on neighborhood public schools. city & community, 12(3), 238–259. https://doi.org/10.1111/cico.12027 kennedy, m., & leonard, p. (2001). dealing with neighborhood change: a primer on gentrification and policy choices. brookings institution center on urban and metropolitan policy; policylink. https://www.brookings.edu/research/dealing-with-neighborhood-change-a-primer-ongentrification-and-policy-choices/ kohli, r., pizarro, m., & nevárez, a. (2017). the “new racism” of k–12 schools: centering critical research on racism. review of research in education, 41(1), 182–202. https://doi.org/10.3102%2f0091732x16686949 ladson-billings, g. (2008). “yes, but how do we do it?” practicing culturally relevant pedagogy. in w. ayers, g. ladson-billings, g. michie, & p. a. noguera (eds.), city kids, city schools: more reports from the front row (pp. 162–177). the new press. leonardo, z., & hunter, m. (2007). imagining the urban: the politics of race, class, and schooling. in w. t. pink & g. w. noblit (eds.), international handbook of urban education (pp. 779– 802). springer. martin, d. b. (2015). the collective black and principles to actions. journal of urban mathematics education, 8(1), 17–23. https://doi.org/10.21423/jume-v8i1a270 martin, d. b., anderson, c. r., & shah, n. (2017). race and mathematics education. in j. cai (ed.), compendium for research in mathematics education (pp. 607–636). national council of teachers of mathematics. martin, d. b., & larnell, g. v. (2013). urban mathematics education. in h. r. milner iv & k. lomotey (eds.), handbook of urban education (pp. 373–393). routledge. matthews, l. e. (2008). illuminating urban excellence: a movement of change within mathematics education. journal of urban mathematics education, 1(1), 1–4. https://doi.org/10.21423/jume-v1i1a20 meyer, b. (2016). a critical dialogue: continuing the conversation about “the collective black and principles to actions.” journal of urban mathematics education, 9(2), 29–32. https://doi.org/10.21423/jume-v9i2a302 milner, h. r., iv. (2012). but what is urban education? urban education, 47(3), 556–561. https://doi.org/10.1177%2f0042085912447516 popkewitz, t. s. (1998). struggling for the soul: the politics of schooling and the construction of the teacher. teachers college press. moldavan editorial journal of urban mathematics education vol. 13, no. 2 25 schaffer, c. l., white, m., & brown, c. m. (2018). a tale of three cities: defining urban schools within the context of varied geographic areas. education and urban society, 50(6), 507–523. https://doi.org/10.1177%2f0013124517713605 stinson, d. w. (2018). celebrating a decade of critical mathematics education knowledge dissemination: a movement of people revolutionaries. journal of urban mathematics education, 11(1&2), 1–6. https://doi.org/10.21423/jume-v11i1-2a353 tate, w. f. (2008). putting the “urban” in mathematics education scholarship. journal of urban mathematics education, 1(1), 5–9. https://doi.org/10.21423/jume-v1i1a19 tatum, a. w., & muhammad, g. e. (2012). african american males and literacy development in contexts that are characteristically urban. urban education, 47(2), 434–463. https://doi.org/10.1177%2f0042085911429471 walker, e. n. (2012). building mathematics learning communities: improving outcomes in urban high schools. teachers college press. wallace, m., & wray, a. (2016). critical reading and writing for postgraduates (3rd ed.). sage. welsh, r. o., & swain, w. a. (2020). (re)defining urban education: a conceptual review and empirical exploration of the definition of urban education. educational researcher, 49(2), 90– 100. https://doi.org/10.3102%2f0013189x20902822 copyright: © 2020 moldavan. this is an open access article distributed under the terms of a creative commons attribution-noncommercial-sharealike 4.0 international license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. microsoft word 5 final leonard & evans b vol 11 no 1&2.doc journal of urban mathematics education december 2018, vol. 11, no. 1&2, pp. 142–151 ©jume. http://education.gsu.edu/jume jacqueline leonard is professor of mathematics education in the school of teacher education, university of wyoming, 1000 e. university avenue, department 3374, laramie, wy 82071; email: jleona12@uwyo.edu. her research interests include computational thinking, self-efficacy in stem education, culturally specific pedagogy, and teaching mathematics for social justice. brian r. evans is professor of mathematics education and associate dean for academic affairs in the school of education – pace university, 163 william street, 11th floor, new york, ny 10038; email: bevans@pace.edu. his research interests include teacher quality in alternative teacher certification pathways and traditional preservice teacher preparation programs, mathematics history, urban and international mathematics education, and mathematical problem solving. revisiting the influence of math links: building learning communities in urban settings jacqueline leonard university of wyoming brian r. evans pace university t has been a decade since the article “math links: building learning communities in urban settings” (referenced throughout as math links) was published in the inaugural issue of the journal of urban mathematics education (jume; leonard & evans, 2008). the math links study, as reported in that article, investigated teacher interns’ attitudes and beliefs about their interactions with urban students in a community-based setting. in that article, it was acknowledged that changing teacher attitudes and beliefs can be challenging, but nonetheless accomplished. the goal for the teacher interns was to transform attitudes and beliefs about teaching mathematics to urban students from routine and decontextualized ways to classroom practices in which culturally based and social justice oriented methods framed instruction. here, we reflect on that work for the 10th anniversary issue of jume and how that project has influenced our teaching, research, and scholarship in mathematics education over the past decade. math links influence on jacqueline (jackie) and brian jackie’s narrative after completing the community-based study in philadelphia, pennsylvania, which was supported by temple university and the united methodist church, office of urban ministries, i realized that grounding this work was the way to engage urban youth from underrepresented backgrounds in culturally relevant and place-based science, technology, engineering, and mathematics (stem) education. in the math links study, preservice teachers provided students with culturally relevant, hands-on stem instruction (leonard, 2002; leonard, moore, & spearman, 2007). the sites included mount zion united methodist church and tindley temple united methodist church. children and youth who participated in i leonard & evans revisiting math links journal of urban mathematics education vol. 11, no. 1&2 143 math links, as well as its predecessor, earth links, were eager to attend classes at the community-based sites. anecdotal records revealed one african american boy insisted that his mother get up early on saturday to ensure that he arrived on time. when certificates of attendance were provided during a sunday church service, an african american girl did not feel the need to be recognized as an outsider stating, “i already go here.” through community-based stem programs, underrepresented minority students from urban settings developed a stem identity as well as individual and community agency as members of the congregation and community helped them to learn and achieve educational goals in culturally relevant ways. since leaving temple university and philadelphia in 2010, this model was transferred to my new setting at the university of colorado, denver where i worked from 2010 to 2012. while in colorado, i worked in collaboration with iglesia y vida, shorter african methodist episcopal church, campbell chapel african methodist episcopal church, and the oleta crain enrichment center in denver. the project entitled dinosaurs, denver, and climate change (d2c2) was funded by the national science foundation (nsf). it provided urban children with the opportunity to learn about paleontology and environmental science within the context of place-based education. twenty-four youth and young adults served as interns (also called near-peer mentors) in the summer of 2013 or 2014 (leonard, chamberlin, bailey, verma, & douglass, in press). more than 60 children from african american and latinx backgrounds participated in the study (djonkomoore, leonard, holifield, bailey, & almughyirah, 2018; leonard, chamberlin, johnson, & verma, 2016). in this informal learning environment, students assisted in maintaining a compost for one week. they also learned about climate change by taking field trips to a science museum, botanical garden, and national park. several students responded positively to the study: the coolest thing was starting a compost … i watch tv shows; composts aren’t that big. but now that we are doing compost, it takes a lot of work. my favorite [sic] day was at the nature and science museum because when me and master chief [near-peer mentor] stuck our hand in a hole, and we were both scared, and we looked inside it, and it was deer hide. my favorite part of the [field] trips so far is the botanic garden. first, the japanese garden was beautiful. there were many lily pads with wonderful flowers. second, was the survivor. we saw many plant-like [banana] trees. third, there were pretty flowers. all of the flowers i saw were glamourous like queen anne’s lace. all in all, my favorite part of the trip so far is the botanic garden, and going there makes me feel like a calm bee. not all of the students enjoyed every activity that took place during the summer camp. some of the constructive comments made by students include: leonard & evans revisiting math links journal of urban mathematics education vol. 11, no. 1&2 144 i didn’t really like the botanic gardens because it was kind of boring, and there was a lot of walking. and nature, to me, is basically plants. it’s just plants, so they really don’t interest me. it was scary to me a little bit by the mountain because we saw a coyote…. it wasn’t that cool because it was limping. when we went to the mountains, i didn’t like the scat because it was everywhere…. scat is poop. the researchers used the feedback received from students to prepare future urban students for stem learning in informal settings and to develop rich cohesive programs for preservice teachers and students. after working with students in denver, two additional projects were funded by the nsf in 2013 and 2014, respectively, during my role as director of the science and mathematics teaching center from august 2012 to july 2016 at the university of wyoming. the first project was funded under the robert noyce teacher scholarship program. thirty summer interns and 24 scholars were recruited for the wyoming interns to teacher scholars (wits) study. summer interns participated in rural (n = 28) and urban (n = 2) settings. ten of these interns identified as racial, ethnic, or language minorities (4 african american, 3 native american, 2 latina, and 1 biracial). however, the majority of the interns identified as white (n = 23). in wyoming, interns worked in numerous settings such as state parks (i.e., sinks canyon), starbase in cheyenne, teton science schools in jackson and saratoga, summer school in lander, and summer camps at the university of wyoming in laramie. these summer research experiences were used to recruit prospective stem majors into teaching (leonard, aryana, johnson, & mitchell, 2015) and to build upon the collaborative and community-based model developed in the math links study. comments from several summer interns include: we learned how to teach using place-based education in a positive learning environment. in jackson, i had the experience of learning place-based education in jackson. i also used place-based education on the field trips that i went on for lander. i worked in the art section and we learned how to make chromatography, teaching the students how to grow their own crystals, experiments with different solutions. i helped teach lessons in an earth science classroom. we built a lot of things with our hands (e.g., pulley systems, ramps, wedges, etc.). the focus of the week was mechanical science. teton science schools allowed me to participate in the developing of lesson plans and the presentation of lessons to the children, which allowed me to understand the efforts leonard & evans revisiting math links journal of urban mathematics education vol. 11, no. 1&2 145 it takes from beginning to end with how a teaching lesson is formed and then presented. in addition to these rural settings, two students requested to complete their summer internship in an urban setting. they worked with predominantly african american students during a stem summer camp at temple university in philadelphia in 2016. some excerpts of children’s comments that were obtained from focus groups in this setting include: what i thought was cool was how we was able to make actual live robots, and i also liked that we were able to program different types of video games. i liked how we got to make the robots out of legos because usually legos take a very long time to put together and to start moving cause you to have to push it, but it is neat how we got to use the ipads to control them. i like how we got to customize our characters and program them, and i also liked how we could program our games and people could actually play them. i did not like my group because they were not very cooperative with everyone. like they had jobs for a builder, operator and then they had a go-getter to find different legos and then they had a me. i didn’t like that it was hard to program your character in your game. while a few students had some level of frustration with roles during robotics and difficulty programming during game design, other students noted enjoying the opportunity to create, tinker with, build, and/or program robots and digital games. the male teacher interns, who identified as minorities, developed robust relationships with the students. although the study was not community-based, it used the math links internship model to develop teaching opportunities for undergraduate students. the second project, ugame-icompute, was conducted in several districts in wyoming and one urban district in philadelphia. it was funded by the innovative technology experiences for students and teachers (itest) program at nsf. in this project, 45 teachers worked with more than 950 students from 2014 to 2016 during afterschool programs to promote stem education using robotics and game design in the contexts of culture and place (leonard et al., 2016). in the urban context, practicing teachers noticed students’ performance (i.e., motivation, effort, and learning) in informal settings was different than their performance in traditional classrooms: the independent learning part was big for me. it actually totally changed the way i teach. period. that the kids were able to go on to the nxt web site and look up how to leonard & evans revisiting math links journal of urban mathematics education vol. 11, no. 1&2 146 build like…different things. it wasn’t just like ok guys today we’re going to build this. they were in charge of their own learning. it’s not all teacher directed. the students are helping each other out and they’re like “let’s find this out together.” there’s really a lot of collaboration between them, and we don’t have to…hands-off on our part and hands-on on theirs…to where we’re not always…having to, “oh, let me help you here, let me help you there, let me help you there.” they’re helping each other, you know, which is really nice! the accountability…kinda comes back to them, in knowing that they have a stake in their own learning, when it comes to a particular math, you know. that they can look through their computation and find out what went wrong knowing that numbers only line up a certain way, and either you made a mistake, or you didn’t program your equation the right way. in this study, teachers learned much about their teaching and students. teachers engaged in active learning and culturally relevant pedagogy rather than teaching by telling and allowed students to engage in critical thinking and inquiry as they drew upon culture and place. during robotics and game design, students demonstrated autonomy and were self-directed. similar to the math links study, teachers’ pedagogy was influenced by the learning and engagement they witnessed during afterschool stem clubs (leonard et al., 2017). the aforementioned studies build upon the math links model. in addition to these research studies, the model has been employed internationally. i was selected as a fulbright canada research chair in stem education at the university of calgary in fall 2018. in calgary, i plan to work with indigenous students in community-based settings on scalable game design (repenning et al., 2015) and 3-d computer modeling. brian’s narrative at the time of the original publication of math links, i was early in a tenuretrack assistant professor position in mathematics education in the school of education at pace university. in 2011, i was tenured and promoted to associate professor, and more recently to full professor. i also transitioned to an administrative position at pace as assistant provost for experiential learning. in this position, i had been developing and coordinating a signature program that combines academics with real-world experiential learning, mentoring, and planning in order to give students a rich and robust college experience. i also became the managing editor of a peer-reviewed research journal focused on alternative teacher certification, which constitutes a large part of the teacher education program at pace. the journal of the national association for alternative certification (jnaac) is the leading research journal for alternative teacher certification. leonard & evans revisiting math links journal of urban mathematics education vol. 11, no. 1&2 147 in each issue of jnaac, i publish an article in the editor’s perspective section. in each article, i either present a research study i have conducted or more typically, a perspective on education. one of my most recent articles (evans, 2017) relates to the importance of affective variables for new alternative certification teachers and alludes to my own shift in viewing successful and effective teaching through this lens and framework. in the article, i provide an example of two different teachers; below is an excerpt from that article: imagine we have two teachers, teacher a and teacher b. teacher a has a very strong academic background and a solid grasp on educational research and theory. teacher a knows the works of vygotsky, piaget, and dewey very well, and teacher a understands how children develop and learn. teacher a’s pedagogical skills are very strong. teacher b struggled with coursework in college (and elementary and high school as well). teacher b does not have a strong understanding of educational research, theory, and pedagogical skills. however, teacher b understands how to connect with people. students simply love teacher b, and they know teacher b has their interests as a top priority. teacher a does not really like young people all that much even though teacher a chose to be a teacher. teacher a has trouble connecting with students and finds it challenging to communicate with them. while the students believe teacher a is very knowledgeable and accomplished, they do not have much respect for teacher a. they find teacher a unnecessarily difficult toward them, and often think teacher a really does not prefer to be a teacher. who do you think will have more success with the students? interestingly, it is likely teacher b. i certainly do not wish to imply that content knowledge and pedagogy are unimportant. they absolutely are. however, content knowledge and pedagogical skills are not the only variables to consider for teaching success and may not even be the most important variables in every situation. lessons such as this one have been critical to my own development as a professor of education. (p. 23) in the article, i reference the experiential learning program that i developed and had coordinated at pace university. this work is related to a finding in the math links study, which emphasized the importance of providing preservice teachers with field-based experience prior to student teaching. early field experience influenced how i have positioned experiential learning at pace. this influence coincides with the overarching philosophy of the school of education’s teacher preparation program that students receive field experiences early in the teacher education program. math links influenced my approach to working toward early involvement and increased field experiences for other majors at pace. the link between teaching mathematics for social justice was an important aspect in the math links study as well, which coincides with the school of education’s conceptual framework at pace: the school of education believes that a fundamental aim in education is to create opportunities for individuals to realize their potential within a democratic community. therefore, we prepare graduates of our programs to be reflective practitioners who pro leonard & evans revisiting math links journal of urban mathematics education vol. 11, no. 1&2 148 mote justice, create caring classrooms and school communities, and enable all students to be successful learners. (pace university school of education, 2017) i have continued to teach mathematics methods to graduate students in alternative teacher certification and undergraduate students in traditional preservice teacher programs. perhaps the most impactful and sustained influence of math links is the shift in emphasis on the affective variables related to high-quality and effective teaching, such as teacher attitudes and beliefs, self-efficacy, culturally responsive pedagogy, social justice, and rapport and trust between teacher and students. during graduate school and early into my tenure-track assistant professor position, i was committed to reform-based mathematics education utilizing a sociocultural perspective within a constructivist framework. i believed that culturally responsive pedagogy was an essential ingredient for effective teaching in urban schools. nevertheless, i continued to prioritize mathematics content and traditional pedagogical knowledge as the two most important variables for successful and effective teaching. by examining research on preservice teachers’ attitudes, beliefs, self-efficacy, and culturally responsive pedagogy, i experienced a major conceptual shift in priorities as it related to successful and effective teaching. while i continue to acknowledge the importance of mathematics content and pedagogical knowledge, such as mathematical knowledge for teaching (ball, 2005), i now believe that affective variables are more important for effective teaching than is perhaps commonly acknowledged. this affective framework is supported in the work of several authors; leonard (2008), ladson-billings (1994, 1995), and martin (2007) are just a few examples. in my mathematics methods courses, i use this affective framework to help preservice teachers understand the importance of their own development as teachers in the critical areas of academic, social, and developmental learning. i learned that in-service and preservice teachers, like i once thought, tended to believe that content and traditional pedagogical knowledge are the most important variables for classroom success. while i have not formally collected evidence, i sense that teachers left my courses with a changed disposition and reformed attitudes toward culturally responsive pedagogy and teaching mathematics for social justice. the math links study emphasized that teacher interns should not consider social justice in a vacuum but rather consider social justice in the real-world context of the classroom. this finding has influenced my instruction and use of fieldwork as the basis for practice in my courses. furthermore, math links contributed to the idea that a formal study on teacher change could build upon my assumptions and practice. finally, i have been traveling to shanghai, china, yearly for the last couple of years to work with teachers on teacher assessment and curriculum development. my focus has been on using the danielson framework (danielson, 2018) to evaluate teaching and the use of data collection to improve instruction. i have used wiggins and mctighe’s (2005) backwards design in order to plan lessons and units leonard & evans revisiting math links journal of urban mathematics education vol. 11, no. 1&2 149 in american and chinese curricula. lessons learned in china have influenced my work with preservice and in-service teachers, expanding the community-based model to a global model. this work has complemented my teaching at pace. closing thoughts while our (jackie’s and brian’s) progressions as researchers and teacher educators cannot solely be attributed to math links over the last decade, there is no doubt that findings from the math links study have been highly influential in our research and pedagogy. for both of us, the math links study, along with other teaching and research opportunities, shaped our perspectives on teacher attitudes and beliefs. perhaps the most important finding from the math links study was the acknowledgment that teacher attitudes and beliefs are malleable. math links provided us with the impetus to continue our efforts to shape teacher attitudes and beliefs toward teaching by providing field experiences that allowed teachers to practice teaching mathematics (or stem) for cultural relevance and social justice in urban contexts. reflecting on math links has helped us realize that the findings from the study provided a catalyst for future research and ideas on how to influence positive teacher attitudes and beliefs about urban students. as we reflected on the changes that have occurred since the publication of math links at the end of 2008, the country was in financial crisis and had just elected its first african american president, who would serve two terms over the next 8 years. brian still recalls where he was while watching the results from a restaurant in brooklyn, new york. after barack obama had been declared the winner, brian walked the streets of brooklyn, and then times square, to experience the historical moment along with the celebrations. jackie had the privilege of hearing candidate obama at progress plaza near temple university during the 2008 campaign. it was surreal to hear a brilliant, young african american senator talk about education, healthcare, and uplift in a black community. jackie personally benefitted from president obama’s educational initiatives for more stem teachers when she received a robert noyce grant. however, there are those who believe obama fell short by not articulating a clear vision to deal with state violence (davis, 2016; taylor, 2016). the deaths of trayvon martin, michael brown, and many others occurred during the obama administration. the uprisings in ferguson and baltimore took place on obama’s watch. attorney general eric holder’s visit to ferguson was historic and comforting, but few policies other than establishing the ferguson commission and adding police bodycams have resulted from the michael brown tragedy. clearly, it takes more than the presidency of one black man to tear down centuries of racism and prejudice. leonard & evans revisiting math links journal of urban mathematics education vol. 11, no. 1&2 150 at the time of this publication, we are more than two years into the presidency of donald trump. it feels like the country has taken a big step backward as the altright has become empowered and emboldened by xenophobic and racist ideology. yet, other important social justice movements have strengthened during, or because of, the trump presidency (e.g., #blacklivesmatter, #neveragain, #timesup, #metoo). along with the protesters in charlottesville and the women’s march in washington, dc, many people have voiced their dissent. teachers across the country took to the streets to advocate for higher wages and greater resources for their students. through public discourse and protests, we have seen the country take a serious look at police brutality, sexual abuse, and gun violence in our nation’s schools. these resistance movements give us the audacity to hope that we will eventually see that “justice rolls down like the waters and righteousness like an ever-flowing stream” (amos 5:24).1 when that happens, trust, which has eroded in black and brown communities, will hopefully be restored as well. references ball, d. l., hill, h. c., & bass, h. (2005). knowing mathematics for teaching: who knows mathematics well enough to teach third grade, and how can we decide? american educator, 29(1), 14–17, 20–22, 43–46. danielson, c. (2018). the framework. retrieved from https://www.danielsongroup.org/framework/ davis, a. (2016). freedom is a constant struggle: ferguson, palestine, and the foundations of a movement. chicago, il: haymarket books. djonko-moore, c. m., leonard, j., holifield, q., bailey, e. b., & almughyirah, s. m. (2018). using culturally relevant experiential education to enhance urban children’s knowledge and engagement in science. journal of experiential education, 41(2), 137–153. evans, b. r. (2017). improving alternative certification teachers’ experiences by developing “soft skills” for the classroom. journal of the national association for alternative certification, 12(2), 21–25. ladson-billings, g. (1994). the dreamkeepers: successful teachers of african american children. san francisco, ca: jossey-bass. ladson-billings, g. (1995). toward a theory of culturally relevant pedagogy. american educational research journal, 32(8), 465–491. leonard, j. (2002). let’s go fly a kite. science and children, 40(2), 20–24. leonard, j. (2008). culturally specific pedagogy in the mathematics classroom: strategies for teachers and students. new york, ny: routledge. leonard, j., aryana, s., johnson, j. b., & mitchell, m. (2015, june). preparing noyce scholars in the rocky mountain west to teach mathematics and science in rural schools. in s. mukhopadhyay & b. greer (eds.), proceedings of the 8th international mathematics education and society conference, (pp. 737–749). portland, or: portland state university. leonard, j., buss, a., gamboa, r., mitchell, m., fashola, o. s., hubert, t., & almughyirah, s. (2016). using robotics and game design to enhance children’s stem attitudes and computational thinking skills. journal of science education and technology, 28(6), 860–876. 1 the holy bible, english standard version leonard & evans revisiting math links journal of urban mathematics education vol. 11, no. 1&2 151 leonard, j., chamberlin, s., bailey, b. e., verma, g., & douglass, h. (in press). broadening millennials’ participation in stem and teaching professions through culturally relevant, placebased, informal science internships. in g. prime (ed.), effective stem education for africanamerican k–12 learners. new york, ny: peter lang. leonard, j., chamberlin, s. a., johnson, j. b., & verma, g. (2016). broadening urban students’ opportunities to learn science and influencing student interest through place-based education. the urban review, 48(3), 355–379. leonard, j., & evans, b. r. (2008). math links: building learning communities in urban settings. journal of urban mathematics education, 1(1), 60–83. retrieved from http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/5/5 leonard, j., mitchell, m., barnes-johnson, j., unertl, a., outka-hill, j., robinson, r., & hestercroff, c. (2017). preparing teachers to engage rural students in computational thinking through robotics, game design, and culturally responsive teaching. journal of teacher education, 69(4), 386–407. leonard, j., moore, c. m., & spearman, p. (2007). teaching science inquiry in urban classrooms: case studies of three prospective teachers. the national journal of urban education & practice, 1(1), 37–55. martin, d. b. (2007). beyond missionaries or cannibals: who should teach mathematics to african american children? the high school journal, 91(1), 6–28. pace university school of education. (2017). conceptual framework. retrieved from https://www.pace.edu/school-of-education/content/conceptual-framework repenning, a., webb, d. c., koh, k. h., nickerson, h., miller, s. b., brand, c., … repenning, n. (2015). scalable game design: a strategy to bring systematic computer science education to schools through game design and simulation creation. acm transactions of computer education, 15(2), 11.1–11.31. taylor, k-y. (2016). from #blacklivesmatter to black liberation. chicago, il: haymarket books. wiggins, g., & mctighe, j. (2005). understanding by design (2nd ed.). alexandria, va: association for supervision and curriculum development. microsoft word 424-2387-1-6-20210727-1 (proof 2).docx journal of urban mathematics education december 2021, vol. 14, no. 2, pp. 105–116 ©jume. https://journals.tdl.org/jume jami c. friedrich is a post-doctoral research fellow with the stem education innovation lab at mercer university, 3001 mercer university dr., atlanta, ga 30341; email: jami.c.friedrich@live.mercer.edu. she has been a middle and high school mathematics teacher and instructional coach. her research interests involve stem education in underserved and underrepresented populations and culturally responsive pedagogy. tynetta jenkins is a graduate research fellow with the stem education innovation lab at mercer university, 3001 mercer university dr., atlanta, ga 30341; email: tynetta.s.yarbrough@live.mercer.edu. she has been a secondary science teacher, elementary grades science teacher, and science curriculum developer for her local school district. her research interests involve the underrepresentation of african american women in stem careers. critical reads leveling the playing field: addressing the culture of urban mathematics education in black, brown, bruised: how racialized stem education stifles innovation jami c. friedrich mercer university tynetta jenkins mercer university n black, brown, bruised: how racialized stem education stifles innovation by ebony omotola mcgee (2020), the author focuses on the stories of underrepresented, racial minoritized (urm) students that have found success in the fields of science, technology, engineering and mathematics (stem) rather than the two-thirds of urm students who begin their education in stem programs and then drop out. understanding experiences of urms who succeed academically “enables a deep appreciation of what it means for students of color to be academically successful in places where their numbers are few and negative beliefs about their ability prevail” (mcgee, 2020, p. 1). as a white cis-gender woman and an african american cisgender woman who are educators and emerging scholars in the field, our discourse is based on honest conversations about our own differing experiences and perspectives. we situate ourselves in a place in which mcgee describes as “a keen desire to work for racial and global justice” (p. 79), and this book review is based on our own understanding of the inequities in the field of stem education and the effects of these inequities on mathematics education. we came together to read this text in the midst of the covid-19 global pandemic, welcoming candid conversations and research initiatives that this text would inevitably inspire. the disproportional impact of the covid-19 pandemic on minority groups illuminated issues of inequity that extend into various realms of reality for minoritized groups of people. for example, we noted the impact that access to healthcare had on the vulnerabilities of minority groups of people to the effects of the pandemic. furthermore, there was an undeniable difference between the ability of some americans to transition to working from home and virtual learning for their i friedrich & jenkins critical reads journal of urban mathematics education vol. 14, no. 2 106 children, compared to others that continued to work in person due to the essentiality of their positions, inaccessibility to childcare, or lack of resources needed for virtual learning. this issue is a social one determined by demographic criteria such as education, income, ethnicity, and marital status. social identities impact the risk and exposure of minority groups to the virus compared to those from more dominant groups. we noted the theme of inequity in reference to the pandemic and how social identities and societal factors similarly affect stem experiences and mathematics education. working together as a part of a stem education research lab team, we found ourselves having many conversations about the underrepresentation of minorities in our field. jami, who just graduated from her phd program, was living in a racially divided southeastern u.s. city and working with primarily black and hispanic middle and high school mathematics students. she saw firsthand the inequities in the k– 12 educational system and spent many nights trying to come up with ways to increase the opportunities for her students. tynetta, a black woman in her third year of her phd program with three young black children of her own, has a research interest specifically focused on black women in stem because of her personal experience while pursuing a stem degree. she experienced the minority-related obstacles firsthand as a person of color and as a woman, an experience characterized as the “double-bind.” we committed to reading this text together, to have open and honest conversations, and to share both personal experiences with ourselves, our children, and our students. we value the importance of discourse between a white woman and a black woman in stem. we invite the reader to join us in this conversation of recognizing the hardships and embracing the success of urms in mathematics and other fields of stem while continuing to learn how we as educators work to alleviate the leaks in the stem pipeline by supporting urms and the unique burdens that characterize their experiences. the leaky pipeline: a thematic analysis black, brown, bruised: how racialized stem education stifles innovation is divided into six chapters, each of which focuses on the personal accounts of urms that have had academic success in stem in order to “hear the voices of scholars of color as they feel their way through a forbidding stem educational landscape” (mcgee, 2020, p. 3). the structure of the book is designed to take readers from understanding the current discriminatory culture of stem to practical suggestions to acknowledge and eliminate structural barriers facing urms. we discuss the book in three overarching themes: dealing with structural racism, equity ethics and community support, and practical guide for leaders. these themes were derived by the authors to organize the six chapters into the three overarching ideas. the authors friedrich & jenkins critical reads journal of urban mathematics education vol. 14, no. 2 107 group the first three chapters of the book within the theme of “dealing with structural racism,” as mcgee uses these three chapters to emphasize male dominance and lack of diversity in stem fields (chapter 1), identify structures in place that perpetuate racism (chapter 2), and describe the building blocks of racial discrimination and the detriments of self-coping mechanisms (chapter 3). mcgee then shifts her focus to what the authors have classified as “equity ethics and community support” in the next two chapters, as she details identityand culture-based strategies in stem that work for minority students and identifies current efforts that have increased urm participation. the final chapter of the book provides stem leadership with the next steps for structural change and is classified by the authors of this review as “a practical guide for leaders.” we have organized this review thematically, identifying points of our critique within a summary of the text. dealing with structural racism in the opening chapter of the book, mcgee (2020) identifies stem culture as “individualist, ultracompetitive, overwhelmingly white (with some tokenized asians), mostly heterosexual, militaristically grounded, middle-to upper-class, nationalist, able-bodied” (pp. 20–21). the discriminatory and racist nature of stem culture is evidenced in hegemonic practices focused on preparation of a stem workforce, anchored in concerns for global competition, national security, and militarization (takeuchi et al., 2020). it is in our own experiences that the white supremacy within stem culture is prevalent. mcgee (2020) argues that this stem culture contrasts a mindset of group solidarity and collective work that is commonly found in black, latinx, and indigenous communities. additionally, the current stem culture and the underrepresentation of black, latinx, and indigenous people in stem leads to missing critical contributors, which permeates the continued pattern of structural bias within the field. as a result, many students of color feel isolated, invisible, marginalized, and lack belongingness (mack et al., 2019). discrimination and reduced participation of urms in stem fields result in limited quantity and diversity of intellectual capital available in these fields (ballenger et al., in press; mckim et al., 2017; nilsson, 2017). consequently, “stem and stemers’ abilities [are hampered, preventing them from being] as ingenious and imaginative as they can be, thereby stifling innovation in these fields” (mcgee, 2020, p. 20). these consequences are not only detrimental to individuals but have greater effects at the macro level. when the voices of the leaders in the field only represent one specific demographic, we wonder how this affects students that are not a part of that demographic. mcgee (2020) points out urm students’ feelings of isolation and marginalization, but we are left wondering how diverse leadership teams support urm students and enhance programs as a whole. furthermore, we want to know the effect it has on stem innovation. while mcgee begins the text with the notion that racism and discrimination prevent ingenuity and the production of products and services beyond friedrich & jenkins critical reads journal of urban mathematics education vol. 14, no. 2 108 our wildest dreams (i.e., flying cars in the jetsons), she fails to consider stem environments that are largely diverse and known for stem minority success. for example, faculty at spelman college have established a stem climate that supports the belief that all students can achieve their goals and all students can be academically successful, even implementing a common curriculum that has been developed to encourage attainment of minorities with changes in instructional approaches that promote success (perna et al., 2009). this inclusive culture exists in stem environments at many historically black colleges and universities (hbcus; mcgee, 2020; perna et al., 2009; tate & linn, 2005). to avoid dismissal of the achievable vision of flying cars or the like, important to consider are the accomplishments and inventions of minorities that inhabit areas where diversity and inclusion exists. while some stem spaces are successful at establishing inclusion, stem culture largely remains discriminatory, marginalizing minorities. to promote anti-oppressive and humane mathematics education agendas, scholars have created funds of knowledge (gonzález et al., 2005; kiyama & riosaguilar, 2018; moll et al., 1992). mcgee (2020) identifies funds of knowledge as “an antideficit framework that unearths and leverages knowledge produced in the cultural-historical experiences of latinx students’ families and communities” (p. 23). the idea of funds of knowledge could have been further explored to examine the communal functioning of other minority groups, and the funds of knowledge that urms bring to the stem field are not recognized. further consideration of urm groups beyond latinx would have strengthened a pedagogy of solidarity (mcgee, 2020), or a unity of urm communities to combat white supremacy in stem culture. additionally, although mcgee recognizes that scholars in the field have come together to leverage their individual and collective expertise, voicing their ideas and concerns related to the field of mathematics, there are no explicit examples supporting the change or lack of change due to their contributions. having an example could empower readers and give them a practical guide to voice their own ideas and concerns in their stem field and institutions. in the second chapter, mcgee (2020) details how underrepresentation of both students and faculty in stem disciplines is just one aspect of structural racism. many aspects of structural racism lead many urm students to leave the field, including but not limited to racial stereotypes, lack of opportunities, and unwelcoming institutional climates (beasley & fischer, 2012; rainey et al., 2018; riegle-crumb et al., 2019). maurice, for example, had a strong interest in engineering and was proficient in mathematics and science but explained that during an engineering summer internship as a high school student, he was treated like an “affirmative action, token negro” and was consistently challenged and questioned about his ability to do his job (mcgee, 2020, p. 34). unfortunately, it was at a young age that maurice learned that his hard work and intellect was not enough to prevent him from experiencing racism. this experience led him away from furthering his education in the field of engineering. friedrich & jenkins critical reads journal of urban mathematics education vol. 14, no. 2 109 according to mcgee (2020), racism in education systems is perpetuated by the structures that protect them. she describes the “feelings of powerlessness, invisibility, loss of integrity, or pressure to represent one’s group” (p. 38) as psychological race-related stressors that lead to racial battle fatigue. although she identifies these feelings, it leaves us wondering how we as educational leaders support urms in urban mathematics education when our structures lead to psychological and behavioral responses that are debilitating. psychological support is critical, as seen through the experiences of successful urms in stem and specifically at hbcus that mcgee shares. it is important that we learn from these successes. mcgee (2020) dives into the personal accounts of urms who have found success in stem despite pervasive structural racism. stereotypes and racial microaggressions not only made it more difficult to be successful but also affected the health of the students that mcgee interviewed. common patterns included “selfblame and self-questioning (imposter syndrome), overworking in the hope of having their competence recognized, going into survival mode, experiencing racial battle fatigue and being unemployed as a direct result of being denied opportunities by white or asian principal investigators” (mcgee, 2020, p. 57). although in survival mode, the success stories of urm students recognized that personal experiences helped them survive, as they had become conditioned to hostile environments throughout their entire academic careers. additionally, those who had success found a way to form functional mathematics and stem identities. mcgee (2020) references tinesha and rob, students who found internal motivation and embraced mathematics, mastering mathematics despite persistent acts of racism and their conditioned ability to cope with them. their coping strategies included working on self-discovery and self-definition. they both joined organizations that celebrated black stem students’ identities and achievements and also associated with like-minded mentors. tinesha and rob were fortunate to operate from a position of strength, minimizing psychological damage (mcgee, 2020). coping mechanisms are common responses to the negative experiences minorities face in stem environments. coping strategies are not interventions at all, though they have shown to be influential to stem persistence (alexander & hermann, 2016; carlone & johnson, 2007; mcgee & bentley, 2017; watkins & mensah, 2019). we wonder what can be done in educational environments to support urms and prevent them from needing coping strategies to lessen the blunt force of racism as they pursue their educational journeys. equity ethics and community support through the examples in the book, we can see that structural racism makes it more difficult for urms to find success in stem. one pattern that mcgee (2020) found with students of color who did find success is that they wanted to serve their communities and the world. she described this as a key motivator and called it an friedrich & jenkins critical reads journal of urban mathematics education vol. 14, no. 2 110 equity ethic, defined as “a set of values that includes a principled concern for justice, particularly racial justice, for addressing racial inequities, and for the well-being of people suffering under various inequities” (p. 76). in the stories shared, she also identified equity ethic as a cultural phenomenon, as black, latinx, and indigenous cultures tend to emphasize the importance of family and community, whereas american culture tends to be more individualistic. supporting this claim, a study found that mathematics classes at two hbcus adopted a communal and kinship structure, resulting in markedly increased student motivation (mcgee, 2020; taylor et al., 2008). adopting a structure characterized by personal interactions, sharing of information, and mutual support of peers supported the cultural experiences of the students and seemed to have increased their motivation. as students of color are motivated by their own equity ethic and become empowered, this communal and kinship structure results in increased stem interest and relatability to stem subjects (hollins, 1996; mcgee, 2020; tytler, 2007). mcgee (2020) concludes that structuring stem education to encourage equity ethic (such as giving students the opportunities to apply their skills to humanitarian projects and incorporate community service learning), establishing learning communities for collaborative learning, and recognizing the cultures and experiences of each unique student will attract more urms to the field. she expands on the idea of equity ethic by identifying effective educational approaches for urms in stem in chapter 5. she begins by describing the importance of a welcoming environment through learning centers, workshops and seminars for study skills and career support, career and financial counseling, and academic counseling. we identify this as a way for the structure of the program to demonstrate equity ethic and provide community support to urm students and believe it is important to learn from programs that have been successful. in three examples identified as successful programs that support students of color, community support was also a common theme. all programs focused on teamwork, collaborative learning, and building a community. the mathematics workshop program at the university of california specifically focused on increasing the number of african american and latinx students. through the emphasis on “group learning, efficient studying, and a community whose members share an interest in mathematics” (mcgee, 2020, p. 103), the program found that participating black and latinx students had more academic success than their nonparticipant urm, white, and asian classmates. although not mentioned in the book, after further research we found that the mathematics workshop program was only in existence from 1978 to 1984. although the program achieved dramatic results, including the first african american and first female student at berkeley to be awarded a rhodes scholarship, it was short lived, leaving us wondering why it is no longer in existence and why mcgee does not mention this. friedrich & jenkins critical reads journal of urban mathematics education vol. 14, no. 2 111 mcgee (2020) also identifies mentoring as a crucial component of urms’ success when implemented appropriately. she defines successful mentoring as when mentors provide “emotional support, accountability, the skills and strategies they needed to negotiate higher education, and a way of understanding themselves in relation to race-and class-based systems of inequity” (p. 106). we found her detailed list a practical guide for institutions. additionally, we want to emphasize the importance for “faculty members from dominant groups [to] acknowledge the existence of unequal power relationships, discrimination, stereotyping, and oppression of urm groups” (mcgee, 2020, p. 110). it is not until this acknowledgement that we feel that white faculty members can build successful relationships and learn about their urm mentees’ personal histories and goals. although the importance for dominant groups to acknowledge the marginalized positions of urms is present in the text, there are no recommendations made for how to address this with the dominant group. we would have liked to have seen mcgee give specific examples for how institutions can work with their faculty to ensure understanding and strategies to best mentor urms. without these specific examples, minority faculty are left with the responsibility to mentor urms along with other designated and voluntary service work. while mentorship between urms and faculty members with common race and gender have been shown to be effective in providing support in stem environments (borum & walker, 2012; hanson, 2004; jackson, 2013; lockett et al., 2018), worth noting is the limited availability of university faculty that are from historically marginalized groups and the burden this places on those that are in faculty positions (armstrong & jovanovic, 2017; mcgee, 2020). learning from the success stories can enhance teacher education programs and ultimately support students prior to higher education. in the success stories, we have seen patterns of efforts to provide support, embrace culture, and promote positive stem identity development. if teacher education programs are training the newest generation of teachers to be successful mentors to urms and to provide a classroom environment that promotes equity by embracing each student’s culture as well as selfdiscovery, there is a possibility that this shift in stem culture will encourage the success of urms. practical guide for leaders throughout the book, mcgee (2020) not only shares the struggles of urms that have been successful in the field of stem but acknowledged the perseverance that each individual demonstrated to be successful in a system that was designed to exclude them. in the concluding chapter, she identifies seven practical steps to encourage and support students of color to enter the fields of stem. she claims the following practices will not only increase their graduation rates but also their comfort, safety, and health: friedrich & jenkins critical reads journal of urban mathematics education vol. 14, no. 2 112 1. hire more faculty of color in stem faculty positions 2. implement identity-conscious stem mentoring programs 3. hire counselors of color who specialize in the trauma experienced by stem higher education students (and faculty) 4. create pathways for people of color to pursue stem entrepreneurship 5. retain stem faculty and industry leaders 6. acknowledge the work of stem research educators 7. respect and properly fund hbcus, hsis, and tribal colleges as the leaders of urm stem student success (pp. 126–132) these are important steps to take because, as mcgee (2020) states, “diversity becomes key to excellence” (p. 134). these steps are relevant to us personally. as members of the stem education innovation lab at mercer university, we see the benefits of implementing these steps. six out of seven faculty members on our stem education research team are faculty of color. all faculty members implement an identity-conscious mentoring framework to mentor the graduate and postdoctoral fellows. additionally, we are given the opportunity to engage in the work of and interact with stem educators and researchers of color. our team of graduate students have had success both in the classroom and with their own personal research interests. we credit this to the support of the faculty on our research team. in addition to providing practical steps for higher education, mcgee (2020) also provides recommendations for the stem community, stem departments, and policy makers. stem communities should self-examine and closely look at existing diversity models and determine how well they are working. stem department leaders must examine their own departments to determine if additional steps need to be taken to “develop, extend, and sustain equity-centered practices” and “work to increase their cultural competency and use their learning to inform programs, initiatives, and decisions” (mcgee, 2020, p. 135). finally, policy makers should promote equitable teaching practices and development by “hold[ing] universities accountable for maintaining quality standards to enact policy that mandates tackling institutional bias” (mcgee, 2020, p. 136). while we see the value in providing practical guides for stem leaders, departments, communities, and policy makers, we note that mcgee (2020) does not discuss how to address these same issues in teacher education programs. we believe that these recommendations should also be applied to urban mathematics education to dismantle structural racism and support the education of urms at all levels of education. specifically, we would like to see mathematics departments self-examine and look closely at their existing diversity models as well. by determining the state of their diversity models, mathematic departments can either ensure that equity-centered practices are in place or develop these practices to increase their cultural competence. we would also like to see mathematic departments working alongside friedrich & jenkins critical reads journal of urban mathematics education vol. 14, no. 2 113 stem departments to provide community outreach to support the education of urms at all levels of education, beginning in elementary school. with a focus on underrepresentation in higher education, we wonder what a focus on recruiting and retaining stem teachers of color might have looked like in this text. if we recruit and retain stem teachers of color and have more representation of urm teachers in k–12 school buildings, how might that encourage younger generations to matriculate into stem programs? how might that encourage urms to form functional stem identities at younger ages? would urms with frequent stem teachers of color develop healthier, more effective ways to cope with racism and hostile educational environments? black, brown, bruised: how racialized stem education stifles innovation sets the groundwork for discourse on urms in stem; we hope that this conversation will expand to include urban mathematics education, k–12 stem education, and teacher education. we believe shifting our focus to critiquing structures currently in place is necessary and promotes questions that have roots far deeper than experiences in higher education. we wonder how to change the culture of stem environments to embrace diversity, acknowledge cultural backgrounds, and encourage the success of urms. as seen throughout the stories shared, it is apparent that systemic racism and structural biases begin far before higher education. conclusion as we began reading the book, neither of us were surprised by what we were reading, as we have seen many of these experiences play out in front of our eyes. however, mcgee (2020) calling out structural and institutional racism, discrimination, eugenics, and oppression opened our eyes to the reality that jami knew existed and that tynetta experienced firsthand. mcgee opens the door to this discourse by sharing the conversations and stories of others through a presentation of commonalities. in this text, readers are forced to see the struggles of urms in stem rather than reading over it with sugar-coated terminology. black, brown, bruised: how racialized stem education stifles innovation encourages readers to have difficult conversations to acknowledge issues in urban mathematics and stem education and to call for equity and change in these fields. we engaged in this conversation ourselves. tynetta shared with jami that the decision to further her own education and pursue a phd was not only because of her love of learning but largely centers upon her desire to accomplish more for her family. as a first-generation college graduate, tynetta paved her own path. jami, on the other hand, grew up in a family with two parents who are both college graduates. when jami was in high school, she was never asked the question, “will you be going to college?” rather, the question was always “where are you going to college?” not only did she grow up seeing college in her future, she also knew she had the support friedrich & jenkins critical reads journal of urban mathematics education vol. 14, no. 2 114 of her parents who had been to college themselves and grew up understanding the process of continuing into higher education. this cultural capital based on the family she was born into gave jami an advantage over tynetta. we engage in this work to help urms gain the cultural capital to level the playing field. additionally, tynetta knows that more education for her unlocks career opportunities otherwise unobtainable without a terminal degree. however, in most intellectual and professional spaces, including faculty meetings, she feels insignificant, ignored, and overlooked. despite being one of the most qualified in the room, she is often the only black woman. she expressed to jami that her hope is that obtaining a phd gives her a seat at the table. she said, “my academic journey continues to show me that even still, sometimes education and qualifications aren’t enough for a seat at any table. it is unfortunate but a reality that many tynettas know.” jami wonders if an idea she has shared would have been shut down if it had come from tynetta. for jami, it is a hypothetical situation, but for tynetta it is an unfortunate reality. it is the structural racism that tynetta has personally experienced and the open and raw conversations she has shared with jami that motivates us both to engage in this work. these motivates us to continue our research and activism at our own university to promote the success of urms and implement change. our hope is that our work is impactful and prevents future urms in stem fields from experiencing marginalization or abandonment of the stem arena and instead fosters them with feelings of inclusion, support, and acknowledgment by all with a seat at the table. we bring this review to the journal of urban mathematics education to invite urban mathematics educators and researchers to join us in this conversation about the structural and institutional biases that are stifling the success of urms in stem disciplines. black, brown, bruised: how racialized stem education stifles innovation brings to light the academic and psychological struggles urms must overcome to be successful in stem education. the ideas expressed should be the starting point for many more conversations that promote action to reimagine and reconstruct the educational system. references alexander, q. r., & 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(1996). culture in school learning: revealing the deep meaning. lawrence erlbaum associates. jackson, d. l. (2013). a balancing act: impacting and initiating the success of african american female community college transfer students in stem into the hbcu environment. the journal of negro education, 82(3), 255–271. https://doi.org/10.7709/jnegroeducation.82.3.0255 kiyama, j. m., rios-aguilar, c. (eds.). (2018). funds of knowledge in higher education: honoring students’ cultural experiences and resources as strengths. routledge. lockett, a. w., gasman, m., & nguyen, t.-h. (2018). senior level administrators and hbcus: the role of support for black women’s success in stem. education sciences, 8(2), 48–57. https://doi.org/10.3390/educsci8020048 mack, k. m., winter, k., & soto, m. (eds.). (2019). culturally responsive strategies for reforming stem higher education: turning the tides on inequity. emerald. mcgee, e. o. (2020). black, brown, bruised: how racialized stem education stifles innovation. harvard education press. mcgee, e. o., & bentley, l. (2017) the troubled success of black women in stem. cognitionand instruction. 35(4), 265–289. https://doi.org/10.1080/07370008.2017.1355211 mckim, a. j., sorenson, t. j., velez, j. j., field, k. g., crannell, w. k., curtis, l. r., diebel, p. l., stone, d. l., & gaebel, k. (2017). underrepresented minority students find balance in stem: implications for colleges and teachers of agriculture. north american colleges and teachers of agriculture journal, 61(4), 317–323. moll, l. c., amanti, c., neff, d., & gonzalez, n. (1992). funds of knowledge for teaching: using a qualitative approach to connect homes and classrooms. theory into practice, 31(2), 132–141. nilsson, m. r. (2017). point of view: diversity in stem: doctor, heal thyself. journal of college science teaching, 46(4), 8–9. https://doi.org/10.2505/4/jcst17_046_04_8 perna, l., lundy-wagner, v., drezner, n. d., gasman, m., yoon, s., bose, e., & gary, s. (2009). the contribution of hbcus to the preparation of african american women for stem careers: a case study. research in higher education, 50(1), 1–23. https://doi.org/10.1007/s11162-008-9110-y rainey, k., dancy, m., mickelson, r., stearns, e., & moller, s. (2018). race and gender differences in how sense of belonging influences decisions to major in stem. international journal of stem education, 5(10), 1–14. https://doi.org/10.1186/s40594-018-0115-6 riegle-crumb, c., king, b., & irizarry, y. (2019). does stem stand out? examining racial/ethnic gaps in persistence across postsecondary fields. educational researcher, 48(3), 133–144. https://doi.org/10.3102%2f0013189x19831006 takeuchi, m. a., sengupta, p., shanahan, m.-c., adams, j. d., & hachem, m. (2020). transdisciplinarity in stem education: a critical review. studies in science education, 56(2), 213–253. https://doi.org/10.1080/03057267.2020.1755802 friedrich & jenkins critical reads journal of urban mathematics education vol. 14, no. 2 116 tate, e. d., & linn, m. c. (2005). how does identity shape the experiences of women of color engineering students? journal of science education and technology, 14(5), 483–493. https://doi.org/10.1007/s10956-005-0223-1 taylor, o. l., mcgowan, j., & alston, s. t. (2008). the effect of learning communities on achievement in stem fields for african americans across four campuses. the journal of negro education, 77(3), 190–202. tytler, r. (2007). re-imagining science education: engaging students in science for australia's future. australian council for educational research. watkins, s. e., & mensah, f. m. (2019). peer support and stem success for one african american female engineer. the journal of negro education, 88(2), 181–193. https://doi.org/10.7709/jnegroeducation.88.2.0181 copyright: © 2021 friedrich & jenkins. this is an open access article distributed under the terms of a creative commons attribution-noncommercial-sharealike 4.0 international license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. journal of urban mathematics education december 2018, vol. 11, no. 1&2, pp. 1–6 ©jume. http://education.gsu.edu/jume david w. stinson is professor of mathematics education in the department of middle and secondary education in the college of education and human development, at georgia state university, p.o. box 3978, atlanta, ga, 30303; e-mail: dstinson@gsu.edu. his research interests include exploring socio-cultural, -historical, and -political aspects of mathematics and mathematics teaching and learning from a critical postmodern theoretical (and methodological) perspective. he is a co-founder and current editor in chief of the journal of urban mathematics education. editorial celebrating a decade of critical mathematics education knowledge dissemination: a movement of people revolutionaries david w. stinson georgia state university cholar and public intellectual michelle alexander (2018) recently cautioned about the uncritical use of the term resistance in her debut the new york times op-ed essay, “we are not the resistance.” as defiable national and international events have unfolded at an escalating pace in the past two years or so, she argues that although there is power in numbers in the resistance there are downsides as well: but the time may have come to take the downsides [of resistance] more seriously. resistance is a reactive state of mind. while it can be necessary for survival and to prevent catastrophic harm, it can also tempt us to set our sights too low and to restrict our field of vision … leading us to forget our ultimate purpose and place in history. … viewed from the broad sweep of history, … [they are] the resistance. we are not. those of us who are committed to the radical evolution of american democracy are not merely resisting an unwanted reality. to the contrary, the struggle for human freedom and dignity extend back centuries and is likely to continue for generations to come.… a new nation is struggling to be born, a multiracial, multiethnic, multifaith, egalitarian democracy in which every life and every voice truly matters.… every leap forward for american democracy—from slavery’s abolition to women’s suffrage to minimum-wage laws to the civil rights acts to gay marriage—has been traceable to the revolutionary river [of people], not the resistance. in fact, the whole of american history can be described as a struggle between those who truly embrace the revolutionary idea of freedom, equality and justice for all and those who resisted. (para. 8–17) with local, national, and international events occurring daily (some days, it seems hourly) that go against the fundamental ideals of decency and humanity which i was taught as a child, i have continued to think about alexander’s essay that outlines the difference between those who are revolutionaries and those who are resisters. her words certainly have come to the fore as i have been thinking about writing this, my last editorial for the journal of urban mathematics education (jume) after a decade s http://education.gsu.edu/jume mailto:dstinson@gsu.edu stinson editorial journal of urban mathematics education vol. 11, no. 1&2 2 as editor in chief. i am proud to proclaim here that this special double issue which marks a decade of critical1 mathematics education knowledge dissemination, in actuality, marks a decade of a movement of people—a movement of revolutionaries. the revolutionary spirit of jume was unquestionably present in the six articles published in the inaugural jume issue on december 11, 2008. those articles are republished in this special double issue,2 with each proceeded by a follow-up essay, if you will, a decade later written by at least one of the authors revolutionaries of the original article.3 that revolutionary spirit which was present in each inaugural article is again unquestionably present in the follow-up essays published here. all in all, jume was born, so to speak, out of a revolutionary spirit. lou matthews (2008), the founding editor in chief, in his inaugural editorial captured that revolutionary spirit in his description of the nearly two-year developmental stages of jume. developmental stages in which the founding editorial team4—the founding revolutionaries—worked through the start-up logistics of a peer-reviewed, online journal; speculated about its long-term sustainability; and struggled with the multiple meanings of “urban.” ultimately, we collectively decided on a mission statement that guided our work then and continued to guide the work of subsequent editorial teams (i.e., subsequent revolutionaries): to foster a transformative global academic space in mathematics that embraces critical research, emancipatory pedagogy, and scholarship of engagement in urban communities. 1 critical is used here in the critical theoretical sense. bronner (2011), in providing a definition of sorts of critical theory, writes: critical theory refuses to identify freedom with any institutional arrangement or fixed system of thought. it questions the hidden assumptions and purposes of competing theories and existing forms of practice. … critical theory insists that thought must respond to the new problems and the new possibilities for liberation that arise from changing historical circumstances. interdisciplinary and uniquely experimental in character, deeply skeptical of tradition and all absolute claims, critical theory…[is] concerned not merely with how things [are] but how they might be and should be. (pp. 1–2) 2 the six inaugural articles are republished in this special double issue as initially made available with only minor formatting and copyediting changes. 3 each of the inaugural articles has a follow-up essay except for the article written by pamela l. paek (2008); unfortunately, she was not able to contribute a follow-up essay, but we honored her contribution to the inaugural issue by republishing her 2008 article here. 4 the original editorial team included lou matthews, the founding editor in chief, and associate editors pier junor clarke, ollie manley, david stinson (me), and christine thomas. stinson editorial journal of urban mathematics education vol. 11, no. 1&2 3 i now like to think that the transformative global academic space in mathematics created in and through jume5 has been in resistance neither to the whitestream journals of mathematics education (gutiérrez, 2011) nor to the institutional space of whiteness of mathematics education (martin, 2013). but rather, as alexander (2018) suggests, has been a revolutionary river of people who have understood that “every life and every voice truly matters” (para. 13) and “who truly embrace the revolutionary idea of freedom, equality and justice for all” (para. 17). for the past decade, i have populated the opening editorial pages of nearly every jume issue with what i hoped would be a thought-provoking discussion representative of the revolutionary spirit of jume. here, in my closing editorial, i wish to just simply and humbly thank, by name, the people—the revolutionaries—who brought jume into existence: the authors, the reviewers, and the editors (see listings below).6 whatever role or roles one has played throughout the past decade, it has truly been an honor to work with each of you. both my personal and professional lives have been enriched by the opportunity. my thinking as a mathematics education scholar, researcher, and teacher educator—and, most importantly, my thinking as a human being—has benefitted greatly through the human connections i have had with each of you during my time as editor. in the end, it is always the human connections, the human relationships in which we accumulate throughout our multiplicitous and fragmented lives that actually bring meaning(s) to living. – thank you for the opportunity to connect! david w. stinson, ph.d. editor in chief journal of urban mathematics education june 2009–december 2018 5 to learn more about jume and its growing impact over the past decade see journal history under about on the jume website: http://ed-osprey.gsu.edu/ojs/index.php/jume. currently, after more than a decade at georgia state university, jume is in search of a new editor in chief, editorial team, and academic home. check the jume website periodically for updates to when jume will resume accepting manuscripts for publication consideration. 6 there is another extraordinarily important group of people to thank that, unfortunately, i cannot thank by name: jume readers/users. with nearly 300,000 web views and counting of jume content and nearly 1,500 google scholar citations and growing exponentially, to thank this group individually is impossible. but it goes without saying, jume readers/users directly and indirectly have strengthened the revolution. http://ed-osprey.gsu.edu/ojs/index.php/jume/about/history http://ed-osprey.gsu.edu/ojs/index.php/jume/about http://ed-osprey.gsu.edu/ojs/index.php/jume https://scholar.google.com/citations?user=dyb3gm0aaaaj&hl=en stinson editorial jume authors gill adams olufunke adefope helle alrø nathan n. alexander charlotte agger julia maria aguirre joel amidon celia rousseau anderson annica andersson cynthia oropesa anhalt glenda anthony robin averill tamika n. ball vanessa pitts bannister david barnes lorraine m. baron tonya gau bartell dan battey michael t. battista robert q. berry, iii kristin bieda jo boaler gareth bond john bragelman andrew brantlinger denise natasha brewley m. lynn breyfogle diane j. briars kanjana brodie liz brown viveka a. brown victor brunaud-vega erika c. bullock jessica hopson burbach susan o. cannon robert m. capraro matthew g. caputo nicole carignan iman c. chahine egan j. chernoff ervin j. china haiwen chu marta civil jere confrey lesa m. covington clarkson sandra crespo helen crompton dionne i. cross erica r. davila julius davis ira david dawson zandra de araujo higinio dominguez lesley dookie corey drake teresa k. dunleavy cyndi edgington mark w. ellis indigo esmonde brian r. evans james ewing mathew d. felton-koestler anthony fernandes cecilia henríquez fernández mary q. foote deana j. ford maisie l. gholson laura m. gilbert imani masters goffney curtis v. goings conrado gómez lidia gonzalez melva r. grant susan a. gregson paula guerra maura varley gutiérrez rochelle gutiérrez eric (rico) gutstein victoria hand shelley sheats harkness jacqueline a. hennings beth herbel-eisenmann keith e. howard rick a. hudson roberta hunter christopher c. jett margarita jimenez-silva kate r. johnson mindy kalchman signe kastberg virginia keen lena l. khisty lonnie c. king richard s. kitchen jennifer kosiak terri l. kurz joan kwako matt larson brian r. lawler mi yeon lee okhee lee shonda lemons-smith jacqueline leonard kate le roux chance w. lewis luis a. leyva woong lim pauline lipman fiachra long dihema longman patricia l. marshal danny bernard martin lou edward matthews jane mcchesney amy roth mcduffie allison w. mcculloch ebony o. mcgee maggie lee mchugh laura mcleman michelle mcnulty mekyah q. mcqueen michael meagher bryan meyer alexia mintos jessica morales-chicas crystal h. morton judit moschovich eduardo mosqueda stinson editorial journal of urban mathematics education vol. 11, no. 1&2 5 eileen c. murray jomo w. mutegi kayla d. myers marrielle myers na’ilah suad nasir megan nickels francis m. nzuki stephen o’brien sarah oppland-cordell daniel clark orey pamela l. paek arnulfo perez joyce piert elijah porter roland g. pourdavood hilary povey angiline powell tamra c. ragland lauren rapacki mary candace raygoza maría elena rodríguez pérez eliana d. rojas martin romero milton rosa amber rose francine cabral roy laurie h. rubel derrick saddler matthew sakow pedro paulo scandiuzzi allison scott walter g. secada tesha sengupta-irving stanley f. h. shaheed james sheldon ksenija simic-muller ole skovsmose beverly s. smith erin smith joi a. spencer megan e. staples kathleen jablon stoehr marilyn e. strutchens dalene m. swanson paola sztajn miwa takeuchi paulo tan daryl a. tate william f. tate clarence l. terry, sr. stephanie timmons-brown mary p. truxaw erin e. turner paola valero eugenia vomvoridi-ivanović lanette r. waddell anita a. wager erika n. walker margaret walshaw john o. wamsted catharine warner dorothy y. white kristin whyte megan h. wickstrom craig willey brian anthony williams candace williams morgin jones williams joycelyn wilson p. holt wilson susan wilson melanie n. woods constantinos xenofontos emily joy yanisko cathery yeh zeyner ebrar yetkiner jamaal rashad young jemimah lea young jan a. yow paul w. yu maria del rosario zavala jume reviewers see reviewer acknowledgment january 2008–december 2018 jume editors lou edward matthews* editor in chief january 2008–may 2009 david w. stinson* associate editor editor in chief special issue editor special issue editor copy and production editor january 2008–may 2009 june 2009–december 2018 november 2012–july 2013 january 2018–december 2018 january 2008–december 2018 pier junor clarke* associate editor january 2008–july 2015 ollie irons manley* associate editor january 2008–december 2011 christine d. thomas* associate editor january 2008–july 2015 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/365/240 stinson editorial journal of urban mathematics education vol. 11, no. 1&2 6 erika c. bullock assistant editor assistant to the editor special issue editor associate to the editor public stories editor copy and production editor january 2010–december 2012 january 2013–may 2013 november 2011–july 2012 june 2013–december 2018 january 2014–december 2018 january 2013–december 2018 christopher c. jett assistant editor associate editor book review editor january 2010–july 2011 august 2011–july 2015 august 2015–december 2018 nermin bayazit associate editor august 2010–july 2015 stephanie behm cross associate editor august 2010–july 2015 iman chahine associate editor august 2010–july 2015 jessica hale assistant editor august 2013–july 2014 morgin jones williams assistant editor august 2013–july2014 alesia mickle moldavan assistant editor associate editor august 2014–may 2018 june 2018–december 2018 susan o. cannon assistant editor august 2017–december 2018 nathan n. alexander special issue editor november 2011–july 2012 maisie l. gholson special issue editor november 2011–july 2012 joi a. spencer special issue editor november 2012–july 2013 * founding editorial team member references alexander, m. (2018, september 21). we are not the resistance. the new york times. retrieved from https://www.nytimes.com/2018/09/21/opinion/sunday/resistance-kavanaugh-trump-protest.html bronner, s. e. (2011). critical theory: a very short introduction. new york, ny: oxford university press. gutiérrez, r. (2011, april). identity and power. in r. gutiérrez (chair), who decides what counts as mathematics education research? symposium conducted at the research presession of the national council of teachers of mathematics, indianapolis, in. martin, d. b. (2013). race, racial projects, and mathematics education. journal for research in mathematics education, 44(1), 316–333. matthews, l. e. (2008). illuminating urban excellence: a movement of change within mathematics education. journal of urban mathematics education, 1(1), 1–4.retrieved from http://edosprey.gsu.edu/ojs/index.php/jume/article/view/20/9 paek, p. l. (2008). practices worthy of attention: a search for existence proofs of promising practitioner work in secondary mathematics. journal of urban mathematics education, 1(1), 84–107. retrieved from http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/6/1 https://www.nytimes.com/2018/09/21/opinion/sunday/resistance-kavanaugh-trump-protest.html http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/20/9 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/20/9 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/6/1 microsoft word 2 final tate b et al vol 11 no 1&2.doc journal of urban mathematics education december 2018, vol. 11, no. 1&2, pp. 31–44 ©jume. http://education.gsu.edu/jume william f. tate is the edward mallinckrodt distinguished university professor in arts & sciences and dean & vice provost for graduate education at washington university in st. louis, one brookings drive campus box 1183, st. louis, mo 63130-4899; e-mail: wtate@wustl.edu. his interdisciplinary scholarship concentrates on two main areas: mathematics, science, and technology education, specifically, in metropolitan america; and geospatial and epidemiological approaches to the study of the social determinants of education and health outcomes. celia rousseau anderson is a professor in the department of instruction and curriculum leadership in the college of education at the university of memphis, 3798 walker avenue, memphis, tn 38152; email: croussea@memphis.edu. her research interests include equity in mathematics education, urban education, and critical race theory. daryl a. tate is an assistant professor and the coordinator of learning systems technology education at the university of arkansas at little rock, 2801 s. university avenue, little rock, ar 72204; email: datate@ualr.edu. his research and professional interest coalesce into four interrelated areas: (a) on-line learning communities, (b) multimedia integration into curriculum and technology pathways, (c) technology professional training and development, and (c) digital methodology to enhance scholarly inquiry. “sum” is better than nothing: toward a sociology of urban mathematics education william f. tate washington university in st. louis celia rousseau anderson university of memphis daryl a. tate university of arkansas at little rock the purpose of this commentary is to serve as a warning that developing and testing theories is central to making urban mathematics scholarship a visible research enterprise. more specifically, i will argue that there are lessons to be learned from the social sciences literature that can inform the advancement of a robust, theoretically based, empirical project in urban mathematics education research. in addition, these fields of social science are part of the rationale for why putting the “urban” in mathematics education scholarship is important. (tate, 2008a, p. 5) early ten years after this call to develop an urban mathematics education research enterprise, anderson and tate (2016) described the conceptual components of a sociology of mathematics education. they argued several questions should inform the development of a sociology of mathematics education. adapted and expanded from their analysis, we submit the following questions to serve as starting points for scholars interested in sociological approaches to the study of mathematics education in urban communities: 1. is geography a factor in opportunity to learn in urban mathematics education? 2. how does mathematics education socialize in urban communities? 3. does credentialing influence opportunity in urban mathematics education? n tate et al. “sum” is better than nothing journal of urban mathematics education vol. 11, no. 1&2 32 4. what societal factors influence urban mathematics education? in the following sections, we explore each of these questions relative to mathematics education in urban communities. we describe in brief what a program of study focused on a sociology of urban mathematics (sum) education entails. we view this as a starting point, rather than an all-inclusive framework. geography of opportunity a regime is a public-private partnership designed to produce a specific goal. by way of law, public policy, and social custom across urban communities in america, segregation regimes framed the nature of opportunity in local economies (rothstein, 2017; tate & jones, 2017). federal lawmakers and policy elites; local municipality leaders; and private sector actors, such as real estate brokers, churches, financial institutions, and neighborhood associations, collaborated to restrict african american families’ access to quality neighborhoods. the segregation regime limited the access of african american families to communities perceived as more desirable for white families and steered many of them to segregated public housing and high poverty neighborhoods. african americans’ opportunities to acquire residential housing, wealth, and to accrue value from community-based related services, such as education and health care in a region, diminished in appreciable fashion. consequently, neighborhood poverty in urban communities concentrated, and for low income african americans and latinos/as endures for multiple generations (sharkey, 2013). given that education has served as the primary mechanism supporting income mobility in the united states (duncan & murnane, 2015), understanding the geography of opportunity in education represents a primary consideration for scholars in mathematics education (tate, 2008a, 2008b). geographic factors within neighborhoods and communities influence the relational, organizational, and collective actions impacting the social formation of opportunity and access in mathematics education (anderson & tate, 2016; tate, 2008a). hogrebe and tate (2015) described the implications of these social formations for conducting research in mathematics education where the school district serves as the unit of analysis: in the case of neighboring school districts, there is clustering created by spatial proximity that produces unique local contexts and concomitant within-group correlation. in many instances, families and students within school districts tend to be somewhat similar on factors such as education, income, and housing. the built environment does not start and stop abruptly at district boundary lines. the argument is that similarity across local contexts functions as a spatial continuum in geographic space. therefore, while unique local contexts certainly exist and the correlation of factors within them must be accounted for, analysis techniques that do not consider the spatial continuum may be inappropriate for spatially clustered data such as school districts within a region or state. (p. 100) tate et al. “sum” is better than nothing journal of urban mathematics education vol. 11, no. 1&2 33 figure 1. local r2 values for the relationship between minority percent and algebra i scores in missouri school districts. to account for regional clustering of school districts, several stem (science, technology, engineering, and mathematics) education studies use geographically weighted regression (gwr), a method designed to incorporate the underlying spatial continuum (hogrebe & tate, 2012; tate & hogrebe, 2015). the modeling technique allows relationships to vary across groups (e.g., school districts in a state), as does multilevel modeling (mlm), but unlike mlm, it does not ignore the similarity among neighboring school districts. the methodology estimates where variable relationships differ across regions and throughout the state. thus, the approach provides important insight into the geography of opportunity in mathematics education. for example, hogrebe and tate (2012) examined the relationship between district composition variables and end of course algebra i scores across the state of st. louis metro area kansas city metro tate et al. “sum” is better than nothing journal of urban mathematics education vol. 11, no. 1&2 34 missouri. figure 1 illustrates r2 values across the state. several distinct cluster patterns emerged with local r2 values ranging from close to zero to .56. school districts in metropolitan st. louis and kansas city formed two distinct clusters of high r2 values. figure 2. statistically significant beta coefficients for the relationship between minority percent and algebra i scores in missouri school districts. negative coefficients indicate higher minority percentage associated with lower algebra i scores. multiple t-tests for beta coefficients corrected for false positive rate by benjaminihochberg (b-h) procedure (thissen, steinberg, & kuang, 2002). figure 2 shows that the statistically significant beta coefficients cover the same two large urban metro regions. the negative sign of the coefficients suggests st. louis metro area kansas city metro ar tate et al. “sum” is better than nothing journal of urban mathematics education vol. 11, no. 1&2 35 that lower algebra i scores were associated with higher percent minority in the school district. in contrast, the southwest border cluster had positive beta coefficient signs that associate higher minority percent with higher algebra i scores. estimates of the relationship between minority percent in the school district and algebra i performance across a majority of the state did not elevate to statistical significance. the missouri case offers an important lesson for theory building in the sociology of urban mathematics education. the negative relationship between percent minority enrollment in a school district and mathematics attainment found in urban communities warrants attention in theory development. using missouri as a case, hogrebe and tate’s (2012) findings affirm the rationale of the national science foundation (nsf) to develop a specific stem intervention designed for urban school districts. williams (2018) reflected on the call to develop the nsf’s urban systemic initiative (usi) program: urban school systems enrolled approximately 50% of u.s. public school k–12 students. moreover, as noted in the usi solicitation, there is a well-documented disparity between the academic performance of these students and that of their counterparts in suburban schools, an achievement gap that is not independent of uneven allocation of resources, including experienced (and highly competent) teachers, as well as the paucity of advanced courses, curriculum materials, instructional equipment and facility. (pp. 12–13) too often in theory building and in the establishment of traditions in education research, the challenges and assets of urban communities fail to emerge as foundational in the research, evaluation, and related reform. charged with oversight for education, states need to account for how the distribution of opportunity to learn factors related to mathematics education vary within their borders. many state agencies lack the infrastructure to realize this goal. read and atinc (2017) call for infomediaries, civil society organizations (e.g., universities), researchers, and the media to translate data into actionable information designed to foster shared learning among parents, teachers, administrators, and policymakers. urbanists focused on researching the geography of opportunity in mathematics education should engage fully this call. geography matters. socialization in their 2016 chapter in the handbook of international research of mathematics education (3rd ed.), anderson and tate examined the existing research addressing the question: how does mathematics education socialize? their review focused on research involving mathematics socialization and mathematics identity. these processes of mathematics socialization and identification have emerged as the foci of significant lines of scholarship in mathematics education, particularly with regard to the experiences of students of color in mathematics (langer-osuna & es tate et al. “sum” is better than nothing journal of urban mathematics education vol. 11, no. 1&2 36 monde, 2017; martin, anderson, & shah, 2017). while a growing body of research related to identity and socialization in mathematics exists, we submit that the role of urban mathematics education in the socialization process has not been addressed substantively to date. we acknowledge that many of the studies of socialization and identity involve students who attend urban schools (martin & larnell, 2014). moreover, these studies have provided insight into the process of racialmathematical socialization (martin, 2006; martin et al., 2017). however, we submit that research in mathematics education has yet to explore the ways in which urban mathematics education socializes. we refer to a study by jackson (2009) to illustrate how this attention to urban mathematics education might expand our understanding of the socialization process. jackson’s research documents the instructional practices of a 5th grade mathematics teacher in an urban school, which she calls johnson middle school, and the role of these teaching practices in the mathematics socialization of two african american students. jackson also interrogates the ways that larger school and community-wide discourses about students shape what happens in the classroom. according to jackson, the teacher’s classroom “as all classrooms, was a nexus of discourses about youth, about mathematics, and about pedagogy. the local practices were influenced by discourses about poor children of color and mathematics that circulated outside of johnson middle school” (p. 195). the study provides important insight into the interactions of the local context in shaping teachers’ conceptions of students and mathematics and the impact of these conceptions on students’ experiences. yet, using jackson’s (2009) study as a starting point, we offer additional questions that might illustrate the potential of a sociology of urban mathematics education in the examination of mathematics socialization. specifically, what features of urban schools and systems influence the mathematics socialization of students in these schools? in thinking about this question in relation to jackson’s (2009) study, we consider it crucial to note that jackson conducted her research in an urban charter school that was part of a larger charter network. according to jackson, although all the students in the school were of color, the staff was predominantly white and most had been teaching at the school for less than two years. she also reveals that the school’s basic-skills mathematics curriculum had been created by the charter school network. the school operated under the assumption that a deficiency existed in the entering 5th grade students’ values, academic backgrounds, and behaviors. as a result, the school implemented strategies to intervene in these areas, including a 3-week summer intensive initiation for new students to learn the school norms, attention to “character building,” and a monetary system of rewards and sanctions for behavior. a sociology of urban mathematics education might focus on the typicality of schools such as johnson middle school and seek to interrogate the ways in which the features of these schools influence mathematics socialization. for example, the tate et al. “sum” is better than nothing journal of urban mathematics education vol. 11, no. 1&2 37 fact that johnson middle school was a charter school is potentially noteworthy from a sociology of urban mathematics education perspective for several reasons. one of the hallmarks of urban education reform has been the growth of charter schools and charter school management organizations. this growth has occurred within a larger market-based approach to educational reform or as part of urban districts’ adoption of a portfolio management model (anderson & dixson, 2016; bulkley, henig, & levin, 2010; dixson, royal, & henry, 2014). these schools have increased levels of autonomy, relative to traditional district schools, with respect to issues such as curriculum and instruction, staffing, budgeting, and so on (berends, 2014; bulkley, 2010). we do not know from jackson’s (2009) study how the specific history or characteristics of johnson middle school might reflect the larger trends of urban education reform. however, given the role of charter schools in urban districts, we consider it worthwhile to examine the ways in which settings such as johnson might shape the mathematics socialization process. for example, we might consider the socialization process at johnson middle school (jackson, 2009) within the larger context of charter school organizations. specifically, some charter school organizations reflect what social scientists refer to as a “no-excuses” approach. according to carr (2013), no-excuses reformers believe that— low-income children must be taught, explicitly and step-by-step how to be good students. staff at a growing number of “no-excuses” charter schools—which are highly structured and emphasize college matriculation—are prescriptive about where new students look (they must “track” the speaker with their eyes), how they sit (up-right, with both feet planted on the ground, hands folded in front of them), how they walk (silently and in a straight line, which is sometimes marked out for them by tape on the floor), how they express agreement (usually through snaps or “silent clapping” because it’s less disruptive to the flow of the class), and, most important, what they aspire to (college, college, college). this conditioning (or “calibration,” or “acculturation,” as it is sometimes referred to inside no-excuses schools) starts with the youngest of students. (pp. 42–43) while we do not know from jackson’s (2009) description whether johnson middle school reflected this no-excuses model, features such as a 3-week initiation for new students, attention to “character building,” and a college prep mission suggest at least some alignment with the no-excuses approach. on the surface, this attention to the “cultural” model utilized by a segment of the charter school sector may seem far afield from mathematics education research. yet, as jackson’s research demonstrates, these organizational perspectives on students can have a significant influence on the ways that schools socialize students into mathematics. thus, we submit that a sociology of urban mathematics education might situate the socialization process within this larger charter context. tate et al. “sum” is better than nothing journal of urban mathematics education vol. 11, no. 1&2 38 the role of the accountability system in the socialization of students in urban schools represents another element of schooling that should be considered in a sociology of urban mathematics education. urban schools often experience conflicts between the accountability system and the time and curriculum needed to support teaching for understanding (gamoran et al., 2003; walker, 2012). depending on the structure of the accountability system in the local context, schools can feel substantial pressure to focus on preparing students for state tests in order to avoid sanctions that can include take-over or closure (anderson, bullock, cross, & powell, 2017). based on jackson’s (2009) description, we do not know what role the accountability system might have played in the approach to mathematics evidenced at johnson middle school. however, the focus on “basic skills” and procedural knowledge represents a common response to high-stakes accountability (gamoran et al., 2003; walker, 2012). understanding the nature of the accountability system in the local context and the pressures around student testing represent potentially important considerations within a sociology of urban mathematics education. finally, situating cases such as johnson middle school (jackson, 2009) within the larger trends of urban education reform also requires consideration of the role of issues such as teacher shortages, teacher turnover, and teacher capacity in the mathematics socialization process. for example, according to jackson, only a few of the teachers had worked at johnson since its opening 2 years before the start of her study. again, we do not know what the rate of turnover had been at johnson. however, it is well-documented that urban schools are characterized by higher rates of teacher turnover than schools in other settings and that this turnover is a key factor driving shortages of qualified and effective teachers (ingersoll & merrill, 2013). moreover, research on the impact of turnover has pointed to the negative impact not only on the achievement of students whose teachers leave but also on other students within the school. this finding suggests that the harmful effect of teacher turnover is at least partially related to issues of school-wide culture (ronfeldt, loeb, & wyckoff, 2013). given the prevalence of turnover in many urban schools, we submit that a sociology of urban mathematics education would attend to these issues in an examination of the socialization process. in this section, we sought to outline potential areas of focus for research on mathematics socialization grounded in a sociology of urban mathematics education. this is not to suggest that research on mathematics socialization involving urban schools and students does not exist (martin & larnell, 2014; walker, 2012). rather, we sought to highlight ways in which scholarship on mathematics socialization might engage what we know from sociological research around urban education. we utilized jackson’s (2009) study as a reference point because her research clearly links mathematics socialization processes to larger discourses within the school and community. our purpose was not to suggest that her research was lacking in any way but simply to illustrate how this work could be bridged with other bodies of tate et al. “sum” is better than nothing journal of urban mathematics education vol. 11, no. 1&2 39 scholarship to craft an account of mathematics socialization informed by a sociology of urban mathematics education. credentialing and societal factors in addition to its socializing role, mathematics education involves the credentialing of students and their teachers (anderson & tate, 2016). the credentialing problem space requires both a geospatial perspective and racial lens. toldson (2014) reported that (a) black children received access to less cognitively demanding coursework with greater misalignment to college admission requirements, (b) schools assign black children the lowest paid and least experienced teachers at higher rates, and (c) black students are the most likely to have teachers credentialed through alternative teacher certification programs. education as a credentialing mechanism dates back in history several centuries. adam smith (1776) implicitly framed the role of education as credential, stating, “the public can encourage the acquisition of those most essential parts of education by giving small premiums, and little badges of distinction, to the children of the common people who excel in them” (p. 352). operating as modern day “badges of distinction,” college-preparatory tracks in high school serve a credentialing function for students. this track offers academic preparation in mathematics aligned with access to post-secondary education and variety of competitive vocational opportunities. debates related to the design, organization, and content of the college preparatory math track remain central to opportunity to learn in urban communities. over the past 3 decades, calculus served as the mathematics credential required for students seeking entry into the nation’s top colleges and universities. thus, the pathway to calculus completion in high school drove access and opportunity discourse. math track trajectories gained perceived value in the credentialing process in relationship to the calculus entry point. school districts reverse-engineered their mathematic programs putting great emphasis on building the prerequisite knowledge associated with readiness for calculus. thus, early entry into algebra i and algebra ii emerged as guides for determining student excellence. this perception persisted despite signals that calculus completion in high school did not consistently result in students placing out of calculus in college. in fact, many high school calculus completers start their college mathematics education in calculus readiness coursework. noting these college placement trends and the racial and socioeconomic disparities in calculus enrollment in high schools, reformers call for an alternative post-secondary pathway with statistics as the content driving the new learning opportunity (burdman et al., 2018). the urban sociology of mathematics education project corresponds with documenting the alignment of any new credentialing pathway and determining access to the post-secondary education system. tate et al. “sum” is better than nothing journal of urban mathematics education vol. 11, no. 1&2 40 statistics as the primary secondary school credential also stems from changes in the scientific enterprise and social sciences including the increased importance of data science, bioinformatics, and modeling. moreover, statistical modeling and computational science continue to grow in their importance as a tool to survey and to police the urban underserved: the advocates of automated and algorithmic approaches to public services often describe the new generation of digital tools as disruptive. they tell us the big data shakes up hidebound bureaucracies, stimulates innovative solutions, and increases transparency. but when we focus on programs specifically targeted at poor and work-class people, the new regime of data analytics is more evolution than revolution. it is simply an expansion and continuation of moralistic and punitive poverty management strategies that have been with us since the 1820s. data scientists push high-tech tools that promise to help more people, more humanely, while promoting efficiency, identifying fraud, and containing costs. the digital poorhouse is framed as a way to rationalize and streamline benefits, but the real goal is what it has always been: to profile, police, and punish the poor. (eubanks, 2018, pp. 37–38). social equity calls for urban mathematics education researchers to evaluate how students come to understand the use of mathematics in society. the design of credentialing pathways should link not only to post-secondary education opportunities but also to how society uses mathematics in context. in a speech given on march 31, 1968 at the national cathedral in washington, dc, martin luther king, jr. warned us about this era: there can be no gainsaying of the fact that a great revolution is taking place in the world today. in a sense it is a triple revolution: that is, a technological revolution, with the impact of automation and cybernation; then there is a revolution in weaponry, with the emergence of atomic and nuclear weapons of warfare; then there is a human rights revolution, with the freedom explosion that is taking place all over the world. yes, we do live in a period where changes are taking place. (para. 7) ever the prophet, today dr. king’s words ring true, with automation being used as a weapon against the poor living in urban america. preparing informed citizens requires a secondary mathematics credentialing pathway that supports students’ opportunities to understand how automation influences their experiences, support structures, and life course. tate (1994) argued that economics drives the reasons situations are mathematized. the first goal is to maximize the return on information with respect to organizational aims. the second goal is to minimize the challenges and responses to how the mathematizing informs the decision-making process. in the age of automation and big data, mathematics represents both an opportunity accelerant toward the economic advantages associated with attaining a post-secondary education and the foundation of a technological world assigning merit to citizens’ requests for health and other developmental supports. understand tate et al. “sum” is better than nothing journal of urban mathematics education vol. 11, no. 1&2 41 ing this duality and tension represents a central problem space in a sociology of urban mathematics education. the importance of credentialing and digital advances in society extends to teachers. the preparation, credentialing, and distribution of teachers have been demonstrated to influence urban students’ opportunities to learn and to achieve in mathematics. according to carver-thomas (2018), approximately 40% of new los angeles unified school district teachers fall short of completing the preparation and requirements for a preliminary teaching credential. in the stockton unified, over 50% of the new teachers lack the credential and associated preparation. severe shortages exist in mathematics and bilingual education across the state of california. shortages exist nation-wide. in the state of texas, more than half of the students reside in school districts where teacher certification requirements have been waived (dugyala, 2018). the potential consequences of these credentialing challenges on the teaching and learning conditions in urban schools represent another area of study in the sociology of urban mathematics education. some scholars suggest digital advances associated with online learning offer a remedy to credentialing inequalities. christensen, hor, and johnson (2010) projected that 50% of all high school classes will be offered online by 2020. an ambitious projection if framed as an actual count of classrooms, nevertheless the ability to offer online content across the secondary mathematics curriculum exists. the primary advantage of online learning involves how the delivery method levels the playing field for students experiencing geographic disparities with respect to educational resources and quality learning experiences (kuo, 2014). online learning provides anytime, anywhere learning experiences, while allowing access to interactive format offering varied demonstrations and practice sites, emerging communication innovations, and responsive assessment practices. in addition, advocates of online platforms promise that digital learning increases the numbers of students that can be reached as districts struggle to find and to pay for credentialed teachers. evidence suggests that a blended option consisting of face-to-face and online opportunities produces learning outcomes similar to traditional classrooms, yet better than online only (escueta, quan, nickow, & oreopoulos, 2017). the role of digital learning as an equity project to address teacher distribution disparities aligns with the sociological study of urban mathematics education. as digital learning formats developed and matured, online teacher education programs emerged to address the teacher shortage disparities. for example, a large state university in a major metropolitan region of texas offered an online post baccalaureate program designed to improve teacher candidate diversity and to increase the candidates in areas of shortage such as mathematics (harrell & harris, 2006). the 18-credit hour program consisted of 12 credit hours of fully online study and 6 credit hours of practicum. early results indicated the program attracted more career changers and minority candidates, while increasing the institution’s candidates in tate et al. “sum” is better than nothing journal of urban mathematics education vol. 11, no. 1&2 42 teacher shortage fields of mathematics and science. online teacher education represents a potential growth industry as geographic disparities in teacher distribution fail to dissipate. the need to evaluate the quality and efficacy of online teacher education represents another area to prioritize in the sociological study of urban mathematics education. sum why invest into the development of a sociology of urban mathematics research? if we sum up the country’s population, over 60% of the u.s. population lives in an urban city, yet urban cities make up less than 4% of the landmass in the country (cohen, 2015). while our essay focused on the united states, mathematics education research is a global enterprise. unicef (2012) estimated that over one billion children live in cities and towns worldwide. additionally, they reported that by 2050, 7 in 10 people will live in cities across the globe. the massive volume of children calls for understanding the scale and nature of poverty and how it influences opportunity to learn mathematics in urban areas. in addition, the need to identify barriers and to remove obstacles constraining opportunities for urban students in mathematics will grow as the population in cities and metropolitan regions increases. a clear understanding of the state of the infrastructure and of the delivery services to support mathematics learning in urban cities is connected tightly to reducing inequality and poverty reduction. credentialing functions operate at different levels of government, corporate sector, and nonprofits. evaluating how these institutions partner in urban communities to form opportunity regimes in mathematics education or flounder is a priority. sum effort on this front is better than nothing. acknowledgments the authors acknowledge and thank edna j. cash and the special issue editors for their thoughtful editorial support. references anderson, c., bullock, e. c., cross, b., & powell, a. 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(2018). the national science foundation’s urban systemic initiative: a retrospective assessment. in. l. s. williams & m. cozzens (eds.), projecting forward: learning from educational systemic reform (pp. 1–22). medford, ma: comap. microsoft word 372-article text no abstract-1914-1-6-20200331 (proof 2, page no.).docx journal of urban mathematics education may 2020, vol. 13, no. 1, pp. 15–33 ©jume. https://journals.tdl.org/jume michelle t. lo is master of arts and teacher candidate at stanford teacher education program, graduate school of education, 485 lasuen mall, stanford, ca 94305-3096; e-mail: mlo3@stanford.edu. her research interests include student development of authority and agency in the classroom, teacher action research, mathematical student identities, and kinesthetic learning. jennifer l. ruef is assistant professor of mathematics education in the university of oregon’s department of education studies, 5277 university of oregon, eugene, or 97403-1215; email: jruef@uoregon.edu. her research interests include how students identify themselves, or are identified by others, as being “good at math” as well as the ways in which students and teachers learn to make sense of mathematics and construct convincing and powerful arguments. student or teacher? a look at how students facilitate public sensemaking during collaborative groupwork michelle t. lo stanford university jennifer l. ruef university of oregon as institutions strive to improve teaching and learning for all, educators must consider equitable instruction. this includes equitable distributions of authority and agency among students. the authors define distribution of authority as shared opportunities for decision-making in enacting mathematical tasks and agency as the power to carry out these self-made decisions. equitable distributions of authority and agency can be enhanced in mathematics public sensemaking classrooms where students participate in discourse as active members of the classroom. in public sensemaking classrooms, students understand and acknowledge one another’s ideas, present and revise arguments, and take risks by sharing ideas. this study investigates one group of students and how they positioned one another during mathematical groupwork in a public sensemaking classroom as well as how the positioning impacted the distribution of agency and authority. at the time of data collection, the students attended a school for grades 6–12 situated in an urban public school that focuses largely on preparing students for careers in the health sciences and other health-related fields. data was collected as video footage and analyzed using a priori codes. analyses indicate that one student replicates the role of teacher by redistributing authority and agency back to other students through selfremoval. this is contrasted with other motivations for self-removal while doing mathematics in a team. the findings inform researchers and classroom teachers of potential metacognitive awareness of equity between students, positioning patterns that may occur during collaborative groupwork, and the effectiveness of public sensemaking classrooms on distributing authority and agency equitably during groupwork. keywords: groupwork, authority, agency, mathematics education, public sensemaking lo & ruef public sensemaking during groupwork journal of urban mathematics education vol. 13, no. 1 16 introduction istorically, particular groups of people, such as women, african americans, hispanics, indigenous peoples, and southeast asians, have been marginalized and underrepresented in science, technology, engineering, and mathematics (stem) fields. this restriction of access presents an equity problem and impoverishes stem fields by limiting participation. the requirements of a 21st century workforce call for increased collaboration and problem solving (national research council, 2011). thus, the ability to work productively with diverse groups is critical. as collaboration, problem solving, and diverse perspectives are required for future success, educators must shift their practices to ensure all students are gaining access to rigorous mathematics, especially those who have been traditionally excluded and marginalized (burris et al., 2006, 2008; darling-hammond, 1995; gamoran, 2009; jett, 2019; langer-osuna, 2011; oakes, 1990). because mathematics is evident in the other stem fields and beyond, an increasing number of mathematics educators are concerned with establishing equity in classrooms and implementing equitable practices to keep students, especially those who have been historically marginalized, interested in mathematics throughout their k–12 education (national council of teachers of mathematics [nctm], 2018). when students become active partners in creating learning opportunities for each other, the power of an equitable classroom manifests in a network of actors. no longer the sole province of the teacher, distributions of agency allow students to create opportunities for their colleagues to succeed. but how might teachers foster such distributions of equity among students? how might teachers and researchers recognize such agency when it occurs? our study examines this problem space and expands knowledge of how students might co-construct equitable teaching and learning practices. our study was situated within an urban public school. the students and their teacher are thus part of a larger context that is often framed in terms of deficits (e.g., hand, 2010; horn, 2007; jackson, 2010; martin, 2012; tate et al., 2018). this is evident in policies that punish schools and districts for low performance on highstakes tests or impose impoverished forms of teaching and testing based on beliefs that urban students are less capable of engaging with challenging curriculum and pedagogy than their suburban peers (cabana et al., 2014). our study joins a growing body of research that frames historically minoritized and urban students in terms of assets (e.g., dunleavy, 2015; langer-osuna, 2018; leonard & martin, 2013; ruef, in press; watanabe & evans, 2015). this research is, at its roots, focused on equitable teaching and learning practices. we share an example of how one group of students mirrored their teacher’s model for distributed authority and agency. distributions of authority and agency are core elements of teaching for equity. we define equity in teaching and learning mathematics in terms of learning environments that are inclusive and comprehensive. inclusive learning environments provide h lo & ruef public sensemaking during groupwork journal of urban mathematics education vol. 13, no. 1 17 students “a fair distribution of opportunities to learn” (esmonde, 2009, p. 249) and comprehensive learning environments “open up interactional space for a broad range of competent ideas” (dunleavy, 2015, p. 62). effective mathematics classes should support students in becoming collaborative and critical problem solvers prepared to leverage productive change in the world. gutiérrez (2009) framed equity in terms of access, achievement, identity, and power. viewed through this lens, equitable mathematical teaching and learning practices enhance access to challenging curriculum and engaging learning environments. they support achievement in mathematics by widening the ways students can perform competence—when there are more ways to be “good at math,” more students will be competent. this definition of competence draws from research on complex instruction (cohen & lotan, 2014; horn, 2012; ruef, in press). seeing oneself as mathematically competent increases identification with mathematics, both individually and collectively. power, in turn, refers to relational equity within the classroom or, more specifically, how students and teachers treat one another (boaler & staples, 2008). power must also be considered beyond the classroom, and we must recognize how politics impact school policy and even how classroom achievements can impact policy (e.g., gutstein, 2003). for the purposes of this paper however, we focus on equity within a classroom where teaching and learning mathematics are inclusive and comprehensive. although educators may generally agree that teaching and striving for equity are important, these practices are often difficult to enact. this article shares a success story that sheds light on how a teacher and her students co-constructed an equitable teaching and learning culture. equitable distributions of authority and agency can be enhanced in mathematics classrooms where students participate in discourse as active members of the classroom. thus, one way to investigate equity in action is through the degrees to which authority and agency are equitably distributed among members of a classroom. authority is the amount of “given opportunities to be involved in decision-making,” which may include “establishing priorities in task completion, method, or pace of learning” (gresalfi & cobb, 2006, p. 51). we define agency as the ability to carry out those self-made decisions in the classroom. these decisions are not necessarily restricted to a mathematical task but may extend to how students physically situate themselves in order to complete a task (ruef, in press). in this paper, we focus primarily on authority and agency during the completion of a specific task. note that authority and agency are not limited to the students in the classroom. teachers are generally recognized as the primary holders of authority and agency and often must actively work to share these with students (cobb et al., 2009; cohen & lotan, 2014; horn, 2012; ruef, in press). educators are increasingly utilizing collaborative learning through groupwork in mathematics classrooms (nctm, 2018). previously, transmission-based models of teaching and learning were frequently utilized in united states mathematics classrooms. in this form of learning, students often passively participate as learners by lo & ruef public sensemaking during groupwork journal of urban mathematics education vol. 13, no. 1 18 quietly taking notes or reproducing ways to solve problems (ruef, 2013). transmission-based learning may reinforce typical representations of what it means to be “good at math,” which suggest that those who are able to absorb mathematical knowledge through lecture and notetaking, sitting still, being quiet, and doing well on exams are seen as being “good at math” (e.g., boaler, 2002). transmission-based instruction is problematic both in terms of learning mathematics and fostering equity. in contrast, sensemaking-based instruction requires students to be active learners. building a sensemaking culture in the classroom can unlock new ideas about what it means to be “good at math” for students, especially when students learn by participating in public sensemaking (ruef, 2016, in press). features of public sensemaking include striving to respect, acknowledge, and understand each other’s perspectives; welcoming mistakes and productive struggle as aspects of learning; and taking risks by “sharing one’s thinking; presenting, critiquing, revising, and refining arguments” (ruef, 2016, p. v). these aspects of public sensemaking clearly contrast the common images of what it means to be “good at math” in transmission-based instruction. few classrooms land squarely at either end of the transmission-sensemaking continuum. most contain elements of both ends of the spectrum with teachers and students working hard to create productive learning environments. the nctm (2018) publication catalyzing change details the current general state of high school mathematics teaching and learning in the united states and offers a vision for its evolution. one of the equitable teaching practices suggested in catalyzing change is for teachers to “facilitate meaningful mathematical discourse” among students (p. 30). productive groupwork (e.g., complex instruction) is one tool that brings forth meaningful discourse. when students engage in groupwork, they have opportunities to collaborate. during this work, students are often tasked with collectively solving mathematical problems, sharing or double-checking solutions from assignments, or putting together presentations. in all three cases, students are asked to publicly communicate their ideas, questions, and solutions as they engage in public sensemaking. groupwork allows teachers to delegate authority towards the students, empowering “students to argue, evaluate, and confirm the validity of their mathematical ideas” (dunleavy, 2015, pp. 63–64). the result is a broader definition of what it means to be “good at math” in the classroom. conceptual framework our work is situated within related work attending to student identity and learning opportunities. both of these are informed by distributions of agency and authority and broader definitions of competence. to determine distributions of authority and agency, our conceptual framework draws from positioning theory (esmonde, 2009; lo & ruef public sensemaking during groupwork journal of urban mathematics education vol. 13, no. 1 19 van langenhove & harré, 1999), and it is framed by esmonde’s (2009) and dejarnette & gonzález’s (2015) classifications of positions and actions in systems of negotiation. we extend this classification with the addition of the positions of contributor and remover. positioning theory this study framed the students’ mathematical interactions in terms of positioning. positioning theory utilizes speech and actions to identify someone’s rights, obligations, and duties (van langenhove & harré, 1999). the researchers analyzed students’ interactions during groupwork. specifically, the current study centered on interactive positioning, which occurs when students position one another in relation to each other (davies & harré, 1999). students can position themselves or be positioned by others. additionally, positioning can be “intentional or unintentional, explicit or implicit” (ruef, 2016, p. 11). this study examines the types of positioning that occurred in one public sensemaking classroom in terms of evident positions created during groupwork. esmonde (2009) described three types of positioning that may be present during groupwork: expert, novice, and facilitator. the expert is often deferred to mathematically and granted or ceded authority to decide whether an idea or work is correct. novices often defer to experts, asking for and receiving help from others. novices position themselves as less competent, but they may sometimes challenge the expert. facilitators regulate group activity and participation from group members, and they actively elicit group members' contributions to joint problem solving and include them in discussions. in addition to esmonde’s (2009) three types of positioning during groupwork, we define two additional types of positions that may occur. the first is the contributor, who participates in groupwork non-mathematically but increases the group’s productivity in some way. this might include keeping track of time or gathering necessary supplies. the second position we define is the remover. this position is evident when a student removes themselves from the task at hand, leaving other students to complete the task. the reasons a student is positioned as a remover may vary. for example, a student might remove themself from the task when home or family life present distractions from classwork. although each type of positioning is common during groupwork, each one is not always present or obvious. ideally, in an equitable classroom, positions are impermanent as an equitable classroom culture would produce relatively frequent shifts in positioning. this is because each student brings a different level of prior knowledge or expertise to the task at hand. a student who recalls a strategy used in a similar problem may be viewed as an expert on one task and a novice on another where the student has less mathematical insight. regardless of prior knowledge, any student may step into the role of facilitator, guiding the group’s social interactions and mathematical work. similarly, any student may play the role of contributor in moving the task forward through non lo & ruef public sensemaking during groupwork journal of urban mathematics education vol. 13, no. 1 20 mathematical work. finally, students might remove themselves from the task at hand, stepping into the role of remover. some students may rarely change their role either by choice or because they are excluded by peers (cohen & lotan, 2014; esmonde et al., 2009; horn, 2012). this limits their opportunities to sensemake and is an indicator that authority and agency are not equitably distributed in the classroom. system of negotiation to classify speech or actions produced by students, it is important to discuss how certain moves may map onto the five positions (i.e., expert, novice, facilitator, contributor, and remover). according to halliday (1984), an interaction is an exchange that has two variables. the first variable is a commodity that is exchanged; this includes either information or goods and services. the second variable is the role of the speaker/actor, which involves giving or demanding a commodity. for this study, the role of the speaker/actor could also include rejecting or ignoring a commodity. this study employed a system of negotiation that includes five moves that a speaker/actor can employ during an interaction, all of which are combinations of rejecting, giving, or demanding information or services. table 1 provides the five moves that may occur during an interaction and an example of what the move might look like in a mathematics classroom. the moves are divided into those made by primary (k1) and secondary (k2) knowers, primary (a1) and secondary (a2) actors (berry, 1981), and primary deflectors (d1). in terms of authority and agency, note that k1 moves hold high levels of authority because primary knowers create opportunities for themselves to provide or make decisions on mathematical knowledge. a2 moves hold high levels of agency because secondary actors initiate decisions to be carried out. lo & ruef public sensemaking during groupwork journal of urban mathematics education vol. 13, no. 1 21 table 1 types of negotiation moves and examples negotiation move definition example primary knower (k1) provides mathematical information. “area is length times width.” secondary knower (k2) requests mathematical information. “what’s the formula for area of a rectangle?” primary actor (a1) provides, or offers to provide, an action related to the activity of doing mathematics. performs or offers to perform activity. [reads the math problem aloud] secondary actor (a2) requests or commands an action related to the activity of doing mathematics. directs activity. “can you read the problem out loud?” primary deflector (d1) deflects a request for information or knowledge or request to present information or knowledge. “i’m not telling you the formula for area.” the study adapted and added to the mapping created by dejarnette & gonzález (2015) to relate the system of negotiation and the types of positioning that appear during groupwork (figure 1). in addition to the mapping used by dejarnette & gonzález, we created two more mappings such that each negotiation move maps directly onto a position, creating a one-to-one mapping. k1 moves are mapped onto the expert because the student providing information is also one that is most likely to be deferred to during mathematical groupwork. k2 moves, negotiations that request information, align with the position of novice. by definition, a novice holds less knowledge about the mathematical problem at hand and, as a result, needs more information to build understanding. next, a1 moves are mapped onto the contributor because the contributor participates nonmathematically in such a way that moves the group’s productivity forward. a2 moves are mapped onto the facilitator role because those requesting action are inherently prompting all students to participate in completing the mathematical task in some way. finally, through performing d1 moves by rejecting, deflecting, or ignoring requests by other students, the remover excuses themself from participation in the groupwork. lo & ruef public sensemaking during groupwork journal of urban mathematics education vol. 13, no. 1 22 figure 1. one-to-one mapping: negotiation moves to position assigned to a student by self or other. research questions prompted by the prevalent and growing use of groupwork in mathematics classrooms, this study examined the different ways a group of students positioned each other in a classroom that encouraged public sensemaking. we also considered how positioning affects the distribution of agency and authority among students in a group. these goals translate to the following research questions: 1. how do students negotiate positioning during mathematical groupwork in a public sensemaking classroom? 2. how does interactive positioning impact distributions of agency and authority among students? the following study provided a snapshot of the ways students positioned one another as well as how agency and authority were distributed in a public sensemaking classroom during collaborative groupwork. methods this embedded case study (yin, 2017) draws from a larger study and body of data that took place across the 2015–2016 school year in three sixth-grade mathematics classes all taught by one teacher (ruef, 2016). the larger study made use of lo & ruef public sensemaking during groupwork journal of urban mathematics education vol. 13, no. 1 23 presurveys, postsurveys, ad hoc exit tickets (i.e., short answer questions), participant observation collection of field notes and video footage, and formal and ad hoc interviews with all participants. analysis from the larger study was ongoing and iterative, making use of both quantitative (i.e., linear regression analysis) and qualitative (i.e., a priori and emergent coding; analytic memoing) methods (emerson et al., 2011). participants & setting the participants in this study were students who attended city school (a pseudonym) in california. city school is housed in a large urban area and focuses largely on preparing students for careers in the health sciences and other health-related fields. additionally, city school educates students in grades 6–12, and many of its graduates meet the criteria to become first generation college students. over 91% of city school’s students qualified for free and reduced lunch at the time of data collection, and statistics from 2009 demonstrate that 74% of the students identified as latinx/hispanic, 11% as african-american, 11% as asian, 2% as filipinx, 1% as native american, and 1% as white. the participants in this study were sixth graders taking a mathematics class taught by ms. isabella mayen. ms. mayen was chosen for her focus on public sensemaking. all names are pseudonyms chosen or approved by the participants. at the time of the study, isabella mayen was a second-year teacher of mathematics. she identifies as latina, had attended a well-known teacher education program, and is well-versed in how to facilitate public sensemaking. the tables in her classroom were arranged in five or six groups of four, and the groups were known as “teams.” in ms. mayen’s classroom, working in a team implicitly meant that “everyone has equal status as a sensemaker, and including everyone in a team is an important social function” (ruef, 2016, p. 58–59). the message of teamwork attempted to prevent opportunities for exclusion, thus promoting equity in learning mathematics. team membership changed every two to three weeks and was randomly assigned (e.g., cohen & lotan, 2014; horn, 2012). it is important to note that ms. mayen and her colleagues had control over the content and pacing of the curriculum they taught. they selected, composed, and adapted curriculum to best fit the learning trajectories of their students and prepare them for success in stem-related studies and careers. this study centered on one particular team in september of 2015, about one month into the academic year. the students in this team were brooklyn, kazaly, flor, and elena. at this point in the school year, ms. mayen was leading a three-day lesson on finding the areas of polygons. she tasked the teams with finding the area of an irregular trapezoid (figure 2) accounting for the number of unit squares without using standard formulas. lo & ruef public sensemaking during groupwork journal of urban mathematics education vol. 13, no. 1 24 figure 2. irregular trapezoid with area of 60 square units. ms. mayen’s goal was to have the entire team agree on one solution and for representatives to present the team’s method to the class. one or more students from each team were expected to present, and the teams decided who would represent them. data sources this study utilized preexisting classroom video footage of students at city school from 2015. the footage was originally recorded by ruef. video recording focused heavily on the first four weeks of the school year, including 13 of the first 18 days. because ruef collected the primary data, she was able to choose when and which groups to film during class. the accumulation of video footage was captured from the “perspectives and lenses” (ruef, 2016, p. 34) of the original researcher. according to ruef, she “strategically and purposively” picked moments where there was “strong evidence of positioning by both students and the teacher” (ruef, 2016, p. 34). for the purposes of the current study, a 26-minute video was excerpted from the total video footage to be analyzed as a case study. this portion of the accumulated video footage from ms. mayen’s class was selected for analysis because it displayed salient attributes of distributing agency and authority among students during mathematical groupwork. data analysis analysis of the video footage was completed using maxqda 18.1.1, a qualitative data analysis software. the videos were first transcribed, then coded using the negotiation and positioning moves (see table 1) as a priori codes. table 2 provides an example of how student interactions during one continuous conversation were coded. lo & ruef public sensemaking during groupwork journal of urban mathematics education vol. 13, no. 1 25 table 2 example of interaction using negotiation moves as codes line speaker transcription code 1 elena okay, let's get back to work. a2 2 flor what are we getting back to do? we're finished! d1 3 brooklyn ¡no sé! (i don't know!) k2 4 kazaly how do you got 66? k2 5 brooklyn you add them all together and you get 66. k1 6 flor you add them altogether once you get all these numbers right here. k1 7 brooklyn and even if you and if you count them one by one too, you still get 66. k1 8 flor yeah. k1 note that not all interactions were coded with the five moves if the interaction did not fit any of the defined codes. the unit of analysis was a talk turn. as it evolved, our analysis was discussed in regular research meetings with colleagues to invite “outside eyes” into the process. we discussed existing and emergent codes to determine face validity, asking if the codes mapped to what others observed in the data. we also considered additional codes suggested by colleagues. given the focus of the research questions and the data we were analyzing, we were satisfied that we had reached saturation of observed phenomena in relation to both codes and the data set. the authors further discussed and refined the coding scheme. we then selected 20% of the video footage to calculate interrater agreement, resulting in a cohen’s kappa value of 0.616. satisfied with this level of interrater agreement, lo calculated the percentages of individual negotiation moves out of the total number of negotiations (see table 3). results the percentages in table 3 revealed several findings. on this team and for this task, brooklyn was positioned as expert-facilitator-remover. brooklyn accounted for 54 percent of the total number of k1 moves, 55% of the total number of a2 moves, and 47% of the total d1 moves. flor was positioned as contributor because 47% of lo & ruef public sensemaking during groupwork journal of urban mathematics education vol. 13, no. 1 26 the total a1 moves belonged to her. there was no prominent novice because none of the students exhibited a high percentage of the total k2 moves. table 3 percentage of frequencies of negotiation moves among students negotiation move elena brook-lyn kazaly flor total # of negotiations primary actor (a1) 2 (12%) 5 (29%) 2 (12%) 8 (47%) 17 secondary actor (a2) 9 (27%) 18 (55%) 3 (9%) 3 (9%) 33 primary knower (k1) 25 (17%) 81 (54%) 14 (9%) 31 (20%) 151 secondary knower (k2) 7 (19%) 13 (36%) 6 (17%) 10 (28%) 36 primary deflector (d1) 4 (21%) 9 (47%) 2 (11%) 4 (21%) 19 distribution of agency recall that removers can vary in their reasons for excluding themselves from groupwork. although brooklyn was positioned as the expert-facilitator of the group, her self-positioning as the remover provides evidence that she distributed agency back to the other team members. table 4 shares an instance where all four students worked to decide which one of the two methods they had developed for finding the area of an irregular trapezoid to present to the class. although elena and flor initially volunteered to present to the class before this discussion, both deflected possible opportunities to make decisions about the method they were to present. at one point, elena removed herself from the role of presenter saying, “i don’t wanna go up ‘cause they ask so many questions!” note that this decision to choose not to present is evidence for agency. the team was thus in flux regarding who would present their method to the class. lo & ruef public sensemaking during groupwork journal of urban mathematics education vol. 13, no. 1 27 table 4 deciding who will present to the class line speaker transcription code 1 elena which one should we do? k2 2 flor brooklyn, which one should we do? k2 3 brooklyn you guys are going up there, so you guys decide, d1 4 brooklyn but remember you still have to count the little ones k1 in the transcript in table 4, elena attempted to broadly ask the team for help (line 1), but flor explicitly funneled the opportunity to make a decision to brooklyn (line 2) perhaps due to brooklyn’s established position as expert. however, brooklyn redistributed agency back to elena and flor by reminding them they volunteered to present and pressing them to choose (line 3). this move positioned brooklyn as the remover. by not making the decision on which method to use, brooklyn removed herself from the decision-making process and redistributed agency back to her teammates in the process. brooklyn also retained her position as expert by reminding them to count “the little ones,” or the partial square units (line 4). distribution of authority in addition to distributing agency, brooklyn also distributed authority to the other members of the group by positioning herself as remover in addition to expertfacilitator. by the end of the lesson, brooklyn, elena, flor, and kazaly had discussed a second new method on how to find the area of the trapezoid. specifically, the team discussed the idea that the diagonal line through two square units divides the two squares into one square unit each on the top and bottom half of the rectangle. table 5 makes it clear they had yet to confirm which agreed method to present to the class. lo & ruef public sensemaking during groupwork journal of urban mathematics education vol. 13, no. 1 28 table 5 deciding which method to present to the class line speaker transcription code 1 brooklyn i think you guys should do this one – and then explain – remember it's half? half of two is one? k1 2 elena i don't know how to explain that. k2 3 elena or you should go. a2 4 brooklyn i’m not going. d1 5 flor i'll go up with you, brooklyn. a2 6 brooklyn i don't want to go. d1 7 elena everybody goes. a2 brooklyn exhibited traits of a participant in a public sensemaking classroom by acknowledging alternative ideas. however, elena refrained from risk-taking and presenting the newer method (line 2). because she does not yet know how to explain it, it is highly probable that elena rescinded her offer to present in front of the class in fear of embarrassment. next, brooklyn turned down elena’s request for brooklyn to go to the board to present the team’s method (line 4). by rejecting elena’s request, “or you should go,” brooklyn attempted to redistribute authority back to her teammates, elena and flor, through a deflector (d1) move. brooklyn increased the number of opportunities for elena and flor to act as experts when she rejected the invitation to present to the class. in response, flor, who originally volunteered with elena to present the team’s method to the class, took up the new opportunity to present with brooklyn (line 5). had brooklyn readily accepted the invitation to present the newer method, the opportunity for flor to participate in public sensemaking may have been diminished, overshadowed by brooklyn's implicit authority through her position as expert. finally, it is interesting to note that as brooklyn redistributed authority to her teammates, elena declared, “everybody goes” (line 7). this statement created an opportunity to present as a team and reinstated elena as a presenter. discussion this paper aimed to answer two questions. first, 1) how are students positioned during mathematical groupwork in a public sensemaking classroom? from this snapshot drawn from video footage of brooklyn, kazaly, elena, and flor, one lo & ruef public sensemaking during groupwork journal of urban mathematics education vol. 13, no. 1 29 combination of positioning that may occur in public sensemaking classrooms resulted in a clear expert, facilitator, contributor, and remover, but no clear novice. identifying positions during groupwork can help determine what equity issues may be present in mathematics classrooms, especially when teachers are looking out for and circumventing shifts from temporary positioning to calcified positioning. in this case study, the lack of a clear novice did not raise any red flags. however, asking questions and requesting information to make sense of mathematics, a critical role of the novice position, is important in public sensemaking. in this instance, the teacher of the classroom, ms. mayen, would need to ensure that there is never a permanent absence of a novice, as that would indicate that the class has no need for mathematical sensemaking. on the other hand, in order to maintain equity in the classroom, ms. mayen would also need to look out for potential calcifications of positioning during groupwork. analysis of the video footage also aimed to answer the following question: 2) how does interactive positioning impact distributions of agency and authority? the two instances discussed above highlight the ways in which two students committed deflection (d1) moves and removed themselves from participation in presenting their strategy to the class for different reasons. elena refused to present because she was apparently anxious about answering questions while at the board. brooklyn, while positioned as an expert-facilitator, redistributed authority and agency to her teammates by “refusing to be the source of authority” (ruef, in press) and so positioned herself as a remover. by taking this action and others like it, brooklyn stepped into the role of their teacher, ms. mayen. brooklyn exhibited the traits of a teacher of a public sensemaking classroom by encouraging risk-taking as she pressed elena and flor to present while removing herself from the discussion. this allowed others to make decisions while they were also attempting to make sense of the mathematics they were engaged with. teaching and learning become more equitable when students do not permanently position each other. temporary positioning, fluid across activities, creates space for students to fluctuate between expert, facilitator, contributor, novice, and remover positions. even though brooklyn is positioned as the expert-facilitator on the team, she attempts to dissolve her position by allowing others to decide on a method to present and providing space for elena and flor to take risks, thus positioning herself as a remover. in other words, her self-positioning as remover limited her positioning as the expert-facilitator as temporary. she accomplished this through selfrefusal and invitations to peers to participate in presenting the group’s method. brooklyn’s imitation of ms. mayen’s role as a teacher surfaces several interesting points. first, as brooklyn took up actions similar to those of a classroom teacher, there are new questions about whether students are metacognitively aware of equity issues in their own classrooms and teams. dunleavy (2015) stated that “striving toward equity in mathematics education invokes constant, purposeful work lo & ruef public sensemaking during groupwork journal of urban mathematics education vol. 13, no. 1 30 from the teacher as she or he seeks to diminish differences in access, opportunities, and outcomes for students” (p. 64). although teachers are typically the ones tasked with noticing and treating inequitable participation in mathematics classrooms, this case study reveals the possibility that students may also have the capacity to address the distribution of authority and agency by monitoring how many opportunities they have to participate in sensemaking. teachers cannot attend to every student individually and simultaneously, but their bandwidth is extended and amplified when students foster equity within groups, as demonstrated by brooklyn. public sensemaking classrooms with established norms for students to take risks, acknowledge ideas, grapple with mistakes, and share their thinking can open pathways for students to maintain equitable distributions of authority and agency within their own teams. limitations and next steps this case study is specific to a particular group engaged in a specific task at a fixed time and place. group dynamics vary by task, participation structures, and participants. thus, generalizing student positioning in the classroom is beyond the scope of this paper. extending this research might include analyzing different groups of students in ms. mayen’s classes to see what kinds of positioning are present and to map their fluidity. student interactions are also more complicated than five types of negotiation moves. future studies and expanded teaching and learning contexts may surface additional types of moves. conclusion in summary, this paper aims to provide a window into how agency and authority were redistributed in terms of public sensemaking when students positioned one another during groupwork. it is heartening to present evidence of students from historically minoritized backgrounds taking up the work of equitable distributions of agency and authority in facilitating public sensemaking for themselves. this study shows, empirically, that historically marginalized students of color are capable not only of making sense of mathematics at a deeply conceptual level, but also in providing opportunities for equitable learning. it also dives deeply into the ways this group of four students positioned each other and challenged one another to present their thinking to the class. it was relatively early in the school year, and the norms for public revision of work were still under development (e.g., jansen, 2020; ruef, in press). public sensemaking thus required bravery and persistence as the team did not know what questions their classmates would ask. teachers who facilitate public sensemaking develop lenses through which they notice various classroom phenomena. they notice risk-taking, productive discourse, and potential status issues. teachers who notice, and praise, students who act as lo & ruef public sensemaking during groupwork journal of urban mathematics education vol. 13, no. 1 31 brooklyn did—creating opportunities for classmates to step into agentic roles by stepping back—can communicate new ways of being competent to their students. one can be good at math by deflecting if that deflection invites a colleague into agency and authority. teachers must occasionally support students who remove themselves from instructional activity for a variety of reasons. although the reasons for “checking out” are often legitimate, removal has real consequences (e.g., hand, 2010). when students do not participate, at best they deprive their group of an engaged thought partner in sensemaking. however, the researchers argue that removal can be a positive and supportive act. both elena and brooklyn deflected requests to present at the board but for different reasons. elena’s motivation appeared to stem from fear: what would happen if she were at the board and her colleagues asked her challenging questions? brooklyn’s motivation appeared to be sharing power: who would step up if she refused the role of presenter? the answer appears in the results; brooklyn’s self-removal created a vacuum. her encouragement to elena to present provided a push, and elena stepped back into the role of presenter. elena’s temporary removal and her explanation for it allowed her colleagues opportunities to rebut her reasons and encourage her to take up the challenge. this study supports equitable teaching and learning practices by expanding frameworks for positioning. this framework can be applied in a systematic manner in researching student and group interactions. our findings support teachers in examining group and class dynamics for fluidity in status (i.e., expert, novice, facilitator, contributor, and remover). the increased presence of collaboration and problem solving happening in urban public sensemaking classrooms gives us hope for the future of diversity and success in stem-related fields. acknowledgements we wish to thank anthony muro villa iii, jill baxter, melynda casement, and alecia magnifico for their assistance in creating this paper. we also wish to thank ms. mayen and her students for allowing us to learn from their classroom for this study. this research was supported by the stanford graduate school of education dissertation support grant, the stanford vice provost for graduate education dissertation research opportunity grant, 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(2017). case study research and applications: design and methods. sage. copyright: © 2020 lo & ruef. this is an open access article distributed under the terms of a creative commons attribution-noncommercial-sharealike 4.0 international license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. microsoft word author approved reviewer acknowledgment vol 2, no 2.doc journal of urban mathematics education december 2009, vol. 2, no. 2, pp. 72–73 ©jume. http://education.gsu.edu/jume journal of urban mathematics education vol. 2, no. 2 72 reviewer acknowledgment january 2008–december 2009 reda abuelwan, sultan qaboos university dan battey, arizona state university joanne becker, san jose state university clare bell, university of missouri robert berry, university of virginia tonya bartell, university of delaware angela brown, piedmont college evelyn brown david brown, texas a&m university, commerce joan bruner-timmons, miami-dade county public schools lecretia buckley, jackson state university leonides bulalayao gustavo bermúdez canzani theodore chao, university of texas, austin carrie chiappetta, stamford public schools karen cicmanec, morgan state university marta civil, university of arizona raeshaun costley ubiratan d'ambrosio ella-mae daniel, florida a&m university julius davis, university of maryland donna dodson marilyn evans, national council of teachers of mathematics cassie freeman, university of chicago joseph furner, florida atlantic university mary foote, queens college, city university of new york imani goffney, university of michigan lidia gonzalez, york college, city university of new york susan gregson, university of illinois, urbana-champaign barbro grevholm, university of agder victoria hand, university of colorado, boulder shandy hauk, university of northern colorado daphne heywood, university of toronto crystal hill, indian university purdue university indianapolis leslie hooks, forth worth independent school district tisha hyman christopher jett, georgia state university alanna johnson jason johnson, middle tennessee state university martin johnson, university of maryland shelly jones, central connecticut state university karen king, new york university richard kitchen, university of new mexico steven kramer, baltimore freedom academy brian lawler, california state university, san marcos della leavitt jacqueline leonard, temple university dorothy lewis-grace, dekalb county public schools sarah lubienski, university of illinois, urban champaign danny martin, university of illinois, chicago donna mccaw, western illinois university jennifer mccray, erikson institute allison mcculloch, north carolina state university regina mistretta, st. john’s university kamau mposi mayen nelson, houston independent school district madeline ortiz-rodriguez, interamerican university of puerto rico pamela paek, national center for the improvement of educational assessment angiline powell, university of memphis journal of urban mathematics education vol. 2, no. 2 73 arthur powell, rutgers, the state university of new jersey malik richardson, charlotte mecklenburg schools fatimah saleh, universiti sains malaysia walter secada, university of miami stanley shaheed, dekalb county public schools joi spencer, university of san diego megan staples, university of connecticut william tate, washington university james telese, university of texas, brownsville la mont terry, occidental college carmen thomas-browne, art institute of pittsburgh thomas thrasher david wagner, university of new brunswick erica walker, teachers college, columbia hersh waxman, texas a&m university desha williams, kennesaw state university maya wolf microsoft word 389-article text no abstract-2131-1-6-20200906 (galley proof 2).docx journal of urban mathematics education december 2020, vol. 13, no. 2, pp. 1–16 ©jume. https://journals.tdl.org/jume susan ophelia cannon is an assistant professor of elementary and middle grades education in the tift college of education at mercer university, 3001 mercer university drive, atlanta, georgia, 30341; email cannon_so@mercer.edu. she works the boundaries of the fields of mathematics education, statistics education, qualitative inquiry, and teacher education in order to disrupt takenfor-granted modes of thinking and to radically reimagine different futures. editorial a call for field disruptions and field connections in mathematics education research susan ophelia cannon mercer university t the opening plenary of the 39th annual conference of the north american chapter of the international group for the psychology of mathematics education, rochelle gutiérrez (2017) suggested that the mathematics education research community should “think not only about more ethical ways of applying mathematics in teaching and learning but to question the very nature of mathematics, who does it, and how we are affected” (p. 2). she proposed that “interaction between different knowledges, different ways of knowing and different knowers” (p. 2) could serve to respond to and perhaps address the precarious state of our planet and our relationship with it. gutiérrez is not the first leader in the field of mathematics education research to call on the community to think about mathematics education research differently. twenty-five years ago, tate (1995) called on the community to rethink policy in relation to equity in mathematics education. he stated that he found the “paradigmatic boundaries of mathematics education somewhat narrow,” and he intentionally modeled his work after scholars who “crossed the epistemological boundaries of their fields to provide a more cogent analysis of important issues facing african americans” (pp. 425–426). the newly conceived field disruptions and field connections section of the journal of urban mathematics education (jume) specifically seeks contributions that cross epistemological and methodological boundaries to open up space in the field for new and different kinds of research that does not reify and reproduce white supremacist and settler colonist ways of being and knowing (stinson, 2017). the field of mathematics research and mathematics education research more specifically is not known for its readiness to adapt new ways of thinking (hansen, 2019). adler and lerman (2003) proposed that the way that mathematics education researchers describe their research and determine if a piece of research is acceptable within the community is an ethical matter. they posit that these actions are particularly difficult in a field such as mathematics education, which developed into a social science from the disciplinary fields of mathematics and psychology. because of this, mathematics educators may find it difficult “to problematize current images of good mathematical practice when the focus of the zoom lens is tightly on mathematical a cannon editorial journal of urban mathematics education vol. 13, no. 2 2 activity” (adler & lerman, 2003, p. 445). as such, they advocated for a broadening of what counts as mathematics education research, stating, all judgements of what is acceptable at any time as mathematics education research by the various gatekeepers are value judgements. what matters ethically is that those values are made explicit and are constantly under challenge and review by the community. this demand places great responsibility on journal editors, presidents of organizations such as the international group for the psychology of mathematics education, and the like, but also on all of us as reviewers, ph.d. examiners, and so on. (pp. 451–452) the goal of this section is to make the aims and values of jume explicit. we value field disruptions and field connections as well as different ways of thinking, researching, and doing mathematics with and for students, teachers, and in-school and out-of-school spaces characterized as “urban.” using the traditionally accepted methods and ways of producing knowledge most often continues to produce a status quo that is racist, sexist, and disproportionately disadvantages people with low-socioeconomic status. our ways of researching in mathematics education must allow new ways to think and rethink mathematics and mathematics teaching and learning. it is past time for dismantling, reconfiguring, and rewriting mathematics education research if we as a community are really serious about creating and implementing new ways of being mathematical and new ways of mathematics teaching and learning that motivate means of inclusion rather than exclusion (see skovsmose, 2009). this call for change and reimagining is not new. in a 2010 jume commentary, danny martin, masie gholson, and jacqueline leonard (2010) responded to two significant events that attempted to severely delimit the boundaries of the field of mathematics education and what counts as mathematics education research. the first delimiting event was the publication of m. kathleen heid’s march 2010 editorial, “where’s the math (in mathematics education research)?,” in the journal for research in mathematics education (jrme), which called on readers and potential authors to look for the mathematics in mathematics education research. heid reminded readers that jrme publishes mathematics education research articles that “focus on critical features of mathematical understanding … in which mathematics is an essential component rather than being a backdrop for another area of inquiry” (p. 103). this editorial was a surveilling and disciplining move (see foucault, 1977/1995) that regulated mathematics education manuscripts that prioritized equity, race, social constructs, and research methods outside the boundaries for publication in what is arguably the leading u.s. mathematics research journal. in essence, the demand for “critical features of mathematics” removed space for critical takes on mathematics and mathematics teaching and learning. the second delimiting event occurred in that same year when the national council of teachers of mathematics (nctm) at its 2010 research conference sponsored a session entitled keeping the mathematics in mathematics education cannon editorial journal of urban mathematics education vol. 13, no. 2 3 research (teppo et al., 2004). d. b. martin and colleagues (2010) argued that both events “marginalize[d] scholarship within particular areas of focus,” including sociopolitics, identity, power, and race (p. 13). they did not claim that mathematics is not important but rather called on the community to continue “efforts to add needed complexity to the understanding of learners, their social realities, and the forces affecting these realities” (p. 15). in particular, they pointed to the lack of progress in the area of equity despite rigorous empirical research over the past several years on and about minoritized and underserved populations. this lack of progress seemed to, in their words, “demand that we pursue all promising areas of inquiry informing us about how to help [minoritized and underserved populations] experience mathematics in ways that allow them to change the conditions of their lives… now is not the time for restricting the production of knowledge” (p. 17). this new jume section aims to provide a different space for all promising and innovating theories, texts, and methodologies to be presented to and explored with the urban mathematics education audience, often times prior to any direct empirical work with the theory or method in mathematics education. this different space is for urban mathematics education researchers to ask, what if? a focus on equity is not enough if systems and structures do not change. the nctm has at times directed attention to issues of equity through policy, most recently with the principles to actions: ensuring mathematical success for all (nctm, 2014) document. although it was a public proclamation of the organization’s commitment to equity, d. b. martin in a 2015 jume response commentary directed toward the larger mathematics education community called for a revolution of values, a new way of thinking, and a radical decolonizing of education for the collective black: the hard truth is that the outcomes and inequities lamented over in principles to actions and previous documents are precisely the outcomes that our educational system is designed to produce. equity oriented slogans, statements about idealized outcomes, and tweaks to teaching or curricular practices within this system do not change this fact. (p. 21) in a response to d. b. martin’s commentary directed toward the nctm community, the then nctm president, matt larson (2016), acknowledged that “significant structural obstacles, including tracking and teacher assignments that disadvantage students who have been marginalized, remain unacceptable practices in too many schools” (para. 3). but as d. b. martin pointed out, although it seems that the larger mathematics education community is beginning to acknowledge structures and obstacles that marginalize, the call for more equitable practices has been met with intense standardization in research and an increase in measurement and assessment of students and teachers (see attick & boyles, 2016; biesta, 2016). these solutions cannon editorial journal of urban mathematics education vol. 13, no. 2 4 attend to a positivist, linear, cause-and-effect pathway for producing change in mathematics. but what if we researched differently? i, similar to many others in the field (e.g., brown & walshaw, 2012; de freitas & walshaw, 2016; ernest, 1999; stinson & walshaw, 2017), believe in the power of theory to shift the boundaries and borders of our field. de freitas and walshaw (2016) describe their approach to theory as impacting their thinking and meaning-making, explaining that “the act of defining or creating new concepts is precisely what theory has the potential to do. thus, theory is a creative tool, an inventive approach to making meaning, as well as being an intervention into current cultural practices” (p. 4). theory, then, is not merely something that a researcher thinks about prior to research or something that is applied to research data but rather an integral and inevitable component that directly and indirectly effects the possibilities and impossibilities of meaning-making through research. de freitas and walshaw do not privilege a particular theory as better, stating, “there is no perfect incontestable theory” (p. 2); rather, they advise that mathematics education researchers consider how theory functions on the possibilities and impossibilities for how research and mathematics can be thought. in the field disruptions and field connections section of jume, i echo calls in the mathematics education research community that a different space should be opened up (see de freitas & nolan, 2008) to consider research in mathematics education that crosses epistemological boundaries and works to open up spaces for mathematics educators, teachers, and students to think about themselves, mathematics, and schools differently. in other words, this new section will be a space with the aim of opening up “the fictions, fantasies, and plays of power inherent in mathematics education” (walkerdine, 2004, p. viii). i also want to create space for theories, knowledges, and methods that have potential to shift our field but may not yet fit in the confines of more traditional mathematics education spaces. i welcome theoretical and methodological essays that connect underutilized theories or methods for the purpose of bringing potential change in urban mathematics education research. i welcome data excerpts and narratives that might not fit in empirical manuscripts yet matter for how they allow educational researchers to think and rethink mathematics, students, teachers, and in-school and out-of-school educational spaces. to illustrate the potential for shifts in knowledge production as a result of thinking with underutilized theories and methodologies, in this introductory and invitational editorial, i offer some examples of poststructuralist studies in mathematics education that have shifted the way that mathematics education researchers take up the subject. next, i turn to agential realism and inclusive realism as theories that are beginning to be taken up in mathematics education and may have potential in urban mathematics education. i then come back to calls by gutiérrez (2017) and d. b. martin (2015) to consider how possibilities for ethical action are structured by the ways we do and think research, thus offering new possibilities for ethical action by doing and thinking research in new or different ways. finally, i open the invitation for cannon editorial journal of urban mathematics education vol. 13, no. 2 5 creative, radical, and innovative submissions to this new section of the journal of urban mathematics education. poststructuralism/postmodernism and mathematics education poststructural theories have functioned to allow us to think and research differently in mathematics education. poststructuralism1 has intersected with mathematics education for decades, and mathematics educators’ use of poststructural theories have made it possible to consider how meaning and knowledge get made and whose “interests are privileged, marginalized, or silenced” (stinson & walshaw, 2017, p. 148). in general, poststructuralism refuses generalizations and questions the taken for granted assumptions of stable human subjects, the transparency of language and meaning, and the separation of human subjects as independent of discourses and social structures. poststructural theories move from a conception of a student as an independent, rational, and stable subject to an evolving subject constituted by the discourses and disciplining practices of schools and societies. this move from an idea of identity as stable and fixed to the concept of subjectivity has changed the types of research questions that can be asked, allowing a different view of teachers and students as subjects that are constituted through interactions with the powerful discourses of school mathematics, education, and gender. valerie walkerdine (1994) drew on poststructural and critical theory, particularly those of homi bhabha and michel foucault, to consider the production of the “appropriate” mathematical subject, which “regulates or polices itself” (p. 63). she traced the development of girls’ mathematical subjectivity in classrooms to societal pressures and discourses about inherent sexual difference. walkerdine argued, 1 i use the term poststructuralism here although the term postmodern is at times used interchangeably with poststructuralism and they connote similar ideas for many readers. nonetheless, there are differences in the ways the terms are used. these differences are not easily marked, especially because poststructuralism works against fixed meanings and signifiers. preissle (2006), in discussing the history of qualitative inquiry and paradigms, consistently refers to “poststructural and postmodern” (p. 689), never listing one without the other. skovsmose (2012) uses the term postmodern “as a reference to a critique of modernity” (p. 233) and its privileging of science, knowledge, progress, and education. within the same article, skovsmose categorizes foucault as postmodern although foucault is in other spaces labeled a poststructuralist. it should be noted that the majority of philosophers/thinkers who are labeled as poststructuralist did not self-identify in that way and in fact rejected the label. peters and burbules (2004) differentiate between postmodernism and poststructuralism, explaining that poststructuralism takes the place of the theoretical object structuralism, and postmodernism takes the place of the theoretical object modernism. further, they advise, “when discussing poststructuralism it is important to recognize it as a movement (perhaps construed in the musical sense of the term)––as a complex skein that intertwines many different strands and also conceals important differences among the thinkers identified as poststructuralist” (peters & burbules, 2004, p. 30). this description, not definition, of poststructuralism is the one that i like to think with. cannon editorial journal of urban mathematics education vol. 13, no. 2 6 “theories of the development of reasoning when incorporated into education become ‘truths’ which actually serve to produce the desired kinds of subjects as normal and pathologizes differences” (p. 65). there were truths operating about girls who performed well in mathematics. walkerdine found that girls were described as hardworking while boys, even ones who did not perform particularly well in school, were described as bright. walkerdine pointed to the ways in which girls were positioned as compared to boys. she asked, “what then are the fears, phobias and fetishes in which the girls are inscribed, what are the stories about girls that have to be endlessly repeated as to make them true?” (p. 67). she found that underperforming boys were more likely to be positioned as having potential, while underperforming girls were taken up as lazy or incapable. she stated of her interactions in classrooms, “it was almost harder for a camel to go through the eye of a needle than for a girl to be called ‘bright’ (walkerdine, 1989)” (walkerdine, 1994, p. 67). her attention to the practices of subject formation as an ongoing process outside the control of a single individual opened the doors to exploration of how the ways that teachers interact with students in classrooms promote particular kinds of mathematical subjectivities. she warned, “when we treat the world as abstract in this way we forget the practices which form us, the meanings in which we are produced, we forget history, power and oppression. this universalizing and abstracting approach forgets colonization, patriarchy” (p. 71). walkerdine, drawing on bhabha and foucault’s writings, dismantled the idea that practices of schooling and the production of mathematical subjects are neutral and removed from intersecting issues of power, discourse, and gender. her work has influenced the continued dismantling of taken-for-granted assumptions about gender and mathematics education (see langer-osuna, 2011; leyva, 2016; mendick, 2005; walshaw, 2001). ideally, the dismantling is followed by revisioning and reimaging, finding other ways to represent, other ways to tell stories, other ways of living and being. in addition to gender, mathematics education researchers have taken up poststructuralist theories to consider how race interacts with mathematical identity. stinson (2013) used poststructural theories to decenter prevailing narratives about the achievement of african american male students in comparison to their white counterparts. stinson drew on lyotard’s concept of the metanarrative to explain the white male math myth before drawing on foucault to examine the discourses that four male african american students worked with and against to develop and perform robust mathematical identities as an act of subversive repetition (see butler, 1990). in his study, poststructural theories allowed stinson to not only think of identity and discursive practices in relation to mathematics differently but also to rethink the way that he conducted research. stinson conceptualized a research method that he did with participants rather than to or on them. importantly, stinson acknowledged that he borrowed “theoretical perspectives from the disciplines of anthropology, cultural/social psychology and sociology” (p. 75). in addition, stinson also reconfigured cannon editorial journal of urban mathematics education vol. 13, no. 2 7 martin’s (2000) framework for analyzing mathematics socialization and identity among african american students with poststructural theory to allow for a different kind of complex analysis of his participants, which acknowledged them as discursive subjects negotiating sociocultural discourses. this poststructural reconfiguration talked back to the generalizations being made about african american students’ achievement as less than that of their white counterparts. within a traditional research and theoretical paradigm, stinson would have been disciplined to keep an objective distance from his participants and not provoke or disturb their conceptions of their identity. by drawing on different disciplines and reconfiguring established frameworks with other theories, stinson created a new method of knowledge production with participants, ending his study with disruptive questions for the field. materialisms/new materialisms and mathematics education poststructuralism’s “analytical edge” has already made particular cuts in mathematics education research. it has taken on the humanistic stable subject and the power of discursive formations; however, poststructuralism has been critiqued for its focus on the linguistic and lack of attention to the material. new materialism2 was born out of a lack of attention to the material in poststructuralist and feminist writings. educational research has entered the materialistic turn, where the question of what matter matters has been raised. new understandings and theorizations of quantum mechanics and environmental concerns have come together to produce theories that undo the nature–culture divide and decenter the human as privileged caretaker or dominator of the earth. new materialism has many spin offs and nomenclatures (speculative realism, object-oriented ontology). dolphijn and van der tuin (2012) explain: new materialism is fascinated by affect, force and movement as it travels in all directions. it searches not for the objectivity of things in themselves but for an objectivity of actualization and realization.... it is interested in speeds and slownesses, in how the event unfolds according to the in-between. (p. 113) the key tenets of these new materialisms, similar to poststructuralism, function to trouble binaries and distinctive boundaries. in addition, new materialist theories take seriously what matter matters and how it comes to matter. new materialisms and in particular karen barad’s (2007) agential realism are beginning to be taken up by mathematics educators and are effecting/affecting the types of knowledge that are being produced through research (e.g., ferrara & ferrari, 2017; roth, 2017; wolfe, 2019). barad’s conception of onto-epistemology and the 2 i use this term here as heckman uses it; however, i also want to honor the critique of the new in new materialism that tuck and mckenzie (2014) put forward. cannon editorial journal of urban mathematics education vol. 13, no. 2 8 collapsing of knowing and being drew me to her as a theoretical guide for my dissertation research (cannon, 2019) and how i continue to think through my ongoing research agenda. given gutiérrez’s (2017) and d. b. martin’s (2015) demands, it is barad’s inclusion of ethics and her view on responsibility that could really matter for students and researchers in mathematics education. her concept of intra-action demands a relational ethics; as being and knowing are entangled, so, too, is living well and in respons-ability to all others. barad (2007) proposes, ethico-onto-epistem-ology—an appreciation of the intertwining of ethics, knowing, and being—since each intra-action matters, since the possibilities for what the world may call out in the pause that precedes each breath before a moment comes into being and the world is remade again, because the becoming of the world is a deeply ethical matter. (p. 185) in other words, educational researchers cannot separate ontologies, epistemologies, and ethics. they are entangled in the production of our worlds and our lives. hillevi lenz taguchi (2009), a childhood educator and researcher who has been taking up new materialist theories and particularly barad’s concept of intra-action in her thinking of school-nature-childhood entanglement, asks, “what reality is invoked and materialized before us depends on what ontological and epistemological position we take?” (p. 160). the responsibility then of ethico-onto-epistem-ological choices that are made in intra-actions become paramount as educational researchers are reconfiguring the world as we move with it. elizabeth de freitas and nathalie sinclair, scholars who bridge the fields of philosophy, mathematics, and feminism, have taken up barad’s agential realism, among other theories, in their ambitious text mathematics and the body: material entanglements in the classroom (2014; see cannon & myers, 2016, for a review of the book and a brief introduction to inclusive materialism and minor mathematics that de freitas and sinclair offer). in their book, de freitas and sinclair note four crucial aspects of inclusive materialism: 1. it is not reductive, seeing all matter as the same; instead it privileges “difference and multiplicity” (p. 42). 2. the socio-political and the material are seen as “inextricably entangled” (p. 42), and in this viewing inequity issues in education can be addressed within a broader framework. 3. affect, aesthetics, and nonsense are central, and rationality is not privileged. 4. humanist notions and human agency are decentered. agency, additionally, is distributed across the assemblage. cannon editorial journal of urban mathematics education vol. 13, no. 2 9 drawing on these aspects, de freitas & sinclair put forth the concept of a minor mathematics that is “not the state-sanctioned discourse of school mathematics but that might be full of surprises, non-sense and paradox” (de freitas & sinclair, 2014, p. 226). they believe that a radical reconfiguration of school mathematics is necessary to engage students in more expansive ways. in their rethinking and re-rethinking of mathematics, they use inclusive materialism to question the ways we think about what counts as a mathematical concept and the ways in which curriculum is invented and organized. de freitas and sinclair (2014) disrupt the traditional view that “learning is assumed to have a teleological trajectory toward fixed and immovable mathematical concepts. concepts are said to emerge through activity, but there is no troubling of the specificity of the concepts. in other words, “the mathematical concepts (multiplication, cube, zero) are taken for granted, while students collaboratively move towards them” (p. 40). this traditional, taken for granted view is upturned in favor of sensational (not just sensible) learning that is inventive and intra-active. instead of approaching the scenario of a child learning how to count on an ipad by trying to discern the effect of that technology on achievement, they viewed the child/ipad/finger/table/number/researcher as an entangled phenomenon. in this framework, educational researchers cannot separate out technology as a single variable that has a direct and measurable impact on the equally measurable learning of the child. the technology in intra-action with the child, mathematics, concepts, researcher, materials, and so on co-constitute each other through that intra-action. as with poststructural theories, de freitas and sinclair’s connections to agential realism and theories from within mathematics (e.g., châtelet, 2000) and other fields allow a disruption in the field of mathematics education research. what if urban mathematics education researchers thought with inclusive materialism? ethico-onto-epistem-ology in mathematics education research inclusive materialism is not the theory. it is one theory that allows a different perspective on urban mathematics education. the ideas within it are not particularly new to mathematics education research. gutiérrez’s (2017) mathematx was born out of field connections. she draws on examples from biology, ethnomathematics, postcolonial theory, aesthetics, and indigenous knowledge. she acknowledges the material and the socio-political. d. b. martin (2019) has intense concern for the material effects of knowledge-making practices as well. let me be clear, the point that i am making is not that urban mathematics education research needs these particular theories (poststructuralism, agential realism, inclusive materialism) but rather that different theories open up different possibilities and impossibilities for urban mathematics education research. they simply help us to rethink our own re-rethinking yet again (see foucault, 1969/1972). thus, making a dedicated space for theory and innovation cannon editorial journal of urban mathematics education vol. 13, no. 2 10 in mathematics education research demonstrates the value of theory and the need to extend connections to other fields. the choices we make—the theories and methods we use—as researchers create the field of urban mathematics education. we become educational researchers with the field as we create the field (cannon, 2020; mazzei, 2013). educational researchers are engaged in boundary-making practices that categorize and classify: “cuts are enacted not by willful individuals but by the larger material arrangements of which ‘we’ are a ‘part’” (barad, 2007, p. 178). these cuts have material effects. for example, in gutiérrez’s (2008) work on the fetishizing of the “achievement gap,” she points to cuts that are made around black and brown students that produce them as deficient and lacking. this gap is not in black and brown students; rather, the gap is produced in and through the gap gazing research and reified each time another researcher cites it. cathy o’neil (2016) argues that the way that statistics and mathematical models are used have material effects on people’s lives and discriminatively negative effects on the poor. she shows how mathematical models are “not only deeply entangled in the world’s problems but also fueling many of them” (p. 2), and the models used extensively today “tend to punish the poor” (p. 8) and perpetuate cycles of poverty, causing “widespread damage that all too often passes for inevitability” (p. 200). far from being an abstract and static discipline that it is sometimes assumed to be, mathematics is intimately entangled in our lives as it continues to serve as a proxy for truth and privilege. the way that data/mathematical models are used and the way that we do research matters. the models that we set up—that is, the apparatus within which we are entangled (barad, 2007)—determine reality (o’neil, 2016). in each intra-action, then, researchers determine reality and reconfigure the world. these determinations cannot be made ahead of time and cannot be rule-bound or universalized. as lenz taguchi (2009) explains, “such universal ethics will not be understood as universally ethical by all, and second, such questions exclude the possibilities of asking ourselves how can or might we all live in different and other ways?” (p. 178). this question brings us back around again to gutiérrez’s (2017) call for different knowledges to be privileged in mathematics education research, d. b. martin’s (2015) call for a rethinking of equity for the collective black, and de freitas and sinclair’s (2014) call for radical reconfiguration of school mathematics. as educational researchers, how do we work to continually pose questions to ourselves and each other that take into consideration how we might all live differently? what if schools were structured differently? what systems might be dismantled and radically reimagined? cannon editorial journal of urban mathematics education vol. 13, no. 2 11 ethics and living differently theory is ever present in mathematics education research. we are always already operating with theory whether we are explicit in naming it or not. in mathematics education research, which theories are allowed to count (martin et al., 2010), how we count them (martin & lynch, 2009), and who disciplines the field is driven by the political, social, and material. stinson and walshaw (2017) asked the following as they ended their chapter in the compendium: [will] the battles over the nature of knowledge, truth, reality, reason, power, science, evidence, and so forth…continue indefinitely. or might the battles wane, as mathematics education researchers, funding agencies, and policy makers come to a different understanding of “what works”? how do we, the community of mathematics education researchers, learn to evaluate science across paradigms? how do we learn to use science that produces different knowledge differently? (p. 147) in drawing on different theories and methodologies, i hope that mathematics education researchers not only produce different knowledge differently but also give rise to more ethical and just ways of living and being in the world. as hultman and lenz taguchi (2010) explain, our engagement with the world, as researchers has real consequences. these are consequences that might evoke new realities and new ways of being, which in feminist and political perspective is of vast importance. what we do as researchers intervenes with the world and creates new possibilities but also evokes responsibilities. (p. 540) hostetler (2005), when considering the question of good educational research, states, “to each of those scenarios, we can and must say, ‘okay, but how does that serve people’s well-being?’ and to answer that question, we have to venture wide-eyed and strenuously into the ‘bewildering complexities’ of human good” (p. 19). read alongside barad and gutiérrez, hostetler’s question increases in complexity with the understanding that we “learn from other-than-persons, which, in turn, may change our relationships with them” (gutiérrez, 2017, p. 2). being human now requires a more than human awareness to create radically reconfigured realities. research over the last few decades has shown that reconfiguration is possible, and we may have to unlearn some of what we know to achieve the radical reconfigurations. we may need to ask, what if? the call in this introductory and invitational editorial, i have traced how two particular theories—poststructuralism and new materialism—have brought about disruptions in the field of mathematics education and allowed for different, more complex cannon editorial journal of urban mathematics education vol. 13, no. 2 12 conceptions of mathematical subjects and the questioning of mathematics itself. these serve as illustrations of what this new jume section might hold. too often theories and methods such as these, which might be impactful in disrupting and dismantling mathematics education as usual, are relegated to special issues or do not make it to press at all. if mathematics education researchers wish to understand differently, they must research differently. d. b. martin (2013) has argued that mathematics teaching and learning are racialized projects and that the mathematics education enterprise in general is “an instantiation of white institutional space” (p. 328). in relying on traditional theories and methodologies, mainstream mathematics education researchers too often have minimized race or erased race altogether in their analyses (martin, 2009). historical tracings of mathematics education show that there is deep and embedded structural racism in mathematics education (e.g., berry, 2018; martin, 2019). the particular theories that i present here do not adequately or directly address race and racism. they are not enough to change urban mathematics education. we need more and different ways to reconfigure mathematics education. mathematics education researchers can look to other disciplinary fields that are taking up other theories, concepts, and methodologies, such as black feminism (evans-winters, 2019), abolitionist teaching (love, 2019), anti-racist education (evans-winters & hines, 2019), posthumanism (taylor & bayley, 2019), and others. i am confident that there are mathematics education researchers already thinking and re-rethinking with these and other new and newly considered theories and methodologies that extend and connect mathematics and mathematics teaching and learning to other disciplinary fields. i invite potential authors to bring your connections and disruptions here. in light of the aforementioned shifts made possible through new or newly considered theories and methodologies, the section aims to make a dedicated space for the introduction or mapping of theories and methods that have the potential to matter for urban mathematics education research. while the journal for research in mathematics education has been making moves through its editorial pages to “firm up” what counts as mathematics education research (e.g. cai et al., 2019), it is especially important that the journal of urban mathematics education make moves through this section and throughout its pages to re-affirm its commitment to different ways of doing science in mathematics education. doing so is not a criticism of traditional ways of doing science per se but rather an acknowledgement of a widening of the boundaries for mathematics education researchers to radically reconfigure the field with and for students, teachers, and humanity more broadly. in this section, i seek texts that call on the jume audience to question ideas and practices that are taken for granted (field disruptions) or to consider the potential of a theory used in other fields (field connections). in other words, manuscripts submitted for consideration might question the taken-for-granted discourses and discursive practices that prevail in mathematics and mathematics teaching and learning cannon editorial journal of urban mathematics education vol. 13, no. 2 13 or introduce readers to a theory and/or methodology that is not prevalent in mathematics education research but has potential to shift the field (see davis & jett’s, 2019, opening chapter in critical race theory in mathematics education). authors should carefully outline the theory and/or methodology and make distinct and direct connections to issues in urban mathematics education. i also welcome alternative research texts—for example, autoethnography, blackout poetry, found poetry, ethnodrama, fictions, and so on––that are most often excluded from traditional mathematics education research outlets and venues. of course, there are other forums for this type of work, but by creating this section specifically in a mathematics education journal, i hope to bring these different possibilities for conducting and presenting science into the field of mathematics education more fully to see how they might help us think differently. i cannot predict nor promise exactly what this section might become; it will depend largely on the disruptions and connections brought by the jume community. the priorities for acceptance are that the theories/methodologies/alternative texts are seriously considered and are offered in ways that are informative to the broader mathematics education audience and that the authors communicate the relationship to urban education either directly through the work or in an accompanying cover letter. i encourage potential authors to reach out to me with ideas or suggestions for this section, as i hope it will reflect a wide and inclusive set of ethico-onto-epistemologies. references attick, d., & boyles, d. 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(2001). a foucauldian gaze on gender research: what do you do when confronted with the tunnel at the end of the light? journal for research in mathematics education, 32(5), 471–492. https://doi.org/10.2307/749802 wolfe, m. j. (2019). smart girls traversing assemblages of gender and class in australian secondary mathematics classrooms. gender and education, 31(2), 205–221. https://doi.org/10.1080/09540253.2017.1302078 copyright: © 2020 cannon. this is an open access article distributed under the terms of a creative commons attribution-noncommercial-sharealike 4.0 international license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. microsoft word 382-article text no abstract-1942-1-6-20200418 (galley proof 2).docx journal of urban mathematics education december 2020, vol. 13, no. 2, pp. 42–59 ©jume. https://journals.tdl.org/jume robin k. henson is chair and university distinguished teaching professor in the department of educational psychology at the university of north texas, 1155 union circle #311335 denton, tx 76203-5017; e-mail: robin.henson@unt.edu. his research interests include applied general linear model analyses, measurement, and self-efficacy theory. genéa k. stewart is a phd student and teaching fellow in the department of educational psychology at the university of north texas; email: genea.stewart@unt.edu. her research interests include multilevel modeling, issues in educational equity, mental health, and help-seeking attitudes in college students. lee a. bedford is a phd candidate in the department of educational psychology at the university of north texas; email: leebedford@my.unt.edu. his research interests include measurement, trauma, and military psychology. guest editorial key challenges and some guidance on using strong quantitative methodology in education research robin k. henson genéa k. stewart lee a. bedford university of north texas university of north texas university of north texas ducational researchers often struggle to draw conclusions from quantitative methods in ways that honor local contexts (casad et al., 2017; gutiérrez, 2002; valero, 2008). some in the field of mathematics education have called for more specificity when developing statistical models and assessments that can inform the field about why, how (cai et al., 2019), and for whom (adler et al., 2005; connolly et al., 2018) interventions and assessments work, even down to the lesson level (cai et al., 2020). such concerns elevate the need to evaluate carefully, with strong methods, which curricular interventions and assessments should be brought to scale. in educational research, idiographic realities often detract from our ability to see consistent results over time and across sites. berliner (2002, p. 19) pointed out that a “ubiquity” of interactions is what helps to make educational research the hardest science of all. still, the field seeks to implement research designs that help to identify practical effects of curricular interventions despite any variations between individual students, teachers, schools, and districts in which the study is conducted. appropriate use of any methodology requires extensive, sound decision making to warrant conclusions drawn from the method. indeed, it can be considered somewhat of an art to parse out the influences of situational, organizational, and environmental factors in order to assess impact. attention to good quantitative practice is worthwhile however painstaking. strong methodological practices may help mathematics teachers and practitioners avoid lamenting the “whiplash” of starting over every few years due to frequent educational reforms (cai et al., 2020, p. 134), which can be based on weak, over-generalized evidence. e henson, stewart, & bedford key challenges and some guidance journal of urban mathematics education vol. 13, no. 2 43 the thread of context should be woven through all stages of the research process, from design to reporting, and this focus should include whether the methods used are appropriate for the purpose of the research. unfortunately, methodological errors (or perhaps less-than-optimal decisions) can be common. the peer-review process helps adjudicate research quality, but it can still result in research with common flaws. though typically highly knowledgeable in their respective fields, reviewers are human. they are often very busy researchers bound by time, competing obligations, and varied methodological expertise. moreover, the field of quantitative methodology is constantly evolving (see aiken et al., 2008; henson, 2006; hughes et al., 2010; thompson, 1999) regardless of the misconception that the field is static. all of these factors, and others, contribute to the importance of maintaining a current and appropriate understanding of quantitative methodology while considering the context surrounding its use and interpretation. failure to do so may result in flawed research practices that can distort applications of theory, misinform policy and budgetary decisions, or even result in negative research funding decisions. as we continue to refine our understandings of best practices for quantitative methodology, it is incumbent on researchers, reviewers, and journal editors to stay well-versed in new developments. toward this end, the purpose of the current article is to review several common areas of focus in quantitative methods with the hope of providing journal of urban mathematics education (jume) readers with some guidance on conducting and reporting quantitative analyses. our intent is to challenge and stimulate strong methodological thinking. after providing some background for the needed discussion, we will review briefly the nature of recent jume articles and then comment on several quantitative issues that deserve our attention while referring readers to resources for more comprehensive treatments. where are we now? a brief review of some common errors unfortunately, examinations of analytic and reporting practices underscore the prevalence of errors and omissions. kesselman and colleagues (1998) conducted a comprehensive review of 17 education and behavioral science research journals for articles that contained at least one of the following: analysis of variance (anova), multivariate analysis of variance (manova), and analysis of covariance (ancova). they found that statistical assumptions are often not reported or are even violated, effect sizes are rarely reported, and sample sizes are not regularly based on power analyses (kesselman et al., 1998). both inadequate sample sizes and non-random sampling introduce bias in the interpretation of results in the form of increased sampling error, which decreases the accuracy of findings (tabachnick & fidell, 1996). in a broader review focused specifically on education research, zientek et al. (2008) examined 174 articles cited by the american educational research henson, stewart, & bedford key challenges and some guidance journal of urban mathematics education vol. 13, no. 2 44 association panel on research and teacher education (see cochran-smith & zeichner, 2005), finding only 13% of the articles reported score reliability, 4% reported confidence intervals, and 39% reported effect sizes. furthermore, in a review of the journal of applied psychology, courville and thompson (2001) found that 94% of articles contained discrepancies between beta weights and structure coefficients when ranking predictors in regression analyses. the authors also highlighted other common errors and misinterpretations of regression analyses. there are common mistakes to be found in studies that use exploratory factor analyses (efas) as well. henson and roberts (2006) conducted a review of 60 articles that used efas and found that some studies contained less than the recommended sample size (median n = 267); tabachnick and fidell (1996) would consider this to be below the minimum sample size for an efa. many studies also contained less than the recommended amount of variance explained by the factors, and often researchers likely did not extract the correct number of factors. furthermore, 65% of studies did not report which matrix of association was analyzed, 13% did not report the extraction method used, and nearly 57% used the default extraction method in their efas. henson et al. (2010) identified and emphasized deficiencies in quantitative and research methods training in education doctoral programs that may lead to usage errors and reporting problems in research articles. they also argued that researchers rely too heavily on traditional research designs and statistical analyses, resulting in limited learning and application of new advances in quantitative methodology. the authors suggested several ideas for advancement, such as additional training and consulting with methodologists early in the process when designing studies. the current article can be reasonably considered as one small educational step in that direction. our approach is holistic in the sense that we fully comment on the research process in multiple areas (e.g., design, data analysis, and reporting). it is impossible to provide comprehensive guidance in one article, but we address some key challenges and best practices in quantitative research in the following domains: causal inferences, measurement, handling missing data, testing for assumptions, addressing nested data, and evidence for outcomes. these domains were selected based on discrepancies consistently identified in methodological reviews of educational research (aiken et al., 2008; connolly et al., 2018; courville & thompson, 2001; enders, 2010; henson et al., 2010; henson & roberts, 2006; kesselman et al., 1998; peugh & enders, 2004; vacha-haase et al., 1999; zientek et al., 2008). an empirical review of recent jume methods to provide a baseline for the discussion, we reviewed all jume publications categorized as research articles (n = 24) from 2014–2017 to evaluate the types of research methods typically employed in the journal. the 2018 volume was not henson, stewart, & bedford key challenges and some guidance journal of urban mathematics education vol. 13, no. 2 45 included because it consisted of reprints from 2008 as a 10-year follow-up. the 2019 volume was not yet available online. all articles classified under the editorial, commentary, response commentary, or research impact categories were excluded. regular readers of jume would likely not be surprised that the journal has historically tended toward publication of qualitative-oriented articles. fully 75% of the articles reviewed could be classified as qualitative, whereas only 16.7% utilized quantitative designs and 8.3% used some form of a mixed methods approach. table 1 provides a summary of the review and also illustrates the types of analyses employed. because our focus is on quantitative research, we did not delineate types of qualitative approaches. table 1 methodological approach in jume research articles (2014–2017) type of method n primary method qualitative methodology 18 ü quantitative and mixed methodology 6 cross-tabulations 1 paired sample t-tests 1 ü independent samples t-tests 1 propensity score matching 1 ü multiple regression 1 ü meta-analysis 1 ü item response theory 1 analytic weights 2 chi-square 1 coefficient alpha 3 descriptive statistics 5 henson, stewart, & bedford key challenges and some guidance journal of urban mathematics education vol. 13, no. 2 46 although the nature of jume’s historical focus may have gravitated toward a qualitative emphasis, the policies and procedures of the journal indicate an interest in publishing “data based qualitative and quantitative studies, action research, research syntheses, integrative reviews and interpretations of research literature” (journal of urban mathematics education, n.d.-a). in many cases, qualitative methods allow for strong idiographic exploration of phenomena at the local level (berliner, 2002; demerath, 2006). however, appropriate integration of strong quantitative methodology could broaden the journal’s goals of publishing data-based research and promoting diverse applications to “foster a transformative global academic space in mathematics” (journal of urban mathematics education, n.d.-b). of course, in any study, the research question should drive the methods used. if appropriately applied and done well, quantitative methods can help bolster research inquiry with measurable and distinct evidence for effects of interventions and other correlational questions. some key challenges and comment on good practice below are suggestions and resources for addressing problems in the following six quantitative domains: causal inferences, measurement, handling missing data, testing for assumptions, addressing nested data, and evidence for outcomes. these suggestions are based on previous methodological reviews in the fields of education (e.g., henson et al., 2010) and psychology (e.g., aiken et al., 2008). it is our hope that researchers in the field of urban mathematics education will heed warnings identified in these related fields in order to produce higher quality manuscripts. recent empirical studies in urban mathematics education are referenced throughout this paper to provide support for methodological recommendations. causal inferences researchers are often concerned with making causal claims between variables. the field of mathematics education, in particular, would benefit from an increased understanding of causal processes within the classroom that influence instructional approaches across explicit conditions (cai et al., 2019; maxwell, 2004), such as student perceptions of equity and access to participation (vogler et al., 2018). it is not uncommon to find statements of causal inference and generalization in qualitative studies. although the claims may be presented tentatively, causal inferences are either explicit or implied far more often than is warranted by our designs — whether qualitative or quantitative. researchers may take for granted the following essential elements needed for causation: (a) the independent variable and dependent variable must be correlated, (b) the independent variable should take temporal precedence, or come first, over the dependent variable, and (c) there must be no effect of extraneous variables (shadish et al., 2002). therefore, the ideal approach for henson, stewart, & bedford key challenges and some guidance journal of urban mathematics education vol. 13, no. 2 47 researchers to determine causation is through manipulation of variables where any outcome differences at post are attributable to the treatment and not pre-existing systematic differences between groups (e.g., self-selection). experimental designs are considered very strong because they allow the researcher to observe “phenomena which are made to occur in strictly controlled situations in which one or more variables are varied and the others are kept constant” (zimney, 1961, p. 18). random assignment of participants to conditions helps researchers rule out rival or competing hypotheses because theoretically confounding variables are leveled through randomization. it follows that experimental research lends itself well to formation of causality arguments. still, experimental research is not often feasible for much of educational research, or even desirable. quasi-experimental studies are not as strong as experimental but are still preferred over correlational studies because conditions can be manipulated so that (a) covariance between the intervention and outcome can be observed and (b) the intervention precedes the effect (johnson & christensen, 2019). however, quasi-experimental studies fall short of the third causal criteria, as they do not fully account for confounding variables. this is often because researchers cannot randomly assign students to conditions (e.g., classrooms or schools) of their choosing or have no ability to include a control group. collecting data necessary to address competing explanations can strengthen causal arguments with this design. in educational research, randomized control trials (rcts), studies in which individuals or groups of individuals are randomly assigned to treatment conditions, are becoming an increasingly popular approach toward the development of evidencebased practices and theories of change (connolly et al., 2018). however, to avoid overgeneralization of cause and effect claims, connolly et al. (2018) advised that rcts in education should involve some sub-group analyses and incorporate process evaluations as a component of the research. finally, in observational studies where random assignment is simply not possible due to either program criteria or the practical logistics of the setting, propensity score matching (psm) offers a valuable alternative (austin, 2011; henson et al., 2010; morgan et al., 2010). psm seeks to reduce selection bias by approximating a randomized experiment with “treatment” and “control” groups based on participant covariates and evaluating whether differences are likely due to treatment (austin, 2011; rosenbaum & rubin, 1983). thus, the psm process can serve as an analog to the random experiment and is superior to drawing conclusions from observation alone. one of the major drawbacks to adopting psm is that it fails to balance unobserved, or unmeasured, characteristics in the statistical model and it rests on the core assumption that all confounders are measured (austin, 2008; hill, 2008). rcts work better to balance measured and unmeasured covariates across intervention groups (austin, 2008). regardless of method, researchers should take care to qualify their findings with clear indications as to whether any causal claims are supported. as an henson, stewart, & bedford key challenges and some guidance journal of urban mathematics education vol. 13, no. 2 48 example, howard et al. (2015) demonstrated good practice in tempering causal language (i.e., qualifying conclusions) within a psm study using a secondary longitudinal dataset: it is important to note that it cannot be inferred that the more negative psychological scores were necessarily caused by the failing grades. it is feasible that the students failed because they already had more negative dispositions toward mathematics, or conversely that negative dispositions emerged following the failure. regardless of the direction of causality (if any), the existence of more negative dispositions towards mathematics indicates that psychological dispositions are somehow associated with mathematics performance in the eighth grade, and placing these students in advanced mathematics courses without addressing these dispositions may be inadequate in terms of the support they may need. (p. 54) the reporting in this particular jume manuscript (included in our empirical review) was comprehensive and informed. further elaboration on limitations of design in papers such as this will help to move the mathematics education field toward an even deeper understanding of the extent to which we can draw causal inferences across study designs. measurement issues researchers need to either use instruments that yield scores with strong psychometric properties or create measures themselves that yield scores with strong reliability and validity. further, researchers should use caution when creating their own measures (e.g., surveys). scores from every measure should be deemed valid and reliable prior to being used in published research, as poorly written surveys that may appear to only have face validity can weaken the foundation of entire bodies of research (borsboom et al., 2004; lissitz & samuelson, 2007). measurement is often the achilles’ heel of quantitative research. if we do not measure what we are interested in well, then it may not matter what else we do in the study! in a classical test theory framework, test scores are considered reliable when the test’s “observed scores are highly correlated with its true scores” (allen & yen, 1979, p. 72). this theoretical concept is often operationalized and assessed with testretest reliability correlations and, most commonly, with coefficients measuring internal consistency (i.e., correlation) between items (hogan et al., 2000). henson (2001) provides a primer on the meaning and interpretation of internal consistency reliability coefficients, such as coefficient alpha (see cronbach, 1951). unfortunately, researchers often ignore reliability or incorrectly assume that a measure will yield reliable scores in a current study just because it has in prior samples. vacha-haase et al. (1999) reviewed 839 articles in three counseling and psychology journals and found that only 35.6% of the articles reported reliability coefficients for the data being analyzed. these coefficients are essential for evaluating good henson, stewart, & bedford key challenges and some guidance journal of urban mathematics education vol. 13, no. 2 49 measurement because scores, not tests themselves, are either reliable or unreliable (vacha-haase et al., 2002). regarding measurement validity (i.e., measuring what you claim to be measuring), researchers should expect evidence for construct, content, convergent, and discriminant validity before assuming that scores on a measure may be valid in a current or future sample (borsboom et al., 2004; cronbach & meehl, 1955; lissitz & samuelson, 2007). in order to consider a measure good enough to be included in one’s research study, there should be sufficient evidence for score validity from prior work in similar samples. failure to evaluate and report the validity evidence for scores on a measure amounts to rolling the dice on whether one’s obtained scores in a current study will mean anything at all. score reliability and validity cannot be assumed even if an instrument was previously published in a reputable journal and used widely in the literature (see henson et al., 2001). not only do researchers need to select measures that will yield reliable and valid scores, they must also be aware of measurement invariance between samples or groups (i.e., there may be systematic biases in measures based on the sample being tested; millsap, 2011). researchers often believe that “reliability coefficients from previous samples or test manuals are psychometrically applicable for their current published work,” but this is simply not true in an absolute sense (vacha-haase et al., 2002, pp. 563–565). factor structures of the measures should be assessed (henson & roberts, 2006), and researchers should “investigate the invariance of the measures implemented before comparing the results from the measure in a study” (henson et al., 2010, p. 234). millsap (2011) provides a review of measurement invariance testing. of course, understanding these measurement implications can be difficult when not regularly part of researcher training. in a survey of psychology doctoral programs in the united states and canada, aiken et al. (2008) found that only 64% of all departments provided a doctoral course in measurement. furthermore, the researchers found that 40% of departments taught sections that included item response theory (irt) as an element every 2 years, with only 9% teaching a full semester of irt. irt is an advanced measurement approach that allows for item parameters and test-taker ability to be assessed on the same scale (reise et al., 2005). for example, one study focused on understanding the flynn effect was able to separate out general intelligence, an ability parameter, from item parameters (e.g., discrimination and difficulty) on a mathematics achievement test (beaujean & osterlind, 2008). irt is a key method in modern test development (embretson & reise, 2000), and recent studies have utilized irt to create and validate scores on mathematics assessments (e.g., the probabilistic reasoning scale; see primi et al., 2017; the abbreviation math anxiety scale; see sadiković et al., 2018). henson, stewart, & bedford key challenges and some guidance journal of urban mathematics education vol. 13, no. 2 50 missing data techniques missing data is a common occurrence in educational research, yet modern missing data imputation techniques, such as multiple imputation (mi) and full information maximum likelihood estimation (fiml; schafer & graham, 2002), are not regularly taught in doctoral programs (aiken et al., 2008). researchers often address missing data by either removing the cases that contain missing values (i.e., listwise deletion) or with mean, median, or regression imputation of the missing values (peugh & enders, 2004). listwise deletion leads to biased parameter estimates if the data are not missing completely at random (mcar; little, 1988; peugh & enders, 2004). furthermore, even if the data are mcar, mean imputation can produce biased parameter estimates (e.g., inaccurate r2 effect sizes in regression models). moraleschicas and agger’s (2017) math achievement study highlighted the limitations introduced when data are missing and reported the listwise deletion of cases. a description of the missing data and a statement about whether the data were mcar would strengthen the reporting and allow readers to better interpret the final results. newer methods of missing data imputation, such as mi and fiml, can produce unbiased estimates for missing values that are either mcar or missing at random. it should be noted that values that are missing not at random (i.e., systematically missing) will tend to produce biased parameter estimates regardless of which missing data technique is utilized (peugh & enders, 2004). howard et al. (2015) provided transparent and detailed reporting of their missing data decision-making process wherein the authors compared results from multiple imputation data and the original data but ultimately used the original data due to negligible differences between the two: little’s mcar statistic (spss missing values 22.0) revealed that the missing data met the assumption of mcar, χ 2(39) = 52.84, p = .07. there were no systematic patterns of missing data when compared to the observed values for all of the matched covariates, the prior mathematics assessment and psychological measure scores, and the collegebound variables. (p. 48) fiml uses a maximum likelihood estimation instead of the least squares estimation that other imputation techniques utilize. mi creates multiple imputed datasets and pools the estimates in a two-step process (enders, 2010). the primary benefit of both techniques is that they require less strict assumptions regarding the missing values (peugh & enders, 2004). fiml is slightly more accurate (schafer & graham, 2002), but mi can be more versatile in the imputation process (peugh & enders, 2004). see enders (2010) for a full guide to missing data techniques. testing for assumptions because inferential statistical analyses use data from samples to generalize to populations, each analysis is accompanied by a set of assumptions (cohen et al., henson, stewart, & bedford key challenges and some guidance journal of urban mathematics education vol. 13, no. 2 51 2003; tabachnick & fidell, 1996). thus, the key assumptions need to be tested and reported in published articles. in the event that an analysis’ assumptions are violated, researchers should evaluate the implications based on the severity of the violation and consider either data transformations, seek out correction methods, or switch to nonparametric tests to ensure the most accurate results (osborne, 2013). researchers should also be aware that assumptions change based on the analysis used. for instance, the assumption of multivariate normality is a prerequisite for an analysis that involves the simultaneous prediction of multiple outcomes (henson, 1999). in turn, homogeneity of variance is assumed for between-group comparisons, and sphericity is assumed when testing change within subjects over time (tabachnick & fidell, 1996). the assumption of homoscedasticity, wherein all values of x share the same scatter around a regression line, should be met in order to draw accurate conclusions from analyses under the general linear model, such as multiple regression (cohen et al., 2003). yet, in a review of 61 articles that utilized between-subject analyses, kesselman et al. (1998) found that only one study reported testing for both normality and homogeneity of variance. furthermore, onwuegbuzie and daniel (2005) conducted a review of articles submitted to research in the schools and noted that 91% of submitted manuscripts did not discuss model assumptions. it is not uncommon for reporting of these assumptions to be omitted in published multiple regression articles in the field of mathematics education (e.g., irvin et al., 2017; lee, 2018; morales-chicas & agger, 2017; smith & hoy, 2007). reporting assumptions is essential to good quantitative practice, for when we report results without checking or meeting the assumptions, we risk publishing results that are not replicable (kesselman et al., 1998). addressing nested data the complex research questions that drive our analyses of program effectiveness are often concerned with data at multiple levels (e.g., classrooms, schools, districts, and states). people or students in these naturally occurring hierarchical groupings, also referred to as clusters, often share variance on outcomes because of their common experience, setting, and so forth. therefore, when researchers treat clustered data as independent, they are at an increased risk of making type i errors (ferron et al., 2008). instead, we can consider the data as nested (i.e., students at level 1 nested within schools at level 2) and adjust our model to account for these cluster effects (e.g., non-independence of data). these multilevel models, also referred to as hierarchical linear models and variously as mixed models, mixed effect models, and random coefficient models (mccoach, 2010; raudenbush & bryk, 2002), allow for interpretation of both fixed effects, which may be estimates of slopes and intercepts that do not vary by cluster, and random effects, which are allowed to vary by organization unit or by individual (woltman et al., 2012). henson, stewart, & bedford key challenges and some guidance journal of urban mathematics education vol. 13, no. 2 52 multilevel research questions related to cross-sectional analyses often seek to account for variability stemming from student-level differences and any cluster effects of the aggregate unit (e.g., does class size moderate the relationship of student self-esteem and mathematics achievement?). to illustrate the benefits of multilevel modeling for addressing complex research questions in mathematics education, consider a study by young (1997). the author employed multilevel modeling to decompose science and mathematics achievement by simultaneously investigating predictors at the teacher/school level (e.g., student support or mission consensus), classroom level (e.g., student cohesiveness or task orientation), and student level (e.g., self-concept and satisfaction). in that study, class and school effects were present, but ultimately student self-concept explained the most variance in achievement. this important finding informs educators’ ongoing engagement with students. in urban mathematics education, lekwa et al. (2019) also used multilevel modeling, this time with students nested in classrooms, to measure “the predictive relationship between teacher practices, as measured by the csas-o, and gains in student achievement in reading and mathematics” (p. 15). use of multilevel modeling in longitudinal designs can additionally allow exploration of differences in student growth rates (e.g., to what extent do boys and girls differ in their rate of change in self-confidence based on program involvement?). multilevel modeling also offers the advantage of increased statistical power to detect growth effects when student data is missing at various waves (kwok et al., 2008). in order to determine the degree of dependence of the data and provide some preliminary guidance on whether a multi-level model would even be appropriate, researchers can compute an intraclass correlation coefficient (icc). the icc provides a proportion of variance that can be explained by hierarchical groups, and it is computed as a ratio of between group variance divided by the sum of the between group variance and within group variance in a model with no specified predictors (raudenbush & bryk, 2002, p. 71). the icc can also be more simply explained as the degree of similarity we might expect for any two randomly selected student scores within the same organizational unit. in an instructive chapter on good reporting practices in hierarchical linear modeling, mccoach (2010) explained, “an icc of 0 indicates independence of observations, and any icc above 0 indicates some degree of dependence in the data” (p. 134). for instance, matthews’ (2018) study on urban adolescents’ cognitive flexibility and views of mathematics exhibits use of the icc to make statistical decisions: the intraclass correlation coefficient for year-end attainment value was .042, which indicated that the large majority of variation in attainment value existed between students and very little, 4.2%, between classrooms. little variation at the classroom level reduces the need for multilevel modeling (bryk & raudenbush, 1992). (p. 6) henson, stewart, & bedford key challenges and some guidance journal of urban mathematics education vol. 13, no. 2 53 evidence for outcomes perhaps the most central question regarding quantitative analyses is the degree to which they provide information about evidence for study outcomes. any study can typically yield multiple pieces of information that can be consulted as evidence that helps communicate the story found in the data. it is generally important to consider several elements when evaluating outcomes, such as effect size (i.e., practical significance), confidence intervals, power, and statistical significance. regarding traditional null hypothesis statistical significance testing, onwuegbuzie and daniel (2005) found that 86% of studies did not include any discussion of a priori or post-hoc statistical power analyses, while 14% clearly lacked adequate statistical power. if null hypothesis statistical significance testing is going to be used, sufficient power is necessary to provide context for interpreting obtained p values (cohen, 1983). furthermore, 33% of the studies had omitted one or more appropriate statistics (e.g., degrees of freedom, p value), and 33% confused statistical with practical significance. the majority (65%) of articles had no discussion of limitations and legitimacy of findings. these omissions are not uncommon in the literature, and they reflect a general lack of understanding of the importance of providing clear evidence to support claims regarding study outcomes. for example, it is now commonly accepted that multiple pieces of evidence should be reported and that this should include effect size interpretation and confidence intervals beyond traditional statistical significance testing. henson (2006) argued for stronger meta-analytic thinking across the literature when evaluating study outcomes with emphasis on reporting and interpreting effect sizes. young et al. (2019) demonstrated careful interpretation of results in their study on effects of urban mathematics teachers’ professional development and stated, “although isolated effect size results suggest an overall positive outcome for the professional development, meta-analytic thinking can contextualize the results and provide a broader interpretation of the professional development effectiveness” (p. 322). along with other meta-analytic considerations (see cumming & finch, 2005; quintana & minami, 2006), henson (2006) suggested reporting confidence intervals around obtained effects (see also thompson, 2002). indeed, the most recent publication manual of the american psychological association noted that “for readers to appreciate the magnitude or importance of a study’s findings, it is recommended to include some measure of effect size in the results section” (american psychological association [apa], 2020, p. 89). past language also stressed, “it is almost always necessary to include some measure of effect size in the results section” (apa, 2010, p. 34). effect sizes can be reported in original units (e.g., a regression slope) or standardized units (e.g., cohen’s d value), and researchers should use their best judgement about which approach is preferred for better interpretability when communicating magnitude of effect (apa, 2020). henson, stewart, & bedford key challenges and some guidance journal of urban mathematics education vol. 13, no. 2 54 common effect sizes include variance-accounted-for measures (e.g., r2, h2) and standardized mean differences (e.g., cohen’s d, hedges’ g). finally, in many analyses in which interesting effects are observed with sufficient evidence for the study outcome, researchers must evaluate the role of variables in the model. for example, an interesting multiple regression model would necessitate interpretation of which predictors were most impactful in explaining variability in the outcome. in such cases, a traditional approach would focus only on the standardized coefficients, or perhaps unstandardized versions, of the predictors. however, comprehensive researchers will also include interpretation of structure coefficients in conjunction with beta weights to evaluate potential impacts of multicollinearity, the relative strength of predictors, and possible suppressor effects. readers are referred to courville and thompson (2001), henson (2002), and kraha et al. (2012) for detailed explanations of structure coefficients. summary our primary goal as researchers is to produce good research that implements strong research designs with results that are understood by both researcher and practitioner in a way that can be practically applied. although not always the case, we also often care about whether outcomes are generalizable and replicable. before particular methods are considered and employed for these ends, however, we should be asking strong research questions that are grounded in theory and contextualized in appropriate settings. only then should our consideration turn to the appropriate use of methodology to help answer our questions. this article highlighted key challenges and some best practices we find in quantitative research in several areas. decisions made in these domains should be reported in manuscripts whenever possible for transparency. we also offered resources to pursue in order to strengthen quantitative research designs and practices. editors and peer-reviewers are essential to this process, as they are gatekeepers to the publication of good research. when relevant, peer-reviewers should understand these quantitative practices and apply that understanding with thoughtful and detailed feedback to authors. it is also very helpful to provide resources and relevant citations to assist authors with the publication process. aiken et al. (2008) noted glaring weaknesses in the quantitative training provided in psychological doctoral programs. many programs do not teach modern measurement analyses, advanced techniques for handling missing data, or complex inferential analyses. henson and williams (2006) found similar outcomes in education doctoral programs. conducting quality quantitative research is obviously a complex process, but training in terminal degree programs is certainly a piece of the puzzle. it is also reasonable to think that the professional development of researchers would play a role in whether or not quality methodology is used in the literature. like henson, stewart, & bedford key challenges and some guidance journal of urban mathematics education vol. 13, no. 2 55 other fields, quantitative practice changes and improves through methodological and technological advancements. if we are going to conduct and publish quantitativeoriented research, we need to do it well, and doing it well requires that we make good, contextualized decisions throughout the entire research process. references adler, j., ball, d., krainer, k., lin, f., & novotna, j. 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(1961). method in experimental psychology. ronald press. copyright: © 2020 henson, stewart, & bedford. this is an open access article distributed under the terms of a creative commons attribution-noncommercial-sharealike 4.0 international license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. microsoft word final chu and rubel vol 3 no2.doc journal of urban mathematics education december 2010, vol. 3, no. 2, pp. 57–76 ©jume. http://education.gsu.edu/jume haiwen chu was a high school teacher of mathematics in new york city for 7 years and is currently a doctoral candidate in urban education policy at the city university new york graduate center, 365 fifth avenue, new york, ny 10016; email: achu@gc.cuny.edu. his research lies at the intersection of mathematics education and multilingual education, particularly, for latina/o and immigrant students. laurie h. rubel is an associate professor in the school of education at brooklyn college of the city university of new york, 2900 bedford avenue, brooklyn, ny 11210; email: lrubel@brooklyn.cuny.edu. her research focuses on mathematics teacher education and professional development in an urban schools context. learning to teach mathematics in urban high schools: untangling the threads of interwoven narratives haiwen chu graduate center city university of new york laurie h. rubel brooklyn college city university of new york in this article, the authors explore learning about equity pedagogy in mathematics by focusing on the experiences of a teacher and teacher educator within the centering the teaching of mathematics on urban youth project. one teacher’s story is interwoven as a counterpoint and specific trajectory within the broader narrative provided by the teacher educator. key themes addressed include the nature of teaching mathematics, identity and position, and developing culturally relevant mathematics pedagogy. the authors’ goal is not to report on the effects of a mathematics teacher professional development program per se, but rather to open the conversation, between teacher and teacher educator, to a broader audience. keywords: culturally relevant pedagogy, mathematics education, teacher professional development, urban education ince the publication of its principles and standards of school mathematics (2000), the national council of teachers of mathematics (nctm) has widely disseminated its message of “mathematics for all,” stressing the need for equity in mathematics education. leaving aside the elusiveness of a definition of equity, what does it look like to zoom in, from the landscape of national priorities to a small learning community of mathematics teachers, and then to the trajectory of an individual member of that community? in this article, we provide a pedagogical framework about equity in mathematics education and describe a multiyear professional development project organized around this framework, ultimately reaching the scale of a single teacher. research literature about professional development typically offers readers conceptual frameworks, analyses, and summative results. the voices of practicing teachers who participate in the professional development are often absent, or are present in fragmented ways that do not capture the formative nature of growth. s chu & rubel learning to teach journal of urban mathematics education vol. 3, no. 2 58 therefore, here, in an attempt to capture the formative nature of growth for teacher and teacher educator alike, we interweave our voices—a teacher educator/researcher and a high school mathematics teacher—as part of our collaboration in a multi-year professional development project. (details of the professional development project are described elsewhere; see rubel, 2010.) for reading ease, the change in voice (i.e., who is speaking) is noted with different fonts: laurie rubel, the teacher educator/research and project organizer, presents her narrative in times new roman font; haiwen chu, the high school mathematics teacher and participant in the project, presents his narrative in arial font. this two-font format enables us to be in conversation with the reader as we tell a story of teacher and teacher educator learning, a conversation in which we react, disagree, elaborate, confirm, qualify, instantiate, and generalize. as we represent the back-and-forth nature of the conversation, laurie, the teacher educator, connects her conversation to existing literature and to the anticipated learning outcomes of the professional development project. she reflects not only on haiwen’s experiences but also on the intended learning outcomes of the professional development project. haiwen, the mathematics teacher, provides his personal reflections in less formal, more conversational language. our goal here is not to report on the effects of a mathematics teacher professional development program. instead, we seek to open our conversation to a broader audience. at one level, our descriptions of and reflections about our collaborative work demonstrate the synergistic potential of teacher education, research, and practice. more specifically, this conversation, in narrative form, is about mathematics educators—a teacher educator and high school mathematics teachers—working together to develop equity pedagogy. we open our narrative with brief introductions to contextualize ourselves and our work. next, we outline the theoretical framework guiding our teacher learning community and broadly describe our collaborative activities. as we share our respective narratives, we untangle the threads of our individual stories as they interweave over time, around our learning, as mathematics teachers and teacher educators, about equity pedagogy in mathematics. brief introductions i am a second-generation american woman in my early 40s. i hold the privileges of being a united states citizen with a ph.d. who “passes” as a white woman. other aspects of my identity have given me a disposition to notice and be reflective about “otherness,” not only of my own but also of those around me, especially in relation to societal power structures. my academic research focuses on learning how to work with mathematics teachers to develop their abilities to do the same. i developed the centering the teaching of mathematics on urban chu & rubel learning to teach journal of urban mathematics education vol. 3, no. 2 59 youth (ctmuy) project as a way to collaborate with mathematics teachers in urban high schools. i am in my late twenties and a child of immigration. my parents came to the united states from taiwan to complete graduate degrees. i grew up in miami, where i attended an ethnically and socioeconomically diverse honors program housed within a larger school that was majority black, consisting primarily of caribbean and haitian students. we later moved to the midwest, and i had to readjust in my last 2 years of high school to what i would come to call “the final frontier of white flight.” i went to an elite college and majored in mathematics. after college, i moved to new york city and began an alternative teacher certification program. just 3 months later, i was the mathematics teacher of record in an alternative, transfer high school. i completed a master’s degree in mathematics education in the evenings and was a high school teacher for 7 years. i changed schools twice, eventually finding a home in a high school for newly arrived immigrant students, a school at which my ability to speak mandarin chinese and spanish was put to good use. learning to teach mathematics in urban schools when i began my work as a mathematics teacher educator and researcher in new york city, most of my students, like haiwen, were in their first and second years of teaching. as part of an alternative teacher certification program, my students were completing their teacher preparation and certification coursework at night and during the summers. their “student teaching” experience consisted of being full-time teachers of record, mostly in schools located in highly underserved, urban neighborhoods with “students of color” from various communities. the teachers, and i along with them, came to know the challenges of teaching in so-called “hard-to-staff” schools, all in the context of the no child left behind act and the near-hysteria about performance on standardized tests. as my first cohort of students neared completion of their graduate work, many of them expressed eagerness to continue to study together with the goal of improving their practice. i was intrigued by the framework of culturally relevant pedagogy, described as instruction that emphasizes students’ academic success, encourages the development of cultural competence, and facilitates the students’ development of critical consciousness (ladson-billings, 1995). i read examples of culturally relevant pedagogy in elementary and middle school mathematics teaching (e.g., gutstein, lipman, hernandez, & de los reyes, 1997; gutstein, 2003; matthews, 2003; vithal, 2003) and began thinking about how to build on this work with high school teachers. i mapped ladson-billings’ framework onto chu & rubel learning to teach journal of urban mathematics education vol. 3, no. 2 60 the subject area of mathematics and arrived at a framework of culturally relevant mathematics pedagogy (curemap), which consists of three inter-related tiers. culturally relevant mathematics pedagogy (curemap) curemap’s central, or first, tier is teaching mathematics for conceptual understanding. this orientation to teaching mathematics prioritizes the connections between mathematical concepts, procedures, and facts (hiebert & carpenter, 1992), instead of focusing strictly on skills and procedures. we can also think of teaching for understanding in terms of the sociocultural view of understanding, implying engaging in sense making of problematic situations (wenger, 1998). thus, not only must the curriculum support an emphasis on the connections between mathematical ideas but also classroom social and sociomathematical norms (cobb, yackel & mcclain, 2000) must facilitate opportunities for students to participate in mathematical sense making. the second tier of curemap is the inclusion of relevant or meaningful realworld contexts as a regular aspect of mathematics instruction (moses & cobb, 2001; silver, smith, & nelson, 1995), or, as tate (2005) suggests, instruction should be “centered” on students’ experiences. one form of “centering” is contextualizing the lesson’s mathematical task in aspects of students’ everyday experiences, such as traveling by public transportation (moses & cobb, 2001), or other aspects of local, neighborhood life. alternately, the mathematical task might remain in abstract terms, but the representation used to solve the problem might be one that builds on students’ experiences. we can also view centering in terms of opportunities for students to participate in mathematics. while the lesson may not make explicit connections between the mathematics at hand and students’ lives, centering on students implicates the creation of classroom norms and participation structures that invite and sustain student participation, so that students are central participants in the development of mathematical understanding. the third tier of curemap challenges teachers to develop students’ critical consciousness. one approach to developing students’ critical consciousness in mathematics is to foster critical thinking with mathematics, which corresponds with the teaching mathematics for social justice literature (cf., gutstein, 2003; turner, 2003). teachers can create classroom investigations that address local or societal issues of power or fairness as objects of mathematical analysis. for instance, mathematics can be used to precisely quantify unequal distribution of resources (staples, 2005). we can also view the development of students’ critical consciousness in terms of thinking critically about mathematics (skovsmose, 1994). for instance, students can be given the opportunity to think critically about who creates mathematics and for what purposes; this shift in perspective inverts the more typical power dynamic between mathematics and students. chu & rubel learning to teach journal of urban mathematics education vol. 3, no. 2 61 the “cultural” in culturally relevant the “cultural” in culturally relevant mathematics pedagogy extends across all three of its tiers and has tremendous implications for teacher education. first, teachers should be aware of how students participate in their multiple communities and out-of-school activities to create classroom social and sociomathematical norms and participation structures that support students’ development of mathematical understanding. second, teachers’ knowledge of students’ out-of-school experiences is the basis of their selection of meaningful and relevant contexts for mathematization. and third, teachers need to be connected enough with students’ experiences in order to identify relevant themes or topics that might then be analyzed or described with mathematics. curemap takes as a starting point that teachers need mathematical knowledge and mathematical knowledge for teaching. curemap, however, also challenges teachers to build knowledge about their students and their students’ communities. sometimes when i think about the word “cultural,” i’m not sure if it means everything or nothing. as a mathematics teacher, i typically use “culture” to refer to “classroom culture.” implicitly, the connotation is positive, with the ideal classroom culture being one where students work independently and also cooperate with one another. in this sense, a teacher is responsible for creating and sustaining classroom culture. it is much more difficult for me to look beyond my classroom, beyond my own locus of control, and into students’ experiences outside of my classroom and the school. when i was a novice teacher, i was focused on establishing classroom culture. i did not think deeply about how students’ lived experiences and culture could inform my instructional practices, offer me contexts for mathematization, or provide openings to discuss social justice. centering the teaching of mathematics on urban youth (ctmuy) i created a professional development program for high school teachers, with culturally relevant mathematics pedagogy as the guiding framework. my guiding philosophy for this program was that it would be a collaborative program for teachers, outside of their schools, in which we would explore the three tiers of curemap and support one another in our learning. thirteen teachers, from 10 different schools, participated in a range of ctmuy activities in the 2005–2007 school years. a central component of ctmuy was the 5-day summer institute, which took place in each of the summers of 2005 and 2006. (details about the content of the summer institutes can be found in rubel, in press.) all of the participants became teachers through the same alternative certification program, all but two were younger than 30 years of age, and all had chu & rubel learning to teach journal of urban mathematics education vol. 3, no. 2 62 been teaching for fewer than 5 years. the group, however, was diverse across an array of other salient categories. five of the teachers were women, and eight were men. only one of the 13 teachers was raised in new york city; 10 were transplants to new york city from various regions across the united states and two were raised outside of the united states. six of the teachers self-identified as white, three as afro-caribbean, two as asian, and two as african american. they also had diverse experiences with mathematics: seven had undergraduate degrees in mathematics (or the credit equivalent) to be certified by new york state to teach in grades 7–12 and, because of more limited mathematics coursework, six were certified only to teach in grades 5–9. we were teachers at a range of schools, across different neighborhoods in the same city, with very different student communities, and, although we had all attended the same graduate program, few among us had visited each others’ classrooms or schools. we shared mutual respect at a distance and a foundation of experiences in our first 2 years of juggling teaching with coursework that could be alternately too demanding and too simplistic. as a group, we also had stronger backgrounds in mathematics than most other mathematics teachers i knew. rarely would i ever “do mathematics” with other teachers at my school, but, with this group, we built our relationships through mathematics. i began participating in the project as a way to stay in touch with classmates, fellow teachers i had known from my very first days of teaching. in the school years between summer institutes, we held monthly meetings on weeknights. these meetings were often held at my home, over dinner that i prepared for the teachers. later in the project, the teachers took turns hosting the meetings, also over dinner. eating together, we discussed our work, struggles, successes, stories, and hopes in an atmosphere of companionship and camaraderie. i found it helpful to listen to and talk with teachers i respected and knew well at a good distance from my day-to-day routine. we had to distill our experiences into a series of snapshots, to tell and re-tell our stories about teaching. we would come straight from school, but could also leave school behind, even as we were talking about it. the regularity of our meetings, in each others’ homes and over dinner, gave us space and an open structure. we publicly proclaimed ourselves failures and simultaneously gave abundant evidence of steady progress through small successes. we took turns crying over the magnitude of the problems we were trying to solve, and laughing at the same old jokes and stories. as part of the research associated with the project, i conducted interviews with individual participating teachers and regularly visited their schools to observe their teaching and their students’ learning. these school visits typically chu & rubel learning to teach journal of urban mathematics education vol. 3, no. 2 63 included preand post-observation discussions. this format enabled me to develop one-on-one working relationships with each of the teachers, and provided me with first-hand knowledge of their schools, their interactions with students, and the neighborhood contexts of their schools. in the 2005–2006 school year, i visited haiwen at his school six times and during the 2006–2007 school year, another four times. my most satisfying experiences of reflection as a teacher have come right after laurie’s visits to my classroom. her feedback, in those intense sessions right after class, is helpful, not just because of her knack for providing useful advice but also her understanding of the mathematics at hand. the critique of my work as a teacher is both general and specific: suggesting, for example, in lessons that are structured with an inductive approach that i should first give students a glimpse of the “big picture” rather than having the pieces unfold in a way that might make sense only to me. unlike observations by others, these visits and debriefings feel nonevaluative. we talk not just about the mathematics and the teaching but also about what she sees going on among the students in the classroom. the “relevant” in culturally relevant one way that we explored the notion of “relevance” in ctmuy was to use mathematics to describe and analyze societal themes from a perspective of social justice (gutstein, 2006; gutstein & peterson, 2005). for example, in one summer institute, we examined data about death penalty rates for murder convictions, disaggregated by race of defendant and also by race of victim (yates, moore, & mccabe, 1999). looking at the data in this way generates an example of simpson’s paradox: a higher percentage of convicted whites, overall, received the death penalty than convicted blacks. a paradox emerges if the numbers are further disaggregated according to race of the victim as well. the death penalty rate for convicted blacks is greater in the case of black victims and the death penalty rate for convicted blacks is also greater in the case of white victims. i brought this investigation and others to the teachers as an example of how ideas in the high school mathematics curriculum (in this case, simpson’s paradox) could be contextualized in issues of social justice that implicate issues of race and power. however, in this particular example, the context pertains to the united states justice system and might only be relevant to students in general rather than personal terms. as one of my first attempts to design a meaningful project for my students, i created a project about taxation systems. i based this curricular unit on a single factoid: in 1944, the top marginal income tax rate in the united states was 94%. through the 1970s, this top rate was still as high as 70%. it was not until the reagan administration that the marginal rate chu & rubel learning to teach journal of urban mathematics education vol. 3, no. 2 64 for the top bracket dropped to 28%. there is plenty of mathematics here. because income brackets are taxed at different rates, the tax paid is a piecewise linear function of income, and the effective rate is a piecewise rational function. mathematics and connections to social justice were there, but the students were left out. my students do not pay income taxes, and the hypothetical incomes they researched for years before they were even born were anything but “relevant.” the project had became an elaborate exercise set in an alien context, and it flopped. “relevance” can also be viewed in highly localized terms, at the scale of students’ daily lives or the local experiences of their families and communities. knowing how students participate in their multiple, local communities (extended family, church, basketball team, summer youth jobs, etc.) can help teachers to create classroom norms that support the forms and depths of student participation necessary for developing mathematical understanding. we also need to know enough about our students to be able to select meaningful and relevant mathematics problem contexts and to identify meaningful issues or themes that can be analyzed or described with mathematics. the urban context poses a challenge in terms of learning about relevance because urban high schools typically serve a great diversity of students in terms of their race, ethnicity, culture, socioeconomic class, and home language. the ctmuy teachers, including the “teachers of color,” were outsiders to communities where they taught, further complicating this notion of learning about relevance. my students’ ethnic and linguistic diversity, in addition to their emergence from adolescence into young adulthood, make it difficult for me to address the broad spectrum of their interests with contexts in my mathematics curriculum. the two largest groups of students at my school are spanish speakers, from a wide range of latin american countries, and chinese students who speak mandarin in addition to other home dialects. there are smaller groups of students whose first languages are bangla, polish, or tibetan, and five or fewer students whose first languages are thai, vietnamese, west african languages, farsi, or haitian creole. not only do my ninthand tenth-grade students come from different countries and languages but also they live in a range of local neighborhoods. even if i had asked or known then the names of the specific neighborhoods from around the city that they came from, i wouldn’t have been able to locate those places on a map, or imagine the students’ paths from home to school with public transportation. also, because the school is located in an industrial area, i’m not “local” to the school area either. chu & rubel learning to teach journal of urban mathematics education vol. 3, no. 2 65 community walks because curemap necessitates that teachers accumulate knowledge about their students, the ctmuy professional development offered teachers various tools with which to begin or continue this process. one activity was a community walk, which i also describe in rubel (2010) and rubel (in press). my goals for the community walk and associated discussions were twofold. first, a physical neighborhood can function as a context for a variety of mathematical investigations. by physically exploring a neighborhood, teachers can discover information about that place and its residents and practice posing mathematical questions that pertain to those discoveries. second, the community walk is a tool that can help expose teachers to aspects of their students’ lives. examining a neighborhood in terms of the resources it offers to its residents and the challenges it poses to its residents can enable teachers to better identify students’ community resources, funds of knowledge (moll, amanti, neff, & gonzález, 1992) and out of school activities, all of which could contribute toward the goal of culturally relevant teaching of mathematics. in one iteration of the community walk, in the summer of 2005, teachers were grouped in pairs, and each pair was assigned a street map of a distinct, nearby census tract. depending on its population density, a city’s census tracts are typically small in area (8–10 blocks) and all of its streets can be readily explored in 2 hours. these particular census tracts surround our urban campus, and contain a combination of residential and commerical spaces, with a primarily low-income, afro-carribean resident population. none of the participating teachers lived or taught in this particular neighborhood. we shared our findings and also explored a variety of electronic resources that contain information linked to those census tracts. the rationale of this activity was for teachers to become familiar with the process of doing a mini-ethnography about a neighborhood and to learn how to supplement a physical experience in a space with a variety of quantitative information about that space. i was struck by how the teachers avoided talking with people as a way to learn about this area and also by how they “played it safe” in terms of where they visited or what they chose to focus on. my experience of that neighborhood, up until that first community walk, was practically a bee-line from the subway station to the education building. i literally had never gone on the “other side of the tracks.” i hadn’t thought about the neighborhood in terms of its residents as a place that people lived or as a neighborhood separate from the university. yet, when a colleague and i went on our walk through the tract defined not by those who lived there but by the census bureau, we did not leave our mathematical habits behind. we were very precise in terms of quantifying everything we saw. and what we saw were things, not processes: objects rooted in place, not practices or people moving through space and over chu & rubel learning to teach journal of urban mathematics education vol. 3, no. 2 66 time. we counted houses, floors of houses, windows, trees, and businesses. in doing so, we were likely thinking about the types of things that we would be able to have our students count. in a sense, perhaps we were jumping ahead to our roles as mathematics teachers, rather than acting as observers immersed in this setting. it was habitual for us as mathematics teachers to quantify, rather than describe. we used our eyes to see and our hands to record, but did not interact with or talk to anyone along the way. we returned to the community walk activity a year later, in the summer of 2006. this time, each pair of teachers was directed to a single census tract, but each pair was asked to adopt one of the following data collection strategies: (a) choose a specific location within the tract, remain silent in that location, and observe the people and activities in that location; (b) select a specific theme and walk through the tract looking for data that might connect to that theme; or (c) gather information by interacting with people in the tract. remembering the teachers’ avoidance of exploring issues of race and class in the previous year, this time, i also participated in the walk and shared my experiences with the group. another facilitator and i identified establishments within this census tract that seemed to cater to a low-income community: a pawn shop, a check-cashing store, a rent-to-own furniture outlet, and an off-track betting branch. given that we both knew little about these businesses, this represented an opportunity to try to understand how these stores operate, as well as to discover who seems to frequent these particular stores, to gain an understanding of their role in this particular neighborhood. although these and other businesses oriented toward a low-income population were in prominent locations in this census tract, on both iterations of the walk, all of the participating teachers avoided entering unfamiliar territory. a colleague and i decided to do a “community sit:” instead of walking around the census tract, we would sit in silence in the public library within the tract and observe the people and activity there. we arrived before it opened and were surprised that there was such a long line outside. when the doors opened, everyone rushed in and signed up for computers with internet access. there were not enough computers, so some kids would intently watch other kids until their turn came. with the snapple machines, the social networking on websites that i had never heard of, people of several different generations, the ways that the actual books were in the periphery, and the open arrangement of the space all in one large room, i realized that “going to the library” was an activity that i had not really tried to understand beyond my own experience. looking back on how we ended up in the library and not some other place, i realize now that a library was still in our zone of comfort, and still a place where we could just “hang out,” to observe without being observed. i’m glad we went to the chu & rubel learning to teach journal of urban mathematics education vol. 3, no. 2 67 library because it was public and not a space where people went to get some business done and then leave. i may have learned more had we pushed past our boundaries, but i still think that understanding how youth approach libraries could inform how to make classrooms more open spaces for students to learn on their own while also interacting with others. in the 2005–2007 school years, i noted that the impact of the community walk experience seemed to be limited to the teachers’ developing curricular projects using contexts of local data and local maps (see rubel, 2010 and rubel, chu, & shookhoff, in press, for description of some of these projects). haiwen and other participating teachers created highly creative projects in which students used maps, or physical, quantifiable aspects of neighborhoods, as ways to explore mathematical concepts such as oneand two-variable statistics, ratio, scale, and proportion. these curricular innovations were products of our collaboration and demonstrated that the participating teachers had created new types of opportunities for students in terms of these mapping projects. however, the essential goal of the community walk activities was for teachers to replicate or adapt the community walk activity as a way to learn about their students. the teachers were not doing community walks in their own school contexts as a means of learning about their students’ worlds. after the first community walk, i developed a project for students that directed them to research census data about their own neighborhoods to produce a neighborhood profile. students then went out individually on community walks to identify and measure their own variables for developing a “scale of importance” for places significant to them. i was trying to make the curriculum unit student centered, while also giving me a sense of students’ perspectives. they then represented these data as cartograms that remapped their neighborhoods. what they found important could include factors left out by the census profile, such as the number of spanish speakers or the number of young people who frequent specific places, such as stores and libraries. what i learned, beyond how to better design the curriculum for the next iteration, was how little i knew about where my students live and how they spend their time outside of school. in some cases, students also expressed how they don’t visit many of the places near where they live; they did not necessarily identify with their officially designated census tract. because we teach students for two consecutive years, teachers at my school would often revise and repeat a unit only in alternate years, and so i did not immediately re-design the community mapping project. i also felt that students would connect better with a project explicitly centered on their home countries. this time, my students first examined cartograms on internet sites (i.e., www.worldmapper.org), which laurie had shown us in chu & rubel learning to teach journal of urban mathematics education vol. 3, no. 2 68 ctmuy. then, we restricted our focus to countries represented by students in each class. each group was comprised of three students from different countries to research a statistic of interest. students used these data to make large world maps scaling each country according to their variable of interest, while trying to preserve the shapes and relative positions of the countries. after completing the big world map in their groups, each student recentered and rescaled the map so that his or her country would be at the center and have an area of one square inch, a new unit for measuring the world. this act of unitizing scaffolded comparisons relative to their home country—it became natural to talk of countries with 1.5 times the population, for instance. we used these maps to try to see connections between variables, such as literacy rates and airports to cellphones and poverty rates. together, the maps formed an “atlas of origins.” one of their large maps, comparing the literacy rates in a variety of countries such as pakistan, nepal, vietnam, mexico, and el salvador, is shown in figure 1. figure 1. atlas of origins. in this case, it was especially interesting to hear some of my students’ resistance—”my country’s not at the center!” “that’s not what it chu & rubel learning to teach journal of urban mathematics education vol. 3, no. 2 69 looks like!” at the same time, the available statistics, such as gross domestic product or poverty rates, condense a great deal of information— and often suffering—into a tidy figure that we then turned into neat, little, colorful polygons. i felt hardly qualified to tackle all of these complicated issues in my mathematics class. haiwen drew upon his own knowledge of mathematics to create and implement projects that related to local or global maps and corresponding descriptive data. in doing so, he, along with many of the other participating teachers, was indeed working on aspects of teaching for understanding as well as using relevant contexts for mathematization. at the same time, the participating teachers did not seem to recognize the potential impact of getting to know the cultural contexts of their students and their students’ communities. as one of the project advisors reminded us, “how we know our students and their families and the community, the better we do, the easier it is, and that doesn’t make it easy but the easier it is to try and figure out how to mathematize people’s social reality and build from there” (e. gutstein, personal communication, august 25, 2006). there are several explanations as to why the task of learning about students was challenging for this group of teachers. gaining a deeper knowledge of this neighborhood and their students’ neighborhoods would require that teachers negotiate multiple “border crossings” (anzaldúa, 1987). for example, teachers belong to a different generation than their students and know more school mathematics than their students. the teachers were raised and schooled in different geographical regions than their students, live in different parts of the city, and, in most cases, speak different home languages and dialects than their students. as a result, and perhaps most important, they experience a different racial-ethnic and social class reality than their students. these different racial realities clearly play an important role in shaping how mathematics might be relevant, both for whom and for what purposes, and also present challenges to teachers in terms of their openness to try to learn about their students’ experiences. although i started my teaching career in schools that were more homogeneously black and latina/o, my current school is truly “diverse.” my school is diverse in that its students represent different nationalities, languages, and cultures, and not in the way that diversity is oftentimes used as a euphemism for just “different” from the white mainstream. yet, i’ve found that, in all of the schools in which i’ve taught, i have rarely met other teachers who are willing to talk about race and class as they play out in schools themselves. as one of the few teachers of color in a school for students of color, i find myself multiply conscious: not only what it is like to be “other” but also how teachers think of and talk about those who are other. the barriers i feel are not visible, but palpable, especially when crossed. my students assume that i am also an immi chu & rubel learning to teach journal of urban mathematics education vol. 3, no. 2 70 grant, even though, as a chinese american whose parents immigrated from taiwan, i assert to them that i am american. i see no contradiction in this assertion. i know that others often think i’m just chinese, not seeing the layers of experience that come from being taiwanese and born in the united states. yet, this persistent misperception has reinforced my selfconsciousness. race is not something we can put aside, even when we try. i sometimes find myself restraining my enthusiasm for mathematics because i worry that i will end up perpetuating my students’ racialized notions of mathematical competence. a second interpretation of the difficulty for teachers in learning about their students’ communities relates to dual challenges posed by urban diversity. in cities, even a single school neighborhood is likely to have a web of different racial, ethnic, or culture communities. for instance, one of the associated schools has an overwhelming majority latina/o student population. while the use of the term “latina/o” may suggest a homogenous student population, there is great variation at this school in the length of time students and their families have spent in the united states. some of the students are recent immigrants, and other students are second-, third-, or fourth-generation american. in addition, students and their families might come from puerto rico, the dominican republic, ecuador, and mexico as well as other spanish speaking countries. diversity is geographically and densely layered such that any physical boundaries between communities are fluid and/or difficult to identify. these characteristics of urban diversity complicate learning about students because of the multiple cultural communities at play. furthermore, these characteristics of urban diversity also complicate the pedagogical goal of using mathematics to analyze relevant social or political themes—relevant to whom? in project meetings, some teachers indicated that focusing only on abstract mathematics, and not also on developing students’ critical consciousness with or about that mathematics, was a convenient way to avoid confrontation or negotiation of local, racially, and culturally charged tensions or conflicts. although i share the experience of being “other” with my students, i don’t automatically gain insight into their multiple identifications. it’s taken me time to learn how my students aren’t just latina/o, but from mexico, and not just from mexico but puebla, and not just from puebla but huaquechula. at the same time, i have also learned to resist the myth of the single origin: other students i’ve taught are iraqi, but lived in turkey or the netherlands before coming to the united states. students’ origins are thus as personal, local, and multiple as their present lives, and this complexity is multiplied by the number of students i teach. layered on top of those older loyalties are the new places students are navigating, which are often narrow swatches according to regular daily rhythms. these are chu & rubel learning to teach journal of urban mathematics education vol. 3, no. 2 71 not things that we as teachers could understand by looking at a map. it has taken me a long time settling into this city to develop a sense of my students’ worlds. often, i don’t know what questions to pose to students to surface any of these issues. and so i am often reticent or silent when i could be bringing up issues that might then lead to deeper conversations about social justice. but my hesitation is thoughtful, perhaps in contrast with the self-imposed blindnesses of some fellow teachers, who declare: “i try not to read what’s on their t-shirts.” “the great misunderstanding” as my work with this group of teachers drew to a close, i labored over how to clarify with them the distinction between creating curriculum for students that contextualizes mathematics in local geography or local data and teachers using community walks or other tools to learn about their students, their families, and their communities. earlier that year, i made this distinction in an academic presentation, sharing it with mathematics education research colleagues. although i was positioning my project as a teacher–researcher collaboration, i found myself, instead, adopting a stance of a researcher who reports findings about teachers. i needed to share these ideas with the participating teachers as well. so in one of our monthly meetings, i noted to the teachers that they seemed comfortable with the first interpretation but seemed hesitant, or even resistant, to tinkering with the second. in interviews and later meetings, the participating teachers referred to this moment as “the bomb” or “the great misunderstanding.” the story i tell about that evening goes something like this: it’s a dark tuesday evening in the middle of mid-winter break. we gather at one teacher’s apartment over dinner. we are the “old guard” of fifth-year teachers, with laurie, our former professor. our assembled feast grows cold as we go over slides for a presentation on the mapping project i am going to give in a few days. our feedback gets stuck between our summer institutes and the actual curriculum we subsequently implemented. laurie interjects—my point in having you go on community walks was not necessarily to have you make projects for your students. what i was hoping was that you would go out into your students’ neighborhoods and use walks like these to get a better sense of their lives. that walk might not turn into a project, but it might just help you to better understand your students. we object, variously: we’re classroom teachers, that’s our job; who has time to go out into the neighborhood?; i already know a lot about my kids; what you’re saying is what we already do as good teachers; where is the rigor and the generalizability?; hey, i just moved to my kids’ neighborhood; i spend plenty of time with my students and they tell me all chu & rubel learning to teach journal of urban mathematics education vol. 3, no. 2 72 about their lives; sure, i have been to my students’ communities; it’s hard enough just being in the classroom. i had wrestled for months about how to give the group of teachers this feedback. these teachers, who had been star students in my graduate mathematics education courses, were accustomed to my support and applause. they always enjoyed mathematical challenges and seemed genuinely committed to a lifetime of becoming better teachers, but this was a difficult moment in that the subtext of my message was that there was something, something that i felt was very important, that they did not yet seem willing to try to understand. i was the first to leave that particular meeting, and i knew that they would continue discussing my feedback and need to process my message. it was a difficult trip home for me that night; i worried that i had damaged my working relationships with the group and even more importantly, i was struck by how strenuously some of them objected to the notion of spending dedicated time to deeply get to know one’s students. after laurie leaves, we linger a while longer. we wrestle with why all of this sounds like news to us. some of us fixate on generalizability: if we were to, as laurie suggests, shadow a single student for an entire day to see what that student’s experience of school is like, what would we then be able to say about all of the students? some worry that it would be unscientific to generalize from our observed experience of one student to all students. some of us see the advantage of going to other teachers’ classes, but when we do so, we focus on the teacher’s practices and don’t seem to assume the students’ perspective and experience in those classes. laurie has been very specific over the years: she told us how she learned about her own students on camping trips with them, or how she better understood the participation patterns of classes after she attended a church service with a student. maybe we misunderstood what laurie was asking us to do because we were going in with a different primary objective and intention: we had all done non-mathematical activities with our students, but perhaps we hadn’t done these things from the explicit perspective of learning about students. i think i interpreted the community walk activities in terms of my primary roles as a teacher of mathematics, albeit a progressive, project-based, and inquiry-driven teacher. i don’t think that i viewed myself as a teacher of young adults. i hadn’t realized how i had already generalized about my students, without having done the work of observation and participation with them in a range of activities, including mathematics. strangely, even though the message to get to know one’s students had been a constant theme in ctmuy over the years, it felt to me as if the teachers were hearing me that night for the very first time. why had they not heard me before? one reason might be my non-didactic teaching style. i pose problematic situations chu & rubel learning to teach journal of urban mathematics education vol. 3, no. 2 73 for people to make sense of—in mathematics or about teaching. even in the face of resistance from my students, i do not present the solution or the strategy; i am mostly interested in people sharing all of the possible approaches and working together to compare these approaches in terms of their efficiency, ease, or elegance. i approach teacher education in the same way. i am slow to prescribe to a teacher exactly what to do; instead, i can try to provide experiences for them that prompt reflection about their students, about mathematics, or about teaching. ultimately, there are many choices for them in terms of how they use new knowledge to inform their practice. although this non-didactic pedagogical approach to teacher education facilitated an atmosphere of collaboration in this phase of ctmuy, clearly, in this case, there was a need for me to provide more direction. a second contributing factor could be the day-to-day working reality for this group of new teachers. they had an immediate need to develop curricular projects for their students, and our work together on the topic of urban communities sparked many ideas for them in this direction. these curricular projects focused on their students and likely facilitated new opportunities for students to build mathematical understanding. in contrast, getting to know one’s students and the circumstances of their lives, is an extended process that might not have immediate or obvious connections to teaching mathematics. teaching teachers to value knowledge about students and their communities is challenging given that the connections between this knowledge and one’s teaching practice are, perhaps, harder to pinpoint. after some time elapsed, with more conversations and meetings to process the great misunderstanding, i noted that the feedback i had provided seemed to function as a catalyst for teacher learning. high school mathematics teachers often view teaching as the process of transmitting their own knowledge of mathematics to others (mclaughlin & talbert, 2001). with this view of teaching, it is understandable that the teachers initially responded to the community walk activities by focusing on the mathematics content and trying to find creative ways to connect that mathematics to their students’ worlds. in some cases, they did so in very literal terms by using local maps. however, after the great misunderstanding and ensuing conversations, some of the teachers demonstrated that they had moved to a new way of thinking about what it means to be a culturally relevant mathematics teacher. in addition to developing mathematics curricula, they began to recognize the importance of learning about their students and their students’ families and communities. for instance, one teacher quickly created an action plan to “shadow” one of his students, through the school day and into after-school activities. the experience of learning about one student in this way was so rich for him that he continues to do this activity once a year. similarly, in a group meeting about six weeks after the great misunderstanding, haiwen described a shift in his chu & rubel learning to teach journal of urban mathematics education vol. 3, no. 2 74 view about teaching, to a teacher as a builder of relationships with students, and among students, about mathematics (lampert, 2003). i still think in terms of knowing things about the world that i might develop with the students rather than knowing things about the students. even “centeredness,” i’ve taken in my work as “let’s make it about you” rather than “let’s find out about you.” i mean, that’s the question, right? where am i getting my knowledge? i don’t know. it is in classroom settings, it’s from sources other than going to their communities. it’s the difference between being and doing. i mean, you’re asking us to really— we’re comfortable with doing things, writing lessons, teaching them. but to really be, in a different way, is hard. i failed to realize that being culturally relevant was not possession of some body of knowledge, but rather is a matter of relating to students, being involved with them, and engaging them. i now see how i had assumed that projects were a good enough point of departure, and that i knew what real-world problems mattered to my students. i had also assumed that problems were the right place to begin. i realized that i need to learn a lot more about my students, their neighborhoods and communities, and what they value, first. i don’t live or spend much time with my students outside of school, or know, literally, where they are coming from: both each morning or before they came to this country. creating safe spaces in our classrooms for difference is not enough: we must push ourselves out of our zones of comfort and into the places our students live. it takes long, slow work to get to know our students and their lives. curriculum and projects are just one step, and maybe not even the first. final thoughts in writing this article together—that is, in telling our stories together—we have begun to untangle some of the threads of our thinking about teaching and learning. collaborating with teachers about culturally relevant mathematics pedagogy in urban high schools consists of two interrelated strands. teachers need to know how to “find the mathematics” from the high school curriculum in urban contexts, like the transportation systems, the architecture, or the arrangement of housing and resources. teachers also need to know their students well enough to organize instruction to maximize students’ participation or to be able to identify contexts of potential relevance. in other words, working with teachers on culturally relevant mathematics pedagogy cannot just focus on how to create curricula that contextualize school mathematics in experiences that are relevant to one’s students. alongside this effort must be an effort to teach teachers about the chu & rubel learning to teach journal of urban mathematics education vol. 3, no. 2 75 importance of building relationships with their students and about learning about their students as a necessary part of this process. i have changed my focus. i realize that mathematics is so much a part of who i am and how i view my role that i need to work explicitly to find other ways of relating to and connecting with students. these ways include learning how to talk to students, how to listen to them, how to ask the right questions, and then how to listen to them some more. although my inclination is still to insert mathematics into every conversation, i now restrain myself. i have come to see mathematics teaching, in addition to being cognitive and academic, as highly personal and social. building relationships with students only occurs over time and is deeply connected to teaching them mathematics. i have come to enjoy developing those relationships and learning from my students just as much as i enjoy teaching and doing mathematics. and as with mathematics, part of the joy of learning about students is finding out how much more there always is to learn. acknowledgments contributions of the two authors to this article were equal. this material is based upon work supported by the national science foundation under grants 0742614, 0333753, 0119732 and the knowles science teaching foundation. any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the national science foundation or of the knowles science teaching foundation. references anzaldúa, g. 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(2003). critical mathematical agency: urban middle school students engage in mathematics to investigate, critique, and act upon their world. unpublished doctoral dissertation, university of texas, austin. vithal, r. (2003). in search of a pedagogy of conflict and dialogue for mathematics education. dordrecht, the netherlands: kluwer. wenger, e. (1998). communities of practice: learning, meaning, and identity. cambridge, united kingdom: cambridge university press. yates, d., moore, d., & mccabe, g. (1999). the practice of statistics. new york: wh freeman. microsoft word 427-article text no abstract-2487-1-6-20211005 (proof 1).docx journal of urban mathematics education may 2022, vol. 15, no. 1, pp. 31–53 ©jume. https://journals.tdl.org/jume seonhee cho, ph.d., is an associate professor at the college of mount saint vincent, 6301 riverdale avenue, bronx, ny 10471; email: seonhee.cho@mountsaintvincent.edu. her research interests involve preand in-service teacher education for english language learners. hea-jin lee, ph.d., is an associate professor at the ohio state university at lima, 4240 campus dr., lima, oh 45804; email: lee.1129@osu.edu. her research areas are related to mathematics teacher education, including designing and evaluating professional development programs, assessing teacher growth, and culturally responsive mathematics teaching. leah hernan-patnode, ed.d., is an associate professor at the ohio state university at lima, 4240 campus dr., lima, oh 45804; email: herner-patnode.1@osu.edu. her research interests include culturally responsive teaching in mathematics, teaching preservice teachers to teach equitably, and inclusive practices. mathematics lesson design for english learners versus non-english learners from perspectives of equity and intersection seonhee cho college of mount saint vincent hea-jin lee the ohio state university at lima leah herner-patnode the ohio state university at lima calls for “mathematics for all” or “mathematics for social justice” bring light to the importance of equity issues within and through mathematics education. employing the theoretical perspectives of equity and social justice for mathematics education, and the intersection of language, culture, and mathematics, this study examined how a group of in-service teachers working in inner-city settings designed mathematics tasks and strategies for english learners (el) in comparison with non-els. the data, 23 sets of lesson design responding to two learner profiles, was analyzed using inductive content analysis. findings suggest that meaningful opportunities to learn the same mathematics concept were presented less often to els, who also were more likely to experience a lack of active participation and engagement than non-els. from the intersectional perspective, teachers heavily relied on els’ native language support rather than exploring the complexity of mathematical discourse. the lack of cultural integration in their lesson designs was also notable. these findings imply that the attention of mathematics education for els needs to be redirected from language support per se to the interplay between language, culture, and mathematical concepts in order to create a level playing field. keywords: culturally responsive mathematics teaching, english learners, equity of mathematics education, mathematical discourse, mathematical instructions for english learners cho, lee, & herner-patnode mathematics lesson design for english learners versus non-english learners journal of urban mathematics education vol. 15, no. 1 32 alls for “mathematics for all,” or “mathematics for social justice,” demonstrate the efforts to address persistent disparity issues within and throughout mathematics education. the issue of inequity within mathematics education lies in its failure to provide historically marginalized students with meaningful opportunities to learn mathematics (van de walle et al., 2019), perpetuating gaps in learning outcomes between students of diverse backgrounds and their mainstream counterparts (abedi & herman, 2010; barajas-lópez, 2014; bartell, 2013; mosqueda, 2010; welner & carter, 2013). as such, the national council of teachers of mathematics (nctm) recommends mathematics teachers to ensure high-quality mathematics instruction for all students, including students whose first language is not english. in order to support english language learners’ (els) mathematical understanding and proficiency, research suggests that mathematics teachers provide els access and opportunities to learn mathematics through accommodated instruction and extended learning opportunities (abedi & herman, 2010; kersaint et al., 2013; moschkovich, 2013). furthermore, every student’s diverse cultural and linguistic backgrounds should be respected and included in the learning environment (nctm, 2014). similarly, english as a second language (esl) education provides prek–12th grade proficiency standards that explicitly state that esl education should help els successfully communicate information and mathematical concepts (teachers of english to speakers of other languages [tesol], 2006). thus, both the fields of mathematics and esl education stress that teaching mathematics to els should be a collaborative effort between esl teachers and mathematics teachers (ewing et al., 2019; kurz et al., 2017; nutta et al., 2012; song & coppersmith, 2020). furthermore, instructional approaches such as content-based instruction (cenoz, 2015) and content and language integrated learning (moore & lorenzo, 2015) underline the importance of content integration in language learning. furthermore, there is a growing understanding that mathematics has its own complex language that could further challenge traditionally marginalized students with different language and cultural backgrounds. research has shown how language and culture intersect with mathematics (de araujo et al., 2018; o’halloran, 2015; zahner et al., 2018). despite the importance of equity in mathematics education and teachers’ shared responsibilities for els’ mathematics education, it is less known how teachers are prepared to meet the unique needs of els in mathematics lessons. thus, this study investigated how a group of in-service teachers in a graduate tesol program responded to two learner cases involving els and non-els with strategies and tasks to teach a 5th grade mathematics standard. c cho, lee, & herner-patnode mathematics lesson design for english learners versus non-english learners journal of urban mathematics education vol. 15, no. 1 33 theoretical perspectives and literature review the theoretical perspectives of equity and social justice as well as the intersection of mathematics, language, and culture provided us with a critical lens to examine the present study. the equity and social justice perspectives underpin culturally responsive mathematics teaching in that it promotes equal access to meaningful learning opportunities and empowers traditionally underprivileged students through academic engagement and success. the perspective that views mathematics education as an intersection of mathematics, language, and culture enabled us to study mathematics education for els in a more inclusive and comprehensive manner, as will be delineated in the ensuing sections. mathematics education from an equity perspective mathematics education from an equity perspective concerns equal opportunities and access to mathematics learning for all learners. research indicates that students of color are disproportionately placed in tracking and lower level mathematics courses, further preventing them from advancing their studies (bartell, 2013; gonzalez, 2009; larnell et al., 2016). welner and carter (2013) argued achievement gaps are the direct result of opportunity gaps. the practice of tracking affects mathematics performance, but also low expectations and low quality of instruction leads to low student engagement (larnell et al., 2016). culturally relevant/responsive pedagogy, which gloria ladson-billings (1995, 2014) pioneered and gay (2010) expanded into curriculum, highlights the ways that teaching enables equity, social justice, and empowerment of students from diverse backgrounds. specific to mathematics education, incorporating students’ lived experiences, interests, and cultural backgrounds into mathematics lessons and curriculum is crucial (banse et al., 2017; driver & powell, 2017). when students are able to make connections with mathematics concepts, learning improves. thus, the equity perspective highlights the importance of meaningful opportunities for disadvantaged students. this is one way to manifest social justice in mathematics education. another way to address social justice is incorporation of mathematics lessons that help students understand, question, and critique social equity and justice issues (felton-koestler, 2020; gower, 2015; leonard et al., 2009). felton-koestler (2020) gave an example about a response to the events in charlottesville where students discussed the difference between individual and institutional racism and were given graphs with topics such as median income by race from 1967–2014 and white and black people’s views about how far our country has come in addressing equal rights. the students then responded to specific questions about their reaction to the graphs. students could also do further work examining the difference in median incomes (morin et al., 2017) or examine the rates of segregation in large cities where cho, lee, & herner-patnode mathematics lesson design for english learners versus non-english learners journal of urban mathematics education vol. 15, no. 1 34 demographic data is easily accessible (felton-koestler, 2020). other examples that might be easier for elementary students include calculating revenue for fair trade and non-fair-trade chocolate bars or using population figures to determine the probability of being born in a different country (gower, 2015). intersection of mathematics, language, and culture unlike commonly held misconceptions that mathematics consists of numbers and words and therefore els can catch up to grade-level mathematics quickly (e.g., simpson & cole, 2015), a number of studies have explained that mathematics uses its own complex mathematics language and that different cultures may have different ways of presenting mathematical concepts (see de araujo et al., 2018; o’halloran, 2015). supporting this claim, recent research on the mathematics education of els has expanded from focusing on only cognitive aspects—“how language might be a barrier for els’ mathematics learning”—to a sociocultural aspect—“how els gain access to the mathematics register through teaching and learning processes” (de araujo et al., 2018, p. 880). although different studies have adopted different terms, such as mathematics register, multi-modal, or multi-semiotic, to describe approaches to mathematics language, they confirm that mathematical discourse has complexity and density with symbolic notations and images that convey meanings and commands (civil, 2018; moschokovich, 2015; o’halloran, 2015). we further explore how languages and cultures have been presented in mathematics in other literature in the following sections. mathematics intersecting with language. gee’s (1989) concept of discourse views discourse as special ways of thinking, speaking, writing, and interacting with each other within the community of an academic field. simpson and cole’s (2015) meta-analysis of the language of mathematics defines that mathematical discourse not only includes academic vocabulary but also grammar features and the ways that mathematics concepts/problems are organized and asked. the discourse of mathematics is also multi-semiotic in that numeric numbers, mathematical symbols, images, graphs, and charts have their own communicative meanings that could be understood and interpreted within the mathematics discourse community. for example, x in the problem x = 2(3+1) means “unknown number,” which is to be solved, while the parentheses regulate the sequence of problem solving and “+” denotes a command to add. furthermore, mathematical discourses include written text and spoken form, such as explaining multi-step problems, justifying the answer, and demonstrating the reasoning process. thus, mathematical discourse is interlinked with the mathematical concepts rather than a discrete decontextualized language to learn (de araujo et al., 2018; peng et al., 2020). cho, lee, & herner-patnode mathematics lesson design for english learners versus non-english learners journal of urban mathematics education vol. 15, no. 1 35 similarly, zahner et al. (2018) identified languages that intersect in mathematics as “mathematical explanations, justification, generalization,” “specialized notation and symbols,” and “everyday language with content-specific language” (p. 34). thus, some studies have suggested utilizing specifically focused instructional strategies to support mathematical language, particularly for els. for example, els can connect with these practices best when teachers use multiple resources and communication modes while encouraging students to use home languages as a resource (chval & pinnow, 2018; sorto et al., 2014). teacher modeling through think-alouds, reformulation or expansion of students’ responses, and questioning also add to the effectiveness of the applied standards (banse et al., 2017; driver & powell, 2017; hansen-thomas, 2009). such mathematical discourse practices represent how students and teachers can process mathematical information and engage in teaching and learning of mathematics. without the engagement of mathematical discourse, simple translation turned out to be ineffective in els’ learning (turkan & de jong, 2018). mathematics intersecting with culture. the ways that culture intersects in mathematics can include integrating students’ culture and cultural resources into mathematics instruction. students’ funds of knowledge and out-of-school mathematics-related activities could be incorporated into mathematics curriculum and lessons to help engage and empower students (civil, 2014; gonzález et al., 2005; rios-aguilar et al., 2011; wagner, 2012). more specifically, wagner’s (2012) study examined how a group of mathematics teachers incorporated students’ out-of-school mathematics into their lessons, and she argued that the gaps between students’ out-of-school mathematics practice and school curriculum have increased. in the endeavor to reduce this gap, wagner provided a professional development program for in-service mathematics teachers through which she discovered four types of practices that the teachers had adopted. among them, the most frequently employed practice was using cultural contexts in mathematics problems. for example, a teacher might use the image of a soccer field to introduce the concept of area. although the authenticity of such activities is questionable because in real life students use soccer fields to play soccer rather than to measure the field, the practice is easily employable and could motivate students. wagner (2012) further proposed that mathematics-embedded practice in out-ofschool contexts such as family grocery shopping could be the most authentic practice of mathematical concepts. however, it takes effort and time to identify authentic mathematics practices in students’ households. nevertheless, her study demonstrated how out-of-school activities representing students’ culture, resources, and lived experiences could be well integrated into mathematics lessons. another illustration of cultural differences in mathematics education is when different cultures stress different skill sets. for example, u.s. mathematics education emphasizes demonstration of reasoning and proof processes, while some other cho, lee, & herner-patnode mathematics lesson design for english learners versus non-english learners journal of urban mathematics education vol. 15, no. 1 36 cultures assume mental mathematics and do not necessarily emphasize students showing their thinking processes and reasoning steps (yang & huang, 2014). as another example, the methods of presenting decimals in thousands units by using commas and periods vary in different countries. furthermore, how to do division can differ in other cultures (son & senk, 2010). different metrics systems could further confuse students who are not used to u.s. measurement systems, such as “ounce,” “pound,” “inch,” “feet,” etc. the collective results from these studies indicate that cultural aspects in mathematics education should not be overlooked. rather, they should be actively utilized and integrated into mathematics lessons to help students learn. hence, one of our focused analyses entailed how mathematics, language, and culture intersected in teachers’ mathematics lesson development. research methods for this study, we employed a qualitative descriptive study to seek how a group of in-service teachers approached mathematics instruction design for non-els versus for els. the main research question guiding our study was the following: how does a group of in-service teachers in a tesol graduate program design mathematics lessons responding to learner-specific cases? specifically, how different or similar were their instructional support and procedures for els versus non-els in their mathematics lesson? research contexts and participants the participants of this study were a group of in-service teachers who were already certified in areas other than esol, such as childhood education and secondary level subject areas. they were taking tesol graduate courses for an additional certificate in esol at the time of study. among the participants (n = 23), 14 teachers reported that they were certified in childhood education (grades 1–6), while seven teachers reported that their certification area was in secondary subjects. two teachers indicated that their initial certifications were in the category of “other,” such as bilingual education or special education. twenty participants had received some form of training related to esl from graduate and/or undergraduate courses as well as professional development workshops. in addition, all of the participants reported that they had worked with els. their years of teaching experiences ranged from two to 20 years. all participating teachers in the study were teaching in school districts in inner city settings where els make up about 15% of the entire population. according to 2018 state-wide mathematics assessment data, els performed approximately 31% to cho, lee, & herner-patnode mathematics lesson design for english learners versus non-english learners journal of urban mathematics education vol. 15, no. 1 37 33% lower than their non-el counterparts (new york state education department, 2019). although it is not clearly known to what degree the teachers had learned how to teach els, all of them had worked with els at the time of this study. thus, it is fair to say that these participating teachers had been exposed to cultural and linguistic diversity on a daily basis and were familiar with esl students. it is of note that teachers with more field experience and exposure to els better understand the interconnection of language and mathematics teaching than their counterparts (mcleman et al., 2012). data source and instrument the study was conducted through a case study teaching approach, which allows teacher educators to present targeted experiences that connect the prek–12 classroom and methods courses (jeffries & maeder, 2011; turkan & de jong, 2018). according to biza et al. (2007), teachers are invited to answer highly focused mathematical and pedagogical case study questions that allow them to generate best practice with scenarios similar to what they are facing or will face. thus, the classroom scenario-based case study teaching approach has a clear purpose in that it helps teachers develop a deeper understanding of teaching mathematics to diverse students (turkan & de jong, 2018). teacher educators find that classroom scenario-based cases are a valuable research data collection method because they approximate the situations and students they encounter in real classrooms (erickson et al., 2021; kennedy, 1999). upon irb approval, the participating teachers enrolled in a tesol graduate course that covered how to teach els across content areas, such as mathematics, science, social studies, and english language arts. the participants were asked to respond to two learner profiles with tasks and strategies right before they learned how to support els in mathematics lessons. first, a standard titled “numbers and operations—fractions” was given to them with the following word problem: “if each person at a party will eat 1/4 of a pound of roast beef, and there will be 5 people at the party, how many pounds of roast beef will be needed? between what two whole numbers does your answer lie?” following this, two learner profiles were provided— one for a general education non-el, whom we named “lenny,” and one for an el, whom we called “marcella.” following this, the participants were asked to design the main body of the lesson with instructional strategies and tasks that serve students like lenny and marcella (see appendix a). the word problem that was given within the standard was intentionally created because word problems present opportunities “to study culturally and linguistically responsive mathematics instruction, because of the role of context as well as linguistic complexities inherent in problems” (driver & powell, 2017, p. 43). first, the provided word problem contextualized the mathematics standard that the participants needed to teach. second, we wanted to examine how cho, lee, & herner-patnode mathematics lesson design for english learners versus non-english learners journal of urban mathematics education vol. 15, no. 1 38 the participating teachers handled the challenge of presenting a social situation, in this case a dinner party, and a measurement system that might not be relevant to learners. this particular word problem was not to trick the participating teachers but intended to find out how the teachers responded to the context of the story problem. the nature of the task was diagnostic with the intention of finding what these teachers already knew and what their needs were rather than measuring the impact of instruction. data analysis we adopted inductive content analysis to examine tasks and strategies that the teachers developed to teach a 5th grade mathematics standard on fractions to els and non-els. content analysis has been used “to classify written or oral materials into identified categories of similar meanings” in descriptive research (cho & lee, 2014, p. 3). our approach was inductive in that we drew the patterns and themes through multiple coding processes beginning with open coding as detailed below (cho & lee, 2014; schreier, 2012). thomas (2006) defined the development of categories into a model or a framework with the outcome of an inductive analysis while conveying key themes and processes. the inductive reasoning approaches to open-ended response data involves finding patterns while analyzing the data without testing any pre-existing hypotheses or theories (creswell, 2013; thomas, 2006). this study followed the general inductive approach, coding steps, and the development of key themes processes (creswell, 2015, elliott, 2018; richards, 2015). the final themes of the study were developed through multiple steps (see table 1). during the initial coding process, each of us identified processing codes in teachers’ lesson designs (first-level code). these processing codes were a word or a phrase, such as “home language,” “student interests,” “manipulative,” “explain,” and “stepby-step breakdown.” for example, when teacher #23 responded to lenny, “i would use a number line and manipulatives such as fraction tiles to help lenny solve the problems and master the skill of multiplying with fractions,” our processing code was “manipulatives” for “fraction tiles” and “number line.” in comparison, when the same teacher responded to marcella, “she can join another group in the class in order to be assisted by her classmates who may be on a higher level of english than her, but working collaboratively with her peers will help her even more,” our processing codes were “peer tutoring” and “home language.” second, we, as a group, compared our own processing codes with each other and reached a consensus on a set of categories through discussions. for example, we categorized the processing code “manipulatives” as “instructional resources” and “home language” as “language support.” these categories were “language support,” “cultural support,” “instructional resources,” “instructional procedures,” and cho, lee, & herner-patnode mathematics lesson design for english learners versus non-english learners journal of urban mathematics education vol. 15, no. 1 39 “collaborative learning.” in this phase, we also analyzed frequencies of these patterns. although our initial intention was not to quantify qualitative data, similar recurring remarks afforded us the opportunity to quantify for clear comparison between els and non-els. next, we reread the original data to confirm and/or find further nuanced patterns. for instance, we found “language support” could be divided into general language (english or home language) and mathematical discourse. finally, we developed these patterns into a few themes from two main theoretical perspectives—equity and intersection—as we will describe them in the following sections. as far as credibility of data and analysis is concerned, the participating teachers were assured that their lesson designs would not be graded and completion itself would be considered as participation in class. there were no time and space restrictions in their responses (see appendix b for a single entire example). at the analysis level, the researchers, who specialize in mathematics, esl, and special education, respectively, went through multiple steps of coding individually and together. cho, lee, & herner-patnode mathematics lesson design for english learners versus non-english learners journal of urban mathematics education vol. 15, no. 1 40 table 1 coding steps of lesson design data analysis step 1 step 2 step 3 initial code examples common codes/ categories frequency (non-el vs. el) & percentage emerging themes from equity perspective emerging themes from intersection perspective home language, labeling, directions verbally, numerals, vocabulary, simple words, tiered/leveled texts, pre-prepared notes, sentence starters/stems language support english language; mathematical discourse 2 vs. 15 (9.5% vs. 71.4%) participation in learning expectations and quality of instruction language support cultural support mathematical support student interests/likes, real-life connection cultural support 4 vs. 3 (19% vs. 14.3%) anchor chart, manipulatives, number lines, visual aids, examples/multiple problems, videos instructional resources (digital resources) 13 vs. 10 (61.9% vs. 47.6%) centers/stations, step-by-step breakdown, procedural explanation (step-by-step solution), conceptual explanation, additional time, slower pace, modeling, challenging/higher order thinking, have students explain, student-centered, start with inquires, i do, we do, you do instructional procedures *discovery 3 vs. 0 (14.3% vs. 0%) pairing, grouping, peer tutoring collaborative learning 7 vs.11 (33.3% vs. 52.4%) note. although 23 teachers participated in this study, two teachers who did not design the lessons were removed from the analysis. some teachers adopted multiple resources, such as charts, graphic organizers, and manipulatives. instead of counting these as three separate categories, they were counted as one. cho, lee, & herner-patnode mathematics lesson design for english learners versus non-english learners journal of urban mathematics education vol. 15, no. 1 41 findings the results showed the patterns of “language support,” “culture support,” “instructional resources,” “instructional organization/procedures,” and “collaborative learning.” what follows is our quantified analysis in each category to see how els were supported differently or similarly from non-els. furthermore, results from content analysis of the teachers’ lesson tasks and strategies for els and for non-els will be presented while taking a few supportive examples of the data in order to corroborate our claims. language support findings show teachers chose two types of language support: general language and mathematical discourse. general language support includes translations, simplified english sentences and words, and labeling in a home language. mathematical discourse concerns mathematical reasoning, questioning, concluding, and showing understanding the steps in mathematical problems. more than 70% of teachers in this study adopted translation, labels, sentence starters and stems, and home language use to support marcella. in contrast, about 10% of teachers adopted mathematics discourse for lenny (non-el). for example, teacher #13 stated, “i would provide her [marcella] with a vocabulary list of mathematical terms in both english and spanish, so she can learn key vocabulary in english… mathematics has many cognates in spanish, and therefore i can make connections.” in addition to providing terms in the els’ home language, teachers used sentence stems or starters, as is shown in the following excerpt. for marcella, i would have sentence strips for her that says ____ multiple by _____ equals to __________. i would also break down the story problem into parts guiding marcella through it and modify the language of the story problem (teacher #12). it is noteworthy that teachers who explicitly introduced mathematics vocabulary and discourse to lenny failed to do the same for marcella. for instance, teacher #12 (childhood education certificate) stated in the design for lenny, “…first explain vocabulary words: multiplication, fraction, whole number, product and equation.” another example is teacher #6, who mentioned, “i would have lenny work with a small group to share 3 strategies and engage in active mathematics discourse using domain vocabulary.” the very same teachers, however, decided to provide to marcella “a mathematical dictionary and have her work with a push-in esl teacher to work on her basic knowledge of english.” although mathematical discourse is different from language translation and simple vocabulary instruction, teachers’ lesson design tended to focus on general language support rather than mathematical discourse. cho, lee, & herner-patnode mathematics lesson design for english learners versus non-english learners journal of urban mathematics education vol. 15, no. 1 42 cultural support culturally responsive teaching encompasses integrating students’ experiences, interests, likes, and real-life connections in mathematics lessons. despite this broad view of cultural support in this study, approximately 20% of the teachers integrated the student’s culture into their mathematics lesson for lenny, whereas less than 15% of the teachers mentioned incorporating marcella’s culture. among them, two teachers (#4 & #16) mentioned culture both for lenny and marcella but slightly differently. for example, teacher #4 proposed that “i would create a story with elements that marcella may be familiar with based on my conversations with her about her culture. so maybe instead of using pizza as an object that can be cut into pieces, i could refer to a soft taco or maybe a plantain.” the same teacher used pizza slice examples for lenny. teacher #16 similarly claimed that having real-life examples such as food is a great way to engage marcella in the lesson because she would feel “more involved.” aside from these two examples, teachers did not consider students’ cultural backgrounds overall in lesson development. instructional resources instructional resources include not only traditional hands-on materials, such as manipulatives and drawings, but also digital materials, such as internet interactive activities and instructional video clips. a variety of instructional resources could engage students’ learning and enhance their mathematics learning. in particular, handson approaches coupled with visual aids and multimedia sources help els learn mathematical concepts while they are still developing english language skills (hur & suh, 2012). in the present study, the majority of teachers adopted a variety of instructional resources, such as number lines, manipulatives, visual aids, and anchor/reference charts, in their lesson development. however, only two teachers indicated they would use digital resources (e.g., instructional video clips) for both lenny and marcella. for instance, teacher #2 and teacher #15 both used similar manipulatives and described how they would use them: “she [marcella] would be given fraction bars and divide them into fourths. she will then color in ¼ of them. she will also cut and paste colored squares on ¼ of a fraction bar until she has colored manipulatives to use.” another example came from teacher #18 and regarded lenny: “using a graphic organizer, draw out the fractions that are needed to solve this problem. then make a model with manipulatives once the drawings are complete.” the same teacher stated, “provide manipulatives along with other necessary visuals before beginning the story problem. have students work with a peer to ‘act out’ story problem with manipulatives.” it was obvious that all participating teachers were cognizant of the importance of handson materials when students learn mathematics, whether they be an el or non-el. cho, lee, & herner-patnode mathematics lesson design for english learners versus non-english learners journal of urban mathematics education vol. 15, no. 1 43 an intriguing result, however, is that approximately 62% of the teachers adopted a variety of instructional resources for lenny, while 48% the teachers considered using instructional resources for marcella. although it is expected that marcella would need far more instructional resources to increase comprehensibility of new mathematical concepts, the results indicate that this did not happen. collaborative learning collaborative learning in the form of group or pair work has been promoted for student engagement in learning. learners can accomplish more when they work together with more experienced/advanced peers than when they work independently. an additional benefit for els is they feel less anxiety in group settings than in whole class or individual situations, which yields positive influences on their lesson engagement (takeuchi, 2016). although teachers in this study embraced collaborative learning strategies for marcella more than for lenny (52.4% vs. 33.3%, respectively), their reasoning for implementing collaborative work was not always the same. teachers chose group work for lenny to encourage the sharing of solutions or collaborative problem solving. for instance, teacher #3 stated, “after they have a few minutes to work on their own i would then put them in a group so they can discuss their different answers and strategies they used.” another teacher (#22) proposed using the group activity of genuine collaboration for coming up with a solution rather than for simply sharing ideas: “group must identify the key pieces of information and set up a solution using given information and solve.” on the other hand, the main purposes of collaborative learning for marcella were to provide language support, as teacher #20 put: “i would also have her work with a partner that speaks spanish as well as english if i had a bilingual student in my class who could assist with translating and partake in mathematics discussion with her.” similarly, teacher #23 stated, “marcella can join another group in the class in order to be assisted by her classmates who may be on a higher-level english than her.” these examples confirm that teachers tend to view group work as an opportunity to help marcella with language rather than true collaboration to solve the problem and to share solutions. an additional thought-provoking finding in collaborative learning was peer support with different expectations. for example, two teachers, #10 and #16, mentioned asking lenny to be a tutor to help other students. teacher #10 noted, “he [lenny] will work with another student who may be struggling, explaining steps he took to solve the problem.” in contrast, most teachers viewed marcella as a help receiver in group work, rationalizing that she needed help in language, vocabulary, and content skills. cho, lee, & herner-patnode mathematics lesson design for english learners versus non-english learners journal of urban mathematics education vol. 15, no. 1 44 instructional organization while analyzing the data, we noticed how teachers organized their instruction differently for els versus for non-els. one was the discovery learning approach versus explicit explanations as well as the use of guided scaffolding. guided scaffolding utilizes step-by-step procedures to increase student independence in learning. at least three teachers adopted inquiry-based and discovery learning for lenny, but no teacher attempted to employ a discovery learning strategy for marcella. for example, these teachers began the lesson with an exploratory challenge, as teacher #3 stated: “for this student [lenny] i would use the upside-down mathematics approach where i would present the class with an open-ended question based on fractions.” interestingly however, the same teacher chose teacher-led explicit instruction for marcella: “i would introduce multiplying fractions. i would present marcella with the rule on multiplying fractions using an example.” aside from the three exceptions that adopted discovery-oriented instruction for lenny, there was a pattern that teachers chose to explicitly explain and model mathematical concepts rather than encourage learners to discover them. discussion while we analyzed how the participating teachers designed a mathematics lesson when instructional cases were given for lenny (non-el) and for marcella (el) in their instructional support, approaches, and procedures, we discovered a few salient themes drawn from the emerging patterns. they include different quality of instruction and teachers’ different expectations for els versus non-els as well as language support that has little connection with mathematical discourse and culture for els. in this section, we will discuss our findings around themes while comparing them with previously published studies. quality and expectation from an equity perspective participation in learning. collaborative learning has its own merits in that students often feel safe and confident in group settings and improve engagement and eventually understanding (nebesniak & heaton, 2010). however, the patterns of participation and engagement, which were based on teachers’ expectations, paints marcella as a passive learner. for example, some teachers viewed lenny (non-el) as an active learner and contributor to class, as he could serve as a tutor to other students. in contrast, they perceived marcella (el) as needing assistance. it is of note that marcella has the grade-level mathematics and literacy skills in her first language. however, her limited english skills put her in a passive learning role, reducing her participation and power. els’ mathematical contributions are not considered, and cho, lee, & herner-patnode mathematics lesson design for english learners versus non-english learners journal of urban mathematics education vol. 15, no. 1 45 their passive role is cemented as a receiver of help (turner et al., 2012). although our study did not specifically investigate how marcella’s gender interplays with her being an el, teachers may inadvertently underestimate female students’ abilities under the assumption that they may not be good at mathematics (see amador, 2018; clark et al., 2014). quality of mathematics instruction. there were very little in-depth explanations of mathematics strategies for teaching fractions with conceptual models, unlike moschkovich’s (2015) view of mathematical academic literacy that encompasses proficiency, practice, and discourse. although teachers adopted instructional strategies such as an anchor chart, fraction strips, and modeling, their lessons focused more on resources rather than how to use the resources to teach the mathematics concepts. for example, teacher # 16 had a higher expectation for lenny: “since he is at grade level, giving him challenging word problems or higher order thinking problems, will challenge him into developing higher skills in that content area.” although the same teacher acknowledged marcella’s strong first language literacy skills and average mathematics skills, her instructional strategy was to use translation until marcella became comfortable. very similarly, another teacher, teacher #20, showed she had high expectations for lenny and stated, “i would challenge lenny by asking him to represent his answer on a line plot and also in some type of picture or chart.” these examples imply that the participants’ quality of mathematics instruction suffered from different expectations of what lenny and what marcella can do. this finding confirms abedi and herman’s (2010) study that reported that els had lower levels of learning opportunities compared to non-els, which, in turn, negatively influenced their performance. teachers often adapt and modify lessons for students with special needs, including els. the problem, however, is that they tend to choose something “easier” for them and “provide fewer opportunities for students to connect ideas and build knowledge—thereby impeding, not supporting, learning” (van de walle et al., 2019, p. 26). our findings are consistent with previous research that has found limited meaningful learning opportunities for els. language-heavy support from an intersection perspective from a perspective that mathematics education for els intersects with culture and language (aguirre & del rosario zavala, 2013; van de walle et al., 2019), we found that most teachers employed translation-based and simplified language support for marcella. language proficiency of els is different from mathematical knowledge and discourse (turkan & de jong, 2018). multiple ways to scaffold mathematics discourse include using questioning, repetition, elaboration, think-aloud, and consistent incorporation of mathematical discourse using modeling and elicitation rather than simple vocabulary instruction or translation (banse et al., 2017; hansen-thomas, 2009). song and coppersmith’s (2020) study also found that teachers who had cho, lee, & herner-patnode mathematics lesson design for english learners versus non-english learners journal of urban mathematics education vol. 15, no. 1 46 completed a professional development training on how to support els still did not engage els with mathematical discourse in their classroom applications. what these studies commonly point out is that simple translation or english language proficiency itself fails to view the complexity of mathematical discourse that could be taught within the mathematical conceptual understandings. yet, the teachers of this study often chose simple translations and labeling to support els and how mathematical concepts, culture, and language are interlinked in mathematical problem solving was little considered. another compelling finding was the lack of cultural support and presence in mathematics problems and instruction. to our surprise, none of the participants raised questions about the word problem, the context of which was a dinner party, or the fraction example being based on a pound of roast beef. this word problem is not relevant to students’ cultural backgrounds or interests. furthermore, non-u.s. countries use metric-systems that do not include pounds or ounces. however, no teachers pointed out these discrepancies. conclusion this study investigated how a group of teachers in a tesol graduate program developed strategies and tasks in response to two learner profiles—el and non-el. teachers more actively reasoned why they chose certain instructional strategies in the case of marcella and cared about providing language support in their lesson designs. however, from the equity perspective that concerns equal meaningful learning opportunities for els, their lessons fell short of creating these opportunities. furthermore, their heavy language support with little mathematical conceptual development or cultural integration proved that the teachers did not have a full understanding of the intricate interconnections between mathematics, language, and culture despite their exposure to esl training and experience working with els. limitations and implications this study has a few limitations. the number of participants is small with convenient sampling, which makes the findings difficult to generalize or translate into other contexts. in addition, the teachers were not instructed to design an actual lesson plan but to respond to the case with instructional strategies and tasks. this might have hampered their lesson development. despite the merits of the case approach that could be used in a methods course to approximate the actual teaching contexts, there might be discrepancies between what teachers plan to do and what they actually do in a real classroom. in spite of these limitations, our study shed light on a few critical issues that teacher education programs of mathematics education and esl education should cho, lee, & herner-patnode mathematics lesson design for english learners versus non-english learners journal of urban mathematics education vol. 15, no. 1 47 consider. as our findings show, teachers tended to have different expectations for els as passive learners and mistake lack of language proficiency for lack of mathematical understanding. this prevented genuine engagement opportunities. simple translations and providing native language support are not enough for els to improve mathematical discourse. high expectations and quality mathematics instruction should be ensured while educating traditionally marginalized students. conspicuously, the lack of cultural integration into 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possible story problem: if each person at a party will eat 1/4 of a pound of roast beef, and there will be 5 people at the party, how many pounds of roast beef will be needed? between what two whole numbers does your answer lie? 1. lenny is a 5th grade student who is performing at grade level in mathematics. 2. the english language learner, marcella, is originally from the dominican republic. the student is from a middle-class background. she arrived in the u.s. three weeks prior to enrolling in the 5th grade. she has had formal schooling in her native country and has developed strong literacy skills in her first language, spanish. marcella appears to have average mathematics skills, but her english is at a kindergarten level. please respond to the following questions as best as you can. 1. i am certified in____________. 2. i have taught for____________years altogether. 3. i have had english learners in my classes. ___yes____years ___no 4. i have taken (workshops, undergraduate courses, graduate courses) to learn about instructional strategies for english learners (circle all that apply). 5. please feel free to make any comments or your thoughts on this task. thank you for your participation! cho, lee, & herner-patnode mathematics lesson design for english learners versus non-english learners journal of urban mathematics education vol. 15, no. 1 52 appendix b instructional scenario you want to teach a math lesson on multiplication with fractions to 5th graders. please design a main body of the lesson such as instructional strategies and tasks that serve students like lenny and marcella. standards: grade 5, number and operations fractions apply and extend previous understandings of multiplication to multiply a fraction by a whole number. a. understand a fraction a/b as a multiple of 1/b. for example, use a visual fraction model to represent 5/4 as the product 5 × ( 1/4), recording the conclusion by the equation 5/4 = 5 × ( 1/4) or 5/4 = 1/4 + 1/4 + 1/4 + 1/4 + 1/4 5 x 1/4 = ? possible story problem: if each person at a party will eat 1/4 of a pound of roast beef, and there will be 5 people at the party, how many pounds of roast beef will be needed? between what two whole numbers does your answer lie? 1. lenny is a 5th grade student who is performing at grade level in mathematics. please design a main body of the lesson such as instructional strategies and tasks that serve students like lenny. i would teach lenny this concept by showing several examples on the board, and then allow him to use a reference sheet with the math formula to assist him with independent work. his independent work would be similar fractions examples for which he would complete on his own. some of these problems would include word problems, and lenny would be asked to explain his work and thinking with words at the end of the word problem. i would challenge lenny by asking him to represent his answer on a line plot and also in some type of picture or chart. 2. the english language learner, marcella, is originally from the dominican republic. the student is from a middle class background. she arrived in the u.s. three weeks prior to enrolling in the 5th grade. she has had formal schooling in her native country and has developed strong literacy skills in her first language, spanish. marina appears to have average mathematics skills, but her english is at a kindergarten level. please design a main body of the lesson such as instructional strategies and tasks that serve students like marcella. cho, lee, & herner-patnode mathematics lesson design for english learners versus non-english learners journal of urban mathematics education vol. 15, no. 1 53 i would show marcella several examples of these fractions problems on the board. i would provide marcella with a fractions vocabulary list with common words translated in spanish since she has strong spanish literacy skills. she could refer to this sheet during the lesson to help her understand the discussion and teaching. i would have directions written on the board in spanish and english before giving marcella the independent math activity. i would make sure that her worksheet was differentiated with sentence starts and a word bank for her to use when describing her math work for the words problems. i would also translate the word problems into spanish to support her in understanding what is expected of her within the problem. since marcella seems to be an average math learner, i would ask her to complete similar examples to the rest of the class, but focus on differentiating for her language needs. i would also have her work with a partner that speaks spanish as well as english if i had a bilingual student in my class who could assist with translating and partake in math discussion with her. feel free to share any thoughts and opinions. copyright: © 2022 cho, lee, & herner-patnode. this is an open access article distributed under the terms of a creative commons attribution-noncommercial-sharealike 4.0 international license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. journal of urban mathematics education december 2009, vol. 2, no. 2, pp. 6–11 ©jume. http://education.gsu.edu/jume candace l. williams is a mathematics teacher at redan high school, stone mountain, ga 30088, and part-time doctoral student of mathematics education in the department of middlesecondary education and instructional technology, in the college of education at georgia state university, atlanta, ga 30303; email: cwilliams136@student.gsu.edu. public stories of mathematics educators  my intimacy with pedagogy of the oppressed candace l. williams redan high school in [cultural invasion], the invaders penetrate the cultural context of another group, in disrespect of the latter‘s potentialities; they impose their own view of the world upon those they invade and inhibit the creativity of the invaded by curbing their expression. – paulo freire, pedagogy of the oppressed during my undergraduate studies, there was a course that was required of all freshmen for completion of any degree program. the class was very large and met in an auditorium-styled seating lecture hall. psychology 101 was considered one of the basics, and it was unavoidable. there was only one professor that taught the course during my first semester, a tenured member of the faculty, with a reputation for being really tough. on the first day of class, he began his lecture with a look at how the world “really” is. he pontificated about which race of people in the class were genetically inclined to do the best, and which ones would struggle, bordering failure. he pointed out who was more inclined to lead fruitful lives, and how many of us would not be at the university by the end of the semester. we heard of how some of our cultures forbade us to excel in certain aspects of life, and who those persons were that would ultimately be successful in whatever they endeavored, adding ever so often quotes and citations from literature that supported these notions. after grunts and sighs grew in the lecture hall, he proclaimed, “this is not my opinion, these are the facts.”  lou matthews, in his editorial in jume, 2(1), argued that one of the greatest challenges for mathematics educators has been in defining a people-centric mathematics education, claiming that to do so would require that we begin to tell our stories in the face of perplexing times in urban education. the ―public stories of mathematics educators‖ section of jume is a newly created section to provide an intellectual space for k–16 urban mathematics teachers and teacher educators to tell their stories as they reflect on and transform their pedagogical philosophies and practices and, in turn, the opportunities to learn for the students they serve. williams public stories journal of urban mathematics education vol. 2, no. 2 7 he then began to read to the class from a storybook, an american childhood classic, the little engine that could. stressing to the class that the small engine was pressing on to deliver toys to the good little boys and girls, he then turned the text around to show the class that the children illustrated in reference were white, blonde-haired, and blue-eyed. many of the minority students left at this point. fear of leaving and being perceived as oppositional to his authority, i kept my seat and continued to listen attentively. in a setting of strangers, before any coursework commenced, i thought, “why was i left out of the story?” most important of all though, i thought, “i am definitely going to fail this class.” he aforementioned and what follows is my own, personal story—my counterstory (solórzano & yosso, 2002), so to speak. i went to every class, and studied especially hard. after the first exam, i made an appointment with the professor to discuss my grade. i was extremely nervous as i approached the closed door and knocked. his voice commanded that i enter, and i pushed open the door to find him outside on a small terrace, smoking a cigar. before i uttered a word, he said, ―let me guess. you‘re here about your exam.‖ my voice cracked, ―yes,‖ as he motioned me to enter. he began with explaining how the first exam always identifies the students who are ill-prepared, and alluded to his first day of class lecture. i nodded in agreement, and then pulled the test from my black, mesh jansport book bag to reveal a large, red ―a‖ at the top. looking stunned, he asked, ―well, what are you here for?‖ i replied, ―i‘m here to ask that you consider revising your lecture for the first day of class.‖ as i reflect on the lecture given that first day of class, having been on campus at the university for only a few days, i remember how i had never been in any space like that before. prior to taking the psychology course, i had excelled in an environment that deemed me ―at-risk‖ and ―underprivileged.‖ i was a product of an urban, lower-classed, minority upbringing, being schooled in centers where there were poor resources, dilapidated conditions, and second-rate educational opportunities. i had ultimately succeeded to become a part of an ―elite‖ class of students at a highly selective, prestigious, private university, and here i was, being reminded that it was a huge mistake. as i sat there, at an institution of higher learning that statistically reported a total of less than 3% minority students (non-white), being taught in a massive lecture hall on the first day of class that my existence was doomed due to the very nature and forces of oppression, i felt defeated for the first time in my life. not only having my utter existence compared to one‘s interpretation of a children‘s storybook but also having that professor ground his theory in supposed fact, reaffirming himself in the notion that certain minorities were fashioned to ―run faster, t williams public stories journal of urban mathematics education vol. 2, no. 2 8 jump higher, and be physically stronger,‖ i sat overcome, broken-spirited, and crushed under the fatal blow of oppression. my psychology professor‘s ―task …to ‗fill‘ the students with the contents of his narration‖ (freire, 1970/2000, p. 71) had succeeded, and he had undoubtedly ―us[ed] science…as [an] unquestionably powerful instrument for [his] purpose: the maintenance of the oppressive order through manipulation and representation‖ (p. 60). how dare he? this professor had only set his eyes upon this 300-plus class approximately 20 minutes before we were all the buttocks of a classic american children‘s story, ironically of which was supposed to confirm that hard work and optimism ends in success for all, reifying the american dream. had i just been punk’d? was this classroom experience some mind trip inflicted by a coy professor of psychology, whose ulterior motive was to simply challenge our thinking and charge us to reason more critically than what we absorbed on the surface? had this professor been an advocate of paulo freire‘s ―problem-posing education‖ (p. 79), where he would infuse a critical discourse of teaching methodology through intentionality in order to promote communication and embody consciousness in his pedagogy? as i want to believe that his teaching methods were purposefully driven by helping every student in his class to progress toward success, it was sadly not so. unfortunately, after his response in my private meeting with him regarding the first test, after he constantly questioned my upbringing and what my parents had done for a living, i quickly knew that he had not intended to ―transform the world‖ (freire, 1970/2000, p. 88), but only to inflict his oppressive nature and assert his authority. the result was a sordid dehumanization; something that made me, and the other minority students, feel less than deserving to be amongst the others. what i now understand of my feelings in that moment is unsettling because what he had successfully done was manipulate the minds of the students in a way that made us question our own identities. how often do educators, intentionally or unintentionally, socially manipulate students? if the students in my class did not band together, and ―cut the umbilical cord of magic and myth which [bound us] to the world of oppression‖ (freire, 1970/200, p. 175), we would surely all be persuaded by him and his views—that we were less than human. i agree with freire that ―no one can be authentically human while he prevents others from being so‖ (p. 85); therefore, the audacity of my psychology professor to even remotely preclude our existence was adversarial to the educational opportunities that he was bound to keep as a professor of the university. dehumanizing others in the name of academia is not only detrimental to a learner‘s confidence but also it may cause lasting, more serious developmental issues. an educator can either enhance or impede a learner‘s value of knowledge, and that is the most serious result of oppression. williams public stories journal of urban mathematics education vol. 2, no. 2 9 like freire, my position in critical theory and the importance of having an awareness of teaching (mathematics) for social justice is posited in my lived experiences. it happens to occur in the space that freire most eloquently demands we reside in if we want to ―apprehend and comprehend the object of knowledge‖ (macedo, 2000, p. 19). as i continue to develop my own personal theory on certain issues of inequity in my mathematics classroom, i am constantly reminded of how i was made to feel in that psychology course. at the most critical time in my academic development, having been met with the task of transitioning not only academically into a foreign world but also being surrounded by others who looked and sounded different from me, i needed a sense of solidarity. my professor had obviously enacted the role of the oppressor, and refused us the necessary dialogue to create an effective learning environment. in response to the mutters of disagreement, he continuously used his own authoritarian, oppressive pedagogy and proved that he indeed confused his ―authority of knowledge with his…own professional authority, which …he [set] in opposition to the freedom of [his] students‖ (freire, 1970/2000, p. 73). like the others, i would not stand up and walk away from the deliberate, classicistic oppression. my race, gender, or socioeconomic status would not perpetuate ―that certain groups in any society are privileged over others, constituting an oppression that is most forceful when subordinates accept their social status as natural, necessary or inevitable‖ (crotty, 1998, p. 158). the apparent lack of cohesiveness was present in my classmates‘ shared cause, but i would not allow the professor to appropriate my freedom of creating dialogue with him about it. in the moment i stepped into his office, i constituted what freire (1970/2000) has professed as the ―dialogical character of education‖ (p. 93). he states, ―without dialogue there is no communication, and without communication there can be no true education‖ (pp. 92–93). when confronting my oppressor, i not only possessed the liberating feeling of having proved that his theory was wrong about me but also i had engaged him in the act of education as well! even if he did not agree with my own personal stance, or suggested pedagogical practice for his course in the future, i am inclined to believe that in that instance, i provided a lasting and life-changing perspective for him. i am immensely baffled by the fact that my past lived experience has been revealed to me, now more than 10 years later, as an act of emancipation. if i had read freire‘s (1970/2000) pedagogy of the oppressed back then, it would have all made perfect sense. i would have recognized that what the professor was saying to me and the class was utter close mindedness, probably a result of his own oppressive upbringing, and i could have been comforted by the proposition that if students are not able to transform their lived experiences into knowledge and to use the already acquired knowledge as a process to unveil new knowledge, they will williams public stories journal of urban mathematics education vol. 2, no. 2 10 never be able to participate rigorously in a dialogue as a process of learning and knowing. (macedo, 2000, p. 19) in some ways, my not-knowing back then serves as my euphoric knowing today. critical pedagogy and teaching (mathematics) for social justice has unveiled many things for me, but none more lasting than being able to ―read the world‖ with my experiences. in a strange way, my education in the psychology course has affected the way i have internalized the work of freire (1970/2000) in pedagogy of the oppressed. i think it is because of how freire conceptualizes the pedagogy of the oppressed: a pedagogy which must be forged with, not for, the oppressed (whether individuals or peoples) in the incessant struggle to regain their humanity. this pedagogy makes oppression and its causes objects of reflection by the oppressed, and from that reflection will come their necessary engagement in the struggle for their liberation. and in the struggle this pedagogy will be made and remade. (p. 48) freire‘s work is timeless, and i have been truly transformed through it. my teaching practice and philosophy of education continue to be transformed, and with a look at this stance on critical pedagogy and an awareness of teaching mathematics for social justice (see, e.g., gutstein, 2006), my lived experience has made the text pedagogy of the oppressed exceedingly intimate. the curriculum that we use in our mathematics classrooms often categorizes the approach to the education of youth as a list of ―best practices,‖ where teachers are expected to follow the script in order to produce the highest quality, most efficient product: the autonomous, self-efficacious being. this type of curriculum model is problematic in that it alienates students that do not identify with the current models, and who are often ill-equipped to assimilate into the ―normal,‖ ―mainstream‖ social constructs. many of the individuals who are struggling with these issues are often students of color, or those who come from urban, minority environments. venezuela (1999) refers to this curriculum approach to learning as subtractive schooling (as seen through my experience in the psychology course) and iterates that these policies and practices are designed to divest certain students of their culture and language. in lieu of this best-practice approach, a critical pedagogical perspective could be useful in breaking the cycle of the one-size-fit-all (reyes, 1992) recipe for learning. exercising unconventional teaching practices, as suggested by freire (1970/2000), where the students enact life in the classroom that resembles their lived experiences could also encourage the oppressed to feel (and become) more empowered. ―by encouraging [my students] to question, investigate, and interpret their experience of the world‖ (sylvester, 1994, p. 311), my students and i, collectively, have begun to seriously explore pedagogical possibilities that might con williams public stories journal of urban mathematics education vol. 2, no. 2 11 tribute to transforming existing oppressive and dehumanizing structures within my mathematics classroom into liberating and humanizing structures. references crotty, m. (1998). critical inquiry: contemporary critics & contemporary critique. in the foundations of social research: meaning and perspective in the research process (pp. 139–159). thousand oaks, ca: sage. freire, p. (2000). pedagogy of the oppressed. (m. b. ramos, trans., 30th ann. ed.). new york: continuum. (original work published 1970) gutstein, e. (2006). reading and writing the world with mathematics: toward a pedagogy for social justice. new york: routledge. macedo, d. (2000). introduction. in p. freire, pedagogy of the oppressed (30th ann. ed., pp. 11– 27). new york: continuum. reyes, m. de la luz. (1992) challenging venerable assumptions: literacy instruction for linguistically different students. harvard educational review, 62(4), 427–446. solórzano, d. g., & yosso, t. j. (2002). critical race methodology: counter-storytelling as an analytical framework for education research. qualitative inquiry, 8, 23–44. sylvester, e. s. (1994). elementary school curricula and urban transformation. harvard educational review, 64(3), 309–331. valenzuela, a. (1999). subtractive schooling: u.s.-mexican youth and the politics of caring. albany, ny: state university of new york press. journal of urban mathematics education december 2018, vol. 11, no. 1&2, pp. 152–174 ©jume. http://education.gsu.edu/jume pamela l. paek is a research associate at the charles a. dana center, university of texas at austin, 2901 north ih-35, suite 2.200, austin, tx 78722; e-mail: pamela.paek@mail.utexas.edu. her interests include developing practitioner–researcher partnerships, secondary mathematics, the impact of policy and practice on teaching and learning, and issues of equity and access in education. practices worthy of attention: a search for existence proofs of promising practitioner work in secondary mathematics1 pamela l. paek university of texas at austin the goal of the practices worthy of attention (pwoa) project was to surface innovative practices currently in use by urban schools and districts that show promise of improving students’ secondary mathematics performance. each school and district explored has a different perspective and a unique set of practices in place to improve secondary mathematics achievement. the goal of this project was not always to discover innovations in how practitioners address similar issues, but rather to document what practitioners are doing to strengthen secondary mathematics education. thus, although the practice highlighted might be commonplace, the specific structures and strategies being employed by the school or district to implement it are worthy of attention. a cross-case analysis of the 22 practices revealed two main categories: raising student achievement and building teacher capacity. keywords: secondary mathematics, student achievement, teacher capacity n the last half of 2006, i led a national search for practices in urban schools and districts that show promise—on the basis of early evidence and observation—of increasing student learning in secondary mathematics. i call these “practices worthy of attention” (pwoa), and my work on them had three overarching goals: 1. to better understand existing initiatives, innovations, and programs that are being used to improve secondary mathematics teaching and learning around the country, and mark these for further scientific inquiry. 2. to identify common themes in these practices that can be used to strengthen student achievement in urban systems across the country. 3. to provide research support to help the practitioners more rigorously evaluate how well their practices are working, which in turn can help to strengthen their methods of operation. 1 originally published in the inaugural december 2008 issue of the journal of urban mathematics education (jume); see http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/6/1. i http://education.gsu.edu/jume mailto:pamela.paek@mail.utexas.edu http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/6/1 paek practices worthy journal of urban mathematics education vol. 11, no. 1&2 153 theoretical framework and connection to the literature recent federal and state education policies call for a substantial increase in the breadth and depth of mathematical knowledge that students must acquire in order to graduate from high school. for example, a growing number of states that once required knowledge only of middle-school-level mathematics for high school graduation have, over the past 5 to 7 years, begun to require that all students demonstrate mastery of algebra i and geometry content (center on education policy, 2006). to give students opportunities to take higher-level mathematics courses in high school, which will better prepare them for mathematics in their postsecondary lives, many states and districts have policies encouraging students to take algebra i in the 8th grade. these policies have had an effect: the national assessment of educational progress (naep) shows that in 2000, only 27% of eighth-grade students nationwide took algebra i, whereas by 2005, 42% of eighth graders nationwide had taken algebra i (mathews, 2007). outside of policy requirements, improving student access to and achievement in mathematics is important because students’ performance in middle school and high school mathematics correlates with their overall academic success in high school and beyond. the national educational longitudinal study (nels) indicated that students who took rigorous high school mathematics courses were much more likely to go to college than those who did not take such courses (u.s. department of education, 1997). research suggests that specific mathematics courses, like algebra i, serve as gatekeepers to more advanced mathematics courses and can affect mathematics enrollment and achievement in high school, which in turn affects enrollment in college and completion of a 4-year degree (adelman, 2006; ma, 2001). the nels study showed that 83% of students who took algebra i and geometry enrolled in college within 2 years of graduating from high school, whereas only 36% of those who did not take these courses enrolled in college. therefore, understanding the factors that contribute to improved student learning in algebra i and a successful transition to geometry is a critical first step toward increasing the postsecondary opportunities available to students. unfortunately, few school districts in the nation have the capacity to help their students meet these rigorous mathematics requirements. nationaland statelevel reports document a critical shortage in the supply of appropriately trained and certified mathematics teachers as well as a high rate of attrition among those teachers, especially in urban areas (national science board, 2006). many secondary mathematics teachers lack deep knowledge of the mathematics content they are expected to teach (barth & haycock, 2004; massell, 1998). in fact, ingersoll (1999) found that a third of all secondary school teachers of mathematics nationwide had neither a major nor a minor in mathematics. moreover, research shows inconsistencies in instruction across classrooms within the same district and even within the paek practices worthy journal of urban mathematics education vol. 11, no. 1&2 154 same school. teachers interpret the same instructional ideas in various ways (marzano, 2003; stigler & hiebert, 1998, 1999) and accordingly make independent decisions about whether to ignore, adapt, or adopt policymakers’ recommendations for instruction (spillane, reiser, & reimer, 2002). in urban districts faced with these and other difficult issues—including heavy turnover among administrators, administrators who do not understand what is needed to support a high level of mathematics learning, and low expectations for student performance from both teachers and administrators—mathematics instruction has proven very difficult to improve (bamburg, 1994; beck-winchatz & barge, 2003; tauber, 1997). as a result, all too often, students in urban school districts are not given adequate opportunity to enroll and succeed in challenging mathematics courses in their secondary years (national science board, 2006). the pwoa project was inspired by these challenges and by the need for education systems to invest resources wisely. thus began the work of identifying practices in secondary mathematics education that might merit further attention, greater investment, and wider dissemination. defining practices worthy of attention research on pwoa differs from other work describing “best practices” or “promising practices” in that the pwoa work starts from where schools and districts presently are, focusing on work and ideas currently in progress. starting by investigating practices that have not yet been identified as “best” or “promising” through specific national criteria, such as those of the what works clearinghouse or the national center for educational achievement, means that there is often little or no documentation of how a practice is being implemented and scarce evidence of the practice’s impact or effectiveness. therefore, the first step in researching a practice is spending time with the practitioners in each school or district to discover the theory-of-action behind the practice and to document the implementation of the practice and the evidence of its effectiveness so far. this step not only provides a historical record of activities but also honors the work, giving practitioners a chance to see their ideas and efforts documented in a way that shows a picture of the work to date. this step also provides a starting point for researchers to continue to work with practitioners to better measure the effects of the practices on secondary mathematics teaching and learning. developing methods to accurately and comprehensively measure and assess the impacts of these practices on mathematics teaching and learning helps to meet a current need of urban districts and schools. ironically, just as policymakers and district leaders are looking to raise the evidentiary standard for adopting a school improvement practice, the size of district offices—including that of their research and evaluation staff—is being greatly curtailed, or staff is being diverted to deal with paek practices worthy journal of urban mathematics education vol. 11, no. 1&2 155 the reporting exigencies related to no child left behind act of 2001 (nclb).2 thus, many urban districts do not have the staff and financial resources to clearly determine what data are needed by each person in the system and how such data can be used. most important, these districts have not yet worked out how to translate the knowledge gained from the data into effective decision making at each level of the education system. methods initial selection of programs, schools, and districts the first step in the pwoa project was to interview administrators and teachers at schools and districts across the united states that embody diverse educational systems but that primarily serve students classified as economically disadvantaged and/or as racial and ethnic minorities. i contacted networks of mathematics leaders and teachers known to staff at our institution and partner institution and drew on my knowledge of schools and districts to develop an initial pool of administrators and teachers to interview regarding practices in their schools and districts that were potentially worthy of further examination. protocol for initial interviews at the june 2006 urban mathematics leadership network (umln)3 meeting, i used a four-question protocol to interview mathematics administrators from the 12 participating umln districts. the interview protocol included the following definition of what constitutes a practice worthy of attention: a practice worthy of attention (pwoa) is a practice being used in your district that shows promise of improving mathematics education within your district and across districts. the pwoa i seek specifically look at the grade range of middle school through college. a pwoa is an example of how you have solved problems or challenges your district faced, ideally with tools that measure the effects of change. the first question asked the administrators which practices they would nominate for their district. the second asked what types of documentation of the practice existed (e.g., training protocols or documents describing school or district initiatives) and what evidence was used to show the effectiveness of the practice (e.g., 2 no child left behind act of 2001, public law 107-110, 20 u.s.c., §390 et seq. 3 umln serves as a vehicle for rapid dissemination of advances and promising practices, and enables state mathematics leaders and the leaders of large urban districts to work together to better align their mathematics improvement efforts and thus raise student achievement. paek practices worthy journal of urban mathematics education vol. 11, no. 1&2 156 school/district evaluations of students and/or teachers, third-party evaluator reports, improvement in test scores). the final two questions were logistical, concerning scheduling a site visit and establishing a contact person at the school or district. follow-up phone interviews with umln district staff were conducted with two main goals in mind: (1) to get more details about the nominated practice, including documentation or evidence of effectiveness available to date, and (2) to schedule a site visit. on the basis of these interviews, eight practices were chosen for further investigation. for non-umln schools and districts, most initial interviews were conducted by phone, although in a handful of cases, i was able to learn about the practices by attending presentations on them at conferences. the protocol for these phone interviews was a combination of the two protocols already discussed. ultimately, i gathered information on about 30 programs, schools, and districts, and scheduled site visits with 22 of them. the remaining eight were not followed up on either because the practice did not fit the goals of the project or because the site did not respond to requests for a visit. site visits and profiles of the practices i visited most of the 22 sites to develop a fuller picture of how the practices were actually being implemented and evaluated. during most of these visits, i attended a professional development workshop centered on the practice being studied; this allowed me to get more detailed information about the practice by witnessing how schools and districts were explaining and teaching it. the visits also included time to talk further with the person interviewed on the phone and the opportunity to gather any materials related to the practice. i also had informal, face-toface conversations with other staff members to learn what they thought about the practices. for a few sites, an actual visit was not feasible, but enough information about the sites’ practices was available to write a profile, with feedback from the district or program to ensure that the profile correctly reflected the practice. practices that exemplify these categories are described next in two separate cross-case analyses, with snapshots of each practice. results on the basis of the site visits, the interviews with teachers and campus and district leaders, and the documentation of the practices, i concluded that the innovations within the practices could be classified into one of two main categories: (1) approaches to raising student achievement and improving student learning in mathematics, and (2) approaches to increasing teacher capacity. paek practices worthy journal of urban mathematics education vol. 11, no. 1&2 157 raising student achievement through academic intensification all of the schools, districts, and programs profiled in this study have increased their expectations for student achievement, but some of them focused particularly on academic intensification strategies to help students meet the higher expectations. the types of practices that emerged in support of academic intensification include: building summer bridge programs, requiring and supporting more rigorous mathematics courses, and providing intense and ongoing support throughout the school day. summer bridge programs two of the practices deemed worthy of attention involve summer bridge programs, which help students transition from middle school to high school mathematics: academic youth development (ayd) initiative and step up to high school (a chicago public schools program). these programs are not remedial programs; rather, they focus on developing problem-solving skills that form a foundation for success in algebra i. both programs are based on the demonstrated efficacy of social interventions on student engagement and academic success. step up to high school, for example, models its format on the emerging scholars program, a college-level program developed to improve minority and female participation in mathematics. academic youth development (ayd) is an algebra i readiness program being implemented by many urban districts in the united states (e.g., chicago, atlanta, new york city) that focuses on helping students better understand content by presenting it from multiple perspectives and applying it in real-life situations. at the heart of ayd is a 3-week transitional summer school and yearlong follow-up program. rather than focus on the behavior of all students, the initiative focuses on the beliefs, attitudes, and behavior of a cadre of student allies upon whom the algebra teachers can rely to model respectful engagement and academic success and thus help shape the classroom culture during the regular school year. teachers nominate for the program students who not only are at risk of failing a future algebra i course but also who have good attendance and show potential leadership skills. in addition to mathematics problem solving, ayd concentrates on teaching students persistence and giving them the power to be in charge of their own learning. for instance, students who view intelligence as a factor that can be improved with learning and habits of mind are more likely to persist through initial failure (dweck, 2002). ayd gives students information about the changing nature of intelligence and encourages them to see failure not as a sign that they cannot learn, but as a signal to change strategy. step up to high school, in the chicago public schools, is a 4-week literacy and mathematics program for students in the summer before their ninth-grade year. paek practices worthy journal of urban mathematics education vol. 11, no. 1&2 158 step up targets students who are likely to be overlooked by other programs—their low test scores indicate that they are at risk for academic failure as they transition into high school, but their scores are not quite low enough for them to be placed automatically in other academic support programs. in addition to building the academic skills in reading and mathematics that are key to high school success, step up focuses on helping students build teacher– student relationships and student–student relationships around shared academic interests. step up includes orientation seminars and activities, information about high school resources, and discussions of study skills, such as organization and time management. students attend step up at the high school they will attend and are taught by teachers, who teach at that school in the regular academic year, ideally by the teacher who will be their first-year algebra teacher. this arrangement gives the students the opportunity to meet teachers and classmates before high school begins and to learn to navigate through their new physical surroundings. both ayd and step up to high school show promise for improving teachers’ understanding of student learning processes and for supporting students’ mathematical learning and academic engagement. preand post-surveys in both programs show gains in students’ confidence about their ability to do well in challenging academic courses. more rigorous course requirements three sites profiled in this study set specific course completion goals for their students and then backward-mapped the curriculum to better prepare students on the strands and topics they would later be required to know. each site also found ways to support students and help them do well in the more advanced courses. el paso collaborative for academic excellence (epcae) has built and implemented a cohesive k–16 mathematics program for all 12 of the school districts it serves in the greater el paso, texas area. epcae leaders realized that if students could successfully complete algebra ii in high school, they could usually avoid remedial mathematics courses in college and enter college algebra fully prepared. large-scale collaborative effort: the 12 districts that epcae serves collaborate with the local community college, the local 4-year university, and the entire el paso community in an effort to achieve coherence in their curricula, promote success for students past high school graduation, and establish a common vision for a k–16 effort. curriculum alignment: epcae formed a k–16 mathematics alignment initiative composed of mathematics educators—elementary, middle, and high school teachers and college and university faculty—who spent 2 ½ years producing a curricular framework that aligned high school and first-year college mathematics. this group then backward-mapped the curriculum to prepare students for successfully completing algebra ii before high school graduation. after the curriculum frame paek practices worthy journal of urban mathematics education vol. 11, no. 1&2 159 works were developed, epcae provided teachers with professional development to use the frameworks as the foundation for algebra ii in high schools. grant high school in portland, oregon set the goal of having all students pass geometry by their sophomore year of high school. the school’s mathematics teachers set this goal themselves when they became frustrated with what seemed like two schools within one building—one in which students who were predominantly racial or ethnic minorities took the pre-algebra courses, and another in which predominantly white students took the precalculus courses. the teachers felt that this unequal access to higher-level mathematics courses would limit some students’ postsecondary opportunities. four teachers developed an intensive mathematics program, and the school started a freshman academies program to help students transition successfully into high school. intensive mathematics program: grant’s intensive mathematics program is for students who enter high school behind in mathematics. teachers intensified mathematics instruction by providing double periods of mathematics for 2 years, in effect giving the students 3 years of mathematics—pre-algebra, algebra i, and geometry—in just 2 years, beginning in their freshman year. one goal of the 2-year program is to allow students to have the same mathematics teacher both years. this arrangement has helped teachers create a culture of learning and support that students can benefit from in their two periods of mathematics and in their first 2 years of high school. norfolk public schools in norfolk, virginia want to ensure that their students have every opportunity not only to take geometry in high school, but also algebra ii and other higher-level mathematics. school leaders believe that getting students through algebra i earlier—in 8th grade—creates greater opportunities for students to take and excel in the higher-level courses in high school. the district developed the algebra for all project, which requires students to take and pass an algebra i course and the state’s end-of-course algebra exam in 8th grade. norfolk knew that the project could not only consist of changing enrollment patterns, but also needed to involve an improvement in the quality of mathematics instruction. to that end, the district is focusing on curriculum and extending the time students spend working on mathematics each day. curriculum: norfolk focused on vertical articulation and coherence of mathematics across grades. the district realized that a foundation of algebra content was needed in all grades preceding algebra i. mathematics content staff integrated algebraic reasoning across all topics in the k–7 curriculum in a coherent content strand involving patterns, functions, and algebra. the new articulation ensures a progression of concepts, so that when students reach algebra i, they are prepared with basic algebraic ideas and concepts. extended instructional time: mathematics is taught for a minimum of 90 minutes per day at all grade levels. the district provides teachers with an instruc paek practices worthy journal of urban mathematics education vol. 11, no. 1&2 160 tional manual that shows how they can use those 90 minutes to fully engage students in learning mathematics. teachers also help students learn mathematics in “academic success sessions” during the school day or after school. epcae, grant, and norfolk all show promise in helping students meet rigorous mathematics course requirements. epcae has seen an increased number of students enrolling and passing algebra ii as well as increased graduation rates. at grant, the enrollment of black students in algebra ii has increased from 8.9% to 17.9% since the first cohort completed the intensive 2-year course; 100% of students in the 2-year course plan to enter college. in norfolk, the percentage of middle school students enrolled in algebra i has increased, as has the percentage of students passing the course and exam: from 41% to 69%. embedded student support within the school day schools and districts that engage in academic intensification must find ways to support students who come to the mathematics classroom with diverse experiences. two small schools, eastside college preparatory school and high tech high, have found ways to embed such student support in the daily schedule as a regular part of students’ schooling. this scheduling is especially important given that most students do not arrive at eastside and high tech adequately prepared for high school. in these schools, a low student-to-teacher ratio helps teachers give students more individualized attention, and the school culture includes planning for college as a regular part of students’ schooling. larger schools, like evanston township high school, are challenged by large classrooms and high student-toteacher ratios, so these schools must rely on strategies like tutorial programs and extra time for mathematics instruction. eastside college preparatory school in east palo alto, california is an independent school serving students in grades 6–12 from populations that are historically underrepresented in higher education. enrollment is just over 200 students. eastside’s goal is to provide a strong, student-centered academic environment and requires that, at a minimum, students have completed precalculus before graduating. students receive various forms of support that are embedded into the school day, including daily tutorials and individual advising, to meet these high expectations. tutorials: two 90-minute tutorials are built into the school day to ensure that students are getting support to understand the core course content (english and mathematics) and are completing their homework. the tutorial sessions come immediately after the targeted course and are led by the same teacher, who tutors around 20 students. the tutoring sessions ensure that students receive timely help on concepts and ideas. the framework ensures consistency of instruction and allows teachers extra time to work with students who are struggling and to provide more intensive opportunities for students to engage with the academic content. paek practices worthy journal of urban mathematics education vol. 11, no. 1&2 161 advisory system: students meet daily with an advisor, who works with them specifically on their personal and academic challenges and issues. advisors are teachers assigned to a group of 6–8 students with whom they work closely over the 4 years of high school. advisors also provide students with resources for extracurricular activities that can help support their academic interests and portfolios for applying to college. academic support: additional academic courses that focus on reasoning and analytical skills as well as topics in college admission and transitioning to college are required. these courses provide students with a strong foundation in the skills and habits that are necessary for academic success in high school and beyond. high tech high (hth) in san diego, california is a charter school that focuses on solutions for dealing with student disengagement and low academic achievement. the school develops personalized, project-based learning environments and expects all students to graduate well prepared for college. the school’s enrollment is just over 500 students. hth encourages student learning through project-based learning and close work with advisors and mentors. project-based learning: hth offers hands-on experiences in mathematics through project-based learning. after mathematics teachers provide a lesson and tasks for students to engage in, students break into small learning groups to work on projects that require them to apply the mathematics concept to a hands-on activity. because the classrooms are grouped by grade level and students come in with differing levels of mathematical proficiency, classes are taught in ways that cover the span of several mathematics courses; for example, algebra i, geometry, and calculus are taught in the same class, and the teacher focuses on a mathematics strand and differentiates the difficulty in the project activity for students. students work within and across groups to gain advice and input for their projects, and the teachers check in with each group to monitor the projects and provide support and guidance as needed. advising: the advisory program was designed to support students in their academic preparation for college. each hth student is assigned a staff advisor who also acts as a liaison to the student’s family, so parents are aware of their child’s growth and challenges at hth. advisors work closely with students to help them plan for their futures, navigate the college admissions process, and apply for financial aid and scholarships. internships: beginning in their junior year, students work as interns two afternoons a week for at least one semester at local businesses, schools, nonprofit organizations, or professional associations. each student works on a specific project overseen by a mentor who understands and supports hth’s design principles and works individually with the student to cultivate a productive learning experience that exemplifies the project-based learning in school in an actual work-related setting. paek practices worthy journal of urban mathematics education vol. 11, no. 1&2 162 evanston township high school in evanston, illinois has an enrollment of over 3,100 students. the school is working on building student success in algebra i and has taken steps to ensure that students receive daily, individual support in mathematics. intensive daily support: algebra i classes are structured to provide more instructional time for all students. students work in small groups to discuss an idea and then share their findings with the whole class; students feel comfortable asking questions of each other and of the teachers when they do not understand a concept. students in upper-level mathematics courses have been recruited to assist in algebra i classes, helping students understand concepts and serving as teachers’ aides. in addition, to make sure struggling students receive support, the chair of the mathematics department meets individually with students who have failing grades to discuss their performance and talk about what kind of help they need. algebra i teachers also have 30 minutes each morning to work with struggling students. eastside, high tech, and evanston township all have programs in place that show promise for supporting students on a daily basis to ensure their long-term success. in the two small schools that mainly serve economically disadvantaged, firstgeneration college-bound students, 100% of students graduate high school and enroll in four-year universities. in evanston, students are passing algebra i at higher rates. summary for raising student achievement raising student achievement requires changes in the attitudes and practices of administrators, teachers, and students. in summer bridge programs, students learn about the value of academic effort and build peer and teacher relationships that will support them throughout high school. success in these programs necessitates firm belief on the part of teachers that their students really can succeed in high school mathematics and that collegial student peer groups can be a strong support for that success. requiring rigorous courses of all students demands both a change in how districts and schools think about student ability and much more support for both students and teachers. intense, embedded daily support, for example, constantly reiterates the idea that mathematics is important and that, with hard work and a strong network of teacher and peer support, all students can take and pass rigorous mathematics courses. building teacher capacity all of the schools, districts, and programs profiled in this study have increased their expectations for what teachers should do, but some of them have focused intense attention on improving teacher practices. the practices designed to build teacher capacity provide opportunities for teachers to interact with other paek practices worthy journal of urban mathematics education vol. 11, no. 1&2 163 teachers in focused and specific ways, share knowledge, and thus improve and expand their current practices. the practices designed to build capacity also increased individual support for teachers and expanded their access to resources. these practices require support from administrators if the traditional ways teachers have interacted are to be overcome. as teachers are asked to support students with various experiences and backgrounds, districts and schools are asked to support teachers the same way, instead of providing all teachers the same training and expecting all of them to perform the same way. three main approaches to building capacity emerged: redefining mathematics teacher roles and responsibilities, making instruction public, and having new, customizable tools for teaching. redefining mathematics teacher roles and responsibilities four districts focused on broadening the sphere of mathematics teachers’ roles and responsibilities in two main ways: by improving the teaching of specific subpopulations and by increasing teacher participation at the district level. improving teaching for specific subpopulations. in new york city and denver public schools, mathematics teachers work closely with teachers who specialize in teaching students with special needs, learning how to maintain rigorous content standards while supporting students learning english or students in special education. the practices encourage good teaching by focusing on the types of instructional tasks that teachers can use for differentiating instruction to meet the diverse needs of students, encouraging the use of academic vocabulary, and providing various entry points for students to learn the mathematical concepts. these practices also provide teachers with feedback on specific ways that some students may struggle as a result of language acquisition issues or cognitive impairment. denver public schools developed a collaboration between mathematics and special education teachers. the district believes that special education teachers often do not have expertise in mathematics and thus have difficulty supporting their students in higher-level mathematics. mathematics teachers do not always know how to accommodate special education students’ individualized education plans without “dumbing down” the mathematics content. denver saw a need to broaden teachers’ roles by having mathematics and special education teachers work together to best support all of their students in secondary mathematics. in denver’s program, teachers are matched in pairs (one special education teacher and one mathematics teacher) for the academic year. the whole group meets about every 6 weeks. in each meeting, each pair of teachers writes a single mathematics lesson plan, working together to build in accessibility and accommodations to address the range of their students’ individual challenges and needs. the goal is for teachers to maintain the integrity of the mathematics while also following a process for planning accessibility strategies that address learning barriers. to paek practices worthy journal of urban mathematics education vol. 11, no. 1&2 164 make their work concrete, the teachers each choose three students who represent a range of mathematical abilities and write their lessons with those students in mind. built into each meeting are opportunities for teachers to reflect on their use of specific strategies and share their goals and cautions regarding accessibility strategies. this type of sharing builds a supportive group that shares ideas and actual practices in the field, giving the teachers a common set of goals to aim for and cautions to keep in mind. new york city department of education created the english language learners (ell) mathematics initiative to raise the academic achievement of ell students through a strong network of district and school-based mathematics and ell leaders. the initiative is designed to raise the quality of mathematics instruction while providing for the diverse needs of students with various language and academic backgrounds. at the core of the initiative is a professional development program for mathematics teachers that emphasizes techniques specifically geared to teaching students whose first language is not english. at the core of the program is the belief that mathematics is not “language-neutral”—meaning that mathematics pedagogy depends on the language of instruction—and therefore the professional development opportunities focus on how teachers must teach in ways that incorporate students’ native languages, english, and academic mathematics language. teachers are trained in wested’s quality teaching for english learners (qtel), which helps them develop a theoretical foundation and corresponding strategies for effectively teaching academic language to ell students. the tools and processes taught in professional development modules focus on developing adolescent students’ abilities to read, write, and discuss academic texts in english. reflection activities for teachers provide opportunities to think about past lessons and plan how to address specific challenges. teachers also analyze case studies and videos that show a range of teaching styles, in order to better understand some obstacles to their own as well as their students’ understanding. additionally, teachers are asked to develop resources and lesson plans and to problem-solve specific teaching and learning situations. district roles and responsibilities. in lamoille south supervisory union and portland public schools, mathematics teachers are taking on leadership roles and working with district leaders to learn more about specific district mathematics needs; this in turn improves their own practices. in the partnership for high achievement, district leaders and teachers work to communicate common goals and sustain them with concrete steps for improving classroom practices. lamoille south supervisory union in morrisville, vermont consists of three school districts serving students in grades k–12. lssu is creating a local, balanced assessment system in mathematics that is aligned with the k–12 curriculum. to support that work, teachers’ responsibilities now include developing assessments at paek practices worthy journal of urban mathematics education vol. 11, no. 1&2 165 the district level. teachers receive training in assessment development and assessment for learning, which helps them understand how assessment can provide the information they need to improve their practices. lssu incorporates the use of ongoing and embedded professional development structures that broaden teachers’ knowledge and understanding of the development, use, and analysis of assessment. lssu leaders involve teachers in writing assessment items because they believe that, to affect instruction at the classroom level, teachers need to understand what is expected at the district level. they also believe that teachers need to be involved in the kinds of conversations that help them reflect on their practice. as they develop assessment items, teachers talk about different types and uses of assessments (formative, benchmark, and summative), learning how to make judgments about student learning depending on the type of student work or data they have available. in addition, given that teachers use the same assessments, they can collaborate to analyze the results and then plan interventions and modifications together. portland public schools in portland, oregon has developed a set of districtlevel leadership opportunities for all interested mathematics teachers. the district mathematics specialists believe that developing local leaders at each school as agents of change is the most effective way to sustain a common set of mathematics goals across the district. they hope that this leadership development will increase teacher capacity at each school and lead to better and more consistent mathematics teaching so that students have equal opportunities for mathematics achievement. leadership opportunities are organized within a large group of teachers and district mathematics specialists. each year, the large group divides into subgroups focused on different ways of approaching mathematics education improvement. one year, the topics the subgroups focused on were determining the content for a new, third year of high school mathematics graduation requirement; supporting the transition of students from eighth-grade to high school mathematics; and developing and piloting districtwide common formative assessments in grades 6–8. the next year, the third-year math and transition to high school topics remained, and two new topics were added, one focused on implementing the college preparatory mathematics program and the other on using technology in mathematics classrooms. the subgroups and topics change shape as the responsibilities and needs of teachers change. the subgroups generate guidelines for interaction to support individual teacher voices and develop a clear set of steps to meet goals. teachers volunteer to facilitate monthly meetings, and the district mathematics specialists help them plan the agendas. in their teacher-leader roles, teachers feel they have the power to make a difference beyond their own classrooms, and leading and participating in these dis paek practices worthy journal of urban mathematics education vol. 11, no. 1&2 166 trict-level groups is a way for them to be directly involved in district improvement in student mathematics learning. the partnership for high achievement (pha) is a program designed to strengthen the capacity of leaders and teachers in texas school districts to implement a research-based instructional support model to continuously improve teaching and learning. the model integrates leadership development for department, school, and district leaders with support for classroom teacher development. pha’s strategy is to provide technical assistance and professional development to a district’s teachers and leaders to support the district in ensuring that every student has access to the same curriculum. to implement this strategy, a leadership advisor works with the district leadership team, and a mathematics advisor works with designated teacher teams. the advisors teach district leaders and teachers about the instructional support model and how to implement it, and provide supplementary resources based on the unique needs of the district. the advisors work with the district leadership team and teacher teams throughout the school year to ensure that the elements of the instructional support model are accomplished. in denver, new york, lamoille south, portland, and pha-partnered districts, the broadened teacher roles and responsibilities promise to increase teacher skill sets and renew investment in student learning. certainly, the teachers seem to be embracing their new roles. in denver, reflective feedback collected from the participating teachers indicates that they are learning more about content and improving their teaching strategies. new york city teachers appear receptive to improving their practice to accommodate ell students. in lssu, teachers are having epiphanies about the role of assessment in learning and are eagerly engaging with one another and their students. in portland public schools, 36% of secondary mathematics teachers are involved in a mathematics leadership subgroup. in pha, participating districts’ mathematics and science scores have gone from below the texas average to above the texas average. making instruction public deprivatizing instruction, or making instruction public, is a powerful means for changing teacher practice. this process requires teachers to open up their classrooms, trusting that observers are not evaluating them but are providing valuable feedback to help them reflect on their practices. making instruction public allows teaching and learning to be captured in multiple ways from multiple sources, giving teachers regular feedback so they can continually work on improving their teaching. three districts and one multi-district initiative have made open classrooms a major part of their mathematics improvement plans. bellevue school district in bellevue, washington has set the goal of “getting rid of walls of classrooms” and building a culture of openness and sharing among teachers and the district mathematics curriculum coaches. the curriculum coaches paek practices worthy journal of urban mathematics education vol. 11, no. 1&2 167 observe classrooms, learn what teachers are doing successfully, share the successful practices with all mathematics teachers, and help teachers with their concerns and challenges. although some teachers were defensive at first, feeling that observations were a threat to their autonomy, they soon saw the value in sharing their successful practices, especially when they were working together toward the same goals. bellevue further encourages collaboration by sharing among teachers the results of common assessments, so that teachers can see how all students are performing on the same types of tasks and discuss how their practices contributed to their students’ performance. the district develops common assessments for every unit at every grade level, and teachers are required to administer the assessments, score students’ work, and post results on the district’s intranet. with assessment results accessible to the entire professional community in bellevue, the hope is that teachers will seek out and share best practices with each other in the ongoing effort to improve work with students. further, the operations and results of teacher practices are available to greater numbers of people, including parents, because the district requires all teachers to have a classroom website that includes the course syllabus and/or grade-level goals and expectations. the website also includes online access to grades. columbus public schools in ohio has made classrooms public by instituting a peer observation program for teachers. at each school, a teacher leader, trained at the district level to support professional learning communities, conducts weekly meetings to help other teachers work as a team to address challenges. most of the time in these meetings is spent developing specific strategies for addressing student needs, but the work also involves reviewing progress on school-specific action plans, student testing results, and teacher-student survey results. these meetings have helped encourage teachers to stop working in isolation and to open their classrooms and their practices to observation. teacher leaders have developed and refined a data collection tool they use in observing classrooms and collecting information about instructional strategies. the teacher leaders use the data they collect to promote discussions with teachers about how to learn from these observation experiences; the culture surrounding these discussions is collaborative, not evaluative. principals observe classrooms to see if there is systematic use of the standards-based mathematics curriculum guides. most principals do classroom walkthroughs daily, as required by the district. the principals have been trained to ask reflective questions of teachers and have also learned how to focus on what they should be seeing in mathematics classrooms. district-level administrators also visit classrooms, and several mathematics curriculum specialists spend at least a half-day per week visiting schools and monitoring the implementation of the mathematics curriculum. paek practices worthy journal of urban mathematics education vol. 11, no. 1&2 168 yes college preparatory school in houston, texas has embedded into the teaching culture a teacher feedback and evaluation system that includes regular observations by coaches, mentors, peers, and supervisors. this system supports teachers with goal setting and reflection, providing feedback to improve teacher practices throughout the school year as part of their ongoing professional development. at the beginning of the year, teachers set goals, using a summative rubric as a guide. the rubric covers four domains: classroom management and culture, instructional planning and delivery, yes responsibilities, and yes values. each domain has multiple indicators, so observers rate teachers on each indicator to develop a composite domain rating. this detailed rubric helps observers identify the areas in which teachers need the most assistance and support, which enables them to customize mentoring and coaching to improve teacher pedagogy. throughout the year, teachers receive feedback from their peers, from supervisors, and from students. at the end of the school year, the summative rubric, along with a teacher’s course material, progress on professional development goals, self-reflections, self-evaluations, administrator evaluations, student performance, and student feedback, is used to evaluate the teacher’s performance. phoenix union high school district in phoenix, arizona uses professional learning communities to create a culture that focuses on how to change the way teachers engage with students. teachers in phoenix union began to change the culture of their practice by opening their doors to peer review and learning from one another about best strategies for improving student learning in mathematics. when teachers opened their doors to each other, no teacher worked in isolation. teachers began to share what worked well and went to one another for help when they struggled with a concept or topic. they make all student work public so they can analyze what students really know and what they are struggling with. teachers began to change their thinking about classroom observers, no longer assuming they were evaluative and critical; instead, teachers learned ways of improving their practice through observation of their peers. these changes resulted in more consistent instruction and assessment strategies across the district. the district also asks teachers to work in teams to provide meaningful lessons and assessments that are congruent with the curriculum. although methods for building lessons and assessments are discussed in teacher preservice and inservice workshops, the teams allow teachers to help each other better understand the development process as they look at specific instructional examples, resources, and strategies. by developing and working with common lessons and assessments, teachers can learn from one another and develop more consistent methods of delivering instruction. silicon valley mathematics initiative (svmi) in the san francisco bay area believes that the key to improving student achievement is improving instruction through intensive, hands-on professional development for individual teachers. to paek practices worthy journal of urban mathematics education vol. 11, no. 1&2 169 that end, the initiative has mathematics coaches frequently observe classrooms and discuss their observations with teachers one on one. this practice makes teachers’ instruction open to outside feedback while providing a structure for teachers to learn how to improve their instruction. the main job of the coaches is to assist the teachers they work with to focus on student thinking and mathematical pedagogy. coaches visit the classrooms of each of their teachers about 20 times per year. the general structure of each visit includes a pre-conference, observation of a lesson, and a post-conference. coaches encourage teachers to reflect on the lesson, examining student work as evidence, to help inform and adjust future instruction. the mathematics coaches tend to relate to their teachers in one of three ways—as collaborators, models, or leaders. in the collaborator role, coaches are a resource to the teacher, providing materials, information, and encouragement, and collaborating with the teacher to plan lessons. in this role, coaches do not give direct feedback about the teacher’s pedagogy, but focus more on student work, which makes the teacher feel less defensive about being evaluated or criticized. in the model role, coaches model instruction of deep problem-solving tasks for students. teachers can use this model lesson as a guide for developing their future lesson plans. as a leader, the coach guides the teacher in nonevaluative ways. for instance, the coach’s comments are grounded in what was just observed—what the teacher understood about how well the lesson went and what students seemed to learn. the coach then assists the teacher in figuring out how to address the content the students did not seem to understand well. the various strategies for making instruction public practiced in columbus, at the yes school in houston, in the phoenix union district, and svmi schools are helping teachers better understand their own practices and improve their teaching. teachers in these districts have found that deprivatized instruction encourages collaboration and allows them to support each other. in bellevue, teachers are much more comfortable now sharing their information with each other and with parents. in columbus, teachers indicated that the weekly meetings were useful for establishing collaboration and consistency of instruction, and they are now accustomed to regular visitors in their classrooms. at yes, all teachers are meeting a minimum standard for providing quality teaching to their students. in phoenix union, teachers have an open-door policy that fosters consistent observation and learning from one another. teachers involved in svmi coaching are using evidence of what students have learned rather than anecdotal information to gauge students’ understanding. new tools for teaching an issue in training teachers in the use of new tools and resources is that professional development is usually the same for all teachers in a given school or field. the success of such strategies and tools, however, differs significantly in different paek practices worthy journal of urban mathematics education vol. 11, no. 1&2 170 cases, because teachers come into professional development workshops with different knowledge, experiences, and pedagogical practices. to remedy this problem, one program and three districts provide customizable trainings to assist teachers appropriate new tools and strategies to improve their teaching practices. agile mind is an online tool that supports and models sustainable teaching in secondary mathematics courses (from middle school mathematics through ap calculus). curricula are aligned to state standards in the states in which agile mind is used, the national council for teachers of mathematics (nctm) standards, and various mathematics textbooks so that teachers can use agile mind to support the textbooks they are required to use. instructional resources are available for teachers to use in planning and delivering instruction and effective assessment. each course includes several topics, and within each topic, an online instructional guidance system provides teachers with specific resources for instruction planning, teaching, assessment, addressing various teaching challenges, and alignment to state standards and textbooks. teachers can use all of these resources or select specific ones. within each online resource, teachers have the option of adding their own notes, which helps them customize their practice. agile mind provides instructional guidance for all aspects of the lesson, from opening questions that enable teachers to introduce key concepts and engage students in discussion to framing questions that support teachers in helping students apply the lesson to real life. further questions are suggested to help probe students’ thinking and to uncover misconceptions. teaching tips offer strategies for dealing with possible challenges students might face. assessments are built into each topic, with different types of reports available so teachers can review both what the entire class understands and what individual students understand. teachers are offered a range of resources they can use in secondary mathematics courses, giving them the flexibility to choose the resources best suited to their instructional goals. anchorage school district in alaska has developed its own assessment reporting system, a comprehensive database system that follows students longitudinally with all the data that was previously kept in their paper cumulative folders. the purpose of this system is to give teachers access to data on their students at any time. for instance, if a student transfers to another teacher or school within anchorage, that student’s data are immediately transferred electronically into the new classroom, so teachers have up-to-date access to all the student information they need. data are available for individual student performance on district and state assessments across several years. while teachers can view their own classroom data, school administrators can view an entire school or any classroom within their assigned school. the system allows the district to customize professional development opportunities to the needs of individual teachers and schools. district-level paek practices worthy journal of urban mathematics education vol. 11, no. 1&2 171 mathematics curriculum specialists work with individual teachers and schools that have lower than average performance in the district. the assessment reporting system allows users to sort students’ proficiency on various mathematics assessments by demographic information like race/ethnicity and gender according to the entire assessment or selected mathematical strands. the four proficiency levels are color coded to give teachers a visual snapshot of where students need the most help, allowing them to target specific students struggling in each strand. the format of all data output has been customized based on teachers’ requests, and the reports continue to be revised in response to teacher feedback. because the system is homegrown, not an off-the-shelf product, anchorage has the flexibility to further customize the system to improve its usefulness as a tool to inform teacher practices. the assessment reporting system also features a grade-level expectation item bank. teachers can pull items from this bank that are linked to the grade-level expectations they are focusing on and use those items to develop customized miniassessments. the data from these items can then be used as part of the instructional cycle for measuring and improving student learning on different mathematics expectations. boston public schools’ secondary mathematics coaches use asset-based instruction to develop teacher capacity. asset-based instruction encourages teachers to focus on students’ strengths rather than on their deficits. coaches model the asset-based approach for teachers by emphasizing instructional experiences they observe that enhance teachers’ understanding of and competence in teaching mathematics. this approach builds on teachers’ strengths, helping them see how they can then use those same techniques to engage their students. the asset-based approach allows teachers to customize their instruction and allows coaches to customize their approaches to teacher professional development. because coaching is at the individual teacher level, coaches can customize the training to emphasize what they believe a teacher needs to work on. after observing a teacher’s classroom, a coach talks with the teacher about student-centered coaching and the strategies teachers can use to take advantage of the known strengths of each student and the class as a whole. the coach usually focuses on the interaction of the teacher with a particular student to exemplify the techniques. the teacher and coach discuss the importance of both affective and cognitive experiences in helping motivate students, again from the perspective of building on students’ strengths. they also talk about how to improve ability beliefs. together, the teacher and coach also identify patterns of students’ strengths by analyzing student work and assessments. the coach reinforces how to motivate students with genuine positive support and encouragement as often as possible. the teacher and coach also identify places in the curriculum where students are current paek practices worthy journal of urban mathematics education vol. 11, no. 1&2 172 ly successful and map out a lesson that guarantees at least one successful experience for each student. cleveland municipal school district is using a program called keeping learning on track (klt) in its 10 lowest-performing k–8 schools. klt is a formative assessment program developed by educational testing service. klt focuses on using evidence of learning to adjust and customize instruction as it is taking place so that teachers can immediately address students’ learning needs. because teachers’ instructional styles vary, klt provides a variety of ways for teachers to measure student learning on the fly, giving teachers the flexibility to choose the strategies that best allow them to make instructional adaptations right at that moment. these types of formative assessment checks can provide teachers the feedback they need to change their daily practice, and that small change might result in large changes in teacher pedagogy, the classroom culture, and student learning. teachers using klt meet regularly to reinforce and build upon the techniques, strategies, and ideas behind the program. teachers use these meetings to discuss the implementation of assessment-for-learning techniques in their classrooms and to refine their understanding of klt techniques. agile mind and the practices in use in anchorage, boston, and cleveland all show promise for improving teacher practices. agile mind users tend to increase the implementation of the resources each year they use it, and schools tend to expand the courses that can be supported by it. in anchorage, teachers report that they appreciate the assessment reporting system and use it to analyze and understand how their instruction affects student performance. in boston, teachers appreciate the individual coaching and modeling they receive and recognize how asset-based instruction changes the culture of their classrooms. in cleveland, teachers report that they regularly use assessment-for-learning techniques; the schools using klt have seen substantially greater gains in student achievement than have non-klt schools. summary for building teacher capacity building teacher capacity requires changes in district and school attitudes about how to best support teachers as they improve their teaching. with broadened roles and responsibilities, teachers redefine how they think of teaching and what they can contribute. they learn that they can gain the expertise to work successfully with subpopulations of students in need of their help, be part of a development team for building common assessments at the district level, or participate as leaders in the district for promoting change in mathematics. when instruction is public, teachers learn about the power of collaboration for improving their practice and lose the fear of having observers in the classroom. with structured observation protocols and regular opportunities for feedback, teachers forget about working in isolation and paek practices worthy journal of urban mathematics education vol. 11, no. 1&2 173 focus more on the ways they can work together to improve student achievement. finally, with new tools and customized support, teachers can access the individual training and feedback they need to make good practices part of their daily instruction. discussion and next steps the practices i have identified address challenges that virtually all american school districts must face. in too many cases, however, school districts create their solutions to these challenges from scratch and in isolation. the practices worthy of attention project is designed to offer a more effective approach to collaborative learning and to the dissemination of creative solutions to difficult educational problems. to successfully tackle the challenges faced by all educators and leaders in improving mathematics teaching and learning, researchers must spend more time in schools and districts, observing and analyzing how the broad approaches and big ideas are actually codified, implemented, and assessed within and across districts. this project is a first step toward creating a nationwide group of practitioners who can share specific strategies with and learn from one another, which will serve to open doors across districts much as classrooms have been opened within schools. by taking the time to observe and evaluate actual practices, researchers can find out directly how research is interpreted and implemented and therefore advise mathematics leaders and teachers in ways that directly affect their work. the next phase of this work is to partner researchers with schools and districts to raise the standards of evidence by which the researchers measure the effectiveness of these practices. this partnership will allow for the fulfillment of a key purpose of this work: not only to identify common themes in these practices that can be used to strengthen teachers’ practices and student achievement in urban systems across the country, but also to determine the effects of districts’ initiatives for improving teacher practices and, in turn, the effects of those practices on students’ secondary mathematics progress and achievement. references adelman, c. (2006). the toolbox revisited: paths to degree completion from high school through college. washington, dc: u.s. department of education. bamburg, j. (1994). raising expectations to improve student learning. oak brook, il: north central regional educational laboratory. barth, p., & haycock, k. (2004). a core curriculum for all students. in r. kazis, j. vargas, & n. hoffman (eds.). double the numbers: increasing postsecondary credentials for underrepresented youth (pp. 35–45). cambridge, ma: harvard education press. beck-winchatz, b., & barge, j. (2003). a new graduate space science course for urban elementary and middle school teachers at depaul university in chicago. the astronomy education review, 1(2), 120–128. paek practices worthy journal of urban mathematics education vol. 11, no. 1&2 174 center on education policy. (2006). state high school exit exams: a challenging year. washington, dc: center on education policy. dweck, c. s. (2002). messages that motivate: how praise molds students’ beliefs, motivation, and performance (in surprising ways). in j. aronson (ed.), improving academic achievement: impact of psychological factors on education (pp. 37–59). san diego, ca: academic press. ingersoll, r. m. (1999). the problem of underqualified teachers in american secondary schools. educational researcher, 28(2), 26–37. ma, x. (2001). a longitudinal assessment of antecedent course work in mathematics and subsequent mathematical attainment. journal of educational research, 94(1), 16–28. marzano, r. j. (2003). what works in schools: translating research into action. alexandria, va: association for supervision and curriculum development. massell, d. (1998). state strategies for building local capacity: addressing the needs of standardsbased reforms. philadelphia, pa: center for policy research in education, university of pennsylvania. mathews, j. (2007, march 12). adding eighth-graders to the equation: portion of students taking algebra before high school increases. washington post. retrieved from http://www.washingtonpost.com/wpdyn/content/article/2007/03/11/ar2007031101438.html?noredirect=on national science board. (2006, january). america’s pressing challenge: building a stronger foundation. nsb 06-02. washington, dc: national science board. spillane, j., reiser, b., & reimer, t. (2002). policy implementation and cognition: reframing and refocusing implementation research. review of educational research, 72(3), 387–431. stigler, j. w., & hiebert, j. (1998, winter). teaching is a cultural activity. american educator. retrieved from https://www.kentuckymathematics.org/docs/teaching_is_a_cultural_activity_teachingwinter_98-stigler.pdf stigler, j. w., & hiebert, j. (1999). the teaching gap: best ideas from the world’s teachers for improving education in the classroom. new york, ny: free press. tauber, r. (1997). self-fulfilling prophecy: a practical guide to its use in education. westport, ct: praeger. u.s. department of education. (1997). mathematics equals opportunity. washington, dc: u.s. department of education. retrieved from https://files.eric.ed.gov/fulltext/ed415119.pdf http://www.washingtonpost.com/wp-dyn/content/article/2007/03/11/ar2007031101438.html?noredirect=on http://www.washingtonpost.com/wp-dyn/content/article/2007/03/11/ar2007031101438.html?noredirect=on https://www.kentuckymathematics.org/docs/teaching_is_a_cultural_activity_teaching-winter_98-stigler.pdf https://www.kentuckymathematics.org/docs/teaching_is_a_cultural_activity_teaching-winter_98-stigler.pdf https://files.eric.ed.gov/fulltext/ed415119.pdf microsoft word final mosqueda vol 3 no 1.doc journal of urban mathematics education july 2010, vol. 3, no. 1, pp. 57–81 ©jume. http://education.gsu.edu/jume eduardo mosqueda is an assistant professor in the department of education at the university of california, santa cruz, 1156 high street, santa cruz, ca, 95064; email: mosqueda@ucsc.edu. his research interests include the mathematics education of english language learners and equity in mathematics education. compounding inequalities: english proficiency and tracking and their relation to mathematics performance among latina/o secondary school youth eduardo mosqueda university of california, santa cruz in this article, the author examines whether disparities in mathematics performance might be exacerbated by the track placement of native and non-native latina/o english speakers in the education longitudinal study of 2002. the effect of track placement on the mathematics performance of english learners (el) differed as a function of their level of english proficiency. the scores of latinas/os with low levels of english proficiency in the general track were similar to the scores of students in the college track with comparable levels of english proficiency. the scores of non-native english speakers in the college track with high levels of english proficiency, however, were higher than those of their peers in the general track and nearly as high as those of native english speakers in the college track. implications for the potential development of the mathematics language register of els are discussed. keywords: english language learners, english learners’ mathematics achievement, urban mathematics education, urban schools fter the passage of the immigration act of 1965, which eliminated quotas for immigrants entering the united states from foreign countries, a surge in immigration contributed to a pronounced increase in the number of students who were not proficient english speakers in u.s. public schools. the u.s.-born children of these immigrants number an additional 30 million. more than half of these immigrant and u.s.-born children are of latin american descent (u.s. census bureau, 2000). consequently, during the 2001–2002 school year, about 1 in 10 u.s. public school students was not proficient in english and was designated limited english proficient (lep).1it is this student population that is the primary fo 1 the term english learner (el) is used interchangeably with the term non-english proficient. both terms are used in place of the federal designation of limited english proficient (lep) (unless citing from another source) because el more appropriately represents students in the process of developing english proficiency as a second language rather than as having linguistic deficits or limitations. a mosqueda compounding inequalities journal of urban mathematics education vol. 3, no. 1 58 cus of this article—non-native english-speaking latinas/os, particularly those with low-levels of english language proficiency (i.e., english learners, whom i refer to as els). among latinas/os, the majority of immigrants and, to some extent, the children of immigrants are typically non-native english speakers. non-native english speakers are confronted with unique challenges in school related to their level of english proficiency. unfortunately, research specifically documenting the academic and linguistic needs of els is limited, with the majority focusing on english language literacy in the elementary grades. little research focuses on the achievement of els in content area courses, particularly in the area of mathematics and at the secondary school level (august & hakuta, 1997; meltzer & hamann, 2004). a focus on secondary school mathematics is important, particularly for latina/o els, because it is one of the strongest indicators of high school graduation and college matriculation. research has shown that high school students who complete advanced mathematics courses (i.e., algebra 2, trigonometry, pre-calculus, or calculus) are more likely to graduate from high school and are twice as likely to attend college compared to students who are enrolled in lowlevel mathematics courses (adelman, 1999, 2006). the purpose of this study is to explore whether disparities in the mathematics performance between latina/o2 native and non-native english speakers might be exacerbated by their academic track placement. more specifically, i examine whether placement in a general or college preparatory track might have a differential effect on mathematics performance and, in turn, whether this relates to the level of english proficiency of non-native english speakers (compared to native english speakers). a distinguishing feature of this study is the focus on latina/o els and whether the effect of track placement on their mathematics performance differed as a function of their level of english proficiency. this study expands on research that has quantitatively examined the relationship between english proficiency and academic tracking using school district level data (cf. callahan, 2005; wang & goldschmidt, 1999), and analyzes this relation using a nationally representative subsample of tenth-grade latina/o students in the education longitudinal study of 2002 (els: 2002). here, i focus exclusively on latina/o students because they make up the largest language minority group in the united states and because of the enduring underperformance in mathematics of too many latinas/os students. latinas/os comprise nearly 80% of the el population in the united states (kindler, 2002). among latinas/os, els not only perform poorly on standardized tests in mathe 2 throughout this article, when reporting aggregated data on specific racial, cultural, and/or ethnic groups (e.g., latina/o, non-native english speakers, english learners, etc.), i acknowledge the significant within group variation embedded (and made invisible) in such data, specifically in regards to academic achievement/performance. mosqueda compounding inequalities journal of urban mathematics education vol. 3, no. 1 59 matics compared to their english-speaking peers (abedi, 2004; abedi & lord, 2000) but also they are the lowest achieving group on the national assessment of educational progress (naep) among all racial and ethnic groups (u.s. department of education, 2004). in addition, latina/o students, and els in particular, are no longer concentrated in a handful of the states with high proportions of latina/o immigrants (i.e., california, florida, illinois, new york, and texas). current trends show that latinas/os are rapidly expanding to new destinations in the united states such as midwestern and southern states where there is often a limited understanding of how to best meet the academic needs of the recent school-aged latina/o population (ruiz-de-velasco & fix, 2000). therefore, the low performance of latinas/os in mathematics (as measured by aggregate standardized test scores), especially among els, should be a national concern. too many latina/o students, including those who are english proficient and non-english proficient, repeatedly underachieve in u.s. public schools. the test scores (as an aggregate) for latinas/os are often described as pervasively, disproportionately, and persistently low over time relative to similar outcomes for whites (valencia, 2002). a performance disparity on the naep appears as early as the 4th grade and persists through high school for latina/o students (again, as an aggregate) (smith, 1995). while scores for latinas/os collectively have increased over the last 15 to 20 years, the differences in mathematics achievement between latinas/os and whites have remained steady over the same time (smith, 1995). in a recent meta-analysis, latina/o performance demonstrated increases but so too did the performance of the other comparison racial and ethnic groups, with latina/o gains lagging those of their comparison peers (capraro, capraro, yetkiner, rangel-chavez, & lewis, 2009). the low achievement of many latinas/os at the secondary school level, however, is more profound than what mere aggregate mathematics standardized test scores imply because disproportionate numbers of latina/o students are denied access to rigorous content (capraro, young, lewis, yetkiner, & woods, 2009). a study by ortiz-franco (1999) revealed that latinas/os made only small gains in their basic mathematics skills between 1970 and 1990, while their performance in mathematical problem solving requiring high-level, problem-solving skills did not improve over the same period. scholars agreed that the lack of improvement in the application of complex mathematics concepts is a cause for concern and merits continued and persistent investigation (gutiérrez, 2002, 2007; khisty, 1995; moschkovich, 1999; secada, 1992, 1996; tate, 1997). presently, over 43% of all teachers in u.s. public schools have at least one el student in their classrooms (zehler, fleishman, hopstock, pendzick, & stephenson, 2003). yet, few of these teachers are adequately prepared to educate els. a national survey showed that teachers with at least three els in their classroom had received, on average, a mere 4.0 hours of lep inservice training within the last 5 mosqueda compounding inequalities journal of urban mathematics education vol. 3, no. 1 60 years (zehler et al., 2003). unless effective research-based strategies are developed to meet their linguistic needs, els are likely to continue to underachieve academically at disproportionate rates nationally. the rapid growth of latinas/os and the dearth of research on how to best meet their academic and linguistic needs contributes to the widespread application of misguided beliefs about the role of language in instruction of els, which, in turn, negatively affects their achievement (flores, 1997; khisty, 1995; secada, 1992). there is a long-standing myth in mathematics education that english proficiency is not an issue because mathematics is a “universal language.” as a result, many educators believe that students’ english proficiency has a minimal effect on their mathematics learning (mather & chiodo, 1994). however, longstanding empirical research that documents the relationship between english language skills and mathematics achievement refutes this myth (cuevas, 1984; cocking & mestre, 1988; de avila, 1988). this body of research has found strong positive correlations between the english proficiency of els and their mathematics achievement on standardized tests. this correlation suggests that the failure to meet the linguistic and academic needs of els in mathematics classrooms can hamper their potential to further develop their mathematics language register. english proficiency and developing a mathematics language register research shows that english proficiency plays an important role in learning mathematics, specifically because of the complexity of rigorous secondary school mathematics content (august & hakuta, 1997; cuevas, 1984; khisty, 1995) and the differences between the language used in mathematics courses and everyday language. in these advanced mathematics courses, both the language of instruction and the content are highly abstract and complex. therefore, students’ english proficiency must be considered in order to ensure that els are provided with opportunities to learn3 (and comprehend) the complex mathematical concepts that mathematics teachers present (garrison & mora, 1999). for instance, research suggests that els need an “advanced level of control” of english to convert word problems into mathematical sentences and perform operations within abstract settings (wong-fillmore & valdez, 1986, p. 663). the inability to comprehend instruction in their non-dominant language can create confusion and stifle an els’ ability to learn mathematics content (barnette-clark & ramirez, 2004). 3 according to croom (1997), opportunities to learn involve the equitable treatment of students from diverse racial and ethnic groups and female students in general. all students must be afforded equal access to learn high-level mathematics concepts as their upperand middle-class white and/or asian male counterparts. moreover, classrooms should be non-threatening and supportive places that encourage all students to explore, conjecture, reason, and make decisions. mosqueda compounding inequalities journal of urban mathematics education vol. 3, no. 1 61 considerable research demonstrates that mathematics alone is a language that is more complex than everyday english (cuevas, 1984; garrison & mora, 1999; gutiérrez, 2002; khisty, 1995, 1997). the language of mathematics is described as a “register” of words, expressions, and meanings that differ from those of everyday language (secada, 1992; cuevas, 1984; mestre, 1988). for example, the language of mathematics has specialized meanings for words and phrases such as “horizontal,”“vertical,”“subtract,”“difference,”“equivalence,” and “inverse,” to name a few, that differ from the everyday conversational and academic meanings that els are learning in their english-language arts courses (ron, 1999). given the important differences between the language of mathematics and the everyday (english) language, non-native english speakers with low levels of proficiency face the added difficulty of becoming proficient in english while they also develop proficiency in the language of mathematics. therefore, simply becoming proficient in english is not sufficient for students to become successful in mathematics. cummins (1986) argues that, as students develop proficiency in english, it is necessary to distinguish between the language used in informal, everyday situations and the language necessary for communication in academic situations. his work suggests that the existence of a minimal threshold level of proficiency in english students must reach—a level of cognitive academic language proficiency (calp)—to function effectively on academic tasks that are cognitively demanding (cummins, 1986). similarly, other recent research expands on the importance of students developing proficiency in academic english in order to experience success in content courses with english-only instruction (meltzer & hamann, 2004; valdés, 2001). expanding cummins’ (1986) notion of academic language proficiency even further, mathematics education researchers argue that, to be successful in advanced secondary school mathematics courses, non-english proficient students must reach a “technical threshold” of english proficiency that is beyond the calp threshold (burns, gerance, mestre & robinson, 1983). dawe (1984) terms this cognitive academic mathematics proficiency (camp), which consists of cognitive knowledge (mathematics concepts and how they are applied) that is embedded in a language specifically structured to express that knowledge (as cited in spanos, rhodes, corasanti dale, & crandall, 1988). in other words, camp is a level of proficiency that demands a high-level of competence in both english and in the language of mathematics. other research supports this conclusion, but also adds that els require considerable proficiency in both their first and second language—spanish and english—if they are to cope with the linguistic and cognitive demands of learning advanced mathematics (cuevas, 1984). mosqueda compounding inequalities journal of urban mathematics education vol. 3, no. 1 62 other factors influencing mathematics achievement while english proficiency is important, it is but one factor that contributes to decreased academic attainment for many non-native english speakers. institutional factors, such as tracking, play an important role in structuring the academic success and failure of latinas/os in general, and latina/o els in particular (conchas, 2001, 2006; gándara, 1995, 1997; lucas, henze, & donato, 1990; mehan, villanueva, hubbard, & lintz, 1996; olsen, 1995). recent research focuses on mathematics tracking of latinas/os, particularly those who are not proficient english speakers. a growing body of research analyzes the (unjust) practice of placing els in low-track classes and consistently finds low-track placements negatively affect students’ achievement in mathematics (callahan, 2005; katz, 1999; wang & goldschmidt, 1999). language proficiency interacts with other factors creating a compounding effect that further diminishes achievement. research on latina/o els suggests that english proficiency significantly factors into decisions about latina/o els mathematics placement (callahan, 2005; gándara, 1999; harklau, 1994a, 1994b; lucas, 1997; walqui, 2000). in their research on schools with large numbers of latinas/os in the southwest, for instance, donato, menchaca, and valencia (1991) found that track assignments were strongly influenced by students’ level of english-language proficiency, and resulted in remedial or vocational track placements. furthermore, the placement of all els, including latina/o els, in low-track classes is often justified by the assumption that these classes are not as difficult linguistically, compared to higher-level courses (harklau, 1994b; katz, 1999). the research on tracking has illuminated differential opportunities to learn for latina/o students, as a result of differences in access to challenging curriculum, low student expectations, and well prepared teachers resulting from low or high-track placements (oakes, 1985). such inequities are found to disadvantage students in the low track and advantage students in the high track. the placement of latina/o els in lower-track classes raises important questions about the rigor of the curriculum to which they are exposed, given the research that has found that low-level track curriculum is cognitively undemanding and focuses on memorization and repetition (heubert & hauser, 1999; oakes, 1985; oakes, gamoran, & page, 1992). for instance, research found that students placed in low-tracks worked at a low cognitive level on tasks that are profoundly disconnected from the skills they need to learn because instruction is often geared predominantly toward multiple-choice tests (darling-hammond, 1991). in contrast, the literature on tracking documents several educational advantages for students placed in high-track courses. oakes (1985), for example, found that a primary advantage for students in high-track courses is the curricular emphasis on high-status knowledge (i.e., the knowledge required for students to take mosqueda compounding inequalities journal of urban mathematics education vol. 3, no. 1 63 more advanced mathematics courses and attend college). additionally, high-track courses emphasize higher-order cognitive tasks and focus on more open-ended types of discussions that lead to richer learning opportunities (gamoran, 1987). moreover, high academic tracks (most often) offer more highly qualified and better-prepared teachers (oakes, 1985). finally, research on tracking has shown that once students are sorted into groups—those who receive high-quality education and those who receive inferior curriculum and teaching—that students often become “locked in” to these arrangements (heubert & hauser, 1999; murphy & hallinger, 1989; oakes, ormseth, robert, & camp, 1990; oakes et al., 1992). thus, latinas/os placed in remedial mathematics courses that produce lower and slower rates of learning (oakes et al., 1992) have a lower probability of receiving better track assignments in the future (heubert & hauser, 1999). the long-term effects of permanent placement in low-track courses are linked to lower academic achievement and higher dropout rates for latina/o els (medina, 1988; rumberger & larson, 1998; romo & falvo, 1996). nonetheless, even acquiring a higher level of english fluency is rarely a guarantee for promotion into high-track courses (olsen, 1997). a study of high school students in low-track courses found that after students became proficient in english they were only promoted horizontally in the tracking system (valenzuela, 1999). therefore, students with high levels of english proficiency are removed from the low-track english as a second language (esl) classes and reassigned to the english-only, low-level track courses. the use of english proficiency as a prerequisite for enrollment in rigorous mathematics courses is a source of unjust inequity for latina/o els. such practices are questionable because placement decisions are often made without an accurate assessment of a student’s level of english proficiency (duran, 2008; martiniello, 2008; solano-flores, 2008; valdés, 1998), mathematics background (dentler & hafner, 1997; gándara, rumberger, maxwell-jolly, & callahan, 2003) or prior educational background in their native country (ruiz-de-velasco & fix, 2000; lucas, 1997). these practices continue to have negative long-term effects on the learning and subsequent performance of els because they delay the entry of els into those courses until they reach an “academic” level of english proficiency. given that academic proficiency may take anywhere from 3 to 7 years to develop (cummins, 1986; ovando & collier, 1998), by the time els develop a level of english proficiency deemed “appropriate” to handle the linguistic complexity of high-level secondary school mathematics content, it may be too late for them to take and to meet the mathematics requisites for graduation and college attendance. as stated previously, in this study i build on other studies that have analyzed whether tracking impacts achievement and examine how track placement and english proficiency relates to mathematics achievement for latina/o native and mosqueda compounding inequalities journal of urban mathematics education vol. 3, no. 1 64 non-native english speakers at a national level. this investigation intends to characterize the breadth and scope of the challenges faced by latina/o els in mathematics to highlight the urgent need for improvements that can assist in redressing disparities in els educational outcomes and, most important, els’ opportunities to learn. this research is guided by the two research questions: 1. why do native and non-native english speaking latina/os who are placed in the general (low) academic track have lower mathematics achievement than their latina/o peers in the college preparatory track? 2. how does the relation between academic tracking and mathematics achievement differ for native english speakers and non-native english speaking latinas/os with high and low levels of english proficiency? methods the data from the first wave of the educational longitudinal study (els) of 2002 (els: 2002), a large nationally representative dataset provided by the national center for education statistics (nces) are used. the base year of the els: 2002 represents the first stage of a longitudinal study that will ultimately provide policy-relevant trend data about critical transitions experienced by a national probability sample of students as they proceed through high school and into college or their careers (ingels, pratt, rogers, siegel, & stutts, 2004). the first wave, from which the subsample for this article is drawn, contains a sample of students in the 10th grade in 2002 and includes 15,362 students from a random sample of 752 public, catholic, and other private schools. the dataset contains assessments of students in reading and mathematics performance in addition to measures of important student, family, teacher, classroom, and school characteristics. it also contains information on students’ immigrant status, language proficiency, and track placement. the analysis is based on the subsample of 2,234 first-generation latina/o immigrants and u.s.-born secondand third-generation latinas/os present in the els: 2002 dataset. statistical power analyses (light, singer, & willett, 1990) suggest that this sample size provides high power (.90) to detect small effects at typical social science levels of statistical significance. variables in the models question predictors. non-native (nonnativij) is coded as 1 = non-native english speaker, 0 = native english speaking latina/o. about 50.5% of the latina/o students in this sample reported being non-native english speakers. in order to differentiate among the level of english proficiency of non-native english speakers, the cross-product nonnativij*engprofij is used. nonna mosqueda compounding inequalities journal of urban mathematics education vol. 3, no. 1 65 tivij is described above. engprofij is a composite that ranges from 3 to 8 (low to high), based on each student’s self-reported level of english proficiency.4, 5 this weighted composite score is comprised of students’ responses to four ordinal dimensions of self-reported english proficiency that includes how well students: “understand spoken english,”“speak english,”“read english,” and “write english.” for each of these dimensions of english proficiency, students provide one of the following four ordinal responses: “very well,”“well,”“not well,” or “not at all.” the inclusion of this interaction in a hypothesized regression model allows for the comparison between non-native english speakers with varying levels of english proficiency and their native english-speaking peers. finally, the variable gentrackij is a dichotomous predictor that indicates whether a student is placed in the general/vocational academic track or in the college preparatory track (1 = general/vocational track, 0 = college preparatory track). approximately 52.6% of the students reported general track placement, and the remaining 47.4% reported being placed in the academic track. outcome variable. mathematics achievement (mthachij) represents an item response theory (irt) scaled mathematics achievement score (ingels et al., 2004) variable for each student i in school j. the els: 2002 assessment itself contains items in arithmetic, algebra, geometry, data/probability, and advanced topics (ingels et al., 2004). these scores are standardized to a mean of 50 and a standard deviation of 10 in the complete els: 2002 sample (ingels et al., 2004). the test score for the subsample of latinas/os is 45.7, with a standard deviation of 9.6 points on the els: 2002 assessment. irt scores are used because they simplify the interpretation of the impact of predictors on the outcome. a one-point difference associated with the outcome variable equals one item correct on the els: 2002 assessment. see table 1 for a more detailed description of all of the variables included in analysis. control predictors. the analyses include a series of control predictors to account for individual background and school context variation that might impact the outcomes, and to assess the potential impact of selectivity bias. these controls include individual level gender, ses, parental education, and each immigrant student’s prior level of education in their native country. also included are controls for the instructional conditions of the classroom that pertain to the mathematics 4 english proficiency level of latinas/os in the els: 2002 dataset was not assessed by a specialized instrument and instead was based on a self-reported measure. a limitation of such measures is that they suffer from self-report bias due to students over or under reporting their perceived level of english proficiency based on their own social and linguistic context. however, the english proficiency of all non-native english speakers in the sample was on the same “metric.” while not ideal, this approach is one of a limited number of options available for nationally representative samples. 5 widely cited large-scale sociological studies of immigrants using similar types of datasets have used these same self-reported english proficiency measures and find that they are relatively reliable measures (portes & rumbaut, 2001). mosqueda compounding inequalities journal of urban mathematics education vol. 3, no. 1 66 teacher’s professional preparation—whether a student’s mathematics teacher had a mathematics degree and whether the teacher is certified. at level 2, are a set controls for selected aggregate measures of the school context such as the percentage of students that are placed in the general track, the percentage of all el students, and the percentage of poor students within each school (the number of students within each school that qualify for free or reduced-priced lunch is used as a proxy for poverty). see table 2 for a description of the definitions and coding of each variable in analysis. table 1 descriptive statistics of all variables variable description n mean s.d. min. max. student background ses standardized ses composite 2222 8.12 2.50 1 13 female sex (0 = male and 1 = female) 2222 0.50 0.50 0 1 non-native non-native english speaker (0 = native english speaker) 2222 0.51 0.50 0 1 immigration status firstgen first generation immigrant 2222 0.15 0.42 0 1 secgen second generation immigrant 2222 0.25 0.43 0 1 thirdgen third generation immigrant 2222 0.60 0.48 0 1 teacher preparation mthmajor teacher has degree in mathematics and/or related field 2222 0.51 0.50 0 1 mthcert teacher is certified in mathematics 2222 0.76 0.43 0 1 school context measures pctlep percent of 10th graders that are lep students in the high school 524 0.78 0.81 0 4 pctlunch percent of 10th grade students that qualify for free lunch 524 3.42 1.96 1 7 public school control (1= public high and 0 = catholic or other private) 524 92.35 17.31 0 1 english proficiency engprof level of english proficiency 2222 6.90 1.22 3 8 track level placement gentrack general track placement 2222 0.53 0.50 0 1 mosqueda compounding inequalities journal of urban mathematics education vol. 3, no. 1 67 table 2 data and coding of all variables variable definition notes/coding student background ses standardized composite measure: family income, parent education, and occupational status range: -1.98–1.79 female students gender 1 = female, 0 = male nonnativ indicates whether the student is a non-native english speaker or a native english speaker 1 = non-native; 0 = native immigration status firstgen indicates whether both the student and parent are foreign-born 1 = yes; 0 = no secgen indicates whether the student is u.s.-born while at least one parent is foreign-born 1 = yes; 0 = no thirdgen indicates whether both the student and parents are u.s.-born 1 = yes; 0 = no teacher preparation mthmajor indicates whether student’s mathematics teacher has a bachelor’s degree in mathematics and/or related field 1 = yes; 0 = no mthcert indicates whether student’s mathematics teacher is certified 1 = yes; 0 = no school context measures pctlunch proxy for school ses measured by the percentage of 10th-grade students eligible for free or reduced lunch 1 = 0–5%; 2 = 6–10% 3 = 11–20%; 4 = 21–30% 5 = 31–50%; 6 = 51–75% 7 = 76–100% pctlep percentage of 10th-grade lep students enrolled 0 = none; 1= 1–10% 2 = 11–25%; 3 = 25–50% 4 = 51% or more public indicates whether the school is public or catholic/other private 1= public 0= catholic/private english proficiency engprof weighted composite of self-reported level of english proficiency: ability to understand, speak, read, and write english range: 3–8 track level placement gentrack academic track placement of each respondent 1 = general/vocational 0 = college mosqueda compounding inequalities journal of urban mathematics education vol. 3, no. 1 68 missing data and sample weights multiple imputation is used to account for the common problem of missing data on surveys (rubin, 1987). the multiple imputation procedure (proc mi) in the sas statistical software package uses information from the sample distributions of the variables themselves to replace missing values with randomly generated but contextually appropriate values. using multiple imputation, five subsidiary datasets are generated, each with different randomly imputed values for the individual level. the hypothesized regression models are then fitted separately in each of the imputed datasets, and the results are averaged and corrected for the inclusion of the random variation in each of the imputed datasets. because the imputed datasets have no missing values except for the dependent variable (which was not imputed), sample size is preserved. this process provides the best estimates given the stability of other factors for the true effect of the given variables. the els: 2002 student-level panel weights and school weights were applied to the analysis according to the guidelines provided for the hierarchical linear models (hlm) software (raudenbush & bryk, 2002). data analysis using hlm, the following four fitted multilevel models were evaluated in which the mathematics achievement of latina/o students is modeled as a function of the control and question predictors. multi-level modeling is well suited for this analysis as it can account for the clustering of students within schools. the first fitted model (m1) was the null or unconditional model that contained no predictors, and estimated the average mathematics achievement for the subsample of latina/o 10th graders in the els: 2002 dataset. the second fitted model (m2) was the baseline control model, and included all the individual-level and school-level control predictors. the third model (m3) added the first of the key question predictors and presented the main effect of both english proficiency and track placement on mathematics achievement. the fourth fitted model (m4) examined the interaction effects between english proficiency and academic track placement. to address the research questions, hlm was used to examine the main effects of how english proficiency and tracking might relate to the mathematics performance of latina/o native and non-native english speakers with varying levels of english proficiency. the fitted multilevel regression model corresponding to the first research question was: mthscoreij =β0 + β1nonnativij + β2(nonnativij*engprofij) + β3gentrackij+γ1zij 6 + γ2zj 7 + (εij + uj) 6 γ1 is a parameter vector describing the impact of the individual-level controls zij 7 γ2 is a parameter vector describing the impact of the school-level controls zj mosqueda compounding inequalities journal of urban mathematics education vol. 3, no. 1 69 to address the second research question, the main effects and statistical interactions between the english proficiency and tracking predictors were added to the multilevel model in the previous equation, as follows: mthscoreij =β0 + β1nonnativij + β2(nonnativij*engprofij) + β3gentrackj +β4(gentrackij * nonnativij) + β5(gentrackij * nonnativij*engprofij) + γ1zij 4 + γ2zj 5 + (εij + uj) findings the findings are reported for models 3 and 4 because they correspond to the research questions guiding this inquiry. model 3: english proficiency and tracking model 3 (see table 3) addresses the first research question, which evaluated the main effect of native or non-native english status (nonnativ), a student’s level of english proficiency (engprof), along with the academic track placement on the mathematics performance on the els: 2002 assessment. the parameter estimates indicate that latina/o native english speakers, on average, scored higher than most non-native english speakers, and students in the college preparatory track scored higher than latinas/os in the general track with some exceptions. the results show that the aggregate test scores of latina/o non-native english speakers with low levels of english proficiency, the latina/o els, were lower than the aggregate test scores of non-native english speaking latinas/os with higher levels of english proficiency.8 thus there was a positive relationship between mathematics achievement and english proficiency (β = 1.14, p< .01). more specifically, a one unit positive difference in the level of english proficiency of non-native english speakers is associated with a 1.14 positive difference in their aggregate mathematics score on the els: 2002 standardized test, all other predictors being equal. this difference in performance is equal to one-tenth of a standard deviation for every unit difference in english proficiency. additionally, the results from model 3 show that placement in the general track had a negative effect on mathematics achievement. on average, general track placement is associated with a 2-point lower difference (or about one-fifth of a standard deviation) in aggregate mathematics test scores for both latina/o native and non-native english speakers, compared to their peers in the college preparatory track. 8 in the previous section, i described predictor engprof as a cross product of non-native status and english proficiency (nonnativ*engprof). mosqueda compounding inequalities journal of urban mathematics education vol. 3, no. 1 70 table 3 final estimated hierarchical linear models model 1 model 2 model 3 model 4 fixed effects coef. (se) coef. (se) coef. (se) coef. (se) intercept 45.46*** 44.28*** 44.31*** 45.46*** (0.46) (0.73) (0.74) (0.77) student background socioeconomic status 2.89*** 2.72*** 2.72*** (0.66) (0.66) (0.72) female (male omitted) -0.26 -0.51 -0.49 (0.59) (0.59) (0.57) non-native speaker -2.05* -9.94*** -14.19*** (0.95) (2.06) (2.86) immigration status first generation 0.33 (0.95) second generation 2.84** 2.28** 2.28** (1.00) (0.81) (0.80) teacher preparation mathematics and/or related 2.07** 1.94** 1.97* (0.66) (0.71) (0.72) certified 0.94 0.94 0.75 (0.87) (0.95) (0.90) school context measures 10th-grade % free lunch -1.07*** -1.07*** -0.95*** (0.27) (0.27) (0.26) 10th-grade % lep -0.88 -0.89 -0.54 (0.54) (0.54) (0.48) public school -1.11 -1.13 -1.25 (catholic/other private omitted) (1.82) (1.83) (1.62) english proficiency level of proficiency 1.14** 1.62*** (0.39) (0.42) track level placement general/vocational -2.07** -2.61** (.64) (.93) interactions non-native x gen/voc 5.67** (2.05) mosqueda compounding inequalities journal of urban mathematics education vol. 3, no. 1 71 non-native x proficiency x gen/voc -.64** (0.21) random effects within schools (τ00) 18.30 11.19 11.97 11.19 between schools (σ2) 66.20 62.01 60.20 59.90 chi-square 1165.74*** 995.34*** 1028.03*** 1004.99*** note. 1variance component indicating whether there are differences between schools.2 indicates the amount of residual (unexplained) variance within schools. * p< .05; ** p < .01; *** p<.001 figure 1. main effects plot of mathematics performance on the els: 2002 assessment by english proficiency and tracking because the interrelationships of the variables were complex, they are illustrated graphically on the fitted plot in figure 1 for students with a teacher who was both certified to teach and had a degree in a mathematics or related field, and with all other control predictors set to their mean in the latina/o subsample. it is important to note that, because there was no measured variation in the english proficiency of native english speakers, in figure 1 the horizontal line in the figure represents their mean mathematics achievement. thus the horizontal fitted (dashed) line serves only as a reference for comparison with the mathematics 0
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 mosqueda compounding inequalities journal of urban mathematics education vol. 3, no. 1 74 due to the complexity in the interpretation, these findings are represented graphically in figure 2, for students with credentialed teachers who also have a background in mathematics, and holding all other control predictors constant. in figure 2, latina/o non-native speakers in the college preparatory track with high levels of english proficiency scored as high as native english speakers in the same high track. however, the mathematics scores of latina/o els in the college preparatory track were as low as the scores of els with the same low level of english proficiency in the general track. discussion present mathematics achievement patterns continue to reflect disparities in the mathematics performance of latinas/os. performance outcomes generally show that latina/o students have disproportionately low standardized test scores compared to white students, and that the scores of latina/o els are even lower. in spite of the extant research, we only have a limited understanding of the reasons for the persistence of these patterns of low performance for latina/o els. this study examined multiple factors related to the mathematics performance of latina/o native and non-native english speakers, including english proficiency and academic tracking while controlling for individual characteristics and important aspects of the school context. particular attention was placed on the lowest achieving students, latina/o els, to examine whether the relation between low levels of english proficiency and academic tracking exacerbated their already low mathematics scores on standardized assessments. the findings revealed that the relationship between academic tracking and the level of english proficiency of non-native speaking latina/o students is indeed an important predictor of their performance on standardized mathematics tests. the analysis showed that the mathematics test scores of latina/o els are considerably lower than the test scores of both non-native english proficient students and native english speakers. for example, at the lowest level of english proficiency (engprof = 3), latina/o els score nearly 7 points—equaling over twothirds of a standard deviation—below both latina/o english proficient (engprof = 8) and latina/o native speakers. a test score difference of this proportion is substantial and alarming. the findings also showed that tracking has a negative effect on the mathematics achievement of both latina/o native and non-native english speakers. this finding suggests exposure to rigorous mathematics content plays an important role in mediating the mathematics test scores of all latinas/os. not surprisingly, and consistent with the aforementioned literature, latina/o native and non-native english speakers with varying levels of english proficiency who are placed in the general academic track have lower mathematics achievement scores than their mosqueda compounding inequalities journal of urban mathematics education vol. 3, no. 1 75 peers in the college preparatory track (callahan, 2005; wang & goldschmidt, 1999). for all latina/o students, the effect of low-track placement is linked to a lower test score of 2.6 points (or one-fourth of a standard deviation) on average, compared to students in the college preparatory track. this difference in test scores is also sizeable. this study also examined whether the effect of tracking on mathematics performance differs by the level of english proficiency of non-native english speakers relative to the scores of native english speakers. these findings indicated that the level of english proficiency of non-native english speakers is more important for predicting the mathematics performance of els in the college track compared to els in the general track. the analysis also showed that the impact of tracking varies as a function of the level of english proficiency of non-native englishspeaking latinas/os. these findings revealed that while having a low level of english proficiency can disadvantage els in the college preparatory track, when non-native english speakers acquire a high level of english proficiency, they outperform their english-proficient peers in the general track, and score as high as native english speakers in the college preparatory track. although this study was not designed to identify the specific processes that explain the ways in which english proficiency matters more for latina/o els in the college preparatory track compared to their peers in the general track, there are possible explanations and implications for this finding. for example, the unexpected finding that latina/o els in the college preparatory track scored as low as els in the general track highlights how english proficiency is a more important factor for latinas/os in the college track than for those in the general track. this finding, however, is explained in part by research reviewed earlier, which argues that sophisticated mathematics-specific discourse and the complexity of the rigorous mathematics content itself demands a high degree of english proficiency (cuevas, 1984; garrison & mora, 1999; gutiérrez, 2002; khisty, 1995, 1997). these findings also raise questions about the long-term effects of general track placement for latina/o els after they reach a high level of proficiency in english that should be investigated in future research. the english proficiency level of latina/o els will invariably improve over time, but if els are relegated to remedial mathematics instruction in the general tracks they will not benefit from the potential higher performance advantages associated with both high levels of english proficiency and access to college preparatory mathematics content that are reported in this study. findings from this study challenge the practice of making english proficiency a prerequisite for enrollment in rigorous mathematics courses. given the complexity of mathematics language, deficit theories that limit els’ opportunities to take challenging courses based on their level of english proficiency will con mosqueda compounding inequalities journal of urban mathematics education vol. 3, no. 1 76 tinue to have long-term negative effects on their learning and test performance because this practice delays their entry into rigorous courses until they reach an “academic” level of english proficiency. given that academic proficiency may take anywhere from 3 to 7 years to develop, by the time els develop a level of english proficiency deemed appropriate to handle the linguistic complexity of secondary school mathematics content, it may be too late for them to take and meet the mathematics requisites for graduation and college attendance. furthermore, deficit-oriented practices that lead to the placement of els in low-level mathematics courses solely based on their english proficiency seem to unjustifiably use english proficiency as a proxy measure of an els’ mathematics capacity. conclusion this research lends support to the argument that the mathematics performance of latina/o students on standardized tests is mediated by factors at the institutional level, along with individual-level characteristics. therefore, explaining the achievement differences in mathematics test scores among latina/o native and non-native english speaking students solely in terms of individual characteristics can lead to inappropriate conclusions given the impact that institutional factors, such as low-track placement, have on lower mathematics test scores. the results of this study have important policy implications because of the dispersion of latinas/os and latina/o els to states without experience with how to best educate these students. a critical first step is that english proficiency not be used to limit their access to rigorous courses. states with increasing populations of english learners in particular must avoid reproducing practices that diminish the potential performance of els by not restricting their access to challenging mathematics courses. beyond policy implications, the higher mathematics performance of latinas/os in the college preparatory track reported in this study suggests that future research should analyze the impact of instructional approaches that simultaneously promote learning the rigorous mathematics content of the college track as latina/o els also develop both their proficiency in english and their mathematics language register. in other words, future research needs to investigate whether the provision of native language support can mitigate the negative relationship among the low english proficiency of els, track placement, and the mathematics performance patterns reported in this article. acknowledgements the research reported in this article was generously supported by the spencer foundation doctoral dissertation fellowship, and by the center for the mathematics education of latinas/os (cemela). i wish to thank john willett, thomas hehir, vivian louie, patricia gándara, robert ca mosqueda compounding inequalities journal of urban mathematics education vol. 3, no. 1 77 praro, mary capraro, kip telléz, laura lopez-sanders, and maria rendon for feedback on earlier drafts of this article. the opinions expressed in this article are mine and do not necessarily reflect the views of the funding agencies. references abedi, j., & lord, c. 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(2003). descriptive study of services to lep students and lep students with disabilities (no. 4 special topic report: findings on special education lep students). arlington, va: development associates. microsoft word author approved clark jerome vol 16 no 1.docx journal of urban mathematics education june 2023, vol. 16, no. 1, pp. 72–95 ©jume. https://journals.tdl.org/jume daniel l. clark is an assistant professor in the department of mathematics at western kentucky university, 1906 college heights boulevard, #11078, bowling green, ky 42101; email: daniel.clark@wku.edu. his research focuses on policy, program structure, communication, and equity in mathematics education. angela m. jerome if a professor in the department of communication at western kentucky university, 1906 college heights boulevard, bowling green, ky 42101; email: angela.jerome@wku.edu. her research agenda focuses on organizational rhetoric, particularly in the areas of crisis communication and image repair. applying symbolic convergence theory to pre-service teachers’ responses to mathematics education organizations’ statements on racial violence daniel l. clark angela m. jerome western kentucky university western kentucky university in june 2020, the national council of teachers of mathematics (nctm) and the association of mathematics teacher educators (amte) released statements regarding racism, racial violence, and mathematics teaching. pre-service elementary teachers (psts) in a mathematics content course wrote reactions to the organizations’ statements. after using an emergent coding process to code the reactions for major themes, the authors used a theory from communication studies called symbolic convergence theory (sct) to analyze how closely the psts’ understanding of the statements aligned with the vision espoused by the organizations in the statements. psts largely understood the need to make their classrooms safe and supportive spaces; however, they struggled to connect antiracist ideals specifically to mathematics teaching. the authors discuss potential ways nctm and amte can address this disconnect. keywords: symbolic convergence theory, mathematics education, antiracist teaching, organizational communication in june 2020, following the deaths of george floyd, breonna taylor, and ahmaud arbery, the national council of teachers of mathematics (nctm) and the association of mathematics teacher educators (amte) released social justice statements regarding racism (see appendix a & b for transcripts of the statements). even a cursory reading of the statements makes it clear that both organizations were working to advance the following rhetorical vision: the math classroom must be an antiracist space. for example, the nctm statement argues, “as mathematics educators, we must engage in anti-racist and trauma-informed education in our daily practices as processes of learning and adjustments” (2020, para. 2). in a similar vein, the atme statement contends, clark & jerome field disruptions and field connections 74 we must actively work to be anti-racist in our acts of teaching, research, and service. today we call on you to not simply express allyship, but to engage with a new resource to strengthen your own ability to see and to act in ways that are anti-racist and to critically examine your own practices and the biases implicit within them. (2020, para. 5) though several mathematics education scholars have advocated this vision for some time, prominent mathematics education organizations have only more recently made this an explicit part of their mission (battey & leyva, 2016; gutiérrez, 2017; martin, 2009, 2013). how well this vision has chained out or will chain out to new members of the math education community, k-12 pre-service teachers (psts), is unclear. yet, understanding this phenomenon is particularly important for urban mathematics education and k-12 students of color for a number of reasons. first, before teachers and psts can begin to address the problem, they must realize how race has influenced mathematics education to the point of making it a white institutional space. martin (2013) discussed how mathematics education is used to reproduce the current state of capital and market-based ideas that are largely biased in favor of white people. furthermore, battey and leyva (2016) devised a framework to study the impact of whiteness on mathematics education that both addressed this reproduction of racial privilege and documented the harm caused to minoritized students. the framework focused on three dimensions of mathematics education as a white institutional space: institutional, labor, and identity. leyva (2021) later offered another framework for studying whiteness in mathematics education along institutional, ideological, and relational dimensions. one example is the ideological dimension of leyva’s (2021) framework in which he states that this dimension concerns how mathematics achievement is often viewed as a meritocracy. as goffney et al. (2021) noted, teachers often compare “students’ academic performance without accounting for inequities in schools and districts” (pp. 13–14). such comparison results in the perceived existence of an achievement gap, which, in turn, only serves to cause further harm to minoritized students (gutiérrez, 2008; martin, 2009). some teacher beliefs also impact on the institutional dimension of the framework (leyva, 2021). battey et al. (2021) showed how racial bias among teachers can result in differential evaluation of students’ mathematical thinking. the psts in their study were prone to evaluating “black students’ thinking less favorably compared with white students” (p. 62). even if unintentional, this ultimately leads to a phenomenon known as the “soft bigotry of low expectations” (rubel & mccloskey, 2019, p. 120). when mathematics teaching and learning is racialized, the effects on minoritized students in general and black students in particular are many and varied (martin et al., 2019). moreover, though this process begins at the elementary school level, it continues and grows throughout a student’s educational experience. it affects choices regarding course selection and feelings of belonging in upper-level stem courses clark & jerome field disruptions and field connections 75 (leyva et al., 2021). nishi (2021) also found that at the university level, white college algebra students would often exclude minoritized students even if they were assigned into a working group together. the disparities throughout students’ mathematical educations have profound impacts on students’ economic prospects in their lives after exiting formal education (battey, 2013). once teachers recognize the problem of racism in mathematics, teachers can take actions to reduce the harm caused by racist structures. instances in the literature show mathematics teachers and educators critically examining their beliefs about race and how they affect their instruction so that they can work to make antiracist changes (anderson et al., 2020; fonger, 2022). one example of an action teachers can take at the classroom level is considering problem contexts. many mathematical problem contexts come from the perspective of consumers or employees, whereas few mention race or inequality (rubel & mccloskey, 2021). battey and coleman (2021) provide an example of having students use police data in a mathematics assignment as a way to engage in antiracist mathematics teaching. at a broader level, mathematics education researchers have also called for the enactment of a “more race conscious mathematics curriculum” (battey, 2013, p. 332), the creation of mathematics curriculum content that centers black people (matthews et al., 2021), and moves from being merely inclusive toward creating a “black liberatory mathematics education” (martin et al., 2019, p. 45). of course, none of these changes, small or large, can be enacted by teachers on a broad scale without making them accessible to and usable by teachers (gutiérrez, 2008). and before that, teachers must desire to enact antiracist mathematics education and want to seek out information on how to do that. the purpose of this manuscript is threefold. first, in her call for field disruptions and field connection in mathematics education research, cannon (2020) challenged researchers in the field to seek out underutilized theories and methods to advance research in areas like the one outlined above. we argue that theories developed by scholars in the communication discipline offer one potentially fruitful starting point for such endeavors. as such, we introduce mathematics education researchers to one such theory, symbolic convergence theory (sct). second, we apply sct to the statements of nctm and amte and examine the responses of psts to those statements to explain their effectiveness in persuading psts that being antiracist must be central to future work. third, we offer theoretically grounded strategies for improving future messaging. literature review sct explicates the development of group consciousness (bormann, 1985). to understand how/why this theory is useful in analyzing not only the messages offered by nctm and amte but also whether or not the messages were successful in clark & jerome field disruptions and field connections 76 persuading psts to share their rhetorical vision, it is important to thoroughly outline the five-decade development of the theory before proceeding. studying the processes of classroom groups, bales (1951) discovered that classroom groups, particularly zero-history groups, dramatized communication such that they developed fantasies that chained out (as cited in bormann, 1972). fantasies, in the realm of sct, are not figments of the imagination as in fairytales; rather, they are the means by which “communities of people create their social reality” (bormann, 1982b, p. 52). though fantasies may contain imaginary characters, they regularly address real-world events, arouse emotions, and move people to action (bormann, 1985). bormann (1972, p. 397) termed “the composite dramas which catch up large groups of people in a symbolic reality” rhetorical visions. bormann (1972) extended bales’ work by applying it to other types of groups because dramatizations, once they have chained out in small groups, are integrated into speeches and media content, which then may chain out to mass audiences. bormann (1982a) also noted, “our studies of consciousness raising communication theory and of such groups in action revealed that the technique was essentially that of intentionally dramatizing highly emotional fantasies drawn from an established rhetorical vision to induce neophytes to share them” (p. 291). further, he noted that other studies found that analyses of target audiences were sometimes used to plan what fantasies could be used to persuade them. therefore, he contended that a rhetorical analysis of the fantasy themes that chain out to create the rhetorical visions of groups provide rhetorical critics “a way to examine messages for insight into the group’s culture, motivation, emotional style, and cohesion” (p. 396) and that fantasy theme analysis is powerful because of “its ability to account for the development, evolution, and decay of dramas that catch up groups of people and change their behavior” (p. 399). such analyses, he argued, are useful because they help scholars understand movements. building on the work of cragan and shields (1998), bormann et al. (2001) provided a comprehensive summary of the history, development, and anatomy of sct. as the theory developed, its terms and definitions shifted. as such, bormann et al.’s (2001) definitions of the technical concepts of sct relevant to the current study are used herein for consistency. bormann et al. (2001) argued the first important aspect of the anatomy of a theory is to identify its basic concept or “the thing one must be able to find and identify” to understand a particular theory (p. 282). they claimed that the basic concept clark & jerome field disruptions and field connections 77 of sct is the fantasy theme.1 a fantasy theme “is a dramatizing message that depicts characters engaged in action in a setting that accounts for and explains the human experience” (p. 282). according to bormann (1982b), this could be stories about living or historical persons or about an imagined future. in essence, fantasy themes allow rhetors2 “to make visible (understandable) a common experience and shape it into social knowledge” (1982b, p. 52). when a rhetor decides which person(s) to include in the fantasy theme, what the scenic elements will be, places events in a certain order, and/or assigns motives to persons, they are working to form a specific rhetorical vision (bormann, 1985). the principle of explanatory power is central to the process of developing a new consciousness. according to bormann et al. (1996), “the principle of explanatory power asserts that, when events become confusing and disturbing, people are likely to share fantasies that provide them with a plausible and satisfying account that makes sense out of experiences” (p. 3). the second anatomical aspect laid out by bormann et al. (2001) are the message structure concepts. they argued that rhetorical vision is the central message structure concept of sct and retained the definition of rhetorical vision (bormann, 1972) as outlined above. however, they also noted that there are four substructural elements of a rhetorical vision: “dramatis personae or characters, plot lines or action, scene or setting, and sanctioning agent or legitimizer for the rhetorical vision” (2001, p. 285). rhetorical visions have a five-stage lifecycle: consciousness-creating, consciousness-raising, consciousness-sustaining, consciousness-decline, and consciousnessterminus (bormann et al., 1996). the first stage is central to the current study. the consciousness-creating stage “involves the sharing of fantasies to generate new symbolic ground for a community of people” (p. 2). bormann et al. argued that consciousness-creating, or the creation of a new rhetorical vision, are grounded in novelty, explanatory power (defined above), and imitation. they argued, the principle of novelty asserts that, when established visions lag behind changing hereand-now conditions, they will often fail to attract members of the second and third generations of those who inherit them. as old visions lose their vitality, rhetoricians who use an innovative set of dramatizations will find fallow ground among substantial segments of the lukewarm inheritors of the older visions. (p. 3) 1 though symbolic cues, fantasy types, and sagas are also basic concepts of the theory, no evidence of their use was found in the messages analyzed for this study. therefore, they were not outlined in this literature review. 2 rowland (2012) asserts, “rhetoric involves the use of symbols (primarily language) to persuade or inform” (p. ix). therefore, a rhetor is one who persuades or informs. clark & jerome field disruptions and field connections 78 further, “the principle of imitation asserts that, with boredom or confusions, people begin to share fantasies that give some old familiar dramas a new production” (p. 3). visions are tagged here with cues like new and neo to help people draw on familiar, historic heroes and values. third, bormann et al. (2001) outlined the dynamic structure concepts of the theory or the tensions/wars that underlie the theory. in sct the tension/war occurs between three competing rhetorical visions (sometimes referred to as master-analogues): righteous, social, and pragmatic. righteous rhetorical visions “stress correctness, the right way, morality, and so forth” (p. 288). social rhetorical visions “stress such elements as humaneness, social concern, family, brotherhood and sisterhood, and so forth” (p. 288). pragmatic rhetorical visions “stress such elements as the bottom line, what will work, what is expedient, and so forth” (p. 288). fourth, bormann et al. (2001) laid out the communicator structure concepts of sct; these concepts “focus on the names given to communicators from the lens of a particular theory” (p. 288). for sct, the communicator structure concepts are “fantasizers and rhetorical community along with their attributes such as propensity to fantasize and dramatistic communication style” (p. 288). central to sct is the investigation of fantasizers. the current study also allows for an examination of the existence (or lack thereof) of a rhetorical community. rhetorical communities, as defined by bormann (1985), are “the people who participate in a rhetorical vision” (p. 53). therefore, assessing how psts communicate about the messages put out by different math associations can help us understand whether or not they buy into the rhetorical visions being laid out by the associations. fifth, bormann et al. (2001) laid out the medium structure concepts of sct, noting that media are the “propagating substance” of a theory or what makes it grow (p. 290). growth of fantasies are beholden to group or public sharing. without substantial fantasy chaining, the creation and maintenance of rhetorical visions are stunted because there is little or no shared communication and, therefore, no shared consciousness in the rhetorical community. last, bormann et al. (2001) overviewed the evaluative concepts of sct. these are: fantasy theme artistry, shared group consciousness, and rhetorical vision realitylinks. fantasy theme artistry “concerns the rhetorical creativity, novelty, and competitive advantages of fantasy theme, symbolic cues, fantasy types, rhetorical visions, and sagas” (p. 291). it is also important to note that when examining fantasy theme artistry, scholars must assess closeness of fit, meaning a fantasy theme must “fit its rhetorical community and be consistent with other fantasy themes of the rhetorical vision” (p. 292). shared group consciousness is simply looking for evidence of symbolic convergence. according to bormann (1985), when elements of the group fantasy recur in group meetings and other contexts, it is evidence of symbolic convergence. reality-links “tie rhetorical visions and fantasies to the objective reality of the public record and material facts” (p. 293). as noted by bormann et al. (2001), studies clark & jerome field disruptions and field connections 79 that found a lack of symbolic convergence have attributed that failure in the absence of artistry, reality-links, novelty, channel access, and/or “competing, symbolic consciousnesses that provide a better accounting of the phenomena being explained” (p. 293). though the bulk of sct studies look at fantasy themes, rhetorical visions, and the chaining of messages sent by one organization or individual, sct’s conceptualization of rhetorical communities and prior studies (e.g., kroll, 1983; huxman, 1996) afford the rhetorical space for scholars to use the messages of more than one entity to examine the existence or shift in rhetorical visions, particularly when they are tied to the early stages of the life cycle of rhetorical visions outlined above. for example, using the rhetoric of two different women’s movement organizations, kroll (1983) explained how the fantasy themes, types, and rhetorical visions of the women’s movement in the twin cities of minneapolis and st. paul required subtle but necessary shifts to advance the movement beyond the consciousness-raising stage. in doing so, kroll thoroughly outlined how the fantasy themes and types advanced by each organization led to the success or failure in building support for or opposition to the organizations’ rhetorical visions. huxman (1996) argued that cross-movement rhetorical analyses using sct as a guide also have a place in advancing knowledge because they allow scholars to find the conceptual core of multiple movements. to make her point, huxman used the rhetoric of three pillars of the women’s movement, operating in different eras of the movement, to demonstrate how their problem congruence as well as their congruence on ideational and stylistic levels undergirded and helped advance the rhetorical vision “codified in the expression of rights at seneca falls” (p. 26). therefore, she argues that examining cross-movement rhetoric allows scholars to examine more deeply and celebrate the achievements of individuals in a movement. although the work of kroll and huxman is encouraging, and other studies look at how sct can be used to improve organizational and small group communication (e.g., bales, 1951; bormann, 1975; bormann et al., 1978; cragan & shields, 1992), more work is needed. as indicated in a study by gilmore and kramer (2019), which analyzed how public school teachers used fantasy themes to navigate the changing nature of education in the united states and develop shared identity, education is an excellent site for advancing this mission. as mathematics educators who have pushed for the mathematics education field to take on a more antiracist social justice perspective have been subjected to significant online and media backlash and harassment (e.g., amte, 2018; gutiérrez, 2017, 2018), the current study offers a way to both fill gaps in the sct literature and provide mathematics educators who support the antiracist social justice rhetorical vision with tangible feedback on how their rhetorical vision is being received and interpreted by psts. this is central to predicting whether future members of the rhetorical community in question would rally around clark & jerome field disruptions and field connections 80 the new rhetorical vision and, if not, offering guidance for adapting the themes to be more persuasive to this audience in future communications. positionality statement understanding the authors’ positionality can be helpful in understanding how the data below were collected, viewed, analyzed, and discussed. both authors are professors at a university in the south of the united states. the first author is a white, u.s.-born mathematics education scholar who has expertise in mathematics education policy, in particular how different stakeholders in mathematics education view their work, communicate with each other, and understand their responsibilities to others. he led the data collection and original coding and analysis process. the second author is a white, u.s.-born communication scholar who has expertise in organizational communication and rhetoric, particularly in the areas of persuasion, sport, crisis communication, and image repair. as such, she led the analysis of the nctm and amte statements using sct as the framework for analysis. given our commitment to diversity, equity, and inclusion and our interests in communication in general and within the mathematics education community, we were well-positioned to explore this project. site and participants the present study included the participation of 27 elementary and/or special education psts enrolled in the third of three mathematics content courses in their program during the spring 2021 semester at a large, regional, public university in the mid-south. the university is classified as a doctoral/professional university with very high undergraduate enrollment (american council on education, n.d.). twenty-six of the psts were female, and one was male. three students were african american, and the remainder were white. given the sensitive nature of the assignment described below, the african american students were given the option to do a different assignment for the same amount of credit. all three chose to do the assignment described below. method the authors of the present study sought to understand what psts understood and took away from the nctm’s and the amte’s june 2020 statements on the deaths of george floyd, breonna taylor, and ahmaud arbery. as part of a larger assignment, the psts were given two brief paragraphs describing the missions of nctm and amte and the two statements mentioned above. the psts were asked to write a paragraph summarizing the nctm statement, a paragraph clark & jerome field disruptions and field connections 81 summarizing the amte statement, and to write a half page responding to the following prompt: “considering your role as someone who is preparing to become and will soon be a math teacher, write at least half a page in reaction to these two statements. the reaction is yours to write as you wish, but feel free to include any questions or uncertainties you may have.” to identify whether or not symbolic convergence had occurred among the participants, the nctm and amte statements first had to be rhetorically analyzed using fantasy theme criticism. according to foss (2017), a fantasy theme criticism requires three steps prior to writing the research essay: selecting the artifact(s), analyzing the artifact(s), and formulating a research question(s). she asserted that the artifact(s) selected should demonstrate that at least some symbolic convergence has occurred. the nctm and amte statements were chosen for student response because of the two organizations’ prominence in the mathematics education community. nctm is the world’s largest mathematics education organization, boasting tens of thousands of members, including k-12 teachers and other mathematics education professionals. amte is america’s leading organization for mathematics teacher educators at colleges and universities. though both statements were written by the leadership of the organizations, there is evidence in each that a number of current members agree with the statements. for example, the amte statement ends with, “while the words in this statement were assembled and edited by president mike steele, presidentelect megan burton, and executive director shari stockero, they originate in large part from the lived experiences of educators of color within amte leadership who contributed their perspective and wisdom” and then it lists the names of all who contributed. while nctm’s statement is not as explicit, it harkens back to the work of other mathematicians who made similar statements after events in charlottesville, va, in 2017 and references its own catalyzing change series, which has already begun addressing these issues. further, because these statements were put forth by associations with large memberships, it stands to reason that many of their members agree with the statements. to analyze the artifacts, foss (2017) established that researchers must first code for fantasy themes, carefully examining the artifact(s) to discover settings, characters, action themes, and sanctioning agents (if they exist). then, the researcher must construct the rhetorical vision from the fantasy themes, which requires one “to look for patterns in the fantasy themes” (p. 113) in order to “come to some conclusions about the worldview constructed by the rhetor…” (p. 114). this project was guided by three, overarching research questions: 1. what were the fantasy themes and rhetorical visions developed in each statement? 2. where, if at all, did the two statements converge on themes and vision? clark & jerome field disruptions and field connections 82 3. how, if in any way, did those themes and visions chain out to the participants? to answer the first two questions, the second author performed a fantasy theme criticism on each message, looking for themes in each message and then any thematic convergence or divergence between the messages. the first author then verified those findings. to answer the third question, the pst’s reactions were coded using an emergent coding process. two researchers began the analysis process by individually reading all of the psts’ reactions. during that initial reading, each researcher made notes about general themes they saw in the reactions. then both researchers met to develop and agree upon a coding scheme for the reactions. once the coding scheme was complete, the researchers individually reread all of the psts’ reactions. during this reading, the researchers coded approximately sentence-sized pieces of the psts’ reactions according to the coding scheme. then both researchers met, discussed the reactions and the coding scheme, and resolved all differences in codes in the psts’ reactions. in addition to this process, reactions were also coded for whether the pst identified their race or their hometown within the body of the reactions. in the larger analysis, 13 codes emerged and were applied to the psts’ reactions by the researchers. for the purposes of this article, the focus will be on a subset of those codes: statements having to do with teaching mathematics for social justice, statements generally about mathematics, statements of concern about their own classroom practice, and statements about whose responsibility it is to take action regarding the content of the statements as well as the proposed actions. collectively, these are five of the 13 codes in the coding system. the codes regarding taking action comprised the most often used codes in the coding scheme and had the most students with comments coded as such. therefore, they seemed to be the most important to discuss. similarly, considering the purpose of the organizations’ statements, psts’ views on teaching mathematics and social justice were important to include in this analysis. most of the remaining codes were either sparsely used or highly concentrated in the reactions of just a few psts. message analysis it is important to note here that many members of nctm and amte may be in the consciousness-raising and consciousness-sustaining phases of the life cycle of the rhetorical vision being advanced in these statements as these organizations have been advancing this rhetorical vision for at least half a decade (e.g., amte, 2017). however, the psts were likely unaware of this fact. the psts studied were nonmembers of either organization; they are potential future members. therefore, most, if not all, participants of this study inhabited the clark & jerome field disruptions and field connections 83 consciousness-creating phase of the sct life cycle, meaning these messages represented new symbolic ground for them to process. even if the psts were aware of the rhetorical vision being advanced, the deaths of george floyd, breonna taylor, and ahmaud arbery clearly provided a new setting in which to advance their novel rhetorical vision. these tragic events were reality-links for the rhetorical vision advanced by nctm and amte. these three instances of objective reality were facts upon which these organizations could argue that old rhetorical visions for the math classroom were lagging behind current conditions, establishing a solid foundation for this novel rhetorical vision. as noted above, it is clear the nctm and amte represent similar rhetorical communities and are advancing the same righteous rhetorical vision: the math classroom must be an antiracist space. in fact, both explicitly state as much. fantasy themes while each statement differed in rhetorical style and fantasy theme artistry, both organizations, acting as sanctioning agents, shared three primary, action-oriented themes in support of this vision. math classrooms (and their affiliated organizations) must be supportive/safe spaces. first, nctm’s statement couples antiracism with trauma-informed education, indicating it understands the link between the two. not surprisingly, nctm’s third position statement directly addresses the need for math classrooms and affiliated organizations to be safe spaces; it states, “we encourage all educators to create socially and emotionally safe spaces for themselves, their students, and colleagues” (para. 3). amte’s statement on this issue begins, as mathematics teacher educators, each of us must be cognizant of the lived experience of black americans by reading the history of the united states through a social justice lens. next, we must learn ways to empower and provide access to students who often are judged by the color of their skin and not by their knowledge and abilities. (para. 2) further, amte hits on this theme, in whole or in part, in four of its qualities describing well-prepared math educators; it notes these educators should “recognize their responsibility to cultivate positive math identities with their students,” “identify and implement practices that draw on students’ mathematical, cultural, and linguistic resources/strengths,” “understand the roles of power, privilege, and oppression in the history of mathematics education,” and be “knowledgeable about, and accountable for, enacting ethical practices that enable them to advocate for themselves and to challenge the status quo on behalf of their students” (para. 2). clark & jerome field disruptions and field connections 84 while analyzing the data, the researchers coded two areas with respect to taking action: who psts said was responsible for taking action and what actions the psts proposed. overall, 20 psts (74%) stated that some action was needed in response to the organizations’ statements. among those, 19 psts (70%) made 45 statements noting that teachers, schools, districts, and/or the field of education should act, whereas only 14 psts (52%) made 29 statements saying they themselves specifically needed to act. in general, this theme chained out to the psts significantly more than the other two fantasy themes. thirty-three of the psts’ proposed actions (almost half of the total proposed actions) were along the lines of creating a safe environment, leaving no one out, and/or holding students accountable for racist remarks. when psts proposed these actions, though, the context tended to be either nonspecific, superficial, and/or related to classroom management. extremely few connections were made to the four amte qualities describing well-prepared math educators quoted in the previous paragraph. antiracism is everyone’s job (not just the job of those in the bipoc community). nctm’s second position addresses this notion: “we encourage all educators to challenge systems of oppression that privilege some while disadvantaging others” (para. 3). it also hits on this theme by using the inclusive term “we” when discussing those who must act. for example, nctm asserts, “anti-racist and trauma-informed education not only raises our awareness of racism and trauma experienced by black, latinx, asian, and all marginalized peoples, but it also recognizes that we must be purposeful in addressing racism and trauma” (para. 3). amte’s statement is more explicit in this regard: “we cannot look at what is happening to black americans and other oppressed groups as problems that they alone need to solve” (para. 1). further, after its bulleted list of attributes of well-prepared educators, amte argues, “we as an organization strengthen our own ability to serve as advocates for those whose voices have been muted and prepare a generation of teachers who are willing to address the systemic problem of inequity in our schools, nation, and world.” (para. 3). the statement also notes that its specific call to action is aimed at white members of the math education community, asserting “it is long past time that we assume the burdens that have been largely left to mathematics educators of color” (para. 5). this theme of antiracism being everyone’s responsibility chained out significantly less with the psts than the theme regarding creating safe and supportive spaces. nine statements were made that noted the need for advocacy, with some psts specifically citing their own white privilege as a reason they need to be involved in advocacy. eight psts (30%) noted a need to educate themselves to clark & jerome field disruptions and field connections 85 mitigate bias, and seven psts (26%) noted a need to educate themselves on the black experience. seven psts (26%) discussed the need to be culturally inclusive in their classrooms, including things like having representative classroom libraries. one pst specifically mentioned the homework gap, how that specifically affects students of low socioeconomic status and otherwise minoritized students, and how they would address it. in this study, none of the psts mentioned taking any action that involved mathematics or mathematics teaching specifically with respect to antiracism. while analyzing the data, the researchers coded two areas with respect to taking action: who psts said was responsible for taking action and what actions the psts proposed. overall, 20 psts (74%) stated that some action was needed in response to the organizations’ statements. mathematics is an appropriate site to challenge power, privilege, and oppression. not surprisingly, both statements call for math educators to go far beyond the provision of equal access, calling for them to use their classrooms and careers as a site to challenge power, privilege, and oppression. in fact, nctm’s first sentence is “as president and past president…we are committed to a position of social justice that challenges the roles of power, privilege, and oppression” (para. 1). further, its statement reiterates calls made by larson and berry (2017) following events in charlottesville, va. the first of its positions states, “we support the use of mathematics as an analytical tool to challenge power, privilege, and oppression” (para. 3). further, it states, “as nctm’s catalyzing change series advocates, we need to engage in critical conversations that urge educators to create structures where each and every student can be fully engaged in our democratic society” (para. 4). likewise, amte notes that well-prepared math educators will be able to “recognize the difference between access to and advancement in mathematics learning and work to provide both access and advancement for every student,” “challenge policies and practices grounded in deficit-based thinking,” and “question existing education systems that produce inequitable learning experiences and outcomes for students” (para. 2). notably, amte goes a bit further than nctm within this theme, arguing that it and its members need to go beyond the math classroom, partnering with organizations like black lives matter, to dismantle systemic racism in this country. this theme chained out least among the psts studied. in the 27 psts’ reactions, four psts (15%) each made a single nontrivial statement generally about mathematics or mathematics teaching. here, nontrivial means a mention of mathematics itself beyond simply as part of the organizations’ names. the four statements mentioning mathematics were: clark & jerome field disruptions and field connections 86 • “i realize that mathematics knowledge is a powerful tool.” • “i really like how the nctm’s statement went about what they were saying. i completely agree that math can be a tool to challenge and empower.” • “as a future educator i am up for the challenge of making the world a better place one math problem at a time.” • “i know it is a math class … but everybody could benefit from reflecting on these two statements and learn something.” in terms of what nctm’s and amte’s statements were advocating for, there is not much in the psts’ reactions related to mathematics or mathematics teaching. two of the statements note that mathematics can be a powerful tool but go no deeper than that, the third makes a superficial reference to math problems, and the fourth merely notes that the statements were important to read and reflect on despite such reading happening in a mathematics content course. there were no statements in any pst’s reaction that attempted to apply antiracist ideas specifically to mathematics teaching or that discussed how mathematics could be used to challenge power, privilege, and oppression. furthermore, in the 27 psts’ reactions, three psts (11%) each made a single statement about concerns they have regarding incorporating the aims of the statements into their own classroom practice: • “i am concerned a bit because as a teacher, relating with your students is an important part of your effectiveness and i can recognize that there are some things concerning this topic that i will never truly understand.” • “parents of students may not agree with supporting and standing with african americans and supporting the black lives matter movement so it is important that as a teacher we can create a warm, welcoming environment for all students.” • “some uncertainties that i have are the colleagues that i have that do not believe that racism is a problem or do not see themselves as having white privilege. i am not sure what to do if i have teachers who are so adamant that there are not these problems or do not see their privilege or even worse do not believe in black lives matter. what would i do in this situation?” in all, one pst was concerned about being able to live up to the ideals of the organizations’ statements, while two others expressed concern about perceptions by others outside their classrooms. those concerns provide insight into why nctm and amte may have trouble bringing psts into the rhetorical community. as noted above, rhetorical visions rooted in righteousness “stress correctness, the right way, morality, and so forth” (bormann et al., 2001, p. 288). these concerns indicate that while psts might agree with the rhetorical vision being advanced, clark & jerome field disruptions and field connections 87 they fear the backlash they may face as a result of being an open part of the rhetorical community. when considering the general statements about mathematics and the statements of concern about incorporating antiracism into classroom practice together, it becomes clear that none of the psts truly considered both antiracist teaching practices and mathematics together in a single thought. given this lack of coordination of the two ideas, it is unsurprising that psts found the theme regarding creating safe and supportive spaces to be more salient than the other two themes. discussion there was little evidence that the psts studied symbolically converged with the rhetorical vision being advanced by these organizations. thus, these psts are not yet fully part of the rhetorical community. however, there are indicators that show that the first two fantasy themes are, at least partially, salient for the psts. because they are psts, one could argue they do not have a propensity to fantasize about this issue quite yet. however, the results above show that the psts view racism as they understand it negatively, and that they see the need to combat racism in their teaching. what is absent from the results is evidence that the psts were able to meaningfully connect the ideas of fighting racism and teaching mathematics as envisioned by the statements’ authors. part of the disconnect may be due to some psts’ narrow understanding of what racism is. if a pst merely views racism as intersecting their future classroom in the form of overtly racist remarks a student might make, then it would follow that their proposed actions would take the form of classroom management strategies. racism consists of much more than overt remarks, though. nctm’s (2020) and amte’s (2020) statements both clearly indicate that. furthermore, many of the psts demonstrated that they understand the expansive nature of racism as well, yet they were still unable to make meaningful connections to mathematics teaching in their reactions to the organizations’ statements. another part of the disconnect may be due to the sample of psts being overwhelmingly white. as noted at the outset of this article, mathematics education is a white institutional space (battey & leyva, 2016; leyva, 2021; martin, 2013). since this was the first experience many of these psts had being exposed to thinking of mathematics education in that way, it is not wholly surprising that they were resistant to the more ambitious themes (warburton, 2015). moreover, even if they are open to teaching mathematics for social justice, psts may inadvertently reinforce whiteness as they are learning to do so (harper et al., 2020). that said, the small but promising number of pst interactions with the second fantasy theme, antiracism being everyone’s job, give us hope that more white psts are taking up the mantle to do this work rather than engaging in the clark & jerome field disruptions and field connections 88 problematic behavior of leaving that work for the bipoc community (beard et al., 2021; han & leonard, 2017; miles et al., 2019). while we recognize that the target audiences for these statements were current organizational members, psts are future members. as this study indicates, psts are potentially open to engaging in antiracist mathematics teaching. thus, nctm, amte, and similar organizations may do well in advancing this novel rhetorical vision by targeting psts more directly in future messaging. to do this well, sct suggests ways to enhance future organizational statements aimed at psts to assist them in making the connection. the good news is both organizations are advancing the same rhetorical vision. thus, they are not advancing competing rhetorical visions, and the fantasy themes they have established to advance the vision are well aligned. what seems to be missing from the perspective of psts is a strong nod to the principles of explanatory power and imitation. in a few psts’ reactions, they stated that they wanted more than “just words.” those psts were reacting almost as they might to a soft drink commercial that promotes antiracism but awkwardly does nothing to connect the product to the idea, leaving viewers to wonder how a soft drink can be antiracist. by offering psts links to resources that can help them more concretely make sense of the world around them, psts’ propensity to fantasy may increase, which would aid in getting them to take up this vision and begin making statements that more clearly indicate fantasy chaining. for example, the organizations could be more specific about antiracist mathematics teaching strategies in the statements themselves. we acknowledge both nctm and amte have and promote a variety of resources for antiracist mathematics teaching. in their recent statements on anti-asian violence, for example, nctm, amte, and others (2021) directly referred and linked to sources to promote equitable mathematics teaching. such resources were absent from the statements analyzed in the current study. such direct referrals could be useful for psts. including links to resources in the statements could greatly assist psts in making the connection between antiracism and mathematics teaching. the organizations could even offer focused professional development opportunities to psts, practicing teachers, and university mathematics educators on how to incorporate materials like bartell et al.’s (2022) or activities like battey and coleman’s (2021) into their work to help them envision what an antiracist mathematics classroom can be. as demonstrated by other studies of sct, the 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(2019). the “soft bigotry of low expectations” and its role in maintaining white supremacy through mathematics education. occasional paper series, 41, 113– 128. https://doi.org/10.58295/2375-3668.1280 rubel, l. h., & mccloskey, a. v. (2021). contextualization of mathematics: which and whose world? educational studies in mathematics, 107(2), 383–404. https://doi.org/10.1007/s10649021-10041-4 warburton, t. t. (2015). solving for irrational zeros: whiteness in mathematics teacher education (publication no. 10010629) [doctoral dissertation, university of utah]. proquest dissertations publishing. clark & jerome field disruptions and field connections 92 appendix a june 1, 2020 a statement on george floyd, breonna taylor, and ahmaud arbery as president and past president of the national council of teachers of mathematics (nctm), we are committed to a position of social justice that challenges the roles of power, privilege, and oppression. we extend our heartfelt sympathies to the loved ones of george floyd, breonna taylor, and ahmaud arbery. as a mathematics education community, we must not tolerate acts of racism, hate, bias, or violence. many of you, your students, and colleagues watched the events in minneapolis, louisville, and brunswick, georgia, unfold on television and social media and have been affected by those incidents and the public reaction to them. the trauma of these developments has an impact on the social and emotional well-being of students and teachers in daily life and in classroom learning. our colleague matt larson reminds us that we teach more than mathematics. as mathematics educators, we must engage in anti-racist and trauma-informed education in our daily practices as processes of learning and adjustments. anti-racist and trauma-informed education not only raises our awareness of racism and trauma experienced by black, latinx, indigenous, asian, and all marginalized peoples, but it also recognizes that we must be purposeful in addressing racism and trauma. in august 2017 larson and berry made several calls to the mathematics education community in their response to the unrest in charlottesville, virginia. in this message we renew these calls. as educators, teachers of mathematics, and a council, we reiterate our position: o we support the use of mathematics as an analytic tool to challenge power, privilege, and oppression. o we encourage all educators to challenge systems of oppression that privilege some while disadvantaging others. o we encourage all educators to create socially and emotionally safe spaces for themselves, their students, and colleagues. as nctm’s catalyzing change series advocates, we need to engage in critical conversations that urge educators to create structures where each and every student can be fully engaged in our democratic society. we owe this not only to our students but also to the society we wish to inhabit both now and in the future. clark & jerome field disruptions and field connections 93 one either allows racial inequities to persevere, as a racist, or confronts racial inequities, as an antiracist. there is no in-between safe space of “not racist.” the claim of “not racist” neutrality is a mask for racism. (ibram x. kendi, author of how to be an antiracist, p. 9) trena l. wilkerson nctm president @trenawilkerson robert q. berry iii nctm past president @robertqberry clark & jerome field disruptions and field connections 94 appendix b to the membership of the association of mathematics teacher educators, the association of mathematics teacher educators (amte) stands in solidarity with black americans in the face of racial injustice. we are dismayed by the inhumane and unjust treatment of black americans by law enforcement personnel in recent months with the deaths of george floyd, breonna taylor, and ahmaud arbery. we acknowledge the inequities that the pandemic has illuminated related to health care, economic standing, and education. as an organization, amte believes that racism must be interrogated in this country. we cannot look at what is happening to black americans and other oppressed groups as problems that they alone need to solve. as mathematics teacher educators, each of us must become cognizant of the lived experience of black americans by reading the history of the united states through a social justice lens. next, we must learn ways to empower and provide access to students who often are judged by the color of their skin and not by their knowledge and abilities. we must ensure that we foster well-prepared teachers of mathematics who: • recognize the difference between access to and advancement in mathematics learning and work to provide both access and advancement for every student, • recognize their responsibility to cultivate positive mathematical identities with their students, • identify and implement practices that draw on students’ mathematical, cultural, and linguistic resources/strengths, and challenge policies and practices grounded in deficit-based thinking, • understand the roles of power, privilege, and oppression in the history of mathematics education and are equipped to question existing educational systems that produce inequitable learning experiences and outcomes for students, and • are knowledgeable about, and accountable for, enacting ethical practices that enable them to advocate for themselves and to challenge the status quo on behalf of their students (amte, 2017, p. 21–24). through these actions, we as an organization strengthen our own ability to serve as advocates for those whose voices have been muted and prepare a generation of teachers who are willing to address the systemic problem of inequity in our schools, nation, and world. clark & jerome field disruptions and field connections 95 in our lives as citizens, we must look to organizations whose mission is to respond to continued racial injustice and to eradicate white supremacy and build local power to intervene in violence inflicted on black communities by the state and vigilantes. we must stand with organizations like black lives matter that seek to elevate awareness of the lived experiences of americans of color and dismantle systems of continued racial oppression. we must act. we (mike and megan) issue this call to action to white mathematics teacher educators, including ourselves. it is long past time that we assume the burdens that have been largely left to mathematics educators of color. all of us must affirm and support the lived experiences of our students and colleagues of color who are and have been suffering. we must actively work to be anti-racist in our acts of teaching, research, and service. today we call on you to not simply express allyship, but to engage with a new resource to strengthen your own ability to see and to act in ways that are anti-racist and to critically examine your own practices and the potential biases implicit within them. a list of resources specifically related to our work as mathematics teacher educators is available on the amte member bulletin board. we invite others to submit additional resources to this site to be shared with our mathematics education community. to our colleagues of color, to our students, and to all who are suffering in this moment: we see you, we love you, and we support you. together as mathematics teacher educators, we will bend the arc towards justice. yours in service, mike steele, amte president megan burton, amte president-elect while the words in this statement were assembled and edited by president mike steele, president-elect megan burton, and executive director shari stockero, they originate in large part from the lived experiences of educators of color within amte leadership who contributed their perspectives and wisdom. this statement includes contributions from amte past presidents marilyn strutchens, christine thomas, and randy philipp; avp for equity carlos lopez leiva; avp for advocacy zandra dearaujo; vp for professional learning jennifer suh, and vp for publications babette benken. microsoft word ms 406 (proof 1).docx journal of urban mathematics education december 2022, vol. 15, no. 2, pp. 41–63 ©jume. https://journals.tdl.org/jume roland g. pourdavood, is a professor of mathematics education at cleveland state university, 2121 euclid ave., julka hall room 325, cleveland, oh 44115; email: r.pourdavood@csuohio.edu. his research interests include mathematics teachers’ dialogues and reflections for transformation and school reform, cultural diversity, socio-cultural aspects of education, and emancipatory action research for personal and social praxis. meng yan is a ph.d. candidate of learning and development in urban education at cleveland state university, 2121 euclid ave., julka hall, cleveland, oh 44115; email: m.yan@vikes.csuohio.edu. her research interests include curriculum development and instruction, parenting style and student well-being, student motivation, second language acquisition, language and thought, quantitative methods, and statistics. teaching mathematics and science through a social justice lens roland g. pourdavood cleveland state university meng yan cleveland state university teaching mathematics and science embedded in social justice is not a familiar concept for many teachers, especially pre-service teachers. this qualitative, descriptive, and interpretative study examines the experiences and reflections of 26 middle grade and secondary pre-service mathematics and science teachers on teaching and learning mathematics and science through the social justice lens as they took a semester-long course concurrently with their student-teaching. the primary research question was, “how may a semester-long course focusing on teaching and learning mathematics and science with social justice awareness provide pre-service teachers with opportunities to reflect on and change their teaching practices?” data included researchers’ field notes and participating pre-service teachers’ verbal discussions, written reflections, and classroom presentations. the findings suggest that teaching mathematics and science in the context of social justice enhanced the participating pre-services teachers’ awareness of educational opportunity and equity. the findings also indicate teaching mathematics and science from the social justice perspective requires a paradigm shift in teaching and learning. furthermore, the study exposes the limitations of the current school structure and culture for meaningful learning, the limitations of existing curricula and state-mandated texts, and the lack of adequate resources in teaching mathematics and science in social justice contexts. keywords: mathematics and science, social justice, teaching and learning pourdavood & yan teaching through a social justice lens journal of urban mathematics education vol. 15, no. 2 42 s the demographics of the united states become more diverse than ever, concerns remain about the growing number of racially and ethnically diverse populations. with most of the curricula, textbooks, and teaching materials focusing on the values and beliefs of the mainstream culture, the diverse groups receive little attention (banks, 2013; nieto, 2000, sleeter & grant, 1999). students from racially and ethnically diverse communities cannot see their lived experiences in the school curricula and establish personal connections with the content being taught. this makes them feel the school does not belong to them, and they seem to always ask, “why do i need to learn this?” historically, mathematics and science curricula have been presented from an ethnocentric perspective in the united states. students growing up in urban communities with low socioeconomic status, female students, immigrants, and lgbtqi+ students are too often overlooked within the classroom setting. understanding mathematics and science from a social justice perspective is not the way students across the country experience these subjects. as many scholars argue, teaching a traditional, western-minded mathematics or science course in a classroom with diverse learners is neither beneficial nor engaging; it not only hinders the utilization of students’ unique characteristics and funds of knowledge (gonzález et al., 2006) but also deprives students of the opportunity to use mathematics to expose and confront obstacles to their success (gutiérrez, 2002; gutstein, 2003; martin, 2003; tate, 1995). educating young citizens to become more engaged and critical of the environments they are living in and experiencing is a step toward social justice and equity in education. according to garii and appova, (2013), social justice is grounded in the daily realities of people as they experience and witness inequality and injustice. students can engage with social justice when the teaching of mathematics and science is done in an integrated way that promotes thinking in terms of relationships, connectedness, and context, which helps students form integrated knowledge and experience meaningful learning relevant to real life (drake, 2000). according to larnell et al. (2016), teaching and learning mathematics and science for social justice “represents an ideological commitment to using mathematics as a means of encouraging young students to leverage their own content learning toward redressing sociopolitical injustices in our society” (p. 20). in this sense, preparing pre-service teachers for teaching young citizens from a social justice perspective is a moral obligation of higher education institutions. based on the argument of felton-koestler (2017), pre-service teachers must experience mathematics for social justice as a learner, see examples of what this could look like in practice, and reflect on their own beliefs about mathematics. literature around teaching mathematics and science for social justice draws attention to the fact that if pre-service teachers do not experience explicit exposure to examples of these lessons and opportunities to plan lessons for themselves, they are unlikely to investigate social justice lessons in their own classrooms (myers, 2019). a pourdavood & yan teaching through a social justice lens journal of urban mathematics education vol. 15, no. 2 43 learning mathematics and science through a social justice lens provides students opportunities to analyze their world critically and ultimately promote a democratic society in which everyone can get an opportunity to participate fully (frankenstein, 1995; skovsmose, 1994). mathematics and science can be used as meaningful tools to teach and learn about issues of social injustice and to support arguments and actions aimed at promoting equitable change (bartell, 2013). the first step to remedy the archaic form of instruction is to understand students and the different sociocultural backgrounds they come from. teaching mathematics and science embedded in social justice contexts provides an alternative pedagogy to present content related to students’ backgrounds (bartell, 2013; wager & stinson, 2012). however, many teachers, rather than educating students in sociocultural and sociopolitical contexts, often teach mathematics and science in the way they were taught, such as covering the required curriculum handed to them by their school district, administering an exam, and promptly continuing to the next unit of study. although this approach is considered to be effective for state testing purposes, it lacks a serious reflection of whether instructors are teaching for testing or teaching for learning. the key to solving this issue is through introducing activities related to students’ lives that are meaningful and can provide students with a sense of purpose. therefore, teaching mathematics and science in a social justice context must engage learners in critical thinking, multiple representations, argumentations, discussions, and debates. this is an act of love for teaching and learning within a caring and belonging learning community. it is further important to note that the experiences and practices of middle grade and secondary mathematics and science teachers cannot be understood in isolation. these experiences and practices must be studied within the sociocultural contexts the instructors live in and interact with (frankenstein, 2012, 2013; gutierrez, 2013, gutstein & peterson, 2013; tate, 2013). to this end, this study explored 26 middle grade and secondary pre-service mathematics and science teachers’ engagement in a semester-long course through their participation in classroom discussions, critical reflections, and presentations. the research question was, “how may a semester-long course focusing on teaching and learning mathematics and science with social justice awareness provide pre-service teachers with opportunities to reflect on and change their teaching practices?” review of literature just as apple (1995) designated social justice as “sliding signifiers,” there exist multiple interpretations or definitions of teaching through the perspective of social justice. according to apple (1995), “what social justice teaching actually means is struggled over, in the same way that concepts such as democracy are subject to different senses by different groups with sometimes radically different ideological and educational agendas” (p. 335). freire (1968/1970) termed the pedagogy of the oppressed as a pedagogy which “must be forged with, not for, the oppressed (whether pourdavood & yan teaching through a social justice lens journal of urban mathematics education vol. 15, no. 2 44 individuals or peoples) in the incessant struggle to regain their humanity” (p. 30). our conception of teaching through a social justice lens in the current study is mainly guided by freirian epistemology (freire, 1968/1970), which refers to such pedagogy that aims to establish a society where people actively engage in the transformation of it to achieve a better one. as bartell (2013) stated, “the purpose of education is not to integrate those who are marginalized into the existing society but rather to change society so that all are included” (p. 131). the shift to social justice may be part of an evolving and perhaps more recent sociopolitical shift in the area (gutierrez, 2013; stinson & bullock, 2012). there has been a large amount of research addressing teachers learning to teach from a social justice perspective in teacher preparation programs and disciplines other than mathematics and science (see, for example, adams et al., 1997; ayers et al., 1998; barton, 2003; cochran-smith, 1999, 2000; darling-hammond et al., 2002). de freitas (2008), however, argued that “addressing social justice issues should be a primary goal of all education, including mathematics [and science] education” (p. 43). with social justice receiving more and more attention in mathematics and science education, and with many of the perspectives having been adopted in mainstream mathematics education discourse, martin (2003) noted that the formulation of these projects is narrowly based on “modifying curricula, classroom environments, and school cultures absent any consideration of the social and structural realities faced by marginalized students outside of school and the ways that mathematical opportunities are situated in those larger realities” (p. 7). although extant research documenting teaching mathematics and science in a social justice context is sparse, an increasing number of scholars believe that this pedagogy can provide support for the ongoing struggle for equity in mathematics and science education (e.g., frankenstein, 1995; gutstein, 2003; skovsmose & valero, 2002; tate, 1995). sociocultural and sociopolitical contexts embody pedagogical and organizational forms and methods that inform mathematicians and scientists how concepts and ideas were conceived and evolved from distant antecedents (gutierrez, 2013; kokka, 2019; swetz, 1995). the evolution of mathematics and science pedagogy and practices cannot be attributed to a single group; they are primarily of babylonian and chinese descent and have influenced modern education (kelly & lesh, 2012). culturally relevant teaching supports academic success, cultural competence, and the development of critical consciousness; it also facilitates reflection, participatory engagement, and transformative knowledge, beliefs, and attitudes (mcdonald 2005, 2008; nolan, 2009). further, its practice can offer opportunities to learn “in ways that are deeply meaningful and influential to the development of a positive mathematics [and science] identity” (leonard et al., 2010, p. 261). developing a strong mathematics and science identity through culturally responsive teaching and social justice pedagogy is critical for students’ academic success and understanding of their world (bartell, 2013; gay, 2010; ladson-billings, 2014; leonard et al., 2010; tate, 2013). pourdavood & yan teaching through a social justice lens journal of urban mathematics education vol. 15, no. 2 45 there are multiple contemporary empirical studies that provide examples of this work in different contexts. brown et al.’s (2019) study reported on teachers’ understandings of cultural relevance and practices learned in professional development, moving culturally responsive teaching and cognitive apprenticeship from theory to application. the findings of this study indicated that, ideologically, teachers were well aware of culturally responsive teaching as an educational construct but struggled to explain how it existed pedagogically and in translation from theory to practice (brown et al., 2019). teachers need to connect theory and practice with the principles of culturally responsive teaching—providing support in the form of collaboration to promote success in planning, creating opportunities to adopt languages that elicit the role of educators to engage students in knowledge, as well as finding personal relevance and developing critical awareness to deepen students’ understanding (gay, 2010; ladson-billings, 2014; mensah, 2011; tate, 2013). combining culturally relevant instruction and teaching mathematics for social justice, leonard et al. (2010) conceptualized ways to offer opportunities for marginalized students to learn mathematics and develop a positive mathematics identity. to determine the nuances and complexity of coupling teaching mathematics for social justice and culturally relevant pedagogy, these researchers investigated four case studies across racial dynamics, grade levels, class backgrounds, and school contexts. they concluded that when teacher educators model teaching mathematics for social justice and culturally relevant pedagogy in their methods courses and professional development sessions, mathematics teachers can envision how to implement this pedagogy in their daily classroom instruction. in contrast, upadhyay (2010) conducted a study documenting the perceptions articulated by two middle school science teachers on social justice and how they implemented it in their instruction. schools are built on mainstream structures and values, and that complies with the cultural norms and expectations of the majority population in society. consequently, those students “from low-income and ethnic minority communities who do not fit these norms are either excluded or marginalized in science classes” (upadhyay, 2010, p. 67). this makes it difficult to teach science for social justice in urban schools that consist of mostly poor, immigrant, and minority students. upadhyay determined that the concepts of social justice frameworks were enacted in three forms—attending to individual students’ needs, valuing and recognizing individuals’ experiences, and working against institutional oppression and inequities. the study concluded that teachers must understand the greater social structures of inequality that impact students’ lives in order to teach mathematics and science for social justice (see also gutstein & peterson, 2013; kokka, 2019). a similar study was conducted by gonzalez (2009), who explored the developing identities of seven new york city public high school teachers as both teachers of mathematics and agents of change. the researcher met with the teachers weekly for 10 weeks and prepared them to act as agents of change through their practice of pourdavood & yan teaching through a social justice lens journal of urban mathematics education vol. 15, no. 2 46 teaching mathematics for social justice. they developed a unit of study connecting high school mathematics standards to a range of social justice issues that affected the lives of urban students. gonzalez (2009) found that the teachers’ awareness of the importance of infusing social justice into their teaching increased as the study continued, and the data analysis indicated that the teachers became keenly aware of the injustices their students face, the living conditions of students’ family, inadequate academic preparation, and lack of opportunities for students and their families. gonzalez (2009) recommended future research on the development of teachers' understandings of teaching mathematics in social justice contexts and the ways to provide support for teachers to move from simply understanding teaching mathematics for social justice to implementing it in their classrooms. other studies have indicated that such implementation can be a difficult process and often falls short of an instructor’s intended goals. for example, garii and rule (2009) conducted a content analysis of integrated mathematics and science lessons incorporating social justice to determine the efficacy of such lessons, stressing that social justice pedagogy contextualizes mathematics and science into the lives of students and the communities they live in. four pedagogical approaches—data analysis, discussion, modeling, and library/internet research—were addressed in this study. garii and rule (2009) ultimately found the integration of social justice and academic content was not yet complete in the analyzed lessons, which tended to focus on one or the other. importantly, the lessons were seriously compromised because the teachers failed to recognize the importance of including social justice education into the curriculum. recommendations of this study call for practical opportunities for preservice teachers to understand how social justice can enhance classroom practices. science education for social justice is premised on three broad assumptions: “having the opportunity to learn science as content knowledge, discourse, and practice is a civil right, teaching and learning science involve critical activism and citizenship, and the goals of science literacy involve personal, social, and economic empowerment” (barton & upadhyay, 2010, p. 5). it provides students with access to traditional knowledge and practices as well as the opportunity to question, challenge, and reconstruct existing theoretical structures. social justice pedagogy further offers the possibility for transformation, not only in the lives of the learners but in the social, political, and historical contexts in which it takes place as well. it gives the learner a sense of identity and power for changing their living conditions (barton & upadhyay, 2010; frankenstein, 2013; gutierrez, 2013; gutstein, 2007; kokka, 2019; nolan, 2009). thus, teaching and learning science for social justice must uphold sociopolitical contexts to contribute to a socially just society at both the individual and community levels. the history of social justice research in fact indicates that we, as a society, have the potential to lay the foundation for equitable mathematics and science education (barton & upadhyay, 2010; kokka, 2019; mcdonald, 2005, 2008). pourdavood & yan teaching through a social justice lens journal of urban mathematics education vol. 15, no. 2 47 theoretical assumption the theoretical assumption of this study is grounded in sociocultural and sociopolitical perspectives. according to the sociocultural perspective, all learning is related to our social and cultural activities (gutierrez, 2013; radford et al., 2018; rogoff, 1990; roth & walshaw, 2015; rowlands, 2010; saxe, 1991; schmittau, 2003; vygotsky, 1978). there are four main assumptions in sociocultural theory. the first is regarding the active construction of the knowing and understanding of individuals through social interaction with their environments and others. in this sense, learning involves creating a mental representation of the information provided by experiences. the second assumption argues that learning could lead to higher mental development. this is what vygotsky (1978) called the zone of proximal development (zpd), which is the distance between individuals’ actual knowledge development without assistance and their potential knowledge development with the assistance of their more capable peers or teachers. the concept of zpd has important implications in teaching because it impacts the way educators plan for activities, assess students’ performance with and without assistance, and apply developmentally and culturally appropriate practices. the third assumption states that knowledge development cannot be separated from its social and cultural contexts. for vygotsky (1978), a child is embedded in their cultural context and society, which has a significant impact on how they think and learn. therefore, the child’s mind is in both their own head and their society. although people differ from culture to culture, vygotsky (1978) believed that there is a similar mind structure in all humans. the fourth assumption relates to the importance of language in mental development. vygotsky (1978) believed language is an essential part of individuals’ thinking processes. it allows people to make sense of their world. language acts as a mediator that allows people to carry their cultural and social experiences for self-regulation and self-actualization. in this regard, social interaction and critical reflection play an important role in knowing and understanding (radford, 2015; stemn, 2010). consistent with the sociocultural perspective, the sociopolitical perspective focuses on the notions of identity and power. gutierrez (2013) framed identity as something that a person does rather than something that a person is. she argued that all learning and knowing is inherently situated in social interaction. in this sense, identity is a reflexive relationship between the individual’s lived experiences and the sociocultural and sociopolitical milieu the person interacts with. for example, one of the researchers of the current study is male, a political refugee, and a mathematics educator, and he does research relative to transforming mathematics teaching and learning, teachers’ critical reflection, and emancipatory action research. it is these actions that define this researcher’s identity. another important idea related to the sociopolitical perspective is the idea of power. gutierrez (2013) defined power as active participation in and reconstruction of our everyday activities. pourdavood & yan teaching through a social justice lens journal of urban mathematics education vol. 15, no. 2 48 the sociopolitical turn signals the shift in theoretical perspectives that see knowledge, power, and identity as interwoven and arising from (and constituted within) social discourses. adopting such a stance means uncovering the taken-for-granted rules and ways of operating that privilege some individuals and exclude others. those who have taken the sociopolitical turn seek not just to better understand mathematics education in all of its social forms but to transform mathematics education in ways that privilege more socially just practices. (gutierrez, 2013, p. 4) the sociocultural and sociopolitical perspectives provide a basis for understanding, describing, and interpreting pre-service teachers’ experiences, discussions, and reflections on the teaching of middle grade and secondary mathematics and science through a social justice lens in urban settings. context of the study this study was conducted in a state-supported urban university in the midwestern united states. the course titled perspectives in mathematics and science was developed for middle and secondary school pre-service mathematics and science teachers to take concurrently with their student teaching requirements during the spring semester each year. the three-credit-hour course took place once a week for three hours. there were 26 pre-service teachers in the class during the spring semester of 2020 (13 with mathematics backgrounds, nine with science backgrounds, and four with middle school mathematics and science specialization backgrounds). nineteen of the pre-service teachers were female and seven were male. participants self-identified with the following ethnic backgrounds: african american (3), latinx (4), asian (1), middle eastern (1), native american (2), eastern european (4), and western european (11). eighteen participants were undergraduates and eight were post-baccalaureates. their age varied from 22 years to 39 years. all the pre-service teachers were placed in urban middle and secondary school settings for their student-teaching practice. the perspectives in mathematics and science course has four broad, interlocking goals: 1) provide an overview of the history of mathematics and science, 2) enable future teachers to enact these historical perspectives and contexts throughout their pedagogy, 3) promote intellectual curiosity and sharpen critical thinking skills, and 4) improve verbal and written communication. throughout the semester, the pre-service teachers completed the following required assignments and activities: a) reading, reflecting, and discussing issues regarding their reading assignments from two required textbooks; b) participating in and contributing to the classroom activities; c) choosing, preparing, writing, and presenting a project that contains two interconnected pieces—historical development of a mathematics or science topic and a lesson plan and presentation connecting part 1 to part 2; d) selecting and communicating with another pair from the classroom and providing critiques of their peers’ lesson pourdavood & yan teaching through a social justice lens journal of urban mathematics education vol. 15, no. 2 49 plans; and e) writing a final reflective paper related to their experiences throughout the semester. the critiques of their peers’ lesson plans were more constructive and suggestive than evaluative, and this portion of the course activity intended to promote a caring learning community in which risk-taking, trust and belonging, common interests, and meaning-making were encouraged and celebrated. the first required textbook was rethinking mathematics: teaching social justice by the numbers by gutstein and peterson (2013), which proposes an alternative perspective from which mathematics is taught in a way that helps students understand concepts in connection with their own life and as an approach to solving social injustices. the book also shows the importance of equity in mathematics teaching, emphasizes the significance of teacher-student relationships, and tries to enable teaching and learning to be contextualized around students’ lived experiences in their communities. furthermore, this book illustrates the viability of critical thinking, allowing students to recognize mathematical power as an analytical tool that connects their cultural and community histories and equips them for playing a more active role in society. the second required textbook was the story of science by hakim (2007), which explains the evolution of science beginning in the 1500s through the present. one underlying theme of this book is that progress in one area of science directly or indirectly leads to progress or discovery in another, and they are interrelated. the book celebrates the contributions of many scientists around the world, arguing that the evolution of science cannot be attributed to a single person or group. it also emphasizes the relationship between the history of science and the evolution of science, which has been vastly shaped and influenced by the sociocultural activities of people around the world. the book goes beyond providing a basic understanding of the content; it explains science as a study of patterns and relationships. methodology this qualitative, descriptive, and interpretative study is guided by constructivist inquiry (guba & lincoln, 1994). therefore, the study is context specific. one measure of trustworthiness is the acceptance of the findings by the participants. the primary researcher of the current study acted as both participant-observer and facilitator of the classroom activities and discussions. at each stage of the semester, he communicated with the participating pre-service teachers concerning his understanding and interpretations of their stated beliefs and actions (i.e., triangulation of data processing). another measure of trustworthiness is the provision of thick description. in this study, the participating pre-service teachers’ voices and concerns are the focal points. to accommodate all the participants and make the assignments and activities meaningful and doable within the time constraints of a semester, the primary researcher paired them so that one pre-service science teacher and one pre-service pourdavood & yan teaching through a social justice lens journal of urban mathematics education vol. 15, no. 2 50 mathematics teacher worked together collaboratively to fulfill their common goal of a semester-long project. furthermore, each pair would select another pair in the classroom to critique their project. the reciprocal peer critiques provided the participants with an opportunity to reflect, modify, and re-plan their projects and presentations. a third way of triangulation occurred between the two researchers. they met every other week to exchange ideas and share their ongoing understanding and interpretations of the data. data sources included the researchers’ classroom observations and field notes as well as participating pre-service teachers’ verbal and written responses to class discussions, reading assignments, and course activities. data analysis occurred alongside data collection, which began with coding, a process charmaz (2001) described as the critical link between data collection and meaning interpretation. open coding, which was referred to as descriptive codes by saldaña (2016), was assigned to identify primary themes of the data. this allowed us to explore the understanding of the participating pre-service mathematics and science teachers through a social justice lens. once open coding was concluded, we moved on to the inductive sorting of codes, identifying recurrent codes, metaphors, and contradictions. the data were then integrated and sorted into categories according to links between the codes, with the focus on the participants’ experiences and realizations as well as the challenges, problems, and possibilities of teaching and learning mathematics and science embedded in social justice contexts. we iteratively moved between data and the coding framework and refined codes into consistent and discrete categories. along with the coding process, reflexive and analytical memos were written to “document and reflect on the coding process and code choices” (saldaña, 2016, p. 41), which helped us achieve reflexivity on the data corpus and at the same time provide documentation and transparency about our methodology. once coding was completed, important factors were identified as considerations in promoting the integration of mathematics and science content into social justice contexts. researchers’ role as dollard (1949) stressed, the researcher “must pay the price of intense awareness of self and others and must constantly attempt to define relationships which are ordinarily taken for granted” (p. 20). researchers with different backgrounds might push or support participants differently. coming from different ethnic backgrounds, the two researchers of the current study were acutely aware that their roles could influence the participants. as such, they both managed their roles cautiously due to their various social identities throughout the research. the primary researcher, who was also the instructor of the course at the center of the current study and has teaching the course for over 10 years, came to the united states as a political refugee from iran years ago and has been teaching at his current pourdavood & yan teaching through a social justice lens journal of urban mathematics education vol. 15, no. 2 51 institution for 25 years. his mediating role as facilitator in the participating pre-service teachers’ conversations and his position as a researcher in the classroom called his attention to the importance of maintaining a reflexive lens on his relationship with the participants. being an immigrant professor, he kept cautious not to let his unique race and social class influence his role as facilitator in the participants’ conversations. as a veteran educator of this course, he was sufficiently capable of fully eliciting the participants’ self-reflections, but he also regularly reflected on himself, trying not to shape participants’ discussions with possible pre-assumptions or stereotypes. in this sense, he was an observer of himself. he shared his understanding and interpretations of the findings with the participating pre-service teachers and the other researcher on a regular basis (every two weeks) to ensure transparency and trustworthiness. the other researcher, a third-year doctoral student of urban education, came from china. she had taught in high school settings for nearly 20 years in china before coming to the united states. as a former teacher in china where historical background and student demographics are very different from those in the united states, she was a complete “outsider.” she was clear that her subjectivity might constrain her ability to see phenomena in the field and hear participants’ conversations, so she constantly probed to gain a more precise understanding of any concepts whose meanings she did not feel fully certain of. as a qualitative researcher, she reflected constantly on her positionality and subjectivity, communicating her understanding and discussing her uncertainty with the primary researcher each time they met. being both the documenter of events and co-constructor of the meanings, it was easy to slip from the role of documenter to the role of co-constructor; she balanced her roles and goals through timely and constant reflexive memos. findings as the participating pre-service teachers engaged in classroom discussions, critical reflections on teaching and learning, as well as lesson plan presentations, several themes emerged. these included the importance of teaching mathematics and science through a social justice lens, the participants’ realizations and conscious awareness, and the challenges, problems, and possibilities of teaching and learning embedded in social justice contexts. importance of teaching mathematics and science through the social justice lens the data analysis revealed a consensus that teaching mathematics and science embedded in social justice contexts would make learning purposeful, interesting, and engaging. the participating pre-service teachers stated it is important to challenge the false eurocentric viewpoint that has dominated mathematics and science education in the united states for a long time. pourdavood & yan teaching through a social justice lens journal of urban mathematics education vol. 15, no. 2 52 traditionally mathematics and science have been taught from a eurocentric or western perspective. this is not fair for every student and especially unfair to underrepresented and minority students. this narrative does not tell the truth and can lead those students to believe that mathematics and science are not part of their own cultural background or their own selves. this is quite untrue. mathematics and science are a part of every culture and part of all humankind. teaching from a eurocentric perspective limits the student’s opportunities to learn about diversity, mathematics, and science from a different perspective. i believe that teaching content from a multicultural perspective embedded in social justice contexts will engage all students and create a classroom where all students can thrive. (cory,1 an african american male teacher) the pre-service teachers mentioned the importance of teachers’ awareness of their own bias and prejudice, which they thought would create a more inclusive classroom for their students belonging to underrepresented cultures. they concluded that this cultural awareness would allow students to find their own identities in the classroom. while i had a decent understanding of multicultural education, i was not expecting to find out that so much stuff had been taken from other cultures and written off as a european discovery. the incredible achievements that the babylonians and the egyptians made many years before any european would begin thinking about such things were something that i had never realized before. despite my awareness, i too fell into the eurocentric trap. i just accepted what i heard, never really caring about where the theories came from. it was not until i came into this class and read more about how long this eurocentric takeover has been going on that i realized just how strong this cultural impact is. it is important to make the classroom a beacon for cultural and historical education so that every student will know how important their cultures are to the development of the modern world, which helps them find their identities in the classroom. (sophia, an eastern european female teacher) the participants also stressed how vital it is to teach from the social justice perspective. they indicated that this approach to teaching would allow students to better understand the world around them. they discussed that making mathematics and science relevant and personal to students would enable them to learn how to apply the learned content to the real world, which is much more important and meaningful than rote memorization of facts. social justice in the curriculum is not just a suggestion; it is a must for students and teachers alike. the important point is that social justice will enrich lessons, making them more influential and extend beyond the classroom. students who feel empowered through lesson plans will have the confidence necessary to apply their knowledge to social issues. without teaching mathematics and science in a way that highlights these issues, students lose out on a critical part of their education and teachers lose possibly the most important aspect of education: to empower their students. (emma, a native american female teacher) 1 all names are pseudonyms. pourdavood & yan teaching through a social justice lens journal of urban mathematics education vol. 15, no. 2 53 data analysis further suggested that the participating pre-service teachers had repeatedly proposed access and equity in mathematics and science learning in their classroom discussions, presentations, and writings. we as educators should not be the gate closers in education. we should be the gate openers for our students instead. this allows our students to play a part in the world around them. understanding the meaning behind numbers helps students spot injustices in their own communities. a deeper understanding of mathematics and science also enables students to defend their reasoning behind their points of view and beliefs. this skill can improve the confidence of the students, help find their voices, and show them that they can change the world around them. this makes mathematics and science alive and gives real meaning to the subjects. (benjamin, a middle eastern male teacher) as the participants gradually became aware of and tried to avoid their own biases, they became more convinced that teaching mathematics and science through a social justice lens would result in more engaging, meaningful, and democratic classrooms. by empowering themselves with social justice awareness for teaching, they became more prepared to design lessons geared toward their students’ empowerment. below are three examples of lesson plans developed by pre-service teachers that relate to social justice goals and objectives. these three lessons were selected to show various aspects of teaching mathematics and science through a social justice lens, including health care justice, environmental justice, and pandemic-related justice. the first lesson relates to social justice through its focus on the health care issues that arose during the aids epidemic: this lesson is designed for a mathematics classroom when students are learning to graph and solve different forms of equations or when students are learning about using exponential equations. the lesson would follow students’ ability to graph and solve exponential equations. since the aids epidemic can follow the logistic style of equations, the lesson is meant to include a focus on an important social issue that impacts so many and that so many have no idea about it. the video engagement activity helps to start discussions on the topic of hiv, with 3 true or false statements that are commonly misconstrued. this fuels the fire of curiosity of what students really know, what misconceptions they have, and what they might be interested in knowing or learning. the personalized approach allows students to focus on researching what is interesting to them while filling out a worksheet to gain the knowledge of basic hiv facts as supported by the [state] science standards as well as the mathematics they have been learning. the follow-up lesson for tomorrow will include modeling various exponential equations and solving different components of the equation. we will look at different epidemics that occur in our society and have student teams to choose one scenario to model in the classroom, ending with a short research paper assignment to discuss the cultural and social effects of what would happen if the epidemic scenario were to occur in our state. (noah, a western european male teacher) pourdavood & yan teaching through a social justice lens journal of urban mathematics education vol. 15, no. 2 54 the second example focuses on environmental justice and requires critical analysis of climate change, its impact on our living conditions, and what can be done to change this situation. it is presented as follows: the goal of this lesson is for students to understand the issue of climate change: what causes climate change, what impacts it has on our environment, and what we can do to help reverse the effects of climate change. the objective of the lesson is for students to investigate the issue of climate change through a powerpoint presentation and a wholeclass discussion. students will also be able to identify the effects of climate change. they will also use a kwl chart to document what they already know about climate change prior to the lesson, what they want to know about it, and what they have learned after the lesson is complete. at the end of this lesson, students will brainstorm ideas and move toward a team-based project of their choice to apply their knowledge. the significance of this lesson is the students’ awareness of how global warming is affecting people of every part of the world (i.e., of diverse backgrounds). more locally, the factors contributing to global warming are being manifested (e.g., pollution of cuyahoga river, unsustainable private and public practices, etc.). it is also important for students to learn this lesson to meet the [state] learning standards and pass the state test. (mary, an asian female teacher) the third lesson plan example concerns the current covid-19 pandemic, which is impacting the lives of people almost all over the world, particularly the lives of minorities, people of color, and low-socioeconomic status populations. as a response to this current social issue, one pair of the pre-service teachers initially selected a topic concerning the ongoing pandemic. however, due to many unknowns regarding the nature of the virus and the lack of a viable vaccine for controlling the spread of the disease at the time of the study, they decided to change their topic to a similar but very well-known disease, as their lesson plan aimed at involving their students in critical thinking about health care justice. below is how they presented it: this lesson is the start of a new mathematics unit and also a review of the concepts taught in this school year and previous years. students have learned various mathematics graphs, but this lesson will go into more detail and cover some new graphs. the preassessment will be used to see how much they remember what they have learned and what they have not yet learned. during the lesson, students will be given data concerning the chickenpox virus to learn about the scientific process for data processing and presentation. they will work in groups to find a way to represent the data and then present their graphs to the class. each group will have different data representing a different aspect of chickenpox. they will need to tell the class their group’s original hypotheses about a unique issue, what they think the given data means as depicted in their graphs, and their opinions about social concerns and future issues regarding chickenpox based on the data. students will work cooperatively and respect different perspectives. the follow-up activity after this 40-minute class will occur on the following day. during the next class, students will continue working on their graphs and their worksheets and will also prepare to speak in front of the class. presentations should have smooth transitions and each group member should have a chance to present a different aspect of the information. once all the presentations are complete, one class period will be dedicated to a test that pourdavood & yan teaching through a social justice lens journal of urban mathematics education vol. 15, no. 2 55 includes scientific inquiry, statistics, and graphs. additionally, each student will fill out a “peer evaluation” for each member of their group. this evaluation will show how well the group worked together. (lucas, a latino male teacher) social consciousness through critical reflections provided the pre-service teachers with opportunities to carry out their situated instructions in their own classrooms during their student-teaching. these above praxes (i.e., dialectical relationships between actions and reflections) are clear examples and testaments of the importance of teaching mathematics and science embedded in social justice contexts. participating pre-service teachers’ realizations and conscious awareness throughout the semester, the participating pre-service teachers realized several issues concerning teaching and learning mathematics and science, such as racial justice, gender equity, the importance of teacher-student relationships, and the notion of caring and belonging in education. a major realization that has come about from observing patterns in mathematics and science education is the marginalization of female contributions in those fields. female students need to see themselves reflected in those fields when learning. but because the history of female contribution is omitted or because females are underrepresented in teaching the historical contexts of mathematics and science, there is a gender disparity within the stem fields. i further realized that science and mathematics do not lack female contributors; rather, typical k-12 science and mathematics education simply failed to acknowledge these contributors. (william, a western european male teacher) the female pre-service teachers in the classroom came to the realization that women are underrepresented in mathematics and science education and that there is a large gender gap in science, technology, engineering, and mathematics (stem) fields for several reasons. their reflections also revealed these trends in their own educational life. another important realization mentioned by several pre-service teachers was the importance of teacher-student relationships and the sense of belonging. as one of them put it: the degree of students’ engagement in the classroom is related to their sense of belonging at the school. students with a low sense of belonging, often marginalized students, are at risk of disengaging and falling behind their peers. if the school system does not attempt to remedy the cause behind this sense of belonging, it could have lifelong impacts on the students. research shows harmonious teacher-student relationships can increase students' sense of belonging. the sense of belonging is crucial for the development of adolescents as they go through many changes socially and intellectually. the needs of students of these marginalized groups must be met for their overall improvement in content areas and furthermore, in their professional life. (isabella, an african american female teacher) pourdavood & yan teaching through a social justice lens journal of urban mathematics education vol. 15, no. 2 56 the participating pre-service teachers also asserted the significance of caring communities for equitable education. some of the teachers shared their classroom interactions with their high school students. for example, a teacher shared her classroom teaching activities that were based on a community project and meant to help students understand the concept of percentage in mathematics: from this class, i learned to check my biases at the door before going into my own classroom, which helped me view students equally without discrimination. as a teacher, this is a very important quality, helping me build my classroom into a caring learning community. once my students were doing a project based on their communities’ needs. at first, i asked them about their concerns in their communities, and then each student chose a concern and wrote down their thoughts about it. the students then looked up the percentages of different concerns and compared them with their peers’. this helped them understand the meaning of percentage and the method of calculating it. (mia, a latina female teacher) project-based learning was very common for these participating pre-service teachers to use as a teaching strategy to help their students. for example, they would focus on topics initiated by students and try to guide and facilitate their completion of the projects. as the students were engaged in project-based activities, they were learning significant integrated mathematics and science content within the contexts of their communities. challenges and possibilities of teaching mathematics and science embedded in social justice contexts data analysis revealed that the current school structure and culture, the existing curriculum, state-mandated tests, limited resources, and the lack of meaningful teaching experience were all challenges the pre-service teachers stated that they faced or might face in the future as they attempt to teach mathematics and science in social justice contexts. for example, one major obstacle is called the culture of silence, which occurs when a teacher is not fully committed to overcoming the structural systems of inequality in education. it is critical that teachers strive to achieve full awareness of the five social factors—race, social class, gender, culture, and disability status. it is also important that teachers reflect on cultural and linguistic diversity to gather necessary information to develop culturally responsive instruction. teaching mathematics and science for social justice should instill students with new knowledge that encourages them to become social change agents. some of our participating preservice teachers were not comfortable confronting the challenges of injustice and inequality in urban settings. for example, one of them presented this: while i think it is beneficial to connect the content to student life, i feel like that discussing these topics, especially in urban areas where there are typically high concentrations of poverty, might be too personal for some students and may result in them feeling pourdavood & yan teaching through a social justice lens journal of urban mathematics education vol. 15, no. 2 57 ashamed, embarrassed and even angry. i am uncertain as to if it will work in a middle school classroom where some students already feel self-conscious. (katie, a native american female teacher) these feelings of uncertainty and being passive in the face of the “hot topics” surrounding the educational community were expressed by several participating teachers. however, this perspective was challenged by most of the pre-service teachers during classroom discussions. as one of them put it: something i have realized over the semester is the difficulty in actually teaching mathematics in a way that integrates history and social justice into the classroom. this is because it is not the way i was taught growing up and is new to me and many of my colleagues. but i have also realized the importance of teaching in this way because it will open a door for the low-status adolescents in urban settings to become both aware of the historical background of the learning materials and actively engaged in communitybased activities that are of interest or importance within their daily life. (charlotte, a western european female teacher) another teacher supported the above statement by saying that sometimes being uncomfortable confronting challenging issues is positive rather than negative. he stated this: i enjoy this course because it opened my eyes and changed my thoughts about education. i believe that mathematics and science are best taught from a multicultural perspective. i do not think it is easy to implement this approach into a classroom, but i believe those uncomfortable feelings would motivate and help us to overcome all the difficulties. i also believe our efforts are worthwhile for the better education of our students. (liam, an african american male teacher) despite all these challenges, obstacles, and problems, most of the participating pre-service teachers were convinced that it is worth fighting to teach mathematics and science in the context of social justice. discussion this study examined 26 pre-service middle and secondary school teachers’ experiences and reflections on teaching and learning mathematics and science from a social justice perspective as they took a semester-long course concurrently with their student-teaching. the findings of the study suggest that a semester-long discussion, critical reflection, and lesson plan development provided the participating pre-service teachers with opportunities to question their own learning experiences in mathematics and science classrooms and to become active members of transforming learning communities. moving away from traditional ways of teaching mathematics and science toward developing engaging mathematical and scientific content embedded in pourdavood & yan teaching through a social justice lens journal of urban mathematics education vol. 15, no. 2 58 social justice contexts requires the transformation of consciousness, which is consistent with the findings of previous research (bartell, 2013; felton-koestler, 2017; myers, 2019). this transformation is more than changing teaching strategies; it is a change in ways of knowing and reading the world through critical lenses to become change agents. teaching mathematics and science from an integrated perspective embedded in social justice contexts served the pre-service teachers a sense of purpose. it allowed them to present academic content in meaningful ways by making connections between students’ interests and state standards. by developing, preparing, and delivering activities grounded in project-based pedagogical contexts, the pre-service teachers were able to better understand the contributions of underrepresented groups and diverse cultures in the development and evolution of mathematics and science. moreover, classroom discussions and critical reflections regarding culturally responsive teaching were essential for them to address the unique learning needs of students in diverse classrooms. this is how university courses help prepare pre-service teachers for their future teaching. collaboration and support, critical reflection, and personal relevance are important elements for pre-service teachers to adopt in preparation for mathematics and science instruction embedded in social justice in their classrooms. marginalized students have already fallen behind historically and will continue to do so unless major changes are made in the way stem courses are taught. sociocultural and sociopolitical perspectives must be introduced into the classroom to encourage students who have been hindered from engaging in mathematics and science that they too have a voice about. without the effort of teachers to learn their students’ needs, students will be further alienated to the point where they feel they have no discourse in their study. these pre-service teachers, although convinced regarding the vitality of culturally responsive pedagogies, may face significant challenges, problems, and obstacles as they start teaching their own classrooms within the existing school structure and culture. however, as many of the participating teachers stated, it is worth fighting, and perseverance will help them achieve the integration of social justice into their classroom activities. as one of the participating teachers expressed: we must not make our students feel as though they have no voice in the classroom; we must give them that voice. we must give them the voice they need so as to instill changes they want within their communities. the only way that voice can be found is through the integration of cultural-historical perspectives in the classroom. without this crucial element in instruction, the problem of marginalized students falling behind in stem education will only become even worse. (john, a latino male teacher) with classrooms becoming increasingly diverse, it is the duty of educators to make sure students from all backgrounds feel included both in the physical classroom and in the content being taught. there are deep sociocultural, sociopolitical, and pourdavood & yan teaching through a social justice lens journal of urban mathematics education vol. 15, no. 2 59 sociohistorical contexts surrounding mathematics and science content that need to be integrated. although traditional lectures may be easier to accomplish, the job of teachers is not to look for easy ways to conduct classes but to ensure educational standards are met and that students establish the conceptual knowledge being taught as well as critical thinking skills necessary to tackle issues they will face in their lives. there might be some degree of pushback, but this work can be highly rewarding and entirely achievable as pre-service teachers tend to be more involved in real-world problem contexts and new mathematical and scientific methods (aguirre, 2009; ensign, 2005; mistele & spielman, 2009; rodriguez, 2005). conclusion the problem of systemic injustice in our society is deeply rooted in the notion of identity and power. this problem cannot be solved from the same mindset where it originated. in the context of education, there ought to be a shift of mindset from considering preparing students to live in the existing world to considering preparing them to reconstruct those current social systems and to remove obstacles experienced by minorities, women, and others (secada, 1989). the implication of this transformation is significant. it denotes that school reform cannot be actualized without active participation in and reconstruction of school systems by teachers and students. they are the beacons of hope and the beacons of change. due to the intricate connections between education and economic, political, and social power structures in society, which contribute to inequity in both schools and society (apple, 1992; kozol, 2005), achieving social justice in mathematics and science education remains a huge challenge for educators. however, if mathematics and science teaching and learning are situated within sociocultural and sociopolitical contexts combined with viable organizational support, there is hope. teaching through a social justice lens is not a methodological issue but a process that requires teachers to adapt to the particular contexts they and their students belong to (cochran-smith, 1999). learning to teach mathematics and science from a social justice lens is a “lifelong undertaking” and a complex process that requires effort, perseverance, and reflection (darling-hammond, 2002, p. 201). it also requires teachers to see it as such (gutiérrez, 2009). although a one-semester university course cannot make learning to teach for social justice “happen,” such efforts increase our understanding of the broader goals of mathematics and science education. this study contributes to the process of educating pre-service teachers for the implementation of social justice into their mathematics and science instruction and the development of their new conscious awareness. more importantly, this study is a story of hope. we hope the study encourages future professional development programs and future studies on both pre-service and in-service mathematics and science teachers’ engagement in social justice issues. pourdavood & yan teaching through a social justice lens journal of urban mathematics education vol. 15, no. 2 60 references adams, m., bell, l. a., & griffin, p. 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(2012). teaching mathematics for social justice: conversations with educators. national council of teachers of mathematics. copyright: © 2022 pourdavood & yan. this is an open access article distributed under the terms of a creative commons attribution-noncommercial-sharealike 4.0 international license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. microsoft word 377-article text no abstract-1820-1-18-20200206 (proof 2, author).docx journal of urban mathematics education may 2020, vol. 13, no. 1, pp. 1–14 ©jume. https://journals.tdl.org/jume sarah theule lubienski is the associate dean of graduate studies and a professor of mathematics education at the indiana university school of education, 201 n. rose avenue, bloomington, in 47405-1006; email: stlubien@iu.edu. her research focuses on inequities in students’ mathematics outcomes and the policies and practices that shape those outcomes. weverton ataide pinheiro is a ph.d. student and associate instructor of mathematics education at indiana university, department of curriculum and instruction, 201 n rose ave, bloomington, in 47405-1006; e-mail: wpinheir@iu.edu. his research is focused on critical mathematics studies (mathcrit), college-level students' mathematics generalizations, and students' mathematical identities. gender and mathematics: what can other disciplines tell us? what is our role? sarah theule lubienski indiana university weverton ataide pinheiro indiana university in this article, we begin by taking stock of broad trends related to gender and mathematics, focusing primarily on patterns within the united states and considering how these patterns may vary by social class and race. this article is not a traditional, empirical piece but instead pulls together evidence from quantitative and qualitative studies to argue that gender remains an issue worthy of our attention within mathematics education. given that much recent work on gender and mathematics has been situated outside of traditional mathematics education frames, we consider how ongoing work in psychology and gender studies can contribute to our understanding, and we use this work in conjunction with interview data from a study of women's experiences in a mathematics ph.d. program. ultimately, we argue that mathematics education researchers bring unique expertise to the table and have a particular role to play in building upon findings in other fields to further the work on gender and mathematics. keywords: gender, psychology, race, socio-economic status gender—still an issue within mathematics education? everal decades ago there were marked disparities between women and men in key metrics of concern to the mathematics education community, such as completion of high school mathematics courses and bachelor’s degrees in mathematics, but these disparities ultimately narrowed (dalton et al., 2007; perez-felkner et al., 2014). perhaps because of this, gender has received less attention since the turn of the century within the mathematics education community than it did in the 1970s– 1990s (grevholm, 2011); however, work has continued in other fields, such as psychology and sociology, with dedicated journals such as psychology of women quarterly and sex roles (lubienski & ganley, 2017). moreover, the idea of gender itself has been challenged. we use the term “gender” throughout this article to reflect our primary focus on socially constructed norms associated with being a woman or man in our society as opposed to a focus on fixed s lubienski & pinheiro gender and mathematics journal of urban mathematics education vol. 13, no. 1 2 biological traits (hall, 2014). we acknowledge the limits of our use of binary categories, but we find these categories useful for taking stock of equity concerns for our purposes here; as discussed below, this very conversation is something that gender scholars have grappled with in recent decades as well. overall, with the advances in equity in gender and mathematics, along with the deconstruction of gender as a concept, it is not surprising that attention to gender has declined within the mathematics education community (grevholm, 2011; lubienski & ganley, 2017). still, our aim in this paper is to argue that mathematics educators have a responsibility to attend to gender and mathematics education and that, despite the contributions of scholars in other fields, we have particular expertise to contribute. according to some key metrics, progress toward gender equity in mathematics has stalled. for example, in the united states gender disparities in performance among k–12 students continue to favor boys, who are substantially overrepresented at the top of the mathematics achievement distribution (cimpian et al., 2016). with some exceptions, significant gender differences also tend to favor boys internationally, with boys in organisation for economic co-operation and development (oecd) member countries outscoring girls by an average equivalent of 5 months of schooling (oecd, 2015). although gender gaps in performance are often viewed as relatively small, gaps in mathematical confidence are larger, both in the u.s. (ganley & lubienski, 2016) and among other oecd countries (oecd, 2014). major gender disparities in mathematics persist beyond k–12 schooling as well, and the collegiate and professional mathematics settings have in fact experienced some regression in women’s representation since the turn of the century. although u.s. women are more likely than men to attend college, the percentage of bachelor’s degrees in mathematics earned by women was 46% in 1997 and only 42% two decades later; additionally, women earn less than 29% of mathematics doctoral degrees and only 19% of bachelors’ degrees in computer science, down from 27% in 1997 (national center for science and engineering statistics, 2019). furthermore, the median salary for women who are college graduates and employed fulltime is only 74% of the median salary of men with a similar education level (american association of university women [aauw], 2018), and a substantial portion of this disparity is attributable to men’s overrepresentation in mathematics-intensive professions (corbett & hill, 2012; ryan, 2012). this imbalance in mathematics-intensive fields diminishes women’s pay and status as well as the pool of skilled professionals contributing to these fields. hence, despite some progress toward gender equity in the field of mathematics, concerns remain. gender differences in mathematics outcomes are specific to our field, as they do not occur in reading or even in science majors that are not mathematics intensive. additionally, although differences in mathematics outcomes by race and social class tend to be at least partially attributable to differences in schools attended (fryer & levitt, 2004), gender differences appear among boys and girls lubienski & pinheiro gender and mathematics journal of urban mathematics education vol. 13, no. 1 3 learning within the same mathematics classrooms. because of this, mathematics educators arguably have a particular responsibility to examine and address issues of gender equity. gender patterns by ses and race although national data provide overall averages by gender, they can also mask the ways in which patterns vary across contexts. hence, it is important to examine gender issues with an intersectional lens. gender and ses gender disparities in mathematics are larger in high-socioeconomic status (ses) schools (reardon et al., 2018). given that higher ses students tend to have higher achievement than their less advantaged peers (reardon, 2011), this ses pattern is consistent with gender gaps being larger at the top of the distribution (cimpian et al., 2016). these patterns may seem surprising given that high-ses parents have been found to espouse more egalitarian views than do lower ses parents (marks et al., 2009). an explanation for this pattern may be found in a study that used national data from the early childhood longitudinal study, which found that high-ses children were more likely than their low-ses peers to participate in parent-initiated activities falling along traditional gendered lines, such as dance lessons for girls and organized sports for boys (lubienski et al., 2013). although these differences in activities themselves did not explain emerging gender differences in mathematics performance, the study suggests that high-ses children may experience other gendered parenting practices that could contribute to the early gender gaps we see among high mathematics achievers. ellison and swanson’s (2010, 2018) analyses of american mathematics competitions (amc) data provide a different look at girls’ opportunities to excel in mathematics within varied ses contexts. the amc is a series of competitions held at over 3,000 u.s. high schools. although 43% of the amc test takers examined were girls, boys were six times as likely to score among the top 1% (n=1,200) of amc scorers. ellison and swanson (2010) observed fairly consistent gender gaps in scores across most high schools but found that while high-scoring boys came from a wide variety of schools, high-scoring girls were concentrated in a very small number of elite, high-ses schools. it is important to note that although gender disparities might be larger among high-ses students, girls who do excel tend to come from high-ses contexts. these results highlight both the importance and possibility of schools doing more to nurture all girls’ mathematical talents. specific to considerations of urban education (the focus of this journal), these results also raise questions about the resources teachers and schools need in order to foster all students’ talents, and how lubienski & pinheiro gender and mathematics journal of urban mathematics education vol. 13, no. 1 4 schools’ geographic contexts may affect teachers’ and students’ access to such resources. gender and race just as students’ gendered experiences and outcomes can vary by ses contexts, they also have been found to vary by race. for example, mcgraw et al. (2006) found that black girls tended to score higher than black boys on the national assessment of educational progress (naep). the most recent data from naep, from 2015 and 2017, confirm this pattern. specifically, small gender differences among black students tend to favor girls across the 4th, 8th, and 12th grades, and there are reverse patterns among asian, hispanic, and white students (data pulled from the naep data explorer at https://www.nationsreportcard.gov/ndecore/xplore/nde). this trend among black students in grade school does not mirror the trend of black students in college pursuing science, technology, engineering, and mathematics (stem) majors however. riegle-crumb and king (2010) analyzed educational longitudinal study data and found that among u.s. students who attended a four-year college, black and white men were far more likely to pursue physical science and engineering majors than both black and white women. moreover, black men were more likely to pursue such a major than white men when academic preparation was accounted for. it is critically important to remember that these numbers are restricted to those in fouryear colleges and also do not consider the relatively high departure rate of black students in stem majors (riegle-crumb et al., 2019). still, the data support gholson’s (2016) argument that intersectionality is important and that black women should not be overlooked by mathematics education researchers and support programs. interpreting women’s experiences in graduate mathematics decades ago, fennema and colleagues (1998) published a small-scale study in which elementary school girls were found to approach problems with more taught strategies while boys more often used invented strategies. the researchers then opened the work up to consideration by scholars in other fields to interpret the findings. we think much can be gained by such an interdisciplinary approach, and so we embark on a small version of multi-disciplinary conversation here, highlighting how recent work in psychology and gender studies can complement—but not replace the need for—mathematics education research on gender and mathematics. we use these two fields as examples because our own thinking about gender and mathematics has recently been informed by these fields. we do not mean to suggest that other fields are any less important for informing such work or that gender-related work at the intersection of these fields and mathematics education has not been done before. lubienski & pinheiro gender and mathematics journal of urban mathematics education vol. 13, no. 1 5 to illustrate how frames from different disciplines can enrich our thinking about gender and mathematics, we use data from an ongoing study of how women who are ph.d. students in mathematics perceive themselves in mathematical spaces. we see these spaces as “sites where mathematics knowledge is developed, where induction into a particular community of mathematics doers occurs, and where relationships or interactions contribute to the development of a mathematics identity” (walker, 2012, p. 67). using a “life story interview approach” (mcadams, 2008), the second author of this article interviewed six women in mathematics ph.d. programs at a large midwestern university in fall 2019. these students were contacted via email, and they completed an online survey providing information about their demographics and mathematical background. intentionally, the researcher chose two students in the beginning of their studies, two in the middle, and two close to the end. this strategy was used to examine students’ experiences throughout different phases of doctoral study (pinheiro, 2019). only students who identified as women and who had exemplary grades in high-level graduate mathematics courses were chosen to participate in the project. among these six students, four identified as white women, one identified as an asian american woman, and one identified as a latina from ecuador. each woman was interviewed twice for a total period of two hours. each person was interviewed on two consecutive days. overall, the women tended to report that they (1) had more interest in teaching than doing research, (2) considered family-related concerns when making career plans, and (3) had lost confidence in their mathematical abilities as they reached higher-level mathematics. these findings are not new; earlier studies found that women often decide to study a ph.d. in mathematics because of their love of teaching (herzig, 2010), their departure from mathematical studies can be connected to a variety of family responsibilities (herzig, 2004b; lovitts, 2001; sonnert & holton, 1995), and women tend to be less confident in mathematics than men (etzkowitz et al., 2000; herzig, 2004a). the current study suggests that women’s experiences and the way they feel in mathematical spaces may remain similar to what they were a generation ago. because of space limitations, we focus specifically on kelly’s1 experiences, as quotes from her interview succinctly convey these themes. kelly is a white 5th-year student who decided to pursue a ph.d. in mathematics because of her interest in teaching mathematics: when i was in, you know, middle school, i wanted to teach middle school math. when i was in high school i wanted to teach high school math, and then in college i was like, no this is way better, i want to teach college math. and then i knew i needed a ph.d. in order to do that. 1 kelly is a pseudonym. lubienski & pinheiro gender and mathematics journal of urban mathematics education vol. 13, no. 1 6 although kelly had worked for years toward her goal of becoming a college mathematics teacher, she expressed a willingness to compromise her desired career path for the sake of her family: i consider myself a feminist, but as much as i really do want to teach at a small school, my career is not the most important thing to me. i am much more family oriented. and so i would be much more willing to give up that career in order to have a job close to a city that works well for my husband. and so sometimes i feel odd about vocalizing that to people, because i don’t want to be perceived as not empowering to women. despite the mathematical accomplishments of the six women in the study, each of the participants, including kelly, said that their mathematical confidence in graduate school was lower than it was in elementary/middle school, high school, and college. for example, kelly said she was “definitely less confident in math” in graduate school than she was previously. when asked to describe her confidence on a scale of 1–10, she explained that it started high in graduate school, but fell by her second year: kelly: started off at 9 or 10. and then by year 2, five maybe. interviewer: why is that? kelly: i think a variety of reasons. i think part of it was suddenly it got way harder. part of it, too, was not being sure that i wanted to finish the program. and… those moments, when i felt like i didn't belong as much, was sort of a similar thing, where there are some people in this program who just, they read something and they remember it, they can rattle off a theorem. like it's nothing. and i, i always thought that i had a really good memory, up until i came here. and all of a sudden, i, i don't have a memory like theirs. kelly also talked about her desire to have children, along with her questions about timing: … my husband and i have thought about kids more, because we both do want kids. but… i don't know if i can handle that during grad school. but also they say it doesn't get easier. so that's kind of where i'm at right now. how might we make sense of kelly’s statements? we turn now to consider how work in psychology and gender studies might inform our interpretations. we then close by considering what mathematics education researchers might see in these quotes that others do not and argue that mathematics education research is needed to complement research on gender and mathematics in other fields. lubienski & pinheiro gender and mathematics journal of urban mathematics education vol. 13, no. 1 7 perspectives from psychology and gender studies psychology kelly pursued a ph.d. because she wanted to teach college mathematics. however, when the math became “way harder” in doctoral study, she lost confidence and considered leaving the program. she continues to grapple with how to reconcile her prioritization of family with her feminist identity. within psychology, kelly’s statements can be viewed in light of expectancyvalue theory, which emphasizes the role of individuals’ values and expectations for success when making career choices (eccles, 2009). instead of focusing on women’s avoidance of mathematics, expectancy-value theory focuses our attention on why women choose the careers they do. for example, eccles and wang (2016) found that women’s greater prioritization of working with people and meeting family needs helps explain gender disparities in 12th graders’ career plans, including why women are more likely than men to pursue non-mathematical careers. students’ mathematical self-concepts and expectation for success in mathematics also predict students’ pursuit of math-related careers (lauermann et al., 2017). through this lens, we notice that kelly became unsure of whether to remain in the program once her expectations for success (mathematical self-concept) faltered. kelly continues to struggle to reconcile her career goals with her family aspirations, particularly because “they say it doesn’t get easier,” meaning that she is not sure that she can balance motherhood with an academic career in a mathematics department even after graduate school. a lack of role models in her current program (e.g., women professors who are mothers) may exacerbate this sense of conflict in her values as well as diminish her expectancy for success as a mother-mathematician. another recent line of research in psychology highlights that mathematics and a few other fields (e.g., philosophy) are associated with innate “brilliance.” consistent with stereotypes about white men being “naturally smart” as opposed to needing to work hard to succeed, the fields associated with brilliance have relatively high percentages of ph.d.s who are white men (leslie et al., 2015). with this lens, we might wonder if kelly has internalized the idea that some students inherently have a greater chance at success as research mathematicians than others (i.e., those who “can rattle off a theorem” are simply more mathematically “brilliant” than she is). gender studies whereas psychology helps us understand kelly’s choices as shaped by her individual values and expectations, gender studies helps us focus more on broader societal influences. gender studies is an interdisciplinary field that has attracted increased attention recently, given evolving views on gender and the ways in which these permeate society and impact people’s lives. here we consider ideas from lubienski & pinheiro gender and mathematics journal of urban mathematics education vol. 13, no. 1 8 feminist and queer theories, both of which fall under the umbrella of gender studies and can provide distinct views of kelly’s experiences. feminist theorists focus on injustices that have historically shaped women’s oppression, striving to understand the subordination of women in order to take action to address it (mccann & kim, 2013). in looking at kelly’s statements, feminist theories would help us notice ways in which her views and experiences have been shaped by her gender in a male-dominated society. although kelly considers herself a feminist, she is willing to prioritize her husband’s career over her own. this is consistent with donovan (2012), who argued that in societies where masculinities are still dominant, women fill the role of caring for men. additionally, korth’s (2003, 2005) methodological research on caring helps us consider ways in which kelly’s desire to both care for her family and to be a teacher may be connected to her socially constructed gender. unlike feminist studies, which is focused on women, queer theory is not attached to any specific gender identity. instead of a focus on commonalities among women versus men, queer theory deconstructs the ideas of gender normativity and the heteronormative, prompting us to think beyond our current binary notions of gender and to consider gender as a socially constructed, performative attribute of an individual (i.e., it’s what you do, not who you are) (butler, 1993). amin (2017) and other queer theorists would argue for research from multiple historical perspectives to examine what makes kelly’s current experiences possible. such examinations could help us consider, for example, how our historical emphasis on women and men as categories have perpetuated gendered patterns in mathematics outcomes. in summary, psychological research has informed us about factors shaping individuals’ career choices and how these can collectively result in the underrepresentation of women in mathematics, while work in gender studies foregrounds considerations of gender within broad historical and social contexts, pushing us to think more deeply about the way in which societal gender norms have shaped gendered role expectations over time. within gender studies, feminist theorists emphasize the importance of interpreting and improving kelly’s experiences as a woman, whereas queer theory urges us to look beyond current notions of gender and envision what kelly’s experience might be if such notions were dismantled. what can mathematics education researchers add? after considering kelly’s experiences from the perspectives of psychology and gender studies, one might wonder if there is anything else for mathematics education researchers to add. in particular, how might a more explicit focus on mathematics and mathematics education influence what is noticed and queried about kelly’s experiences? lubienski & pinheiro gender and mathematics journal of urban mathematics education vol. 13, no. 1 9 first, although both gender studies and psychology’s expectancy-value theory help us understand that kelly’s desire to care for others is consistent with larger societal gender norms, we, as mathematics educators with a fuller appreciation of the discipline, might question whether kelly understands how mathematics can be used to help people in a variety of professions beyond teaching. second, given our primary concern for improving mathematics teaching and learning, mathematics education scholars might wonder how the mathematics instruction that kelly encountered shaped her mathematical identity and confidence. more specifically, although expectancy-value theory suggests that it makes sense for kelly to consider exiting a field in which she lacks confidence, mathematics educators might choose to examine whether poor mathematics instruction contributed to her drop in confidence and/or if there may be a way for high-quality mathematics instruction to ameliorate this decline. finally, mathematics educators would likely be quite troubled by kelly’s perception that memorization is key to success in higher-level mathematics. this last point returns us to fennema et al.’s (1998) study. in the two decades since fennema and colleagues observed a greater tendency among boys to use invented strategies, several studies have found similar patterns (gallagher et al., 2000; goodchild & grevholm, 2009; lubienski et al. 2018; zhu, 2007). these studies point to several hypotheses: are girls socialized to be “good girls” who memorize and implement procedures given by the teacher or text, whereas boys are rewarded for becoming “outside the box” problem solvers? does a “bolder” approach to problem solving have greater payoff on nonroutine, complex problems, thereby explaining larger gender gaps at the top of the distribution as well as the decline in women’s confidence when reaching higher-level mathematics? how can mathematics teachers interrupt these patterns? such questions have risks. for example, as leyva (2017) asks, might the portrayal of mathematics as primarily involving independent problem solving align with norms of middle-class, white men, perhaps thereby perpetuating stereotypes within the field? gholson (2016) complicates this a bit further in noting that black girls tend to exhibit some traits associated with mathematical success and masculinity, including confidence and independence. her work prompts us to ask how the mathematical experiences of women of color might differ from those of kelly, as well as whether more independent approaches to school mathematics might have more payoff for some students than others. musto’s (2019) longitudinal study in a public suburban middle school near los angeles pertains to the questions of differential payoff and how engagement can be viewed differently in different contexts. musto found that in higher track classes, which contained more affluent white and asian american students, teachers allowed boys to break rules in class without consequence. however, in the lower track classes, where lower ses latinx students were overrepresented, those same teachers tended lubienski & pinheiro gender and mathematics journal of urban mathematics education vol. 13, no. 1 10 to punish boys for breaking rules. while boys (especially white boys) in the higher track classes ended up dominating conversation and being perceived as exceptionally smart, the lower tracked boys pulled back from participation and were ultimately seen as less smart than the girls. sociological studies such as musto’s raise questions that provide additional opportunities for mathematics education research. for example, musto’s work can help us see how teachers might contribute to conceptions of who should express independence and who is smart in school, but the study does not focus specifically on mathematics. mathematics education scholars could take such work a step further to examine how the interactions in classrooms affect the ways in which students identify with and engage with mathematics, including how they approach mathematical problems. mathematics educators would also bring a critical eye to mathematics instruction, looking, for example, at ways in which highly structured instructional approaches may be more or less likely to exacerbate genderand ethnicity-related patterns. still, such studies may be limited by their reliance upon binary gender categories and their focus on differences between these categories rather than their fluidity. some scholars have begun to discuss how gender research that goes beyond such binaries might contribute to the field (e.g., mcgraw et al., 2019; walshaw et al., 2017). additional studies might also examine how urban or rural school contexts can shape the genderand ethnicity-related patterns examined by musto in suburban schools. we also wonder how theories not focal in this article, such as black feminist theory (borum & walker, 2012; collins, 1990) and queer crit perspectives (misawa, 2012), might help us ask new questions and examine data with new lenses. conclusion in this brief essay, we could only scratch the surface of gender issues to be considered. the compendium for research in mathematics education chapter on gender (lubienski & ganley, 2017) further discusses evolving perspectives of gender and mathematics, international patterns, and additional potential factors shaping gender differences in mathematics outcomes. clearly, there is more work to be done, and mathematics education scholars have a role to play in both informing and building from work grounded in other fields. given the costs of underrepresentation of women in math-intensive careers to both women and those fields, such work merits our efforts. acknowledgements the research reported in this article 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(2017). beyond the box: rethinking gender in mathematics education research. in a. chronaki (ed.), proceedings of the ninth international mathematics education and society conference (vol. 1, pp. 184–188). mes9. https://www.mescommunity.info/mes9a.pdf zhu, z. (2007). gender differences in mathematical problem solving patterns: a review of literature. international education journal, 8(2), 187–203. copyright: © 2020 lubienski & pinheiro. this is an open access article distributed under the terms of a creative commons attribution-noncommercial-sharealike 4.0 international license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. putting the “urban” in mathematics education scholarship journal of urban mathematics education december 2018, vol. 11, no. 1&2, pp. 26–30 ©jume. http://education.gsu.edu/jume william f. tate is chair of the department of education, director of the st. louis center for inquiry in science teaching and learning, and edward mallinckrodt distinguished university professor in arts & sciences at washington university in st. louis, one brookings drive, campus box 1183, st. louis, mo 63130-4899; e-mail: wtate@wustl.edu. he is immediate past president of the american educational research association. tate’s interdisciplinary scholarship concentrates on two main areas: mathematics, science, and technology education, specifically, in metropolitan america; and the social determinants of education and health disparities. commentary putting the “urban” in mathematics education scholarship1 william f. tate washington university in st. louis ow many candidates running for local, state, or federal office in the 2008 elections in the united states highlighted the importance of urban america as a site of opportunity, or even challenge? briggs (2005) argued that the geography of opportunity in education, employment, safety, health, and other vital areas of the next generation are invisible in the nation’s public life and agenda. in her classic book titled the death and life of great american cities (1992), the late jane jacobs argued a successful city neighborhood is a place that is sufficiently aware of its problems so it is not defeated by them. in contrast, an unsuccessful neighborhood is a place that is engulfed by its deficiencies and is increasingly more powerless before them. she argued we americans are poor at managing city neighborhoods as documented by the long collection of failures. her treatise is one of numerous scholarly projects that underscore the unique importance of recognizing the urban context as a powerful influence on human development broadly defined (orfield, 2002; pattillo, 2007; rusk, 2003). the purpose of this commentary is to serve as a warning that developing and testing theories are central to making urban mathematics scholarship a visible research enterprise. more specifically, i will argue that there are lessons to be learned from the social sciences literature that can inform the advancement of a robust, theoretically based, empirical project in urban mathematics education research. in addition, these fields of social science are part of the rationale for why putting the “urban” in mathematics education scholarship is important. perhaps there are some scholars who accept the notion of research focused on the urban context as relevant and of great consequence. they understand that urban cities and communities are unique contexts that require research and policy evaluation to support their governance function. not everyone accepts this notion. is there a growing research literature related to urban communities in mathematics education? unfortunately, the 1 originally published in the inaugural december 2008 issue of the journal of urban mathematics education (jume); see http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/19/2. h http://education.gsu.edu/jume mailto:wtate@wustl.edu http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/19/2 tate commentary journal of urban mathematics education vol. 11, no. 1&2 27 answer is clear. too many education researchers ignore geospatial considerations. my hope is that the journal of urban mathematics education will create a new marketplace where theories related to urban cities and metropolitan regions across the world can be empirically tested and evaluated. a major point of emphasis for the scholar interested in urban mathematics education is theory building and empirical evaluation. if there are no theories (small or grand) to test and evaluate akin to the efforts in other social science research, then the field will yield little more than polemic and empty ideology. to date, both are plentiful. a brief review of the social sciences literature will illustrate the importance of geography as part of theoretical construction and testing. urban economics is a branch of microeconomics that examines urban spatial structure and the location of households and firms (o’flaherty, 2005). the urban economics literature includes the study of industrial clusters and technology-based hubs in metropolitan communities across the globe (gordon & mccann, 2000; sorenson, 2003). how industries cluster is directly related to a range of social factors including employment opportunities and tax capacity. incidentally, these two factors influence the quality and financial support for education (orfield, 2002). employment rates and tax capacity are important constructs in school finance. moreover, employment opportunities and tax capacity are a part of an expanding literature in urban sociology. the point is that economic theories make it possible to test the nature and extent of relationships within economics and across fields of study including sociology. urban sociology is the scientific study of social relations, human life, and human behavior in metropolitan areas. in this field, the chicago school has been a major influence. for example, both social disorganization theory and the spatial mismatch hypothesis have been tested and studied as a part of this urban research tradition (bursik, 1988; foster-bey, 2006; wilson, 1996). the point here is not to review these two theories; rather, the intent is to make clear that there are important theoretical projects being tested and vigorously debated. if urban mathematics education research is to be taken seriously, this kind of theoretical and empirical interaction should be the norm. theory-driven, empirical research is the norm in other fields of social science as well. the political science literature includes a sub-field in urban politics (brunori, 2003; judd & swanstrom, 2008). urban regime theory is prominent in this field (stone, 1993). the urban public health literature includes the examination of medical resources, risk factors, and disease (airhihenbuwa & liburd, 2006; douglas, esmundo, & bloom, 2000; jones-webb & wall, 2008). epidemiological theories and method are central to urban public research. the literatures of community psychology and the developmental sciences examine child and adolescent development and cognitive outcomes in a variety of urban settings (lee, 2008; spencer, 2008; spencer, dupree, cunningham, harpalani, & muñoz-miller, 2003). in a range of research fields the study of social interaction in the urban metropolitan http://en.wikipedia.org/wiki/microeconomics tate commentary journal of urban mathematics education vol. 11, no. 1&2 28 regions of the world is ongoing. this study of social interaction has been the case in education research as well. journals such as urban education and education and urban society have articles that are retrievable in electronic databases dating back to the 1960s. urban education published a special issue focused on mathematics education (tate, 1996). in sum, there is a long history of research that has taken seriously urban geography and related social interactions. this history suggests there is an intellectual space for urban mathematics education research. this intellectual space calls for scholars to fill the void. urban mathematics education is a rich topic with significant policy implications. during the 1980s and 1990s, both the ford foundation and national science foundation invested in mathematics education reform efforts and related evaluation studies in cities across the united states (campbell, bowden, kramer, & yakimowski, 2003; kim, crasco, blank, & smithson, 2001; silver & stein, 1996; webb & romberg, 1994). these large-scale interventions and evaluation studies brought attention to the topic of research and urban mathematics education. there is other mathematics education research focused on course-taking, teacher quality, and assessment practice that has a spatial dimension (anderson & tate, 2008). the geography of opportunity has been central to the mathematics education research involving urban communities. however, there are two interrelated challenges that must be addressed if this scholarship is to flourish going forward. the first challenge involves theory. there is a need for theory building, testing, revision, and retesting. there are important lessons to be learned from closely examining the history of research in urban economics, urban sociology, urban health, urban politics, and community psychology. a second challenge is related to collective cognition. in their award-winning book titled building civic capacity: the politics of reforming urban schools, urban regime theorists, stone, henig, jones, and pierannunzi (2001) contended that collective cognition matters when the goal is to take on the task of problem solving in urban school reform. to this end, i have argued elsewhere that urban communities are in desperate need of research consortiums where the distinguishing features are comprehensive data archives that provide sustained opportunities to study and learn about human development in the region (tate, 2008). the data archives should include at minimum the theoretically important measures related to urban mathematics education. in addition, this intellectual space is where researchers and practitioners should test and retest the theoretical project and push the boundaries of new knowledge. the challenge is to build theories and models that realistically reflect how geography and opportunity in mathematics education interact. if this challenge is addressed, the field will be one step closer to making scholarship in urban mathematics education visible. tate commentary journal of urban mathematics education vol. 11, no. 1&2 29 acknowledgments a special thank you is extended to celia keiko anderson, debra barco, richard milner, and dorothy white for their feedback on this commentary. this article is based on research and development supported by the national science foundation under award no. esi-0227619. any opinions, findings, and conclusions or recommendations expressed here are those of the author and do not necessarily reflect the views of the national science foundation. references airhihenbuwa, c. o., & liburd, l. 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(1996). when work disappears: the world of the new urban poor. new york, ny: knopf. title of your article journal of urban mathematics education december 2018, vol. 11, no. 1&2, pp. 78–102 ©jume. http://education.gsu.edu/jume ole skovsmose is a professor in mathematics education at the department of education, learning and philosophy, aalborg university, fibigerstraede 10, 9220 aalborg east, denmark; e-mail: osk@learning.aau.dk. he has a special interest in critical mathematics education and has investigated the notions of landscape of investigation, mathematics in action, students’ foreground, and ghettoising. pedro paulo scandiuzzi is a professor at the department of education, mathematics pratical education, university são paulo states, brazil. he has a special interest in ethnomathematics education, indigenous education, mathematics in different social-cultural groups, and teacher formation. paola valero is an associate professor in mathematics education at the department of education, learning and philosophy, aalborg university, fibigerstraede 10, 9220 aalborg east, denmark; e-mail: paola@learning.aau.dk. her research interests are the political dimensions of mathematics education at all levels. helle alrø is a professor in interpersonal communication at the department of communication and psychology, aalborg university, kroghstraede 1, 9220 aalborg east, denmark; and professor ii at bergen university college, norway; e-mail: helle@hum.aau.dk. she has a research interest in interpersonal communication and learning in helping relationships. learning mathematics in a borderland position: students’ foregrounds and intentionality in a brazilian favela 1 ole skovsmose aalborg university paola valero aalborg university pedro paulo scandiuzzi university são paulo states helle alrø aalborg university bergen university college in this article, the authors introduce a theoretical framework for discussing the relation between favela students’ life conditions in relation to their educational experiences and opportunities. a group of five students from a favela in a large city in the interior of the state of são paulo in brazil was inter-viewed. the students were invited to look into their future and explore whether or not there could be learning motives relating mathematics in school and possible out-of-school practices, either in terms of possible future jobs or further studies. four themes were identified: discrimination, escape, obscurity of mathematics, and uncertainty with respect to the future. students in a favela could experience what the authors call a borderland position, a relational space where individuals meet their social environment and come to terms with the multiple choices that cultural and economic diversity make available to them. keywords: brazilian favela, borderland position, students’ foregrounds 1 originally published in the inaugural december 2008 issue of the journal of urban mathematics education (jume); see http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/4/4. http://education.gsu.edu/jume mailto:osk@learning.aau.dk mailto:paola@learning.aau.dk mailto:paola@learning.aau.dk mailto:helle@hum.aau.dk http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/4/4 skovsmose et al. brazilian favela journal of urban mathematics education vol. 11, no. 1&2 79 he permanent growth of shanty towns (favelas in brazil, invasiones in colombia and ecuador, townships in south africa, or grecekondu in turkey) is characteristic of the unequal growth of modern society in many countries in the world. the brazilian word favela refers to an urban area formed when large groups of people moved from the rural areas to the big cities in search of work, took possession de facto of large empty extensions of land, and started constructing dwellings by putting together plastic, cardboard, wood, concrete, or whatever material could offer shelter from the inclemency of sun and rain. when the roof is completed, the house is finished. a favela is always transient and in permanent construction, even if time seems to have regularised it. the older red brick houses are rough, built side by side and on top of one other in layers that remind one of a fragile domino rack. the red bricks remain exposed and uncovered as they are; they never get dressed with cement, and the walls never get painted. the entrenched network of small, almost impenetrable streets is a labyrinth where vulnerable electricity installations meet flying water pipes and open sewers. for an outsider, a favela signals resignation. the film cidade de deus (city of god) provides an impression of life, and of criminal life in particular, in one of the most famous favelas in rio de janeiro.2 that is the picture that many people have when thinking of a favela. however, favelas in other cities in brazil, with different conditions, look more like slums where disadvantaged people struggle to make a living. the metropolis of today includes a patchwork of neighbourhoods and economic extremes. one finds squatter settlements beneath highway junctions where the passing of speedy, fashionable new cars almost blow poverty away. rich neighbourhoods and favelas are separated by only a few streets. the patchwork of diversity is kept together by invisible threads that also maintain radical forms of separation. rich condominios (gated communities) are surrounded by high walls topped with electric wires. a guarded gate separates the outer reality from the apparently protected, wealthy life inside a condominio, which looks more like a small city surrounded by a wall than a neighbourhood. here, unlike most houses in brazilian cities, no walls separate the houses and windows are not barred. green lawns and gardens, crystal blue swimming pools, and well-dressed families certainly contrast with the air of messiness that emanates from a favela only a few streets away from the outer walls of the condominios. that students coming from different neighbourhoods experience different educational opportunities is no new eye-opener in educational research. many studies focusing on students’ backgrounds and their influence on education have provided 2 cidade de deus is an oscar-nominated brazilian film, released in its home country in 2002 and worldwide in 2003. it was adapted by bráulio mantovani from paulo lins’s novel city of god (1997/2006), which is based on the true story of the parallel lives of two young men from a favela in rio de janeiro. t http://en.wikipedia.org/wiki/academy_award http://en.wikipedia.org/wiki/brazil http://en.wikipedia.org/wiki/2002 http://en.wikipedia.org/wiki/2003 http://en.wikipedia.org/wiki/paulo_lins http://en.wikipedia.org/wiki/city_of_god_%28novel%29 skovsmose et al. brazilian favela journal of urban mathematics education vol. 11, no. 1&2 80 evidence of the fact that there is a strong relation between students’ material and cultural life conditions and their experience in an educational system. it is beyond the scope of this article to provide a thorough account of research documenting this relationship because there have been many in different countries in the world. researchers such as cooper and dunne (1999) in england, zevenbergen (2001) in australia, vithal (2003) in south africa, and oakes and collaborators (2004) in the usa, have provided an analysis of this issue in operation in mathematics and science education. our intention in this article is to bring into the discussion a set of different theoretical tools to cast light on the relation between students’ life conditions and their educational experiences and opportunities. students coming from different neighbourhoods can experience and foresee very different life opportunities. students belonging to disadvantaged and marginalised social groups are faced with the stark question of who they are and who they can become. students’ perceptions of their future life possibilities are full of conflicting experiences, realities, dreams, and hopes for the future. all of these can impact students’ motives for engaging in schooling and learning in general, and in learning mathematics in particular. in what follows, we start by introducing the notions of foreground, intentions for learning, and borderland position. we explore the potentiality of these notions by relating them to a conversation with a group of brazilian students in a favela. we highlight some of the issues that we see emerging from the interview in relation to the notions, and we conclude by discussing the potentialities of the concepts in relation to mathematics education. foregrounds, intentions for learning, and borderland position we have been developing the notions of students’ foregrounds and intentions for learning over a longer period of time, while only recently have we try to explore the notion of borderland position. we define a person’s foreground as his or her interpretations of life opportunities in relation to what appears to be acceptable and available within the given socio-political context (see, e.g., alrø & skovsmose, 2002; skovsmose, 1994, 2005a, 2005b). this notion emphasises that students’ engagement in learning is deeply rooted in the meaning they attribute to learning with respect to their future life. in this sense, the intentions for learning might be connected not only to the “past” or the background of a student but also to his or her “future” or foreground. seeing meaning in learning as related to the future, rather than to the past, emphasises that students’ making sense of schooling in general, and of mathematics education in particular, is not only cognitive in nature but also skovsmose et al. brazilian favela journal of urban mathematics education vol. 11, no. 1&2 81 socio-political. meaning given to learning is bounded by the learner’s social, political, cultural, and economic conditions and how the learner interprets them.3 the notions of students’ foregrounds and intentions for learning have been used in interpreting a variety of educational phenomena. some educational research has located certain groups of students as having problems with mathematics. a grotesque example of such a stigmatization is found in the so-called white research on black education, conducted during the apartheid period in south africa (see khuzwayo, 2000, for a critical discussion of this research). this research identified black children as low achievers in mathematics and suggested an explanation of this observation in terms of a deficit discourse. this discourse could take different racist formats: the weak performances of black children are due to their biological origin; or: the weak performances are due to the structures of black families. however, do we consider the black children’s foregrounds in apartheid south africa; it simply appeared ruined due to the very apartheid regime. a socio-political and economic destruction of opportunities for a certain group of people is a tremendous obstacle for learning. considering the students’ foregrounds might reveal the limitations of deficit interpretations of school performances, and turn the attention to the sociopolitical and economic formatting of life opportunities, and, as a consequence, of conditions for learning. in previous studies, we have illustrated how the way students experience learning may relate to their foregrounds. in alrø, skovsmose, and valero (in press) we interviewed 8th grade students in a multicultural school in denmark. in one inter-view, razia, an iraqi refugee, clearly points out how, in her perception of her school mathematics experience and her hopes for the future, discrimination is present. her reaction to this discrimination is incarnated in her head-scarf, a symbol of muslim womanhood that she herself has decided to keep and defend fiercely as a way of showing who she is, where she comes from, and what she wants to become. valero (2004) illustrates how the mathematical school experience of colombian students in poor public schools is deeply rooted in the socio-political context where the students act as human beings. escaping a harsh life might be a reason to learn, however, not powerful enough to give full meaning to school mathematics. in skovsmose, alrø, and valero (2007), we have explored how a group of indigenous students in brazil see their foregrounds, and the meaning they attribute to the experience of learning mathematics. the apparent lack of significance of mathematics is replaced mainly with an instrumental significance. baber (2007) has studied how 3 such a definition of foreground allows thinking about the similarities and differences with other powerful notions such as “identity,” which has been increasingly used in educational research. the discussion of identity and foreground deserves an article on its own. suffice it to say, here, we see similarities and differences with the notion of identity as presented by, for example, sfard and prusak (2006). skovsmose et al. brazilian favela journal of urban mathematics education vol. 11, no. 1&2 82 pakistani families in denmark see mathematics as playing a central role in their participation as citizens of the country, and he points to the uncertainty about the future that characterises their current situation. we see the students’ foregrounds and their intentions for learning as closely related. furthermore, we find that both foregrounds and intentions are structured differently for different groups of students. here, we will pay particular attention to the notion of borderland position, which refers to a position from where the individual can see his or her current life conditions in relation to other life possibilities. the “borderland” metaphor has been used in research dealing with cultural diversity to signal the vicinity and overlapping, as well as the conflict between people’s participation in different cultural worlds.4 we see the borderland as a space of individual and social exchange where the meaning of difference is negotiated. a borderland position is a relational situation where individuals meet their social environment and come to terms with choices that diversity makes available for them, as well as with the many choices that are beyond reach. borderland positions exist for all people. for a person placed in a marginal position in relation to the dominant culture or establishment, however, the borderland position shows the sharp and clear contrast between his or her world and other worlds, particularly those belonging to the participants in the dominant culture. being in a borderline position allows that person to experience social, cultural, and political differentiation and the stigmatization that operates through the stories that the dominant culture constructs about his or her life. focusing on people in borderland positions allows us to have an insight into how exclusion/inclusion mechanisms operate and, more important, are experienced by those deeply affected by them. we now turn to the streets, houses, and people in a brazilian favela with the intention of illustrating the significance of the notions in relation to how a group of youngsters experience their mathematical learning. inter-viewing students in a brazilian favela in what follows, we will meet five students from a favela located in a large city in the interior of the state of são paulo in brazil. pedro paulo scandiuzzi has known them for some time and invited them to look into their future: how would they like to see themselves in the future? could there be any “learning motives” relating mathematics in school and possible out-of-school practices, either in terms of possible future jobs or further studies? ole skovsmose has also met the five students and spoken with them. paola valero and helle alrø have never met the stu 4 for further discussion of related notions see chang, 1999, and macdonald and bernardo, 2005. skovsmose et al. brazilian favela journal of urban mathematics education vol. 11, no. 1&2 83 dents personally but have read pedro paulo’s inter-view transcript and ole’s accounts of his meetings with the students and with pedro paulo. the five students that pedro paulo inter-viewed5 were: júlia, mariana, natália, argel, and tonino.6 mariana was 14 years old at the time, while júlia and natália were16 years old. the two boys, argel and tonino, were both 16 years old. argel was eager to present what he wants in life, while tonino remained quieter. mariana and natália talked rather freely, while júlia was normally relatively quiet. but given that the inter-view took place in júlia’s house, she might have taken upon herself the responsibility of being a hostess, and in this respect, she participated eagerly. júlia, mariana, natália, and argel attended a public school called floriano paixoto. this school is comprehensive, containing primary, secondary, and upper secondary levels. the school is surrounded by high walls. the gate of the school is locked and watched by a guard who ensures that only those who are supposed to enter, in fact do enter. in this city, even a poor school is in danger of being robbed. the walls might also help to protect the students when they are in school, as well as preventing them from escaping before they are allowed to leave. the school is located in a densely populated and rather poor area of the city. part of the area includes the favela cidade de são pedro, where four of the students come from; tonino is from a nearby favela. tonino does not attend floriano paixoto but an agricultural school called esperança verde, which is located on the outskirts of meiadia, a neighbouring town. this school is surrounded by fields and has a variety of animals. the students have the opportunity to learn farming through the praxis of farming. the agricultural school applies an alternative educational programme, where students have to be at the school for 2 weeks, and then work at home for another 2 weeks. this alternating attendance ensures better possibilities for students from poor families to go to school, given that their financial support could be needed at home. in esperança verde, 5 hours per day are dedicated to regular school subjects, while 4 hours are reserved for practical subjects. pedro paulo and ole visited the schools, floriano paixoto and esperança verde. the head of floriano 5 following previous studies where our task was the empirical exploration of students’ foregrounds (see skovsmose, alrø, & valero, 2007; alrø, skovsmose, & valero, in press), pedro paulo orchestrated a conversation with the students where questions about their life, their imaginations for the future, their like for mathematics, and their perception of mathematics in their current and future life were discussed. we use the term inter-view, inspired by kvale’s (1996) concept of a semi-structured inter-view that develops as a conversation about selected topics. thus, a semi-structured inter-view is “an interview whose purpose is to obtain descriptions of the life world of the interviewee with respect to interpreting the meaning of the described phenomena” (p. 5). this description also implies an active asking of questions and exploring of answers between inter-viewer and inter-viewee that emerge through the conversation. 6 all names of students, schools, and locations are pseudonyms. skovsmose et al. brazilian favela journal of urban mathematics education vol. 11, no. 1&2 84 paixoto showed them around and told about the stressful life of directing a school. at esperança verde two students showed them around and talked about the organisation of the school. pedro paulo has had contact with people from cidade de são pedro for a long period of time. he knows many people there, and he is known by many. because the neighbourhood is nearby the university where he works, and the university library is available for schools and students from the neighbourhood, pedro paulo has had the chance to help these students when they have needed a hand with homework or an activity in the university. in this way, pedro paulo has become a friend, a person who is allowed in the favela even though he does not live there. he has often visited júlia’s family, and júlia was happy to invite her friends to her house for the inter-view. the inter-view was scheduled for the evening to make it possible for the students to participate. in what follows, we turn to the inter-view and listen to how the students describe their situations and their expectations and hopes for the future, and their wishes for further education.7 what do you not want to do with your life? the small room in júlia’s house accommodates enough chairs to seat everyone. some of the chairs have seats made of braided plastic strings, originally of different bright colours. time and use, however, have made them appear the same. pedro paulo breaks the ice and tells a bit about himself: pedro paulo (pp): when i was your age, 14 and 16, i studied in a public school in a town close to here [...] i went to school, played ball, went fishing, took small jobs, and dreamed of travelling, and that’s why i studied a lot. i dreamed of attending good schools. and that was my life. i studied a lot. and afterwards, i left and went to work in ubatuba8 as a mathematics teacher. now i have returned, and i’m working here at the university […] they say that i’m at the end of my life, being over 50. so, i’m getting to the end. 7 after pedro paulo conducted the inter-view, a transcript of the session was produced and translated into english. readings of the transcript were discussed between ole and pedro paulo, who provided additional information and contextualization about the students’ ideas, based on pedro paulo’s knowledge of them and their situation. all members of the research team discussed different interpretations of the students’ words and of what seemed to be behind them. we do not use the interview as an empirical documentation of the students’ actual thinking, motives, and intentions. we use what they express as a window into a reality that triggers our reflections on the concepts that we want to explore. the quotations from the inter-view are presented in the original order. parts of the transcription, however, have been omitted. 8 ubatuba is a town along the coast between são paulo and rio de janeiro. skovsmose et al. brazilian favela journal of urban mathematics education vol. 11, no. 1&2 85 the first to be addressed is argel, who is in his 2nd year of upper secondary school. in addition to his regular classes, he takes a course in electronics and a course to prepare for a military career.9 such a career includes much competition, but argel is ready to face the challenge. he says that he likes geography, history, and biology, and also art education, although less so. he also likes mathematics and portuguese a little. he prefers physics and chemistry, however. let us listen to his remarks about mathematics: pedro paulo (pp): what are you learning in mathematics? argel (a): uh, i’m studying, now at this moment, matrixes; i’m studying matrixes and i’m studying all the definitions—reverse functions, inverse of the matrix […] pp: and what have you thought about doing with these matrixes? a: well, last year, other calculations appeared; this year for me, what am i going to do with this […] another course that i’m taking is electronics. the matrixes i’m going to use—they have a binary sequence. argel is working with matrixes: their definition and formal properties. he also refers to possible connections between matrix calculations and the electronics course he is taking. the calculus of matrixes might well be included on the exam argel needs to pass in order to get started on his military career. he tries to clarify connections between matrixes and binary numbers. it is obvious, however, that the possible applications of matrix calculus are not clear to argel. crucial to argel is his choice of career. he is interested in the military, and this priority provides meaning to many other activities in school: a: i’m taking a preparatory course for the military. pp: you want to be in the military? a: i like it, the army or the naval air force. pp: the army or the naval air force? a: i’m not sure yet, where i’ll go. pp: is that what you want to do with your life? a: yes. pp: what do you not want to do with your life? a: hanging out here without doing anything, making a living doing what? i’m not going to keep depending on my parents for the rest of my life. argel has not made up his mind if he prefers the army or the naval air force. but his overall decision is made: he wants to pursue a military career. argel certainly does not want to hang out in the neighbourhood doing nothing. and he does not want to be financially dependent on his parents. argel’s comments touch upon the notion of meaning. learning about matrixes might not be experienced as meaningful because of applications that he knows 9 the course prepares students for the entrance exams of the military schools. skovsmose et al. brazilian favela journal of urban mathematics education vol. 11, no. 1&2 86 about; but rather, it might have instrumental significance, if it is significant for passing tests crucial for his future career. in fact, students might be ready to accept an instrumental significance as a preliminary resource of meaning as they assume that what they learned could later turn out to be relevant. to escape from the city a little tonino opted to study at esperança verde, near meiadia. but why did he choose to do so? tonino (t): ah, to escape from the city a little. pp: escape from the city a little? are your parents agricultural workers? t: my mother is a seamstress, and my father works in a factory. pp: yes, but were they agricultural workers before? t: my mother lived in the country. i don’t know about my father. pp: what led you to want agricultural school? t: employment, you know—leave there with employment guaranteed. tonino wanted to escape from the city. however, he does not seem to have connections to rural life, except that he knows that his mother once lived in the country. it might not be the content of agricultural work that provides the main attraction for tonino. it seems important to him to change location, and maybe, first of all, to be able to secure a job. this choice could provide stability in life, different from being a seamstress or a factory worker. tonino seems to believe that an agricultural education would lead to “guaranteed employment.” people from certain neighbourhoods in the city are not considered to be “reliable,” and they have difficulty getting a job. so, in order to get a job, it is not only important to get an education that could lead to a permanent job; it might also be important to change location, in order to get rid of the stigmatization that people from certain neighbourhoods, like cidade de são pedro, suffer.10 which school subjects does tonino like the best? could his preferences in school have something to do with the choices he has made? pp: what are the courses you like the least? t: history and portuguese. pp: do you like mathematics? t: more or less. pp: what have you studied in mathematics? t: i don’t remember. pp: you don’t remember? what are you going to do with this subject matter that you don’t remember? t: i don’t remember anything. 10 students comment on this issue later in the inter-view. skovsmose et al. brazilian favela journal of urban mathematics education vol. 11, no. 1&2 87 tonino might be referring to the mathematics from the secondary school. he might also be referring to mathematics at esperança verde. as mentioned before, the schedule in the school is organised around 2 weeks of work in school and 2 weeks of work at home. but mathematics, as well as any other school subjects, is out of tonino’s memory. he does not remember anything. then the conversation includes argel again: pp: argel plans to do what in the military, be a soldier? a: yeh, i suppose, you know […] there you start out as a soldier and then you pass the exams, tests, to rise in rank to captain, sergeant […] something to grow there within— but it’s a course there. you go through a public exam, and if you pass, you have technical courses, and after you leave there, you can even work in a large factory. because the highest salary is for colonel, retired earning double, in these industries. pp: you want to work in a factory? a: no, i want to take the course, because in addition to the military high school, i have technical courses in the morning, and in the afternoon, i practice and earn a salary like the ita [instituto técnico da aeronautica – technical institute in aeronautics], the espcex [escola preparatória de cadetes do exercito – preparatory school for the army], or aman [academia militar das agulhas negras – special force academy]. pp: you want to do one of those? a: i want to do the ita. pp: do you study a lot? a: at least two hours a day; if not, i don’t pass the tests. pp: two hours a day. you work, too, or just study? a: not me, i don’t have time. i study during the week and on saturday. i only have sundays free. argel knows about career possibilities and about how to obtain them: studying, despite the fact that studying “at least two hours a day” seems to be considered a lot. argel expresses his interest clearly. but what about tonino? is his interest limited to getting out of the city and getting a job? are there more reasons for argel to “remember” mathematics in light of his desire to enter the military, than for tonino who wants only to get a stable job? it might well be that stronger desires for the future bring better reasons to want to remember school mathematics. what do you remember? at this moment, the girls enter the conversation; first júlia, who is kind of a hostess. the subjects she likes include art education and physical education, while she does not like portuguese, which she finds to be very difficult. pp: do you like mathematics? júlia (j): more or less. skovsmose et al. brazilian favela journal of urban mathematics education vol. 11, no. 1&2 88 pp: what are you studying in school right now in mathematics? j: i’m reviewing the subject matter from the 3rd quarter for the test. pp: what subject matter, do you remember? j: delta, sets, images, things like that. pp: in the future, what do you plan to do with this mathematics that you are learning? j: i don’t know what i’m going to choose as a profession. i think it [mathematics] will help. júlia’s first answer to what she is studying does not concern the mathematical content. she studies for the test. asked directly about the subject matter, júlia refers to topics like delta, sets, and images (of functions). delta is the expression ∆ = b2 – 4ac used when solving the 2nd-degree equation ax2 + bx + c = 0. and what to make of this when one thinks of future education? júlia, certainly a polite hostess, confirms that although she does not know what she will choose as profession, she thinks that mathematics will turn out to be helpful. thus, júlia also seems to believe in the instrumental significance of mathematics. later in the conservation, júlia emphasises that she does not want to become a housewife and do housework. she does not want to stay at home preparing food for her husband. she says that she might want to study healthcare or medicine. these are ambitious wishes, and it might well be that júlia knows that mathematics composes part of such studies, although she does not know in what way mathematics will be useful. a housewife, in my opinion, is a slave natália is 16 years old. she is in the 2nd year of the upper secondary school. júlia and natália are in the same grade, although they are not in the same class. pp: but you’re not in the same class? what are you studying in mathematics? natália (n): we’re doing […] seeing some things about 2nd-degree functions, the delta. these 2nd-degree things. pp: what do you like least about school? n: the teachers. natália remembers the “2nd-degree things.” she seems to remember more than tonino, but a bit less than júlia. natália expresses clearly her dislike for teachers. then she is asked what she would not like to be: n: a housewife. pp: you don’t want to be a housewife? n: a housewife, in my opinion, is a slave. pp: even if she owns her house? n: even if she owns her own house. pp: why do you think that? skovsmose et al. brazilian favela journal of urban mathematics education vol. 11, no. 1&2 89 n: ah! because everything you tell her to do, she does. she doesn’t avoid doing it. even if she doesn’t want to, she does it. it’s like being a slave; you’re giving the order, and she’s following it. in natália’s view, a housewife is given orders and follows orders. this compliance is like the life of a slave, even if she is the owner of the place. natália’s words resonate with júlia’s. in the favela, girls have seen many women, starting with their own mothers, and they express their positive rejection of a life as a housewife. studying and choosing a profession seems to be a way of escaping that frightening scenario. natália therefore dreams of becoming a psychologist or a veterinarian. she likes animals very much, and she likes psychology because she likes to listen to people talk about their lives and to give them advice. when asked if mathematics has anything to do with veterinary medicine or psychology she answers: n: nothing. pp: it has nothing to do with it? júlia, tonino, argel, do you know what psychology and veterinary medicine would have to do with mathematics? t: i don’t have the faintest idea. pp: no idea. so that means that what she’s learning in mathematics will not be very useful to her? n: i think it will, because when you go to a university, you have to study all the subjects. natália seems not to see the instrumental significance of mathematics with respect to psychology and veterinary medicine. she, however, sees clearly that when one gets to the university, one must “study all the subjects,” including mathematics. that perspective might be reason enough to engage in school mathematics, above all, to avoid being a housewife. delta is just a formula mariana wanted to be the last to talk. she lives in a neighbourhood near by. she goes to the same school as argel, júlia, and natália. mariana is 14 years old and she is in the 8th grade, the last year of secondary school. she likes the school and the teachers, and she likes to study. but she does not like the school when there is much quarrelling and disorder. mariana intends to study law and become a lawyer, or maybe she wants to study medicine. and what about mathematics? mariana (m): ah! i’m in 2nd-degree delta, these 2nd-degree things. pp: and what will you do with these 2nd-degree things in medicine, or as a lawyer or a judge? m: ah! i think, for sure i’ll need it to go to the university. i’ll need it. pp: to go to the university. in your profession, you don’t think you’ll use it? skovsmose et al. brazilian favela journal of urban mathematics education vol. 11, no. 1&2 90 m: ah! i don’t understand it a lot. but i don’t think so. i don’t know. like everyone else, except for argel, mariana does not know what to do with the delta. well, it might be necessary knowledge for entering the university or the faculty of medicine. mathematics per se does not seem to be considered important. later in the inter-view, mariana mentions that she does not like portuguese, and grammar in particular: “what sense to make of issues like subordinate clause and punctuation?” mariana does not think of portuguese as being important for studying law. however, if it turns out that it is, she will be ready to study it. then pedro paulo returns to mathematics, and the students comment again on the delta formula. a: […] delta is just a formula. but you use it for the rest of your life. m: you keep deepening it, complicating it, more and more. pp: the delta gets complicated, just like life? m: i think so. j: more or less. pp: more or less? j: all is the same. pp: as times goes it gets more complicated. j: yes. m: in first grade you learn 2 + 2 and then it gets more complicated, you learn to divide. delta is just a formula, but it seems to stick with you, as argel emphasises: “you use it for the rest of your life.” it will appear in more and more complex situations, as all mathematics do. you start with simple things like addition, but it always gets more and more difficult. but as things get complicated, it seems as if the meanings of mathematical expressions and techniques do not emerge in the context of learning. their meaning might (or might not) be revealed later in school or in life. students seem to be struggling with what we could call the “delta syndrome,” a weird kind of disease in which the patients are presented with some mathematical formula or technique, which they are supposed to master in order to get on with their education, but whose significance will not be revealed until later. the inter-view then turns to a discussion of what the students’ parents are doing. it is clear that tonino, argel, júlia, natália, and mariana are hoping they will not become like their parents. mariana does not want to become a maid or cleaning woman, a type of job that many women in the favela have, doing housework in other neighbourhoods. mariana would like to become a housewife, however. she cannot follow júlia and natália who think that a housewife is a slave, even in her own house: m: but a housewife, yes, because i like to do the housework at home. i’m the only one who does it because my mom and dad work. skovsmose et al. brazilian favela journal of urban mathematics education vol. 11, no. 1&2 91 mariana’s mother works in the butcher shop owned by mariana’s father. natália’s father works as a truck driver, and her mother is a seamstress. júlia’s father works as a driver for the local government. part of his work is to assist in repairing the roads. júlia’s mother works as a kitchen assistant. tonino’s father is working in a furniture factory, while his mother is a seamstress. argel’ father has retired, and his mother is a housewife. pp: and she likes being a housewife? a: she likes it, because she didn’t know her mother and father. she was raised by her aunts, so she was their slave. at our house, we tell her not to do stuff, but she ends up doing it. she likes to do things. i want to help her, but she doesn’t let me. students’ expressions of their future profession are far from being inspired by their parents’ current occupations. even when júlia and natália express their dislike of the housewife life, they seem to do so in relation to the situation of their own families and relatives. they hope for something different, probably better. the exams are very complicated the students come to talk about the possibility of realising their dreams. they believe it is possible to achieve what they hope for, but that there are many difficulties. one is the tuition at private universities; another is the cost of the preparatory courses for the college entrance exams. it seems particularly difficult for those who dream of enrolling in some of the most expensive programmes (such as medicine). for example, a driver like natália’s father earns about 800 reais per month and one could expect that the study costs for natália would be around 400 reais per month; having children engaging in higher education puts a huge economic demand on a family. a student could do some work in addition to their studies, but a student’s salary would cover only a minor part of the study costs. only if one chooses to study at night and work during the day is it possible to make a reasonable amount of money. another option is to enrol in shorter technical or vocational programmes; however, those programmes are less prestigious. it is also possible to get some kind of scholarship; but then one must be an exceptional student and have very good grades. anyway, the cost of engaging in further studies is certainly a huge obstacle for making the students’ dreams come true. the public universities are free but very difficult to enter. in brazil, each university applies its own entrance exam, which applicants are charged for. they can register (and pay) for as many exams at different universities as they want, which are typically administered during the months of december and january. the results, often published in early february, take the form of a ranking list of all students that participated in the test. on the internet, one can see one’s position and also where the cut-off for entry was made. naturally, the most attractive public universities are skovsmose et al. brazilian favela journal of urban mathematics education vol. 11, no. 1&2 92 the most difficult to enter. many hopeful applicants take the exams, and the most attractive universities need only to select the top 10% of applicants. if one does not succeed one year, one can pay for a one-year study programme to prepare for the next year’s exams. and so on, until one enters, or until one gives up on the idea of doing further studies. again, the need for good exam results seems to go against the realisation of their hopes and future expectations. a: … the exams are very complicated. n: there aren’t many people admitted, either. students from a public school like floriano paixoto are unlikely to be as wellprepared for the college entry exams as students from private schools. brazil has a large number of private elementary and high schools, always better equipped than the public schools, and usually more focussed on ensuring their students good possibilities for pursuing further studies. so, the private schools provide the very best preparation for students entering the attractive public universities. the situation could be very different with respect to the public school, as argel explains: a: the classes they give, they are the same in the private high schools, and in the public schools it’s the same. but the teachers are slow. they’re not too concerned. some are concerned; others don’t even care about you. you, who are from the public or municipal school. n: in the public school, the teacher doesn’t care about what he does. a: in public universities, it’s very difficult to find people like us who studied in the public schools. in the public universities, they only have daddy’s little kids going there. they are in no need of going to public universities. pp: so, what are you going to do? you are in the public schools. you depend on a salary, and the salary isn’t high. you have the wish to get into a good program. what are you going to do? are you going to say, like, we’re just going to stop here? a: we have to study, to fight. n: we have to make an effort. the problem is clearly formulated by argel: in public universities, there is no room for many students from public schools. it is mostly well-off students who manage to get in. the students really find their opportunities restricted by their economic situation. some try to compensate by doing some extra courses. thus, júlia does extra studies in english, and argel takes a course in electronics, including computation. the lack of access to computers at home is a problem, so it is important for students to take courses where they are able to get experience with computers. the situation at home does not facilitate any form of study. most of the time there are not adequate resources to study; normally there are many people around, and it is difficult to find a quiet place to concentrate on studying. besides, many other characteristics of life in a favela—such as violence, struggles related to drug trafficking, skovsmose et al. brazilian favela journal of urban mathematics education vol. 11, no. 1&2 93 and even sexual assaults—are not the most nurturing for youngsters who want peace of mind and who are probably in need of getting rid of the “delta syndrome.” we’re discriminated against it is not difficult to list obstacles that these students have to face in their life. but are they able to find reasons for optimism as well? pp: do you guys see this desire of yours with optimism/excitement or not? j: ah! i get pretty excited when i think about what i want to be. pp: and you, tonino? t: you have to go after it. pp: argel? a: you have to fight. and if you get discouraged, feel down—you can’t get discouraged. pp: why do you get discouraged, argel? a: well, it’s kind of different when it’s time to study there. i feel discriminated against. j: sometimes people fail; give up, too. you have to persist. m: because public schools, the teaching is weak. not that it’s weak, it’s that the teachers don’t care, and the students even less. […] n: we keep getting left behind. m: i have a friend, he studies in the seta. he’s in the 8th grade. he knows five times more than i do. seta (sociedade educacional tristão de andrade – educational society tristão de andrade) is an expensive private school, located in the city centre. according to mariana and júlia, who know people attending this school, the students there are far ahead of those who study in public schools, including those who attend floriano paixoto. for them, it is really necessary to fight. as emphasised by argel, even during education, one gets discriminated against. they perceive schooling as a form of establishing and maintaining inequalities, rather than promoting equity. pp: argel, you said that you feel discriminated against sometimes. why do you feel discriminated against? a: ah! because they feel—they’re better than us, you know? pp: who? a: these people who are daddy’s little kids and are protected by their parents. then they want to give us the cold shoulder. they think they’re better than we are. the students experience discrimination, not only in terms of attitudes, like the “daddy’s little children” who think they are better; they are also discriminated against in real measures. in the private school, there are better teachers with more commitment, and the students have better conditions for learning. pedro paulo, however, points to a fact that might serve as a counter-balance to their experience of being left behind: skovsmose et al. brazilian favela journal of urban mathematics education vol. 11, no. 1&2 94 pp: did you know that in the universities a lot of people are entering that studied in public schools? and those students are getting into a good study habit, facing all those people who had college preparation courses. a: they’re the bigger schools there downtown, aren’t they? […] those schools downtown where the teachers are stricter. j: they get after you more, demand more. pedro paulo points out that one finds many students from public schools studying in the universities. argel, however, stresses that they are from the bigger public schools in the city centre, where teachers are more strict, demand more and, therefore, prepare students more adequately for further studies than schools in a poor favela neighbourhood. then mariana and argel add: m: they [the students from downtown schools] don’t have the needs that we have. they [the teachers] discourage us, too. a: because of two or three in the class, she discriminates against everyone […] everyone pays for it; everyone is a trouble-maker. this is not true. just because of two or three that are like that, everyone gets into trouble. mariana emphasises that there are differences among students in public schools. different students could have different needs. she indicates that teachers discourage students from poorer neighbourhoods to try to pursue further studies. argel follows up by pointing to teachers who exercise discrimination and stereotyping. there might be some students from their neighbourhood who might cause trouble for the teacher, but “everyone pays for it” and all are discriminated against. then pedro paulo turns to tonino who attends the agricultural school in meiadia. how are things experienced in this place? pp: is it like that in meiadia, too, tonino? t: we’re discriminated against in meiadia. pp: you’re discriminated against in meiadia. t: it’s the agricultural school they talk about. leaving to go to another city is difficult. pp: and why did you choose a school that is discriminated against? t: i didn’t know, either, right—i arrived there believing it was a wonderful place. pp: ah! did they take you to visit? t: it was my mom who visited the school. pp: what school there has a good reputation? t: ah! i don’t know. now tonino realises that esperança verde might be a school that is also regarded as having a very low status. it is a rural school, and according to tonino, they are discriminated against in meiadia. the same is the case for floriano paixoto, located in cidade de são pedro. skovsmose et al. brazilian favela journal of urban mathematics education vol. 11, no. 1&2 95 j: they think, like, that it’s poor suburbs. even we who live in the poor suburbs, we’re discriminated against, if you look. a: there when i arrive at home—cidade de são pedro is the worst neighbourhood in our city, a favela. to get a job, it depends on courses. you get there to enrol, and they’re even afraid to meet you. j: because of two or three, we get it because of that. i already tried to get a job, and i didn’t get one. the students address not only the problem of being stigmatised by coming from the favela of cidade de são pedro, but also teachers might exercise discrimination. it might be difficult to get a job in other parts of the city. people in general might feel afraid of someone coming from cidade de são pedro, as argel says. the stereotyping of favela life as portrayed in the media falls on all its inhabitants. júlia expresses it clearly: a few people get in trouble, but not all; still, that affects her own possibilities for employment. issues of life, learning, and mathematics in a favela the inter-view between pedro paulo and argel, júlia, mariana, natália, and tonino reflects different aspects of the life conditions of students in a favela as they perceive and experience them. let us highlight some themes that we see emerging from the inter-view. these themes are related to the students’ foregrounds being in a borderland position and they seem to influence their motivation for learning mathematics. the first theme is discrimination. the students feel they are being discriminated against due to the fact that they come from a favela, a poor neighbourhood. there is no doubt that the socio-economic conditions strongly limit the possibilities for people from cidade de são pedro. favela life is a life in poverty, and poverty stigmatises people. it affects many aspects of life: the clothes one is wearing and one’s habits (young people from a favela do not go to the cinema, but they might hang out at a gas station convenience store). it affects possibilities of doing homework, of accessing books and other resources for doing homework, and of studying. however, poverty not only sets a range of life conditions; it also frames the way others look at one. based on their experience, the students feel it is better not to reveal that they come from cidade de são pedro. they could be discriminated against, not only economically speaking but also in terms of attitude: people could look down on them, look at them as potential criminals. somehow poverty also frames the way one looks at oneself. the students fear being trapped in some stereotype, and there could be good reasons for this fear. a dominant theme of the news in brazil is violence, often associated with the favelas, particularly the famous ones in rio de janeiro and são paulo. as already mentioned, cidade de deus (the city of god) is both a name of skovsmose et al. brazilian favela journal of urban mathematics education vol. 11, no. 1&2 96 a favela in rio de janeiro and the title of a film about the meeting of life, crime, and violence in this favela. this violence includes the wars between gangs, the war to expand or keep the drug markets, and the war against the police. but it also includes the everyday life of thieves who systematically assault the trucks that deliver tanks of propane gas to the households in other areas of the city. it certainly includes the struggles of many workers like the students’ parents to make a living, and the struggle of the students themselves to have a chance in the future. all such common “knowledge” about life in a favela is the basis for the construction of stereotypes that stigmatise favela inhabitants. so, when the students react to the possibility of being discriminated against, they might well have good reasons for doing so. the second theme is escape. there is a strong motivation to begin a new life away from the favela. however, it is not clear to what extent this “new life” is experienced by the students as something they, realistically speaking, could work for, or as just something they dream about. there is a strong motive for escaping the neighbourhood. it could be taken in a strict sense as expressed by tonino. but “escape from the city” could also be taken as a metaphor for getting out of the life conditions the students know all too well, such as júlia’s and natália’s reactions towards being a favela housewife. they all acknowledge that the best way to escape is through further education. therefore, the discussion of tuition fees for entering the university becomes at the same time important and fatal. a third theme concerns the obscurity of mathematics. it seems clear to everyone that education is relevant for ensuring a change in life. the role of mathematics in changing life, however, is less visible. mathematics lessons do not provide any clue of how mathematics might function in this respect. one could see an instrumental significance of mathematics, while the content of mathematics in itself appears meaningless. the mathematics curriculum in brazil is a manifest representation of the school mathematics tradition. this tradition defines the curriculum with strong references to mathematical ideas, notions, and structures. everyday examples might be included, but mainly to illustrate mathematical conceptions, and not as situations to be explored in greater detail. the school mathematics tradition places a particular emphasis on the teacher’s presentation of the mathematical content (and not on, say, communication among students about mathematical problems). naturally, the teacher’s presentation takes on a particular significance when the students have no textbooks, and have to rely only on the notes they take themselves. and in order to make reliable notes, what could be better than carefully copying down what the teacher writes on the blackboard? a nice pedagogical contract could be established between teacher and students. as long as the teacher makes a careful presentation and students copy down the presentation, then everybody has done their job properly, and good order can prevail in the classroom. still the obscurity of mathematics prevails. skovsmose et al. brazilian favela journal of urban mathematics education vol. 11, no. 1&2 97 the inter-view indicated clearly that it was very difficult for the students to point to any relationship between mathematics and their future studies and work. argel gave it a try but was not very successful. the only relationship they could openly express was instrumental: mathematics is a necessary ingredient for passing required university entry examinations. at the same time, they did not deny that mathematics might turn out to be significant; they were just unable to see what this could be like. the “delta syndrome” was part of their experience. this brings us to a fourth theme, namely the uncertainty with respect to the future. the students are remarkably aware of what they do not want from the future: argel does not want to hang out and be financially dependent on his parents, tonino does not want to stay in the favela, and natalia does not want to become a housewife. and they agree that education could be an entry point into another kind of future life. the students find that they might have difficulties in competing with privileged children. they find that the differences are established because of differences in schooling, their teachers and the resources available to them. if one considers the ranking of the different schools in brazil, there is no doubt that wealthy private schools top the list with respect to ensuring their students’ access to private and public universities and colleges. schools located in favelas are very seldom found on such lists. the students also felt that teachers might treat them as inferior; as someone who is not capable of completing further studies. the students could easily formulate very optimistic but almost unattainable aspirations, while reality might set some heavy limitations. how, then, to get out of such uncertainty? one way of getting out is simply to stop dreaming and hoping, and instead become “realistic” and renounce one’s ambitions. one could simply face that one is doomed to a poor modest life. so, it may be better to get out of school and get a job, a permanent job, if possible. leaving school, however, is not what the students want to do and actually seem to do: “you have to fight,” “you have to go after it,” “make an effort,” “persist” are all expressions of their feelings that they can influence their future life. what is the significance of these issues of discrimination, escape, obscurity of mathematics, and uncertainty with respect to the future for understanding the way in which students decide to engage in learning mathematics? in what follows, we explore a bit further the notion of students’ foregrounds, intentions for learning, and borderland position in order to address this question. generating intentions for learning while constructing foregrounds in a borderland position we consider learning as an act, and as such it requires intentional engagement on the part of the learner. this claim does not apply to all forms of learning; thus, many habits may be adopted without much intentional engagement, and some skovsmose et al. brazilian favela journal of urban mathematics education vol. 11, no. 1&2 98 forms of learning may be forced on people. when we see learning as action we have in mind forms of learning as they might take place in school, for example, learning mathematics. students might get involved in solving mathematical problems or be engaged with mathematical investigations; but they might also find the classroom activities to be without meaning and occupy themselves with other things. a decision about being involved in the mathematical tasks, or not, is not simply the result of a conscious individual choice, but rather a decision that is strongly associated with the intricate relationship between the student, the teacher, and the context for learning in the social-political-cultural environment. the meeting between the individual and the social is a space where intentions for learning emerge and grow, or might be destroyed. in that space, the individual constantly constructs and re-interprets both previous personal experiences and actual life conditions in dynamic relation to his or her wishes for life and dreams for the future. in other words, the individual’s consideration of his or her background in relation to his or her foreground is a powerful source of reasons and intentions to decide to engage in learning as well as a cause for giving-up to be engaged in learning.11 while the notion of background has been central in much research trying to establish a connection between students’ learning experiences and students’ social environment, the notion of foreground is relatively new. we find that the notion of foreground has a close relationship with intentions for learning, which in turn represents the broader meaningfulness that students might associate to processes of learning. the students’ foregrounds are constructed through different social processes. in a profound way they are constructed through economic conditions; thus, poverty is a highly influential factor. the construction of foreground, however, includes many other elements. in this article, our interest has been focused on students who are constructed by others, and even by themselves, as marginalized and excluded from dominant cultural practices and forms of life. when students experience discrimination, they perceive that it will be difficult, if not impossible, for them to cross the line and become part of the dominant culture. this experience strengthens their awareness of their own stigmatized position. we find discrimination to be a powerful social factor, which might ruin the foregrounds of certain groups of people. for the five students from cidade de são pedro, argel, júlia, mariana, natália, and tonino, their borderland position allows them to constantly weigh a set of favela-life opportunities against, for example, a set of “city-centre life” opportunities, or “condominio-life” opportunities. they can see what it would take for them and for their education to cross the line to enter other ways of life. one reaction to the experienced discrimination turns into a dream of escape. education is clearly 11 for an in-depth discussion of the notions intentionality in learning, background, and foreground see skovsmose (1994, 2005); see also alrø, skovsmose, and valero (in press). skovsmose et al. brazilian favela journal of urban mathematics education vol. 11, no. 1&2 99 one possible way of doing so and, therefore, learning (mathematics)—even if the reasons are purely instrumental—makes sense and represents a more or less meaningful investment in the future. at the same time, however, they can also see and experience the enormous barriers to a successful jump over the border. their borderland position makes evident the harshness of social division, stratification, and stigmatization. we could imagine a borderland school as the site of learning that provides an opening for radically different life opportunities. (it might also be that such a school would jail students in their current positions.) borderland schools should be able to establish opportunities for a transition from one way of life to a different one. at least students in a borderland school might consider such transitions to be possible. what transitions, realistically speaking, a borderland school might be able to prepare for is another question. the obscurity of mathematics has a strong implication for the students’ experience of the opportunities a school might provide. there seems to be some agreement among the five students from cidade de são pedro that mathematics might play a role in further education, but it is not clear to them what role mathematics in fact could play. this lack of clarity means that it is simply impossible for students to relate their activities in the mathematics classroom to any more specific features of their foregrounds. as students’ foregrounds are associated to their construction of meaning, the activities in the mathematics classroom remain meaningless, or, as best, instrumental. this construction is a huge learning obstacle for students in a borderland position, who experience an uncertainty with respect to their future. in previous studies, we analysed brazilian, indigenous students’ perceptions of their educational possibilities and priorities.12 one student had made a clear choice: he wanted to study medicine. completing such study would certainly establish a new life situation for him. however, his priority did not include a break with his indigenous background and life in the indigenous village. he wanted to study medicine with the particular aim of being able to return to the village and contribute to the effort to improve the health situation of the indian community. therefore, one has to be aware that possible transitions can be thought of in very different ways. when one talks about transitions, one should not assume any simplistic scale of preferences. for example, it should not be assumed that white, middle-class priorities and life opportunities are, by definition, “better” than some other forms of priorities. one should not assume that the scale of priorities reflects a scale of economic wealth. nor should one try to romanticise poverty. we try to avoid assuming any simplistic scaling, and instead to listen to how priorities might be expressed, how students might think of possible transitions, and how they can be related to their learning motives. 12 see skovsmose, alrø, and valero in collaboration with silvério and scandiuzzi (2007). skovsmose et al. brazilian favela journal of urban mathematics education vol. 11, no. 1&2 100 postscript: the fragility of dreams almost three years after the inter-view, pedro paulo and ole again visited júlia’s family. júlia, natalia, argel, and tonino were all there; mariana had moved to another city. it was a nice evening. the four students told about what had happened to them during the past 3 years and about their current situations. they told about what had become of their dreams and aspirations. júlia’s family had moved to a house in the countryside. three dogs barked and wagged their tails welcoming the visitors, together with chickens and ducks, while cows grazed in the field next door. the garden had vegetables, and pedro paulo picked a small bag of lemons from a tree to bring home. there were also some other friends around in the house. júlia’s mother had cooked the food, and her father showed the guests around. júlia was not talking very much, and when her boyfriend arrived—he was even more taciturn than júlia—they spent the rest of the evening holding hands. júlia had stopped her studies, and she was now working as an assistant in a lawyer’s office. she was considering starting studying again in order to become a nurse specialized in radiology, which is a programme that can be completed in only 2 years. tonino had left the agricultural school. he had become much more talkative, and now he wondered why he had started at the agricultural school at all. farming was not really something he found interesting, which he expressed while pressing his long thin fingers firmly together. he liked the city, and he had found a job. he was working in a goldsmith’s shop, and one of his jobs was to put together different components of the jewellery. did he like the work? he was not sure. he said that he would like to become a policeman. he believed this profession would bring him better opportunities in life. natália had begun studying to become a nurse. she helped her mother with the housework. she also helped her mother in her work as a seamstress. natália had entered a private institution, and she had to pay for her programme of study. she was receiving a small scholarship, but the largest portion of the money she needed came from her parents. during the evening, argel was the one who spoke the most. he had stopped his studies and was no longer considering a military career. he had arrived that evening with his wife and their small baby. it was a smiling baby who, in a good mood, said hello to everyone who wanted to touch and tickle him to make him smile, which he did. it was a happy family, and argel took perfect care of his son. he was considering moving to a city in the neighbouring state of minas gerais where he saw some better opportunities for getting a job. he hoped to work with computers. skovsmose et al. brazilian favela journal of urban mathematics education vol. 11, no. 1&2 101 when one considers students’ learning of mathematics in a borderland position, one sees many factors in operation. we have pointed to discrimination, escape, obscurity of mathematics, and uncertainty with respect to mathematics. meaningfulness (or lack of meaningfulness) of learning cannot be analysed if one concentrates on particular elements of the situation. intentions in learning have to be related to students’ backgrounds as well as to their present situation and foregrounds. argel, júlia, mariana, natália, and tonino are still on their way, seeking a better future. the complexity of the situation, however, renders their dreams fragile. acknowledgments this paper is part of the research project “learning from diversity,” funded by the danish research council for humanities and aalborg university. we want to thank the students for participating in the inter-view, luiz carlos barreto for transcribing the inter-view, annie aarup jensen for commenting on the completed manuscript, and anne kepple for translating the inter-view into english and for making a careful language revision of the completed manuscript. references alrø, h., & skovsmose, o. 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(2007). “before you divide you have to add”: inter-viewing indian students’ foregrounds. in b. sriraman (ed.), international perspectives on social justice in mathematics education. the montana mathematics enthusiast, monograph 1, 151–167. valero, p. (2004). postmodernism as an attitude of critique to dominant mathematics education research. in m. walshaw (ed.), mathematics education within the postmodern (pp. 35–54). greenwich, ct: information age. vithal, r. (2003). in search of a pedagogy of conflict and dialogue for mathematics education. dordrecht, the netherlands: kluwer. zevenbergen, r. (2001). mathematics, social class, and linguistic capital: an analysis of mathematics classroom interactions. in b. atweh, h. forgasz, & b. nebres (eds.), sociocultural research on mathematics education: an international perspective (pp. 201–215). mahwah, nj: erlbaum. racism, assessment, and journal of urban mathematics education december 2018, vol. 11, no. 1&2, pp. 45–68 ©jume. http://education.gsu.edu/jume julius davis is a doctoral candidate in mathematics education in the school of education and urban studies at morgan state university, 1700 east coldspring lane, jenkins building 421, baltimore, md 21217. his research focuses on understanding how issues of race and racism shape the lived realities, schooling, and mathematics education of african american students. danny bernard martin is chair of curriculum and instruction and an associate professor of mathematics education and mathematics at the university of illinois at chicago, college of education (mc 147), 1040 w. harrison, chicago, il 60607; e-mail: dbmartin@uic.edu. his research has focused on understanding the salience of race and identity in african americans’ struggle for mathematics literacy. dr. martin is author of the book mathematics success and failure among africanamerican youth (lawrence erlbaum associates, 2000) and editor of the forthcoming book mathematics teaching, learning, and liberation in the lives of black children (routledge, 2009). racism, assessment, and instructional practices: implications for mathematics teachers of african american students1 julius davis morgan state university danny bernard martin university of illinois at chicago couched within a larger critique of assessment practices and how they are used to stigmatize african american children, the authors examine teachers’ instructional practices in response to demands of increasing test scores. many mathematics teachers might be unaware of how these test-driven instructional practices can simultaneously reflect well-intentioned motivations and contribute to the oppression of their african american students. the authors further argue that the focus of assessing african american children via comparison to white children reveals underlying institutionally based racist assumptions about the competencies of african american students. strategies are suggested for helping teachers resist test-driven instructional practices while promoting excellence and empowerment for african american students in mathematics. keywords: african american students, assessment, instructional practice, racial hierarchy, racism lthough the phrase “teaching to the test” has been spoken in hallways and teachers’ lounges throughout the nation’s public schools for decades, with the passage of the no child left behind act of 2001 (nclb),2 the phrase has become somewhat of a formalized instructional practice. the first author taught and conducted research at a middle school in the baltimore city public school system that 1 originally published in the inaugural december 2008 issue of the journal of urban mathematics education (jume); see http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/14/8. 2 no child left behind act of 2001, public law 107-110, 20 u.s.c., §390 et seq. a http://education.gsu.edu/jume mailto:dbmartin@uic.edu http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/14/8 davis & martin racism and standardized testing journal of urban mathematics education vol. 11, no. 1&2 46 utilized teaching to the test as the dominant instructional approach with its african american students. our conceptualization of teaching to the test is characterized by classroom practices that emphasize remediation, skills-based instruction over critical and conceptual-oriented thinking, decreased use of rich curriculum materials, narrowed teacher flexibility in instructional design and decision making, and the threat of sanctions for not meeting externally generated performance standards. reflecting low-level expectations for african american children, these teaching-tothe-test approaches often require teachers to make use of remedial mathematics plans and strategies that focus on lower-level mathematical content. while mastery of this lower-level content is necessary, it often becomes the ceiling of the mathematics that students learn because it allows students to meet minimum standards for what counts as success. in baltimore, district and school administrators and teachers supported this approach by developing and implementing a supplemental saturday mathematics program dedicated to preparing african american students for the state administered standardized test. the administrative staff at the school also developed and implemented an additional plan devoted to increasing african american students’ performance on the test. the district and school administrators selected students to participate in these remedial programs based on their having standardized test scores that were at basic and near-proficient levels. students were required to participate in both the in-school and saturday school mathematics program. nearly 25% of the student body at the researched middle school was required to participate in these special programs. during the school day, students were taken out of their elective courses twice a week to participate in the in-school mathematics program. in the regular mathematics courses at the school, administrators instituted an additional remedial mathematics plan that required teachers to spend the first 30 minutes of their 90-minute class period reviewing mathematical concepts taught to students in previous mathematics courses. the remainder of their class time was spent focusing on the state administered test in mathematics. students were taught from textbooks that focused on this test. they were also inundated with worksheets, board work, test-taking strategies, and other materials devoted to the state administered standardized test in mathematics. because our conceptualization of teaching to the test is based largely on the first author’s observation in a single middle school, it is clearly not exhaustive of the instructional practices found throughout baltimore. we believe, however, that these practices are not isolated to the first author’s experiences. the practices that were observed bear a striking resemblance to those documented in the larger literature (see, e.g., kozol, 1992; lipman, 2004; noguera, 2003; oakes, 1990; oakes, joseph, & muir, 2004) on school inequality and propelled us to use this example to begin a conversation among mathematics educators about such practices and approaches. the literature reveals that teachers with large numbers of african ameri davis & martin racism and standardized testing journal of urban mathematics education vol. 11, no. 1&2 47 can students reported more often that test scores were used to evaluate students’ progress, select textbooks, provide students with special services, and make curriculum and instructional decisions (madaus, west, harmon, lomax, & viator, 1992; strickland & ascher, 1992). in these classrooms, teachers indicated that emphasis was placed on test content, teaching test-taking skills, teaching topics known to be on the assessment, and preparing students for the test more than a month before the test (madaus et al., 1992). these practices force teachers to rush instruction and provide students with little to no opportunity to learn more advanced-level mathematical concepts (madaus et al., 1992). teachers of african american students who focused mainly on preparing these students for tests in mathematics spend a significant amount of time on rudimentary levels of mathematics (madaus et al., 1992). in her book high stakes education, lipman (2004) provided evidence of this approach to educating african american and latino/a children in the chicago public school system (cps). she documented widespread remediation and testfocused instruction in the schools where she conducted her research. lipman stated: cps leaders contend that the harshness of accountability is offset by new remedial “supports”…including after-school remedial classes, mandatory summer “bridge” classes for failing students, and transition high schools. however, these remedial programs are explicitly aimed at the [statewide assessment test]. … the impoverishment and redundancy of this basic skills education for students the district has defined as “behind” can hardly be construed as an antidote for the inequities of the system, particularly as african american and latino/as are disproportionately assigned to this type of schooling. mandating a rudimentary curriculum that few middle-class parents would choose for their own children publicly signals that low-income children and children of color are deficient. (p. 47) lipman’s (2004) analysis is especially powerful because she also highlighted the voices of the teachers who carried out these practices on behalf of the district. the following comments come from interviews that lipman conducted: grover teacher: i’ve been at this school for five years, and the emphasis on standardized tests weighs more heavily than it ever has in my career. (p. 77) westview teacher: we are test driven… everything is test driven. (p. 77) eighth-grade teacher: with all the teaching strategies—teaching them how to take tests. i have tested them to death to tell you the truth. (p. 78) we concur with lipman (2004) when she stated the following about this narrow approach to educating african american children: the emphasis on analyzing and preparing for standardized tests; the immense pressure on administrators, teachers, and students to raise scores; the substitution of testpreparation materials for the existing curriculum; practice in test-taking skills as a legit davis & martin racism and standardized testing journal of urban mathematics education vol. 11, no. 1&2 48 imate classroom activity—these constitute a meaning system that reinforces the definition of education as the production of “objective,” measurable, and discrete outcomes. (p. 80) without an awareness that what was observed in a single middle school in baltimore is also taking place in other locations around the country, one might easily conclude that the approach to teaching mathematics to the students in baltimore was appropriate. there would be no linking of the practices in one context to similar practices in another. hence, the institutionalized nature of african american students’ mis-education would be lost. one might argue that these students needed support to help raise their level of achievement to that of white students and that the district and school officials were simply providing them with that support. many administrators and teachers, however, might be unaware of how such practices can, on one hand, reflect well-intentioned goals but simultaneously contribute to the oppression of their african american students. reflecting on the experiences of the first author, both of us agree that although this approach resulted in increased test scores for sixthand eighth-grade students, these increases do not mitigate the oppression. we unequivocally oppose such a narrow instructional approach and conceptualization of mathematics education for african american children. our opposition is based on our scholarly analysis, our respective teaching experiences in diverse african american contexts, and our willingness to advocate, as african american scholars, on behalf of african american children. yet, as lipman (2004) argued, it is insufficient to analyze such practices in isolation of the larger ideologies and political movements that undergird them. utilizing a race-critical perspective (martin, 2009), we argue that such test-driven instructional practices, particularly within hyper-segregated african american schools, like those in baltimore and elsewhere, must be situated within a larger system of assessment that has “scientifically” (a) supported the social construction of african american children as intellectually inferior, and (b) facilitated the development of ranking systems that reify these negative social constructions. although a full deconstruction of this assessment system is beyond the scope of this article, we offer a partial deconstruction that links scientific racism, race-based ranking systems, and instructional practice in classrooms predominated by african american children. in concert with this deconstruction, we suggest strategies for helping teachers resist narrow, test-focused, instructional approaches while promoting excellence and empowerment for african american children in mathematics. assessment and scientific racism historically, testing and assessment has been linked to larger eugenics and white supremacy efforts that have tried to prove, through science, that african americans and other non-whites are intellectually and culturally inferior (gould, davis & martin racism and standardized testing journal of urban mathematics education vol. 11, no. 1&2 49 1981; herrnstein & murray, 1994; jensen, 1969; ladson-billings, 1999). intelligence testing has evolved alongside various racist beliefs about african americans. according to ladson-billings (1999), “throughout u.s. history, the subordination of blacks has been built on ‘scientific’ theories (e.g., intelligence testing), each of which depends on racial stereotypes about blacks that makes the conditions appear appropriate” (p. 23). from the seventieth century to the first half of the twentieth century, the scientific community participated in validating the so-called inferiority of african americans when compared to whites and the formation of a racial hierarchy in both intelligence and culture (gould, 1981; herrnstein & murray, 1994; jensen, 1969; montagu, 1997; norman, 2000). montagu (1997) argued that few members of the scientific community spoke against the notion of a hierarchy of races. instead, the shared beliefs, values, and techniques exhibited by the scientific community formed the basis of scientific racism (norman, 2000). scientific racism can be defined as the use of scientific methods to support and validate racist beliefs about african americans and other groups based on the existence and significance of racial categories that form a hierarchy of races that support political and ideological positions of white supremacy (gould, 1981; herrnstein & murray, 1994; jensen, 1969; montagu, 1997; norman, 2000). gould (1981), montagu (1997), and norman (2000) asserted that the science establishment invested a considerable amount of resources into advancing scientific racism. according to montagu, “virtually every scientist writing during the nineteenth century was…caught in an inexorable web of racist beliefs” (p. 32). similarly, norman argued, “despite an impressive array of eminent scientific advocates, scientific racism had, from its inception and even up to its modern-day manifestations, been nothing more than the uncritical couching of popular racist beliefs in the idiom of science” (p. 3). there are three faulty assertions guiding scientific examinations of race and intelligence that conceal and couch racist beliefs about african americans and other groups (gardner, 1995; gould, 1981). first, there is widespread belief that intelligence can be described by a single number. gould (1981) contended that converting abstract concepts such as intelligence into numerical entities is a fallacy. gardner (1995) argued that the belief in a single, standardized, and inherent human intelligence or g (general intelligence) ignores the concept of multiple intelligences. second, these faulty assertions fail to take into consideration jones’s (1995) arguments about the long-forgotten justifications of slavery and segregation that rest on beliefs about african american intellectual inferiority and the alleged intellectual superiority of whites; in that, there exists a faulty belief that intelligence can be used to rank social groups in some linear order. gould argued that such ranking requires a criterion that takes the form of an “objective number” to assign all individuals to their proper status. the assumption is that “if ranks are displayed in hard numbers obtained by rigorous and standardized procedures, then they must reflect reality, davis & martin racism and standardized testing journal of urban mathematics education vol. 11, no. 1&2 50 even if they confirm what we wanted to believe from the start” (gould, 1981, p. 26). nonetheless, we concur with gould, who also argued: “science must be understood as a social phenomenon, a gutsy, human enterprise, not the work of robots programmed to collect pure information…. science, since people must do it, is a socially embedded activity” (p. 21). third, the most conservative research on intelligence suggests that it is genetically based and immutable. this suggestion “invariably [leads] to [the conclusion] that oppressed and disadvantaged groups—races, classes, or sexes—are innately inferior and deserve their status” (gould, 1981, p. 25). several scholars abandoned the word intelligence to avoid debates and endless arguments associated with intelligence testing, biological determinism, scientific rationalism, and scientific racism (gould, 1981; herrnstein & murray, 1994; jensen, 1969). herrnstein and murray (1994) suggested that scholars use more neutral terms such as cognitive ability to subside criticism. essentially, the discourse about intelligence testing was minimized throughout the 1970s. we argue that contemporary, race-comparative analyses began to flourish on the heels of this changing discourse, however. since that time, a number of comparative analyses of mathematics achievement have been conducted, typically supporting and serving as evidence for so-called racial achievement gaps (see, e.g., lubienski, 2002; strutchens & silver, 2000). these analyses have consistently normalized white student performance and portrayed african american children as lacking in mathematics skills and ability. politics and purposes of standardized tests apparently neutral assessments are not objective at all, but rather ‘objects of history’— created to fulfill particular social functions, which have shaped the assessments in particular directions that are not readily apparent. the seemingly innocuous requirement for the results of a test to be reliable requires that the test disperses individuals along a continuum so having the effect of placing a magnifying glass over a very small part of human performance, and this is particularly marked in mathematics. (williams, bartholomew, & reay, 2004, p. 58) the history, politics, and purposes of standardized testing, particularly in mathematics, are rooted in the research and discourse revolving around race and intelligence that we outlined above. we contend that deconstructing this research and discourse on intelligence testing is important in understanding the racist underpinnings of contemporary standardized testing not only in mathematics, but also in every discipline. as the quote by williams, bartholomew, and reay points out, one of the purposes of assessment is to create hierarchies. we claim that the use of a single form of assessment or a single statistic to describe mathematical ability is limited in explanatory power. nonetheless, this happens for two main reasons. the davis & martin racism and standardized testing journal of urban mathematics education vol. 11, no. 1&2 51 first—to convey certainty and absolute truth—stems from the fact that numbers and statistics represent a special form of being “objective” (gould, 1981, 1995). they carry the weight of proof. the analysis of all types of data, including statistics, however, involves interpretation that cannot be divorced from social and political contexts. how data is chosen and used depends on who is doing the choosing and their purpose for conducting the analysis. it is not uncommon for statistical reports to be presented in the absence of important qualitative and contextual considerations. even when inclusion does occur, misunderstanding of these contextual forces occurs in deference to supporting the validity of the statistics. for example, race, which usually appears in achievement studies as an undertheorized independent variable, is said to cause measured achievement differences among socially constructed racial groups. yet, this faulty use of the concept of race usually reflects an inadequate understanding of racism and racialization and their impact on educational outcomes. socioeconomic status, typically described by income, is also said to determine achievement outcomes but is made causal without a nuanced understanding of wealth differentials (e.g., property ownership, investments, inheritance) within the same socioeconomic (income) strata and how forces like racism and discrimination, in turn, account for those wealth differences (conley, 1999). similarly, neighborhood effects are used to construct theories about opposition, disengagement, and resistance to schooling that leads to academic failure. yet, these analyses fail to account for student success in these very same neighborhoods. the second reason is that statistics allow students to be ranked and sorted along what are thought to be racial lines (gould, 1981, 1995; tate, 1993). because of how test scores are used in race-comparative analyses (lubienski, 2002; strutchens & silver, 2000; u.s. department of education, 1997), african american students are frequently constructed and represented as being inferior to white and asian students in mathematics (martin, 2007, 2009). in mathematics education, these rankings and sortings have been used to produce what martin (2009) has termed the racial hierarchy of mathematical ability. this racial hierarchy results in white students being positioned at the top and african american students at the bottom. this “ranking” proves to be particularly interesting because asian students, collectively, perform better than white students. yet, it is white students who are used as the barometer for african american students’ performance. for example, a commonly cited research finding has suggested that african american 12th graders perform at the same level as white 8th graders (lubienski, 2002; national research council, 1989; thernstrom & thernstrom, 1999; u.s. department of education, 1997). such findings provide pseudoscientific support for racist assumptions (tate, 1993) that suggest african american students are intellectually inferior to white students and located at the lowest levels of a racial hierarchy. davis & martin racism and standardized testing journal of urban mathematics education vol. 11, no. 1&2 52 a belief in racial hierarchies undergirds all forms of intelligence testing, including school-based achievement testing (gardner, 1995; gould, 1981; herrnstein & murray, 1994), and is aligned with the same racist assumptions that have allowed african americans to be exploited within the laws and practices of the united states more generally (ture & hamilton, 1992; wilson, 1998). we suggest that teachers who engage in teaching to the test and other shortsighted remediation must necessarily accept the existence of this hierarchy as evidenced by their subsequent efforts to relocate african american students within it. assessment in mathematics education: foregrounding race and racism invoking a race-critical perspective, we claim along with others (see, e.g., hilliard, 2003; ladson-billings, 1999) that, beyond any knowledge that might be gained about student thinking and development, the larger political effect of standardized testing, particularly in the area of mathematics education, is to maintain white supremacy in one form or another (e.g., u.s. international standing and competitiveness, normalization of white student behavior).3 ladson-billings (1999), for example, has argued that the school curriculum suppresses multiple voices and perspectives while simultaneously legitimizing the dominant, white, male, upper-class ways of knowing and being as the “standard” that all students should be required to emulate (see also swartz, 1992). this dominance is evidenced by the fact that schools serving african american students typically adopt curriculum from predominantly white school districts (davis, 2008; martin, 2007). such choices are often not done in response to the authentic needs of african american learners but suggest that what african american children need is determined by what is best for white children. we also claim that commonly used race-comparative analyses are one small piece of the larger structural and institutional mechanisms that support this goal. it is this larger structural effect, above and beyond the efforts and intentions of individual white scholars and policy-makers, that continues to drive instructional practices for teachers of all children but especially african american children when they are viewed as less than ideal learners and mathematically illiterate (martin, 2007, 2009). in tate’s (1993) critical race analysis of standardized testing practices in poor school districts serving large numbers of african american students, he used the voluntary national mathematics assessment as a platform to discuss the racist un 3 by this statement, we mean that the goals for testing are often framed in terms of improving the “standing” of the united states relative to other countries in international comparisons. many highachieving asian countries are often discussed as threats to the standing of the united states. the goals for african american children are often framed as increasing test scores for the purpose of having outcomes match those of white children. davis & martin racism and standardized testing journal of urban mathematics education vol. 11, no. 1&2 53 derpinnings of standardized testing. tate argued that standardized tests are “scientifically” constructed to socially reproduce the most negative aspects of african american students’ lived realities. he also argued that standardized tests were designed to prepare poor african american students to replicate their parents in the division of labor by providing them with instruction in mathematics suitable for this purpose. tate further claimed that policies governing standardized test were designed to ensure that poor african american students did not receive the same instruction in mathematics as middleand upper-class members of society. he believed that test scores are not intended to provide feedback for the purposes of educational improvement in mathematics, but to rank students and to determine their economic potential. in other words, standardized tests shape the lives of poor african american students in more significant ways than middle-class or affluent students. we agree with tate’s (1993) analysis. the current environment of high-stakes testing engendered by nclb has caused many states and local school districts to shift their instructional approaches in ways where satisfactory outcomes on state assessments—not authentic learning and development—become the primary goal. these pressures have also positioned administrators and teachers to appropriate much of the underlying ideology that characterizes african american children as mathematically illiterate, using white and asian student performance as the standard. the current environment of high-stakes testing is not only just a contemporary phenomenon but also one that has historical ties to intelligence testing and the construction of racial hierarchies. nclb has repositioned state and local policies and instruction and standardized testing efforts in public schools, specifically, in mathematics, to carry out the construction of these hierarchies. there are two aspects of nclb that shape our discussion of standardized testing in mathematics education. first, one of the main goals of nclb is to close the so-called racial achievement gap in reading and mathematics. martin (2009) argued that plans to move african americans and other marginalized groups from their perceived positions of being mathematically illiterate to being mathematically literate, an intellectual space supposedly occupied by white and asian students, is rooted in racist beliefs about these students. the underlying assumption is that african american students’ performance in mathematics must conform to that of white students in order for these students to be considered mathematically literate (martin, 2009). in our view, the performance of white students as the benchmark for african american students sets an artificially low standard for african american learners given that the collective averages of white students on many large-scale mathematics assessments are less than the highest levels of proficiency (secada, 1992; strutchens & silver, 2000; tate, 1997) and ignores the needs of african american children as african american children. connecting the discourse on african amer davis & martin racism and standardized testing journal of urban mathematics education vol. 11, no. 1&2 54 ican students in mathematics education to intelligence testing, the assumption is that “black inferiority is purely cultural and that it can be completely eradicated by [mathematics] education to a caucasian standard” (gould, 1981, p. 32). the accountability measures dictated by nclb require states to publicly identify low-performing schools. this practice has played a major role in subjecting african american students, their schools, and school systems to inferior labels as a result of failing to meeting standardized testing goals (davis, 2008; lattimore, 2001, 2003, 2005a; sheppard, 2006). this practice was clearly evident in the baltimore city school and district discussed in the introduction of this article. currently, this african american school district is in its second year of system improvement and the students are considered the lowest performers in mathematics in the state. in addition to these labels, the failure to meet standardized test goals places their schools in danger of losing federal dollars to finance their education. second, nclb has explicitly attempted to standardize what constitutes highly qualified teachers for all students.4 the policy mandates the use of standardized tests to quantify what constitutes a highly qualified mathematics teacher. this policy treats the instruction african american students receive in mathematics as a generic set of teaching competences that should work for all students (ladsonbillings, 1999; martin, 2007). when these approaches to teaching fail to produce the desired results, african american students are deemed deficient—not the approaches used to teach these students (ladson-billings, 1999; martin, 2007). later in this article, we will briefly revisit arguments made by martin (2007) who problematized notions of highly qualified mathematics teachers by asking: “who should teach mathematics to african american students?” we use the characteristics described by martin as a catalyst to provide mathematics teachers with strategies to resist contributing to the oppression of african american students. african american students’ experiences in mathematics education martin (2007) discussed how achievement has served as the dominant discursive frame used to talk about the competencies of african americans in mathematics within the context of mainstream mathematics education research and policy. he demonstrated how a framework of color-blind racism, in turn, supports this achievement-focused discourse. martin challenged mathematics education researchers to construct an alternative discursive and assessment frame focused on how african american learners experience mathematics education, and suggested that future research should focus on mathematics learning and participation as ra 4 in this article, we do not give extensive attention to the forms of assessment used for teacher certification. we do, however, claim that the same logic applies to these tests. the nclb policy document defines highly qualified as a teacher who holds at least a bachelor’s degree and has passed statecertification or licensing exams. davis & martin racism and standardized testing journal of urban mathematics education vol. 11, no. 1&2 55 cialized forms of experience, not only for african american learners but also for all learners. analysis of the relevant literature reveals two important insights about african american students’ experiences with standardized testing (berry, 2005; corey & bower, 2005; lattimore, 2001, 2003, 2005a; lubienski, 2002; moody, 2003, 2004; strutchens & silver, 2000; u.s. department of education, 1997). first, school districts serving large numbers of african american students often implement remedial strategies to comply with state and federal regulations surrounding standardized testing in mathematics (davis, 2008; lattimore, 2001, 2003, 2005a; tate, 1993). in support of this strategy, african american students are inundated with practice materials that include worksheets and in-class practice tests devoted to state assessments (lattimore, 2001, 2003, 2005a). the mathematics instruction that these students are exposed to emphasizes repetition, drill, right-answer thinking that often focuses on memorization and rote learning, out-of-context mathematical computations, and test-taking strategies (davis, 2008; ladson-billings, 1997; lattimore, 2001, 2003, 2005a). this type of instruction often leaves african american students disengaged and viewing mathematics as irrelevant and decontextualized from their everyday experiences (corey & bower, 2005; davis, 2008; ladsonbillings, 1997; lattimore, 2005b; tate, 1995). second, standardized tests serve as a “gatekeeper” in providing african american students access to higher-level mathematics, gifted and honors programs, and future aspirations (berry, 2005; davis, 2008; lattimore, 2001, 2003, 2005a; moody, 2003, 2004; oakes, 1990; sheppard, 2006). throughout their schooling experiences, african american students are often denied access to higher-level mathematics and advanced programs because of their performance on standardized test (berry, 2005; davis, 2008; moody, 2003, 2004; oakes, 1990), thereby leaving the majority of african american students in lower-level mathematics courses (corey & bower, 2005; davis, 2008; lubienski, 2001, 2002; moody, 2003, 2004; oakes, 1990; oakes, joseph, & muir, 2004). in high school, state administered standardized tests have also been found to serve as the gatekeeper to african american students receiving a high school diploma (lattimore, 2001, 2003, 2005a). for example, since 2006, students in california must pass an exit exam to graduate. students in maryland will have to pass the state administered standardized test to receive their high school diploma beginning with the graduating class of 2009. re-conceptualizing the assessment of african american students in mathematics: implications for teachers there is very little consideration given to the argument that african american students represent a distinct cultural group (akbar, 1980; ladson-billings, 1994), requiring an education in mathematics that reflects their lived realities and collec davis & martin racism and standardized testing journal of urban mathematics education vol. 11, no. 1&2 56 tive conditions (martin, 2007; thompson, 2008). according to ladson-billings (1999): african american students are a part of almost every social strata and their social context may affect what experiences they have and how they view the world, their cultural knowledge, expressions, and understandings, which may be transmitted over many generations, may share many features with african americans across socioeconomic and geographical boundaries. (p. 699) we argue african american students must receive an education in mathematics that not only prepares them to function effectively in mainstream society but also builds on their cultural knowledge base and value systems (ladson-billings, 1997). this argument implies framing the purpose, structure, and ideology of mathematics education for african american learners in ways that are responsive to their needs as african american learners (martin & mcgee, in press). in reconceptualizing and reframing mathematics education for african american learners, a growing number of african american scholars have begun to advance liberatory mathematics education agendas for african american students (martin & mcgee, in press; moses & cobb, 2001; thompson, 2008). the most notable example of this agenda is the algebra project (moses & cobb, 2001) and its parallel youth development program, the young people’s project (ypp). through the algebra project, civil rights activist and mathematics educator robert moses has led the charge to provide african american students with a liberatory mathematics experiences via a curriculum anchored in culturally relevant activities. moses argued the fight for mathematics literacy is a fight for twenty-first century citizenship and that african american youth must be empowered to fight for their liberation on their own terms. this empowerment was clearly evident in baltimore where african american youth from the algebra project in that city challenged school officials for not providing them with an adequate education (prince, 2006). martin and mcgee (in press) argued, “any framing of the form, philosophy, and content of mathematics education for african americans must address the historical and contemporary social realities that they face.” they suggested that history “compels us to frame african americans mathematics education and mathematics literacy in the same way that education, in general, was framed around their life conditions in the past, for the purposes of liberation.” in defining liberation, they drew on the work of watts, williams, and jagers (2003), who defined liberation as follows: liberation in its fullest sense requires the securing of full human rights and the remaking of a society without roles of oppressor and oppressed. … it involves challenging gross social inequities between social groups and creating new relationships that dispel oppressive social myths, values, and practices. the outcome of this process contributes to the creation of a changed society with ways of being that support the economic, cul davis & martin racism and standardized testing journal of urban mathematics education vol. 11, no. 1&2 57 tural, political, psychological, social, and spiritual needs of individuals and groups. (pp. 187–188) thompson (2008) similarly argued for a liberatory framing of mathematics education for african american learners through what she calls nation building. she defined nation building “as the conscious and focused application of knowledge, skills, and abilities to the task of liberation” (p. 17). thompson further argued that nation building “involves the development of behaviors, values, institutions, and physical structures that elucidate african history and culture” for the purposes of ensuring “the future identity, existence, and independence of the nation” (p. 17). she believed that efforts to increase african americans in mathematics and science should be geared toward the liberation of african people throughout the diaspora and to eradicate systems of racism (white supremacy). these liberatory agendas typically stand little chance of being accepted in mainstream discussions of mathematics education because “african americans seeking equal opportunity in education [specifically in mathematics] will only be granted when the opportunity being sought converges with the economic selfinterest of whites” (tate, 1993, p. 17). bell (1980) has referred to this contingency as interest convergence. critical reflection and advocacy by mathematics teachers of african american students we realize that teachers cannot expect to engage in liberatory instructional practices with african american students and be rewarded by the same system that demands that they contribute to the negative social construction of these students (ture & hamilton, 1992). yet, we would appeal to what is morally correct, given the needs and social realities (ladson-billings, 1997) of these students and frame the discussion that follows as both a challenge and an invitation for teachers. we challenge teachers to engage in critical reflection on their own practices and we invite them to consider the suggestions we make about changes in these practices, where necessary. martin (2007) suggested that teachers should (a) develop a deep understanding of the social realities experienced by african american students, (b) take seriously one’s role in helping to shape the racial, academic, and mathematics identities of african american learners, (c) conceptualize mathematics not just as a school subject but as a means to empower african american students, and (d) become agents of change who challenge research and policy perspectives that construct african american children as less than ideal learners and in need of being saved or rescued from their blackness. we encourage teachers of african american students to reframe their instructional practices by taking the ideas developed by martin seriously. in our view, “teachers who are unable, or unwilling, to develop in these davis & martin racism and standardized testing journal of urban mathematics education vol. 11, no. 1&2 58 ways are not qualified to teach african american students no matter how much mathematics they know” (martin, 2007, p. 25). while we do not offer prescriptive or formulaic approaches for how teachers might utilize martin’s (2007) suggestions, we do point to some important initial steps and underscore that many of these steps should occur simultaneously and throughout teachers’ work with african american students. our conceptualization of the instructional strategies and strategies of resistance offered in this section are in many respects inextricably linked and presented in concert with one another. the strategies are intended to help teachers resist teaching to the test and resist contributing to the negative social construction of african american students. to help resist these efforts, we strongly believe that teachers must continuously engage in critical reflection about their practice, their beliefs about african american students, and their commitment to these students. individually and collectively, teachers must engage in critical reflection on how they conceptualize mathematics education for african american students. in so doing, we believe that issues of race and racism must be at the forefront of discussions of mathematics education for african american students. martin argued that there are several documented cases where failing to do so can stall the progress and design of meaningful mathematics education for these students. teachers have to realize that we are all socialized by institutions (e.g. media, policies, laws, etc.) that support racist views and beliefs about african american children. policies and ideologies associated with high-stakes testing, for example, often position teachers in ways where critical reflection on their practices is deemphasized or derailed by progressive rhetoric. martin (in press) discussed how he has used the following three-question quiz in professional development and research contexts with teachers from various ethnic and racial backgrounds, years of experience, and geographic locations:  how many of you have heard of, and understand, what is meant by the racial achievement gap?  how many of you have, or plan to, devote some aspect of your practice to closing the racial achievement gap?  how many of you believe in the brilliance of african american children? after noting that the vast majority of teachers answer affirmatively to all three questions, martin (in press) goes on to point out how the second and third questions are conceptually and practically incompatible. he pointed out that acceptance of the racial achievement gap rhetoric necessarily requires that teachers, even african american teachers, accept the inferiority of african american children, especially when closing the so-called racial achievement gap is translated as raising african american children to the level of white children. the quiz was a strategy to help davis & martin racism and standardized testing journal of urban mathematics education vol. 11, no. 1&2 59 teachers’ resist and rethink negative social constructions of african american students. in addition to taking the quiz, teachers must ask themselves difficult and uncomfortable questions about african american students and their conditions that include, but are not limited to: do i believe african american students are intellectually inferior? do i believe that issues of race and racism play a role in shaping the lives, schooling, and mathematics education of african american students? do i harbor racist beliefs about african american students? do i believe that history has any bearing on african american students’ contemporary lived realities, schooling, and mathematics education? questions of this nature are reflective strategies of resistance that must be considered, thought about, and answered truthfully. we pose these questions for all teachers but we particularly direct them to those white teachers whom the literature has identified as being particularly resistant to change in their negative or deficit beliefs about african american children (sleeter, 1993). we believe that teachers must decide whether they are willing to be agents of social change for african american students. teachers should ask themselves the following questions: what am i willing to sacrifice for african american students? am i willing to sacrifice or take the risk to provide african american students with a liberatory mathematics education in the face of policies that require me to do otherwise? am i willing to challenge policies that treat african american students as less than ideal learners? once teachers have explored these considerations, we believe that teachers need to spend time seriously thinking about how they envision mathematics education for african american students. in the process of conceptualizing what mathematics education for african american students should look like, teachers should ask themselves the following question: what do i want african american students to be able to do as a result of their mathematics education? we suggest that african american students should be able to use mathematics as a tool to (a) reexamine history and use this history to generate critiques and better understandings of their immediate life conditions and collective group conditions in the world, and (b) gain access to areas in the larger opportunity structure where mathematics knowledge has often been used to keep african americans out. overall, the mathematics education african american students receive should be designed to improve their life and group conditions (martin, 2009; thompson, 2008). in terms of the goals for mathematics learning, teachers might consider adopting the stance that effective teaching should not only produce growth in students’ mathematical skills but also connect to these students’ lives, experiences, and lead them to employ their mathematical knowledge in multiple settings and develop their racial, academic, and mathematics identities. for teachers, this entails thinking of empowerment along three lines: mathematical, social, and epistemological (ern davis & martin racism and standardized testing journal of urban mathematics education vol. 11, no. 1&2 60 est, 2002; martin & mcgee, in press).5 this stance does not mean that students cannot be shown how to carry out procedures and learn to produce correct answers. however, if they do not see themselves as legitimate doers of mathematics, then the acquisition of skills with little personal identification on the part of students is not likely to sustain itself. in other words, we argue that teachers of african american students should consciously attempt to integrate these students’ experiences, home and community lives into their conceptualization of mathematics and teach them how to use mathematics as a means to view and critique the world (lynn, 2001; tate, 1995). several scholars’ work has provided insight into how mathematics teachers have conceptualized mathematics along mathematical, social, and epistemological lines (ladson-billings, 1997; lynn, 2001; tate, 1995). for example, ladsonbillings (1997) described a sixth-grade mathematics teacher of african american students who went beyond the district curriculum by providing her students with an engaging, rigorous, and challenging education in algebra. this teacher’s students were engaged in problem solving around algebra, pushed to think at higher levels, and encouraged and reassured by their teacher that they were capable doers of mathematics. in this class, a student with special needs benefited from this teacher’s belief system and instruction in mathematics. at the end of the school year, this teacher convinced the school principal to remove the student from receiving special education services because of his mathematical performance in her class. lynn (2001) captured the experience of a middle school mathematics teacher who reflected seriously on issues of poverty and racism. this teacher engaged students in a discourse about how the history of lynching and jim crow racism has shaped african americans’ lives. he used this history to teach his students the importance of checking their work and knowing their math facts. this teacher connected the two by making the case that historically african americans have had to prove that injustices actually occurred to them by supporting their experiential claims with facts. in this lesson, the teacher situated this discourse in a historical analysis of african american experiences with racism in society. the teacher pro 5 according to ernest (2002), mathematical empowerment concerns the gaining of power over the language, skills, and practices of using and applying mathematics; that is, the gaining of power over a relatively narrow domain, for example, that of school mathematics. social empowerment through mathematics concerns the ability to use mathematics to better one’s life chances in study and work and to participate more fully in society through critical mathematical citizenship. thus, it involves the gaining of power over a broader social domain, including the worlds of work, life and social affairs. epistemological empowerment concerns the individual’s growth of confidence not only in using mathematics but also a personal sense of power over the creation and validation of knowledge. this empowerment is a personal form: the development of personal identity so as to become a more personally empowered person with growth of confidence and potentially enhanced empowerment in both the mathematical and social senses (and for the mathematics teacher—enhanced professional empowerment). davis & martin racism and standardized testing journal of urban mathematics education vol. 11, no. 1&2 61 vided his students with concrete examples of how to use numerical data presented in the media to critically examine the ways that numbers get utilized in an unjust society. for example, the teacher made the case that a media report describing a decrease in joblessness does not always translate into increasing jobs for the masses of african americans. essentially, this teacher was committed to raising african american students’ social consciousness about the uses and abuses of mathematics in society much like we attempt to do in the article. tate (1995) described a mathematics teacher who engaged her african american students in real problems of social and economic importance for african americans and their community. this teacher asked students to pose problems they felt were important to them and affected their community, to conduct research on one of the posed problems, and to develop strategies to solve that problem. the students were then encouraged to execute the strategy they developed. the students posed a wide range of problems that included the aids epidemic, drugs, ethics in medicine, and sickle cell anemia. in one class, students posed problems about the excessive number of liquor stores in their community and “embarked on an effort to close and/or relocate 13 liquor stores within 1000 feet of their school” (p. 170). the students’ action resulted in “over 200 citations to liquor store owners and two of the 13 stores closed down for major violations” (p. 170). we recommend that teachers spend time developing relationships with their students that extend beyond the mathematics content being taught in their classroom. teachers should not rely solely on secondary sources (e.g., principals, other teachers, cumulative records, etc.) to define their outlook, views, and beliefs about african american students. teachers can spend time learning about african american students’ home life, social realities, childhood experiences, and likes and dislikes. in this way, teachers can show that they are committed to african american children and their families in ways that extend beyond just raising test scores. in his study of african american middle school students, davis (2008) described an african american female mathematics teacher, mrs. rene taylor, who got to know her students by spending time with them in and out of school (e.g., lunch, after school, hallways, taking students to the movies, inviting students to her home, etc.). mrs. taylor spent time listening and talking to students as they spoke about their problems, interests, likes and dislikes. she engaged her students in a discussion about herself that respected the boundaries of her position. mrs. taylor’s relationship with her students inevitably resulted in developing a relationship with their parents. it should be noted that mrs. taylor also allowed two students to move into her home, primarily because one student was homeless and the other student had problems with a drug-addictive parent. we are not suggesting that teachers should do everything that mrs. taylor did, but clearly her actions demonstrate how teachers can develop relationships with students that are genuine and meaningful. davis & martin racism and standardized testing journal of urban mathematics education vol. 11, no. 1&2 62 her actions also illustrated the level of commitment she has to her students and what she was willing to do for them. while getting to know their african american students, teachers should not come to hasty conclusions or generate stereotypical assumptions about their abilities and values. however, teachers should not lose site of the fact that the historical legacy of racism continues to shape african american students’ contemporary lived realities in their community, home life, schooling, and mathematics education despite the absence of overt racist laws and social customs (davis, 2008). research has shown that teachers who know or get to know their african american students provide them with a more enriching educational, mathematical, and social experience (ladson-billings, 1994, 1997; lynn, 2001; moody, 2003, 2004; tate, 1995). in the first author’s research of african american middle and high school students, participants cited the impact that mrs. taylor had on their educational, mathematical, and social experiences. in the midst of conceptualizing mathematics education for african american students and getting to know these students, teachers should spend time learning about and helping to positively shape african american students’ racial, academic, and mathematics identities (see, e.g., martin, 2000; nasir, 2007; nasir, jones, & mclaughlin, 2007). martin (2000) argued that african american students’ racial, academic, and mathematics identities are linked and contribute to these students’ larger sense of self. he characterizes mathematics identity as being shaped by students’ beliefs about (a) their ability to perform in mathematical contexts, (b) the instrumental importance of mathematical knowledge, (c) the constraints and opportunities in mathematical contexts, and (d) the resulting motivations and strategies used to obtain mathematics knowledge. nasir, jones, and mclaughlin (2007) argued that african american students’ racial and ethnic identities vary across individuals and that “the kind of racial identities students hold has implications for their sense of themselves as students, and for their achievement” (p. 3). hence, these scholars’ work indicated that teachers play a significant role in shaping these identities. teachers might question their students about whether they believe that “being african american” and “being a doer of mathematics” are compatible (martin, 2006). ellington’s (2006) study of high achieving african american students found that these students’ racial, academic, and mathematics identities were shaped by how these students saw themselves in the larger african american community. if african american students do not perceive “being african american” and “being a doer of mathematics” as being compatible, then rich and meaningful discussions that affirm students’ racial, academic, and mathematics identities should become an ongoing part of teachers’ practice. this practice would include explicitly addressing (through discussions or journals) and shattering stereotypes about who can and cannot do mathematics and reducing the “stereotype threat” (steele & aronson, 1995) davis & martin racism and standardized testing journal of urban mathematics education vol. 11, no. 1&2 63 that accompanies practices like standardized testing. stereotype threat occurs when a negative stereotype (e.g., african american students are lacking in mathematics ability) becomes salient as a criterion for test evaluation. in that, students become concerned about confirming the stereotype and through various psychological mechanisms; the concern can cause one to perform more poorly than they would perform in a neutral context. we are not suggesting that teachers are not engaging in practices that contribute to the development of their african american students’ racial, academic, and mathematics identities. however, for those teachers who might not have considered this aspect of mathematical development, we strongly believe that these identities are important constructs for teachers to understand and intentionally incorporate into their instructional practices with african american students. for example, as a mathematics teacher, in a high-stakes testing environment at the high school level, the first author engaged in practices that positively shaped his african american students’ racial, academic, and mathematics identities without formal knowledge of these identities. davis (2005) developed a project intended to expose his african american students to the mathematical, technological, and scientific contributions of people of african descent.6 the project required students to do research, write a report, do an oral presentation, create a display, and participate in a school-wide exhibit to expose and engage their school community in a discussion about the person they researched and their contribution to these fields. the students’ oral presentation, research reports, letters, responses to the project, exhibit, and trip ultimately allowed the first author to understand how the project assisted in shaping his students’ racial, academic, and mathematics identities. with respect to african american students’ being characterized as low performers, behavior problems, and disengaged in mathematics setting, teachers should seriously consider alternative reasons for these students’ actions other than the ones commonly cited (i.e., mathematically incapable, uninterested in being doers of mathematics, etc.) by researchers, teachers, and administrators (akbar, 1980; berry, 2005; davis, 2008). akbar (1980) argued that boredom and cultural disconnect of schools are the primary reasons for african american students’ behavior, disengagement, and performance issues in these settings. in mathematics education, berry (2005) and davis (2008) found african american students across achievement levels were bored and disengaged from mathematics and other academic disciplines. berry and davis, similar to corey and bower’s (2005) research, made the case that these students’ mathematics education was disconnected from their cul 6 we use african descent in this context to connote that the racial and ethnic background of the people did not just include african americans or africans but included a wide range of black people from around the world. this inclusion was done to help expand african american students’ conceptualization of what it means to be black or african american into a larger cultural discourse that connects these students to a larger cultural history and heritage. davis & martin racism and standardized testing journal of urban mathematics education vol. 11, no. 1&2 64 ture. in davis’s (2008) research, he found the african american middle school students who he studied in high-stakes testing environments disengaged from mathematics because of the actions of some of their past and present mathematics teachers. these students reported that their mathematics teachers often (a) would not “teach” them; (b) were not able to help them learn mathematical topics and concepts in which they were experiencing difficulty; (c) disrespected, embarrassed, or humiliated them with respect to learning mathematics; (d) did not provide them with challenging and intellectually stimulating mathematics; (e) presented them with mathematical concepts and topics that they had already learned; (f) provided instruction that was centered around worksheets, rote memorization, board work, and test-taking materials and strategies; and (g) maintained classrooms that were in constant disarray. while these students were disengaged from the learning process, they participated in activities (e.g., walking around the classroom and school hallways, horse playing, etc.) that often led them to be further marginalized by district and school rules and policies (e.g., federal, state, local testing policies) and teacher subjectivity when their behaviors were reflective of their resistance to the realities of their schooling and mathematics experience. we strongly urge teachers to continuously engage in critical reflection about african american students, their instructional practice, and conceptualization of mathematics by asking questions such as: do i provide african american students with lower-level coursework because i believe they are incapable of doing higherlevel coursework? do i perceive african american students as being lazy in mathematics because of their racial identities or because they do not engage in mathematics the way white students do? do i believe these students are undeserving of a mathematics education requiring higher-level thinking and coursework? we strongly encourage teachers to consider and make use of our questions, suggestions, and examples, where applicable. conclusion we started this article by framing our discussion about standardized testing practices in a local school district and school serving large populations of african american students where policy initiatives and administrators require teachers to teach to the test in mathematics. our goal for this article was to present arguments about the racist underpinnings of such instructional practices and how federal, state, and local policies institutionalize racist beliefs about african american students. we situated our analysis of these instructional practices within a deconstruction of systems of assessment that seek to create racial hierarchies and offer “scientific” support for african american intellectual inferiority. davis & martin racism and standardized testing journal of urban mathematics education vol. 11, no. 1&2 65 based on our critical analysis, our request to mathematics teachers is simple. mathematics teachers of african american students must stop engaging in teaching-to-the-test and other narrow instructional practices and provide these students with a challenging and intellectually stimulating mathematics education that assists these students in improving their individual and collective group conditions. we are not dismissing the reality that teachers must operate under the conditions created by the oppressive forces of mandates such as nclb. nevertheless, out of genuine concern for african american students, this article is an instantiation of our advocacy for these students to receive the mathematics education they deserve. we appeal to teachers’ moral commitment to african american students by encouraging them to put these students’ well-being over their fear of federal, state, and local sanctions. we urge teachers to take action both individually and collectively at the district, school, and classroom level to provide african american students with a liberatory education in mathematics. in so doing, we have provided teachers with insight and examples of how that might be done. we urge teachers to be agents of social change in their own school and classroom contexts, hopefully driven by beliefs that build on the following: african american children [must be prepared] with the knowledge, skills, and attitude needed to struggle successfully against oppression. these, more than test scores, more than high grade point averages, are the critical features of education for african americans. if students are to be equipped to struggle against racism they need excellent skills from the basics of reading, writing, and math, to understand history, thinking critically, solving problems, and making decisions; they must go beyond merely filling in test sheet bubbles with number 2 pencils. 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(1998). blueprint for black power: a moral political and economic imperative for the twenty-first century. new york, ny: afrikan world infosystems. microsoft word 409-article text no abstract-2157-1-6-20200920 (galley proof 2).docx journal of urban mathematics education december 2020, vol. 13, no. 2, pp. 60–86 ©jume. https://journals.tdl.org/jume kim h. song is professor of tesol in educator preparation and leadership at the university of missouri – st. louis; songk@umsystem.edu. her main research lines of inquiry examine lcr[m] teaching, creativity in technology-mediated content teaching for urban and english learners, translanguaging, and language education trends in the usa, 1960 to present. sarah a. coppersmith is dissertation chair at maryville university, higher education leadership, st. louis, missouri, and adjunct assistant professor at the university of missouri – st. louis; coppersmiths@umsystem.edu. her research interests include linguistically/culturally responsive teaching, teachers’ transformative learning, and ambiguity tolerance utilizing geospatial technology and inquiry via primary sources in k–16 geo-history learning. working toward linguistically and culturally responsive math teaching through a year-long urban teacher training program for english learners kim h. song university of missouri – st. louis sarah a. coppersmith maryville university this qualitative study examined how participating in-service teachers demonstrated linguistically and culturally responsive mathematical teaching (lcrmt) competences after they completed a year-long national professional development program grant-funded project. a two-dimensional lcrmt framework was developed to measure participating teachers’ mathematical and mathematics-related competences. the qualitative data source was from three in-service teachers’ observations and interviews. the interview and observation data were analyzed using open and axial coding and activity systems. three themes emerged: 1) mathematics-related content teaching practices, 2) tools to support mathematics learning, and 3) teachers’ mindsets and attitudes towards english learner (el) teaching. the researchers then compared verbatim examples using activity systems to examine the following research question: how did participating urban in-service teachers apply linguistically and culturally responsive mathematics teaching competences for els learned at a university el teacher training program to their actual mathematics teaching in the classroom? the results, in general, indicated that the urban in-service teachers demonstrated improvement of lcrmt strategies that they used in their actual mathematics teaching after they completed the university training. however, challenges in the areas of mathematical discourse competences and teachers’ sociocultural beliefs toward els revealed the need for ongoing professional support. keywords: activity theory, in-service teachers, lcrmt, teaching competences, urban education song & coppersmith working toward linguistically and culturally responsive math teaching journal of urban mathematics education vol. 13, no. 2 61 here is a need to improve teacher preparation programs for english learners (els) in the united states, as a majority of u.s. teachers and teachers-to-be have little to no training to support els (bunch, 2010). this lack of training is manifested in the secondary school dropout rate of 22.4 % for latino/latina youth, whose population doubles the national el average (janzen, 2008). in mathematics courses specifically, pre-service and in-service teachers struggle to adequately prepare els for academic success. in a study reporting the results of a k–8 mathematics methods course, aguirre et al. (2012) posit the need to develop “robust forms of pedagogical content knowledge (pck) for preservice teachers through a culturally responsive mathematics teaching approach” (p. 113). such a change in educator training has the potential to greatly increase academic success in a large student population. this study uses data from the national professional development grant program project titled excellent teachers for english learners (etel), which collected data from preand in-service teachers in 2015. the etel project aimed at preparing teachers to improve els’ academic achievement with a focus on mathematics. mathematics is one of the major areas els are at risk of failing in school (janzen, 2008). one of the reasons why els have an achievement gap may be a common misleading myth regarding els’ mathematics learning: “the transition from social language to academic language is easier for els in math than in other subjects” (kersaint et al., 2009, p. 60), like reading and social studies. another misleading assumption is that mathematics is a “culture-free” subject (kersaint et al., 2009, p. 60) that may not have to deal with language. however, mathematics education deals both with everyday language and academic or technical mathematical language (schleppegrell, 2007). halliday (1978) points out that counting and calculating for mathematics draw on everyday mathematical language, such as “counting up” and “counting on,” because counting is an everyday language mastered by els and may not need to be taught. on the other hand, els still need to acquire the academic language of mathematics when they enter school. one mathematics content pedagogical approach that seeks to address this is the introduction of ‘‘new styles of meaning and modes of argument and of combining existing elements into new combinations’’ (halliday, 1978, p. 195). halliday emphasizes the linguistic challenges of mathematics education when discussing the “mathematical register,” which he defines as “a set of meanings that is appropriate to a particular function of language, together with the words and structures which express these meanings” (p. 195). the language of mathematics not only requires a list of vocabulary, grammatical patterns, and equations with numbers and/or words with precise meanings, but also requires communicative competence or mathematics discourse sufficient for active participation in meta[cognitive]-mathematical thinking and reasoning procedures (moschkovich, 2012). teachers also need to grasp how mathematics content is t song & coppersmith working toward linguistically and culturally responsive math teaching journal of urban mathematics education vol. 13, no. 2 62 structured with a particular set of mathematical language/discourse and mathematics pedagogy to make their teaching more equitable and effective for els (tran, 2014). understanding the need of teachers’ mathematics discourse and pedagogical competency development in the field of teaching english to speakers of other languages (tesol), this study examines how multidimensional mathematics teaching competences learned at a university el teacher training program have or have not been applied to teachers’ k–6 mathematics teaching for els. a research question guided this study: how did participating urban in-service teachers apply linguistically and culturally responsive mathematics teaching for els learned at a university el teacher training program to their actual mathematics teaching in the classroom? theoretical framework in this section, we review a theoretical framework starting with linguistically and culturally responsive teaching (lcrt) and expanding it to linguistically and culturally responsive mathematics teaching (lcrmt). development of lcrt framework a basic goal of el teacher education programs is to prepare linguistically and culturally responsive (lcr) teachers. highlighted in this framework is linguistically responsive teaching (lucas et al., 2008), recognizing that most research in teacher education has predominantly emphasized culturally responsive/relevant teaching/pedagogy (crt; gay, 2010; ladson-billing, 2014). this predominant crt emphasis has resulted in shadowing or downplaying the role of language in els’ academic achievement (nieto, 2002). we also note that teachers need to develop linguistically responsive pedagogy because “language cannot be separated from what is taught and learned in school” in any content classroom (lucas et al., 2008, p. 362). bonner and adams (2012) emphasize culturally responsive mathematics teaching when teacher education programs “prepare prospective teachers for being able to reach children where they are and make children’s mathematics learning experiences strong” (p. 35). hymes (1972) mentions “linguistic competence as just one kind of cultural competence” (as cited in byram, 1997, p. 8). such a focus on the cultural aspect does not give the due understanding of major roles academic and social languages play in els’ learning (nieto, 2002). in this study, we emphasize linguistically responsive teaching up front, which is tied with culture and power. background of lcrt framework development in 2013, a research-based lcrt framework was developed (song & simons, 2014). the lcrt framework was built on works from van dyne and his associates’ song & coppersmith working toward linguistically and culturally responsive math teaching journal of urban mathematics education vol. 13, no. 2 63 (2012) four factor model of cultural intelligence (cq), which was influenced by van ek’s (1986) sociocultural and communicative competence work. the four factors of their cq model were “metacognitive, cognitive, motivational, and behavioral” cultural intelligences (van dyne et al., 2012, p. 295). metacognitive cq is a strategic factor that involves awareness of oneself and others, planning on how to approach a cultural situation, and checking the process of monitoring what one does and how it actually plays out in relationship to what one expects to happen. cognitive cq is a knowledge factor (i.e., one’s actual knowledge about particular cultural issues and differences, including cultural norms). motivational cq is a drive factor, involving being interested, confident, and driven to adapt to another culture. behavioral cq is an action factor or an actor—an individual’s ability to modify their verbal and nonverbal behaviors in order to interact with others from different cultural backgrounds. thus far, however, cq has not been assessed in relationship to tesol, only in staff development and leadership training (van dyne et. al., 2012). in this study, our aim is not to explore linguistic competence as part of cultural competence but rather the opposite. our focus is on helping teachers develop linguistically (and culturally) responsive teaching competences. in general, teachers lack knowledge on language systems or linguistics, and they do not understand the benefits of valuing the linguistic diversity els bring into the classroom (fillmore & snow, 2002; gonzález & darling-hammond, 1997). teachers with els need to receive more systematic and intensive preparation on what it means to possess and demonstrate linguistically responsive teaching competences, which embrace pedagogical and cultural competences (kim et al., 2018). van ek (1986) introduced a model of communicative ability with six competences (linguistic, sociolinguistic, discourse, strategic, socio-cultural, and social) that emphasizes the role of the native speaker as a model for a language learner. outside of linguistic competence, the other five competences consider the influence of sociocultural and pedagogical aspects in developing communicative competence (song & simons, 2014). lucas and her associates (2008) introduced six linguistically responsive teaching principles for authentic el teaching: a) every day and academic/technical language, b) teachable but challenging content concepts, c) roles of social interaction between els and other peers in the classroom, d) roles of els’ native languages, e) inclusive classroom climate for minimal anxiety, and f) linguistic form and functions. lucas and villegas (2010) designed a linguistically responsive teaching framework that complements van ek’s communicative principles and lucas et al.’s lcrt principles that emphasizes knowledge of languages, second language acquisition principles, and els’ linguistic and cultural diversity. each of these lcrt elements represents a commitment made by teachers to become more aware and considerate of what els can bring to the classroom and challenges that they routinely face (song & simons, 2014). these frameworks represent dimensions of linguistic, song & coppersmith working toward linguistically and culturally responsive math teaching journal of urban mathematics education vol. 13, no. 2 64 pedagogical, and cultural competences, but they are limited by their one dimensionality. in light of this, nguyen and commins (2014) introduced a two-dimensional lcrt framework. nguyen and commins proposed linguistic, pedagogical, and cultural competences as the first dimensional constructs and knowledge in depth; higherorder thinking skills such as application, analysis, and synthesis; and implementation as the second dimensional constructs, which are yet to be elaborated. in 2016, an lcrmt framework, based on nguyen and commins’ (2014) two-dimensional lcrt model, evolved after the first author of the current study explored extensive research on linguistically and culturally responsive teaching as well as mathematics teaching for els (aguirre et al., 2012; fillmore & snow, 2002; gonzález & darlinghammond, 1997; halliday, 1978; janzen, 2008; lucas et al., 2008; schleppegrell, 2007; song & coppersmith, 2017). development and expansion of lcrmt framework the newly developed lcrmt framework has two dimensions: a mathematicsrelated dimension and a meta[cognitive]-mathematics dimension. the first mathematics-related dimension for el teachers includes the following competences: a) mathematics content competence, b) mathematics discourse competence, and c) mathematics pedagogical competence. the second meta[cognitive]-mathematics dimension’s features address larger contexts of mathematics teaching practices, mainly through el teachers’ “how-to-actors”: a) acquiring and demonstrating knowledge in depth, b) developing and applying procedural demands and reasoning skills, and c) examining and developing socio-political teacher beliefs (table 1). the three howto-actors start with the action verb (e.g., acquire, demonstrate, develop, and examine), unlike nguyen and commins’ (2014) noun phrases in the second-dimensional construct. the lcrmt constructs can be used to measure the three competences of the first dimension. the following characterizes details of the lcrmt framework. lcrmt’s first-dimensional mathematics-related competences are interdependent with the meta-mathematics how-to-actors when preparing, delivering, and reflecting on mathematics instruction for els (uribe-flórez et al., 2014). teachers need to maintain a focus on mathematics reasoning (i.e., develop and apply mathematics procedural demands and reasoning under mathematics content competence) as well as language development by recognizing “how els express their mathematical ideas as they are learning english, and teachers can maintain a focus on mathematical reasoning as well as on language development” (moschkovich, 2012, p. 18). teachers need to allow els to use their native languages to understand individual mathematics-related words (acquire and demonstrate knowledge in depth) so they can expand their mathematics reasoning. nguyen and commins (2014) emphasized song & coppersmith working toward linguistically and culturally responsive math teaching journal of urban mathematics education vol. 13, no. 2 65 the importance of teachers’ sociopolitical endeavors because teaching cannot be neutral when they prepare content resources and pedagogies (i.e., examine and develop sociopolitical teaching beliefs under mathematics pedagogical competence). the first author of the current study renamed and rearranged the firstand second-dimensional features of lcrt when developing lcrmt. for the first dimension, the lcrt (d. nguyen & n. commins, personal communication, december 9, 2014) had a) linguistic, b) pedagogical, and c) cultural competences. for lcrmt, the first author added mathematics content competence and renamed linguistic competence to mathematics discourse competence and pedagogical competence to mathematics pedagogical competence to be more inclusive and mathematics-related. lcrt’s cultural competence was incorporated into lcrmt’s third how-to-actor: examine sociopolitical beliefs. the second-dimensional features are “actors” that help teachers acquire, prepare, deliver, assess, and reflect on their mathematics teaching more inclusively to better serve els. these three how-to-actors start with action verbs, such as acquire, develop, apply, demonstrate, and examine, in order to support and measure the first dimension of mathematics-related competences (table 1). for example, when preparing a mathematics lesson on a unit of weight (acquire knowledge in depth under mathematics content competence), an lcr mathematics teacher should explore their els’ academic and cross-cultural funds of knowledge when exploring mathematics pedagogies (examine sociopolitical beliefs under mathematics pedagogical competence; moll, 2015). teachers may need to check if the els are from countries in which they use different metrics (e.g., the metric measurement system rather than the american standard measurement system). teachers also need to recognize the learning needs of els and cultivate strategies “implemented with respect to the [math] languages students bring to the class” (acquire knowledge in depth under mathematics discourse and pedagogical competences; uribe-flórez et al., 2014, p. 236). the new assessments for common core state standards require “students to show their work and explain it” (crouch, 2015, p. a7). in order for els to develop the skillset necessary to do so, teachers need to support them when they solve equations and word problems in class (develop procedural demands under mathematics content competence). in addition, the national council of teachers of mathematics (2000) has set priorities for constructivist teaching, under which students work to problem solve through reasoning and communication (develop procedural demands and reasoning under mathematics pedagogical competence; swars et al., 2015). the term “mathematical register” relates to lcrmt’s second-dimensional how-to-actors by choosing particular language forms to aid understanding about how mathematics content competence is formed and includes multiple “semiotic systems and grammatical patterns” unique to mathematics discourse competence (schleppegrell, 2007, p. 141). song & coppersmith working toward linguistically and culturally responsive math teaching journal of urban mathematics education vol. 13, no. 2 66 el teachers should be aware that the “symbols, oral language, written language, and visual representations such as graphs and diagrams” (schleppegrell, 2007, p. 141) used in the mathematical register may present linguistic challenges (acquire and demonstrate mathematics knowledge in depth under mathematics content and discourse competences; halliday, 1978). el teachers need to conquer these linguistic challenges and portray mathematics concepts by helping els develop procedural demands and reasoning skills. varying ways of illustrating mathematics concepts and ideas through language forms and other avenues (i.e., develop procedural demands under mathematics discourse competence) makes mathematics learning more accessible for els (halliday, 1978; schleppegrell, 2007). as many mathematics ideas and concepts do not translate easily, el teachers need appropriate mathematics pedagogical skills for instructing students in technical mathematics discourses (schleppegrell, 2007). teachers also need to help els understand that there are diverse registers (pimm, 1987) and that they have to learn when to use particular mathematical languages (e.g., everyday versus technical mathematical language) with proper reasoning processes (aguirre et al., 2012). table 1 illustrates the nine constructs of the lcrmt framework with examples and references. song & coppersmith working toward linguistically and culturally responsive math teaching journal of urban mathematics education vol. 13, no. 2 67 table 1 nine constructs of two-dimensional lcrmt framework for mathematics teachers with els math-related vs. meta-math actors math content competence math discourse competence math pedagogical competence 1. acquire & demonstrate knowledge in depth 1a. teachers acquire knowledge of math content, math symbols, notations (+, -, ×, ÷, =, ⁄), math operations, & visual representations (schleppegrell, 2007) 1b. teachers demonstrate knowledge of math content curriculum and standards (grossman et al., 2005). 1a. teachers demonstrate knowledge of math language and math register in teaching els (halliday, 1978) 1b. teachers acquire knowledge of technical content discourse (aguirre, et al., 2012) 1c. teachers acquire natural/ nontechnical math discourse (schleppegrell, 2007). 1a. teachers acquire culturally relevant math content pedagogy knowledge (turner & drake, 2016) 1b. teachers demonstrate teaching strategies to support els’ math language acquisition (richards, 2013). 2. develop & apply metacognitive procedural demands 2a. teachers develop teachers’ understanding of math concepts and procedural demands (turner & drake, 2016) 2b. teachers apply knowledge of math concepts to word problems with reasoning process (janzen, 2008; moschkovich, 2012). 2a. teachers apply meaning of math concepts to procedures and reasoning strategies (what if & therefore; aguirre, et al., 2012) 2b. teachers attribute process of part-to-whole and nonreversible math operation rules (janzen, 2008; turner & drake, 2016). 2a. teachers apply procedural and reasoning skills using cooperative and inquiry-based learning strategies (janzen, 2008; moschkovich, 2012) 2b. teachers create situated learning experiences based on els' linguistic and cultural resources (gee, 2016). 3. examine & develop cross-cultural & sociopolitical beliefs 3a. teachers examine and develop cross-culturally “just” beliefs in teaching math for els (turner & drake, 2016) 3b teachers explore els’ funds of knowledge for academic learning (moll, 2015). 3a. teachers reject an englishonly orientation and apply equitable attitude toward different language use (austin, 2009; liggett, 2014) 3b. teachers develop social discourse for els’ conversational and academic math language advancement (moschkovich, 2012). 3a. teachers integrate equitybased pedagogy into their teaching (flores & rosa, 2015) 3b. teachers create safe, welcoming climate with minimal anxiety (krashen, 2003; pappanihiel, 2002) 3c. teachers examine cross-cultural curriculum knowledge within situated context (ounce & pound vs. gram & kilogram). song & coppersmith working toward linguistically and culturally responsive math teaching journal of urban mathematics education vol. 13, no. 2 68 research methods we chose a qualitative case study design and used the data collected from observations and interviews in which we examined three urban in-service teachers’ applications of el-specific mathematics teaching strategies that they learned at a university el teacher training program to their actual mathematics teaching in an elementary school setting. the research question that guided this study was the following: how did participating urban in-service teachers apply linguistically and culturally responsive mathematics teaching competencies for els learned at a university el teacher training program to their actual mathematics teaching in the classroom? study context the two main activities of the etel project were a) six tuition-free tesol courses for the etel cohort of preand in-service teachers to obtain state tesol certification and b) five full-day summer institute professional development (pd) sessions followed by monthly pds in the area of mathematics. the lcrmt constructs were integrated into each of the six tesol courses: a) linguistics for enhancing teachers’ mathematics content discourse and acquiring mathematics knowledge in depth, b) principles of second-language acquisition to support els’ content and discourse competencies and in-depth knowledge acquisition, c) sociolinguistics and cross-cultural communication to enhance teachers’ discourse competence and examination of sociopolitical teaching beliefs, d) assessment and instructional material development that emphasizes pedagogical competence and development of procedural demands, e) el teaching methods that focus on three mathematics-related competences and three meta[cognitive]-mathematics how-to-actors, and f) practicum for el teachers that is focused on three mathematics-related competences with all of the three meta[cognitive]-mathematics how-to-actors. the participating urban teachers were taking two tesol courses per semester in 2015: two courses in the spring (linguistics and second-language acquisition courses), two in the summer (sociolinguistics and assessment courses), and two in the fall (methods and practicum courses). the participants also attended a five-day pd summer institute that included activities to improve their mathematics content knowledge, content discourse, and content pedagogical competences for teaching els. when taking tesol methods and practicum courses, participants were required to prepare, implement, and reflect on mathematics teaching cases using the lcrmt framework as their guide. a mathematics consultant and a state el specialist from the etel team examined the elementary school mathematics curriculum utilized by the in-service teachers to verify that it followed world-class instructional design and assessment (wida) english language development standards (wida). these standards provided the language supports used during instruction, which were based on the six song & coppersmith working toward linguistically and culturally responsive math teaching journal of urban mathematics education vol. 13, no. 2 69 english proficiency levels present in the four language modalities (reading, writing, speaking, and listening; wida, n.d.). these two mathematics and english-language teaching specialists additionally chose key mathematics concepts, such as place value, number sense, and fractions, for the mathematics pds along with el-specific teaching practices for mathematics content, discourse, and pedagogical competences. some mathematics teaching methods included inquisitive, hands-on, and collaborative activities and backward assessments. tools such as manipulatives, sentence stems, multi-language mathematics vocabulary word banks, word-walls, games, number tables, and graphic organizers were also introduced to and utilized by these participating urban teachers. participants and sampling methods seven urban in-service teachers participated in the 2015 etel grant project. the researchers contacted the seven in-service teachers through emails to arrange follow-up visits in 2016 after their etel graduation in december 2015. three (42.9%) of the in-service teachers agreed to participate in follow-up observations and interviews. the three in-service teachers, who completed the university etel grant project and agreed to participate in this qualitative study, were from two schools. ms. happ is a female sixth-grade teacher who taught at school 1. mr. mack is a male third-grade teacher and ms. bishop is a female second-grade teacher who taught at school 2. ms. happ taught mathematics for the sixth graders in school 1 for five years. mr. mack taught third and fourth graders in school 2 for four years. ms. bishop taught the second graders for two years in school 2. pseudonyms were used for the teachers’ names. the el population in school 1 and school 2 were diverse (more than 10 nationalities and more than nine languages). the els were from south/latin american countries and u.s. territories (colombia, mexico, and puerto rico) as well as bosnia, ghana, iran, nepal, somalia, syria, and vietnam. their english proficiency levels were varied (level 1 to level 4). all three urban in-service teachers were monolinguals, though the two teachers in school 2 tried to use spanish translations for new words that they got from google translate, which was a method introduced in the tesol methods and practicum courses. however, the teachers did not even try to translate other languages, such as bosnian and vietnamese, even though more than 30 percent of the els were from bosnia and vietnam. purposive sampling was chosen for this study in order to gain information “of central importance to the purpose of the research” by “focusing in depth on a small number of participants” (patton, 1990, p. 169). song & coppersmith working toward linguistically and culturally responsive math teaching journal of urban mathematics education vol. 13, no. 2 70 instruments and data collection an observation protocol was designed with three sections: 1) heading (grade, date/time, mathematics content, and school), 2) low inference, which included time and field notes describing what actually happened, and 3) high inference, which contained the observer’s interpretation, comments, and reflection (see appendix a). the observation protocol was developed so the observers (two researchers) could collect low inference data (factual data; i.e., what actually happened) as well as high inference data (opinions and interpretation data; e.g., i think/assume/argue). three observations, each lasting between 30 minutes to one hour, were conducted for each teacher at times convenient for the teachers. these observation periods allowed the researchers to explore if and how the teachers were utilizing what they had learned in the tesol courses and the mathematics pds in the etel learning community. each observation occurred one to two weeks apart, and the researchers took field notes using the observation protocol. an interview protocol was developed to explore the teachers’ mathematics teaching practices with two main questions: 1) what have been some success indicators of your mathematics teaching in the classroom with els? and 2) what have been some challenges when you teach mathematics to els? (see appendix b). two 30minute interviews were conducted with each of the three in-service teachers, taking into consideration the demands on their schedules. the researchers took the notes and recorded the interviews using an ipad. the recorded interviews were transcribed. data analysis this basic qualitative data analysis involved two phases. the first phase was informed by a grounded data analysis approach (charmaz, 2010). the second phase was informed by activity theory (yamagata-lynch, 2010). the grounded data analysis was processed by the two researchers. first, they open coded the observation and interview data individually and compared the coded data regularly to examine the commonality and criticality. then the researchers collaboratively produced a coding book with key codes, descriptions, and categories (saldaña, 2016) after they had open coded individually. using open and axial coding (strauss & corbin, 1998), the researchers explored and obtained verbatim examples to identify emerged themes (saldaña, 2016). each researcher identified individual codes and themes that emerged from the first phase through the grounded theory approach. then, we compared the findings from the collaborative coding process with the elements of the activity systems (yamagata-lynch, 2010). engeström (1987) clarified that two complex human systems (e.g., the university teacher education program versus the public-school classes) can be compared within activity system triangles (figure 1). the activity system originated from song & coppersmith working toward linguistically and culturally responsive math teaching journal of urban mathematics education vol. 13, no. 2 71 leont’ev’s (1978) cultural historical activity theory, which was later refined by kaptelinin and nardi (2009). engeström (2001) clarified the activity system triangles in which human activities take place, such as school environments and job contexts. the elements of activity system triangles are interdependent, and they are subjects (people doing the acting), objects (goals people produce by their acting), and labors (actual labors using tools to reach the goals; hardman, 2015). adopting this activity system triangle framework, the researchers analyzed how the teachers (subjects) performed (labored) in the classroom with els and investigated if they met the learning objectives (i.e., els’ achievement in the content areas). in addition, the researchers analyzed what teachers used as tools when they actually labored to support els. the second phase of data analysis intentionally focused on the data sets for the comparison of the two activity system triangles (figure 1) using analyzed data from the classroom observations for actual teaching context and follow-up interviews for school context. figure 1 was created by the researchers based on engeström’s (1987) model of activity systems, which demonstrated the juxtaposition of the two activity systems (i.e., university system and actual classroom system) using the triangles. the researchers examined if the three teachers utilized their learning from the university pd training context (activity system a in figure 1) when they actually taught mathematics in their classroom context to els (activity system b in figure 1) by adopting engeström’s (1987) “model of interacting activity systems” (as cited in forbes et al., 2009). figure 1. activity systems: from university etel community to actual classroom community song & coppersmith working toward linguistically and culturally responsive math teaching journal of urban mathematics education vol. 13, no. 2 72 through comparisons and contrasts of the key codes from the data using the two activity systems, two patterns converged: (a) transfer of etel lcrmt training to elementary mathematics classrooms and (b) [non] transfer in progress/incomplete. these first and second patterns were related to whether the lcrmt competences that the teachers learned at the university etel community (activity system a) were transferred or not to the elementary classroom community (activity system b), either as a “rule” toward reaching the goal in the elementary school system or the object in the university system. trustworthiness trustworthiness in research represents the rigor of the research procedure and how convincing the finding is (gorard, 2013; shank, 2006). in this study, trustworthiness was achieved by triangulating the qualitative data of observations and interviews through cross-checking between the researchers using open and axial coding. trustworthiness was also checked by triangulating interview and observation data. a descriptive audit trail of data collection and analytical process also ensured that the study kept to the ethical process of data collection, which included appropriate irb procedures, informing participants of their rights, and obtaining consent forms. finally, the thick description of study context and participants’ own words in the result section further established the trustworthiness of this study. results one major phenomenon across the data sets was that the three in-service teachers demonstrated if their lcrmt competences learned from university etel trainings were transferred to their mathematics teaching, including a counter-story that was in progress or incomplete. three themes emerged from the collaborative coding process: 1) mathematics-related content teaching practices, 2) tools to support mathematics learning, and 3) teachers’ mindsets and attitudes towards el teaching. participating teachers’ narratives were analyzed along with the interview and observation data under each of the three emergent themes. each of the three emergent themes were connected to the nine lcrmt constructs and the four main units of the activity system (i.e., objects, rules, tools, and subjects) to measure the goal of the university triangle system as well as of the school triangle system illustrated in figure 1. for example, mathematics-related content teaching practices also address lcrmt’s first how-to-actor (acquire and demonstrate knowledge in depth) under mathematics content, mathematics discourse, and mathematics pedagogical competences (i.e., first dimension of lcrmt [table 1]). tools to support mathematics learning referred to the second how-to-actor under the three competences (develop and apply procedural demands). for example, for a lesson on the part-to-whole song & coppersmith working toward linguistically and culturally responsive math teaching journal of urban mathematics education vol. 13, no. 2 73 fraction method, a teacher leads the els to justify the mathematical concept by using their native languages in the part-to-whole vocabulary (half, one-third, three quarters) and for the multi-language sentence starters (demonstrate knowledge in depth and procedural demands under mathematics content, discourse, and pedagogical competences). teachers’ mindsets and attitudes referred to the third how-to-actor (examine and develop cross-cultural and sociopolitical beliefs) under the three competences of the lcrmt (e.g., els and their families’ funds of knowledge as well as teacher examination of their own attitudes towards els). the emergent themes and related activity system elements were incorporated or integrated when analyzing the three teaching cases of ms. happ, mr. mack, and ms. bishop with the sample narratives and researchers’ overviews and interpretations. the researchers emphasized the areas that contained key discourses and lcrmt competences using italics and parentheses. when reporting the findings, the researchers integrated each of the three teaching cases into the interdependent activity system model with the main units of subjects, objects/goals, rules, and labors/tools (figure 1) and the lcrmt model with nine constructs in its two dimensions (table 1). mathematics-related content teaching practices the following described the first theme: mathematics-related content teaching practices with the main units of the activity system model and lcrmt constructs. mathematics-related content teaching practices can be connected to the activity system object/goal regarding what teachers (subjects) are working towards, as well as lcrmt’s first how-to-actor (acquire and demonstrate knowledge in depth). ms. happ, a sixth-grade mathematics teacher with 14 students, of whom six were els (43%), utilized mathematics content teaching practices to deepen her students’ mathematics content competence. ms. happ’s content and language objectives were presented in a conversation that took place after she frontloaded and taught the foundational mathematical key vocabulary words of a lesson, such as “expression,” “equation,” and “variables,” before introducing the content and language objectives of the lesson so that students could better understand the goal/concept of the lesson. happ: share some of the ideas we discussed yesterday. yesterday we learned “expression” and “equation.” what’s the difference between these two? dan: an expression does not have an equal sign. an equation does have an equal sign. happ: is that right team green? yes. does anybody have a trick? mary: “equation” has “equ” in “equal.” happ: today, i am adding a word, “variables.” why do we use variables in math? mabel: variables, umm, an unknown number. happ: you mean that variables are letters that represent unknown numbers? song & coppersmith working toward linguistically and culturally responsive math teaching journal of urban mathematics education vol. 13, no. 2 74 happ: today, we will use variables in a different way. anyone have a question on expression, equation, and variables? happ: look at the word wall. take a look at objectives here and on the board. content objective: i can write mathematical expressions and equations. language objective: after reading a story problem or a verbal expression, i can represent it with a mathematical expression or equation. mr. mack, a third-grade teacher with 21 students, of whom 11 were els (52%), started reviewing addition (mathematics content) and pronouns. he was trying to teach pronouns before introducing addition by asking questions. this grammar lesson helped students understand the mathematical word problem, which was the goal and objective: mack: jane and i had 4 pencils each last wednesday, and [jane] brought another 4 pencils this morning. how many pencils did we [jane and i] have all together this morning?” mack: who can tell me which pronoun is used for jane? jerry: “she.” mack: can you answer with a complete sentence replacing a pronoun for “jane and i” in the first sentence? jerry: uum. jane and i had 4 pencils each last wednesday, and she brought another 4 pencils this morning. mack: great! who can have a pronoun for “jane and i” in the last sentence? [he was pointing at the last sentence.] mary: we. (pause) how many pencils did we have all together this morning? mack: great. because he is an elementary teacher, mr. mack integrated the grammar lesson of pronouns before introducing addition, which demonstrated an interdisciplinary approach. an interesting phenomenon was that mr. mack did not call on any els to answer his questions. even if mr. mack called on els, they might have failed to have a correct answer if they did not understand what “each” means. because mr. mack did not frontload the lesson with a description of what “each” means, els in his class could have thought that “jane and i” in the question had four pencils altogether rather than four individually. ms. bishop, a second-grade teacher with 19 students, of whom eight were els (42%), taught fraction and parts-to-whole relation as a review. the data did not have any evidence of her covering mathematics content knowledge and mathematics-related discourse (objects/goals), but she only had 30 minutes for the mathematics class. even with only 30 minutes, ms. bishop might have frontloaded the key song & coppersmith working toward linguistically and culturally responsive math teaching journal of urban mathematics education vol. 13, no. 2 75 vocabulary words, but she instead rushed into the content teaching (i.e., fractions with tools [e.g., apple cutouts, orange slices]) without checking if each el understood the content discourses (mathematical academic language and direction/nontechnical language). both mr. mack and ms. bishop demonstrated the typical phenomenon observed in many elementary mathematics classes: teachers prepared some hands-on activities that might work for every student (i.e., one-size-fits-all activities). tools to support mathematics learning. ms. happ talked through each mathematical problem set and allowed students to work in groups (content pedagogical competence) and to express their ideas with reasoning (develop procedural demands) to the whole class. one salient feature in ms. happ’s class was her adoption of teaching tools from the etel training, which encouraged teachers to utilize tools to support mathematics learning by sharing content and language objectives (objects/goals) and explicitly using “i” statements, inquiry-based teaching, and cooperative learning strategies. for example, ms. happ engaged students by using tools such as collaborative learning strategy, which involves setting up group learning (i.e., “green team, blue team, and yellow team”). this was a pattern consistently observed in her lessons and one that matched with an lcrmt construct (mathematics pedagogical competence). furthermore, ms. happ instructed the students by asking them to “give a reason why” when she posed a mathematical word problem (develop procedural demands). to this end, ms. happ tried to emphasize reasoning rather than simply the solving of the mathematical equation. happ: i loved how you said, “four hundredths” and not “point zero four.” how do we know it’s four hundredths divided by n instead of n divided by 0.04? james: you’re dividing four hundredths by n. happ: is your 0.04 the dividend or the divisor? say it on a count of 3 — who can explain? john: i think it is dividend because zero and four hundredths go inside the box. ms. happ regularly used think-aloud, thinking prompts, comprehensible input, scaffolding, and “making thinking visible” to engage students in reasoning through modeling, repeated questioning, and creating instructional structures through collaborative learning opportunities. she kept engaging students by encouraging them to think and express their reasoning through “how, why, and/or because” thinking prompts (develop procedural demands under mathematics discourse competence). mr. mack also demonstrated his lcrmt competences in the emergent theme “tools to support mathematics learning” learned from the etel pd. mr. mack structured mathematics classes with a variety of cooperative learning activities, from song & coppersmith working toward linguistically and culturally responsive math teaching journal of urban mathematics education vol. 13, no. 2 76 whole-class instruction “on the carpet” to pair work, group work, and individual work at computers. mr. mack showed a “phone-a-friend” collaborative strategy as a tool to support mathematics learning. mr. mack’s students also participated in a “i have, who has?” call-out activity during guided practices at the carpet area. students additionally rolled foam dice in pairs and wrote a mathematics addition/multiplication problem along with the answer. despite effectively utilizing lcrmt competences in many of his lessons, mr. mack did not show differentiated teaching strategies for els to improve their mathematics learning. most of the students who were asked for answers were non-els. additionally, when checking their mathematics worksheets, most of mr. mack’s els with low english language proficiency did not have the correct answers to his weight unit, demonstrating that they did not receive adequate support during the unit. mr. mack did attempt to address this weakness in his instruction however. with researchers’ feedback of how els were struggling to acquire the language of mathematics during the follow-up interview, mr. mack attempted several strategies to make mathematics discourse more accessible to els. for example, he infused spanish into a mathematics teaching activity for students following a suggestion from the observer, who had been his previous tesol instructor, about the benefits of using els’ native languages when introducing new vocabulary. these examples illustrated how mr. mack utilized tools to support mathematics concepts and reasoning and tried to adopt a multitude of teaching strategies (phone-a-friend, cooperative learning strategies, and hands-on activities) to support els, as learned at the etel pd. even if this was not enough to demonstrate the teacher’s socially just beliefs, mr. mack utilized subjects’ (engaged els) tools (labored with tools to support mathematics learning) on the second day of the observation. ms. bishop also appropriated a variety of tools to support mathematics learning. on the first day of the observation, she used a children’s book to teach fractions, introducing parts-to-whole. during the lesson, she would read then stop and ask specific questions from the book, in which there was a story related to fractions. ms. bishop also drew a pizza and let six students come up to the board and divide it into the six pieces. the students wrote their names in each of the six slices/pieces of the pizza (tools for understanding mathematics concepts, parts-to-whole). she then used an apple cutout and explained fractions and parts-to-whole. after using the apple cutout, she used actual oranges for the students to peel and portion into segments. in these activities, she tried implementing meaningful mathematics activities with tools using a children’s book and props/realia that were taught during the university mathematics pd. the els in ms. bishop’s class, as well as in mr. mack’s class, struggled to use mathematics vocabulary in sentences to complete their assignments. both teachers tried to introduce mathematics content concepts using tools to help students song & coppersmith working toward linguistically and culturally responsive math teaching journal of urban mathematics education vol. 13, no. 2 77 understand mathematics concepts, but els were not able to compose and/or solve any word problems. it seemed that the teachers might have thought that their onesize-fits-all teaching strategies could work for els. however, when researchers walked around the class while els were completing worksheets, they observed that els were not able to complete assignments with the correct answers. the teachers should have explored the needs of each el and developed their mathematics pedagogy intentionally. in one of ms. bishop’s lessons, some els whose english proficiencies were low did not comprehend the basic content knowledge of fractions with the directions that were given. ms. bishop talked and modeled most of the time, and the students were supposed to follow her directions. this excerpt from ms. bishop’s class showed her attitude and effort; she wanted to prepare for the students, including els, by showing many shapes: you will work on deciding whether pieces are equal or unequal. first, circle “equal” or “unequal.” look at this one. this is a rectangle that is divided into four parts. does each of the four parts, one fourth equal or unequal? [she modeled.] use the circle and other shapes. unfortunately, even in the directions above, there were a number of the mathematical vocabulary words, such as “equal,” “unequal,” “rectangle,” “is divided into,” “each,” and “one fourth,” that should have been taught before telling the els what to do and how to do it. she could have used multi-language (i.e., translated) directions as well as multi-language word banks with the pictures. she could have also shown the final teacher-made sample projects in multiple languages for the els and asked els to demonstrate their understanding of her directions. in summary, from the analyzed observation and follow-up interview data, the three urban in-service teachers demonstrated that they were able to prepare and deliver mathematics lessons that contained mathematics content, discourse, and pedagogical competences by utilizing mathematics-related content practices and tools to support mathematical concepts and reasoning. the three teachers also demonstrated through their efforts and attitudes that they would like to develop professionally to support els. however, the teachers’ efforts and attitudes did not result in differentiating their mathematics teaching explicitly and intentionally enough to produce outcomes that would demonstrate their els’ mastery of mathematics concepts and reasoning skills. the three teachers adopted mathematics teaching tools, such as inquirybased, student-centered cooperative learning strategies, children’s literature, realia, and els’ native languages to show their effort, but their cross-cultural and/or sociopolitical beliefs about els were not evident in their teaching practices. song & coppersmith working toward linguistically and culturally responsive math teaching journal of urban mathematics education vol. 13, no. 2 78 lcrmt competences: incomplete and in progress despite demonstrating multiple levels of transfer of etel training in the elementary mathematics classrooms of three participating teachers, we noticed that there were areas that these teachers needed further support in in order to comprehensively and appropriately apply their learning from the university pd (activity system a) into their individual educational settings (activity system b). first, the teachers’ elspecific lcrmt support was not always consistent. ms. happ utilized content and language objectives, cooperative learning and inquiry-based strategies, pre-vocabulary activities, and cognates as indicators of successful transfer of the etel training. however, it was observed that her els (43% of the class) inconsistently responded to her questions at times. as the excerpts from ms. happ’s classes show, most of the students who responded were non-els. she could have specifically asked the els questions in order to provide more opportunities for them to show their level of mathematics understanding (i.e., tools to support their mathematics content and discourse learning). although ms. happ included mathematics content, discourse, and pedagogical competences by demonstrating depth of mathematics knowledge and procedural demand levels in her lessons, there was not much evidence of her making intentional efforts to demonstrate an equity-based attitude and effort for els. second, the intentional use of language supports, for example, by using everyday mathematical language or els’ first languages (i.e., mathematics discourse competence and teacher’s cross-cultural and multilingual beliefs), was not always observed, which would have helped els with low english proficiency to better comprehend the mathematical language used in class. furthermore, the three teachers indicated more linguistic challenges (halliday, 1978) in the area of mathematics discourse than in the mathematics content pedagogical competence. finally, ms. happ, as well as the other participating teachers, mostly used english-only verbal questioning (i.e., teacher’s attitude toward els’ language and cultural backgrounds) in their teaching strategies. they needed to further examine and develop their own sociopolitical and multilingual beliefs when teaching mathematics to els and prepare multi-language texts, vocabulary banks, and direction using translation applications (e.g., google translate). in mr. mack’s mathematical problem, “jane and i had 4 pencils each last night, and jane had four pencils this morning. how many pencils do they have in total?,” els did not understand what “each” meant (mathematics discourse competence). after a phone-a-friend activity, mr. mack asked the students to answer an addition/multiplication question and had the following interaction: mack: angela [who is an el student], can you answer this question? angela: it is, i think, 8. 4 plus 4. addition. [angela displayed her white board showing 4 + 4 = 8.] song & coppersmith working toward linguistically and culturally responsive math teaching journal of urban mathematics education vol. 13, no. 2 79 shawn: no, it is 12. jane had 4 and i had 4 last wednesday. jane had 4 pencils this morning. 4 + 4 + 4 equals to 12. i can use addition and multiplication. mack: why did you think you can use multiplication? shawn: multiplication because of three same numbers, 4, 4, 4. angela, an el student, did not notice “each” but instead saw two instances of “four” and came up with “eight” for an answer until shawn (a native english-speaking student) listed three fours. it was obvious that angela had understood the concept of addition but did not understand the grammatical meaning of “each” in a mathematical word problem sentence (tools to support mathematics learning under mathematics discourse to develop reasoning and procedural demands). mr. mack used shawn’s reasoning, “because [there were] three same numbers, 4, 4, ,4” to explain multiplication (teacher’s belief) to students. mr. mack did not stop and explain to angela why “eight” could not be a correct answer. as shown in the excerpt above, mr. mack could have guided the students learning addition by asking them to count the number using the number-line up to four then explaining what “each” meant. then students could have counted up to another four, and mr. mack could have asked, “how many pencils did jane bring this morning?” after this, he could have let the els count up to another four. mr. mack could have also explained how this problem of addition or “counting up to” could be used for multiplication by illustrating the three fours and explaining what “each” meant in that specific context. like angela, some els came up with “eight” in this word problem instead of “twelve” because they did not understand the word “each.” without giving time for els to have more practice on the multiplication concept, mr. mack introduced another activity on place value and wrote the two-digit number multiplied by a one-digit number: 14 x 4. mr. mack could have given more opportunities to allow the els to have more practice on word problems that included mathematical discourses that they needed to understand. in another lesson on weight units, mr. mack focused on the unit vocabulary words of “ounce,” “pound,” and “ton” along with their symbols (oz, lb, t, respectively). he displayed a picture of a pencil when explaining 1 oz, a picture of a can of green beans when explaining 1lb, and picture of a small car when explaining 1 t, assuming the els would understand the concepts of these units. mr. mack could have used more resources, such as a comparison chart with the metric measurement system and the u.s. customary system (e.g., gram and kilogram versus ounces and tons). in addition, mr. mack could have brought a scale (realia) to the class so the els could actually measure a pencil, a piece of paper, an empty can, and so forth. that way, even if els were confused with the u.s. customary system, they could guess or estimate the difference between the two measurement systems. song & coppersmith working toward linguistically and culturally responsive math teaching journal of urban mathematics education vol. 13, no. 2 80 in ms. bishop’s classes, transferring the concepts from the read-aloud to the hands-on activity with oranges to writing an equation was a challenge for most of the els. she could have whispered to the els to check for their understanding and, if needed, directly intervened to better assist them in understanding the mathematics concepts under study. most of ms. bishop’s els were newcomers whose english proficiency levels were low. ms. bishop might have provided mathematics-related pre-activities to help beginning level els to understand the main vocabulary words, such as “equal,” “unequal,” “divided by,” and “one fourth,” and used fractions with pictures and number sentences as well as students’ native languages and multi-language word banks. ms. bishop could have also allowed els to write word problems in their native languages and translate them with partners. she might have also allowed the els who spoke the same native languages to use google translate to discuss what these words meant. while walking around the classroom, the researcher found that ms. bishop’s els were not writing mathematics word problems because they did not understand the mathematics concepts and key vocabulary words used in the lesson well enough to do so. the three urban in-service teachers’ mathematics teaching practices indicated that they tried to adopt mathematics-related teaching practices, tools to support mathematics concepts and reasoning, and their attitudes and beliefs to be linguistically and culturally responsive. however, they did not differentiate or alter their mathematics teaching practices based on the needs of their els. additional language teaching supports were needed for the els to meet content and language objectives. as seen in these cases, the teachers did not sufficiently understand the role of developing mathematics discourse and pedagogical competences for serving els, indicating a gap between the etel training activity triangle system and the elementary school activity triangle system (figure 1). explicit language supports using els’ native languages or intentional efforts to call on them during lessons would have provided deliberate opportunities for els to think and respond to numerous questions. to build on els’ funds of knowledge (moll, 2015), teachers should provide intentional opportunities and proper tools to monitor els’ understanding of mathematics concepts and reasoning given proper attitudes and beliefs. finally, the three participating teachers needed to follow the institutionally established protocols in their instruction in terms of classroom management and the curriculum process. when asked during an interview what she would like to change in her teaching, ms. bishop shared a challenge concerning divided instructional time due to recess occurring in the middle of the mathematics class. she said, “sometimes i struggle getting them back on track, especially my els. i have to revisit the vocabulary we’ve just taught. some els asked what an apple is.” this particular interview excerpt exemplified teachers’ linguistic challenges (halliday, 1978) directly related to a tight schedule. more time is needed for els to develop mathematics discourse song & coppersmith working toward linguistically and culturally responsive math teaching journal of urban mathematics education vol. 13, no. 2 81 and vocabulary work. ms. bishop’s teaching case showed that she tried some lcrmt strategies, such as implementing language and content objectives, adopting children’s literature, using realia, and utilizing cooperative learning strategies, but she was limited by institutional routines (time limits from the school building and classroom structures), and a lack of intentional supports with differentiated mathematics teaching practices to serve each of the els whose needs were different. similarly, mr. mack focused primarily on classroom management by using routines and pacing to keep order. while working at tables of four on worksheets to follow up on the day’s mathematics lesson, some els had difficulty understanding the weight unit and accompanying mathematics word problems (mathematics discourses and goals). however, instead of going over challenging areas of the daily mathematics lesson, mr. mack moved the class on to multiplication games, another routine activity, before the class was over despite the fact that some els did not have the correct answers to the weight unit worksheet. mr. mack missed the opportunity to reteach the mathematics concepts discussed earlier in class due to a focus on institutional routines. this routine was also found in ms. bishop’s second-grade class, as “teacher talk” and teacher-driven activities dominated the instruction most of the time. ms. bishop had a tight classroom structure, with routines and school rules for teamwork and student movement. students, especially els in her classroom, responded to questions only by nodding, with/without thinking, reasoning, and/or questioning, and most of the els did not have correct answers to the problems and were unable to write any word problems. specific differentiated language supports based on els’ english language proficiency levels (e.g., bilingual word bank and having an el with low english language proficiency partnered with an el with high english language proficiency who spoke the same native language and could explain the concept in their shared native language) were not observed. in the examples above, we compared and contrasted the activity theory system for the university etel triangle and the actual school triangle. although goals and tools learned at the university pd were maintained and some good practices were transferred, there were noticeable weaknesses that indicated the need for additional teacher resources, awareness, and intentional support in the teachers’ activity system in order to serve els more equitably and effectively. discussion the purpose of this study was to examine whether or not participating teachers applied the linguistically and culturally responsive mathematics teaching competences when they taught mathematics in their classrooms, especially for els. transferring strategies and pedagogies teachers learn during their university studies to the actual teaching context requires intentional effort and commitment. ms. happ said, song & coppersmith working toward linguistically and culturally responsive math teaching journal of urban mathematics education vol. 13, no. 2 82 “yeah, i was trying to use anchors and sentence stems for each mathematics lesson. i think university helped me become more aware of it.” she continued, “for each lesson, i list the language and content objectives. i prepare content and language objectives because of the university courses. i’m applying many teaching activities i did not know before.” during classroom observations and interviews, we took a closer look at how the three teachers incorporated linguistically and culturally responsive mathematics teaching competences learned at the university el teacher training program into their mathematics classes for els. using mixed methods to analyze the research question, “how did participating urban in-service teachers apply linguistically and culturally responsive mathematics teaching competencies for els learned at a university el teacher training program to their actual mathematics teaching in the classroom?,” we tried to examine, triangulate, blend, and/or integrate the qualitative data to support the research question. we also analyzed these integrated data to examine the juxtaposition between what the teachers had learned at the etel trainings and how they applied the university training to their mathematics teaching in their k–6 classes. when applying the two activity system triangles (engeström, 1987; yamagata-lynch, 2010), the data from observations and interviews revealed that teachers employed lcrmt practices learned in the university training (activity system a) but that they did not fully transfer the lcrmt competences learned at the university into their actual mathematics teaching practices (activity system b). in the collaborative coding process, three themes emerged: a) mathematics-related teaching practices, b) tools to support mathematics learning, and c) teachers’ attitudes and beliefs. in summary, we found that teachers used tools to support mathematics-related and meta[cognitive]-mathematics teaching practices, but they still needed to develop linguistically and culturally responsive mathematics teaching activities and assessments in order to truly help els enhance their mathematics content knowledge and its application as well as their critical thinking skills. we also found that they were constrained by personal, classroom, and school management structures. in addition, although the participating teachers demonstrated use of lcrmt strategies, they did not specifically tailor them to the needs of els, assuming instead that good strategies might work for every learner. in fact, ms. happ mentioned, “i strongly believe that els are learning from my math classes because i am applying all of the etel strategies, i mean lcrmt strategies.” what she was saying was that she was assuming that learning would happen to els without checking if they were actually engaged and learned in her mathematics classes. in reality, deep learning did not consistently happen. the qualitative data exhibited teacher support for students’ engagement through the use of cooperative communication strategies, such as quack-quack, shoulder partners, and phone-a-friend, but teachers should have exhibited their song & coppersmith working toward linguistically and culturally responsive math teaching journal of urban mathematics education vol. 13, no. 2 83 competency and commitment for targeting els with equitable attitudes towards els. in general, all of the three urban in-service teachers applied lcrmt strategies, showing mathematical content, discourse, and pedagogical competences along with procedural and reasoning skills when teaching mathematics in their classrooms. the teachers, however, could have prepared linguistically and culturally responsive mathematics resources and provided more opportunities for els in order to support them intentionally with differentiated language anchors and assessment strategies. the participating teachers showed a shift to becoming more competent in mathematics content and pedagogy in terms of “acquiring knowledge in depth” and “developing procedural demands” but also showed the need for more pd in order to fully examine and develop mathematical discourse in sociocultural contexts where els could feel more comfortable or familiar. in addition, the school system (activity system b) did not fully support the lcrmt competences learned at the university etel trainings (activity system a) because the system demanded its own routinized activities and assessments that did not always match with intentional lcr mathematics teaching strategies for els. conclusion with implications despite the limitations of a small sample size and short duration of research, this study provides meaningful insights for linguistically and culturally responsive mathematics teaching. first, the research process shared a two-dimensional lcrmt framework (table 1) that could be adopted by practicing teachers in their own contexts. second, through data collection and analysis, we compiled narrative examples under the three themes that emerged during the data analysis. finally, this study made a meaningful connection with the activity systems model, illustrating the juxtaposition between the university teacher training and its transfer or non-transfer to the actual classroom context for els. we would like to conclude this paper with implications that might emphasize mathematics competences that move away from mastering a list of mathematical vocabulary words with precise meanings to using collaborative and communicative participation (moschkovich, 2012). for such a change to happen, teachers may need to “recognize and support els to engage with the complexity of language in math classrooms” (moschkovich, 2012, p. 22). these recommendations conflate with the findings of this study; the participating teachers implemented the tools to engage the els, which might help els understand the complexity of mathematical language. the findings also revealed the non-transferred competences at the observation sites. for example, our observational data showed that more non-els participated in lessons and replied to the teachers’ queries than els. unfortunately, the teachers on occasion failed to adequately engage els with mathematical language in a way meaningful to song & coppersmith working toward linguistically and culturally responsive math teaching journal of urban mathematics education vol. 13, no. 2 84 them. materials and professional development should be prepared more explicitly to support teachers with knowledge of when to move from traditional to technical mathematical ways of communication and when and how to develop and implement precise mathematical discourse (schleppegrell, 2007). replication studies are needed to examine how els learn to read and understand different mathematical texts in word problems. when working with els, it is important to distinguish between children who are competent readers in their first languages and those who are not. those who are competent in their first languages are able to read and comprehend not only mathematical texts, but also mathematical word problems, whereas those who are not competent in their first languages may be able to read mathematical texts but may take a longer time to read and understand mathematical word problems (moschkovich, 2012). in line with horn et al.’s (2008) research, the findings of this study highlight the importance of understanding that there are gaps between what teachers are learning when in university and what teachers actually do in the field. we have identified transferred learning and non-transferred gaps in the field of lcrmt for els with suggestions for future professional training and research on this topic. references aguirre, j., zavala, m., & katanyoutanant, t. 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(2012). sub-dimensions of the four-factor model of cultural intelligence: expanding the conceptualization and measurement of cultural intelligence. social and personality psychology compass, 6(4), 295–313. https://doi.org/10.1111/j.1751-9004.2012.00429.x van ek, j. a. (1986). objectives for foreign language learning. manhattan publishing company. world-class instructional design and assessment. (n.d.). wida: can do descriptors. https://wida.wisc.edu/teach/can-do/descriptors yamagata-lynch, l. c. (2010). activity systems analysis methods: understanding complex learning environments. springer. copyright: © 2020 song & coppersmith. this is an open access article distributed under the terms of a creative commons attribution-noncommercial-sharealike 4.0 international license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. the purpose of this study is to examine the role of collaboration between and across school based communities in shifting elementary teachers self efficacy using a reform-based math curriculum journal of urban mathematics education july 2010, vol. 3, no. 1, pp. 82–97 ©jume http://education.gsu.edu/jume iman c. chahine is an assistant professor in the department of middle-secondary education and instructional technology in the college of education, at georgia state university, p.o. box 3978, atlanta, ga 30302; email: ichahine@gsu.edu. her research interests include ethnomathematics, situated cognition, problem solving in nonconventional settings, and multicultural mathematics. lesa m. covington clarkson is an assistant professor in the department of curriculum and instruction at the university of minnesota, 159 pillsbury drive, se, peik hall 230, minneapolis, mn 55455; email: covin005@umn.edu. her research interests include urban mathematics education, the gap in mathematics achievement, and the teaching and learning of problem solving. collaborative evaluative inquiry: a model for improving mathematics instruction in urban elementary schools iman c. chahine georgia state university lesa m. covington clarkson university of minnesota in this article, the authors describe the cyclical process of a collaborative evaluative inquiry project and the data collected throughout the project—data that not only informed ―next steps‖ during the project but also show promise in documenting the benefits of such projects. over a period of 18 months, seven elementary teachers from a k–6 urban elementary school collaborated with university personnel using parsons’s (2002) evaluative inquiry model, a 5-stage, cyclical model that includes defining, planning, and investigating challenges; collecting, analyzing, and synthesizing data; and communicating findings that transpire through collaborative inquiry. overall, the project focused on improving the elementary teachers’ skills of inquiry and, in turn, their mathematics instruction and students’ learning outcomes. the long-term goal was to enhance teachers’ roles in their schools by affording them the opportunities to make informed decisions throughout their teaching based on an effective and skillful use of data. keywords: collaborative inquiry, data-based decision making, mathematics instruction, urban schools ttempts to improve mathematics instruction within school-based communities have become an increasingly prevalent topic in the reform era. stories of successful collaborative endeavors within the discipline of mathematics education between schools and professional development institutions have been reported worldwide (see, e.g., kooper, wagner, breen, & begg, 2003). such promising experiences are inspiring and involve collaborative efforts between teacher educators and k–12 teachers within preservice and inservice teacher education and professional development programs immersed in school contexts. in this article, we aim to assist mathematics teacher educators and elementary teachers in planning and engaging in collaborative projects by describing the cyclical process of a collaborative inquiry project and detailing how data were collected throughout the project—data that not only informed ―next steps‖ during a chahine & covington clarkson collaborative inquiry journal of urban mathematics education vol. 3, no. 1 83 the project but also show promise in documenting the benefits of such projects. while a few studies have empirically investigated the effects of collaboration, still fewer have assessed the fidelity of implementation of such efforts and the possible long-term impact (hagen, gutkin, wilson, & oats, 1998). the absence of evaluative research on collaborative projects often makes it difficult to correctly infer such projects’ actual success in schools. consequently, several researchers have endorsed evaluative research that incorporates mixed methods approaches that aim at assessing quantitatively and qualitatively possible changes in student academic performance and shifts in teachers’ attitudes and beliefs as a result of participating in collaborative projects (gable, mostert, & tonelson, 2004). with the various encouraging findings reported about collaboration in school-based settings, one goal is ubiquitous: to evaluate the success or failure of collaboration in improving students’ learning. collaborative endeavors: a brief review over the past 2 decades, a handful of research projects have explored the issues of collaboration worldwide and across several educational contexts. using a variety of research designs across multiple settings, emphasis has been placed on exploring possible ways through which collaboration influences instruction in the mathematics classroom. for example, nelson (2009) examined the effects of using collaborative inquiry on secondary mathematics and science teachers’ learning when immersed in professional learning communities in a u.s. school. a number of critical questions that addressed potential gaps between a communal vision of student learning and achievement were generated and prospective challenges and difficulties emerged, specifically when questions regarding the teachers’ practices unfolded. southwood and kuiper (2003) examined the experiences of primary grade teachers involved in the mutual support project for encouraging and facilitating collaborative support among teachers in south africa. in this naturalistic and biographical case study, southwood and kuiper highlighted several dimensions that emerged during collaboration, including the dynamics and complexities of interpersonal relationships. in a larger-scale project, nisbet, warren, and cooper (2003) investigated potential ingredients for the success of professional development projects on performance-based assessment in australia. approximately 300 teachers serving as facilitators for their peers were involved in 107 professional development courses delivered as workshops and seminars. although only 10% of the teachers continued as facilitators by the end of the project, nisbet et al. reported a number of successful school-based events that they related to essential characteristics for teachers/facilitators, including teachers’ beliefs and knowledge base, skills in per chahine & covington clarkson collaborative inquiry journal of urban mathematics education vol. 3, no. 1 84 formance assessment, and perceptions of improvement in students’ mathematical performance. huffman and kalnin (2003) described the collaborative professional development efforts of using student achievement data to improve the teaching of mathematics and science in minnesota school districts. an explicit goal of the professional development project was to build partnerships between schools and communities through joint involvement in collaborative inquiry that employed data-based decision making to enhance the academic performance of students. despite the growing body of studies investigating collaborative efforts across educational settings, the majority of empirical research remains somewhat porous. interestingly enough, studies on collaborative efforts that focus on teacher professional development often place greater emphasis on the process of collaboration and less attention toward the outcomes of those efforts (gable, mostert, & tonelson, 2004). the collaborative evaluation communities project the collaborative evaluation communities (cec) project is a national science foundation (nsf) funded project that was created through a leadership collaboration between faculty at the university of kansas and the university of minnesota and aimed at building partnerships between schoolteachers, university professors, and graduate students. the cec project has a number of longand short-term goals. for example, one long-term goal, engaging in collaborative inquiry, aims to encourage teachers to develop the necessary skills as teacher action researchers in their respective classrooms. another long-term goal is to provide teachers with the opportunities (and skills) to effectively use data to make informed decisions regarding instructional challenges. in somewhat similar fashion, one short-term goal aims to support teachers in finding potential solutions to immediate challenges posited by inquiry and to develop possible plans of action to resolve impeding issues that occur in their daily practice. the cec project is currently in its final year. collaborative evaluative inquiry in this section, we describe, in detail, how members of the cec project used evaluative inquiry as the foundation for a collaborative project that focused on improving elementary teachers’ skills of inquiry and, in turn, their mathematics instruction and students’ learning outcomes. a fundamental motivation for the collaborative inquiry project was to support teachers as they built collegial relationships through inquiry: learning from each other in their day-to-day practices, assisting each other in solving teaching problems by sharing craft knowledge, and chahine & covington clarkson collaborative inquiry journal of urban mathematics education vol. 3, no. 1 85 celebrating each other’s successes. over a period of 18 months, seven elementary teachers collaborated with university personnel using parsons’s (2002) evaluative inquiry model, a 5-stage, cyclical model that includes defining, planning, and investigating challenges; collecting, analyzing, and synthesizing data; and communicating findings that emerge through collaborative inquiry. the long-term goal of the project was to enhance teachers’ roles in their schools by affording them the opportunities to make informed decisions throughout their teaching based on an effective and skillful use of data. throughout the project, teachers were continually encouraged to practice and improve their evaluative inquiry skills by examining students’ outcomes and reflecting on their daily teaching practices. although the focus was primarily on students’ learning throughout the inquiry process, instructional experiences were continuously examined by collaborative teams to understand how different practices relate to student attainment of the required mathematics skills elicited in the district’s reform-oriented mathematics curriculum. the context banneker elementary school is a k–6 urban magnet school in the midwest with a focus on academic excellence. according to 2005 school records, the school is comprised of a team of 65 staff members; 26 are licensed teachers with varying degrees and experiences. of the 26 teachers, 38% hold a masters’ degree and have more than 10 years of teaching experience. banneker has been implementing everyday mathematics (university of chicago school mathematics project, 2004); a reform-oriented mathematics curriculum adopted by the school district for grades 1 through 5. banneker serves approximately 350 students, from kindergarten through sixth grade with an average student–teacher ratio of 13:1. though the school is regarded as ―diverse,‖ according to the 2005 demographic data reported by the state department of education, slightly more than 80% of banneker’s students were african american in comparison to less than 30% in the school district and less than 10% in the state. additionally, 20% of the students received special education services in comparison to 17% in the school district and 13% in the state. banneker earned a state rating of a 3 (out of 5) stars for mathematics, meeting the federal accountability requirement during the 2004–2005 academic year. the participants two teams were involved in the project: the action team, which had the lead role, and the evaluative inquiry team, which acted as the support team (parsons, 2002). the action team was comprised of four grade 1 and three grade 2 teachers. one of the 1st-grade teachers taught special education students in a self chahine & covington clarkson collaborative inquiry journal of urban mathematics education vol. 3, no. 1 86 contained classroom; mainstreamed special education students were present in the other classrooms included in the study. forty-six grade 1 and 52 grade 2 students filled the classrooms. the primary role of the action team was to identify potential challenges that impede students’ learning and to design and implement sets of learning experiences that address these challenges as well as help achieve intended learning outcomes. the evaluative inquiry team included two university professors, one from mathematics education and the other from educational psychology, and several graduate students in mathematics or science education. responsibilities of the evaluative inquiry team included facilitating the entire inquiry process and providing step-by-step feedback to the action team by analyzing student assessment data, observing mathematics lessons, suggesting possible interventions, and reviewing literature for relevant information. the process the evaluative inquiry model (parsons, 2002) cycle involved five basic tasks delivered chronologically over an 18-month period. the tasks or stages include: position the inquiry, plan the inquiry, collect data, analyze and synthesize data, and communicate findings (parsons) (see figure 1). at each stage, data were collected and analyzed in an attempt to hypothesize and test assertions that surfaced during the cycle. figure 1. evaluative inquiry model (parsons, 2002) task 1: position the inquiry. this stage involved basic orientation for both teams by defining roles and responsibilities within and across teams, brainstorming needs, and developing clear challenge statements to be investigated. a fivescale collaborative evaluation community survey was given to the participating teachers at the beginning of the project. this survey consisted of 41 items and was designed to examine teachers’ initial attitudes and beliefs toward different instruc chahine & covington clarkson collaborative inquiry journal of urban mathematics education vol. 3, no. 1 87 tional practices and various aspects of school climate. the same survey was administered at the end of the 18-month period and preand post-data were compared to assess the impact of the project on teachers’ attitudes, beliefs, and behaviors. over 4 months, both teams attended regularly scheduled meetings. in addition, the evaluative inquiry team regularly visited the school and observed participating teachers’ classrooms. during the meetings, the action team presented preliminary statements of the curriculum challenges to be investigated and delineated potential inquiry plans. such challenges included: determining appropriate pacing of the mathematics curriculum, prioritizing student attainment of learning goals, motivating students to learn mathematics, and meeting the district’s pacing goals by so on and so forth. an important outcome of this stage is the portrayal of an action inquiry map (aim) (parsons, 2002) that includes a clear statement about the theme and target of inquiry. the theme of inquiry, developed collaboratively, was students’ low mathematics achievement and the target of inquiry was students’ attainment of shortand long-term learning outcomes (also known as secure skills in the everyday mathematics curriculum) on school and state-based assessments (see figure 2). figure 2. evaluative inquiry model – task 1: position the inquiry. chahine & covington clarkson collaborative inquiry journal of urban mathematics education vol. 3, no. 1 88 task 2: plan the inquiry. the second task was initiated with a 2-day workshop organized and presented by the evaluative inquiry team. the workshop provided the opportunity for the teams to set the stage for planning the inquiry by revisiting previous efforts, reviewing analysis of teachers’ surveys, and finalizing the challenge statements. tasks and timelines were also developed and decisions on what data were needed and which instruments to be used were discussed and agreed upon by both teams in preparation for the next task (see figure 3). figure 3. evaluative inquiry model – task 2: plan the inquiry. task 3: collect data. building on the workshop recommendations, task 3 was initiated. this task involved collecting data that provided sufficient information on students’ performance, which, in turn, helped establish the basis for informing decisions regarding planning new learning experiences and developing different teaching strategies (i.e., interventions). during the project, we employed a mixed methods research design. quantitative data included students’ preand post-assessment scores as well as teachers’ responses to survey questionnaires. qualitative data included audio-taped and transcribed students’ responses to questions asked during semi-structured clinical interviews, field notes of classroom chahine & covington clarkson collaborative inquiry journal of urban mathematics education vol. 3, no. 1 89 observations, videotaped classroom interactions during the implementation of interventions, teachers’ written explanations and reflections on videotaped classroom interactions, and audio-taped conversations within and across teams’ focus group discussions. in collecting a combination of quantitative and qualitative data throughout the project, the often-discrete separation between data collection and data analysis collapses in favor of a more integrated cycle of actions and reflections that informs and documents progress. in other words, these multiple data sources not only informed next steps during the project but also show promise in triangulating findings when documenting the benefits of the project. it is important to note that, in the discussion that follows, we are not reporting conclusive findings but rather demonstrating how multiple data sources might inform collaborative projects and show promise in documenting the benefits of such projects. task 4: analyze and synthesize data. a data collection and analysis subcycle was developed for each quarter of the academic year. this cycle motivated an interrelated chain of actions and reflections based on the data that evolved as a result of incorporating the evaluative inquiry model (parsons, 2002) in the daily teaching practices of the participating teachers. this sub-cycle was implemented in five stages (see figure 4). figure 4. evaluative inquiry model – task 4: analyze and synthesize data. chahine & covington clarkson collaborative inquiry journal of urban mathematics education vol. 3, no. 1 90 stage one. the first stage included assessing students’ prior knowledge of the secure skills that were required for the designated quarter. written pretests were prepared by the evaluative inquiry team and were administered for both grades. the pretest for grade 1 consisted of 12 items extracted from the everyday mathematics assessments and focused on the six secure skills required for the first quarter of grade 1. the pretest for grade 2 had 18 items and focused on the 13 secure skills required for the first quarter of grade 2. stage two. this stage involved examining students’ performance on the written pretests. pretests for both grades were corrected, scored, and percentages of correct answers were computed for each grade and for each classroom. students’ scoring data were organized and represented in bar graphs. both teams analyzed data representations that indicated the overall results of students’ scores for all classes within the same grade, and individual classroom graphs. stage three. the third stage involved implementing the collaboratively planned intervention derived from the analysis of students’ scoring data gathered in the second stage. members of the action team delivered the intervention during their regular instruction while a member of the evaluative inquiry team observed the lesson. separate intervention activities were developed for each grade. the intervention for grade 1 targeted basic secure skills on money concepts such as showing money with coins; exchanging and using fewer coins; and finding amounts of money using pennies, nickels, dimes, quarters, and so forth. the grade 2 intervention included enrichment activities for measuring length to the nearest inch and ½ inch and to the nearest centimeter and ½ centimeter, using a ruler to measure a specified length in both inches and centimeters. stage four. in this stage, analyses of students’ learning outcomes after the interventions and teachers’ implementation of the interventions were conducted. an evaluative inquiry was carried out on three tiers: students’ learning outcomes in each quarter, teachers’ levels of implementation of the intervention, and the relationship between students’ learning and teachers’ levels of implementation. student learning. an analysis of students’ learning outcomes was conducted on students’ scores on preand post-tests within and across the four quarters. descriptive bar graphs were used to represent data. students’ learning outcomes were computed by calculating percentages of correct answers, percentages of partially correct answers, and percentages of incorrect answers within and across the classes for each grade level. results of the analyses of students’ scores on the pretests varied across classes and across grade levels. of the six secure skills pretested in grade 1, students scored lowest on counting and exchanging money skills. money skills also seemed to be a stumbling block for grade 2 students. of the 13 skills pretested in grade 2, the success rate for identifying the correct amount of money was only 8%. chahine & covington clarkson collaborative inquiry journal of urban mathematics education vol. 3, no. 1 91 based on results of the pretests, a theme for inquiry was initiated and both teams set the stage for providing different learning experiences that might facilitate and support students’ understanding of the secure skills related to money concepts. to obtain a deeper understanding of the obstacles that students faced in acquiring these skills, semi-structured clinical interviews were conducted with a random sample of four students from each grade level. these clinical interviews provided qualitative data on how students approached problems related to money concepts. students were provided with manipulatives (i.e., coins), and without using paper and pencil they were asked to ―talk aloud‖ and explain their reasoning as they solved money problems. some of the questions posed during the clinical interviews were identical to the items on the written pretest. the clinical interviews were audio-taped and transcribed for further analysis. during the analysis of the transcribed clinical interviews by members of both teams, one item of particular interest became clear, namely that students performed significantly better on the oral clinical interviews than on the written pretests. this discovery seemed to imply that students’ had acquired the skills to identify, use, and exchange coins in the ―real world,‖ but lacked the skills to manipulate written symbols. based on the pretest results and findings from the clinical interviews, both teams collaboratively planned a number of interventions to help address gaps on a selected number of secure skills related to money concepts. teacher implementation. the analysis of a particular teacher’s level of implementation of intervention was based on four main data sources: a videotape during the teacher’s implementation of the intervention within her respective classroom, the teacher’s explanations and reflections on the videotape, field notes written by a member of the evaluative inquiry team during the implementation, and within and across teams’ focus group discussions of potential improvements to the observed implementation that might lead to ―best practice‖ instruction on the designated unit or lesson for grades 1 and 2. the rationale behind assessing the level of implementation was to provide some evidence on the extent to which teachers were committed to the outcomes of collaboration and the value they bestow to the inquiry process. as noted, to assess the level of teacher implementation, classroom interactions were videotaped during teachers’ implementations of the interventions. each teacher was then provided with a copy of the videotape from her respective classroom and asked to complete a reflection form to provide insight on how the intervention was delivered and what might be done in the future to further improve the intervention. qualitatively, the videotaped classroom interactions were enhanced as each teacher was provided with the opportunity to explain and reflect on her implementation of the intervention from her perspective. the videos from all participating teachers were then systematically viewed by the evaluative inquiry team for evidence of a teacher’s level of implementa chahine & covington clarkson collaborative inquiry journal of urban mathematics education vol. 3, no. 1 92 tion of intervention in addition to students’ participation and engagement during the lesson. the levels of implementation were classified as high, medium, and low. teachers who engaged students in classroom discourse as a means of scaffolding students’ knowledge during the intervention and used the pre-planned activity sheets explicitly with students were rated as ―high‖ implementers, those who only gave students the activity sheets to work on their own without teacher’s scaffolding were rated as ―medium‖ implementers, and those who proceeded with the lesson without engaging students in classroom discourse nor using the activity sheets were rated as ―low‖ implementers. an analysis of teachers’ explanations and reflections by the evaluative inquiry team revealed two major assertions that are important to note. these assertions illustrate instances in which teachers alter their decisions regarding teaching a specific concept by negotiating alternative strategies that improve the lesson delivery. the first assertion is that, in most cases, teachers exhibited a significant shift from ―explaining away‖ or defending their practices to openly reflecting on and considering alternative ways to otherwise use the intervention in more meaningful ways. when asked about what struck her most after watching her videotaped lesson, a grade 1 teacher commented: i do not believe that all the children were engaged as much as they could have been…. i really didn’t tie the game that was to be played very well with the mini lesson…. everything seemed like an unrelated skill. there did not seem to be a connection to how every coin or dollar fits in to make money. and when asked what to do if given a chance to re-teach the videotaped lesson, a grade 1 teacher wrote: ―not spending so much time on counting and recounting by 25. prepare coins in individual bags so it does not take so long to pass out coins.‖ the second assertion is that, in general, teachers’ expectations and belief in students’ ability to understand and engage in thought-provoking situations increased. a grade 2 teacher was surprised at ―how much extra hands-on experiences students were able to do, using the model ruler for this specific skill.‖ similarly, a grade 1 teacher said, ―[students] do seem to be thinking and helping each other to find a solution to exchanging nickels for dimes.‖ another grade 1 teacher noted: ―students were attentive even though they were sitting for a long time. i did not raise my voice once, amazing!‖ relationship between student learning and teacher implementation. to analyze the link between learning experiences and students’ outcomes, matched comparisons between teacher’s level of implementation and gains of students’ preand post-test scores were conducted (see figure 5). chahine & covington clarkson collaborative inquiry journal of urban mathematics education vol. 3, no. 1 93 figure 5. relationship between learning and implementation. simple regression and correlation analyses were undertaken to provide explanation for the variation in grade 1 and grade 2 post-test scores by varying (a) level of implementation, (b) secure skills per grade, and (b) years of teaching experience. an analysis of the quantitative data collected from classroom observations and students’ scores on preand post-tests revealed a moderate positive pearson correlation (r = 0.468) between students’ gains and teachers’ levels of implementation. the results suggest that a high level of implementation elicited greater gains in students’ scores on the post-tests in both grades (see table 1). table 1 pearson correlation between students’ correct answers on post-tests and teachers’ level of implementation n sig. (2-tailed) level of implementation correct answers 39 .003 .468** note. *p<.01, **p<.001 chahine & covington clarkson collaborative inquiry journal of urban mathematics education vol. 3, no. 1 94 more interestingly, a simple regression analysis using teachers’ demographic data showed that around 49% of the gains in students’ correct scores on secure skills can be attributable to the teachers’ level of implementation and years of experience (see table 2). table 2 anova for the regression equation: students’ correct answers and teachers’ level of implementation and teachers’ years of experience sum of squares df mean square f regression 12771.726 3 4257.242 11.341** residual 13138.040 35 375.373 total 25909.766 38 note. **p < 0.01 furthermore, when testing the relationship between the percentage of correct answers and years of teaching experience, a moderate negative correlation of -0.502 was noted; that is, more than 25% of the variation in percentage of correct answers can be attributed to teachers’ years of teaching experience (see table 3). table 3 pearson correlation between students’ correct answers on post-test and teachers’ years of teaching experience n sig. (2-tailed) years of experience correct answers 39 .001 -.502* note. *p<.01 moreover, a moderately negative correlation (r = -0.359) suggests that around 13% of the change in the level of implementation is accounted for by the change in teaching experience (see table 4). table 4 pearson correlation between teachers’ levels of implementation and teachers’ years of experience sig. (2-tailed) years of experience level of implementation .025 -.359* note. *p<.05 chahine & covington clarkson collaborative inquiry journal of urban mathematics education vol. 3, no. 1 95 stage five. the last stage in the data collection and analysis sub-cycle involved communicating and sharing reflections between the teams. suggestions for best practice on teaching certain secure skills were brainstormed and recommendations for a next cycle were examined. a post-pretest was administered at the beginning of the next spiral of data collection and items from the original pre-test were included in the post-test in addition to new items that were designed to pretest the secure skills for the next quarter. this five stage data collection and analysis sub-cycle was repeated for the second, third, and fourth quarters. throughout the academic year, a total of 11 secure skills were preand post-tested for grade 1 and a total of 38 secure skills were tested for grade 2. task 5: communicate findings. the evaluative inquiry model (parsons, 2002) was concluded in a 1-day workshop that was held in the summer of the second year of the project and involved communicating findings and the experiences of the action team and the evaluative inquiry team that emerged as a result of immersion in the evaluative inquiry project (see figure 6). the immediate purpose of the workshop was for both teams to explore possible challenges and opportunities for initiating the next inquiry cycle by possibly extending it to include other grade levels (e.g., grade 3), or to examine other subjects (e.g., science, social studies, language arts). the long-term goal of the workshop was to instigate and encourage future collaborative efforts between teachers and to help build confidence in their ability as action researchers by integrating the evaluative inquiry model cyclical process into their lesson planning and daily instruction. concluding remarks while it was tempting to focus on common tasks of being involved in inquiry as the essence of collaboration, it was those relationships that evolved among teachers as collaborators that sustained the work. shared values, beliefs, and goals about the nature of inquiry forged strong collegial bonds among members of the inquiry teams and fostered mutual relational trust. throughout the 18month duration of the project, the action team and evaluative inquiry team worked collaboratively on building partnership between the school community and concerned university personnel. this joint venture was expressed by submitting proposals on the project’s achievements to be presented at national conferences to inspire other teachers and teacher educators to experiment and venture into collaborative evaluative inquiry—a step towards building capacities of teachers to become action researchers in their own classrooms. chahine & covington clarkson collaborative inquiry journal of urban mathematics education vol. 3, no. 1 96 the project now is in its final year. new teachers have come aboard, new spirals were studied and new challenges were investigated. enthusiasm has grown among teachers at banneker elementary school to pursue further evaluative inquiry techniques in enhancing the teaching and learning in the mathematics instruction. despite some difficulties that teachers faced across different stages of the project, the moral was high and there was willingness to extend the use of this approach to teaching other subjects and to other grades as well. amidst all the challenges that teachers and students face in this reform era, it is essential to expose teachers to alternative approaches that self-empowers them as decision makers and problem solvers in their classrooms and beyond. figure 6. evaluation inquiry model: task 5: communicate findings. references gable, r., mostert, m. p., & tonelson, s.w. (2004). assessing professional collaboration in schools: knowing what works. preventing school failure, 48(3), 4–8. hagen, k. m., gutkin, t. b., wilson, c. p., & oats, r. g. (1998). using vicarious experience and verbal persuasion to enhance self-efficacy in pre-service teachers: ―priming the pump‖ for consultation. school psychology quarterly, 13, 169–178. huffman, d., & kalnin, j. (2003). collaborative inquiry to make data-based decisions in schools. teaching and teacher education, 19, 569–580. kooper, a. p., wagner, v. s., breen, c., & begg, a. (eds.). (2003). collaboration in teacher education: examples from the context of mathematics education. dordrecht, the netherlands: kluwer. chahine & covington clarkson collaborative inquiry journal of urban mathematics education vol. 3, no. 1 97 nelson, t. h. (may, 2009). teachers’ collaborative inquiry and professional growth: should we be optimistic? science education, 93, 548–580. nisbet, s., warren, e., & cooper, t. (2003). collaboration and sharing as crucial elements of professional development. in a. p. kooper, v. s. wagner, c. breen, & a. begg (eds.), collaboration in teacher education: examples from the context of mathematics education (pp. 23–40). dordrecht, the netherlands: kluwer. parsons, b. (2002). evaluative inquiry using evaluation to promote student success. thousand oaks, ca: sage. southwood, s., & kuiper, j. (2003). a journey towards collaboration. in a. p. kooper, v. s. wagner, c. breen, & a. begg (eds.), collaboration in teacher education: examples from the context of mathematics education (pp. 7–22). dordrecht, the netherlands: kluwer. university of chicago school mathematics project (2004). everyday mathematics: teacher’s reference manual (2nd ed.). chicago: mcgraw hill. journal of urban mathematics education july 2015, vol. 8, no. 1, pp. 14–16 ©jume. http://education.gsu.edu/jume christopher c. jett is an assistant professor in the department of mathematics in the college of science and mathematics at the university of west georgia, 1601 maple st., carrollton, ga 30118; email: cjett@westga.edu. his research interests are centered on employing a critical race theoretical perspective to mathematics education research, particularly, at the undergraduate mathematics level. he is the current book review editor of the journal of urban mathematics education. editorial an urban mathematics education book review?: considerations for jume book review authors christopher c. jett university of west georgia some books are to be tasted, others to be swallowed, and some few to be chewed and digested. – francis bacon, 1625 eading books can be an enriching experience. there are a growing number of books devoted to many topics in the urban mathematics education domain and beyond. as an intellectual enterprise, urban mathematics education borrows from many intellectual traditions. therefore, it is critically important for urban mathematics education that researchers and scholars who have chosen to work in this domain read across genres, disciplines, and traditions. such cross-disciplinary reading exponentially grows the number of books that have potential to influence the field. the aforementioned francis bacon epigraph suggests that books often fall into three categories for readers. my remarks here are for those who seek to “chew and digest” books that speak, both directly and somewhat indirectly, to urban mathematics education. in this editorial, i highlight some important points for potential book review authors to consider when submitting a manuscript to jume. first and foremost, authors must pay attention to urban mathematics education. i encourage book review authors to read tate (2008) to consider the importance of positioning urban in mathematics education scholarship, in particular, and milner (2012) to consider evolving conceptualizations of urban education, in general. of course, book review authors should consider the customary elements associated with a book review such as understanding the context (i.e., cultural, historical, political, racial, social, and so on) from which the book was written, highlighting special features of the book, and providing an overview of its possible contributions to the field, to name a few. my point here is for authors to consider the frontiers of urban scholarship when outlining, brainstorming, and drafting a jume book review. r http://education.gsu.edu/jume mailto:cjett@westga.edu jett book review journal of urban mathematics education vol. 8, no. 1 15 second, authors should personalize, contextualize, and problematize their identities, experiences, and ideologies with respect to the text. it might be appropriate for authors to do this in the beginning of the book review to “hook” or “grab” the reader’s attention, in the middle when justifying what might be important or lacking in the book, or near the end to link the arguments put forth in the book to their own experiences. the urban domain encompasses many facets; understanding the worldview(s) of the writer of the book review in conjunction with the text helps to shed light on the salient features that the text might provide urban mathematics education research, teaching, and learning. for example, in my jume book review of danny martin’s (2009) edited volume mathematics teaching, learning, and liberation in the lives of black children (see jett, 2009), i shared my epistemological and pedagogical stances as an african american male scholar and researcher. i used my positionality as a means to connect with those who i thought would benefit from the book. i encourage jume book review authors, too, to reflect and to recognize where they stand in regards to issues raised in the text. lastly, authors should be willing to critique the text as the pages of jume are intended to push the boundaries of canonical scholarship while illuminating urban excellence (matthews, 2008). being critical is essential, and engaging in the iterative process of critical thinking should urge authors “to think long, hard, and critically; to unpack; to move beyond the surface; to work for knowledge” (hooks, 2010, pp. 9–10). i challenge authors to think critically about the messages the book sends to, for, and about urban students, teachers, parents, communities, and ultimately, to the disciplinary field of urban mathematics education. therefore, be sure to critically evaluate the book and its associated merits (and lack thereof), especially through an urban mathematics education lens. as the jume book review editor, i invite potential authors to submit manuscripts that advance the intellectual enterprise called urban mathematics education. we, the jume editorial team, especially wish for the book review section to be a space for graduate students and early career scholars to grapple with the possibilities, challenges, and opportunities that might arise from reading and engaging with a particular text. as previously noted, there has been exponential growth in published books that have the potential to positively influence how the larger mathematics community thinks about urban research and scholarship and urban teaching and learning. a book review is a great way to introduce the larger community to different ideas. in closing, i leave potential authors with one final consideration. adler and van doren (1972/2014) contend: “every book has a skeleton hidden between its covers. your job as an analytical reader is to find it” (p. 75). in keeping with the sentiments of that statement, i challenge book review authors to find those urban mathematics education skeletons and bring them to the fore in their book reviews in critical and unique ways. jett book review journal of urban mathematics education vol. 8, no. 1 16 references adler, m. j., & van doren, c. (2014). how to read a book: the classic guide to intelligent reading (touchstone ed.). new york, ny: simon & schuster. (original work published 1972) hooks, b. (2010). teaching critical thinking: practical wisdom. new york, ny: routledge. jett, c. c. (2009). mathematics, an empowering tool for liberation?: a review of mathematics teaching, learning, and liberation in the lives of black children. journal of urban mathematics education [review of the book mathematics teaching, learning, and liberation in the lives of black children. journal of urban mathematics education], 2(2), 66–71. retrieved from http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/48/21 martin, d. b. (ed.). (2009). mathematics teaching, learning, and liberation in the lives of black children. new york, ny: routledge. matthews, l. e. (2008). illuminating urban excellence: a movement of change within mathematics education. journal of urban mathematics education [editorial], 1(1), 1–4. retrieved from http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/20/9 milner, h. r. (2012). but what is urban education? urban education, 47(3), 556–561. tate, w. (2008). putting the “urban” in mathematics education scholarship. journal of urban mathematics education, 1(1), 5–9. retrieved from http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/19/2 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/48/21 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/20/9 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/19/2 microsoft word author+approved+jansen+vol+16+no+1.docx journal of urban mathematics education june 2023, vol. 16, no. 1, pp. 10–39 ©jume. https://journals.tdl.org/jume amanda jansen, ph.d., is a professor in the mathematics education program in the school of education at the university of delaware, willard hall, newark, de 19716; email: jansen@udel.edu. she has a joint appointment in her university’s department of mathematical sciences. her current research interests focus on characterizing mathematics teaching practices that are engaging for learners. the members of the center for inquiry and equity in mathematics are mollie appelgate, kristen bieda, martha byrne, theodore chao, jon d. davis, teresa k. dunleavy, maisie gholson, steven greenstein, frances k. harper, pamela harris, hanna haydar, naomi jessup, crystal kalinec-craig, alison marzocchi, marrielle myers, terrance pendleton, randolph philipp, paulo tan, preety tripathi, eva thanheiser, oyita udiani, shanise walker, jennifer a. wolfe, and kamuela e. yong, with facilitation team members: miriam gates, amanda jansen, anne marie marshall, sarah sword, aris winger, and michael young. entangling and disentangling inquiry and equity: voices of mathematics education professors and mathematics professors amanda jansen university of delaware center for inquiry and equity in mathematics this study describes how mathematics education professors and mathematics professors conceptualize relationships between inquiry and equity. after participating in a week-long summer institute, 24 mathematics education professors and mathematics professors were interviewed twice (initial interview and member check). then, participants engaged in coanalysis and co-writing to construct a framework that provides structure to the complex set of stances about how equity and inquiry intertwine. this framework, which extends the work of tang and colleagues (2017), illustrates ways that the process of inquiry could be more equityminded (equity in inquiry) and shows that inquiry could be conducted to seek outcomes of greater equity (inquiry for equity). findings also illustrate that equity opportunities, challenges, and tensions are always present in inquiry. in addition, this study illustrates the value of (and a process for) mathematics education professors and mathematics professors to work together to understand each other’s perspectives during collaborations. keywords: mathematics, inquiry, equity how can inquiry be equity-minded? inquiry is the activity of posing a problem (silver, 1994) or asking a personally meaningful question (brown & walter, 2005) and pursuing an answer to it. mathematics education professors and mathematics professors could value inquiry for at least two professional reasons: (a) as an approach to teaching and learning and (b) for their research and scholarship. as we engage in inquiry, equity opportunities and dilemmas appear. mathematics can be learned through inquiry. in the context of mathematics classrooms, teachers can promote inquiry through approaches such as guided reinvention (freudenthal, 1973), discovery-learning experiences (goldin, 1990), or problem-based learning (roh, 2003). inquiry instruction can have multiple benefits for learners, including developing meaningful understandings of mathematics (goldin, 1990), productive dispositions among learners (e.g., self-efficacy in mathematics; cerezo, 2004), or powerful identities (melville et al., 2013). inquiry & equity 11 as we practice inquiry during collaborative research or in the context of teaching, we may strive to be equity-minded. according to tang and colleagues (2017), teaching mathematics through inquiry could foster greater equity. they aligned inquiry practices in mathematics classrooms with gutiérrez’s (2012) four dimensions of equity: access, achievement, identity, and power. they asserted that when inquiry teaching practices align with active learning (e.g., promoting student ownership, student generation of knowledge, and peer involvement through collaboration), all four of these dimensions of equity can be addressed. if inquiry is not intentionally enacted to be more equity-minded, inequities could be perpetuated. a contribution of this paper is a framework of relationships between inquiry and equity. this framework provides structure for thinking about the complex practice of inquiry and how it can (or might not be) more equity-minded. this analysis extends the work of scholars who have considered the role of equity in mathematics inquiry (tang et al., 2017) and supports others who want to foster inquiry in ways that are more equity-minded. we employed a co-writing methodology (e.g., manning, 2018) for this analysis. dialogue and collaboration between mathematics education professors and mathematics professors can have multiple benefits, including improvements in undergraduate education. this collaborative work is one example of such dialogue and collaboration. thus, another contribution of this study is the co-writing methodology that we describe in this paper. research context what happens when a group of mathematics education professors and mathematics professors come together to engage in inquiry? what if members of this group of professors also express a commitment to equity that they live out in their professional work? we learned that one important step is to seek to understand the range of perspectives held among group members. specifically, we sought to understand different ways that we thought about how equity intertwined with engaging in inquiry. twenty-four professors attended a summer institute to launch the center for inquiry and equity in mathematics. the institute was facilitated by a team of six. the facilitation team included representation from mathematics education and mathematics programs. mathematics education professors were recruited to apply to attend the institute, and 19 participants were selected from applications. below, we share a quotation from the recruitment handout. we know many mathematics education faculty have thought deeply about issues of equity and about what mathematics is; however, we believe that these issues are so core to the work of mathematics education that we as mathematics educators need spaces for continued growth. this program is for those mathematics educators who wish to continue to interrogate what mathematics is and who is welcome to do it – in community with other educators and in service of empowering future teachers to do the same work. five mathematics professors, invited among those in the mathematicians of color alliance, also participated in the institute. across the following school year, the mathematics professors interacted with the mathematics education professors and collaborated to enhance learning experiences for undergraduate teacher candidates. participants conducted inquiry during the summer institute. inquiry projects addressed questions about sociopolitical issues (mathematics of gerrymandering and environmental justice), mathematics underlying how games are played (gambling [craps] and card games or inquiry & equity 12 board games [spot it!]), their wonderments (whether or when 0.999 repeating is or is not equal to one and understanding happy numbers), or questions their students had posed about mathematical relationships (understanding why descartes’ rule of signs works intuitively). participants reflected together about ways that equity was experienced (or not) in their inquiry during the summer institute. during the institute, this group of mathematics education professors and mathematics professors expressed a taken-as-shared understanding that inquiry and equity cannot be separated. we then engaged in ongoing work together, during the institute and beyond, to unpack that idea because people seemed to agree for different reasons. this paper is a product of our ongoing reflection on how inquiry and equity intertwine. in this paper, we also seek to provide insight for those who would like to build collaborations between mathematics education professors and mathematics professors. when we seek to understand one another, we have a place to begin. the process of trying to understand one another is an equity-minded practice. we avoid making assumptions about each other. we put ourselves in each other’s shoes. we open our minds to multiple viewpoints. (the use of “we” throughout this paper refers to our collective perspective, the first author and the participants in this project, with whom the first author endeavored to co-write this paper [e.g., manning, 2018].) the co-writing methodology describes a process for learning to understand one another, and the framework we present offers what we learned from each other about the range of ways that inquiry and equity can intertwine. inquiry and equity in urban mathematics education inquiry as a process of teaching and learning has long been a part of the history of mathematics education reform, such as learning mathematics through inquiry and problem solving (national council of teachers of mathematics [nctm], 2000). the combination of inquiry as an activity and inquiry in mathematics instruction is described below by staples (2007): inquiry is a practice or stance and indicates a particular way of engaging with and making sense of the world…. inquiry into mathematics involves delving into mathematical ideas and concepts and trying to understand the structure, power, and limitations of mathematics. inquiry with mathematics involves using mathematics as a tool to make sense of problem situations and come to some reasonable resolution…. learning results from, and is evidenced by, student participation in both standard disciplinary practices (e.g., justifying, representing algebraically) and an array of other practices of mathematical communities (e.g., questioning, communicating, informal reasoning). (p. 163) the process of inquiry involves learning new mathematics through the activity of engaging in novel tasks, making conjectures, engaging in reasoning, and justifying (rasmussen & kwon, 2007). learners’ ideas are central to learning and developing understanding in inquiry-oriented instruction (kuster et al., 2018). rationales for engaging learners in inquiry have included providing learners access to practices of mathematicians (lampert, 1990), which can support the development of productive identities, or providing access to opportunities to make sense of mathematical concepts (goldin, 1990). whether we view inquiry as promoting equity (or not) depends greatly upon how equity is defined as well as how inquiry is enacted. equity has long been expressed to be a commitment in the field of mathematics education, but the commitments are enacted in various ways. the first guiding principle – equity – in the nctm’s (2000) principles and standards for school mathematics reads, “excellence in inquiry & equity 13 mathematics requires equity – high expectations and strong support for all students” (p. 11). however, this is only one way to view equity in mathematics teaching and learning. gutiérrez (2012) described four dimensions of equity in mathematics teaching and learning – two along a dominant axis (access and achievement) and two along a critical axis (identity and power). the dominant axis refers to commonly held perspectives of providing access to particular learning conditions (e.g., material or human resources, particular learning experiences, or certain mathematics practices) and considering whether achievement, or student outcomes, differ when groups of students are disaggregated by race, gender or sex, socioeconomic class, or by any marginalized group in comparison to the majority (gutstein et al., 2005). regarding access, inquiry-oriented teaching practices could include providing learners with access to opportunities to engage in higher order reasoning and justification (mueller & maher, 2010). perhaps some might value inquiry if it reduces differences in achievement between groups, but perhaps inquiry also has value for achievement when new forms of knowledge can be constructed. gap gazing (gutiérrez, 2008) or examining differences in achievement between groups of students does not question testing systems that generate the outcomes. the critical axis of equity directs attention to how opportunities to engage in doing and learning mathematics can create or shut off possible identities and can have the power to create change, transform society, address injustices, and transform the discipline of mathematics itself. according to boaler and greeno (2000), “students do not just learn mathematics in school classrooms, they learn to be…” (p. 188) mathematics learners. inquiry practices in school affect how students see themselves (and how others see them) as doers of mathematics (aguirre et al., 2013). school practices can narrow boundaries of ways of thinking (de freitas & sinclair, 2020; popkewitz, 2004) or widen them, which can constrain or open possibilities for identities to be enacted and valued. mathematics can be used to investigate sociopolitical issues as a tool toward change (felton-koestler, 2020). through inquiry, students can learn to “read the world” (gutstein, 2003, p. 44) as they use mathematics to learn about social forces that contribute to the marginalization of some people over others (martin et al., 2010). the multiple conceptions of equity provide different insights into whether and how inquiry can be equity-minded. inquiry instruction and efforts toward equity efforts to enact inquiry-oriented instruction may still fall short of achieving equity goals. in a study of an inquiry-oriented abstract algebra course at the undergraduate level (johnson et al., 2020), women did not perform as well as men on an outcome assessment. the differences were attributed to differences in women’s participation rates across classes (reinholtz et al., 2022), which highlights the importance of considering how differential opportunities to participate are constructed in inquiry-oriented teaching. in his research on african american parents’ experiences with mathematics, martin (2006) used the parents’ voices to illustrate the nature of racialized experiences in mathematics learning. the parents spoke of “how the socially constructed meaning for race comes to be a deciding factor in who gets to do mathematics and who does not” (p. 223). an important factor in equity-minded inquiry is attending to who has the opportunity to participate and in what ways. rubel (2017) investigated mathematics teaching practices designed to address equity and identified that some teachers struggled to enact teaching practices that related to identity and power. she argued for greater attention to race and political knowledge in mathematics teacher preparation. mintos et al. (2019) reported that secondary mathematics teacher education inquiry & equity 14 programs do not always address critical dimensions of equity (identity and power), focusing instead on access and achievement. a greater awareness of the role of equity in inquiry involves attention to critical dimensions of equity. urban mathematics education as location whether relationships between inquiry and equity are relevant for urban mathematics education depends on how we conceptualize urban mathematics education. this phrase can refer to a geographic location, specifically a densely populated region with a unique context of human and cultural diversity in race, ethnicity, language, socioeconomic class, religion, disabilities or abilities, sexual orientation, and gender expression (stinson, 2020) as well as its intersections. we recognize that urban has historically been used as coded language for people of color and for people with fewer economic resources (shah et al., 2021). we do not conceptualize urban in this way, nor do we conceptualize it as monolithic. we align with martin et al.’s (2010) assertion of the importance of analyzing urban mathematics education within its complex social, historical, and political contexts. understanding relationships between inquiry and equity is also relevant when we work with and in urban communities because it is important to consider who may be served or not be served in inquiry practices. as we strive for anti-racist and anti-oppressive practices in mathematics education, we aim to serve communities that have been historically underserved and aim to dismantle systems that have allowed racism and oppression to perpetuate. following larnell and martin (2021), we “envision a form of mathematics education that is worthy of those who encounter it, unlike traditional forms of mathematics education that are fundamentally structured to convey privilege to those few students deemed worthy” (p. 358). we view urban spaces as sites where sophisticated and authentic inquiry does and can occur rather than viewing urban contexts through a deficit lens (cf., larnell & bullock, 2018). facilitators of inquiry, including those who work in urban settings, would benefit from awareness of equity challenges and opportunities so that mathematics education focuses on “increasing their opportunities for success without undermining their cultural practices” (larnell & bullock, 2018, p. 52). urban mathematics education as a political project urban mathematics education is a political project focused on envisioning anti-racist and anti-oppressive spaces of mathematics teaching and learning (cf., larnell & bullock, 2018; larnell & martin, 2021; martin & larnell, 2018). our engagement in this work of conceptualizing equity in inquiry is a process of engaging in such work because we collectively shared and developed our political knowledge of mathematics teaching and learning (gutiérrez, 2013). following gutiérrez’s (2013) description of political conocimiento, we worked to deconstruct deficit discourses when engaging in inquiry and engaged in our inquiry to build solidarity with and commitment to our students. according to larnell and martin (2021), the political project of urban mathematics education research incorporates the use of research methodologies that consider how knowledge is constructed and potentially rejects dominant traditions of social science research. antioppressive research “interrogates the logic models of knowledge production” (larnell & martin, inquiry & equity 15 2021, p. 356). in our methodology section of this paper, we elaborate upon how we designed our research activity to co-construct knowledge as a community of scholars. mathematics education professors’ and mathematics professors’ perspectives on equity researchers have begun to address a call from mcleman and vomvoridi-ivanovic (2012) for more research into mathematics teacher educators’ practices to work toward equity; initial findings revealed a focus on access among mathematics education professors. vomvoridiivanovic and mcleman (2015) found that among a group of 23 participants, mathematics education professors expressed varied and multiple views of equity, but almost half of the views on equity emphasized access (to both high quality instruction and resources). suazo-flores and colleagues (2020) conducted a survey (n = 170) on behalf of the association of mathematics teacher educators and found most mathematics educators reported that they conceived of equity as access. the next most frequently mentioned conceptions of equity included disrupting the status quo (system needs to change because it is inequitable for marginalized learners), promoting positive mathematics identities (including k–12 students and teachers seeing themselves represented in mathematics), and cultural ways of knowing (teachers’ backgrounds and cultural ways of knowing are valued). there is a need to work toward greater emphasis on critical dimensions of equity in future teachers’ learning; this endeavor can be shared among mathematics professors and mathematics education professors because they all affect mathematics teacher education. research is limited on mathematics professors’ perspectives on equity in the context of mathematics teaching and learning. bryant and colleagues (2018) wrote about their experiences as mathematics professors attending a summer workshop about equity and participating with mathematics education professors. they illustrated the following examples of understandings about equity that were co-constructed during this summer workshop: (a) equity is more than creating access to learning opportunities and extends to systemic problems such as tracking and placement of students in courses. (b) equity involves recognizing each learner’s humanity and identity. (c) equity involves interrogating where power is located in classroom interactions. this work illustrates what can be learned when mathematics professors and mathematics education professors reflect together about equity. individuals have had unique experiences and knowledge bases that shape their thinking about equity (foote & bartell, 2011). different definitions of equity are likely to be useful for different purposes (gutstein et al., 2005). to explore how inquiry and equity intertwine, a group of mathematics education professors and mathematics professors collaboratively pursued answers to the following research question: how and in what ways do mathematics education professors and mathematics professors describe intersections between inquiry and equity? methods a contribution of this work is that we co-developed and engaged in a process of cowriting (e.g., manning, 2018; short & healy, 2016; siry & zawatki, 2011) to use an equityminded approach to inquire about how equity and inquiry intertwine. we treated this research project as an opportunity to continue dialogue among participants beyond a summer institute and extend our collaborative learning. our endeavor to position the group as a collective generator of knowledge was an intentional choice to move toward anti-oppressive research practices. below, inquiry & equity 16 we elaborate on this process. (a table presenting a timeline of implementation steps for this study is in appendix a.) methodology: co-constructing results by co-writing with participants co-writing can share the power to generate knowledge with participants. cooperative inquiry, or co-constructing research with participants, occurs when everyone involved explores topics of mutual investment. the coordinating researcher is not positioned as the authority with the insights, and participants in the study have voices in the conclusions reached (short, 2018). this is a democratic approach to conducting research and building knowledge (harding, 2020). when participants have more voice and control over what ends up in print, the write-up of the study is more aligned with the view of participants (heron, 1996). the following quote resonated with the first author: i often feel unsettled when writing about experiences of others…. i wonder if i can ethically represent a participant’s experience, if i am presuming too much or wielding too much privilege in the act of trying to write about the experiences of others…. thinking critically and reflexively about our own positionalities and relationships with participants are important parts of doing research in pursuit of social justice. (manning, 2018, p. 745) our co-writing approach draws from feminist and post-structural perspectives on conducting research. from these perspectives, how we represent research participants can be an ethical struggle due to navigating one’s relationships to participants and our responsibilities to each of them (gonick & hladki, 2005). co-writing aligns with a feminist approach to research because the process of conducting research interrogates objectivity and examines epistemic authority and privilege. according to harding (2020), “knowledge produced by feminist research must contribute to pushing back intersectional relations of power, inequality, and oppression felt by those who were the focus of the research” (p. 2). the first author of this study strove to write with rather than about participants. cowriting was an opportunity for everyone involved in the research study to enact compassion for other participants and to heighten awareness of how other people experience the world that we share (short & healy, 2016). goals of co-writing include dismantling or flattening hierarchies between researcher and participants, seeking a plurality of perspectives (rather than a false notion of a single truth), and decreasing the likelihood that those who are a focus of the research will feel unequal and oppressed (harding, 2020; siry & zawatski, 2011). after all, “when you write about yourselves and those close to you, you are constantly aware of the impact of your words” (short & healy, 2016, p. 192). many of the participants had known one another prior to the summer institute, and they would continue to intersect professionally in the future. it was a worthy endeavor to work to foster an ongoing community of learning through co-writing. co-researching and co-writing began during the data analysis stage; specific details are described below. the first author constructed the research question and goals of the study with the facilitation team of the summer institute. data were generated from two interviews: an initial interview and a member check interview. participant recruitment and selection. as described earlier in the paper, 19 mathematics education professors were selected from an applicant pool to attend the summer institute, and, in addition, five mathematics professors were invited to attend the summer institute. opportunities for mathematics education professors to apply to attend were publicized through major inquiry & equity 17 organizations in mathematics education (websites and listservs), and advertisement handouts were distributed at the 2018 fall conference for the north american chapter of the international group for the psychology of mathematics education. in addition to text from the recruitment flyer presented above, prompts from the application appear in appendix b. mathematics education professors were selected based on their specificity of descriptions of their efforts to work on both inquiry and equity. participating mathematics professors were members of the mathematicians of color alliance, and they were invited because of their efforts to engage in outreach to improve access to mathematics inquiry for people of color and their expertise in conducting research in mathematics, often collaboratively with their students. participant demographics. twenty-four professors participated in this study; 79.2% identified as mathematics education professors (n = 19) and 20.8% identified as mathematics professors (n = 5). of the participants, 54.2% worked in education departments (n = 13) and 45.8% worked in mathematics departments (n = 11); both types of departments could employ mathematics educators. in addition, 62.5% of participants identified as women (n = 15) and 37.5% identified as men (n = 9). regarding racial demographics, 25% of participants identified as black or african american (n = 6), 16.7% identified as asian, asian american, or pacific islander (n = 4), 4.2% identified as latinx (n = 1), 4.2% identified as mixed race (n = 1), 45.8% identified as white (n = 11), and 4.2% did not report their race (n = 1). at the time of the summer institute, 4.2% were post-doctoral researchers (n = 1), 62.5% were at the rank of assistant professor (n = 15), 20.8% were at the rank of associate professor (n = 5), and 12.5% were at the rank of full professor (n = 3). the first author was a member of the facilitation team, and she identifies as a white, female, full professor of mathematics education in an education department. initial interviews: data collection. initial interviews of institute participants (n = 24) took place in november and december of 2019. the interview protocol was designed by members of the center for inquiry and equity in mathematics’s facilitation team. the interviews were conducted by members of the facilitation team through video calls, which were audio recorded and transcribed. initial interviews lasted approximately 45–60 minutes. interview questions were written to explore participants’ experiences in the institute as well as their thinking about inquiry and equity. (see appendix b for the set of interview questions for the initial interview and the member check interview.) we targeted a subset of responses to these interview questions for this analysis. (focal questions in the initial interview are indexed with an asterisk [*] in appendix b.) an example focal question was the following: “if you were to consider where equity was in your mathematical inquiry project experience at the institute, what would you say?” initial interviews: data analysis. our first phase of data analysis followed procedures of qualitative content analysis (schreier, 2012). in qualitative content analysis, after identifying a research question, the next step is to select material for analysis. to select material to analyze, members of the facilitation team identified segments in the initial interview transcripts that included data aligned with research goals. we identified line numbers in the interviews when participants spoke about equity in relation to inquiry activity. next, a coding frame was built from the data. members of the facilitation team examined the selected data and constructed initial conjectures naming ways in which participants characterized how inquiry and equity could be intertwined. conjectures for 13 out of 24 participants were collaboratively constructed by multiple members of the facilitation team. conjectures for the rest of the participants’ interviews were constructed by the first author, inquiry & equity 18 informed by the previous collaborative work. these conjectures were a first step toward constructing codes and were useful for member checking. member check interviews: data collection. member check interviews were conducted in the summer of 2020 by the first author. prior to these interviews, each participant was emailed excerpts from their initial interview that represented their thinking about how inquiry and equity were or could be integrated. during this interview, participants were asked to interpret their quotes. the interviewer presented each participant with conjectures about their perspective on how inquiry and equity were integrated and asked them to comment on and share whether they still held this perspective. participants responded by elaborating upon the conjecture and quotes, providing further detail, or by revising their perspective. participants chose pseudonyms at the end of member check interviews. initially, when inviting the participants to engage in research, we did not plan to co-write, so our consent forms indicated that results would be reported using pseudonyms. additionally, not all participants engaged in co-writing to the same degree. to protect the identities of participants who were less active as co-writers, we maintain the use of pseudonyms in results. member check interviews: data analysis. after the member check interview, the first author drafted a set of codes that were applied to both interviews. thus, the coding process was both emergent and deductive. the deductive codes were informed by the literature (e.g., gutiérrez’s [2002] dimensions of equity, such as the roles of access and power in the process of conducting inquiry). emergent codes described themes repeated by more than one participant. examples of codes include inquiry and equity intertwining in socially based mathematics questions (e.g., mathematics for social justice) or opportunities for choice and flexibility. after trying this coding frame, we modified our process, which is another common phase of qualitative content analysis. reducing data analysis to coding appeared to fragment the data in ways that did not illustrate the complexities of participants’ perspectives, which became clearer through dialogue with participants about data analysis in research meetings (fall 2020, described below). instead, we shifted to explanatory narrative inquiry, which is used to seek explanations for why things occur (polkinghorn, 1995). we sought a holistic approach to content analysis that examined participants’ stories as a whole, consisting of parts (lieblich et al., 1998); the whole was a story of inquiry and parts were connections to equity. thus, constructing thicker descriptions of participants’ perspectives at the case level was more illustrative than applying a set of codes. following foote and bartell (2011), in the winter of 2020–2021, the first author wrote research texts for each participant’s interview. a research text summarized a holistic narrative of participants’ stances on how inquiry and equity intertwined and included quotes from both interviews to justify the summarized stances. these research texts were shared with participants for comment; two-thirds of participants replied with edits to language used to characterize their thinking or confirmation and agreement with interpretations of their thinking. stances in the research texts served as a foundation for the framework of the paper, presented in table 1, which was developed and refined in meetings with participants. co-writing co-writing began after member check interviews. participants were invited to attend monthly zoom meetings in august, september, and october of 2020. (two meetings were held in october by participant request.) of the participants, 70.8% attended at least one of these inquiry & equity 19 meetings, and 20.8% attended multiple meetings in the fall of 2020. at the meetings, emerging findings were shared on collaborative documents (e.g., google slides and google docs), discussed, and refined, both verbally and in writing. after each meeting, the first author synthesized ideas about both interpreting data and the process of conducting the research. for instance, at the september meeting, a participant suggested that our work could be framed as being conducted with co-research methodologies. the first author took up this suggestion, read about co-research, and used what was learned to further guide the research process. participants’ asynchronous comments on their research texts, as described above, also supported co-writing. during the next phase of co-writing, in the spring and fall of 2021, multiple drafts of the paper were shared with participants for feedback; 62.5% of participants (n = 15) engaged in cowriting activities during this phase, and 45.8% (n = 11) participated multiple times. participants reviewed and revised a conference proposal asynchronously, met synchronously to discuss authorship principles and practices, and met synchronously to launch a process of providing feedback on drafts. comments were provided on the draft asynchronously in the late spring, and then the group met early in the summer to discuss the feedback. the first author revised the paper, and asynchronous feedback was sought on the revised draft in the fall. the first author and a participant co-presented these findings at a conference (jansen & center for inquiry and equity in mathematics, 2021). we submitted the initial draft of this paper in january 2022, and we received reviews in may 2022. the first author shared the reviews with all participants and invited them to meet and discuss how to revise the paper. two revision meetings were held in the summer of 2022; 25% of participants participated in these meetings, and 4.17% attended both meetings about revising. during these meetings, we generated ideas of how to respond to specific requests for revision and shared additional readings that could inform the paper in light of revision requests. one of the foci of the second revising meeting was to share ideas generated from additional reading. the revised paper was sent to participants to elicit feedback prior to submission. after the revised paper was accepted with minor revisions, the first author made the revisions and sent the text to the participants for asynchronous feedback. results the results represent an aggregated set of ideas across the participants. it is not the case that every person in the group agreed with every one of these themes or the thoughts expressed within them. instead, the results are a union of what the individuals reported. there were three primary themes across the data about the intertwined nature of inquiry and equity. participants recognized that possibilities and dilemmas for equity are always present during inquiry. in addition, distinctions were made between equity-minded inquiry (as a process) and inquiry conducted to achieve equity (as an outcome). 1. equity possibilities and dilemmas are always present during inquiry we can develop greater awareness and attention to equity during inquiry. while engaging in inquiry, there are implications for equity that some participants reported as being constantly at play, whether inquirers are aware of the equity implications or not. inquiry activities are never politically neutral and are imbued with power and status dilemmas. inquiry & equity 20 2. equity in inquiry equity in inquiry is a process of conducting inquiry that is worth striving toward. it is a vision for how participants wanted the activity of inquiry to operate, particularly when conducted collaboratively. they reported that inquiry would be more equitable when collaborators (a) recognized and honored one another’s strengths, (b) had choice and flexibility about choosing an inquiry question and choosing how they would engage in the process of inquiry, (c) had the opportunity to pursue a line of inquiry that is personally meaningful or interesting, (d) had access to powerful mathematics content or practices, and (e) engaged in inquiry in ways that incorporated interdisciplinary perspectives. 3. inquiry for equity inquiry for equity addresses the goal or intended outcome of conducting inquiry such that the practice of inquiry moves us in the direction of achieving greater equity. one way that inquiry could seek equity was (a) in the nature of questions pursued in inquiry. socially based inquiry questions, such as questions aligned with mathematics for social justice, could provide insight on working toward equity. in addition, (b) participants conducted inquiry that would allow them to support their students by improving their teaching, which could help address equity by reaching more students. [insert table 1 here] equity possibilities and dilemmas: always present during inquiry participants described that opportunities for equity or inequity were inherent in any inquiry activity, both structurally and interpersonally. structural: inquiry is not neutral. participants reported that inquiry is culturally situated and never politically neutral, which was a structural reason why inquiry and equity are intertwined. participants saw inquiry as always racialized and genderized, reflecting values and choices about whose mathematics is practiced (including whether it is only mathematics of the dominant culture). participants shared the following: you can’t separate out the way in which mathematics in a social context is a social practice. and so the inquiry itself is mathematical, but it’s also historical and political and socially situated... even the multiple ways of knowing within mathematics could be informing the inquiry approach that the people are using. so it’s not this thing that exists outside of the people who are doing it, when we’re talking about learning and doing mathematics. (katherine, member check interview) there is no decontextualized math task for me…. even if it’s adding two two-digit numbers, i can think about where the equity is in terms of how can i value a child’s invented algorithm that might not necessarily have the spaces to be able to be seen as smart in mathematics? (alexis, initial interview) ways of knowing mathematics reflect the people, context, and situations in which mathematics is practiced and conducted. according to james (initial interview), “the equity lies in these kind of ancestral knowledges, right? that are not tapped into, in favor of the dominant math if you will. so i think that to me is important, right?” participants talked about how mathematical inquiry & equity 21 activity reflects standpoints of ways of knowing and doing mathematics. in his member check interview, charles said, “when one is doing mathematics, one can pose questions about what assumptions are we making about the nature of mathematics, about whose mathematics it is….” whose mathematics is being centered in mathematics classrooms? if it is always mathematics from a dominant cultural perspective, then a wider range of ways of knowing are not being valued. it really matters who gets to create mathematics, not just learn mathematics. and that is the space where i think inquiry and equity come together because inquiry is for me, a big part of it, is around this authoring of mathematics in an agency to creating mathematics, not just, i’ve been told a procedure and i’m going to practice it and master that procedure. (claire, member check) opening up opportunities for learners to inquire creates possibilities for learners to author mathematics, so it is truly theirs. interpersonal: power and status dynamics during interactions. the second way that equity was identified as always present in inquiry was interpersonal. equity-minded inquiry honors participants’ voices and strengths. power dynamics and status dynamics are always a part of interactions with others during collaborative inquiry, so it is important to develop awareness of how interactions can be conducted with equity in mind. participants, such as carmen, eric, and hazelle, reflected about how they think about equitable interactions during inquiry. carmen spoke about how she thought about equity when interacting with students on mathematics research. i think it’s about voice. a lot of the work that i’ve been doing this summer at least, has been whose voice is filling the space.… i’m struggling and trying to figure out how do i bring equity into my practice when i talk to students about inquiry, mathematical research and sharing ideas, that doesn’t continue to center people who are more prone to speaking. (carmen, member check interview) one way to strive toward equity during inquiry is through centering the voices of those who are not always heard. eric shared about equity challenges during small group interactions among students when he facilitates inquiry as an instructor. the equity isn’t naturally going to just happen because of the inquiry. and something i have to do as a teacher when i have students working in small groups is make sure those groups are not dysfunctional.... [sometimes] the groups are really just not working well together. and what happens is there’s one person who takes over control of the group, typically a male student. (eric, member check interview) he voiced a common challenge, which is that small group interactions were less equitable, in terms of unequal participation, when one student behaves more authoritatively. every choice a teacher makes could have effects on the degree of equity in classroom interactions. hazelle, during her member check interview, said, “it [equity] infuses all parts of your teaching, like the tasks you choose and what kind of voice are you giving?” participants reflected that the potential for inequity or greater equity is always present in inquiry when people work together. equity in inquiry and inquiry for equity participants made a distinction between equity-minded inquiry and inquiry conducted to achieve equity. for example, ziad shared, inquiry & equity 22 there are two ways maybe to think about it. and i said whether there’s equity in inquiry versus inquiry for equity... when we say inquiry for equity is like when we’re doing this math of social justice questions…. i don’t give them [students] social justice all the time. i’m doing geometry, just like number theory and something like with my students. okay. is that also an equity element? of course, because that’s all what i do, is how to make this more accessible. (ziad, member check interview) from ziad’s perspective, fostering inquiry in equity-minded ways (equity in inquiry) involved accessibility of the task, such as selecting and enacting a task that was open enough for students to begin working in ways that made sense to them. but he pointed out that some inquiry was conducted for the purpose of achieving greater equity (inquiry for equity), such as mathematics tasks that were designed to address issues of social justice. cheryl emphasized that equityminded inquiry is not always inquiry for equity. i think inquiry is not equity. it can be used in service of, or it can be maybe a pathway along this journey, because i think equity is something much larger. and i think equity is also, it’s an action word. i think something that’s different is where people see that action as being finished. for us [inquiry group at the summer institute], we did not see that action as being finished with the completion of a mathematics problem. for us, in the context of gentrification [their topic explored in the summer institute], the action would be completed once there was something that was produced that could go beyond our classrooms to actually bring about a change, whether it was to stop gentrification, whether it was to go back in and, like i talked about, ungentrify certain communities to give certain people access that had been denied access or had their access taken away. (cheryl, member check interview) if equity is viewed as working toward creating social change (inquiry for equity), then inquiry could be a vehicle toward equity as an outcome. however, equity could also be viewed as creating conditions that support one another during the process of conducting inquiry, which focuses on equity in inquiry. equity in inquiry: recognize and honor strengths in collaborators. equity in inquiry served as a vision for how participants wanted inquiry to operate, particularly inquiring collaboratively. in equity-minded inquiry, collaborators value each person’s brilliance. participants described how they worked together during inquiry in equitable ways. …just me being genuinely curious to learn from others, and in the group acting in such a way that i would share my ideas but also seek out others’ ideas. for me, that looked like bringing genuine curiosity to gerrymandering with fred and antonio and colleagues, and the same for gentrification. for me, that’s how it was present through the inquiry of those tasks. (casey, initial interview) if i’m in a committee meeting and i am the only person of color in this committee meeting full of men and i’ve seen that my ideas get taken up in certain ways that i didn’t intend or they just straight out take my idea or they push back in certain ways, then i’m less likely to speak up because there’s this inequity happening.... so i didn’t feel that there [at the summer institute]. it was more like we honored each other’s voices.... you know that you all are experts in some ways and that you come to the table knowing that you can learn from one another and that’s what i felt. (kaia, initial interview). so, i definitely think, in our group, we did a great job of not positioning one person as more knowledgeable than each other but really seeing the brilliance in everybody and building on their own expertise. so, even in the ways in which we dialogue and have a conversation, i feel like it was... we were building on each other’s ideas, which was very nice. so, i even think of the ways in inquiry & equity 23 which we engage in mathematical ideas and shared them. i think we did it from a very equitable space because we didn’t necessarily shut someone’s thinking. we didn’t interrupt their thinking or shut them down. (rose, initial interview) how we interact with one another sends messages: do we see each other? do we hear each other? do we value each other? do we appreciate each other? when we are truly open to learning from one another, we learn more, and we are stronger together (featherstone et al., 2011). equity in inquiry: choice and flexibility. equity-minded inquiry can include choice such that inquirers pursue their questions and flexibility in ways that work best for them. cheryl commented on choice and flexibility when elaborating on how she saw equity in inquiry during the summer institute. i think allowing us to select our own topic to investigate. allowing us to choose how we communicated that to demonstrate what we learned, to choose our own representation. allowing us to work how we want it to work. so just even in the physical space, whether we were inside, outside, on the floor, just having some flexibility with time. allowing, i think even allowing us to use different methods to look up information. so just having, being able to use our laptops to find stuff. because there were times we’re on our laptops and we’re on our phones, so we’re just looking up stuff any way we could. (cheryl, initial interview) similarly, carmen in her initial interview said, “i think the fact that we could take our own time in discovery, i think was key.” hazelle also connected choice with an equitable experience when reflecting on her inquiry during the summer institute when she said, “my first thought is, i feel like the equity is that we got to develop our own projects. like that it wasn’t you telling us where the direction it had to go. there’s something equitable about that.” seymour agreed, as he said in his first interview, “it was a problem that i had, nobody gave that to me. i posed my own problem and then i got to pursue it.” he confirmed this perspective in his member check interview when he said, “then the choice part, i think that’s the most significant equity component [of the inquiry work during the summer institute].” equity in inquiry: interesting or personally meaningful inquiry. inquiry can be equityminded when learners pursue a question that is personally meaningful or interesting, which can be a powerful emotional experience. i just love that moment of math problem solving, like, i love to do mathematics… and so i got to do something that i cared about. i’m driven by my own question around mathematics. and i guess, well, i know that not everybody appreciates that the way that i do, but whatever it is that they appreciate about mathematics, i hope to provide students with the space to do those kinds of projects. so if it’s about gerrymandering or if it’s about the parabola, either one, you know that this is something that’s meaningful and someone wants to pursue it. (seymour, initial interview) nefti reflected that the opportunities to ask one’s unique questions could bring more people into the field of mathematics, as he said, my interest in [his applied mathematics subfield], like, what drew me to the field was the opportunity for me to ask my own questions. all of the other areas of mathematics, it seemed like i would have to know a lot to even make headway, where in this particular field i could ask my own questions. (nefti, member check interview) inquiry & equity 24 arabella shared, during her initial interview, working on something that everyone in her inquiry group cared about made the inquiry a more equity-minded process. she asked, “why do we care about this? so not just like, ‘here’s a social justice topic that we can explore.’ but why do we care about it? why does this matter to every single one of us?” equity-minded inquiry, then, was inquiry that mattered to the inquirer, because not every learner has experienced mathematics that is meaningful to them. if working collaboratively, the inquiry was more equity-minded when the group decided on a line of inquiry that everyone wanted to pursue. equity in inquiry: access. access was expressed by some participants as another important dimension of a vision for equity-minded inquiry. priya spoke about how she was provided greater access to opportunities to know and do mathematics, so she wanted to provide such access to her students as well. i feel like one of the most important things to me has been this idea of access to even mathematical ideas, which people may have at that point thought that i was not even worthy of. and yet somebody opened the door to me and allowed me to kind of explore those things…. i feel like that’s one of the things that i am able to do: open doors for my students and say, ‘well, here is some math. let’s see if you might find this interesting.’ (priya, member check interview) multiple participants spoke about teaching mathematics by enacting tasks in ways that helped students have points of entry into working on challenging tasks. maybe the question is something that they might find very passionate or personal and they would want to study, but by the sheer nature of them not knowing trig and trig being a requirement, it excludes them. so if you have other types of problems or if you approach the inquiry in a way that you can scale it to different levels… to still make it accessible to them at their level, but also keeping them engaged as they go through more advanced. (antonio, member check) the equity dimension of access was that students would have opportunities to engage in the activity of doing mathematics as a mathematician might. i look at mathematics problems and even abstract problems like that as a playground. how do we get other students and learners of mathematics to see math as a place where you can play? everybody’s got access to that, right…. i think i see it as a math problem as something that people with very different backgrounds could come in and contribute different things and see different things, but that being able to come in and see math as the playground, that’s a really important piece of equity. who gets to play? (lizzie, initial interview) ben shared that he hoped that he could bring students of color into mathematics by modeling his process of pursuing questions for which he did not already have answers. something that i care deeply about is using inquiry to make math more accessible to people, especially students of color…. but what i really try and do is humanize mathematics by kind of walking this sort of line being an expert in my field and being somebody who, oh, this is an interesting problem, but this isn’t my area of research, so i actually don’t know already what the answer is going to be. (ben, initial interview) although access is a perspective on equity that is commonly recognized, it is important to acknowledge that these participants also valued access as a part of equity-minded inquiry. equity in inquiry: interdisciplinarity. when inquiry draws from a wide range of fields, it is more equity-minded, because the integration of perspectives in new ways can challenge inquiry & equity 25 current understandings. when talking about their experiences conducting inquiry in the summer institute, katherine and nefti discussed that incorporating interdisciplinary perspectives made their projects more equity-minded. in exploring different questions that came up within the group, sometimes we were able to identify specific math content that could help us to understand issues of gentrification. and sometimes it was more social sciences or history or other disciplines. and so i think when you start thinking about mathematics education as a social science, then it is inherently interdisciplinary. and so the inquiry that’s happening is a combination of all of these disciplinary ways of knowing that have formed how you approach the inquiry. and so i think as we try to situate the mathematics in even the pre-k-16 curriculum within our context, the social context of doing and learning mathematics, it also is inherently a social science and interdisciplinary in the way that we’re asking students to do inquiry. (katherine, member check interview) katherine’s group’s inquiry involved mathematics, social sciences, history, and other disciplines and built on the collective efforts across these disciplines to understand gentrification. nefti worked with kaia and rose to understand more about how teachers could position their students productively and give more students access to learning opportunities, and they informed their analysis by thinking about nefti’s research on behavior of ant colonies. me, rose, and kaia were playing around with this idea of how you organize classrooms to make sure that you’re not excluding anyone, just talking concretely about this. there’s things that you could do as a teacher, positioning yourself in the classroom, literally the place in which you stand in a classroom can influence the type of information that students are getting from you…. we were then jumping around and talking about potential connections to optimization problems that have been solved in biology, for instance, in ant colonies, in how they distribute themselves in space within the colony to optimize resource flow. then we talked about different types of resources they might be trying to optimize, whether it’s maximize or minimize the amount of time it takes for information to travel from one part of the colony to another, so that everyone is kept up to speed on what’s going on, and how that relates back to the classroom. i think that it’s possible that… you can ask these questions about equity and strategies to achieve it, but very naturally some of it can become tools for teaching, like really cool mathematics, and some of it even connects out into the realm of applied mathematics. (nefti, member check interview) therefore, equity-minded inquiry could be thought of as a process that generates knowledge that is more robust when wider ways of knowing and knowledge bases are incorporated. thus, interdisciplinary inquiry could have the potential to address issues of power because new approaches to learning can generate new knowledge that contrasts with what we have sociohistorically been able to learn. inquiry for equity: pursuit of socially based inquiry questions. the intended goal or outcome of inquiry could be generating greater equity. inquiry could lead to greater insights, which could lead to actions that create a more equitable and just world. when inquiry is conducted for equity, equity is integrated into the problem context, as students explore issues of justice within mathematical work, as explained by emmalee: the purpose for me was to think about ways, when i think about inquiry and equity, i think part of what you can do is embed dimensions of equity and how i would think about issues in justice within the mathematical work these students are doing. for me, it was trying to figure out a space to kind of marry both the curiosity and strategies for engaging in mathematical ideas with a social context that actually helps students think about issues of equity. i wasn’t trying to construct equity in just an intrapersonal dimension, but how inquiry can shine lights on equity within inquiry & equity 26 mathematics…. i mean, the equity for me was embedded in the context of the problem itself. (emmalee, initial interview) nefti agreed, as he shared in his initial interview regarding his collaborative inquiry: “the equity was forefront. and i think it was just based on the questions. the motivating question was from equity.” an inquiry question could align with a goal of working toward equity and justice. some participants reported that inquiry questions that addressed social issues could be more interesting or meaningful for their students. not all mathematical activity in classroom settings is meaningful to students, and inquiry for equity could be more meaningful to them. i was watching this lesson, and i think we would argue in math ed that it was a great lesson. students were making sense of the volume of a cylinder, and they were using blocks, they were engaged, they were talking to each other, and it was definitely inquiry-oriented. but as i was watching the lesson, i was wondering, why do we care? why do these students care about learning this? what does it matter? does any of them know what to do with this knowledge? and i’m thinking, if we taught math in a different way where, i think, kevin said yesterday, ‘math for democracy.’ the way we bring in headlines, especially right now, and make sense of stuff that’s in our real life. would that be more meaningful and would students know why they’re learning it? (arabella, member check) arabella reflected on how mathematics lessons taught in ways that might be valued by the field of mathematics education could still be about mathematics that students might not care about. instead, mathematics could be taught by investigating problems that supported working toward greater equity in our society. priya agreed, and she suggested that inquiry for equity could be supportive for students of color. a lot of times i’ve taught the [general education-required] math course over here, and that actually has a lot of students of color, for instance, in that. and they get quite passionate if they’re talking about things that have to do with social equity. like when we talk about elections and the ways elections are conducted, they get quite passionate about it. so, i think we’re trying to match some of the math projects with social equity issues… that’s an important way to get those students engaged and interested. (priya, member check) inquiry for equity could afford students opportunities to have a more positive connection with mathematics. inquiry for equity: supporting students through conducting inquiry. when we conduct inquiry to support our students, it is another way we could work toward equity. they engaged in inquiry with the intention of providing their students with a similar inquiry experience around the problem they investigated at the institute. because one of my points in doing this was, and i keep saying this, is, like, you know, i’m always thinking about how can i bring this to my students and how can i use this as something that would be helpful to them and thinking about inquiry or equity. (hazelle, initial interview) claire shared that letting the students’ curiosities drive inquiry was a way of honoring and promoting her students’ agency. the question that i was seeking to explore was one that was personally interesting to me mathematically, but it was really driven by what’s interesting to my students mathematically. this was a question that surfaced from our mathematical interactions. and i felt like, again, it’s about inquiry & equity 27 their agency. what is it that they want to know and get clear about? (claire, member check interview) professors could benefit when we put ourselves in the role of students, by experiencing inquiry as they might experience it. this is an empathetic stance. we can support our students through our inquiry by learning about ideas that they would like to understand, to teach them more effectively, or to experience inquiry as they might. tensions between inquiry and equity some participants reflected on tensions between inquiry and equity that they observed or considered. equity-minded inquiry requires intentional support and facilitation with awareness of how equity can be fostered or inhibited. eric strove to create equity-minded inquiry in his university classroom, but he reported that not all students had equal opportunities to participate: maybe that’s one of the problems is i think i have an inquiry-oriented class when really i have four inquiry-oriented leaders of the class and the rest of the students are not engaging in the inquiry. therefore, they’re not having an equitable inquiry or engaging in equity. (eric, member check) cheryl observed during her member check interview that intentional facilitation is needed to help students understand equity-minded inquiry, such as why we might want to offer them choice and flexibility. there’s also this tension between, on the one hand, we value this process, but i think on the other hand, we understand that if we give something that’s just too open to our students, we may have some pushback because they’re used to something specific that fits in this box, i’m going to do this and then i’m going to get a grade. (cheryl, member check) fostering equitable inquiry, then, is an ongoing process of noticing and attending to students’ perspectives, providing support, and having empathy for students’ experiences. inquirers play important roles in navigating tensions on the way toward equitable inquiry. while working collaboratively during the institute, participants created more equitable inquiry as they worked with one another through their openness toward one another, by asking for options such as more time or particular materials, and by honoring strengths in their collaborators. fellows were open with the facilitation team about their experiences. they shared when they experienced more and less equity during the inquiry experience, which raised awareness about the experiences that they and others were having. there were clearly, at least among the more vocal people, really different ways in which they were constructing what the space was and what it could be used for. i felt like there were lots of tensions and discussions, discussions that needed to be had, things that were operating under the surface, and i thought that was productive. (emmalee, initial interview) for instance, early in the institute, the facilitation team engaged the whole group in a mathematics inquiry question that was more abstract, outside of a socially based context. this brought up a conversation during the institute about pursuing questions that supported addressing issues of social justice or conducting inquiry for equity. fellows provided insights that supported inquiry & equity 28 the facilitation team with modifying the experience during the institute, and participants’ openness about their experiences supported the facilitation team with interrogating their efforts. participants also offered tensions experienced when facilitating inquiry related to sharing power and authority with students. ben shared his process of engaging in inquiry with his students. what i really try and do is humanize mathematics by kind of walking this sort of line being an expert in my field and being somebody who, oh, this is an interesting problem but this isn’t my area of research, so i actually don’t know already what the answer is going to be. and i think that’s, especially for somebody like me, i have to be really careful about these kind of two hats that i wear. one, the young, black man, in an institute that is majority white. inevitably, i sometimes get challenged about whether or not i actually deserve or earned my right to be there. i know i don’t necessarily fit the stereotypical look of a math professor. and so one way to overcome that is to show them like, ‘yes, i know all this stuff. i know this stuff forward and backwards. i eat, live, breathe… all this kind of stuff, and it’s great.’ and then once they see like, ‘oh okay, he’s actually not playing no kind of game,’ that’s a way to kind of earn their respect, earn their trust, and then it’s a great experience. on the flip side of that, when i do my research groups, i purposely pick problems that i don’t know what the final solution is going to look like. i don’t know exactly where it’s going to go or how it’s going to shape out. but i want to give students the experience of what it’s like to really be a mathematician, how to think through these problems…. i really sort of latched onto this duality between being this content expert in the classroom and then being this humble mathematician, who sees this really interesting problem, latches onto it, and then works it out like a mathematician. trying things, some things work, some things don’t. (ben, initial interview) ben shared a tension between proving oneself as an expert and inviting students to author their own journeys. he also situated this narrative in the context of being a faculty member of color at a predominantly white institution. it is possible that trying to disentangle the equity aspects from inquiry activity is not necessarily helpful. fred articulated this perspective when he said the following during his member check interview: it’s maybe a sort of hybrid in the spaces of, it is this connecting of people in a loving way, focused on and solving injustice and making people around us better, right? and i think that’s where the two come together for me. but it doesn’t live in a space of just inquiry. it doesn’t live in a space of just equity. (fred, member check) it is possible that fragmenting the equity elements apart from the inquiry activity is a disservice to the wholeness of the experience. discussion a contribution of this study is that we built upon and extended the work of tang and colleagues (2017), which was a theoretical exploration of how inquiry-oriented mathematics teaching could support equity. their analyses most aligned with our second theme: equity in inquiry, or how the process of conducting inquiry in mathematics could be equity-minded. tang et al. (2017) identified aspects of the practice of inquiry-oriented mathematics teaching, such as student-generated knowledge built through collaboration with peers, and they analyzed how these practices reflected gutiérrez’s dimensions of equity. for instance, they identified how these practices could change power dynamics in the classroom when students built knowledge inquiry & equity 29 together, support students’ development of positive mathematical identities if they enjoy mathematics more or feel more confident, and access learning opportunities. both the analysis by tang et al. (2017) and this analysis examined possibilities for how inquiry can be an equity-minded process, but our analysis revealed additional descriptions of equity-minded inquiry processes that are more targeted and additional ways that inquiry can be equity-minded. for instance, our analysis illustrated that collaboration alone would not ensure equity, but equity-minded collaboration could occur when collaborators recognize and honor strengths in one another and attend to status differences. such collaborative practices can build positive mathematical identities, bring more collaborators’ ideas into the knowledge-building process (access), and increase likelihood of power-sharing among collaborators. we also explored how interdisciplinary perspectives can enhance the nature of knowledge generated during inquiry, which was not addressed by tang et al. (2017). doing so connects to the power dimension of equity by challenging current knowledge bases through interdisciplinary perspectives. a contribution of this analysis is that we described dilemmas and tensions between inquiry and equity, which were not explored by tang and colleagues (2017). regarding dilemmas and tensions, we recognized that equity is ever present during inquiry because actions and choices reflect cultural practices and are never politically neutral. in addition, we identified that power dynamics are always a part of interactions during inquiry, and power imbalances can occur if not intentionally addressed. it is important to be aware of pitfalls for inquiry to become less equity-minded if we are to work toward equity in inquiry. another contribution of this analysis is that we highlighted that inquiry can be equityminded if its goals are directed toward greater equity; tang and colleagues’ (2017) analysis did not describe that inquiry could be conducted for equity-minded purposes. for instance, inquiry could seek answers to socially based questions that can lead to insights and actions to change the world around us (power dimension of equity). in addition, some participants conducted inquiry to support their students, which was another way of sharing power with learners and illustrated empathy toward students. contributions to urban mathematics education this analysis of intersections between inquiry and equity contributes to research in urban mathematics education because we join those who strive toward anti-racist and anti-oppressive practices in mathematics education. by identifying dilemmas and challenges with inquiry, we hope to avoid such pitfalls and intend to promote anti-oppressive practices by identifying more equity-minded approaches to inquiry. along with other researchers in urban mathematics education (cf., larnell & bullock, 2018; larnell & martin, 2021), we view urban spaces as places where sophisticated and authentic inquiry can and does occur, and we encourage facilitators of inquiry to see and amplify complexities and strengths in cultural practices in urban contexts. some of our results regarding intersections of inquiry and equity are rooted in anti-deficit perspectives: monitoring power and status dynamics to recognize and center the brilliance of every collaborator and the nature of questions pursued in inquiry. centering the brilliance of one another is inherently anti-deficit because we strive to act on the assumptions that all collaborators have insights to offer. the nature of inquiry questions – whose questions and what questions are pursued – is potentially connected to the lives of students and colleagues. whose questions are inquiry & equity 30 worth pursuing? some of us focused on pursuing inquiry that is relevant to our students’ curiosities. what questions are pursued? some of us focused on pursuing inquiry that is grounded in socially based issues, potentially directly connected with local activists and communities. recognizing students’ questions and community-based questions as worthy of inquiry involves developing knowledge with and building knowledge of students, which is part of political conocimiento (gutiérrez, 2017). our methodological approach of co-researching and co-writing also contributes to research in urban mathematics education. viewing urban mathematics education as a political project, larnell and martin (2021) described the value of incorporating anti-oppressive research methodologies that use alternative models for constructing knowledge. we, the first author and research participants, intentionally sought to co-construct the knowledge in this paper. in other words, we strove to enact equity in our inquiry for this paper. implications our process of exploring intersections between inquiry and equity as a group of mathematics education professors and mathematics professors was a valuable one. as a group, we developed a broader set of perspectives and a more coherent and structured framework for how inquiry and equity could intertwine. the composition of the group of participants influenced what could be learned. as a group, we all strive to act on our commitment to equity through multiple forms of inquiry (inquiry-oriented teaching practices and inquiry through scholarship). if mathematics education professors and mathematics professors are to have equityminded collaborations to improve instruction in mathematics and mathematics education, collectively developing an understanding of intersections between inquiry and equity can be useful. vomvoridi-ivanovic and mcleman (2015) suggested that one way that mathematics educators can navigate challenges in their professional work toward equity was through engaging in professional learning communities or collaborating with colleagues. following vomvoridiivanovic and mcleman (2015), we were not interested in classifying participants’ ideas in a hierarchical manner. instead, we strove to understand a multiplicity of perspectives, which can support others in anticipating these viewpoints in future dialogue and collaboration. this framework provides perspectives to consider when seeking common ground between collaborators who want to work toward equity and pursue inquiry. certainly it is unrealistic to expect collaborators to come to complete consensus on definitions of equity when working on inquiry. future collaborators could examine and reflect upon this framework together for support in trying to understand a collaborator’s point of view. the collaborative effort of these participants to work together in inquiry during the summer institute and to collaborate together on understanding ways of thinking about equity as intertwined in inquiry is an illustration of potential. mathematics education professors and mathematics professors can grow together as they collaborate. pursuing questions of joint interest (mathematics inquiry questions, inquiry about inquiry and equity) is a way to build community and relationships. seeking to understand one another can generate new collective knowledge. we hope that our framework can be used between other mathematics education professors and mathematics professors to understand similarities and differences between how they currently think and how they want to act as they move forward. limitations and future research inquiry & equity 31 a limitation of this study is that we investigated the perspectives of a relatively small number of professors. although the number of participants was smaller compared to those in previous studies (cf., mintos et al., 2019; suazo et al., 2020), we investigated participants’ thoughts in greater detail through multiple interviews and the co-writing method. this approach led to additional insights over previous studies. the present study also relies on self-reports, which could be viewed as limited. however, the self-reports are from professors whose professional work is aligned with commitments to equity, which offers unique perspectives on understanding how inquiry and equity are and can be intertwined. future research can investigate what inquiry looks like – the similarities and differences – when participants try to understand one another’s perspectives. no one participant reported every single viewpoint in this framework. each participant reported at least one of the components of the framework. do participants with different perspectives on how inquiry and equity intertwine engage differently in inquiry when they investigate their unique questions or do they facilitate inquiry-oriented instruction differently? conclusions our framework of how inquiry and equity intertwine offers structure and insight to discussions of conceptions of equity in mathematics education (e.g., gutiérrez, 2012; mintos et al., 2019; suazo-flores et al., 2020) and approaches to conducting inquiry-oriented instruction (e.g., tang et al., 2017). we hope that this framework raises awareness of how equity can be seen as ever-present during inquiry. the distinction between equity in inquiry and inquiry for equity also offers insight. part of developing political conocimiento (gutiérrez, 2017) is to broaden and challenge current knowledge. we made an effort to do this through extending the work of tang and colleagues (2017). developing political conocimiento also involves learning to notice and entertain multiple interpretations. we engaged in doing so when we built our collective framework for intersections between inquiry and equity. it is our hope that our process of learning to understand these multiple viewpoints can support future collaborative efforts between mathematics professors and mathematics education professors as we work toward shared goals of fostering a more equitable and just society. acknowledgement this material is based upon work supported by the national science foundation under grant due-1821444. any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the national science foundation. references aguirre, j., mayfield-ingram, k., & martin, d. 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(2020, winter). mathematics teacher educators’ conceptualizations of equity. connections. https://amte.net/connections/2020/11/mathematics-teacher-educators%e2%80%99conceptualizations-equity inquiry & equity 34 tang, g., el turkey, h., cilli-turner, e., savic, m., karakok, g., & plaxco, d. (2017). inquiry as an entry point to equity in the classroom. international journal of mathematical education in science and technology, 48(suppl. 1), s4–s15. vomvoridi-ivanovic, e., & mcleman, l. (2015). mathematics teacher educators focusing on equity: potential challenges and resolutions. teacher education quarterly, 42(4), 83–100. inquiry & equity 35 table 1 framework: intersections between inquiry and equity categories of intersections between inquiry and equity examples of intersections between inquiry and equity equity possibilities and dilemmas are always present during inquiry. • structural: inquiry is not politically neutral. it is culturally situated, racialized, and genderized. it reflects values and choices about whose mathematics is practiced. • interpersonal: power dynamics and status dynamics are always a part of interactions during inquiry. equity in inquiry: visions for how collaborative inquiry can operate, as a process, to be more equity-minded • recognizing and honoring the strengths of one another in ways that attend to status differences • offering choice and flexibility in the inquiry question selection and inquiry process • engaging in personally meaningful inquiry • accessing mathematics content or practices • engaging in interdisciplinary inquiry inquiry for equity: inquiry conducted toward an outcome of achieving greater equity • engaging in socially based inquiry questions • supporting one’s students through conducting inquiry inquiry & equity 36 appendix a: timeline for study date implementation steps fall 2018 participant recruitment & selection august 2019 week-long summer institute september & october 2019 design of interview protocol with institute facilitation team november & december 2019 initial interviews with each institute participant conducted by facilitation team via zoom spring 2020 initial interviews analyzed by facilitation team summer 2020 member check interviews with each institute participant conducted by first author via zoom august, september, & october 2020 co-interpretation and co-writing meetings, facilitated by first author. first author applied ideas from meetings to improve analysis and writing between meetings and shared progress at next meeting. december 2020 & january 2021 research texts shared with each participant (about their perspective on intersections between inquiry and equity) for feedback spring 2021 initial paper draft generated by first author march 2021 co-writing meeting to revisit and co-construct authorship principles and practices and decide upon journal outlet april 2021 co-writing meeting to discuss and create processes for providing feedback on paper draft summer & early fall 2021 multiple opportunities provided for participants to provide asynchronous feedback on versions of the drafted paper january 2022 first draft submitted to jume may 2022 reviews received and shared with participants june & july 2022 co-writing meetings to discuss revising the paper august 2022 revision draft distributed to participants for asynchronous feedback. revision revised based on the feedback and submitted to jume. inquiry & equity 37 april 2023 draft with minor revisions distributed to participants for asynchronous feedback. incorporated comments, revised, and submitted to jume. inquiry & equity 38 appendix b: interview protocols initial interview questions *focal questions from initial interview that were targeted for analysis framing to share with participant: for the purpose of understanding how the institute unfolded for participants, we have some questions for you about your experience that week. materials needed: application for the institute [the questions on the application included: why is attending important to you? what do you hope to learn and contribute to the community? how will your students benefit from your participation as a fellow?] questions for interviewee: 1. looking back on the institute in august, what stands out for you about what happened during that week? 2. what did you hope to get out of this week-long institute? in what ways did you have an opportunity to work on this? 3. considering the experiences you had during the institute, which ones stand out as being valuable to you? why were they valuable to you? 4. *in what ways, if at all, did your participation at the institute affect your thinking about inquiry or equity or both? 5. what else would you like to share about your participation in the institute? framing to share with participant: throughout our week together, there appeared to be consensus that issues of equity can be or must be directly integrated with engaging in mathematical inquiry. to situate thinking about relationships between equity and inquiry, we would like to think with you by going back to your inquiry project experience during the institute. materials needed: pictures of participant’s inquiry project work from the summer institute. 6. *so, let’s look back on the inquiry project that you conducted during the institute. here are pictures that we have from your inquiry activity [share screen]. could you tell me the story about your inquiry project experience? what happened? what did you learn? a. looking back on how you began your inquiry project, what did you initially want to investigate? what was the purpose of this project for you? b. how did the project change as you worked on it? what did you give up from making changes? what did you gain from making changes? c. who did you collaborate with and how did this collaboration come to take place? d. what did you learn through this collaboration? consider if any of the collaborative interactions were valuable: what ways did you find value in the collaborative interactions you had? 7. *in the field, individuals are often asked, “where’s the math?” when we talk about mathematical inquiry experiences. as a community, how might we play a role in changing the conversation to “where’s the equity?” so, if you were to consider where equity was in your inquiry project experience, what would you say? inquiry & equity 39 8. in what ways did your inquiry project affect your professional work beyond the institute, if at all? framing: now we would like to ask you some additional questions about how you are thinking about your professional work in the upcoming year. 9. in what ways, if at all, are you engaging in your work or thinking as a mathematics educator differently after spending time with colleagues at the institute? what interactions supported you to do this? 10. one aspect of the evaluation follow-up is to understand any collaborative interactions that occur among fellows and between fellows and mathematicians, if any occur. at this point, do you have plans/what are your plans for collaborating moving forward? would this take place in the fall or spring or both? with whom? what is the work? 11. what sort of collaborations do you wish would develop with institute participants that haven’t developed yet? 12. as you consider future online interactions with each other in this group, what might be some content/interactions that would be useful in supporting your needs/professional growth? concluding question: part of this interview was for project evaluation purposes. part of the interview was for research purposes. questions about the intersection of equity and inquiry were for research. would you be interested in being a part of analyzing and writing about this topic with us? member check interview questions (note: these questions and excerpts from their initial interview were sent to participants prior to the member check interview.) 1. what does this interview excerpt represent to you regarding a way that inquiry and equity could intersect, if at all? 2. what more would you like to share to help others understand your perspective on how inquiry and equity could intersect? 3. if someone wanted to read more to understand ideas to understand your perspective on the intersection of inquiry and equity, what would you recommend? microsoft word 375-article text no abstract-1767-1-9-20191216.docx journal of urban mathematics education december 2019, vol. 12, no. 1, pp. 8–14 ©jume. https://journals.tdl.org/jume jacqueline leonard is professor of mathematics education in the school of teacher education, university of wyoming, 1000 e. university avenue, department 3374, laramie, wy 82071; email: jleona12@uwyo.edu. her research interests include computational thinking, self-efficacy in stem education, culturally specific pedagogy, and teaching mathematics for social justice. editorial “…and a little child shall lead them” jacqueline leonard university of wyoming visit to the national civil rights museum in memphis, tennessee, allowed me an opportunity to learn about the african american experience in the united states since 1619. an exhibit entitled “a culture of resistance: slavery in america 1619–1861” drew my attention because of its historical significance as well as its relationship to the research i have been conducting on how black lives matter in mathematics education. this work promotes using data on racial profiling, housing equality, voting rights, elections, and historical data related to the black experience in america (leonard, 2019). for example, black men were not allowed to vote until 1867 even though slavery ended with the emancipation proclamation in 1863. blacks continued to be disenfranchised at the ballot box through jim crow laws (i.e., literacy tests, poll taxes, and grandfather clauses). understanding the struggle for voting rights and using technology to find ancestors who voted in 1867 and beyond are powerful experiences that can be used to teach mathematics for social justice. thus, the museum offered me a new window to continue this work. visiting the museum brought three thoughts to mind as i drove down mulberry street to park in an adjacent car lot. within the context of the black struggle in the united states, my first thought was the boycott, which was used effectively in southern cities like montgomery, alabama, to protest segregation on city buses. it was not uncommon to find whites riding in the front of the bus and blacks in the back. i remember riding in the back of the bus in st. louis, missouri, during the early 1960s. a little-known fact, however, is that the first african american woman in montgomery to refuse to give her seat to a white person was 15-year-old claudette colvin on march 2, 1955 (hoose, 2009). later that same year, the famous boycott of buses began in montgomery. jo ann gibson robinson, an english professor at alabama state college, promoted the boycott by distributing 35,000 flyers in the black community (osborne, 2006). yet, the montgomery bus boycott, with rosa parks as the central figure, lasted from december 5, 1955 to december 20, 1956 and was notably the first large-scale demonstration against segregation in the united states (history.com editors, 2019). the boycott remains an important tool of protest in the black community. on october 10, 2019, there was a booth on mulberry street that called for a boycott of the museum itself (see figure 1). a leonard editorial journal of urban mathematics education vol. 12, no. 1 9 although freedom of speech is a constitutional right, i wondered why black people would want to boycott a museum that promotes knowledge and history of the black struggle in america. the sign seemed to focus on the $10 million spent on the museum and the souvenirs that could be purchased inside. most of the workers inside the museum, however, including the security guards, ticket agents, docents, and cashiers in the gift shops, were black. moreover, black architects, contractors, and donors were involved in building the national civil rights museum. why do protestors think dr. martin luther king, jr.’s name is being desecrated as the sign suggests? dr. king was in memphis primarily to focus on sanitation workers’ wages, safety, and job benefits. would he be opposed to the museum’s proceeds going to support jobs in the black community? those who support the boycott raise the question of who in the community may benefit more from the construction and the proceeds of the museum. surely, the museum serves a purpose solely by sharing knowledge and history about the civil rights movement and, thereby, benefits everyone in the community. i was left wondering what conversations need to take place to bring ideas and people from all sides of the argument together to allow for broader understanding. undoubtedly, what is just or unjust in this particular context is complex. even more complex, were some of the exhibits in the museum, which brings me to my second thought about desegregation in public schools. one of the exhibits focused on brown v. board of education (1954). as part of this exhibit, a film explained the cases brought forth by two prominent black lawyers, charles hamilton houston and thurgood marshall, to dismantle racism in education and housing. the exhibit also explained that the u.s. supreme court decision on may 17, 1954 to desegregate u.s. public schools was not and is not without controversy. the positive impact of the decision was the admission of black students to predominantly white colleges and universities and the desegregation of k–12 public schools. a negative impact of brown v. board of education, however, was the mass firing of black teachers when schools were closed as a result of desegregation, meaning that many of their previous students were now under the instruction of predominantly white teachers (tillman, 2004). moreover, some public schools were shut down for months if not the entire academic year as part of the fallout (beals, 1994). the immediate figure 1: photograph courtesy of jacqueline leonard. leonard editorial journal of urban mathematics education vol. 12, no. 1 10 consequences of brown v. board of education were significant, but the decision’s long-term effects, though just as impactful to society, are more nuanced. for example, the majority of black and brown children today are still attending segregated schools because the neighborhoods in which they live remain highly segregated (leonard, 2019). thus, the decree had both positive and negative ramifications for black students and teachers alike and the communities in which they lived and continue to live. the majority of the teachers who work with children from underrepresented communities continue to be predominantly white (madkins, 2011; villegas & irvine, 2004), as the teaching profession continues to grapple with recruiting and retaining black and brown teachers (evans & leonard, 2013; leonard, burrows, & kitchen, 2019). a survey conducted in 2016 revealed 80% of the teacher workforce is white, while 9% is hispanic and 7% is black (loewus, 2017). these statistics reveal the percentage of hispanics teachers rose 1% from 2012 to 2016 (loewus) while the percentage of black teachers decreased 1% from 2010 to 2016 (evans & leonard, 2013). thus, having a teacher of mathematics who is black or brown continues to be unlikely for most children of color. villegas and irvine (2004) contended that teachers of color who are familiar with marginalized students’ cultural backgrounds are more likely to establish positive relationships between the home and school, thus enriching learning. such relationships are critical to enhancing learning in high-status courses like mathematics, especially in high-poverty schools that serve predominantly black and brown students (clark, frank, & davis, 2013). historically, young black children have been a force for change in education. in the struggle to integrate public schools, black children put their lives and bodies on the line to integrate public schools in the south and to bring about the potential for change in the american culture of education (e.g., clinton twelve, little rock nine). another exhibit at the national civil rights museum in memphis explained that 2,500 children (10 children per minute) were arrested in birmingham, alabama, on may 6, 1963. under the leadership of fred shuttlesworth, james bevel, and dr. martin luther king, jr., elementary and secondary students participated in nonviolent demonstrations by walking from the 16th street baptist church to city hall. there were both praise and criticism over the decision to allow children to participate in the civil rights movement. the indelible images of children being knocked down with high-powered fire hoses and assaulted by the sharp teeth of barking dogs were etched in my brain as a 6-year-old child in 1963. these same harrowing memories came flooding back when i saw the images on full display at the national civil rights museum in 2019. these surreal images moved a nation, president kennedy, and, in turn, president johnson to act in behalf of black (and all) citizens (e.g., civil rights act of 1964 and the voting rights act of 1965). the impact of young protesters during the civil rights movement had effects that reached far beyond birmingham and the united states. in mathematics education, leonard editorial journal of urban mathematics education vol. 12, no. 1 11 the works of gutstein (2006, 2013), martin (2009), and leonard and martin (2013) were grounded in the civil rights movement and revealed the positive impact of children’s efforts to understand and use social justice for empowerment. this work, along with the current call for papers by the journal of urban mathematics education (jume), which asks for other pieces related to teaching mathematics for empowerment, brings me to my third point. with jume’s recent change in editorial leadership, there are some scholars of color who have chosen to boycott the journal. there may be myriad reasons for them to boycott, and the scholars are within their rights as citizens to do so. nevertheless, the implication reminds me of the complexity i witnessed at the national civil rights museum. jume is a free journal that serves as an outlet for doctoral students and junior scholars to write about issues that impact predominantly urban children of color. there are no dollars at stake, so who hurts or benefits the most from a boycott of jume? what would the boycott achieve except to delay progress in our field and create dissention among those most committed to change in our field? if we are on the side of social justice, we should work together to bring about educational reform to benefit the descendants of 400 years of chattel slavery and the progeny of children who stood up against jim crow laws and segregation, as well as the indigenous and migrant students who also suffer from generational racism and oppression that have negatively impacted their education. the plethora of problems plaguing the educational system allows space for mathematics educators and researchers of all backgrounds to unite with one voice to transform the nation’s schools. the severity of the need renders our differences petty in comparison (ladson-billings, 2006). having recently moved to the greater memphis community for a visiting professorship, i am appalled at the poverty i witness but am energized by the opportunity to engage with students in science, technology, engineering, and mathematics (stem) education. i am inspired by the efforts i see from communities and institutions with privilege to help break the cycle of poverty in economically depressed areas of the city. for example, a local methodist church is working with students in a reading program at an inner-city school. furthermore, rhodes college has a new urban education teacher licensure program to address the need for culturally relevant teachers in classrooms of diverse students. many opportunities are denied to students due to their race, language, class, gender, citizenship status, and so forth; this is a barrier to student success that should be eradicated, and it is encouraging to see local organizations work toward making this happen. to personally address the lack of opportunities many students face, i plan to work with university faculty and community-based leaders in memphis, st. louis, denver, and cheyenne to offer weeklong summer camps in 2020 on computational thinking to support stem education among underrepresented, upper-elementary students. in these camps, children will lead the effort to become creators instead of leonard editorial journal of urban mathematics education vol. 12, no. 1 12 merely consumers of technology and stem more broadly (kafai & burke, 2014; leonard, 2019). children will also be engaged in activities that allow them to engage in computer coding to model and create 3d images of interest to them (kafai & burke, 2014). they will also learn about flight, research the men and women of color who were pioneers in aviation and space, and use drones to investigate urban issues of significance to them. the goal is not to “fix” these children, but rather to selfempower them like the children of the 1960s, specifically by allowing them to tell their stories using digital tools (kafai & burke). these stories can become the springboard to engage children of color in stem while demonstrating their brilliance in computational thinking and mathematics (leonard & martin, 2013). the primary goal of this work, supported by the national science foundation, is to increase underrepresented student interest and access to stem. we must work with communities of color to ensure that school curriculum is important and relevant to them. i have learned from my work with indigenous communities that teachers and elders must be integrally involved in the planning, implementation, and delivery of relevant curriculum (leonard et al., 2018). when researchers value the cultures, histories, experiences, and abilities of the communities they study, participants are not only self-empowered with the tools to tell their own stories but begin to change the narrative of how they are viewed in society at large. furthermore, as mathematics educators and scholars, we must also be cognizant of community goals and participant buy-in to ensure that we do not ourselves engage in interest convergence. interest convergence is defined as a condition when racial equality is championed only when it converges with the interests, ideologies, and beliefs of the status quo (bell, 1980). interest convergence is manifested when, for example, minority participation in the stem workforce is connected to national interest, such as international dominance in stem (martin, 2009). if we are honest, we benefit directly and indirectly from the research that we do with children of color in the tenure and promotion process. none of us, whether white male, black female, or lgbtq faculty, for example, are blameless. we are beneficiaries of the institution of higher education and benefit from the elitist system we often criticize. yet, as part of the hegemonic system that oppresses others, we have a responsibility to critique it. although critique and even boycotts are necessary, we should understand the ramifications of our actions and engage in collaborative efforts that lead to justice and systematic change. the children of the civil rights movement provide excellent role models for other children to participate in nonviolence to promote social justice. from ruby bridges to claudette colvin, children engaged in acts of social justice that provoked a nation to change laws and broaden educational opportunities. yet, over six decades later, there is still work to be done. black children have mathematical brilliance that needs to be noticed. i first witnessed their brilliance 35 years ago in project seed classes in dallas, texas (leonard, 2019). black children demonstrated their mathematical leonard editorial journal of urban mathematics education vol. 12, no. 1 13 prowess when they used computational thinking to explain why any number to the zero power is equal to one. they also used call and response and chorus reading to generalize mathematical properties, such as associative and distributive properties. on the contrary, i have observed teachers’ failure to recognize black students in everyday classrooms despite them flailing their arms to be called on to answer a mathematics problem. similar to demands for equal education in the 1950s, teaching children how to use mathematics for social justice can lead to demands for more rigorous and relevant mathematics education. when this happens, mathematics education may truly be transformed. the wolf also shall dwell with the lamb, and the leopard shall lie down with the kid; and the calf and young lion and the fatling together; and a little child shall lead them. (isaiah, 11:6, kjv) acknowledgments the author thanks dr. christopher c. jett, associate professor (university of west georgia), and dr. cara m. djonko-moore, assistant professor (rhodes college), for their comments and suggestions in crafting this editorial. references beals, m. p. (1994). warriors don’t cry: a searing memoir of the battle to integrate little rock’s central high. new york, ny: pocket books. bell, d. a. (1980). brown v. board of education and the interest-convergence dilemma. harvard law review, 93(3), 518–533. brown v. board of education, 347 u.s. 483 (1954). clark, l. m., frank, t. j., & davis, j. (2013). conceptualizing the african american mathematics teacher as a key figure in the african american education historical narrative. teachers college record, 115(2), 1–29. evans, b. r., & leonard, j. (2013). recruiting and retaining black teachers to work in urban schools. sage open, 3(3), 1–12. retrieved from https://journals.sagepub.com/doi/abs/10.1177/2158244013502989 gutstein, e. (2006). reading and writing the world with mathematics: toward a pedagogy for social justice. new york, ny: routledge. gutstein, e. (2013). understanding the mathematics of neighborhood replacement. in e. gutstein & b. peterson (eds.), rethinking mathematics: teaching mathematics by the numbers (2nd ed., pp. 101–109). milwaukee, wi: rethinking schools. history.com editors. (2019, june 6). montgomery bus boycott. retrieved from https://www.history.com/topics/black-history/montgomery-bus-boycott hoose, p. (2009). claudette colvin: twice toward justice. new york, ny: farrar straus giroux. kafai, y. b., & burke, q. (2014). connected code: why children need to learn programming. cambridge, ma: the mit press. ladson-billings, g. (2006). from the achievement gap to the education debt: understanding achievement in u.s. schools. educational researcher, 35(7), 3–12. leonard, j. (2019). culturally specific pedagogy in the mathematics classroom: strategies for teachers and students (2nd ed.). new york, ny: routledge. leonard editorial journal of urban mathematics education vol. 12, no. 1 14 leonard, j., burrows, a. c., & kitchen, r. (2019). recruiting, preparing, and retaining stem teachers for a global generation. leiden, the netherlands: brill-sense. leonard, j., & martin, d. b. (eds.). (2013). the brilliance of black children in mathematics: beyond the numbers and toward new discourse. charlotte, nc: information age publishers. leonard, j., mitchell, m., barnes-johnson, j., unertl, a., outka-hill, j., robinson, r., & hester-croff, c. (2018). preparing teachers to engage rural students in computational thinking through robotics, game design, and culturally responsive teaching. journal of teacher education, 69(4), 386– 407. loewus, l. (2017, august 15). the nation’s teaching force is still mostly white and female. education week. retrieved from https://www.edweek.org/ew/index.html madkins, t. c. (2011). the black teacher shortage: a literature review of historical and contemporary trends. journal of negro education, 80(3), 417–427. martin, d. b. (ed.). (2009). mathematics teaching, learning, and liberation in the lives of black children. new york, ny: routledge. osborne, l. b. (2006). women of the civil rights movement. portland, or: pomegranate communications. tillman, l. c. (2004). (un)intended consequences? the impact of the brown v. board of education decision on the employment status of black educators. education and urban society, 36(3), 280–303. villegas, a. m., & irvine, j. j. (2004). diversifying the teacher workforce: an examination of major arguments. the urban review, 42(3), 175–192. copyright: © 2019 leonard. this is an open access article distributed under the terms of a creative commons attribution-noncommercial-sharealike 4.0 international license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. microsoft word 423-article text no abstract-2465-1-6-20210911 (proof 1).docx journal of urban mathematics education may 2022, vol. 15, no. 1, pp. 9–30 ©jume. https://journals.tdl.org/jume julius davis is the university system of maryland wilson h. elkins associate professor of mathematics education and founding director of the center for research and mentoring of black male students and teachers at bowie state university, 14000 jericho park road, proctor building 233n, bowie, md 20715; email: jldavis@bowiestate.edu. his scholarly interests focus on critical race theory in mathematics education, and he conducts research of black male students and teachers in mathematics education in urban schools. field disruptions and field connections disrupting research, theory, and pedagogy with critical race theory in mathematics education for black populations julius davis bowie state university s i wrote this article, critical race theory (crt) came under attack in the public, school systems, and the law. this is not the first time that crt has been attacked in public. the most recent attack can be traced to christopher rufo, a senior fellow at a libertarian think tank known as the manhattan institute (harris, 2021). in 2020, he received a report from a seattle city employee about race-based training for staff that purportedly condemned white people and created racial divisiveness (klinghoffer, 2020). rufo (2020b) wrote about the training in the manhattan institute public policy magazine. he discussed how the trainings were being used against white people to provoke supporters of himself and the manhattan institute to take action against the trainings. rufo indicated that this type of training was infecting every part of the city government. he wrote that this type of training was a part of a nationwide movement, but he did not explicitly use the term crt. his article led informants from school systems and federal agencies around the country to reach out to him to complain about race-based training of which they had firsthand or secondhand knowledge. later, rufo used the term crt for the first time in an article where he described the concepts of “whiteness,” “white fragility,” and “white privilege” and how these were spreading rapidly throughout the federal government (rufo, 2020a). in september 2020, rufo was invited to tucker carlson’s show on fox news and warned that the theory was being weaponized against americans (code for white people), publicly requesting that president donald trump ban crt trainings in all federal departments. the president heeded his request. shortly after, trump signed an executive order banning federal departments’ and contractors’ use of crt in trainings. the order was immediately challenged in court, and a federal judge ruled against it. president joe biden rescinded the executive order. by the time the order was rescinded, however, crt had been vilified in the media and became the catchall phrase for any trainings or teaching activities involving race, racism, diversity, equity, inclusion, and so on. many politicians in state legislatures, the united states house of representatives, and the united states senate used rufo's flawed information to create a davis field disruptions and field connections journal of urban mathematics education vol. 15, no. 1 10 legislation to ban crt in school systems and government agencies without a proper understanding of the theory or direct knowledge it was actually being used as propagated. critical race theory legislation was intended to scare off and prevent government agencies, school systems, and companies from discussing systemic racism and the adverse educational and societal conditions resulting from it. there has been a push to pass laws banning schools from teaching or providing professional development in crt. opponents of crt have a problem with the theory’s foundational stance that racism is a natural fabric of america. although crt emerged in the law and education due to widespread discontent with issues of race and racism, it is now being attacked because of mostly white political and parental opposition to discussing and addressing issues of race and racism in schools and classrooms. droves of white parents have been protesting the perceived use of crt in schools, classrooms, and with students, creating public and school system uproar over anything that sounds like their views of crt. parker and lynn (2002) warned that “crt in education will come under the same attack it is facing in the legal arena” (p. 17). the mathematics education community is familiar with public attacks against our colleagues whose scholarship focuses on race, ethnicity, equity, whiteness, and other critical topics. in 2017, one of our colleagues, rochelle gutiérrez, was publicly attacked by two media organizations, the campus reform and fox news. the media outlets cherrypicked specific phrases from her scholarship to incite the public to condemn her. gutiérrez was inundated with hundreds of hate-filled emails, voice messages, and social media posts (e.g., twitter, facebook). alt-right groups produced podcasts and more media to slander gutiérrez and her scholarship. interestingly, although gutiérrez (2017) is one of many mathematics education scholars with critical scholarship that has challenged the status quo over the last fifty years, she nonetheless became a central target for antagonists. gutiérrez (2017) and davis and jett (2019) warned that scholars who seek to challenge the status quo and address issues of race, racism, and white supremacy, or who use crt, should be mindful that they may be attacked for their critical scholarship. like critical scholarship in mathematics education, crt has been around for fifty years and has been attacked in the public before, mainly in the legal realm. like the attack on gutiérrez, the public and legislative attack on crt is new in education. crt in law, education, and mathematics education is an established theoretical, methodological, and pedagogical framework that scholars can use to disrupt racism, white supremacy, and generational inequities in education and mathematics education if fully understood and appropriately used (davis & jett, 2019). solórzano and yosso (2002) argued that crt in education “is a framework or set of basic insights, perspectives, methods, and pedagogy that seeks to identify, analyze, and transform those structural and cultural aspects of education that maintain subordinate and dominant racial positions in and out the classroom” (p. 25). however, critical race theory davis field disruptions and field connections journal of urban mathematics education vol. 15, no. 1 11 in mathematics education (crt(me)) is known to a lesser degree by crt scholars in education and mathematics education (davis & jett, 2019) and the public. a small number of mathematics educators have used crt to disrupt and advance research, theory, and pedagogy for black adults and students. some scholars have used crt to promote mathematics and urban education (davis, 2014; jett, 2012; rousseau anderson & powell, 2009; terry, 2011). given their work, there has been a call to usher in a new generation of scholars to advance crt(me) and push the field forward toward greater racial justice for black adults and students. to fully understand or use crt(me), scholars must understand the crt literature in the law, education, and mathematics education and embrace crossing disciplinary boundaries. doing so would help mathematics education scholars develop a more robust understanding of crt's theoretical, methodological, and practical application and minimize its misuse in mathematics education. in this commentary article, i provide a brief overview of my crt(me) journey and the origins of crt in the law and education; examine crt in law, education, and mathematics education; and conclude with a discussion of how to continue to move crt(me) forward in the field. my crt(me) journey over 15 years ago, i entered my mathematics education doctoral program knowing that i wanted to study how race and racism impact black adults and children in my west baltimore community, but i did not know how to do it (davis, 2016). i started by reading literature on racism and black students’ mathematical achievements and experiences. in 2005, the second year of my doctoral program at morgan state university, i was introduced to crt in education by a colleague who shared marvin lynn’s (2004) article, “inserting the ‘race’ into critical pedagogy: an analysis of ‘race-based epistemologies.’” i read the article intently and appreciated the fusion of crt and afrocentricity. it aligned with my core beliefs and centered race, culture, schooling, and black education. at the time, lynn was an urban education faculty member at the university of maryland, college park. i asked him to be a member of my dissertation committee, led by mathematics educator roni ellington, and he agreed. since then, they have both been my mentors and, in many regards, have significantly influenced my crt(me) scholarship. lynn's (2004) article led me to further explore his work and the larger body of crt in education scholarship, most notably the publications of gloria ladsonbillings, william f. tate iv, and daniel solórzano. tate's crt and mathematics education scholarship provided me with a necessary foundation to build my crt(me) knowledge and understanding (see tate, 1993, 1997). it taught me the importance of reading and studying legal cases, critical legal studies, and crt in law and education to fully understand how to use crt in mathematics education and beyond. at the time, only five studies purported to use crt to examine black davis field disruptions and field connections journal of urban mathematics education vol. 15, no. 1 12 students, teachers, and parents in mathematics education (berry, 2003; brown, 1999; corey, 2000; martin, 2006; snipes, 1997). in 2005, i found robert berry's (2005) article utilizing crt to understand black male middle school students’ mathematical experiences. subsequently, i read berry’s (2003) dissertation, which demonstrated how black males’ and their parents’ experiential knowledge was critical to understanding how race and racism impacted their mathematics education through stories that prioritized their voices. his analysis helped me develop a deeper understanding of using crt to study black students’ experiences in mathematics education in urban schools. berry’s research demonstrated how individual acts of discrimination were connected to institutional racism and how this created barriers for black male students in mathematics education. his results illustrated how white adults sought to exclude black students from higher level mathematics courses and gifted education. it became my go-to model for constructing my dissertation as i thought through conceptualizing the theoretical framework, informant narratives, analyzing data through a crt lens, and developing themes. berry's use of crt in his mathematics education scholarship has significantly impacted my understanding and use of the theory. in 2007, as a doctoral student, i attended the first critical race studies in education association (crsea) conference at the university of illinois at chicago (uic). lynn, a co-founder of the conference, was transitioning into a faculty position at uic to join forces with other critical race and race critical scholars like danny martin, dave stovall, and william watkins. i formally met danny martin, another distinguished mathematics education race scholar who impacted my development as a critical black scholar, at the conference. his scholarship has continuously provided critical perspectives on race, racism, and whiteness (see martin, 2006, 2007, 2008). it has ushered in a liberatory paradigm to achieve racial justice for black adults and children. i collaborated with martin in 2008 to produce my first publication as a doctoral student, “racism, assessment, and instructional practices: implications for mathematics teachers of african american students,” in the journal of urban mathematics education. these experiences and others have strengthened my development as a crt scholar. i dedicated my doctoral studies to reading and studying legal cases, critical legal studies, and crt in the law, education, and mathematics education. i completed my dissertation on black students’ lived realities, schooling, and mathematics education in 2010 and have continued to develop my understanding of crt both within and outside of mathematics education since. after my dissertation, i continued to participate in the crsea conference, present my crt mathematics education scholarship at conferences, publish articles, and read crt publications (see allen et al., 2018; davis, 2014, 2016). as a faculty member, i have supported doctoral candidates that used this framework and methodology in their research across disciplinary boundaries. i have also used crt to deliver professional development for educators davis field disruptions and field connections journal of urban mathematics education vol. 15, no. 1 13 and leaders like the ones rufo (2020a, 2020b) demonized while calling for governmental and public outrage, as mentioned in the introduction of this article. i co-edited critical race theory in mathematics education (2019) with christopher c. jett almost ten years after my dissertation. i have used different elements and tenets of crt in education, research, policy, legal analysis, and practice. it is now a common thread that runs through my scholarship and thinking (davis, 2019). origins of crt in the law and education critical race theory emerged out of the critical legal studies (cls) movement and surrounding discourse. in the late 1970s, scholars developed the cls movement to reevaluate “the merits of the realist tradition of legal discourse” (tate, 1997, p. 207). critical legal studies scholars designed a movement to analyze “legal ideology and discourse as a mechanism that functions to re-create and legitimatize social structures in the united states” (tate, 1997, p. 207). although crt was born out of the cls movement, it is separate from the earlier cls movement and discourse (ladson-billings, 1999). critical race theory was a logical outgrowth of discontent with cls scholars’ failure to address racism. therefore, crt operates from the presumption that racism is deeply rooted in american society and is perhaps a permanent fixture. in other words, racism has never ceased to exist in american society; it has simply shifted over time from more overt and blatant forms of racism, as seen in old television footage, to more “subtle, hidden, and often insidious forms of racism that operate at a deeper, more systematic level” (lopez, 2003, p. 70). racism was seldom contextualized as deeply rooted in the fabric of american institutions (e.g., schools, government, etc.) and connected to inequities or a larger system—where individual acts and institutions function as a system. during the 1990s, crt emerged in education because race and racism were untheorized in scholarly inquiry and scholars wanted to bring these issues to the forefront (ladson-billings, 1999; ladson-billings & tate, 1995; tate, 1993). william tate started using crt in the field of education in his 1993 article “advocacy versus economics: a critical race analysis of the proposed national assessment in mathematics,” where he introduced it to the mathematics education community by exposing the racist underpinnings of standardized testing. tate joined forces with colleague gloria ladson-billings to formally establish crt in the field of education, advancing this theoretical framework by connecting race and property rights as an analytic tool for understanding social inequity, educational inequity, and inequity in mathematics education (ladson-billings & tate, 1995). in this article, ladson-billings and tate (1995) illustrated the significance of crt, not only to the field of education, but to mathematics education, in particular, as well. ladson-billings and tate advanced the notion of intellectual property as a form of property rights and whiteness as property. davis field disruptions and field connections journal of urban mathematics education vol. 15, no. 1 14 two years later, tate (1997) provided a comprehensive description of crt and its intellectual underpinnings to provide the educational community with a way to understand crt and its use in education. in 1998, laurence parker and colleagues renewed interest in crt in education with a special issue of the international journal of qualitative studies in education, most of which was later published in a book, race is . . . race isn’t: critical race theory and qualitative studies in education (parker et al., 1999). one of the most notable articles published from this work was ladson-billings’ article (1998), titled “just what is critical race theory and what’s it doing in a nice field like education?” in this document, ladson-billings (1998) argued that crt could be a powerful tool for understanding inequity that people of color experience through the configuration of education. she critically examined curricula, instruction, assessment, school funding, and desegregation as discussion points to examine culture, white supremacy, racist beliefs about black students, teaching competencies needed to teach students, and social constructions of race and racism. simultaneously, in the late 1990s, crt scholars solórzano and colleagues advanced scores of theoretical perspectives, methodologies, and empirical investigations into the experiences of latinos/as and chicanos/as (see solórzano, 1997; solórzano & solórzano, 1995; solórzano & villalpando, 1998; solórzano & yosso, 2002). in 2002, marvin lynn and associates published a crt special issue in qualitative inquiry in which there was discussion of the relationship between theory and method (see lynn et al, 2002). this special issue employed crt to examine the impact of race and racism throughout the entire educational pipeline from elementary, middle, and high school to the university. this body of work represents some of the foundational crt scholarship in education. since then, scores of crt in education scholarly publications have been produced. the handbook of critical race theory in education is a notable advancement of the theory, methodology, and practice (lynn & dixson, 2013). defining elements of crt in education solórzano and yosso (2002) identified five elements that form the basic insights, perspectives, methodology, and pedagogy of crt and critical race methodology in education. first, crt recognizes the centrality and intersectionality of race and racism. this first element recognizes that crt in education begins with the premise that race and racism are endemic and permanent (solórzano & yosso, 2002). social constructions of race and racism are at the center of a critical race analysis in education. crt recognizes that social constructions of race provide a basic perspective for understanding what racism is and how racism works. for critical race theorists, social constructions of race are central factors for describing how racism functions in u.s. society, the law, policies, schools, and mathematics settings. davis field disruptions and field connections journal of urban mathematics education vol. 15, no. 1 15 second, crt challenges the dominant ideology. critical race theorists challenge traditional claims of “objectivity, meritocracy, colorblindness, race neutrality, and equal opportunity” (solórzano & yosso, 2002, p. 26). critical race theorists expose these traditional claims as a disguise for white individuals’ self-interest, power, and privilege in u.s. society, particularly in law, policies, schools, and mathematics settings (ladson-billings, 1999; solórzano & yosso, 2002; tate, 1993). critical race theorists recognize that these traditional claims have shifted over time. for example, research on race and intelligence (gould, 1981, 1995; jensen, 1969) has shifted over time to focus on cognitive abilities (herrnstein & murray, 1994) to mathematics ability (martin, 2007, 2009; tate, 1993), using statistics and numbers as neutral and objective descriptors of ability and standardized testing as the barometer for measuring ability (davis & martin, 2008; ladson-billings, 1999; tate, 1993). critical race theorists challenge traditional claims that operate to protect white privilege by rejecting notions of neutral research or objective researchers and exposing research informed by deficit theories, beliefs, and assumptions about people of color that silence and distort them. third, crt is committed to social justice. the use of crt acknowledges a commitment to social justice, liberation, or transformative solutions to racial, gender, and class oppression (matsuda, 1991; solórzano & yosso, 2002). crt’s commitment to social justice seeks to eliminate racism, sexism, and classism and seeks to empower oppressed racial/ethnic groups. crt recognizes that “multiple layers of oppression and discrimination are met with multiple forms of resistance” (solórzano & yosso, 2002, p. 26). fourth, crt centralizes the experiential knowledge and voices of people of color. crt scholars recognize “that the experiential knowledge of people of color [i]s legitimate, appropriate, and critical to understanding, analyzing, and teaching about racial subordination” and racism (solórzano & yosso, 2002, p. 26). crt scholars use poetry, storytelling, biographies, scenarios, family histories, counterstories, parables, chronicles, narratives, stories, fiction, and revisionist histories to give voice to the experiential knowledge of people of color (delgado, 1995; ladson-billings & tate, 1995; solórzano & yosso, 2002). crt’s voice component “provides a way to communicate the experience[s] and realities of the oppressed” (ladson-billings & tate, 1995, p. 58) to understand the complexities of racism. there are at least three reasons for incorporating the voices of people of color: (1) reality is socially constructed, (2) stories are a powerful tool for shaping mindsets, and (3) stories are instrumental to the development of a common culture of shared understanding and community building (delgado, 1989; ladson-billings, 1999; ladson-billings & tate, 1995). the voices of black adults and students are needed for a complete analysis of their lived realities, schooling, and mathematics education in urban areas. their experiential knowledge is gathered from a shared history of ongoing struggles davis field disruptions and field connections journal of urban mathematics education vol. 15, no. 1 16 with oppression and resistance. in urban areas, black students’ lived realities, schooling, and mathematics education continues to be shaped by white supremacy. fifth, crt takes an interdisciplinary approach. crt scholars use multidisciplinary knowledge, epistemologies, and methodologies to guide and better understand the effects of racism, sexism, and classism on people of color (solórzano & yosso, 2002). crt crosses disciplinary, epistemological, and methodological boundaries by borrowing from several traditions to point out how race and racism shape the experiences of people of color. critical race theorists insist on analyzing race and racism by placing them in historical, contextual, and contemporary contexts (solórzano & yosso, 2002; tate, 1997). solórzano and yosso (2002) acknowledged that these five elements are not new but collectively represent crt in education and challenge the existing traditional claims of scholarship that seek to mask racism in educational institutions and research. crt theoretical, methodological, and practical elements critical race theory offers the mathematics education community many theoretical, methodological, and practical tools to disrupt the field. scholars have used crt tools to center race, racism, sexism, classism, and other forms of oppression in research, policy areas, and practice. they also use it to achieve liberatory outcomes for black populations in mathematics education that disrupt the prevalence of racism, white supremacy, whiteness, and power. most of these elements have originated from crt in the law, and scholars have subsequently applied them to education and mathematics education. the brief description of the crt elements below provides an overview of its utility to scholars. the father of crt, derrick bell, created the concept of revisionist history to “reexamine america’s historical record, replacing comforting majoritarian interpretations of events with ones that square more accurately with” people of color (delgado & stefancic, 2001, p. 20). this theoretical element helps mathematics educators challenge historical accounts and interpretations that support white supremacy and the oppression of people of color. intersectionality was originally coined in 1989 by kimberle crenshaw to address issues of race and gender for black women, as well as structural, political, and cultural intersectionality. over time, intersectionality has emerged as a field of study (cho et al., 2013). this field consists of (a) an intersectional framework and dynamics, (b) a theoretical and methodological paradigm, and (c) political interventions using an intersectional lens (cho et al., 2013). intersectionality provides the mathematics education community with tools to address intersecting issues of race, gender, identity, class, theoretical and methodological analyses, and political activism. critical race theorists are discontent with liberalism. crt’s critique of liberalism concentrates on how race and racism are not adequately addressed, how equality davis field disruptions and field connections journal of urban mathematics education vol. 15, no. 1 17 of opportunity is not afforded to all, and how liberals’ insistence on incremental change prioritizes individuality and individual rights at the expense of addressing institutional and systemic racism and inequity. these elements create and maintain privilege for whiteness under the guise of having a level playing field. liberalism significantly impacts policies, practices, and discourses in education and mathematics settings. “crt maintains that liberalism actually serves as a key mechanism for sustaining and defending the status quo of white supremacy” (castagno, 2009, p. 757). in cheryl harris's (1993) seminal work, “whiteness as property,” she articulates blackness and whiteness as property, illustrating the connection between race and property rights. the property functions of whiteness were based on the idea that possessing white skin was valuable property that only white individuals could possess. harris (1993) describes four main property functions of whiteness: (a) rights of dispositions, (b) rights to use and enjoyment, (c) reputation and status property, and (d) the absolute right to exclude. the right of disposition contends that property rights are fully transferable to students who conform to white norms or sanctioned cultural practices in education and mathematics education. the right to use and enjoyment asserts that having white skin allows white individuals to use and enjoy properties of whiteness academically, mathematically, legally, and socially without question. the status and reputation of schools and programs in education, in general, and mathematics education, specifically, diminish when white people are not associated with these academic activities. the absolute right to exclude is socially constructed as the absence of blackness in schools, higher level education, and mathematics programs and courses. these property functions provide scholars with tools to understand how whiteness and blackness function in mathematics education in multiple settings. bell developed the interest convergence principle to explain how progress for black people is only achieved when their goals converge with the interests and needs of white people (see bell, 1980). the interest convergence principle hinges on the legal case surrounding brown v. board of education—a landmark 1954 supreme court case to end state-mandated racial segregation in schools. this principle serves two dual purposes in bell’s scholarship (tate, 1997). first, it contributes to the intellectual discourse on race in u.s. society, and second, it promotes political activism to achieve racial justice. interest convergence provides a lens for mathematics educators to become aware of and work to disrupt policies, legislation, and practices designed to meet the interests and needs of white people. ladson-billings and tate (1995) introduced intellectual property in their seminal article on crt in education. they underscored the utility of race and property to education and mathematics in their articulation of intellectual property. intellectual property can be used to understand how “the quality and quantity of the curriculum varies with the ‘property values’ of the school” (ladson-billings & tate, 1995, p. 54). property taxes provide the means to understand how “property” or “property davis field disruptions and field connections journal of urban mathematics education vol. 15, no. 1 18 values” are connected to education and mathematics education. the higher the community's property values, the higher the tax base, which is funding used to support schools and mathematics curriculum. in the context of mathematics education, the quality of the mathematics curriculum and availability of college-level or advanced mathematics courses, high-quality mathematics teachers, and resources tends to be better in predominantly white schools and communities with a higher tax assessment. the intellectual property varies based on the socioeconomic status, racial composition, and geographical location of the school and community. urban schools tend to have poor quality mathematics curriculum, lack college preparatory or advanced mathematics courses, and the real property (e.g., teachers) to support a proper education in mathematics because of their racial composition, tax base, and location. ladson-billings and tate (1995) demonstrated how the quality and quantity of the school and mathematics curriculum and real resources impact black students’ intellectual property and their opportunity to learn high-quality mathematics. crenshaw (1988) explained the expansive and restrictive views of equality in antidiscrimination law that helped advance crt. the expansive view describes equality as an outcome and seeks to use the institutional power of the court and legal system to eradicate racial oppression. the expansive view is cojoined with the restrictive view of equality, which explains equality as a process, minimizes the importance of actual conditions, and seeks to stop future acts of wrongdoing rather than addressing the root cause of the racial injustice or correcting it. in education, tate (1993) has used crenshaw’s expansive and restrictive views of equality to discuss educational policies and court rulings impacting black students in mathematics education. critical race theorists have developed and advanced critical race methodologies in qualitative, mixed methods, and quantitative approaches. critical race methodologies seek to accentuate silent or marginalized voices in qualitative data and humanize quantitative data (solórzano & yosso, 2002). critical race methodology is a research approach grounded in crt that merges theory and method. critical race mixed methodology combines crt and mixed methods approaches to expose racism, challenge racialized structures, and advance social justice through research design. gillborn et al. (2018) explained the theoretical and methodological parameters of quantcrit as centering racism in quantification, acknowledging numbers are not neutral, critically analyzing deficit-oriented analyses serving white interests, critically evaluating categories that are neither natural nor given, centering the voices and experiential knowledge of people of color as essential to understanding quantitative data, and acknowledging how statistical analyses can play a role in the racial justice struggle. there has been limited use of critical race methodologies in mathematics education. jett (2019) has called for increased use of critical race methodologies in the field. davis field disruptions and field connections journal of urban mathematics education vol. 15, no. 1 19 counterstories are also a popular crt qualitative method that originated in the law, and crt in education scholars describe counterstories a method of telling “the stories of those people whose experiences were not often told (i.e., those on the margins of society)” (solórzano & yosso, 2002, p. 32). counterstories are tools used for exposing, analyzing, and challenging the master narrative, monovocal and majoritarian stories. there are three general forms of counterstory employed by critical race theorists (solórzano & yosso, 2002). the first form of counterstory involves personal stories or narratives that describe an individual’s experiences with various forms of racism, classism, and sexism. the second form of counterstory involves other people’s stories or narratives that reveal another person’s experiences with and responses to racism, classism, and sexism as told in a third-person voice. the third form of counterstory involves composite stories or narratives that draw on various forms of “data” to narrate the racialized, sexualized, and classed experiences of people of color. terry (2010, 2011) conceptualized and advanced mathematical counterstories as a research and pedagogical tool to better understand the mathematics experiences of black students. yamamoto (1997) urged crt scholars in the law to engage in critical race praxis that bridges the theoretical constructs to everyday life. he argued for crt scholars to focus less on theory and more on “anti-subordination practice” (yamamoto, 1997, p. 873). he described critical race praxis as critical, pragmatic, socio-legal analysis with political lawyering and community organizing to practice justice by and for racialized communities. its central idea is that racial justice requires antisubordination practice. in addition to ideas and ideals, justice is something experienced through practice (yamamoto, 1997, pp. 829–830). david stovall has been one of the leading scholars calling for less critical race theorizing and more critical race praxis in education. stovall (2004) has advanced critical race praxis by calling for more work on the frontlines of racial issues impacting black and latinx populations in educational and community settings. critical race pedagogy was explained by lynn (1999) as “an analysis of racial, ethnic, and gender subordination in education that relies mostly on the perceptions, experiences, and counterhegemonic practices of educators of color” (p. 615). he argued further that “critical race pedagogues are concerned with four general issues: the endemic nature of racism in the united states; the importance of cultural identity; the necessary interaction of race, class, and gender; and the practice of a liberatory pedagogy” (lynn, 1999, p. 615). mathematics education scholars have engaged in critical race praxis and critical race pedagogy to a lesser degree. to advance crt in mathematics educational spaces, scholars can use critical race praxis and critical race pedagogy to disrupt whiteness, white supremacy, issues of race, class, and gender. those scholars who decide to take up these issues in the mathematics classroom must be educated and prepared for the political ramifications of the decision. in this davis field disruptions and field connections journal of urban mathematics education vol. 15, no. 1 20 section, i have provided a brief overview of how crt’s theoretical, methodological, and practical elements developed in law and education but have been advanced in the larger field of education, in general, and made connections to mathematics education. crt in the law and education many crt in education scholars have noted the importance of reading and studying crt legal literature. ladson-billings (1998, 2013) described how she and tate spent considerable time studying the legal literature and crt in law scholarship. they learned from and engaged with legal scholars to better understand how the law functions and how crt could be applied to education before using it in the field (ladson-billings, 1998; tate et al., 1993). ladson-billings (1998) stated, “educational researchers need much more time to study and understand the [crt] legal literature in which it is situated” (p. 22). during the latter part of the twentieth century, tate was a leading education researcher publishing crt in education and mathematics education scholarship. a salient feature of tate's scholarship is the importance of studying the origins and development of cls, crt in law, and education literature (see tate, 1997). in a published interview, tate discussed the importance of having a “sound knowledge of united states history, constitutional law, social theory, and its debates, psychology, education, and political institutions prior to engaging with critical race theory” (davis & jett, 2019, p. 15). academicians developed crt scholarship in the field of education and mathematics education in the late twentieth century. ladson-billings (1998, 2013) has described the early developments of crt in education, starting with a departmental colloquium, conference presentations, and publications dating back to the early 1990s. the culmination of these scholarly endeavors was ladson-billings and tate's (1995) seminal article, “toward a critical race theory of education” in teachers college record. then, in 1997, tate published a comprehensive overview of crt and its influential components in education. during the same year, solórzano (1997) began publishing on crt and developed a scholarship line at the university of california, los angeles (ucla) with other crt legal scholars. tate noted that ucla is a notable institution in the academy to study crt in law and education, primarily because of the faculty knowledge and understanding of the framework (davis & jett, 2019). in 1998, ladson-billings published an article to illustrate crt’s usefulness in education, explaining the framework’s defining elements and deterring educational researchers from delving into crt without adequate grounding in the legal literature. the law and education are intertwined, and education is not a part of the u.s. constitution but is the state government's responsibility. states enact policies, laws, and regulations that govern education. these early crt developments in education are significant because they illustrate how education scholars utilized legal scholars and davis field disruptions and field connections journal of urban mathematics education vol. 15, no. 1 21 crt in law literature to develop the knowledge and understanding needed to apply crt in an educational context. critical race theory in education scholars center race and push the educational community to consider race as more than just a research variable (lynn & dixson, 2013). in ladson-billings’ (2013) view, critical race theorists believe in the normalcy and omnipresence of racism, interest convergence of material determinism, race as a social construct, intersectionality or anti-essentialism, and voice or counternarrative. i would also argue that critical race theorists believe in achieving racial justice, and mathematics education scholars have critiqued social justice approaches in the field that have not adequately addressed racial justice (larnell et al., 2016). in my view, critical race theorists must work toward liberation by finding solutions to identified problems. in the early phases of crt's development in education, ladson-billings (1998) recognized that the framework was in its infancy, which meant that scholars needed to exercise caution to ensure the theory was not misused. she later reiterated this point in a book chapter that describes what crt in education is not (see ladsonbillings, 2013). she stated that just because a scholar examines race in their work or writes about race and racial issues, this does not make them a critical race theorist. lynn and dixson (2013) made similar arguments. they stated, some scholars claim a crt project simply because their sample may be primarily composed of people of color. far too often, scholars have invoked crt in the introductory sections of their paper, never to revisit the theory or even utilize any of its tenets in their analysis. (lynn & dixson, 2013, p. 3) mathematics education scholars must adhere to this guidance to avoid misusing crt in the field because it interferes with the framework's continued progress. many crt scholars warned of the trivialization and misuse of this framework. they have also advocated using crt in education to rigorously examine race, racism, classism, sexism, oppression, laws, and policies to achieve liberatory outcomes for black adults and students (lynn, 2004, solórzano & yosso, 2002; tate, 1997). i concur with lynn and dixson (2013) that “far too many scholars who have an interest in examining race and racism in education misunderstand and misuse crt” (p. 3). mathematics education scholars, too, misuse and misunderstand the theory. some scholars who include crt in their analyses do so in a superficial way. overall, we must safeguard crt(me) to ensure its proper use in the field (rousseau anderson, 2019). davis field disruptions and field connections journal of urban mathematics education vol. 15, no. 1 22 crt in mathematics education mathematics education scholars have contributed immensely to the furtherance of crt in education. tate and rousseau anderson are two notable mathematics education scholars who have played an instrumental role in advancing crt’s prominence in the larger field of education (e.g., dixson & rousseau, 2006; tate, 1997). however, their crt contributions are not widely known in mathematics education because they did not see their scholarship as directly connected to the field. in an interview, tate stated, “i do not view my crt project as a mathematics education project” (davis & jett, 2019, p. 12). in the same volume, rousseau anderson (2019) indicated, i would not describe myself as a mathematics educator who uses critical race theory. rather, i consider myself a critical race theorist who happens to also be a mathematics educator. while this distinction may seem minor, it is one reason why much of my scholarship on crt does not focus specifically on mathematics education. (p. 19) although their mathematics education knowledge has proven beneficial in their crt scholarship, it has not been a driving force. therefore, christopher c. jett and i have called for mathematics educators to merge their crt and disciplinary knowledge for stronger connections to the field and to continue advancing it deliberately. in our edited book (davis & jett, 2019), we described how william f. tate iv pioneered this theoretical perspective in mathematics education and the broader educational research community. tate's scholarship is salient because he has continuously connected mathematics education to crt, even though the discipline was not a part of his crt conceptualization. for instance, tate and colleagues (1993) used crt and his background in mathematics to examine the social problems underlying the brown v. board of education decision. when tate introduced crt to mathematics education in 1993, scholars did not widely use the theoretical perspective in the discipline or larger education field. since crt’s introduction, many mathematics educators have used it, but their scholarship is rarely referenced or connected to the broader crt discipline, nor is it assembled into a collective volume (davis & jett, 2019). in the 1990s, few education researchers focused their work on crt in education and mathematics education (rousseau anderson, 2019). snipes (1997), brown (1999), and corey (2000) are among the few mathematics education researchers who used crt in the early years. their research was heavily cited by ladson-billings, tate, and other crt scholars. in the 21st century, the use of crt(me) increased as scholars used it to advance theory, methodology, and pedagogy to disrupt the field and elevate new knowledge of black adults and students (see, for example, berry, 2003; corey, 2000; davis, 2010; jett, 2009; mcgee, 2009; terry, 2009; wilson, 2018). these scholars continued the tradition in mathematics education by using crt in their dissertation research davis field disruptions and field connections journal of urban mathematics education vol. 15, no. 1 23 to advance the field. they are responsible for bolstering publications in this area (see berry, 2008; corey & bower, 2005; davis, 2014; jett, 2019; mcgee, 2013; snipes & waters, 2005; terry, 2010, 2011). beyond dissertation research, mathematics education scholars have used crt to advance knowledge through journal and book chapter publications. they have challenged deficit views of black adults and students and increased knowledge of the usefulness of this perspective for black populations (see berry et al., 2014; cobb & russell, 2015; larnell et al., 2016; leonard, 2007, 2009; leonard et al., 2013; martin, 2006; martin et al., 2017; rousseau anderson & powell, 2009; russell, 2013; strutchens & westbrook, 2009; terry & mcgee, 2012). this body of scholarship represents crt’s growth in mathematics education, but it does not necessarily represent grounding in the crt legal and education scholarship. the early mathematics education studies were developed alongside crt as a theoretical and methodological framework (brown, 1999; corey, 2000; snipes, 1997), which suggests that the scholars’ understanding of the theory was limited and developed simultaneously. there is limited evidence to indicate whether snipes, brown, or corey extensively read or studied the legal literature, crt in law, and education literature before using it in mathematics education. these studies often present racism as an emergent construct and neglect its omnipresence in their crt analysis. moreover, brown (1999) often presented large chunks of data with superficial, limited, or no crt analysis. part of brown’s crt analysis limitation is the use of three theoretical frameworks (critical theory, crt, and culture practice theory) that are presented as analogous with one another. her crt analysis was often presented as a rehashing or citing of crt scholars’ writings. brown’s research is an example of how mathematics education scholars must avoid misusing crt. in an interview, tate stated that he observed crt mapped onto cases involving mathematics education (davis & jett, 2019). he questioned the awareness of a definitive article, book, or research program that delineates a crt(me) (davis & jett, 2019). tate asserted that he was not aware of an overarching crt(me) body of scholarship or an existing collection of crt(me) publications (davis & jett, 2019). in our edited book critical race theory in mathematics education, i and jett (2019) sought to create a volume for the mathematics education community that demarcates this perspective for the field, identifies the full spectrum of literature, charts a path forward, and encourages serious scholars to use it as a theory, methodology, and pedagogy to advance the discipline. we sought to assemble veteran and emerging scholars to promote crt’s future direction in mathematics education. when we created this edited book, we had no idea that crt in education would come under attack shortly after and impact how mathematics educators use it pedagogically in the classroom. critical race theory in mathematics education (davis & jett, 2019) is an edited book that provides insight into how scholars have used this theoretical, methodological, and pedagogical approach in mathematics education. in crafting the book, davis field disruptions and field connections journal of urban mathematics education vol. 15, no. 1 24 we sought the father of crt(me), william f. tate iv, to contribute to the book but learned that his scholarly interest had shifted away from crt. he viewed his former students as scholars and left it to them to advance crt further. notably, tate considered his former student and prominent crt scholar celia rousseau anderson the savant to forge crt(me) scholarship ahead. in the book, rousseau anderson (2019) articulates crt’s intellectual boundaries in mathematics education using the crt in legal studies scholarship. she explains how crt could be applied to mathematics education. as a volume, critical race theory in mathematics education (davis & jett, 2019) builds on extant critical legal studies, crt in law, education, and mathematics education. it advances new knowledge of crt and ushers in a new generation of crt scholars in mathematics education. the expansion of crt(me) is crucial because it provides a framework for examining the permanence of race and racism, classism, sexism, and other forms of oppression. crt provides the methodological, theoretical, pedagogical, and analytical framework to shed light on how race, racism, economics, laws, policies, and other forms of oppression shape the mathematical and social experiences, trajectories of achievement, and professional experiences of black adults and students. in this book, the contributors posit a historical counterstory; illustrate the use of allegorical storytelling and personal narratives; critique mathematical proficiency, standardized testing, recruitment, and retention of black mathematics teachers; and present a paradigm of liberatory mathematics education for black students. utilizing crt to disrupt and advance the field of mathematics education as critical race theorists, we must disrupt whiteness, white privilege, and white supremacy in legislation, mathematics curriculum, standards, and practice. scholars have challenged whiteness, white privilege, and white supremacy in education and mathematics education legislation and noted that the black community has not benefited from the law (ladson-billings, 2006; martin, 2008). the mathematics curriculum used in classrooms worldwide promotes the idea that white men have been the main contributors to mathematical knowledge (anderson, 1990). in mathematics education, scholars have illustrated and explained that standards give the illusion of meeting everyone’s interests but serve the interest of those in power (i.e., the white community; apple, 1992). most of the practices used in mathematics are based on white middle-class norms and culture. the voices and experiences of the black community have been historically and contemporarily left out of mathematical spaces, curricula, policy, pedagogical approaches, and intellectual spaces. as critical race theorists, we must continue to center the voices and racialized experiences of black adults and children to understand davis field disruptions and field connections journal of urban mathematics education vol. 15, no. 1 25 their mathematical experiences better and challenge the status quo. crt provides the tools to center the voices and experiences of black learners in the curriculum, theory, research, and practices used in mathematics education. critical race theorists in mathematics education must prioritize liberatory and racial justice practices in informal and formal mathematics settings to disrupt race neutrality and racism. in this article, i have described crt tools that scholars can use to achieve liberatory and racial justice outcomes, but we must decide where to focus our efforts. are liberatory and racial justice actions going to be focused on policy, both within and outside of school and classroom settings? we must ask ourselves, “can liberatory and racial justice be achieved in the mathematics education enterprise?” rousseau anderson (2019) provided guidance for scholars who seek to utilize crt(me) to transform the field. thus, jett and i have called for mathematics educators to first develop the proper understanding of crt(me) to use it properly. rousseau anderson (2019) indicated that crt scholarship involves both action in and reflection on mathematics education. she asked several critical questions to move crt forward in mathematics education: what is the goal of utilizing crt in mathematics education? how will we measure the success of crt? who is our primary constituency? should we think of ourselves first and foremost as academics or advocates? essentially, where are we going? (rousseau anderson, 2019, pp. 29). more importantly, rousseau anderson (2019) insisted that “our students are paying a heavy racial tax in schools every day. what are we doing to alleviate that burden?” (p. 29). as a mathematics education community that values racial justice, i submit that we must continue to grapple with and answer these questions individually and collectively in our crt(me) work. although many mathematics educators have used crt, there is still more work to be done to fully utilize this theoretical, methodological, pedagogical, and analytical framework (berry, 2008; davis, 2014; jett, 2011; mcgee & martin, 2011; terry & mcgee, 2012). i assert that all scholars should not use crt(me). it should be used by those who have (a) an operationalized definition of race and racism; (b) access to the critical perspectives of black adults’ and children’s lived realities, schooling, and mathematics education; (c) the sociohistorical context to analyze race and racism; and (d) a sociopolitical perspective (davis, 2019). if mathematics education scholars are to use crt effectively, they must be prepared for possible backlash and fully commit to studying it and achieving racial justice and liberation for black adults and students in the field. davis field disruptions and field connections journal of urban mathematics education vol. 15, no. 1 26 references allen, k. m., davis, j., garraway, r. l., & burt, j. m. 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(1997). critical race praxis: race theory and political lawyering practice in postcivil rights america. michigan law review, 95(4), 821–900. copyright: © 2022 davis. this is an open access article distributed under the terms of a creative commons attribution-noncommercial-sharealike 4.0 international license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. he who feels it, knows it: rejecting gentrification and trauma for love and power in mathematics for urban communities journal of urban mathematics education december 2018, vol. 11, no. 1&2, pp. 11–25 ©jume. http://education.gsu.edu/jume lou edward matthews, ph.d. is the director of mathematics and science at urban teachers, a national nonprofit that recruits, trains, places, and coaches urban teachers in baltimore, md, dallas, tx, and washington, dc in partnership with johns hopkins university. he who feels it, knows it: rejecting gentrification and trauma for love and power in mathematics for urban communities lou edward matthews urban teachers he who feels it, knows it. – african proverb t has been almost 25 years since tate’s (1994) “state of the union address” on mathematics education for black students invoked woodson’s (1933/1990) passage as a clarion call to reject the foreign “pedagogy” of mathematics for black students: and even in the certitude of science or mathematics it has been unfortunate that the approach to the negro has been borrowed from a “foreign” method. (p. 4) having read tate’s article soon after beginning my phd studies (circa 2000), it was one of my earliest awakenings in the field. it had crystalized a prior visit to a wisconsin classroom on culturally relevant pedagogy taught by professor gloria ladson-billings and the admonition she spoke to me after class: “lou, we don’t need another study on functions” (referencing the cognitively guided instruction dominant mathematics culture at my university). i understood this we. our people. my people. it was the felt presence of my community and communities of the african and caribbean diaspora upon whose hopes and dreams this work is pursued. yet, tate’s call was the harbinger of a more intrusive element in the mathematics of our people—the mathematics education reform enterprise itself. he who feels it knows it from the very onset of the founding movement of the journal of urban mathematics education (jume) with my georgia state university colleagues david, christine, pier, and ollie,1 the goal was to take up tate’s (1994) call to challenge the foreign spaces in mathematics. the tipping point of the work was to create a truer, edgier, more caring environment to engage the “urban” than was available. in 2008, mathematics educators in the urban field found it necessary to sidestep 1 the original jume editorial team included me, the founding editor in chief, and associate editors pier junor clarke, ollie manley, david stinson, and christine thomas. i http://education.gsu.edu/jume matthews rejecting gentrification in mathematics journal of urban mathematics education vol. 11, no. 1&2 12 mainstream mathematics journals, risk promotion and tenure, form deeper connections outside the reach and focus of their respective university bases, soul search, and then make broad leaps in search of a promise land of scholarly and professional pasture not yet seen in mathematics education. the scant treatments of urban topics (e.g., equity, culture, race, language) present in special issues and one-offers in mainstream publications masked the stories of rejection, discounting, shaming, blacklisting, and even public humiliation experienced by urban mathematics scholars. luckily, these testimonies were heard out in the safe spaces carved out at conferences, national and regional meetings, summits, lunch rooms and countless happy hours. it was a time when engaging the status quo came with great risk. i recall that even after it was clear jume would be a reality, i was called into the office of the associate dean to be relayed a message by the dean that beginning a scholarly journal was not the work of tenure-seekers—most of us were not tenured at the time. and now, thousands of pages later, a scholarly line of resistance has been drawn in the sand in and through jume. because the very notion of urban and its use in mathematics has been troubled and re-focused (and re-re-focused) as an ongoing concern, the public stories of the mathematics educator have been recognized and installed as critical to producing the urban. even more importantly, more questions and concerns have given rise to what the next 10 years might look like. jume has become a safe space for edgy, moving work. its impact on my own life has been unmistakable and its presence over the last decade has beckoned a far more brazen question than many of us have been able to ask in open quarters: can mathematics reform be trusted? is it a safe space for our people? in the space that follows, i wish to expand on my own reconnaissance around those questions and my current resolve as one experiencing mathematics in urban communities across the diasporas. much of this time in the last 10 years has been spent in public education and mathematics in bermuda, participating in black community action and development and exploring mathematics in the caribbean, united states, and africa. these experiences have led me to black communities in the cayman islands, canada, ghana, bermuda, and the united states. i find it fascinating to include these in the mathematics space of urban for a couple of reasons: (a) to extend the inclusion of urban issues in mathematics past the geographical and editorial boundaries of the united states into the invisible caribbean and african diasporas, and (b) to affirm jume’s “beautiful” vision of the urban as a safe space where the socio-political and -cultural complexities of mathematics teaching and learning in black (brown and “othered”) communities can be discussed. my lived realities have taught me that the (mathematics) experiences of black communities have little to do with geography (as one might think of urban) and more to do with the complexities of colonialism, racism, and socio-cultural politics. these multiple lived geographic spaces have become a good place to think about we. in looking at my own “identity crisis” in an early editorial for jume (matthews, matthews rejecting gentrification in mathematics journal of urban mathematics education vol. 11, no. 1&2 13 2009), i suggested that we must begin to redefine mathematics education as a movement of people (as opposed to content and method). in reality, school mathematics has most often embodied a movement on people. my feeling is that communities appear helpless to resist the endless parade of initiatives driven by various political entities, powered by a limited view of the nature of mathematics (as abstract, independent, and absolute), and an even more restrictive view of who should have access to mathematics. this scenario has played out “for real, for real” in the urban schooling spaces i have been blessed to live, work, and frequent over my life. in short, this mathematics reform “ain’t been for us,” and it feels foreign. and in the words of the african proverb and the legendary bob marley, “he who feels it, knows it.” since that time, i have been grappling with an ever-evolving set of tensions around my pubic leadership narrative in mathematics and stem (science, technology, engineering, and mathematics) education. the public narrative involves the intersectionality and integration of lived experiences and values in and around mathematics (my story); my connectedness to community(ies) within the african and caribbean diasporas (our story); and the relevance, authenticity, and positioning of mathematics reform as a liberation force (our mission). the more time i spend in these communities, the more this mathematics reform feels foreign. because he who feels it, knows it, i offer insight into these tensions here as i share four moments of the last 10 years. mathematics reform as a gentrification force i relocated to a southeast neighborhood in washington, dc in 2017. i had been to dc many times in the last 20 years, but the demographic and economic shifts had never been more apparent than in this southeast neighborhood. once a predominantly black neighborhood shared with naval buildings and a/the shipping yard, the transition into a gentrified space was immediately felt by me. complete with a cornerstone starbucks, a plethora of salad shops, gyms, boutique pizza joints, pet stores, and grooming salons. all of these changes are headlined by a boardwalk along the river with new construction projects of condominiums, a stadium, and residences who have replaced the old neighborhood. remnants of the old are now further across the river where predominantly black neighborhoods and schools exist under the ever-hanging threat of gentrification. my neighbors are mostly white as well as the thousands of people that visit the area daily to shop, eat at the restaurants, or attend the baseball and soccer games. each morning, dozens of latino and black men arrive to work at the construction sites constructing new apartments, while similarly, many of the workers at coffee shops and grocery stores are people of color. the cleaners and concierges of my apartment building are also people of color. in the midst of this, i am never more clear of my conflicted position. admit matthews rejecting gentrification in mathematics journal of urban mathematics education vol. 11, no. 1&2 14 tedly, i am double-minded: wanting to experience the familiarity and safe space of black community yet situated, and consequently pulled by the relative comforts of urban reform. i travel out of the neighborhood each week to my black church (they are scarce) and my black barber (even more scarce). the way mathematics reform happens looks and feels like gentrification, a foreign force which devalues and displaces black and brown communities. each day i am aware of the clear lines of division that are framing my comfort. this compromised space is similar to my growing positionality in mathematics and reform. it happened recently visiting and working with teachers in ghana in one village’s junior high school. even here, the remnants of past mathematics reform efforts to engage ghanaian children were evident as teachers followed a united kingdom styled reform textbook that had been refined to include ghanaian names, while not allowed to use the native ashanti twi to teach (the children spoke this dialect of the akan language in the playgrounds and in informal circles). similarly, a reform narrative had already started, centered around how unprepared and unready students were for higher level of study—a common refrain in urban spaces. for purposes of this conversation, anyone familiar with a gentrification process can trace at least these four characteristics: (a) degradation and devaluation of the existing areas and spaces, (b) displacement of the established people and narrative, (c) replacement with the new design, and (d) dissemination of the new narrative in the “new” place. think about the motivations for how we have been taught to work as reformers. first, neighbourhoods are deliberately, as unintended consequences of larger societal forces, unable to sustain the necessary conditions for economic, physical, social, and communal growth. the neighbourhood becomes the “hood.” in retrospect, every new curriculum, initiative, or program i have ever encountered in mathematics—as teacher, professor, researcher, administrator—has first worked to create or exploit a narrative in which the established paradigm of mathematics and school practices was first demonized. from spartanburg, south carolina to gomoa fetteh, ghana, the work of improving mathematics has focused on addressing achievement gaps, reworking traditional environments, motivating learners to pursue mathematics or stem, and answering national crises (economic mostly) that all headline an urgent narrative communicated powerfully throughout the pipeline. the narrative is set and acted upon. struggling learners, traditionally underrepresented, multi-language speakers, minorities, and black boys and black girls are named, identified as the displaced. the displacement can be physical if one thinks about the tracking and intervention solutions in some school, but it is almost always a displaced mathematical and cultural identity. the displaced become the face of the (equity) campaign and often viewed absolutely and synonymous with the narrative in much the same way distressed neighborhoods are ghettoized in the minds of the police and potential residents. the mathematics classrooms of urban children are likened unto abandoned matthews rejecting gentrification in mathematics journal of urban mathematics education vol. 11, no. 1&2 15 buildings, housing poor quality educators and resources, and hard to staff. in response new initiatives, curriculum, products, and interventions, creations of the mathematics education enterprise, all representing the new design are deployed. finally, the new narrative (when this works) is heralded usually tied to achievement scores and assessment. by the time new initiatives arrive in the community, the deficit narrative surrounding people and practices have usually been frequently articulated. this one-sided interrogation of urban communities works to establish a dominant power position on the behalf of mathematics education reformers. this process of reform is not dissimilar to the first sighting of colonial fleets on the shores of new lands. in this role, mathematics agents often play a policing role—to enforce non-deviation from new narrative. until interrupted, i was being trained to tacitly accept that the very nature of mathematics reform was good for our communities. the promotion of “good” mathematics teaching and rich mathematics tasks and opportunities were at the heart of mathematics for all doctrine. yet, seeing mathematics reform as gentrification is a far more palatable position as it has allowed me to organize for resistance and liberation more effectively. resistance is a natural orientation for black and urban communities. this stance allows community activist to: (a) identify the hidden, unaddressed drivers of seen change; (b) organize protection of hidden or neglected legacy and wisdom that might otherwise be ignored; and (c) galvanize renewed community-building efforts in ways that resist displacement. such a positioning may allow us to place at the forefront the beauty and power of urban environments. returning to that ghanaian school mentioned earlier, after several days, i was blessed to unearth counter stories centered on the use of twi dialect, cultural mathematics experiences (students playing marbles at recess), and a communal identity among students (witnessed in the village)—all, outside of my espoused mathematics education domains. toward math as black (brown and urban) power a few years ago, i was involved in a protest on the cabinet grounds of the bermuda parliament. thousands of civil service workers over the course of several days had left work to protest the forced extension of furlough days, a budget cutting measure offered by government workers to reduce government expenditures. bermuda had not yet recovered from the devastating global recession of 2007-08 and all government agencies were reeling from systemic cuts, early retirements, and attrition. in the absence of cost of living increases, the furloughs were defacto pay cuts felt deeply by the thousands of government workers representing the working middle class. in the days that followed, i wrestled as the acting commissioner for the department of education over whether to join the ranks of the people who were steadily amassing or continue to work as senior management (although we were all matthews rejecting gentrification in mathematics journal of urban mathematics education vol. 11, no. 1&2 16 impacted). i made the decision to join the ranks of this now historical occupation, not without escaping serious professional reprimand. about five years ago, the black diaspora was shaken irreversibly with a series of protest movements in the united states, black lives matter, spurred on by the public deaths of black men at the hands of police officers from ferguson, missouri to new york city, new york. this movement and its impacts have spread throughout urban communities and became a marker for modern protest for black communities around the world. most notably, black lives matter drives particular attention to the status of racial equality and equity across a number of social contexts. it underscored a renewed resolve amongst black communities, black power, to amplify their right to be treated humanely, to be seen, to have access, and to be acknowledged. similar social media campaigns #blackgirlmagic, #blackboyjoy, #blackgirlsrun and facebook groups like black educators rock with over 10,000 members speak to the online ways black communities have taken control of their own spaces of reform. add to this the phenomenal appeal of the movies hidden figures (gigliotti, chernin, topping, williams, & melfi & melfi, 2016) and black panther (feige & coogler, 2018) in the black diaspora and the last few years have been heralded by a wave of empowering resistance narratives from academic to popular culture. in most places i have been, these movements of community, following some crisis, have powered the words strong black community as a collective unifying theme. in bermuda, this cry arose after the gang and gun violence went viral in 2009 shortly after my return home. a group of us formed the group rise above bermuda to organize community to engage the rash of violent deaths among black men. as in much of these experiences, there are at least five elements for strong black community that resonate: community health (are we all okay?), community safety (is this space safe?), community ownership (are we in control of our own resources and destiny?), community sustainability and legacy (can this be held in our institutions?), and community pipeline (is there for all of us a path for current and future generations to follow?). because mathematics reform ideology is not connected to strong black community, it does not adequately consider these questions. for instance, in the most recent gathering of mathematics resources producers and facilitators, i found only two black entrepreneurs. how do we challenge the producers of mathematics goods and services that proliferate the field? does mathematics power mean black power? martin’s (2018) presentation at the 2018 national council of teachers of mathematics annual meeting and exposition planted a stake in the ground for a divergent path from mathematics reform and toward black and urban communities. martin declares: “we define black liberatory mathematics education as the framing and practice of math education that allows black learners to flourish in their humanity and brilliance, unfettered by whiteness, white supremacy and anti-blackness” (para. 4). this statement is what i matthews rejecting gentrification in mathematics journal of urban mathematics education vol. 11, no. 1&2 17 am terming as a declaration of independence from (existing) mathematics education reform and expands on three components critical to black power: (a) prioritizes liberation, (b) rejects mathematics institutions and organizations entrenched in white supremacy culture, and (c) led first and foremost by “liberation-seeking” members of the black community. although increasingly academic work is including liberatory perspectives in some spaces, rejecting mainstream reform institutions and activities that propagate the invisibility of black community in mathematics represent great difficulty. it is not blasphemy to consider whether this thing we call mathematics education operates in the explicit interests of black and urban communities. why are so many publishing, promotional, exhibit, and professional development interests missing people of color in ownership, contracts, and presence? at the last annual gathering of the nctm, the largest mathematics education organization in the world, i found only two black entrepreneurs in a sea of commercial exhibitors (over 200 by my count). while this anecdote is a limited analysis, my felt experience in scores of these conferences and meetings over the years has borne out limited representation of black owners and producers of mathematics. are urban communities capable of producing the products of innovation needed for addressing mathematics excellence? the notion of mathematics reform as black power in black communities is suitable antidote to the gentrification metaphor. in leading mathematics reform, the central unit of operation is the school, or some element of schooling such as an “achievement gap.” the closing of achievement gaps does not mitigate the existence of excellence, or strong, healthy urban community. this omission was made clear to me when i returned home after finishing my doctoral degree. i began my career poised to engage classrooms and schools by addressing the usual “math speak”—rich tasks, orchestrating classroom discourse, connections, problem solving, representations, reasoning, and so on. my community, however, wanted black male empowerment, affordable housing, political leadership, church rebuilding, jobs creating, engaged gang violence entrepreneurship, and remaking public education. it wasn’t that mathematics wasn’t critical. it could not be actualized (or prioritized) outside of real life, the felt impacts of infrastructural and political dimensions of building a strong black community. the reform imperative (universally given to me in illinois) i had long rehearsed was inadequate to interrogate life in bermuda as constructed. thus, in schools the enterprise of mathematics reform, complete with rationales, resources, and initiatives represented an intrusive platform placed onto, not into or out from our community. scan any of the above movements and what stands out is that they are aligned with community priorities, safety and defense, filled with organic, mobile community leadership, aligned with parental and family empowerment, organized around community spaces, and driven by local narratives and legacies. to see mathematics matthews rejecting gentrification in mathematics journal of urban mathematics education vol. 11, no. 1&2 18 as black power is to envision strong black community as the object of affection. largely, mathematics education infrastructure has not drawn from this power base. such a vision of how this might work was imagined in the fictional nation wakanda in the movie black panther (feige & coogler, 2018). the wonder of wakanda is that black excellence is a critical power base for strong black communities. indeed, strong black communities have always been rooted in such. in resistance, we reposition and realign mathematics reform (language, resources, narratives, affect, policy) with strong black communities drawing from a movement of black power, leaving them fortified for the challenges ahead. in short, several elements are critical to black power discussion: (a) black wisdom, (b) black entrepreneurship, (c) black resources, and (d) black narratives. in black wisdom, the base of mathematics reform would draw from the remastered stories of black mathematicians, community activists, historical community problems overcome, efforts. in black entrepreneurship we redouble our efforts to create, own, and distribute mathematics products that impact our children. in black resources we commit to using the tools of liberation and lastly, we create, unarchive, celebrate, and reposition community narratives as mathematics narratives. these are of course incomplete. suffice it to say, we must question whether movements of urban education can be adequately housed in the academy, national organizations, and even our schools. they must be housed in safe spaces of black community. mathematics experiences as racialized trauma i left her office feeling nothing but shame because of my gender. no one encouraged me in stem. i was seen as just a black girl who happened to outperform her peers. my good friend, kristie (pseudonym), shared her experiences with me as i was preparing for a presentation in 2017 on past mathematics experiences in the black community. she had recently connected me to a massachusetts institute of technology professor doing some culturally relevant work in haiti, and as i was probing her personal interest in stem, i asked her to describe her memorable moments in mathematics. she began to discuss a particular episode, a turning point, where she had been declined entrance into a stem program: i declared math as my major in undergrad college applications. a university in boston invited me for an interview. the admissions dean was a woman—i sat in her beautiful office, nice leather chair, and she said without apology that if i were a male applicant, she would’ve accepted the presentation of my application. she left stem soon after and pursued a job in international business. after several years, she has begun to revisit stem projects in her native homeland of haiti. interactions like these with adults in my community occur almost daily, or at least any matthews rejecting gentrification in mathematics journal of urban mathematics education vol. 11, no. 1&2 19 time i share my role in mathematics. the detail and emotion that accompanies these interactions is always evident. this troubling exchange within the mathematical sciences pipeline, and its gatekeepers—educators, schools, colleges, counsellors— is played out every day in our communities. over the last 25 years of listening to stories of struggle and resistance from within and alongside the black community, i have come to realize the secondary impact of these stories on the educators that tell them. they must not only capture but also relive the collection of disappointments in their own personal mathematics experiences. in the way that i have experienced these accounts, they read more like racialized trauma. which is far different from locating these experiences as mathematics anxiety, something my early training emphasized. in the article “healing the hidden wounds of racial trauma,” hardy (2013) compares racial oppression as a “traumatic form of interpersonal violence which can lacerate the spirit, scar the soul, and puncture the psyche” (p. 25). hardy (2013) describes the assaulted sense of self in racialized trauma as overexposure to dehumanizing experiences, or devaluation. the fact that “the source of their hurt is often confused with distracting secondary symptoms ranging from hopelessness to acting out behavior” (p. 25) is but a distraction. one such distraction might be the focus on mathematics anxiety. when i meet people who profess this popular label a deeper probe often reveals hidden stories of oppression. the end product of mathematics anxiety is exclusion. while much is said about mathematics anxiety, little is written on the longitudinal impact of exclusion on urban families and communities. communities of color describe on a daily basis episode of failure, classifications, denials of access, and being overlooked in mathematics. members and the mathematics scholars that tell their stories experience this same hammering at the hands of the mathematics enterprise in schooling. it occurred as easily as with my ghanaian-american uber driver, cofi, in washington, dc during a casual trip to an appointment; he deplored the one-sizefits-all teaching that “killed” his opportunity to do mathematics in ghana. having heard these stories before, it would not have normally stuck on me had it not been for my subsequent visit to some mathematics classrooms in gomoa-fetteh, a small town outside accra, ghana. there, i witnessed mundane, british-text styled lessons taught with little intent in teaching mathematics for use beyond junior high school, and certainly not for self-empowerment. the significance of cofi’s story was punctuated in my visit to the african institute of mathematical sciences (aims) in the coastal town of biriwa, ghana, one of nine across the continent designed to recruit talented african students into careers in the mathematical sciences. the aims in ghana had enrolled 37 students in various post graduate m.sc. programs, far less than its planned complement of 50, with a third being women and one-third ghanaian. the approximate 13 ghanaian students (ghana’s population is approximately 29 million for context), representing matthews rejecting gentrification in mathematics journal of urban mathematics education vol. 11, no. 1&2 20 fractions of a percentage, was a critical confirmation of the lack of opportunity and access in mathematics in the african diaspora—still felt thousands of miles away, and years later by my uber driver. these experiences were soberly contrasted with a visit to the cape coast slave castle, one of many, where millions of lives were traumatized and lost to one of the most atrocious systems of human depravity imaginable. similar to the parallels of slave castles on the coast with colonialism and slavery, mathematics exclusion is a far-reaching system, punctures of which are farreaching and long-lasting for the black community. the story is also similar for stan, my former undergraduate student, an african american man now in his late 30s who recounts how he was tracked (along with countless african americans) in classrooms across the south into non-collegepointing, remedial mathematics. the classes were disguised as a slower journey into algebra over two years (as opposed to one) but rarely every reached past general mathematics. as he remembers, they were designed for students who weren’t going to college. i can’t forget his words, recalling the trauma, as if it had just happened, “…[stammering] but i wanted to go to college.” after several attempts, much tutoring, counsel, and grit, stan not only passed but is now a principal in south carolina and currently completing his doctorate at clemson university. his story is symbolic of countless other stories where urban community members carry with them—stories of hurt and pain associated with interactions with status quo mathematics policy, racist and chauvinistic deliberations, and hostile expectations. none is more powerful than my most recent experiences at the 2018 nctm meeting: a mathematics instructional coach, a black man in his 40s who attended an nctm session i facilitated several months ago. as the audience was asked to examine the experiences that led to their empowerment or un-empowerment in mathematics, he notably left the room and disappeared for a length of time. he returned and sometime after the session he came up to me and revealed that he was in tears as he revealed the trauma of that high school teacher who told him he would not amount to “shit.” he admitted that he went through his entire college experience and early career with those words chasing him. our session allowed him to release for the first time; he left to call his wife midstream to express himself. as the reggae song goes “i could go on and on, the full has never been told.” as he recounted to me, i went back to the words of my own college professor: “maybe you aren’t cut out for this!” taken together these experiences of trauma in urban communities represent the collective assault felt by the mathematics enterprise. the unwritten rules of mathematics and science exclusion and promotion have lasting generational effects in communities of color and are swifter and longer lasting within communities situated in the caribbean, united states, africa, and the african and caribbean diasporas. my encounters with the multitude of instances of trauma in classrooms and schools surrounding mathematics have convinced me that it may be more useful to matthews rejecting gentrification in mathematics journal of urban mathematics education vol. 11, no. 1&2 21 tackle failed urban mathematics experiences as racialized trauma and begin to triage and treat black families and their educators. because of the longstanding experience in other contexts, i find black communities better organized to countermand racialized trauma (vs. mathematics anxiety) in real, meaningful ways. my church, for example, has a social worker on call for every service. what might a similar approach look like across the urban mathematics landscape. hardy (2013) outlines eight steps needed to engage the effects of racialized trauma which i relate to mathematics: step 1 – affirmation and acknowledgement of racialized experiences in mathematics; step 2 – create space for race through focused conversations about race in mathematics (vs. equity or diversity by association); step 3 – racial storytelling where personal experiences of racial trauma in mathematics are amplified; step 4 – validation of black experiences, worldviews, and origins in mathematics; step 5 – the process of naming is employed to “affix words to racially based experiences” that offer “external and consensual validation to racially oppressed youth and helps restore their voices” (p. 28); step 6 – external devaluation factors that inhibit access, representation, and success are openly confronted in mathematics environments and structures; step 7 – resources that counteract devaluation and act as a “buffer against future assaults” (p. 28) in mathematics are harnessed and positioned; and step 8 – rechanneling rage not to rid students of their rage of racialized trauma in mathematics “but instead to help them be aware of it, gain control of it, and ultimately to redirect it” (p. 28). in considering step 7, counteracting devaluation, i think about how bermuda and other caribbean islands have had to organize to prepare for hurricane assault, particularly when evacuation is not an option. in bermuda, the machine is impressive and the loss of life and time to recover is minimal. an emergency measures organization of representatives from all essential services and agencies is activated to centralize, marshal, and direct all critical resources and information with one goal in mind: minimize disruption and facilitate restoration. in a similar way, mathematics reformers would envision potential and recurring mathematics experiences as forces of disruption. a natural by-product of this mindset might be engineering a mathematics pipeline of explicit trauma-responsive policy, programming, curricula, and structures that acknowledge and circumvent mathematics disruption in urban community. it is probably already evident that the aims of jume in its conception directly address several of the eight critical steps to counteract racialized trauma in mathematics. it is also clear that there are growing number of therapeutic micromovements across mathematics that hold similar ideals. a recent meeting of mathematics education scholars of color (mesoc) provided just such healing space for me in the spring of 2018. for years the benjamin banneker association provide great opportunity for my colleagues and me to chase validation of black excellence. there are currently facebook groups, chats, and a growing twitter community ded matthews rejecting gentrification in mathematics journal of urban mathematics education vol. 11, no. 1&2 22 icated to providing just such opportunities. building a trauma-responsive pipeline to combat racialized trauma in mathematics is a next level of work, and of course, one that may place us in direct conflict with the mathematics education enterprise. but they who feel it, know it. toward mathematics as love me: i love you nana nana: love don’t pay the bills! my nana said the above statement to me once when i was little, and later in my adulthood we would reenact the exchange every chance we got. joke aside, the statement was a powerful reminder for me that i lived in an exacting world not ordinarily concerned with my feelings. this final perspective is written as a follow on and admittedly a work in progress. i have always loved mathematics, but i have often wondered if the relationship is mutual. the last powerful moment to acknowledge is found in hearing christopher emdin’s keynote address to several thousand mathematics educators at the 2018 nctm annual meeting and expositions which was a catalyst for a new public perspective of mathematics for me. addressing several thousand mathematics educators, part of the way through his speech, he proclaimed (paraphrasing): i just want a pedagogy of love. this sentence captured his sentiments and forced me to re-boot, to come clean again, in public narrative fashion, with my own voice. in one sentence, i was forced to connect my social, personal story to my professional story. before this moment, the words in my professional discourse always included: “rethink,” “critical discourse,” “resistance,” “oppositional”—but never “love.” i suspect like others, i resisted including an emotional tie to this work, particularly love, as the underlying motivation of the work. yet, it was true. i believed (though cannot remember ever proclaiming) that mathematics could be experienced in a way that brings people together—in love. this belief may have something to do with my first doctoral professor, a stalwart. the first wave of the culturally relevant movement focused on utilizing culture in ways that supported mathematics learning. the second wave of efforts on social justice (an original component of culturally relevant pedagogy [crp] rendered invisible by mass adoption) center around countering the impacts of injustice and aligning mathematics practice in solidarity with selfand collective-liberation. what i have seen in practice are now waves of practitioners who attempt to do social justice, culturally responsive practice as a form of methodology—as a practical response to achievement gaps and other gap-conditions. i fear we have not rooted crp in what is its most powerful driver: a deep abiding love of our people and our communities. to espouse this condition is to re-define or re-humanize mathematics, teaching and learning, and its purposes for urban communities. that is, to articulate matthews rejecting gentrification in mathematics journal of urban mathematics education vol. 11, no. 1&2 23 a vision of mathematics that can reside closer to human condition, rather than simply operating on it. my team is currently reading annual perspectives in mathematics education 2018: dehumanizing mathematics for black, indigenous and latinx students (goffney, gutiérrez, & boston, 2018) for our book study as we frame mathematics coursework for prospective teachers. the work of re-humanizing mathematics lends itself to repurposing mathematics as love because the act seeks “connections, joy, belonging” (gutiérrez, 2018, p. 4) as part of the mathematics experience. mathematics as love is a counter-space to those who overly tout the beauty in mathematics. under this public narrative, mathematics, and particularly, its beauty is often stated referencing exquisite (another mathematics word ported into elite language) abstract shapes, processes, and algorithms are presented as objects of beauty. i have often heard those who have been really successful in professional mathematics brag about the ability to see it as beautiful. there was a movie several years ago entitled a beautiful mind (glaser & howard & howard, 2001) to capture the narrative of one with one such ability. indeed, popular television media has depicted beautiful mathematics and beautiful minds as in good will hunting (bender & van sant, 1997), scorpion (manson et al., 2014) and numb3rs (scott & scott, 2005). it is no coincidence that the main actors cast in fictional hollywood productions are white men, which works to perpetuate them as the only doers of important mathematics. such a focus is incomplete. it promotes internal devaluation because people who cannot experience, understand or see the beauty of this mathematics are led to believe that they do not possess the capacity to do mathematics. the opposite is true. there is beauty in people, in communities, in the activities that they engage in, culturally, politically, economically, and spiritually. these communities do mathematics and there is beauty in that. mathematics as love captures that orientation. mathematics as love is more fanciful speech though. rather, it is a rallying cry. one, not just for the classroom, but a gut-check call for scholars and educators to reassess the aims, motivations, and the public words around our work. it is a call to reassess the inside dealings of our committees, our conferences, our meetings, and our online relationships. the recent attacks on mathematics scholars of color and others was made easier partly because of the frequent hostilities exuding in environments in our personal professional spaces, where many of us call academy. the competitive, restrictive, and elite clubs of mathematics education have not only contributed to perpetuation of the trauma felt by communities of color, simultaneously, this environment have made it easier for racist entities to attack and isolate. believing that our field is bonded to the human experience beyond its mathematics products, problems and conventions can make us stronger in defense of any, and all of us. there are four elements i associate with a love orientation in mathematics: matthews rejecting gentrification in mathematics journal of urban mathematics education vol. 11, no. 1&2 24 1. espousing a public definition of mathematics that expresses its use in improving and connecting the human/black/brown/urban condition; 2. engaging directly in identity work with prospective and practicing teachers of mathematics in building relationships with urban communities and communities of color; 3. building and mathematizing community projects that work to alleviate conditions and foster healing relationships in black, brown, and urban communities (vs. contrived problem solving); and 4. creating professional spaces for mathematics education scholars to affirm and support each other, in contrast to spaces that favor competition among urban scholars. the past 10 or so years have given me greater clarity as one who reforms and yet, like jume, there are more shifts to come as we explore this complex urban space in love and power. upon reflection of what has been written in the preceding paragraphs, i realize that many of the spaces of love, trauma, gentrification, and power remain unresolved in my own journey. and yet because i have lived these moments, i know them, and it feels right to pursue these tensions. knowing that there are so many more of us pursuing mathematics in this vein is more encouraging now than it was 10 years ago. in seeing mathematics reform as gentrification and racialized trauma in our communities, we are more empowered. mathematics as power gives me something to fight with and mathematics as love grounds us in hope—the hope that we can live in a world better served by human mathematics. references bender, l. (producer), & van sant, g. (director) (1997). good will hunting [motion picture]. united states of america: be gentelmen. feige, k. (producer), & coogler, k. (director). (2018). black panther [motion picture]. united states of america: marvel studios. gigliotti, d., chernin, p., topping, j., williams, p., & melfi, t. (producers), & melfi, t. (director). (2016). hidden figures [motion picture]. united states of america: 20th century fox. glazer, b., & howard, r. (producers) & howard, r. (director). (2001). a beautiful mind [motion picture]. united states of america: imagine entertainment. goffney, i., gutierrez, r., & boston, m. (eds.). (2018). annual perspectives in mathematics education 2018: rehumanizing mathematics for black, indigenous and latinx students. reston, va: national council of teachers of mathematics. gutierrez, r. (2018). introduction: the need to rehumanize mathematics. in i. goffney, r. gutierrez, & m. boston (eds.), annual perspectives in mathematics education 2018: rehumanizing mathematics for black, indigenous and latinx students. reston, va: national council of teachers of mathematics. matthews rejecting gentrification in mathematics journal of urban mathematics education vol. 11, no. 1&2 25 hardy, k. v. (2013, spring). healing the hidden wounds of racial trauma. reclaiming children and youth, 22(1), 24–28. retrieved from https://static1.squarespace.com/static/545cdfcce4b0a64725b9f65a/t/54da3451e4b0ac9bd 1d1cd30/1423586385564/healing.pdf manson, s., … higgs, a. (producers). (2014). scorpion [television series]. cbs television studios. martin, d. (2018, june 12). “taking a knee in math education”: danny martin’s nctm talk, partially transcribed. retrieved from teaching with problems: https://problemproblems.wordpress.com/2018/06/12/taking-a-knee-in-math-educationdanny-martins-nctm-talk-partially-transcribed/ matthews, l. e. (2009, july). identity crisis: the public stories of mathematics educators. journal of urban mathematics education, 2(1), 1–4. retrieved from http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/20/9 scott, r., & scott, t. (producers). (2005). numb3rs [television series]. cbs television studios. tate, w. f. (1994, february). race, retrenchment, and the reform of school mathematics. phi delta kappan,, 75(6), 477–484. woodson, c. g. (1990). the mis-education of the negro. trenton, nj: africa world press. (orginal work published 1933) https://static1.squarespace.com/static/545cdfcce4b0a64725b9f65a/t/54da3451e4b0ac9bd1d1cd30/1423586385564/healing.pdf https://static1.squarespace.com/static/545cdfcce4b0a64725b9f65a/t/54da3451e4b0ac9bd1d1cd30/1423586385564/healing.pdf http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/20/9 journal of urban mathematics education july 2009, vol. 2, no. 1, pp. 5–21 ©jume. http://education.gsu.edu/jume celia rousseau anderson is an associate professor in the department of instruction and curriculum leadership at the university of memphis, 401d ball hall, memphis, tn 38152; email: croussea@memphis.edu. her research interests include equity and opportunity to learn in mathematics education. angiline powell is an associate professor in the department of instruction and curriculum leadership at the university of memphis, 411b ball hall, memphis, tn 38152; email: apowell3@memphis.edu. her research interests include urban education and caring mathematics teaching practices. a metropolitan perspective on mathematics education: lessons learned from a “rural” school district celia rousseau anderson university of memphis angiline powell university of memphis in this article, the authors examine the historical and contemporary conditions of two school districts—one urban and the other rural. despite the surface differences between the districts, this comparison reveals several historical and contemporary similarities and connections between the two settings. the authors describe the implications of these relationships for future directions of urban mathematics education scholarship. specifically, they posit the need for a ―metropolitan‖ perspective that would take into account the interrelationships between cities and their suburban or rural neighbors. keywords: critical race theory, mathematics education, metropolitan perspective, opportunity to learn in a commentary that appeared in the inaugural issue of this journal, william tate (2008) noted an ongoing lack of attention to geospatial considerations on the part of educational researchers. in response to this inattention to the unique features of urban contexts, he argued for the need to ―[put] the ‗urban‘ in mathematics education scholarship‖ and outlined a more expansive vision of the theoretical and empirical traditions that are relevant to urban mathematics education. like tate, we are deeply interested in what putting the ―urban‖ in scholarship on mathematics education might look like and believe that such scholarship must push beyond the traditional paradigmatic boundaries of mathematics education. to further explore the meaning of urban mathematics education, we seek in this article to trace our own intellectual journey with respect to this issue and our developing understanding of a specific urban setting. strangely enough, however, this intellectual journey began in what, at least on the surface, would appear to be a distinctly non-urban (in fact, rural) locale. after describing the historical context and contemporary conditions of both the rural and urban settings in which we anderson & powell metropolitan perspective journal of urban mathematics education vol. 2, no. 1 6 were working, we explore the implications of what we learned from these settings for scholarship in urban mathematics education. our introduction to the rural setting in 2005, we became involved in a project to work with the fayette county schools in the state of tennessee. the delta project, 1 which was funded by a grant from the u.s. department of education, targeted three areas in the mississippi river delta and had as its goal the improvement of student achievement in the participating districts. while the particular focus of the delta project differed by implementation site, the fayette county project targeted mathematics. the fayette county schools had been plagued by low overall achievement in mathematics and a substantial achievement gap 2 between white and african american students. during the first year of the project (2005–2006), for example, the ―grade‖ for the district (based on achievement data for grades k–8) was a ―d‖ in mathematics, compared to a ―b‖ for the state as a whole. moreover, a black-white achievement gap was evident in mathematics at all levels. for example, the 3-year average of students scoring ―proficient‖ or ―advanced‖ on the state assessment in grades k–8 was 85% for white students and 69% for african american students. similarly, at the high school level, the 3-year average of students scoring proficient or advanced on the state-mandated algebra test was 72% for white students and 52% for african american students. this data indicated a need for improvement of mathematics education for the district as a whole. 3 in addition, insofar as african american students made up 61% of the district population, the gap also pointed to a specific need to improve the mathematics opportunities provided to the substantial african american student population. 1 the delta project was funded by a grant from the u.s. department of education awarded to arkansas state university (award no. u215k050343). the conclusions reported in this article do not necessarily represent the positions or policies of the department of education or arkansas state university. 2 we point out the differences in achievement for two reasons. first, the delta project in fayette county was explicitly intended not only to improve overall student achievement but also to address these achievement gaps. also, we posit that these differential outcomes are tied to the historical inequities in the district which are the focus of later sections of this article. thus, we wish to be clear that our focus is not on the gaps themselves, but on the opportunity-to-learn factors that they likely reflect. 3 it is important to note that these outcomes have improved since the time the project was initiated. in this article, however, we describe the 2005–2006 data for both districts (as opposed to the most recent outcomes) to set the context for our developing understanding of the relationships between the two settings. anderson & powell metropolitan perspective journal of urban mathematics education vol. 2, no. 1 7 while these data provide some indication of the need for intervention, they shed little light on the factors shaping the district‘s current status in mathematics. as we began to spend time in fayette county, 4 we learned more about the factors influencing students‘ opportunity to learn mathematics. in particular, the findings of an external audit report, 5 commissioned by the state, pointed to several opportunity-to-learn factors previously outlined by tate and rousseau (2007) as important for student learning and achievement. as described by tate and rousseau, factors related to time and quality influence students‘ opportunities to learn, ensuring that students of color and students from low-income backgrounds often receive fewer opportunities to learn high quality mathematics than others. the findings from the external audit report of fayette county raised several concerns related to opportunity to learn, including issues of curriculum quality, lack of quality materials, and concerns over teacher quality. for example, one key finding of the outside audit report involved the failure of the district to implement a curriculum aligned with state standards. a second factor noted by the audit team was a lack of quality materials and resources. according to the auditors, this lack of resources and materials has a particular impact on traditionally underserved students in the district. finally, the authors of the audit report pointed to concerns related to teacher quality. in particular, they noted the high teacher turnover in the district and the fact that few of the district teachers lived in the county. according to the auditors, the large number of non-local teachers meant that students were being taught by persons who were likely to have little understanding of the community and the students‘ lives. our awareness of these and other opportunity-to-learn factors shaping the outcomes in fayette county was important, as it helped not only to provide some insight into the conditions of mathematics education but also to shape the nature of our intervention in the district. yet, despite this deeper understanding of opportunity to learn in this case, we would argue that our insight into this district was still limited without further examination of the historical and contemporary conditions that impact education in fayette county. we did not know, for example, the factors that might explain the apparent lack of investment in instructional materials and a high quality curriculum. in the next sections, we examine the history of this area and the broader contemporary conditions that are likely related to opportunity to learn in the district. from this deeper understanding of fayette county, we were also able to recognize relationships between the conditions in fayette and those of the nearby urban district. 4 our involvement included providing professional development and instructional support to middle and high school mathematics teachers in the fayette county schools. this involvement continued over a 2-year period, from 2005–2007. 5 the report was prepared by millennium learning concepts. anderson & powell metropolitan perspective journal of urban mathematics education vol. 2, no. 1 8 initially, those relationships were anything but clear to us. the local district with which we typically work—memphis city schools (memphis, tennessee)— is a large, urban district with approximately 180 schools serving over 100,000 students. in contrast, the fayette county district serves fewer than 4,000 students in 10 schools. in addition to the difference in size, we also were immediately cognizant of the geospatial differences. for example, fayette county has a population density of 41 persons per square mile. in contrast, the population density of memphis is 2,327 per square mile (u.s. census bureau, 2009). moreover, the drive from memphis to the district high school in fayette county highlights the ongoing role of agriculture in the county, as the highway is lined on either side with fields of cotton and other crops. as a result of these differences, we initially viewed our work in these two locations—urban and rural—as occurring in two distinct and largely unrelated settings. over the course of the time spent working in fayette county, however, our gaze shifted and we began to recognize that our initial view obscured several similarities and interrelationships of importance for understanding opportunity to learn mathematics in both fayette county and memphis city schools. a critical race theory lens a description of the change in our view would be incomplete without a discussion of the perspective that shaped our thinking. in particular, our approach to understanding the educational conditions in fayette county, and subsequently memphis city schools, was shaped by critical race theory (crt). critical race theory originated in legal studies in the 1970s and has come to influence the work of many scholars of education since its first introduction to the field in 1995 (dixson & rousseau, 2006; ladson-billings & tate, 1995). although critical race theory in legal studies is an eclectic movement, there are several key characteristics of scholarship within this perspective: (1) critical race theory recognizes that racism is endemic to american life; (2) critical race theory expresses skepticism toward dominant legal claims of neutrality, objectivity, colorblindness, and meritocracy; (3) critical race theory challenges ahistoricism and insists on a contextual/historical analysis of the law…critical race theorists…adopt a stance that presumes that racism has contributed to all contemporary manifestations of group advantage and disadvantage; (4) critical race theory insists on recognition of the experiential knowledge of people of color and our communities of origin in analyzing law and society; (5) critical race theory is interdisciplinary; (6) critical race theory works toward the end of eliminating racial oppression as part of the broader goal of ending all forms of oppression. (matsuda, lawrence, delgado, & crenshaw, 1993, p. 6) anderson & powell metropolitan perspective journal of urban mathematics education vol. 2, no. 1 9 this perspective on race and opportunity shaped our thinking about the conditions of education in fayette county. in particular, we explored not only the contemporary educational conditions in fayette county but also the historical and contextual backdrop. fayette county historical context fayette county has a long history of farming and cotton production. as a result of the focus on cotton, it had one of the largest concentrations of slaves in the state of tennessee during the antebellum era. following the civil war, the influence of this history of cotton could be seen in the predominantly african american population, as former slaves and their descendants became tenant farmers or sharecroppers in fayette county (hunt, 1981). despite the relatively large numbers of african americans in the county, however, little political or economic power was held within the african american community in the first part of the 1900s. in fact, there were 16,927 african americans in fayette county in 1959, comprising 68.9% of the population. however, only 17 african americans voted in elections between 1952 and 1959. as a result of these disparities, the fayette county civic and welfare league was founded in 1959 ―to promote civil and political and economic welfare for community progress‖ (hamburger, 1973). one of the league‘s first projects was to encourage voter registration. these efforts, however, were perceived as a threat to the white power structure. in 1959, african american registered voters were turned away from the democratic primary, told that it was ―whites only.‖ in addition to threats of physical violence and intimidation, whites used economic power to punish those who registered. registered african americans lost insurance and credit in local stores and were unable to get farm loans that had been readily available in the past. in 1960, black tenant farmers who registered to vote were evicted from the land that they farmed. nearly 300 people were thrown off their farms. as the economic pressure grew, a list of registered african american voters was distributed to white businesspeople. those registered voters on the list were unable to purchase anything anywhere in the county. individuals on the list were forced to drive 50 miles one way to purchase staples in the nearest big city (hamburger, 1971; hunt, 1981). in an oral history of this time in fayette county, hamburger (1973) interviewed several of the key leaders involved with the civic and welfare league. one of the leaders analyzed the reaction of whites in fayette county to efforts on the part of african american citizens to gain economic independence: ―back then i didn‘t know that when a negro in the south goes into business and tries to make substantial gains he is violating the white man‘s civil rights‘‖ (p. 8). another anderson & powell metropolitan perspective journal of urban mathematics education vol. 2, no. 1 10 leader described a similar perspective on the reality of african american status in the county. reflecting on the process of blacklisting by white shop owners he stated, ―i think that really is the psychology of the white man in fayette county— just keep the negro hungry, keep ‗em on their knees, don‘t allow him opportunity‖ (p. 8). the voices of these men reflect the challenges that african americans faced in fayette county in the effort to secure even basic civil rights. the struggle for economic opportunity represented in the image of displaced farmers was connected to larger issues of rural poverty that characterized fayette county. a 1969 newspaper article described fayette county as the third poorest county in the united states. during the 1960s, more than three-fourths of the county‘s residents lived in poverty and the per capita income was approximately $700 (charlier, 2005). moreover, little economic growth occurred over the next decade. in 1978, 43% of the households in fayette county were below the poverty line. in 1981, fayette county was the second poorest county in the state. at that time, 55.5% of county residents qualified as low-income (hunt, 1981). fayette county has been described as ―the very essence of wrenching rural poverty‖ (charlier, 2005). the examples of white resistance to african american efforts to assert the right to vote were repeated with respect to education. although de jure segregation was outlawed with the brown v. board of education decision of 1954, it was 1966 before the board of education of the fayette county schools instituted a desegregation plan involving voluntary transfer (hunt, 1981). only a small number of african american students, however, chose to transfer to the all-white schools. according to the school superintendent at the time, the board of education made a conscious effort to try to prevent african american students from wanting to transfer. this effort included making concessions to improve conditions at the allblack schools: ―we felt we could delay desegregation if we made the black schools more equal‖ (hunt, 1981, p. 146). similarly, an african american educator noted that the board of education, ―started going along with black schools….see they wanted to keep ‗em separate, so what they would try to do was to please you as much as possible to keep you from wanting to go to the white school‖ (hamburger, 1973, p. 182). nevertheless, despite these tactics, the all-black schools were not ―separate but equal‖ in fayette county. parents and students protested the fact that white and black students had different school calendars (the black calendar was still based on a farming schedule). in addition, the quality of facilities, materials, services, and teachers were not the same within the de facto dual system (hunt, 1981). in 1969, students from the all-black high school, reacting to the differences in schooling conditions and educational opportunities, marched to the predominantly white high school in order to register. they were met by the sheriff and a deputized mob. several students were beaten (hamburger, 1973). the student anderson & powell metropolitan perspective journal of urban mathematics education vol. 2, no. 1 11 march was one example of the protests that took place in fayette county during the era of desegregation (hunt, 1981). one of the results of school desegregation was the growth of private schools in fayette county. for example, fayette academy was founded in 1965 when, according to its website, ―a few parents felt that the education of their children should be of higher quality.‖ fayette academy enrolled approximately 100 students before the 1970–1971 school year. after the 1970 federal court desegregation order in fayette county, however, the enrollment of the academy jumped to 700. in fact, in 1971 judge robert mcrae, the judge presiding over the federal case, referred to fayette academy as ―a monument to segregation in fayette county‖ (hunt, 1981). although the number of private schools is not large, their impact on the public school system has been profound. whereas the african american population in the county was decreasing in the years immediately following desegregation, the percentage of african american students in the fayette county school district dramatically increased over the same time period. by 1980, 39% of the white school-aged population was enrolled in private schools (hunt, 1981). according to hunt, the disengagement of many white students from the public school system compounded the difficulties of improving the fayette county schools: ―the white power structure has given the majority of its support and loyalty to the private schools which were founded after court-ordered desegregation‖ (p. 4). the lack of support for the public schools was also related to the perception of poor quality. according to hunt (1981), ―the legislative body [was] reluctant to support education financially in the county because of the white community‘s opinion of public education and poor teachers‖ (p. 251). however, it was not only white citizens who lacked confidence in the quality of public education. in hamburger‘s (1973) oral history of fayette county, one of the african american respondents described the schools in the following way: ―the education system is poor. that‘s all there is to it. they just don‘t know better. the quality is real low.‖ contemporary conditions fayette county has changed in several ways from the time of school desegregation. one sign of change has been a shift in the demographics of the county. whereas fayette county was once predominantly african american, 2005 census estimates place the percentage of african americans in the county at 28.2%. whites now make up 69.6% of the overall population. in addition to this shift in demographics, the county has been rapidly growing in recent years. the 2005 census estimates reflected a population change of 19.7% from 2000 to 2005. in fact, fayette county has led the state in rate of growth (charlier, 2005; waters, 2005). anderson & powell metropolitan perspective journal of urban mathematics education vol. 2, no. 1 12 this growth, however, is not occurring evenly throughout the county. the fastest growth is taking place on the western side of the county (charlier, 2005; sparks, 2007). areas that were previously farmland are being turned into highend residential projects (charlier, 2005), as people move into fayette county from memphis. in fact, some observers have characterized the change as making the western end of fayette county more suburban than rural (sparks, 2007; waters, 2005). sharp increases in property values have been witnessed in the western part of the county as large homes go up in exclusive developments (charlier, 2005). this transition from rural to suburban brings with it the need for the county to provide additional services. as orfield (2002) notes, some rural areas that grow into suburbs do not have a resource base strong enough to invest in infrastructure improvements which come with the transition. how well fayette county handles this transition remains to be seen. one area that has already been cited as a liability to growth, however, has been education. the perception of poor quality public schools in fayette county has shaped the housing market in the process of suburban growth. those moving to the county are primarily retirees or upper-class professionals who place their children in private schools (charlier, 2005). while the overall perception of education in fayette county is largely negative, the picture is not monolithic. differences between the east side, which is still largely rural, and the changing west side of the county can be seen in the data on education. for example, table 1 displays 2006 school-level achievement data from the website of the state department of education. jefferson and east are on the east side of the district, whereas oakland and west are on the west side. the high school is adjacent to jefferson and east. the data in the table shows not only the relatively low performance of the district as a whole in relation to the state, but also the achievement disparities that emerge when school location is considered. these disparities in achievement related to location also mirror racial disparities in the district. table 2 shows the demographic information for the same schools. this data from the state‘s website reflects the bifurcation of the schools along geographic and racial lines. a predominantly white (69.6%) county has public schools that are predominantly african american (demonstrating the ongoing significance of private schooling in the county). within the predominantly african american district, however, the schools with the highest proportions of white students have the highest achievement and are located in the western (more affluent) part of the county. anderson & powell metropolitan perspective journal of urban mathematics education vol. 2, no. 1 13 table 1 mathematics achievement in fayette county mathematics achievement 2006 district jefferson elem oakland elem east jr. high west jr. high fayetteware high state grades k–8 criterion referenced academic achievement (3-year average) d d c f c b grades k–8 criterion referenced test (% proficient or advanced; 3-year average) 74% 65% 80% 65% 79% 87% grades 9–12 algebra test (% proficient or advanced; 3-year average) 68% 74% 57% 82% table 2 fayette county schools demographic data demographics district jefferson elem oakland elem east jr. high west jr. high fayetteware high white 36.1% 6.8% 69.6% 21.1% 46.5% 24.9% african american 61.1% 89.3% 25.4% 76.2% 50.5% 73.4% hispanic 2.3% 2.9% 3.9% 2.3% 2.3% 1.4% asian 0.5% 1.0% 1.1% 0.4% 0.7% 0.3% native american 0.1% 0.0% 0.0% 0.0% 0.0% 0.0% memphis historical context while it is beyond the scope of this article to provide a thorough historical account of either fayette county or memphis, there are at least two critical components of the history of memphis that are significant with respect to understanding the relationship to surrounding rural counties, such as fayette. the first involves the nature of the memphis economy. for much of its history, the cornerstone of the memphis economy has been cotton. as a result, the fate of the city was closely tied to the surrounding rural areas. while the early years of the twentieth century saw the city‘s economy expand to include hardwood lumber, the reliance on agricultural products continued (pohlmann, 2008). as green (2007) notes, the city depended not only on the products of the surrounding rural areas but also on its labor force. although eventually expanding into new manufactur anderson & powell metropolitan perspective journal of urban mathematics education vol. 2, no. 1 14 ing industries based on cotton by-products and hardwood, the city‘s ―reliance on a regular influx of migrants and a racially segmented low-wage labor force belied its ties to the rural delta‖ (p. 7). in fact, growth in the city of memphis from the late nineteenth through much of the twentieth century came largely from the rural areas surrounding the city (pohlmann, 2008). green (2007) also notes, however, that the movement was not always oneway. during the first half of the twentieth century, the boundaries distinguishing urban from rural were porous, particularly for african americans. some worked in memphis while still living in rural areas. other residents of the city worked as day laborers in the cotton fields. in fact, according to green, ―thousands of black migrants who arrived in memphis found it difficult to avoid seasonal work in the region‘s cotton fields and discovered that they were considered field hands even after relocating to the city‖ (p. 79). thus, the economic and human interrelationships between urban and rural make up a key part of the historical landscape of memphis. also significant to an understanding of the history of memphis is the worldview that emerged as a result of these interrelationships. pohlmann (2008) asserts, for example, that the lack of industrial diversification in memphis and the subsequent reliance on agricultural products ensured that the political culture of the city remained largely traditional, more closely resembling that of the surrounding rural areas. green (2007) points to the racial basis of this traditionalism in the form of the ―plantation mentality.‖ this mentality encompasses the ―racist attitudes that promoted white domination and black subservience… reminiscent of slavery and sharecropping‖ (p. 2). according to green, black migrants from the rural areas surrounding memphis found a city in which the plantation mentality was manifested in countless ways: police harassment; job opportunities that were limited to domestic work and unskilled labor, including seasonal work in the cotton fields; poor housing; political disenfranchisement, and so on. moreover, ―migrants encountered racial practices that appeared to recreate, albeit in specifically urban forms, aspects of plantation culture‖ (p. 18). in fact, in the mid-twentieth century, as local and national leaders called for greater freedom and equality in memphis, such calls often referenced a plantation history. as a result, green argues that an understanding of the black freedom struggle in memphis requires acknowledging both the ―urban-rural matrix‖ and the racial equation reflected in the plantation mentality. one manifestation of the plantation mentality can be seen in the history surrounding school desegregation in memphis. the president of the board of education was quoted in the local newspaper at the time of the supreme court decision in brown v. board of education: ―we have been expecting this to happen for a while…we believe our negroes will continue using their own school facilities since most of them are located in the center of negro population areas‖ (―city anderson & powell metropolitan perspective journal of urban mathematics education vol. 2, no. 1 15 schools,‖ 2004). as was the case in fayette county, this de facto segregation was accomplished initially by focusing on building projects to enhance and expand black schools (green, 2007). in fact, the first black students did not begin attending historically white schools in memphis city schools until 1961. through new school construction and school zoning, the board of education of the memphis city schools was able to maintain de facto segregation for several years after the original brown decision. by 1971, fewer than 4% of black students attended predominantly white schools (herenton, 1971; mcrae, 1996). nevertheless, court-ordered busing for the purposes of desegregation began in memphis in 1973. in anticipation of these changes, whites in memphis established an organization called citizens against busing (cab) and began to set up several private schools. the cab schools typically leased space from local churches. in 1972, there were 40 private schools in memphis. that number increased to 90 in 1973 as court-ordered busing began. private school enrollment climbed to 33,000 in 1973 and increased to 35,300 the following year (biles, 1986). students not only enrolled in the newly-formed cab schools but also in other existing private and parochial schools (egerton, 1973). according to biles (1986), the ―private church-affiliated schools mushroomed across the landscape‖ (p. 480). the impact of ―white flight‖ at the time of busing was substantial. in 1970, just a few years prior to the start of busing, the memphis city schools district was 55% black. while there is some question as to the exact number of students who left the system in response to desegregation, the percentage of white students had dropped to approximately 33% by 1973 (egerton, 1973; mcrae, 1996; terrell, 2004). one author who wrote at the time that busing began characterized the white flight from the memphis city schools in the following manner: ―the school system has already lost many thousands of white students, and in all probability it will lose more. the school system is powerless to control that exodus‖ (egerton, 1973, p. 34). after a dramatic drop when busing began, the percentage of white students in the memphis city schools continued to decrease at a steady pace over the subsequent decades (terrell, 2004). by 1981, white students made up only 24% of the enrollment in the memphis city schools. at the same time, the city had one of the largest private school enrollments in the nation (biles, 1986). contemporary conditions data reported for the 2005–2006 school year list the following percentages as representative of the racial demographics of the memphis city schools: 85.1% black, 4.6% hispanic, 8.9% white, and 1.3% asian or native american. as in the days immediately following desegregation, the demographics of the district do not mirror those of the city itself. according to the memphis chamber of commerce, the city‘s population is 34.1% white, 61.2% black, 2.1% asian, and 2.6% ―other anderson & powell metropolitan perspective journal of urban mathematics education vol. 2, no. 1 16 races.‖ in part, this difference reflects the ongoing role of private schools in memphis. there are currently more than 100 private schools serving nearly 30,000 students in the memphis area, and the vast majority of students in these schools (95–97%) are white (pohlmann, 2008). thus, like fayette county, memphis reflects patterns of public-private segregation. moreover, the ―minority‖ populations in memphis city schools (hispanic, white, and asian/native american) are not spread proportionately throughout district schools. for example, 18 schools (or 10% of the total number of schools) serve over 50% of the hispanic students in the district. nine schools (or 5% of the total number of schools) serve nearly 50% of the asian and native american population. nearly 75% of the white students in the district are served by 18 schools (or 10% of the total number of schools). and 101 of the 180 schools have populations that are at least 95% african american. like fayette county schools, district-wide outcomes in memphis city schools are below state averages. for example, the 2006 graduation rate was 67.2%, compared to a state goal of 90%. the 2006 district grade in k–8 mathematics achievement was a ―d,‖ compared to a ―b‖ for the state. similarly, the 2006 3-year district average on the high school algebra test was 65% proficient or advanced, compared to a state 3-year average of 82%. in addition, like fayette county, black-white achievement gaps are evident in district mathematics scores. the 2006 3-year district average for the high school algebra test was 89% proficient or advanced for white students, compared to 61% for african american students. similarities and connections between the “rural” and “urban” contexts as we began our work with fayette county schools, we were aware of certain similarities between the two school districts—memphis city and fayette county. we knew that both districts had demonstrated poor academic performance in mathematics relative to state averages and benchmarks. we were also aware of other similarities in outcomes, including racial achievement gaps and graduation rates that do not meet state targets. as we spent time in fayette county schools, we also became cognizant of the racial bifurcation between schools on the two different sides of the county. this racial bifurcation is similar, in many respects, to the concentration of students in memphis city schools, where more than half of the schools are 95% african american and 75% of white students are concentrated in 10% of schools. these similarities were more easily recognizable. it was only when we began to explore the history and broader contemporary conditions that we recognized additional connections and similarities—relationships that shed some light on the similar outcomes that we had already identified. anderson & powell metropolitan perspective journal of urban mathematics education vol. 2, no. 1 17 for example, the historical account makes clear that the boundaries between ―urban‖ and ―rural‖ have been porous in this case. as a city, the fate of memphis has been tied to the surrounding rural counties, including cotton-producing fayette (green, 2007). economically, the city has relied on the rural areas in terms of both human and material resources. perhaps as significant was a shared worldview grounded in the plantation mentality—a worldview that shaped the opportunities available to african americans who left the rural counties to come to memphis (green, 2007). thus, the history of these two settings makes clear that the distinction between urban and rural was not as clearly defined in this case as initially perceived, given the surface differences between the two locations. a second similarity between the two settings can be seen in the desegregation histories of the two school districts—histories that demonstrate the salience of the plantation mentality. in both school districts, the historical record documents conscious efforts on the part of the white power structure to maintain a dual education system, even after the brown decision. while both districts did eventually take steps toward desegregation, the resulting white flight from public schools has left both districts with significantly larger proportions of african american students than the populations of the respective city or county. moreover, the physical removal of white students from the public schools has impacted the districts in similar ways. for example, kiel (2008) notes that the response to busing and the divestment of white students from the memphis city schools led to a loss of public support (and, therefore, funding) for education. similarly, fayette county schools lost the support of white state legislators following desegregation and white flight from the district (hunt, 1981). this process of wit hdrawal of political and economic support as a result of white flight is certainly not unique to these two districts. nevertheless, this pattern reinforces the ongoing salience of the racial dynamics captured in the plantation mentality—perhaps suggesting a new manifestation of this mentality. as legal scholar, charles lawrence (2005) notes, [segregated schools] build a wall between poor black and brown children and those...with privilege, influence, and power. this wall denies them access to the resources we command: social, political, and economic….the genius of segregation as a tool of oppression is in the signal it sends to the oppressors—that their monopoly on resource is legitimate, that there is no need for sharing, no moral requirement of empathy and care. (p. 1377) the histories of these two districts suggest that the denial of access to a variety of resources and lack of concern on the part of the powerful have played significant roles in the construction of contemporary conditions. a third connection that was not immediately obvious to us as we initially considered the conditions in these two systems is the ongoing relationship be anderson & powell metropolitan perspective journal of urban mathematics education vol. 2, no. 1 18 tween urban and rural that continues today. upon closer examination, we found that the rural setting is intricately intertwined with the conditions in the urban center. for example, city-zoning ordinances and subdivision regulations adopted in the 1980s began the shift from the city to the suburbs (waters, 2005). fueled by high city taxes and other urban strains (charlier, 2005), that suburban growth has begun to significantly change the landscape of fayette county. as a result of the growth, property values in the western portion of fayette county rose 21% between august 2002 and november 2003. over the same time period, total appraised values in memphis dropped more than $145 million (waters, 2005). thus, the changes that are transforming fayette county from rural to ―suburban‖ are related to conditions in the city. the ―pull‖ conditions that contributed to movement from fayette county to memphis a century ago have now been replaced with ―push‖ conditions that have changed the direction of the population flow. lessons learned at the beginning of this article, we noted our interest in what it might mean to ―put the ‗urban‘ into mathematics education scholarship.‖ our goal in this article has been to use the cases of these two settings (urban and rural) to explore what it might mean to take mathematics education scholarship in this new direction. in particular, we sought to outline how our thinking regarding the differences between the two districts shifted as we began to recognize several similarities and connections. we now seek to outline the implications of these cases for the future development of urban mathematics education scholarship. first, we suggest that the cases of these two settings and the relationships between them point to the critical significance of context. while memphis is decidedly urban, the historical context highlights the salience of the urban-rural interface. we would not expect this urban-rural matrix to manifest itself in the same ways in other urban areas. in fact, what is clear from the historical record is that the nature of this urban-rural connection was shaped by conditions that were specific to this context (green, 2007; pohlmann, 2008). thus, as we seek to put the urban into mathematics education scholarship, we must be aware that urban spaces are not monolithic. there are different trajectories of urban history and development with implications for understanding the contemporary conditions of education in general and mathematics education in particular. as we learned from the examination of these cases, educational researchers take on assumptions (perhaps incorrectly) when we identify an area as urban without fully explicating its history and contemporary context. what this case taught us was the importance of understanding the specific context that we were defining as urban. a second lesson that we learned from these cases was the important role that paradigms beyond mathematics education might play in putting the urban in ma anderson & powell metropolitan perspective journal of urban mathematics education vol. 2, no. 1 19 thematics education scholarship. in particular, we examined these cases through the lens of critical race theory. this framework allowed us to recognize two key elements of the overlap between these two cases. first, crt demands attention to the role of race in constructing contemporary social and educational conditions. in this case, this orientation highlighted both the historical conditions (e.g., the plantation mentality and response to desegregation) and the contemporary status of education (e.g., ongoing segregation both within district schools and across public-private sectors) that reflect the past and current salience of race. in addition, the crt framework requires attention to the historical context. it is not enough to simply examine conditions as they currently exist. those conditions must be understood within the historical dynamics that shaped them. green (2007) points to the importance, for example, of ―discerning the significance of history for today‖ (p. 294). we assert that these elements of the crt framework were critical not only in the examination of these two settings but also offer important directions for future scholarship on urban mathematics education. thus, one lesson learned from this case is the need to take a more expansive view of the research traditions and literature, including crt, that can inform our understanding of opportunity to learn mathematics. finally, we submit that another potentially important point made by this case is the need to consider the changing nature of cities and their relationships to the areas that surround them. our goal has been to illustrate the need to look beyond the surface descriptors to examine the interrelationships between cities and the surrounding communities. the growing suburbanization of the west side of fayette county has yet again tied the fate of the formerly rural county to that of the nearby urban area. moreover, the historical similarities, particularly with respect to education and race, point to underlying conditions that should also be considered when examining what it means for an area to be rural versus urban. the connections between memphis and fayette county reflect the need, in this case, for a more ―metropolitan‖ perspective. according to rusk (2003), the real city is the total metropolitan area, both the city and its suburbs: ―any attack on urban social and economic problems must treat suburb and city as indivisible parts of a whole‖ (p. 7). to understand the potential for the improvement of education in the metropolitan area requires consideration of this relationship between city and suburb. in fact, the memphis mayor pointed to this relationship. he said, ―you know when the [school] funding mechanism [for the city schools] is going to change—it‘s when the education of white students in the suburbs begins to suffer‖ (sparks & dries, 2005). similarly, observers have noted that the future of growth in fayette county is not simply tied to the conditions in memphis but also to the quality of education. according to one newspaper editorial (―fayette‘s growth offers a lesson‖, 2005), ―public education in fayette county could limit the community‘s growth eventually, unless county officials decide to invest more anderson & powell metropolitan perspective journal of urban mathematics education vol. 2, no. 1 20 in schools.‖ how this interdependence will shape school funding in the metro area remains to be seen. the interconnectedness of educational opportunities between memphis and fayette county seems clear, however. it is this interconnectedness that we assert should inform future research in urban mathematics education in the form of a metropolitan perspective. this metropolitan perspective represents a significant shift in our thinking about the relationship between memphis and its rural neighbor. we now recognize the historical and contemporary connections between the cotton fields of fayette county and the conditions of urban memphis. references biles, r. (1986). a bittersweet victory: public school desegregation in memphis. journal of negro education, 55 (4), 470–483. charlier, t. (2005, march 20). fayette fever: low taxes, city backlash fuel land rush. commercial appeal, p. a1. city schools integration timeline. (2005, may 16). commercial appeal, p. a18. dixson, a., & rousseau, c. (2006). and we are still not saved: critical race theory in education ten years later. race, ethnicity, and education, 8(1), 7–28. egerton, j. (1973). promise of progress: memphis school desegregation 1972–73. atlanta, ga: southern regional council. fayette‘s growth offers a lesson. (2005, march 22). commercial appeal, p. b4. green, l. (2007). battling the plantation mentality: memphis and the black freedom struggle. chapel hill, nc: the university of north carolina press. hamburger, r. (1973). our portion of hell – fayette county, tennessee: an oral history of the struggle for civil rights. new york: links books. herenton, w. 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(1996). oral history of deliberate unconstitutional separation of negro race by board of education memphis city schools. memphis, tn: university of memphis. orfield, m. (2002). american metropolitics: the new suburban reality. washington, dc: brookings institution press. pohlmann, m. (2008). opportunity lost: race and poverty in the memphis city schools. knoxville, tn: the university of tennessee press. rusk, d. (2003). cities without suburbs: a census 2000 update. washington, dc: woodrow wilson center press. anderson & powell metropolitan perspective journal of urban mathematics education vol. 2, no. 1 21 sparks, j. (2007, february 24). cotton still king? fayette farms yield crops, subdivisions. commercial appeal, p. b1. sparks, j., & dries, b. (2005, february 25). herenton hits racial nerve on schools. commercial appeal, p. b1. tate, w. (2008). putting the ―urban‖ in mathematics education scholarship. journal of urban mathematics education, 1(1), 5–9. tate, w., & rousseau, c. (2007). engineering change in mathematics education: research, policy, and practice. in f. lester (ed.), second handbook of research on mathematics teaching and learning (vol. 2, pp. 1209–1246). charlotte, nc: information age publishing. terrell, j. (2004, may 16). ―white flight‖ from memphis city schools. commercial appeal, p. a19. u.s. census bureau. (2009). state and county quickfacts. retrieved june 9, 2009, from http://quickfacts.census.gov/qfd/states/47000.html waters, d. (2005, april 24). beyond suburban sprawl: memphis urban area needs a new vision of the good life. commercial appeal, p. v1. http://quickfacts.census.gov/qfd/states/47000.html microsoft word 430-article text no abstract-2495-1-6-20210901 (proof 2).docx journal of urban mathematics education may 2022, vol. 15, no. 1, pp. 78–112 ©jume. https://journals.tdl.org/jume benjamin shongwe is a senior lecturer of mathematics education in the mathematics & computer science education cluster at the university of kwazulu-natal, mtb building, edgewood campus, 121 marianhill rd., pinetown, durban, 3605; email: shongweb@ukzn.ac.za. his research focuses on mathematical proof, reasoning and proving, mathematical argumentation, and early career mathematics teachers, especially as they intersect, with deliberate attention to issues of equity and linguistically relevant mathematics pedagogy. the marginalization of 11th-grade urban african students in proof-related pedagogy: an emancipatory perspective benjamin shongwe university of kwazulu-natal the development of urban students’ mathematical proving ability is a goal of several curricula frameworks, including some located in the southern hemisphere. however, in achieving this goal, most curriculum frameworks do so from a western worldview, which is characterized by competition and the role of the individual. the purpose of this study was to use the emancipatory lens to critique the use of a quantitative methodology in favor of the ubuntu worldview, a methodology grounded in indigenous african epistemologies, particularly storytelling. to this end, i analyzed data drawn from the administration of a survey questionnaire to a conveniently selected sample of 135 11th-grade students enrolled in three separate high schools from ethnically and socioeconomically diverse communities in the ethekwini metropolitan area of south africa. the context for the argument in this study was provided by correlating students’ understanding of functions of proof (verification, explanation, communication, discovery, and systematization) with their argumentation ability, two variables often considered as the key limiting factors for meaningful learning of mathematical proofs. the poor results obtained from a quantitative analysis of data using western perspectives highlight the emerging need for finding postcolonial methodologies that are sensitive to ethnic issues in addition to language and gender issues. in addition, the inadequacy of the current mathematics curriculum to serve the linguistic and gender needs of urban african students became apparent. this increases the need for sub-saharan instructors to have knowledge to pursue emancipatory instruction. the key contribution of this study to the field is that it sheds light on the marginalization of african students in learning mathematical proof and related concepts from western perspectives rather than conducting instructional practices in the global south’s terms; the scope of the effort may explain why research efforts in this line of work have not been documented extensively in literature. keywords: correlation, emancipatory perspectives, ubuntu methodology, urban high school students, western worldview shongwe marginalization of 11th-grade urban african students journal of urban mathematics education vol. 15, no. 1 79 he teaching and learning of mathematics takes place in different cultural contexts around the world. the south african mathematics curriculum framework, the setting of this study, does not, in harris et al.’s (2020) terms, foreground the “different cultural ways of knowing” (p. 128), yet urban mathematics education is a rich research area with significant policy implications (tate, 2008). the argument permeating this paper is that the failure to foreground the pedagogy of south african mathematics classrooms in ubuntu philosophy engenders the perpetual marginalization of urban south african students’ linguistic, cultural, and gender issues, especially so in relation to functional understanding of proof and argumentation ability. given that the term ubuntu has proven hard to define for academics because of the existence of partially corrupted versions of the term (seehawer, 2018), a comprehensive philosophical exposition on ubuntu is beyond the scope of this paper, save to say that in this study ubuntu refers to an ontology and way of living that significantly differs from those of a western worldview in that care for others and cooperation are valued more highly than competition and individual advancement (brock-utne, 2016; keane, 2008). like in the united states (matthews, 2008), there is not only a vacuum of urban scholarship in south africa, but programs meant to address the issues of disadvantaged or underprepared students are fraught with contradictions. for instance, while official language education policies in the public school systems seek to accommodate disadvantaged students, the dominance of english has a substantial impact on linguistic minorities (harper, 2011). in situating the study within literature on urban scholarship, i describe the economic, political, and racial conditions under which the study took place as important. this description is important because these conditions continue to affect the achievement of urban south african students in mathematics in general, and in proof education in particular, along racial lines and will put the results of this study into context. the results and implications of this study were numerous. not only was the investigation of the relationship between functions of proof and argumentation important, but the actual research process formed part of the findings. both these aspects of the research relate to the ubuntu worldview. at a broader level, ubuntu is encapsulated by the principle that “umuntu ngumuntu ngabantu,” which in the isizulu1 language means, “to be a human being is to affirm one’s humanity by recognizing the humanity of others and, on that basis, establish humane relations with them” (naicker, 2015, p. 3). the focus of the present study was on the relationship between students’ functional understanding of proof and their argumentation ability, as well as to point to the importance of ubuntu as an indigenous southern african research paradigm in the context of what a eurocentric curriculum does to african students’ performance in mathematical argumentation, in particular. 1 isizulu is one of the indigenous languages of south africa. t shongwe marginalization of 11th-grade urban african students journal of urban mathematics education vol. 15, no. 1 80 urbanization and urban africans to situate this study within the literature on urban mathematics education, i begin by attempting to define the term urban as viewed in the south african context. although no consensus exists for the term urbanization, it is important to at least describe how it is understood for the purposes of this paper. matthews (2008) lamented the relegation of the term “urban” to an umbrella term that indiscriminately denotes african american, hispanic, immigrant, or low-income students. he provided a sanitized, tentative definition of “urban” as not only a term that considers geographical context but also encompasses “the lives of people within the multitude of cultural, social, and political spaces in which mathematics teaching and learning takes place” (matthews, 2008, p. 2). for the context of this study, in south africa when we say “urban students,” we mean students schooling in metropolitan areas, cities, suburbs, towns, and townships, regardless of racial composition (baffi et al., 2018). i join henslin (2017) in defining urbanization as a process that entails a movement of people into cities. the urban population in south africa accounts for about 65% of the country’s total population, making it one of the most urbanized countries on the african continent (turok, 2014). african adults and their families are currently relocating from rural reservations to urban settings to have access to educational training and employment. the end result of this movement is the assimilation of african people into the dominant european culture thus rendering students unable to engage in a mathematics curriculum that reflects ubuntu values and perspectives. urban sprawl resulting from the relocation of african people from tribally controlled land (i.e., reservations) limits the opportunity for ubuntu practices in an urban setting. however, ubuntu has its own problems, particularly in its inability to affirm women; the implications of ubuntu on gender seems to be conveniently ignored. for instance, manyonganise (2015) described ubuntu as partially oppressing women in that, through the ubuntu worldview, when a baby girl is born, it is tantamount to no human being having been born at all. in contrast, baby boys earn humanness from the moment of their birth. nonetheless, manyonganise (2015) goes on to point out that purely regarding ubuntu as patriarchal would not only undermine its transformative potential but also its actualization “so that it ceases to be steeped in the past, especially in gender relations” (p. 6). the urbanization of africans in south africa is fraught with slavery, cheap and docile labor, and racism, all intended to alienate them from their land (vellem, 2014). in emphasizing the material issues underlying urbanization, mabin (1992) argued that the urbanization of africans in south africa was a consequence of the discovery of gold and the resultant economic expansion. however, the trend continues to this day. the mushrooming of shantytowns on the periphery of south africa’s major cities is evidence of this claim. møller (2007) pointed out that, for example, cape town shongwe marginalization of 11th-grade urban african students journal of urban mathematics education vol. 15, no. 1 81 doubled its size over the previous 20 years (1987–2007). he further pointed out that large-scale rural-to-urban migration reshaped the country’s settlement patterns in the early twenty-first century. rather than radically challenging the apartheid urban landscape, the neoliberal policies of the former “liberation movement,” which is now the ruling party, the african national congress, tend to reinforce race and class segregation (maharaj, 2020). in fact, the financial circuits of capital exacerbate capitalism’s intrinsic economic, social, and environmental inequalities in south africa (bond, 2013). similar to trends in indigenous māori schools (trinick & stevenson, 2010), the academic achievement of urban african students who come from poorer socioeconomic backgrounds is lower than that of students from higher socioeconomic backgrounds, such as whites, indians, and coloreds. students from higher socioeconomic backgrounds come from families who can afford to send their children to elite private schools that charge exorbitant fees. these inequalities are perpetuated by a capitalist economic system inherited by the democratic government. although south africa achieved democracy in 1994 after the end of the apartheid (afrikaans for “apartness,” used to describe a system of separate, racial, and ethnic development based on a white supremacy stance) era, the economic system was not changed. as schneider (2003) put it: although the political environment in south africa is vastly improved, economic apartheid still exists: the economic divisions along racial lines created by apartheid are still in place today. despite these divisions, neoliberal economists continue to press for a largely unregulated market system, which is unlikely to improve the lives of most black [african] south africans. (p. 23) in south africa, four official racial groups were declared: (black) africans, whites, indians, and coloreds. “africans” refers to black people indigenous to south africa, “whites” refers to those born in this country but are descendants of immigrants from europe, and “indians” refers to those with origins from the country of india. the term “coloreds,” as explained by petrus and isaacs-martin (2012), is used to identify a specific group in south africa and is most often attributed to students popularly perceived as being of mixed racial and ethnic descent who, over time and due to specific historical, cultural, social, and other factors, have undergone various changes in their perceptions of their identity as coloreds. although south africa has indigenous nations—for example, the koi sans, amazulu, amaxhosa, and so on—its urban classrooms are multicultural; they are not organized along lines of nationality. put differently, there are no urban schools deliberately designated to attend to a specific indigenous community. as a consequence, all schools follow a neoliberal, capitalist urban system whose curriculum is structured along western contexts and delivered in english. however, standard south shongwe marginalization of 11th-grade urban african students journal of urban mathematics education vol. 15, no. 1 82 african english is not the native language of many african students. the absence of mathematics registers in the various indigenous languages of south africa presents a huge challenge for teachers and teacher educators at a time when, as warren et al. (2007) have pointed out, the discrepancy between home and school language directly impacts african students’ achievement in mathematics in the long term. although ubuntu can be viewed as one way of understanding the struggles of urban african students in a western-oriented mathematics curriculum, mathematics education research requires theories that explicitly address the urban african student. in this regard, i echo tate’s (2008) sentiment that “if urban mathematics education research is to be taken seriously, this kind of theoretical and empirical interaction should be the norm” (p. 6). an economic, political, and racial context of the study the still unresolved idea of south africa has a long history whose full account is beyond the focus of the present study. however, its brief description is, nonetheless, possible. the settlement of the dutch (who later became known as the afrikaner nation) at what is now known as the western cape was followed by an agreement with the british to construct south africa into a white state in which africans featured in the discussions as providers of cheap labor (ndlovu-gatsheni, 2018). worth mentioning here is that about 80% of the south african population is african, yet the western worldview dominates the education system. as ndlovugatsheni (2018) asserted, race rather than class is still an invisible but active organizing principle of informing unchanging patterns of inequality, poverty, and a eurocentric curriculum. accordingly, in the europeanization of africans by western education, the ubuntu worldview has been deliberately suppressed, even under democratic rule. put more emphatically, a predominantly african government has embraced european education and economic systems and demands them even in the postapartheid era. the consequences of this has proven to be dire for the african child, especially in the learning of the concept of proof. proving, like mathematics, is a social activity in which, for example, one of its functions is to serve as a means to communicate mathematical knowledge. the communication function of proof means that students must make sense of the arguments embedded in a proof, learn the mathematical language, and “transmit” this knowledge publicly (de villiers, 1990). clearly, they need to present arguments. this is where the problem lies for urban african students. for most of them, english is a second, third, or even fourth language. certainly, this hampers their articulation of mathematical ideas and the elements of the proving process (e.g., patterning, conjecturing, exemplification, and generalization), which in turn affects their achievement in mathematics. an alternative, however, exists. shongwe marginalization of 11th-grade urban african students journal of urban mathematics education vol. 15, no. 1 83 the alternative i am referring to in the preceding paragraph is ubuntu, widely recognized as being the african’s worldview (church, 2012). according to ndlovugatsheni (2018), the use of ubuntu has gained traction in academic literature over the past years, partly because of the move towards foregrounding african constructions about the world and unravelling of the legacy of colonialism. from a mathematical proof and argumentation perspective, urban students need to come together in mutually supportive and respectful ways as they make arguments to prove a theorem. however, i ask a more general research question: how can an urban african student engage in mathematical argumentation successfully when his or her indigenous language has been relegated to an inferior position and thus never been considered to be a suitable medium of instruction? probably the most sensible answer to this question is provided by brock-utne (2016), who argued that things will not change because the use of the former colonial language serves to keep a tiny minority at the top. hence, she further argued, the majority of students who do not speak the colonial language “either drop out of school or sit there year after year learning nothing except self-contempt” (p. 33). a similar concern has been raised by garcia-olp et al. (2019), who argued that the teaching of mathematics to indigenous students through the european viewpoint rather than through an indigenous lens limits their participation in the mathematics field after high school. brock-utne and desai (2010) argued that the majority of school teachers and students prefer the so-called international (i.e., english) language to continue as the language of instruction even though they can hardly understand it. although i tend to agree with brock-utne and desai, the situation is different for students once they are confronted with more cognitively challenging concepts in mathematics. they yearn for their indigenous languages. from my own existential experience in my mathematics education classes, i would often hear students pleading to provide their answers or to express key ideas in isizulu: “ngingayisho ngesizulu?” (can i say it in isizulu?). that said, i want to echo brock-utne’s (2016) sentiments that for urban african students to achieve in mathematics, the indigenous languages these students speak must be brought into the classroom. writing from a problematic positionality learning mathematics is a social activity. as such, the social context in which its teaching and learning takes place shapes an individual's experiences. as is common in many african settings, the context in which communication between children and adults differs from that of their white counterparts. therefore, children’s learning experiences may also vary (mercer & littleton, 2007). however, this variety is impoverished because of the absence of literature on the functional understanding of proof and argumentation from ubuntu perspectives. the present study is written from shongwe marginalization of 11th-grade urban african students journal of urban mathematics education vol. 15, no. 1 84 the perspective of a western trained academic man (not by design though) who lives and works in the multicultural, sub-saharan african continent. perhaps this explains why i did not situate this study within an ubuntu framework but have only done so retrospectively. as far as i can ascertain, this study is the first to incorporate elements of both western and african views on the relationship between functional understanding of proof and argumentation. the adoption of this approach to the study stems from a reminder that an honest investigation was the one that took into account not only my experiences as a south african but also the historical factors that continue to influence my research work. my africanness and previously designed disadvantages have given me limited opportunities in life—malnutrition from infancy, poor health care, poor education, fewer opportunities to see other parts of the world, daily fear and anxiety, and so on. in the recent past, i have come to view the difficulties with which african students learn mathematics—the proof concept in particular—as more of a sociocultural issue than a cognitive matter. this article challenges existing theories to begin to rethink the teaching and learning of the concept of proof as a space of ontological struggle for urban students of african descent. although the vast majority of african students live in urban environments today, what the mandatory curriculum designers ignored is the ontological orientation that such students bring with them to the classroom. as harris et al. (2020) pointed out, this is a deficit approach in that it considers white, middle-class, monolingual english as the norm against which students of all backgrounds are to be measured. as is the case with all researchers, my life experiences informed various aspects of this study. in this subsection i provide insights into the hypotheses derived from my experiences with the concept of proof. all these aspects influenced the research process (for example, research questions, sample, methods, interpretations, and so on). for instance, though some schools with predominately african students tend to achieve a 100% pass rate in their grade 12 examinations, the quality of these passes tend to be weaker than those of higher socioeconomic schools with similar pass rates. hence, i chose to compare the quality of learners’ functional understanding of proof and argumentation with the economic, political, and racial contexts in mind. overall, despite centuries of engaging with proof, mathematicians, philosophers, educators, and scientists still find the notion of proof as nebulous and contested as ever. it can be argued that this is because proof is as much a philosophical construct as it is a mathematical one. evidence of the nebulous nature of proof is found in difficulties in proof construction as encountered by both high school and undergraduate students, including preservice teachers. however, in proving, students need to construct arguments. these two constructs are important to study because they define the mathematical practice. shongwe marginalization of 11th-grade urban african students journal of urban mathematics education vol. 15, no. 1 85 the functions of proof and argumentation under a western perspective attempts to teach proof to high school students (frequently during short periods of time) have been unsuccessful (hadas et al., 2000; pedemonte, 2007). given that the “failure to teach proofs seems to be universal” (hadas et al., 2000, p. 128), functional understanding of proof and argumentation—activities edwards (1997) referred to as the “territory before proof” (p. 188)—need to be part of the mathematical instruction that precedes and supports the development of students’ practice with proofs. along this line, marrades and gutiérrez (2000) argued that it is vitally important for both teachers and researchers in the area of proof to determine students’ understanding of functions of mathematical proof in order to gain insight into their attempts to solve proof problems. the general motivation for this study came from the need to measure students’ understanding of the functions of proof in mathematics and argumentation quality because lack thereof contributes to difficulties with learning proofs meaningfully (e.g., de villiers, 1990, healy & hoyles, 1998). as de villiers (1994) put it: extensive experience with children in interview and classroom contexts seems to indicate that many of their problems with mathematical content and processes often do not lie so much with poor instrumental proficiency nor inadequate relational or logical understanding as in a poor understanding of the usefulness or function thereof. (p. 11) de villiers’ (1990) model describes five functions that proof performs in mathematics: verification, explanation, communication, discovery, and systematization. thus, proving in the mathematics classroom includes not only engaging in its cognitive functions (explanation and discovery) but also in its social (verification and communication) and epistemological (systematization) ones. according to hanna (2000), the explanatory function of proof helps to make mathematics meaningful and understandable. this “enlightening” or “illuminating” function brings argumentation into the arena. support for this view comes from hanna’s (2007) statement that “an argument presented with sufficient rigor will enlighten and convince more students, who in turn may convince their peers” (p. 22). noteworthy is that argumentation may be of low or high quality based on the absence or presence of rebuttals (i.e., counterexamples) in an argument (osborne et al., 2004). however, as brock-utne and skattum (2009) pointed out, one other reason why learning and teaching of proof has been difficult, particularly for african students, which is often not given the attention it requires, is the student’s indigenous language. they further pointed out that despite more than 40 years of emphasis by educational linguists that the use of a language in which students (and teachers) do not possess the necessary proficiency for cognitive academic development has disastrous shongwe marginalization of 11th-grade urban african students journal of urban mathematics education vol. 15, no. 1 86 consequences—as it does for the economic, social, and political development of a country—the schooling system has not placed a premium on this emphasis. thus, students’ battle with optimal performance in proof and argumentation can be ascribed to either their poor understanding of the text or their difficulty expressing themselves clearly in english rather than to their cognitive ability. this calls for a restructuring of teaching and learning of mathematics, in general, according to the ubuntu worldview. this worldview can be harnessed to improve the learning of proof functions and argumentation. however, it is not clear to what extent students’ functional understanding of proof affects the quality of argumentation made by them. in addition, prior research suggests that teaching approaches may vary across countries (alexander, 2020; mercer & littleton, 2007). moreover, the majority of the existing literature on mathematical proof and its functions as well as argumentation stems from studies conducted in european and north american countries, whereas systematic research on proof and argumentation across international contexts remains limited. learning of proof is embedded in and influenced by socioeconomic factors (mercer & howe, 2012). therefore, students’ understanding of proof functions and argumentation ability identified in these countries may not be generalized to other educational contexts, especially urban settings. still, research on the relationship between students’ understanding of proof functions and argumentation ability is limited and, to my knowledge, no research exists on this hypothesized relationship on the african continent, especially in sub-saharan africa. i am yet to find a study that takes into account the interplay among economic, political, and racial conditions under which students learn proof. the current study reported in this article presents an opportunity to test the universality of western education research findings. it is against this background that this study presents a unique contribution to the field. in addition, by analyzing this relationship, the findings expand previous research to indicate how the lack of mathematics registers in indigenous languages impacts urban african students’ understanding of functions of proof and, by extension, development of argumentation skills. the remainder of this paper is presented in six parts. the first deals with a review of literature related to the functions of proof, argumentation, argumentation quality, and ubuntu methodology that can allow the discipline to respond to social change in proof-related pedagogy while still retaining the ideals of scientific rigor. the next considers the emancipatory lens as a theoretical foundation of the study. what follows next is a discussion of the quantitative methodology. then the results are presented and discussed with reference to the emancipatory framework. the article concludes with a reflection on the results. shongwe marginalization of 11th-grade urban african students journal of urban mathematics education vol. 15, no. 1 87 review of related literature the review of literature is conducted for the reader to understand the background to the argument for a need for social change and is done so from the western worldview. put slightly differently, this approach is deliberately taken to provide the basis for arguing for a curriculum that acknowledges the importance of a pedagogy that foregrounds the linguistic and cultural uniqueness of the urban african student. functional understanding of proof the use of the phrase “functional understanding of proof” is meant to refer to understanding the role, function, purpose, or value of proof in mathematics rather than to its application to the real world outside of the mathematics discipline. as far as i could ascertain, only healy and hoyles (1998) have attempted to capture students’ functional understanding of proof. they used an open-ended survey questionnaire on which students were to write about everything they knew of proof and its functions in mathematics. further, they investigated the influence of statutory instruction on the nature of proof following suggestions that such instruction could contribute to deeper understanding of the notion of proof itself and thus improve its didactic treatment in the classroom. they found that the function of proof as a means to verify was prevalent. hanna (1995) posited that learning about the functions of proof in mathematics is of primary importance to mathematicians. i contend that the value of understanding the functions of proof in mathematics needs to be reflected in the mathematics classroom itself if students are to gain insight into the nature of mathematics. however, as already mentioned, argumentation is a process that brings the explanatory power of proof to bear. argumentation in distinguishing between an “argument” and “argumentation,” like blair (2012), i see the former as a “set of one or more reasons for doing something” (p. 72). although pedemonte (2007) correctly argued that there is no common definition for the concept of argumentation in the field of mathematics education, the current study adopted van eemeren et al.’s (2013) definition of argumentation as “a verbal and social activity of reason aimed at increasing (or decreasing) the acceptability of a controversial standpoint” (p. 5), because it is compatible with classroom contexts advocated by reform statements (e.g., common core state standards initiative, 2010; department of basic education [dbe], 2011; national council of teachers of mathematics, 2009). perhaps it is important to note from this distinction that an argument is the product of the process of argumentation. further, argumentation is not used to shongwe marginalization of 11th-grade urban african students journal of urban mathematics education vol. 15, no. 1 88 refer to a debate, although debate is one form of argumentation, but rather to a process of thinking and dialogue in which students construct and critique each other’s arguments (nussbaum, 2011). toulmin (2003) presented a model, generally referred to as toulmin’s argument pattern (tap), to describe the structure of an argument and how its elements are related. the tap model consists of six interdependent components: claim, data, warrant, backings, qualifiers, and rebuttals. in addition, tap is a model that has been extensively used in instructional practices as a tool to construct mathematically sound arguments (osborne et al., 2004; venville & dawson, 2010). in the context of mathematics lessons, the use of tap has mainly concentrated on the description of small group discussions among students (e.g., inglis & mejia-ramos, 2009; knipping, 2003; krummheuer, 1995, 2000; pedemonte, 2007). briefly, the basic idea of this modified model (figure 1) is that a statement, claim, or conclusion is justified by providing a ground (as shared by the mathematical community). according to dejarnette and gonzález (2017), making and justifying a claim is a fundamental aspect of doing mathematics. for this study, “ground” refers to a datum, warrant, or backing provided by the interlocutor in justifying their claim. this stance finds support in osborne et al.’s (2004) assertion that claims, rebuttals, and justifications are the salient features of argumentation that are critical for developing and evaluating practice with argumentation in the classroom. in addition, grouping these elements into “ground” circumvents the ambiguity embedded in them, as any claim, rebuttal, or justification is referred to simply as “ground” (osborne et al., 2004). a warrant is a proposition that connects a datum and claim. “rebuttal” is taken to mean a statement that seeks to show the weakness in a ground. worthy to note is that argumentations with rebuttals are of better quality than those without given that rebuttals make substantive challenges to the grounds as they refute their applicability (osborne et al., 2004). figure 1 depicts a typical argumentation in proving: “the sum of interior angles of a triangle is 180 degrees.” tap has been adapted by many in mathematics education research (e.g. krummheuer, 1995; rasmussen & stephan, 2008; yackel, 2001) as a lens to frame arguments and to analyze the quality of a specific mathematical argument. shongwe marginalization of 11th-grade urban african students journal of urban mathematics education vol. 15, no. 1 89 figure 1. the modified toulmin’s (2003) argumentation scheme not every one of these tap components is used in every argument. for instance, given the tentative nature of mathematical knowledge and the fact that for students the knowledge being constructed is new, qualifying phrases such as “most probably” or “presumably” are omitted and therefore implied in a claim. there is scarcity of empirical evidence on the influence of students’ functional understanding of proof on the quality of argumentation. knipping (2003) commented that it would be interesting if the relationship between functions of proof and argumentation structures were examined. alibert and thomas (1991) discussed the relationship between functional understanding of proof largely from a theoretical basis rather than conducting a systematic investigation. they believed that students’ distorted understanding of the functions of proof is a direct consequence of instruction that presents proof as a finished product, an approach that deprives students of opportunities to be partners in mathematical knowledge construction. the present study, therefore, sought to expand on previous research on functional understanding of proof and argumentation by disaggregating functional understanding of proof and quality of argumentation into their association with a variety of indicators. argumentation quality several studies on argumentation have focused on identifying, creating, and evaluating argument structures (aberdein, 2013; mariotti, 2006; pedemonte, 2007). the underlying theme of the findings of these studies is that the argumentation process enables the shifting of mathematical authority and ownership from the textbook or teacher to the community of students, who become producers of mathematical knowledge (bay-williams et al., 2013; rumsey & langrall, 2016. in addition, the power of argumentation is that it bears resemblance to how mathematical knowledge claim/conclusion rebuttal angle b is equal to angle d ground (data/warrant/backing) • given that bc is parallel to de • since/because the parallel lines are cut by a transversal, then alternating angles are equal unless lines lie on a spherical surface shongwe marginalization of 11th-grade urban african students journal of urban mathematics education vol. 15, no. 1 90 is constructed in the practice of mathematicians. although argumentation is seen by mathematics education documents and researchers in mathematics alike as vital in the learning of mathematics, little research has focused on measuring the quality of argumentation in students across the school grades. given this background, the purpose of this study was to take a step towards addressing this scarcity by examining the quality of students’ arguments along with the mathematics inherent in a proofrelated task. recent mathematics curriculum reform statements have framed investigations as a key feature in the learning of mathematics in high schools (e.g., national council of teachers of mathematics, 2009; dbe, 2011). these efforts are supported by research suggesting that formulating arguments supports learning of mathematics (jahnke, 2008; rumsey & langrall, 2016). in addition, current research in learning, teaching, and assessment has repeatedly pointed to the importance of eliciting students’ preconceptions in instruction (national research council, 1993). i argue that these approaches, which differ from the more dominant knowledge transmission method, are appropriate, as they seek to create classroom environments that resemble the practice of mathematicians (i.e., abstracting, conjecturing, proving, and seeking counterexamples). the knowledge transmission method relates to the notion that the “expert” (teacher) is required to fill students’ minds with information to be memorized and regurgitated when required (thomas & pedersen, 2003). ricks (2010) bemoaned the character of school mathematics by pointing out that it deprives students of the natural socializing appeal of mathematical activity. in contrast, the methods advocated by curriculum reform efforts underscore investigation as a mathematical activity to reflect mathematics as a human activity. however, conducting mathematical investigations involves high levels of mathematical reasoning (desforges & cockburn, 1987). the benefit that accrues with investigations is much more than the sharing of mathematical ideas and strategies in that as students “prepare to present their work, and as they think about how they will communicate their work and anticipate their classmates’ questions, their own understanding deepens” (fosnot & dolk, 2001, p. 3). the ubuntu methodology in mathematics education the preceding review highlights the western epistemologies in mathematics. however, these ways of knowing are in sharp contrast to how an african child constructs their knowledge. therefore, contemporary research methodologies that are used to investigate african students’ knowledge cannot be grounded in the western worldview. as the results section in this study will reveal, it is disingenuous to describe the work of an urban african child from this worldview. one of the aims of this study is to argue for ubuntu as a viable methodology to study the african child’s performance in proof-related concepts, at least. shongwe marginalization of 11th-grade urban african students journal of urban mathematics education vol. 15, no. 1 91 as already mentioned, the essence of ubuntu is that knowledge construction is a communal effort as expressed through indigenous languages. ubuntu defines the individual in terms of their relationships with the community in which the individual exists; in this case, the individual student flourishes but assists, and interacts with, others (muwanga-zake, 2009). the intent of this paper is not to project an ubuntu research methodology as categorically oppositional to conventional methodologies, but rather on using empirical evidence to advocate for an approach to research that is grounded in indigenous african epistemologies. i provide one essential instance to support the notion that the western way of knowing is indeed incompatible with an ubuntu methodology that can allow the discipline to respond to social change in proof-related pedagogy while still retaining the ideals of scientific rigor. although providing a rebuttal in argumentation is viewed as part of critical thinking, defined here as a set of cognitive skills aimed at strengthening a hypothesis or claim in decision making in complex situations (lubben et al., 2010), it is inconsistent with african cultural practice. for instance, it is taboo for children to talk back in their conversations with adults. the ubuntu worldview emphasizes the building of consensus and avoidance of confrontation by opting for utterances such as “i have another idea which i will explain” (lubben et al., 2010). noteworthy is that this approach to argumentation is not confined to students alone. scholtz et al. (2008) found that teachers with an african cultural background avoid rebuttals and use alternative moves in their arguments, such as questions and seeking consensus. in addition, the lack of mathematics registers in indigenous languages perpetuates a mathematics field that adopts the notion that african students only know the functions of proof and argumentation when learning them from western epistemological orientations, thereby masking the ubuntu ways of knowing. this increases the need for sub-saharan instructors to have knowledge to pursue emancipatory instruction. the next section provides a lens through which i approached this paper. the struggles of the urban african student in mathematics classrooms in the context of functions of proof and argumentation are characterized as tantamount to an oppressive regime. as such, it is reasonable to argue for a need to seek emancipation for these students. the results will clearly support this stance. the emancipatory framework the emancipatory framework provided a lens through which the correlations between functional understanding of proof and argumentation quality after controlling for gender was explored and discussed. this lens was used throughout the study to argue for the inclusion of indigenous languages and the ubuntu worldview in mathematics education as well as to critique and raise methodological issues associated with the use of a quantitative methodology. shongwe marginalization of 11th-grade urban african students journal of urban mathematics education vol. 15, no. 1 92 the central feature of emancipatory research is its intent to challenge inequities and disrupt the status quo where necessary; it is responsive to social issues in that it has oppression as its central focus. thus, the key objective of this framework is social change, fostering an ideology based on the paradigm that knowledge is constituted in a sociocultural context in which research is conducted (rose & glass, 2008). the term emancipation defines a process of resistance aimed at the transformation of existing oppressive structures and the creation of alternative structures that are more progressive (ali, 2002; inglis, 1997). as inglis (1997) put it, the process of emancipation involves “critically analyzing, resisting and challenging structures of power” (p. 4). it is incumbent upon the mathematics education research community to argue for social justice because, according to gorelick (1996), the hidden nature of societal oppression can obscure a student's awareness of their marginalization. as ali (2006) aptly noted: self-report quantitative measures can also be problematic regarding the conclusions we may draw from the data they produce. for example, while it is presumed that participants' selfreports are truthful, the reporting in which participants engage may not fully capture their experience. (p. 32) according to this lens, if oppression was more fully understood, necessary social change could be achieved through society taking political action (ramazanoğlu, 2002). thus, an emancipatory lens is important because it raises the consciousness of a people who are located in marginalized and oppressed positions. in the current study, the author’s aim was to focus on the emancipatory potential of the ubuntu methodology as a tool to challenge the current, dominant western quantitative methodology, which is a mechanism for the perpetuation of economic exploitation and thus effectively deleting the identities of students of african origin in mathematics education. given this background, the purpose of this study was to use the emancipatory lens to critique the use of a quantitative methodology in favour of the ubuntu worldview, a methodology grounded in indigenous african epistemologies, particularly storytelling. the context for the argument in this study was provided by correlating students’ understanding of functions of proof (verification, explanation, communication, discovery, and systematization) with their argumentation ability, two variables often considered the key limiting factors for meaningful learning2 of mathematical proofs. the following research questions guided the investigation of this relationship: 1. what does the western method of analysis reveal about the relationship between students’ functional understanding of proof and argumentation quality? 2. what is the extent to which the western worldview marginalizes urban african students’ performance? 2 according to nafukho (2006), meaningful learning takes place in the languages that students speak. thus, learning to prove and argue in a language in which students are not adequately proficient has negative consequences for them. shongwe marginalization of 11th-grade urban african students journal of urban mathematics education vol. 15, no. 1 93 methods the quantitative study presented here is part of a larger research project that aimed to examine the interplay between self-efficacy in proving, functions of proof, proof construction, and argumentation in high school mathematics. the project began in 2015 and reached its finality in 2019. sample design the prediction study reported herein followed a correlational research design, which is important in understanding teaching and learning for several reasons. correlational designs not only enable the preliminary identification of possible factors that influence students’ ability to produce high-quality argumentation in the construction of mathematical proofs but also variables that require further investigations in the relationship (mcmillan & schumacher, 2010). data were drawn from 135 students in cambridge college, ayanda high, and tswelopele high (pseudonyms), randomly selected from a sample of 10 dinaledi schools in ethnically and socioeconomically diverse communities in the urban district of the ethekwini metropolitan area. in the pursuit of increasing the participation and performance in mathematics and physical sciences of historically disadvantaged students in south africa, the dbe established the dinaledi school project in 2001 (dbe, 2009). the initiative involved selecting certain secondary schools for dinaledi status that demonstrated their potential for increasing student participation and performance in mathematics and science (dbe, 2009). these schools were provided with resources (e.g., textbooks and laboratories) to improve the teaching and learning of mathematics and science. the ultimate intention was to improve mathematics and science results and thus increase the availability of key skills required in the south african economy (dbe, 2009). the sample size of the current study was appropriate for a correlational study in that it was higher than creswell’s (2012) estimated threshold of 30 participants. the rationale for selecting dinaledi schools for the investigation was that these schools were monitored by a team that included senior education department officials and individuals with an interest in educational research. sample characteristics are in presented in table 1. the purpose of collecting sociodemographic data was to be able to adequately describe the sample. shongwe marginalization of 11th-grade urban african students journal of urban mathematics education vol. 15, no. 1 94 table 1 sociodemographic characteristics of sample characteristic female male total n = 78 n = 57 (54.1%) (45.9%) age m 16.42 16.85 16.64 sd 1.28 1.37 1.97 school type fee-paying 28 24 52 no-fee 44 39 83 race white 3 5 8 african 44 39 83 indian 18 16 34 colored 7 3 10 socioeconomic status low 46 39 85 middle 23 19 42 high 3 5 8 measures the independent variables that were hypothesized as influencing (predicting) the dependent variable were as follows: verification, explanation, communication, discovery, and systematization. respondents’ functional understanding of proof was assessed with the learners’ functional understanding of proof (lfup) scale, which was adequately developed and validated elsewhere (mudaly & shongwe, 2017). the lfup is a 25-item likert scale questionnaire whose first section contains items for gathering sociodemographic data, as shown in table 2. taking into account kumar’s (2005) guidelines for formulating questions, every effort was made to ensure that simple and everyday language in the questionnaire was used for two reasons. first, english was not the home language of most of the participants. second, there was no time allocated for explaining the questions to the participants. hence, the language used was made appropriate because misunderstanding of the questions would have resulted in irrelevant responses. the overall cronbach’s alpha of the lfup scale was .812. shongwe marginalization of 11th-grade urban african students journal of urban mathematics education vol. 15, no. 1 95 table 2 the structure of the lfup questionnaire category description number of items sociodemographic code; gender; class; home language; school type, race; socioeconomic status 7 verification function five-point likert subscale assessing understanding of proof as a means to verify 3 explanation function five-point likert subscale assessing understanding of proof as a means to explain 5 communication function five-point likert subscale assessing understanding of proof as a means to communicate 5 discovery function five-point likert subscale assessing understanding of proof as a means to discover/invent 5 systematization function five-point likert subscale assessing understanding of proof as a means to systematize 7 the second section of the lfup questionnaire consists of a 5-point scale (ranging from 1 = strongly disagree to 5 = strongly agree) used to judge students’ functional understanding of proof. the scores on the lfup scale were organized in interval categories, which was amenable to parametric statistical analyses (creswell, 2012). negatively worded items received a mean variance value of less than 2.5, neutral items received a mean variance value of 2.5 to less than 3.5, and positively worded items received a mean variance of value of 3.5 or higher. a sample of the items under the explanation function is shown in table 3. table 3 an extract of items of the explanation subscale on the lfup instrument (n = 135) item sd d n a sa t4 a proof explains what a maths proposition means. 1 2 3 4 5 t5 a proof hides how a conclusion that a certain maths proposition is true is reached. 1 2 3 4 5 t6 proof shows that maths is made of connected concepts and procedures. 1 2 3 4 5 t7 when i do a proof, i get a better understanding of mathematical thinking. 1 2 3 4 5 t8 proving make me understand how i proceeded from the given propositions to the conclusion. 1 2 3 4 5 sd = strongly disagree; d = disagree; no = no opinion; a = agree; sa = strongly agree. shongwe marginalization of 11th-grade urban african students journal of urban mathematics education vol. 15, no. 1 96 respondents’ quality of argumentation was assessed with the argumentation frame in euclidean geometry (afeg) using the mathematical statement that “the interior angles of a triangle sum up to 180º.” the afeg operationally is an index of an argument modelled after tap designed to compare the quality of argumentation across individuals. the duration of the questionnaire was 30 minutes. it consisted of prompts as shown in figure 2. to successfully engage in proving, a student requires a variety of strategies for selecting, recalling, and connecting facts drawn from a rich knowledge base related to the specific geometric task (magajna, 2011). the simplified version of the tap model, like webb and webb’s (2008), makes provisions for students to also think about possible rebuttals to their claims. thus, the analysis of written argumentation enables the making of judgments of the quality of the arguments themselves, that is, determining what makes one argument better than the other. students may then provide qualifiers, which is a way to show specific conditions in which the claim is true (toulmin, 2003). please, make any statement or claim from the diagram and justify it. please, think carefully as you argue your points using the guide provided below. my statement is that ……………………………………………………………………. claim my reason for making this statement is that …………………………………………… warrant arguments against my idea might be that ……………………………………………… rebuttal i will show the condition under which the claim is true by stating that ………………... qualifier figure 1. the afeg questionnaire the psychometric properties of this analytical tool were assessed to ensure that it was valid and reliable. to achieve content validity, a discussion on the constituent elements of the scheme, coding, and scoring system of respondents’ data took place to develop an analytical tool. two university experts in the field of argumentation who were from outside the university where the author was based were consulted. the internal consistency coefficient of the instrument was calculated as α = .81. e d b c a 1 3 2 shongwe marginalization of 11th-grade urban african students journal of urban mathematics education vol. 15, no. 1 97 overall, the analytical framework was found to be sufficient for the purpose of the study. further, these psychometric results suggest that this instrument can used as a reliable assessment and diagnostic tool in instructional practices and mathematics education research. i analyzed a student’s written argumentation in which he or she engaged in a “social interaction” with himself or herself, in line with aberdein’s (2009) definition of argument as “an act of communication intended to lend support to a claim” (p. 1). analysis the data contained no outliers, and there was no multicollinearity among the predictors. this was assessed by checking the correlation coefficients, the predictor variables, and argumentation quality (table 4). in line with nunnally and bernstein’s (1994) guidelines, correlations were acceptable if they exceeded .30. some of the correlations were higher than .60. then, the correlation matrix was further examined for multicollinearity. although the items had to be intercorrelated, the correlations were not higher than tabachnick and fidell’s (2013) threshold of .80, because multicollinearity makes the determination of the unique contribution of the items to a factor difficult (field, 2009). as depicted in table 4, the association was positive and linear, which means that low (or high) scores on functional understanding of proof were associated with low (or high) scores on argumentation quality. table 4 correlation matrix of lfup and afeg scores (n = 135) variables 1 2 3 4 5 6 1. asv — 2. ase .589** 3. asc .595** .842** — 4. asd .439** .639** .785** — 5. ass .614** .674** .614** .741** — 6. afeg score .386* .479** .592* .376* .383* — ** correlation is significant at the 0.01 level (2-tailed). * correlation is significant at the 0.05 level (2-tailed). asv = average of sum of construct with highest value being 5 where: asv = verification; ase = explanation; asc = communication; asd = discovery; ass = systematization. shongwe marginalization of 11th-grade urban african students journal of urban mathematics education vol. 15, no. 1 98 the scores on the lfup scale were organized in interval categories, which was amenable to parametric statistical analyses. a five-tiered grading scale was used to assess and characterize students’ functional understanding of proof (table 5). scores were characterized as naïve (belief that the only function of proof is verification), informed (beliefs about the functions of proof are consistent with those held by contemporary mathematicians), and hybrid (mix of naïve and informed beliefs). table 5 the normative map based on lfup mean acores classification general explanation mean score range from to unencultured naïve 0 <1.5 poorly encultured naïve 1.5 <2.5 moderately encultured hybrid 2.5 <3.5 highly encultured informed 3.5 <4.5 extremely encultured informed 4.5 ≤ 5 it is important to note that the task was one that grade 10 and 11 learners with little exposure to formal proofs could understand. my goal was to make the task mathematically accessible to all participants to maximize learners’ levels of response. similar to other seatwork assessments routinely completed by students, this task may not be the most psychometrically sound assessment of student argumentation performance, but it is closely related to the realities of instruction and learning in most classrooms (calfee, 1985). students’ quality of argumentation was the dichotomous (binary; i.e., low or high) dependent variable whose values were to be predicted and therefore only contained data coded as 0, 1, 2, or 3. table 6 describes how the quality of argumentation was assessed. argumentation frame with a rebuttal was coded as high. the analysis was performed with the assistance of spss version 24.0. shongwe marginalization of 11th-grade urban african students journal of urban mathematics education vol. 15, no. 1 99 table 6 coding of argument components argument definition code description score quality my statement is that … (blank)/a claim (c) is a conclusion put forward publicly for general acceptance (toulmin, 2003). uc (uncodifiable/ no reply). 0 low my statement is that … c (claim; conclusion) 1 low my reason is that … a warrant is ground (g) provided in justifying the claim. c+g (providing ground for claim) 2 low arguments against my idea might be that … a rebuttal (r) is a statement that seeks to diminish the strength of a conclusion (pollock, 2001). c+g+r (refutation of claim/ground) 3 high results descriptive analysis (table 7) revealed that the scores on the verification function of proof were spread out about 0.53 above and below the mean. however, this result presented a rather bleak picture of functional understanding of proof among south african grade 11 students in dinaledi schools; only approximately 14% of respondents believed that proof has functions other than verification. table 7 descriptive statistics mean std. deviation n asv 2.430 0.530 135 ase 2.684 0.863 135 asc 2.835 0.916 135 asd 3.286 0.656 135 ass 2.713 1.108 135 afeg score 1.415 0.901 135 shongwe marginalization of 11th-grade urban african students journal of urban mathematics education vol. 15, no. 1 100 attempts to interpret the correlation between functional understanding of proof and argumentation quality were hampered by the possible existence of a third variable that may influence the relationship between the two variables. i used partial correlations technique to statistically control and thus nullify the effects of gender (wilson & maclean, 2011) as the third or secondary variable on the relationship between the primary variables, namely, functional understanding of proof and argumentation quality. the partialling out of gender was informed by research (e.g., geary, 1999; healy & hoyles, 2000) that suggests that student performance in mathematics tends to be a function of gender. because the zero-order correlations have already been analysed above, i considered the section with the partial correlations in table 8. in the previous section, the significant relationship between functional understanding of proof and gender seemed to suggest that gender has influence in explaining the functional understanding of proof-argumentation quality association. however, the partial correlations section shows that controlling for gender further weakens the strength of the significant relationship between functional understanding of proof and argumentation ability (r = .214, p = .013). clearly, controlling for gender was justified given that gender was, as shown in table 8, one secondary variable that seemed to influence the relationship between the two primary variables. table 8 assessing the influence of functional understanding of proof on argumentation, controlling for gender control variables lfup score afeg score gender -none-a lfup score correlation — significance (2-tailed) afeg score correlation .225 — significance (2-tailed) .009 . gender correlation .171 .089 — significance (2-tailed) .047 .302 gender lfup score correlation — significance (2-tailed) afeg score correlation .214 — significance (2-tailed) .013 . a. cells contain zero-order (pearson) correlations. shongwe marginalization of 11th-grade urban african students journal of urban mathematics education vol. 15, no. 1 101 because statistical significance conflates results as it is affected by sample size, effect size was determined. in addition, relying on the size of the effect of a relationship rather than its statistical significance promotes a more scientific approach to the accumulation of knowledge (creswell, 2012). the multiple correlation coefficient between argumentation scores and covariates combined, r, was computed to determine the coefficient of determination (r2), which is the square of the pearson product moment correlation coefficient. this was performed to express the proportion of variability in argumentation that can be accounted for by fucntional understanding of proof (table 9). according to muijs’ (2004) criteria, this model is of poor fit, as it meant that only 6.3 % of the variance in the argumentation scores was explained by the covariates. table 9 a summary of the r, r-squared, and adjusted r-squared in analysis of lfup and afeg model r rsquared adjusted r-squared std. error of the estimate change statistics r-squared change f change df1 df2 sig. f change 1 .252a .063 .056 .87517 .063 9.011 1 133 .003 a. predictors: (constant), t15. b. dependent variable: afeg score. multiple regression was run to tease out which of the functional understanding of proof variables were most closely associated with argumentation quality, adjusting for gender. gaining insight into whether each of the functions explained a significant amount of variance in argumentation quality was necessary because it helps not only to explain argumentation quality but also to understand the significance of the relationship between functional understanding of proof and quality of argumentation. the beta (β) values in table 10 provide interesting information about some of these factors with regard to their relative effects on argumentation. first, whereas knowing that proof explains had the strongest positive and statistically significant effect on argumentation where β = .50 and the level of significance was p = .01, knowing both that proof is a means to verify and discover had a nonsignificant impact on argumentation. second, whereas knowing that proof is a means to systematize and communicate mathematical ideas yielded nonsignificant results, the former had the weakest negative effect (β = –.07) and the latter the strongest negative effect (β = –.33). third, only knowing that proof systematizes had a statistically nonsignificant result at .174 (p > .01) effect on argumentation. the interesting conclusion here was that only having an understanding that proof as a means to explain can be used to predict students’ argumentation ability. shongwe marginalization of 11th-grade urban african students journal of urban mathematics education vol. 15, no. 1 102 table 10 the beta coefficient in regression analysis model unstandardized coefficients standardized coefficients t sig. b std. error β 1 (constant) -.001 .569 -0.001 .999 verification -.140 .182 -.083 -0.770 .443 explanation -.524 .186 -.502 -2.814 .006 communication -.073 .228 -.074 -0.318 .751 discovery -.164 .187 -.119 -0.875 .383 systematization -.266 .195 -.327 -1.368 .174 a. dependent variable: afeg score. the analysis of the respondents’ writing frames, as shown in figure 3, revealed several noteworthy findings. first, the majority of arguments emerging from the data was at a low level (70%). second, though only a small minority, 18% of these arguments included claims that were substantiated. third, particularly discouraging was that only 2% of arguments developed by students were characterized as being of high quality because they consisted of rebuttals. what was important about these findings was that they provided deeper insights into students’ difficulties with constructing and sustaining a mathematical argument. there were no qualifiers. figure 3. distribution of argumentation elements across the three schools 0 5 10 15 20 25 30 35 c c+g c+g+r uc f tap elements argumentation in modified tap cambridge college ayanda high tswelopele high shongwe marginalization of 11th-grade urban african students journal of urban mathematics education vol. 15, no. 1 103 although the data was analysed by two researchers, we used cohen’s (1968) kappa coefficient (κ) to determine the reliability. in addition, this coefficient was appropriate to use on the basis that we adopted a multicategory rubric comprising a ratio scale in which responses were classified into one of four categories. cohen’s kappa coefficients were calculated for each of the five responses using stata, a statistical software that enables analysis, management, and graphical visualisation of data. the very few unanticipated responses received were fitted into the rubric such that the following kappa coefficients were obtained: content = .95 and argumentation = .97. as altman (1991) suggested, these values indicated very good agreement between the raters. discussion the discussion is organized according to the two research questions. the results of the first research question are discussed purely from its quantitative nature to provide a basis for arguing that the teaching and learning of functions of proof and argumentation from the western worldview only serves to perpetuate the struggle of the urban african student. the next subsections discuss the results in turn. the relationship between students’ functional understanding of proof and argumentation quality from a western methodology the result on the extent to which functional understanding of proof is related to argumentation quality is encouraging for various reasons. first, this positive relationship was anticipated because its existence was primarily based on the author’s hunches, as there were no prior studies that investigated it. second, although the result suggested that the association was tenuously significant, the fact that a correlation existed was important. it is hoped that this result spurs mathematics education researchers to further conduct studies that seek to enhance our knowledge on the role that functional understanding of proof and argumentation can play in the meaningful construction of proof. third, this finding empirically corroborates and strengthens knipping’s (2003) suggestion that this relationship is important if promotion of meaningful learning of proof were to be better understood. fourth, the mathematics classroom is a centre of struggle for urban african students. in addition, this study has demonstrated that the understanding of proof as a means to verify the truth of mathematical statements is prevalent. this finding is consistent with the results of numerous other studies (e.g., de villiers, 1990; harel & sowder, 1998; healy & hoyles, 1998; knuth, 2002). there is absolutely nothing wrong with understanding the function of proof as verification; however, students tend to view verification not only as the sole function that proof performs in mathematics but also as merely using a few cases as proof that a conjecture is true. the shongwe marginalization of 11th-grade urban african students journal of urban mathematics education vol. 15, no. 1 104 main point here is to note that verification of a mathematical proposition can take two forms: empirical or deductive; empirical by selecting a few cases and deductive by logically connecting a set of axioms to produce a new result. de villiers (1990) argued that if students see proof only as a means “to make sure” through their own experimentation, then they will have little incentive to generate any kind of deductive proof. in contrast, seeing the function of proof as a means to explain can motivate students to generate a proof of a conjecture deductively (hanna, 2000). the investigation also included a focus on understanding which of the five functions of proof best predicted the quality of students’ argumentation ability. seeing proof as a means to explain why a mathematical conjecture is true not only significantly predicted students’ argumentation quality but also powerfully contributed to the variability of scores on argumentation quality, and understanding this function of proof was more important for students than appreciating that proof is a means to verify the truth of mathematical statements or believing that proof is a means to communicate mathematical knowledge. these empirical results give further credence to the importance of focusing on functional understanding of proof and argumentation as components of the “territory before proof.” the multiple regression results helped in providing an explanation of the impact of functional understanding of proof on the quality of argumentation that students can generate from a south african perspective. however, the communication function of proof was statistically insignificant in influencing argumentation quality. one explanation of this observation was found in respondents’ low quality of argumentation; communication of mathematical knowledge to peers requires the ability to put together sound and convincing arguments, which clearly the respondents were unable to formulate. the results also provided insight into the proportion of variance in students’ quality of argumentation that can be explained by functional understanding of proof. overall, findings indicated that the regression model was of poor fit because the predictor variables only explained 6.3% of the variance in the argumentation scores. this means that functional understanding of proof explains only a small proportion of students’ quality of argumentation. one explanation of this finding is that although in the design of the study gender was removed so that the relationship between functional understanding of proof and argumentation could be more clearly determined, gender could also be a mediating variable (influencing both functional understanding of proof and argumentation). although the effect size was not, in cohen’s (1988) terms, “grossly perceptible and therefore large” (p. 27) to equate to the difference between the heights of 13-year-old and 18-year-old boys, dismissing it as being of little practical significance can be irrational. confidence in the effectiveness of this relationship can only follow widespread investigations in different contexts and countries. shongwe marginalization of 11th-grade urban african students journal of urban mathematics education vol. 15, no. 1 105 support for this stance is found in glass et al. (1981). in criticizing cohen’s (1988) convention of “small,” “medium,” and “large,” glass et al. (1981) argued that effectiveness of a relationship can only be interpreted in relation to an average of estimates of different effect sizes obtained from other studies and that the practical importance of an effect depends entirely on its relative costs and benefits. for instance, if it could be shown that making a small and inexpensive change in proof instruction by paying attention to the “territory before proof” raises students’ ability to learn proof meaningfully by an effect size of even as little as .10, then this could be a very significant improvement in proof education. thus, it is in this light that this result is viewed as being of educational significance. still, it adds to our knowledge of the nature of the relationship between functional understanding of proof and argumentation. having discussed the results, i consider their implication for the urban african student. what is the extent to which the western worldview marginalizes the urban african students’ performance? the disappointing but not unexpected results are contemporary realities of african students; that is, the exclusion of the languages, cultures, and identities of urban african students and teachers account for these results. it my firm belief that these results do any in any way suggest that african students are poor at understanding functions of proof and at making and supporting their claims. rather, the problem seems, more than anything, to be the foreign language in which mathematics instruction takes place. because ubuntu emphasizes relation to each other through storytelling, then ubuntu storytelling is a research methodology (mucina, 2011) that can be utilized to understand the performance of these students when taught in their indigenous languages. very little can be said about some of the students’ argumentation ability, especially african students. they produced incoherent statements in the afeg questionnaire that can only be described as idiosyncratic in their nature. these results were ascribed to these students’ poor command of the language of instruction, english, thus confirming the need to rethink the way the mathematics curriculum is organized for genuine achievement of our students. it is in this light that future research must use ubuntu stories as methodology to encourage ubuntu scholars to make their work not only accessible to our larger african communities but also make the teaching and learning of proof and mathematics in general in specific african languages a flourishing experience. an attempt such as this will go a long way toward addressing the colonial legacy of research conducted in many parts of the world in general and in the sub-saharan context specifically. shongwe marginalization of 11th-grade urban african students journal of urban mathematics education vol. 15, no. 1 106 concluding remarks the positive correlation between functional understanding of proof and quality of argumentation was helpful in confirming the importance of understanding these two domain areas in mathematics. the analysis of the results paved the way for presenting arguments about the marginalization of the urban african student effected by the western worldview, including conducting systematic investigations in sub-saharan africa. specifically, whereas students’ appreciation of proof as a means to explain why a mathematical proof is true was the most powerful predictor of quality of argumentation ability, the communication function of proof exerted the smallest statistically significant influence on argumentation quality. in other words, the explanatory function of proof was found to be the most important determinant of argumentation ability. the analysis provided an image of the urban african student as poorly performing, yet there are compounding factors (language and gender) contributing to their performance. put broadly, the analysis brought to the fore the need for social change that can create conditions for the flourishing of urban african students in proof-related education. future studies will need to be sensitive to african students’ difficulties and design investigations with the question, “how can the learning and teaching of proof and argumentation be conducted through the ubuntu framework?” gobo (2011) reminded us that quantitative surveys or qualitative in-depth interview methods are based on competition and the role of the individual participant. perhaps future research may employ the participatory research paradigm (lincoln et al., 2018) in critical participatory action research, that is, research undertaken with and by people to build knowledge for understanding and transforming current practice (kemmis et al., 2014). this, of course, in my opinion, could be the only hope for a revamp of the ontological and epistemological stance of a “new” mathematics curriculum that transforms a western academic method into a multicultural framework that is sensitive to ethnic issues in addition to language and gender issues. in short, the results highlight that conducting research from a western lens tends to contribute to the perpetuation of eliminable forms of marginalization. such a lens obstructs the acquisition of proof education, which is, like mathematics, a universal human heritage that must be accessible to urban african students too. thus, future research efforts need to design studies whose approach to research is grounded in indigenous african epistemologies (seehawer, 2018). thus, research in mathematics education must take account of the economic, cultural, political, and racial milieu that affect urban african students’ learning of proof and acquisition of argumentation skills. this increases the need for sub-saharan instructors to be empowered to pursue emancipatory proofrelated instruction. the study reported in this paper adds to literature challenging the relationship between conventional, western-oriented mathematics education and indigenous students who have a 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(2011). research methods and data analysis for psychology. mcgrawhill. yackel, e. (2001). explanation, justification and argumentation in mathematics. in m. van den heuvel-panhuizen (ed.), proceedings of the 25th conference of the international group for the psychology of mathematics education (pp. 9–24). international group for the psychology of mathematics education. copyright: © 2022 shongwe. this is an open access article distributed under the terms of a creative commons attribution-noncommercial-sharealike 4.0 international license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. microsoft word 426-article text no abstract-2288-1-6-20210304 (proof 1).docx journal of urban mathematics education may 2021, vol. 14, no. 1 (special issue), pp. 12–23 ©jume. https://journals.tdl.org/jume imani masters goffney is an assistant professor of mathematics education in the department of teaching and learning, policy and leadership at the university of maryland–college park, 2311 benjamin building, college park, md 20742; email: igoffney@umd.edu. her research focuses on mathematics education and teacher education, especially as they intersect, with deliberate attention to issues of equity, justice, and culturally relevant mathematics teaching. jacqueline leonard is a professor of mathematics education in the school of education at the university of wyoming, 1000 e. university ave., laramie, wy 82071; email: jleona12@uwyo.edu. she is a recipient of the fulbright canada research chair in stem education award (2018). her research interests include culturally specific pedagogy, teaching for social justice, and computational thinking/participation. editorial i, too, am america! teaching mathematics for empowerment imani goffney university of maryland– college park jacqueline leonard university of wyoming chance lewis university of north carolina at charlotte his special issue of the journal of urban mathematics education includes articles that affirm the cultural and american identity of those who experience “othering” in america’s k–12 public schools and in society at large. the theme “i, too, am america! teaching mathematics for empowerment” was inspired by langston hughes’ poem “i, too” (1925/1994, p. 46): i, too, sing america. i am the darker brother. they send me to eat in the kitchen when company comes, but i laugh, and eat well, and grow strong. tomorrow, i’ll be at the table when company comes. nobody’ll dare say to me, “eat in the kitchen,” then. besides, they’ll see how beautiful i am and be ashamed— i, too, am america. t goffney, leonard, & lewis editorial chance w. lewis is the carol grotnes belk distinguished professor for urban education and director of the urban education collaborative at the university of north carolina at charlotte, 320 e. 9th street, charlotte, nc 28202; email: chance.lewis@uncc.edu. dr. lewis has over 100 publications and 25 books that focus on the recruitment and retention of african american teachers, urban student academic success, and eliminating opportunity gaps in urban schools. journal of urban mathematics education vol. 14, no. 1 (special issue) 13 through his creative works, langston hughes continues to inspire an imperfect nation to acknowledge and own its shameful past and to embrace the beauty and brilliance of black, latinx, and indigenous people, who are and have always been an essential part of the u.s. fabric. this special issue provides a scholarly platform for the contributing authors to also address the current social and political climate, especially as it relates to mathematics education (e.g., discipline, behavior, adultification of black children, ability grouping/tracking, lack of access to advanced mathematics courses, etc.). since the call for papers was issued in september 2019, the united states leads the world in covid-19 infections and reported deaths related to the worldwide pandemic. according to the johns hopkins coronavirus research center (n.d.), 26.2 million cases and 441,300 deaths related to covid-19 were reported in the united states as of january 31, 2021. covid-19 has had a staggering and immeasurable impact on communities of color for the past year, creating racial and economic disparities in terms of education (nelson, 2020), employment (williams, 2020), and healthcare, specifically in terms of access to a covid-19 vaccine (johnson et al., 2021). moreover, there has also been racial reckoning after the horrific deaths of george floyd and brianna taylor in spring 2020. in the aftermath, black lives matter (blm) protests engulfed the nation, with solidarity marches taking place across the globe. as a result of their struggle against racism, blm was nominated by norwegian parliament member petter eide for the 2021 nobel peace prize (goillandeau & elassar, 2021). blm protests, which have been mainly peaceful, have led to a movement that not only brought international attention to police brutality in the united states but also helped to galvanize voter registration among young people and people of color in the 2020 u.s. presidential election (alter, 2020). as a result of major grassroot efforts, senator kamala harris was elected as the first woman and person of color to become vice president of the united states, offering hope to young girls and children of color that their dreams can become reality. it is with these historical precedents and dilemmas in mind that this special issue of the journal of urban mathematics education emerges. miseducation and historically excluded students there is a long of history of miseducation for black, latinx, and indigenous students, who have been historically excluded in the united states. schooling has become increasingly focused on standards, and policymakers have become obsessed with teacher accountability and standardized testing to the extent that talent and creativity have been stifled (tate, 2019). comparing students’ academic performance goffney, leonard, & lewis editorial journal of urban mathematics education vol. 14, no. 1 (special issue) 14 without accounting for inequities in schools and districts and maintaining a ranking system where it is preferable to be highly ranked has caused tremendous harm to children, especially black children. the foundation of the measurement enterprise was designed with a deficit lens around a set of racist and dehumanizing ideals to “prove” black children were less intelligent than white children (ogbu, 2003). despite efforts over the last 15–20 years to move beyond the racist foundation of iq testing, the breadcrumbs and residual effects of this faulty foundation visibly exist in current testing regimes and contribute to poor performance on these tests in school (rosales, 2018). these measurement tools were never designed to identify or measure the strengths and brilliance of black, latinx, and indigenous children. while black, latinx, and indigenous students in some communities perform well on standardized tests, these tests were developed and normed for white students, who continue to perform quite well because the tests were designed with them in mind. as a result, too many black, latinx, and indigenous children in vulnerable communities continue to receive instruction that is typically steeped in remediation, especially in mathematics (martin et al., 2019). carter g. woodson (1933/1990) clearly articulated in the mis-education of the negro how the u.s. educational system failed to educate black children. he advocated for more rigorous mathematics, as well as other rigorous subject matter options, to ensure that black children received a quality education rather than the cultural assimilation that was taking place that worked to erase black cultural connections and positive black identities in schools (tate, 1995; woodson, 1933/1990). more than 66 years after the decision of brown v. board of education (1954), mathematics education for black, latinx, and indigenous children continues to be inadequate. black students are often isolated and marginalized in mathematics classrooms and experience dehumanizing learning experiences regardless of their economic status, academic proficiency, or academic potential (leonard, 2019; martin, 2000; ogbu, 2003). the frameworks, theories, and research paradigms that perpetuate hegemonic practices are remnants of a colonizing mindset and at best are incomplete and at worst are deliberately oppressive in ways that leverage whiteness to protect structural oppression and systemic racism in schools and society at large (martin et al., 2019). thus, miseducation continues in mathematics, and the other science, technology, engineering, and mathematics (stem) fields more broadly, for black, latinx, and indigenous students. to address this dilemma, we advocate for a paradigmatic shift in teacher education to improve learning among historically excluded and oppressed students in the 21st century. specifically, we argue for radical pedagogical change, including culturally specific (leonard, 2019), liberatory (martin et al., 2019), and communal learning paradigms (coleman et al., 2017), which have been shown to broaden participation among students of color in mathematics and stem (leonard et al., in press). goffney, leonard, & lewis editorial journal of urban mathematics education vol. 14, no. 1 (special issue) 15 equity in mathematics education we recognize that there are many definitions and conceptions of the term equity, even as it relates to mathematics teaching and learning. we even acknowledge that our own understanding and thinking about equity continues to evolve, leading us to offer more precise and nuanced descriptions and definitions of equity. as such, our current conception of equity is anchored to the pillars of access, achievement, identity, and power (gutiérrez, 2012), with deliberate connections to social justice as defined by the national council of supervisors of mathematics and todos: mathematics for all in the following joint statement: the national council of supervisors of mathematics (ncsm) and todos: mathematics for all (todos) ratify social justice as a key priority in the access to, engagement with, and advancement in mathematics education for our country’s youth. a social justice stance requires a systemic approach that includes fair and equitable teaching practices, high expectations for all students, access to rich, rigorous, and relevant mathematics, and strong family/community relationships to promote positive mathematics learning and achievement. equally important, a social justice stance interrogates and challenges the roles power, privilege, and oppression play in the current unjust system of mathematics education—and in society as a whole. (n.d., p. 1) in response to this statement, we see work on equity in mathematics education as being asset-based, assuming and affirming the brilliance of black, latinx, and indigenous students and also leveraging their ideas, interests, skills, and perspectives in mathematics classrooms in ways that support them learning rigorous mathematics and developing a positive mathematics identity (aguirre et al., 2013; celedón-pattichis, 2018; martin, 2000). the featured articles in this issue of the journal of urban mathematics education address equity in mathematics education from multiple perspectives (i.e., antiracism, critical pedagogy, social justice pedagogy, liberation). cunningham’s article, “‘we made the math!’: black parents as a guide for supporting black children’s mathematical identities,” highlights the richness of the experiential knowledge that black parents possess in helping their children develop mathematics identity while also explaining what it means to be black in mathematics classrooms and in america more broadly. in a qualitative study, cunningham recruited eight black parents to examine how they supported their children’s mathematical identities. parents used multiple approaches that included affirmation (i.e., building confidence), pragmatism (i.e., mathematics’ usefulness), aspiration (i.e., black role models), and race-consciousness (i.e., uplift) to support their children’s identity development in mathematics. downing and mccoy’s article, “exploring mathematics of the sociopolitical through culturally relevant pedagogy in a college algebra course at a historically black college/university,” examines the intersection of culturally relevant, anti-racist, and social justice pedagogy in a college algebra course that took place at a historically goffney, leonard, & lewis editorial journal of urban mathematics education vol. 14, no. 1 (special issue) 16 black college/university. qualitative data were collected on eight student participants who responded to anti-racist and social justice topics that ranged from the deaths of unarmed black men by police to whether or not to raise the minimum wage. thus, critical mathematics literacy (frankenstein, 2012) was used as the basis to apply social justice issues to college students’ day-to-day lives (mcgee, 2016). the final featured article by yeh et al., “radical love as praxis: ethnic studies and teaching mathematics for collective liberation,” examines love as an epistemological force and how love can be used to make sense of the confounding contradictory situations that scholars of color in mathematics education often confront. in the current political climate, one reaction to dissatisfaction with political outcomes and disaffection with the electoral process was the violent insurrection at the u.s. capitol on january 6, 2021. on the contrary, violence is never the answer to conflict. yeh et al. contends that love not only liberates but becomes a pretext to emancipate others (freire, 1968/2000). thus, mathematics educators from historically excluded backgrounds must help the educational community to reframe how they see black, latinx, and indigenous students. all of the feature articles call us to do just that. in retrospect, we offer our own set of imaginings to make this point salient with three vignettes. black boy joy shortly after our call for papers, an abc news story of a little boy named aayan went viral (muir, 2019). this poignant news story featured three-year-old aayan and his parents, alissa and elfa. they taught aayan from the time he was born that he is smart, blessed, and can be anything he wants to be. the news report also featured an earlier video taken by his parents when aayan first learned this mantra as a one-yearold. he did not quite have all of the words, but aayan clearly embodied the saying. watching the video of this young, three-year-old child, with beautiful mocha skin and shiny, curly black hair, walking with his mom on his way to school with his backpack on, fruit in hand, saying his daily mantra is a beautiful example of a powerful morning routine for this young black family. each day he repeated: “i am smart. i am blessed. i can do anything!” imagine aayan, dearly loved and full of confidence and hope, walking into a typical elementary school. what types of experiences is he likely to have? will aayan’s teachers see him as smart, blessed, and capable of doing (learning) anything? what will they say to aayan? will the classroom be structured in a way to demonstrate his smartness? what tasks will the teacher assign for students, including aayan, to work on? will these tasks leverage his interests and strengths? will they challenge him, thus helping him to truly see that he is capable of learning anything? how is behavior handled in aayan’s classroom? will his teacher see him as someone who needs to be controlled, creating an environment where each of his actions are tightly goffney, leonard, & lewis editorial journal of urban mathematics education vol. 14, no. 1 (special issue) 17 policed and monitored with significant sanctions for non-compliance? will aayan leave school each day with positive examples to share with his parents of how he was smart, how he was blessed, and the things he did each day or will he end up believing that he is only smart, blessed, and capable at home with his loving parents because of an oppressive school system? these wonderings are not only for aayan and his parents but for the millions of black, latinx, and indigenous children who attend public schools every day in this country. do public schools function to build on the knowledge and confidence that young students of color bring with them to schools? these questions are not commonly asked in this manner, but they are worth considering. the second vignette is food for thought in terms of moving from teaching mathematics in traditional ways and re-imagining how mathematics can be taught for social justice and empowerment. black girl magic kadijah is an 11-year-old african american girl who lives in the suburbs of a major city and attends a middle school with an international baccalaureate designation. there is no formal dress code, and the rapport between administrators, teachers, and students is generally one of mutual respect and appreciation. the demographic makeup of the school is quite diverse, and many students speak multiple languages. as such, the school actively works against tracking; instead, heterogeneous grouping is used for all subjects except mathematics. mathematics courses (e.g., algebra i, geometry, etc.) are credit-bearing courses that offer students specific curriculum. according to many in the community, it is an excellent school. the following example is drawn from a seventh-grade algebra i classroom. a group of students (two white boys [brad and josh], one white girl [becky], and one black girl [kadijah]) were clustered together working on an algebraic project. kadijah is the only black girl in the class and one of only two black students among the 27 students in the class. on day two, kadijah was expressing her frustrations about the peers in her group during a discussion with her mom after school. her mom suggested that instead of being silenced or tattling, kadijah should use a louder voice the next time her ideas were ignored and that action would prompt the curiosity of the teacher who would come to investigate. when the teacher investigates, then the teacher could intervene and hold the students accountable for listening to the ideas of all of the members of their group (as the project directs). on day three, kadijah’s ideas were once again ignored, so she used a louder tone and, in fact, the teacher did rotate over to investigate. as she arrived, the teacher asked kadijah what was going on. kadijah started to explain, and as she was talking, brad also started to talk and was talking over her and giving a different explanation. the teacher responded by asking kadijah to stop talking because she could not hear what brad was saying. kadijah responded by reminding the teacher that she specifically asked for her goffney, leonard, & lewis editorial journal of urban mathematics education vol. 14, no. 1 (special issue) 18 explanation because she was the one who was using the loud tone. the teacher shifted and instead asked brad to stop talking so she could hear kadijah’s explanation. the situation was resolved by the teacher reminding the students that they were obligated to listen and use the ideas of all of the members of the group for the project. when kadijah’s mom picked her up from school, she told her what happened in class. her group learned to listen to her ideas, and as a team they were successful. she was hopeful that her group interactions would be different in the future. although kadijah’s mom reacted with enthusiasm and remarked that she was very proud of her daughter for having the courage to stand up for herself and handle the situation with grace and respect, she was silently heartbroken that her daughter had to face the same battles that she faced as the only black girl in advanced mathematics classes some 30 years ago when she was in middle school. she wondered if things will ever change. nevertheless, kadijah had voiced her own empowering mantra: “i am the living embodiment of black girl magic.” schools could be transformed if teachers saw their work as identifying and contributing to the development of the brilliance that black students bring to school with them. how might classrooms be structured? would the classroom include a behavior chart that provides a public display of the teachers’ interpretations of children’s behavior or a public display of consequences for disengagement? would students have tables or desks? would they be grouped together to foster collaboration or separately because individual performance for ranking purposes is highly valued? what types of tasks would students be given? what would be the role of the teacher? what pedagogical skills are needed to make these changes? what feedback needs to be given to students about their academic performance? what strengths and resources from students’ lived experiences should be leveraged or erased in the classroom? what does it take to be seen as smart and valuable? the third vignette provides an example of how students used mathematics to show their value to society. using mathematics to understand the world and fight for justice on february 14, 2018, a former student using an automatic weapon (ar-15) opened fire at marjory stoneman douglas high school in parkland, florida. three teachers and 14 students, who ranged from ages 14 to 18, were killed in the deadliest high school shooting in u.s. history. seven of the students who were killed were only 14 years old. seventeen more people were injured, many of whom were severely injured by protecting the safety of their fellow classmates. in the wake of this horrific event that traumatized the entire community and frightened students, teachers, and parents across the country, the survivors sought to find a productive response. they later recounted talking with their teachers to find ways to productively use their rage and fear. goffney, leonard, & lewis editorial journal of urban mathematics education vol. 14, no. 1 (special issue) 19 the first planned outcome was a protest that became known as “march for our lives.” students used social media to coordinate with other students, raise funds, and share their message and vision of a productive response to this tragedy. they successfully raised millions of dollars to support the march, which was held on march 24, 2018, a mere five weeks after the deadly shooting. well designed and well coordinated, the only speakers at the event were school-age children from all grade levels. in a brilliant and powerful manner, in front of 800,000 people along with tens of millions of people around the world watching the televised event, these students used mathematics to craft their arguments. mathematics was used to help them make sense of what happened, to communicate their outrage at the injustice, and demand changes that would make their schools and communities safer. for example, the students argued that in their state there were x number of people but x number of registered guns. one of the most powerful examples were the price tags they created and affixed to themselves showing $1.05. the tag read, “politicians like marco rubio receive millions from the nra. don’t put a price on us.” attendees of the march used these tags and posters to demonstrate their support for the movement against gun violence in schools (see figure 1). these students used mathematics to better understand the problem they were facing and to craft an example about how to express their outrage. the students argued that the lives of their friends were worth more than $1.05. in fact, these students demonstrated untapped value and immeasurable potential. they are our future teachers, clergy, business leaders, engineers, doctors, lawyers, and politicians. what mathematics should we teach and how should we teach it to support students in re-imagining our country as a place where black, latinx, and indigenous students can exhibit all of their brilliance and all of their humanity? how can we help them fight for justice and create a better country that lives up to its ideals? figure 1. a tag used by students, such as naima goffney (age 12, right), while marching for justice at the march for our lives protest in washington, d.c. photographs courtesy of imani goffney. for more information of the march, see https://marchforourlives.com/chapter/mfol-washington-dc/ goffney, leonard, & lewis editorial journal of urban mathematics education vol. 14, no. 1 (special issue) 20 conclusion the foregoing three vignettes help to frame the theme for the featured articles in this special issue: i, too, am america! black, latinx, and indigenous students deserve to be heard and to have high-quality mathematics instruction, which has been identified as the key to success (martin et al., 2010). teachers can use data from covid-19, the electoral process, unemployment, and wage disparity to help students understand the problems and dilemmas they face, but more importantly how to use their individual and collective voices to read and “right” the world with mathematics (berry et al., 2020; gutstein, 2006). we encourage readers to fully engage with the featured articles beyond the usual scholarly practice of reading and citing the works in scholarly contributions. we encourage readers to utilize the featured articles and the special issue as a platform to transform mathematics education in educator preparation programs. we must fully prepare future mathematics teachers to meet the needs of all of our nation’s students and expand curricular offerings in k–12 schools to advanced-level mathematics classes that welcome and fully embrace black, latinx, and indigenous students into a classroom environment where equity is not an “add-on” but the new normal in the learning process. it is our hope that this special issue serves as a scholarly contribution that facilitates change far beyond the walls of academia and penetrates the hallways and classrooms of schools and communities across the united states and abroad. we conclude this special issue by acknowledging our children face an uncertain future that has been exacerbated by two pandemics—racism and covid-19, but there is hope. as notable scholar, dr. eddie glaude, reminded us by quoting james baldwin during an interview on the daily show with trevor noah (amira et al., 2020), “hope is invented every day.” glaude (2020) argued in his latest book, begin again, that baldwin insisted that we be true to ourselves—that we tell the truth about our history as a nation so that we can free ourselves from the past and imagine how we might live together differently in the future. leveraging this idea, the principle could be applied in mathematics education. can we be courageous enough to tell the truth about the whitewashing and oppressive history of the discipline of mathematics? can we acknowledge that mathematics educators and researchers have not always challenged racist and oppressive practices? if so, then we, too, can free ourselves to not only imagine but to build a world that is anti-racist and just. though we may have been bruised and battered by social injustice, racial and educational inequality, and covid-19, as poet laureate amanda gorman proclaimed on january 20, 2021, in the excerpt below, we, as americans, are an unfinished work in progress (gorman, 2021, pp. 11–13): goffney, leonard, & lewis editorial journal of urban mathematics education vol. 14, no. 1 (special issue) 21 when day comes, we ask ourselves where can we find light in this never-ending shade? the loss we carry, a sea we must wade. we’ve braved the belly of the beast. we’ve learned that quiet isn’t always peace, and the norms and notions of what 'just' is isn’t always justice. and yet, the dawn is ours before we knew it. somehow we do it. somehow we’ve weathered and witnessed a nation that isn’t broken, but simply unfinished. ~ we will rebuild, reconcile, and recover. in every known nook of our nation, in every corner called our country, our people, diverse and beautiful, will emerge, battered and beautiful. when day comes, we step out of the shade, aflame and unafraid. the new dawn blooms as we free it. for there is always light, if only we’re brave enough to see it. if only we’re brave enough to be it. references aguirre, j., mayfield-ingram, k., & martin, d. b. 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(original published in 1933) copyright: © 2021 goffney, leonard, & lewis. this is an open access article distributed under the terms of a creative commons attribution-noncommercial-sharealike 4.0 international license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. microsoft word editorialrevisiting urban mathematics educationtowards robust theoretical, conceptual, and analytical models (proof 3).docx journal of urban mathematics education december 2022, vol. 15, no. 2, pp. 1–7 ©jume. https://journals.tdl.org/jume jamaal young is an associate professor of mathematics education in the department of teaching, learning and culture at texas a&m university, 4232 tamu, college station, tx 77843; email: jamaal.young@tamu.edu. his focus is culturally responsive mathematics teaching as it relates to african american children’s educational needs, multicultural stem project-based learning, pre-service teacher diversity training, literature synthesis, and meta-analysis methodology. mary candace raygoza is an associate professor of education at saint mary’s college of california, 1928 st. mary’s rd., moraga, ca 94556; email: mcr13@stmarys-ca.edu. her research interests include teaching mathematics for social justice, youth participatory action research, and supporting students to be change agents through mathematics learning. tia c. madkins is an assistant professor in the college of education at the university of texas at austin, 1912 speedway, stop d5000, austin, tx 78712; email: tmadkins@austin.utexas.edu. her research focuses on supporting teachers’ use of equitable teaching practices to transform urban stem learning environments for minoritized learners and specifically examines the relationships between the design of learning environments and learner outcomes. editorial revisiting urban mathematics education: towards robust theoretical, conceptual, and analytical models editorial team jamaal young texas a&m university mary candace raygoza saint mary’s college of california tia c. madkins the university of texas at austin brian r. lawler kennesaw state university thomas roberts bowling green state university here are unique, enduring challenges related to mathematics instruction in urban schools, and the most common concerns are related to issues of equitable access to mathematics learning opportunities. in this editorial, we argue that the complexity of teaching mathematics in urban schools requires the attention of mathematics scholars with unique training, expertise, and experiences. developing mathematics instruction for urban spaces often involves refined research techniques, multi-disciplinary perspectives, and collaboration between diverse stakeholders. specificity is the key to addressing these challenges. however, to date, the journal of urban mathematics education (jume)—out of 69 mathematics education-focused journals (nivens & otten, 2017)—is the only journal venue devoted to mathematics teaching and learning in urban environments. although other mathematics education journals and conferences accept scholarship related to urban schools, due to the vast nature of mathematics education research and limitations in journal space, the nuances of specific issues in urban mathematics education (e.g., preparing mathematics teachers well for high-needs urban schools) are underexamined and poorly understood. t young, raygoza, madkins, lawler, & roberts editorial brian r. lawler is an associate professor of mathematics education at kennesaw state university, 1000 chastain rd., kennesaw, ga 30144; email: blawler4@kennesaw.edu. his scholarship focuses on equity issues in mathematics education, in particular the ways in which power and knowledge intertwine to govern the learner’s mathematical identity. thomas roberts is an associate professor and co-ordinator of the inclusive prek-5 education program at bowling green state university, 529 education building, bowling green state university, bowling green, oh 43403; email: otrober@bgsu.edu. his research focuses on issues of equity in stem education, particularly students’ perceptions of and access to high-quality informal stem learning environments, and integrated stem practices to describe quality stem learning. journal of urban mathematics education vol. 15, no. 2 2 urban mathematics education revisited many urban education journals accept mathematics education scholarship, but the challenge of specificity remains. scholarly venues for publishing mathematics education research can sometimes lack the contextual depth necessary for understanding issues related to urban learning environments, while urban education research venues can lack an emphasis on the mathematics context in both teaching and learning. jume provides a space for mathematics discourse at the intersections of mathematics education and urban education to inform the teaching and learning of mathematics in urban spaces. increasing the capacity of the next generation to realize the utility of mathematics is a shared goal of the broader mathematics education community. yet, there is consensus amongst the leading organizations for mathematics education research and teacher preparation (e.g., the national council of teachers of mathematics [nctm], the association of mathematics teacher educators [amte], the research council on mathematics learning [rcml]) that the realization of this goal is highly dependent on the ability of the field to better serve all learners. according to nctm (2014), “addressing equity and access includes both ensuring that all students attain mathematics proficiency and increasing the numbers of students from all racial, ethnic, linguistic, gender, and socioeconomic groups who attain the highest levels of mathematics achievement” (p. 1). likewise, amte (2015) encourages mathematics teacher educators to strive to do the following: recognize, challenge, and ultimately transform structures and systems of inequity that lead to inequity in mathematics learning and teaching based on race, culture, class, gender, sexual orientation, language, religion, and dis/ability in mathematics education, and empower p–12 teachers to do the same in their own classrooms. (p. 1) together these statements encompass the needs of a large segment of the population of learners historically underserved by mathematics education. however, what remains unseen is a similar commitment to the unique needs of learners in specific environments (i.e., rural, urban, and suburban spaces). here we suggest reframing the equity goals of mathematics education and specifically urban mathematics education to reflect the need for quantitative civic literacy. in this article, we define quantitative civic literacy as the ability to formulate, employ, and interpret situations within and beyond one's community and other societal contexts quantitatively. in 2019, raygoza questioned, “as we do the work of reimagining mathematics classrooms young, raygoza, madkins, lawler, & roberts editorial journal of urban mathematics education vol. 15, no. 2 3 as interdisciplinary, problem-posing spaces that connect to students’ lives, communities, and the world, how can we help prepare young people to develop as civic actors, using their mathematical knowledge and skills to build their quantitative civic literacy?” (p. 26). we contend that answering this question should be at the heart of urban mathematics education research and practice. according to martin and larnell (2013), scholars of urban mathematics education should place attention on issues of power, race, and identity while addressing geospatial concerns locally and internationally through theoretically and empirically sound research. this can inform teaching, learning, and policy in urban mathematics classrooms. importantly, urban mathematics education requires a geospatial perspective and approach to research and practice. a geospatial view “increases our understanding of education… by framing research in the context of neighborhoods, communities, and regions” (tate et al., 2012, p. 426) and is important for urban mathematics education. although mathematics teacher shortages, teacher quality, and standardized testing are common challenges, the influences of these factors are highly contingent upon the environment involved. national mathematics achievement trends indicate that learners in urban spaces continuously underperform based on data from the national assessment of educational progress (naep). for instance, only 31% of fourth graders and 27% of eighth graders in urban schools are proficient in mathematics based on data from the naep trial assessment of urban districts (national center for education statistics, 2017). mathematics education scholars also recognize that persistent learning differences present in urban spaces require further attention from the mathematics education research and teacher education community (capraro et al., 2009; varelas et al., 2012). however, conducting mathematics education research in urban spaces is only the first step toward urban mathematics education research and fostering quantitative civic reasoning. for example, mckinney et al. (2009) examined the pedagogical and mathematics instructional skills of 99 in-service teachers serving in a high-needs urban elementary school. the researchers used nctm’s (2000) principles and standards for school mathematics as a framework to characterize the practices of the teachers in their study. the results indicated the mathematics teachers in this urban school strictly adhered to the curriculum and pacing guides, with minimal deviation, therefore failing to address unique student needs and/or attenuating instructional differentiation. additionally, the mode of instruction was often lecture with an emphasis on memorizing algorithms and procedures through drill and practice. although this is only one depiction of teaching and learning mathematics in urban settings, the authors recognized that given the unique challenges facing urban schools, these practices were problematic. the practical significance of improved mathematics teaching and learning in urban schools is critical and should not be understated. yet, the actualization of specific pedagogical trends, policies, and practices in urban schools has been historically hindered by a lack of robust theoretical, conceptual, and analytical models to move the field forward. young, raygoza, madkins, lawler, & roberts editorial journal of urban mathematics education vol. 15, no. 2 4 in the inaugural issue of jume, one of the foremost scholars in urban mathematics education, william tate, urged the field to “build theories and models that realistically reflect how geography and opportunity in mathematics education interact” (tate, 2008, p. 7). tate’s rationale was that if urban mathematics education scholars and educators do not develop and test theories pertinent to urban environments, they cannot adequately inform classroom practices in urban schools. as mentioned above, jume is the lone scholarly outlet dedicated to urban mathematics education scholarship and thus remains the steward of the theory development, knowledge construction, and practical application related to urban mathematics education research. jume is highly regarded within the field of mathematics education, yet urban mathematics education cannot support instructional practices in high-needs classrooms if robust theoretical, conceptual, and analytical models are not at the forefront of the knowledge generation process for the academic gatekeeper of urban mathematics education. therefore, as we onboard the new jume leadership team, we aim to continue emphasizing “theory building and empirical evaluation” (tate, 2008, p. 6) while placing increased emphasis on scholarship that unpacks, critiques, and reframes our current understandings of urban mathematics education. these works should foster new scholarly directions for experts and provide a deeper understanding of urban mathematics education for novice urban mathematics education researchers. these intellectual interactions are essential to the development of robust theories, practices, and policies that support the teaching and learning of mathematics in urban schools. towards robust theoretical, conceptual, and analytical models as the new editorial leadership of jume, we are dedicated to the nexus between theory and practice. thus, we are soliciting manuscripts that promote robust theoretical, conceptual, and analytical models, as well as manuscripts that inform the development of quantitative civic literacy for urban teachers and learners. to this end, the new editorial team is seeking the following submission genres to support the development of theory, policy, and practice to inform urban mathematics educational praxis in quantitative civic literacy: 1. evidence synthesis and systematic reviews a systematic review provides empirically derived summaries of the research literature that can be either quantitative (i.e., meta-analysis), qualitative (i.e., metasynthesis), or hybrid. arguably, one of the most important aspects of a systematic review is the ability to characterize the effectiveness of interventions on important outcomes related to teaching and learning mathematics in urban spaces. these data are essential to the knowledge construction process as it relates to theory and model young, raygoza, madkins, lawler, & roberts editorial journal of urban mathematics education vol. 15, no. 2 5 building. comprehensive systematic reviews are essential to the development of theories and models to support mathematics teaching and learning in urban schools. for instance, a systematic review of the literature outlining the development and characterization of quantitative civic literacy within mathematics education would help to move the field of urban mathematics education forward. unfortunately, jume has had limited success publishing evidence synthesis and systematic reviews (i.e., research synthesis, metaanalysis, and metasynthesis). there are many promising new directions and approaches to systematic reviews, such as mixed-methods research syntheses, network meta-analyses, and bayesian meta-analyses. ultimately, the new jume editorial team asserts that urban mathematics educational researchers need to synthesize the literature to move the field forward by assessing trends and identifying unknowns. 2. broader use and application of different theoretical and conceptual approaches to mathematics mathematics and the teaching and learning of mathematics is not limited to the mathematics classroom (berry, 2021). as such, how students interact with and learn mathematics across contexts as well as how teachers can find opportunities to teach mathematics should be investigated. this work can draw on informal learning environments accessible to learners in urban contexts. explorations of integrated learning, such as high-quality ste(a)m learning experiences (e.g., roberts et al., 2019), in both formal and informal settings are also necessary. researching how mathematics is meaningfully learned, taught, and/or applied in other subject areas is also important. by expanding the focus beyond the formal mathematics classroom, mathematics education research will contribute to the goals of catalyzing change (nctm, 2020) by exploring how various theoretical and conceptual approaches allow us to broaden the purposes of learning mathematics and develop deep mathematical understanding (berry, 2021). informal mathematics learning experiences are uniquely suited to examine quantitative civic literacy, as the informal nature of these settings provide a natural conduit to community and students’ lived experiences. 3. participatory action and youth participatory action research finally, participatory and youth participatory action research is necessary to move robust theoretical, conceptual, and analytical models into mathematics education classrooms to deconstruct power dynamics, challenge authority, and restore dignity for all in mathematics classrooms. according to gutiérrez (2017), we cannot claim as our goal to decolonize mathematics for students who are black, latinx, and aboriginal while also seeking to measure their ‘achievement’ with the very tools that colonized them in the first place. when we consider the relationship of power to mathematics, we cannot be content with notions of power that are limited to solving difficult problems in mathematics classrooms. we must be open to deconstructing power dynamics, challenging authority, restoring peace and dignity, repairing settler colonialism, and positing new questions that need to be asked. (p. 12) young, raygoza, madkins, lawler, & roberts editorial journal of urban mathematics education vol. 15, no. 2 6 participatory and youth participatory action research in mathematics classrooms are characterized by researcher-initiated inquiries that provide unique opportunities to center the inquiry around issues of equity and social justice (desai, 2019), partnerships developed by in-service teacher-researchers within schools (raygoza, 2016), and researcher-student partnerships within out-of-school time activities (mackey et al., 2021). we argue that partnerships between mathematics education researchers, teachers, students, parents, and the broader community are essential for the actualization of change in urban mathematics education through the development of quantitative civic literacy. conclusion teaching mathematics in urban schools requires specialized knowledge and skills, which is more than just knowledge related to fostering classroom access, equity, and diversity. on the contrary, knowledge of access, equity, social justice, and diversity are requisite to successful teaching in all mathematics classrooms. thus, these skills are necessary for teaching in urban schools but insufficient. in this commentary, we argue for a reframing of urban mathematics education in a manner that promotes quantitative civic literacy. specifically, given our dedication to urban mathematics education, it is imperative that we foster the ability of students and teachers in urban spaces to formulate, employ, and interpret their lived experiences through a critical quantitative lens in order to place attention on issues of power, race, and identity that can influence teaching and learning in urban environments. moreover, we hope to realize the initial goals of urban mathematics education by soliciting manuscripts that promote the development of robust theoretical, conceptual, and analytical models to support the field. as the new stewards of jume, we will work diligently to reach this goal by continuing the dedication to excellence set forth by the editorial teams of the past and moving the journal forward to increased success in the future. references association of mathematics teacher educators. 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(2012). content learning and identity construction: a framework to strengthen african american students’ mathematics and science learning in urban elementary schools. human development, 55(5–6), 319–339. copyright: © 2022 young, raygoza, madkins, lawler, & roberts. this is an open access article distributed under the terms of a creative commons attribution-noncommercial-sharealike 4.0 international license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. microsoft word 403-article text no abstract-2424-1-6-20210414 (proof 2).docx journal of urban mathematics education december 2021, vol. 14, no. 2, pp. 1–15 ©jume. https://journals.tdl.org/jume melva r. grant, ph.d., is an associate professor of mathematics education at old dominion university, department of teaching & learning, 4301 hampton blvd, norfolk, va 23529; email: mgrant@odu.edu. dr. grant is an associate editor for jume. her research interests are broadening participation for minoritized persons and improving the effectiveness of prospective mathematics teachers and leaders. her research primarily uses post-structural qualitative methods from critical theoretical stances and is sometimes self-based. yvonna s. lincoln, ed.d., is a university distinguished professor emerita of higher education and administration at texas a&m university, department of educational administration & human resource development, 511 harrington tower, college station, tx 77843; email: ysl@tamu.edu. dr. lincoln’s research focuses on neoliberal and corporatization shifts in faculty work life and university administration, and also in the development of qualitative methods. she has written extensively and taught qualitative research methods and built a legacy that has influenced most if not all qualitative researchers for several decades. editorial a conversation about rethinking criteria for qualitative and interpretive research: quality as trustworthiness melva r. grant old dominion university yvonna s. lincoln texas a&m university his editorial shares a conversation about qualitative and interpretive research quality between friends. the journal of urban mathematics education (jume) is being relaunched after moving to its new home at texas a&m university. the jume mission as stated on the jume website’s “about the journal” page is to foster a transformative global academic space in mathematics that embraces critical research, emancipatory pedagogy, and scholarship of engagement in urban communities. here, the view of the urban domain extends beyond the geographical context, into the lives of people within the multitude of cultural, social, and political spaces in which mathematics teaching and learning takes place. (journal of urban mathematics education, n.d., para. 1.) given this mission statement, we anticipate jume will attract many mathematics education scholars interested in these types of inquiries using qualitative and interpretive approaches. the vast majority of qualitative researchers learn about trustworthiness and its relationship to research quality and rigor during their formal training; however, qualitative research practitioners do not typically ascribe to or feel compelled to conform to strict frameworks of quality. nonetheless, that does not suggest that there should not be evaluative criteria of quality for qualitative research for those inclined to consider them. additionally, our theorizing about qualitative research quality criteria does not claim that these are the only criteria or that they stand above any others; however, these criteria are building upon criteria established over time by thoughtful qualitative researchers interested in quality. t grant & lincoln editorial journal of urban mathematics education vol. 14, no. 2 2 as one of the inaugural jume associate editors, dr. melva grant engaged dr. yvonna lincoln as a colleague and friend in an extended conversation theorizing about the trustworthiness of qualitative research, which yvonna wrote about in the first volume of qualitative inquiry (lincoln, 1995) in an article titled “emerging criteria for quality in qualitative and interpretive research.” during this conversation, we endeavored to consider changes to the landscape of quality of qualitative research as trustworthiness. context dr. yvonna lincoln, university distinguished professor emerita at texas a&m, has been a pioneer in the field of qualitative and interpretive inquiry research. i asked her to provide an overview of her work. she offered the following: i have been doing this kind of work since i was a graduate student and worked for bob wolf, who brought naturalistic inquiry, very unformed, from bob stake from upstate at [the university of] illinois. so, i was noticing that it was unformed and unsystematic and began to work with egon guba. we invited bob to work with us, but bob was busy doing other things . . . but egon and i went ahead with this project of trying to systematize and create a metaphysics for qualitative research which was different from the metaphysics of rationalistic research, otherwise known as scientific method, and i’ve been working in that area as well as working in higher education administration for now close to 40 years. (conversation 1, 2020) yvonna’s brief and modest description failed to mention her extensive scholarship and mentorship. after a brief internet search, i found a short description on the texas a&m university (n.d.) website that described her as having “written over 100 peerreviewed journal articles and chapters, and written, edited or co-edited more than a dozen books. . . . chaired over 100 doctoral committees . . . [and] won the presidential citation from the american educational research association.” there are other notable contributions, such as her being co-editor for five editions of the seminal sage handbook of qualitative research, being co-founder and co-editor for the qualitative inquiry journal, having held leadership roles in several professional organizations in her field, and being the recipient of many prestigious awards for scholarship and teaching. i assert that most if not all qualitative researchers have read something authored or edited by yvonna lincoln during their research education journey. the purpose of this paper is to share yvonna lincoln’s contemporary thinking about quality criteria for qualitative and interpretive inquiry research and to make it available to mathematics educators who conduct qualitative research in urban settings. this manuscript emerged from eight phone conversations that averaged about 35 minutes and took place for just shy of two months as well as quite a few text and email messages that governed our collaborative writing efforts. our conversations started out by unpacking or increasing my understanding of lincoln (1995), a very grant & lincoln editorial journal of urban mathematics education vol. 14, no. 2 3 theoretical endeavor. for those interested in a less theoretical conversation and are seeking guidance for writing up qualitative research for publication review, see levitt et al. (2018). however, if you seek to understand theoretical articulation about criteria for qualitative research that is trustworthy and inclusive of emerging qualitative approaches, we invite you to read on. as a mathematics educator and qualitative researcher who is interested in understanding dr. lincoln’s perspectives and learning more about qualitative research quality, i found these conversations with her interesting and informative. in the beginning, between conversations, i wrote questions to focus our discussions, but over time the dynamic changed; we no longer needed questions to focus our conversations and the writing, which included a revised and eventually expanded table, became its own focusing artifact. during periods between conversations, we both added to or edited the evolving manuscript. i sometimes listened to conversations, took notes or reviewed prior notes from our conversations, and transcribed portions of the recorded conversations for inclusion in the manuscript. we both read literature related to the expansion and past jume qualitative publications in search of examples. yvonna wrote, edited, and collaborated with other peer qualitative researchers, as well as read new things related to our conversations (conversation 2, 2020). i believed that the table in lincoln (1995) would benefit jume readers if we could reformat it to improve its readability; this table describes criteria of trustworthiness of qualitative research over time and method. our discussion about table 1 (see lincoln, 1995, p. 277) shifted because i wondered if the table, referred to hereafter as the trustworthiness table, needed an additional column for post-qualitative research, and i asked yvonna her thoughts. this inquiry transformed our conversations and made the trustworthiness table a central theme of our focus for several weeks as an impromptu cooperative study of reading, outside consultation, and discussions (see table 1). our conversations took place by phone, and i talked on a land line and used an app on my smartphone to make recordings. i did not share audio recordings with yvonna because she prefers paper. however, i shared portions of transcribed conversations to ensure accuracy of interpretation. i did not code or analyze the audiotaped conversations because this was not a research study but an editorial that shares an account of our conversations. i did use the recordings for capturing meanings to support writing, refresh recall, and to produce a coherent account of our conversations, including accurate capture of yvonna’s comments. the trustworthiness of our work, if it were an inquiry, was continuous in the sense that one might characterize what we were engaged in as relational research (see reinharz, 1977). this manuscript was written collaboratively, and we exchanged it multiple times throughout the writing process. we worked this way until we were both satisfied that our voices and perspectives had been captured and represented. in this research analogy, our process blurred the lines between the researcher (i.e., first grant & lincoln editorial journal of urban mathematics education vol. 14, no. 2 4 author) and researched (i.e., co-author) and exemplified reinharz’s (1977) notion of relational research where the researcher and researched were entwined in a caring relationship, which she characterizes as a “lover model” of research. our entanglement was enhanced and enabled because of our friendship that manifested as mutual respect as well as our shared desire to hear the voice of the other within the final document. let me speak on our friendship briefly. in this manuscript, it is clear that i hold yvonna in high esteem and there is a significant power difference between us. however, i believe our friendship is unique and adds to this work and its trustworthiness. our friendship has grown over several years and has moved beyond a professional acquaintance; it has evolved into the personal, which nurtured openness and honesty. these experiences have afforded me a level of privilege that i would not otherwise claim, and it allows me to engage in this work as yvonna’s “peer.” after typing this sentence, i chuckle because even on my best day, i am not her professional peer, but i believe she considers me an equally dear friend. this privilege afforded me to speak truth to her powerful professional presence and to ask questions bravely that emerged through this work without fear of retribution. definition of qualitative and interpretive inquiry research the first exchange during our initial conversation was about yvonna defining qualitative and interpretive inquiry. in lincoln (1995), yvonna wrote about this: consequently, as its acceptance has been debated, it has been involved in intense crossdisciplinary discussions of what constitutes its quality criteria. i prefer to think of this issue of quality as a dialogue about emerging criteria. i label this discussion that way because i believe that the entire field of interpretive or qualitative inquiry is itself still emerging and being defined. (p. 275) this passage led me to ask yvonna to define qualitative research and to comment on if and how the field continues to emerge. vastly simplifying and paraphrasing, she defined qualitative inquiry as research that is centrally focused on people and their lived experiences using their authentic social constructions while recognizing that there is no single accepted reality that can be used for comparison, which renders judgements of goodness or righteousness invalid. social constructions are shaped by people’s social interactions and influenced by their attitudes, values, beliefs, prejudices, biases, stereotypes, and the like. yvonna summarized the definition as, “so, qualitative or interpretive inquiry seeks to explain why people act the way they do based on how people go about constructing their own reality” (conversation 2, 2020). grant & lincoln editorial journal of urban mathematics education vol. 14, no. 2 5 quality criteria for qualitative and interpretive inquiry our conversations regarding the state of qualitative research methodologies and criteria for quality or trustworthiness in the present moment was the focus. the trustworthiness table presented within this manuscript is a revision and expansion of the original table presented in lincoln (1995, p. 277).1 yvonna and her late husband, egon guba, had created the original table in response to a critique by john k. smith, a scholar and critic who suggested that quality criteria for qualitative research ought not to merely focus on the “foundational” parallels to conventional or positivist inquiry, although the criteria she and guba created were drawn primarily from classical anthropology and sociology, and speak to careful methodological rigor (see table 1, column 2). rather, smith suggested that there should be criteria that were drawn from the premises, axioms, and metaphysics of constructivist/naturalistic inquiry itself, and thus the third column of the original table was created. this third column represented quality criteria directly growing from the axioms of naturalistic and constructivist inquiry. i asked yvonna if there was a relationship between the two columns of quality criteria for qualitative and interpretive inquiry (i.e., table 1, column 1 versus column 3). she responded by saying that the five quality criteria (column 3) were deemed responsive to the axioms of constructivist inquiry and were in no way parallel to the methodological criteria for conventional inquiry (column 1). when i queried her as to what she might assent to in this present moment for quality criteria, she commented that she had thought a lot about the criteria and came to the conclusion that she and guba had created the five criteria intuitively and had unconsciously connected them to other streams of research. so, for instance, ontological authenticity not only references how individuals come to understand their own tacit positions and recognize how they “own” them, but it can also be extended to refer indirectly to freirean notions of “true” and “false” consciousness. this is so because the research process can often uncover hidden feelings, beliefs, attitudes, values, and other systems of thought of which a stakeholder might be formerly unaware. consequently, ontological authenticity has the power to sometimes unlock subconscious motivations and beliefs that may be somewhat less than productive or may simply be surprising to the individual. in the same vein, educative authenticity connects strongly to the premise of qualitative inquiry that information, data, and interpretations do not belong solely to those who hold power and/or money but rather are the right of the stakeholders to have and to work with. thus, researchers have the responsibility to share information and interpretations with the stakeholders who have provided the data from which interpretations were drawn and to negotiate collaboratively about what the interpretations and context mean. 1 dr. lincoln quickly pointed out a typographical error in omitting one of the criteria, ontological authenticity, from the table in the original 1995 manuscript. this omission was corrected in the revised version of the table presented in this manuscript. grant & lincoln editorial journal of urban mathematics education vol. 14, no. 2 6 the sharing of data between researchers and stakeholders serves two purposes. first, it permits a more sophisticated and nuanced understanding of stakeholders’ own circumstances and a richer appreciation of the context (e.g., that others in their context might think, believe, or construct as their realities). second, and more importantly, it is the desire of researchers that having this knowledge, having access to data, and reflecting on the interpretations that are proffered by researchers will lead to catalytic authenticity, or the genuine desire to act upon stakeholders’ own circumstances, and to move toward agency in their own lives (see, for example, toness, 2002). it is the case, however, that sometimes stakeholders are unsure of how to act upon this new knowledge and understanding. in this instance, researchers can, and should, work with stakeholders to demonstrate how to access the levers of power in order to act with efficacy on their new understandings. table 1 criteria for assessing qualitative research’s trustworthiness or quality (revised and expanded from lincoln, 1995, p. 277) qualitative research/interpretive inquiry/constructivist/ naturalistic paradigm (trustworthiness) expanded social consciousness paradigms (quality) qualitative methodological criteria – positivist parallels (reliance on data) methodological criteria publication standards for judging quality criteria (reliance on community consensus) paradigm beliefs & norms authentic/ethical/ relational criteria (reliance on ethical system) paradigm axioms & metaphysics transformative/deconstructive criteria (reliance on positioned theorizing/power structure deconstruction) multiple emerging paradigms extrinsic extrinsic or intrinsic intrinsic a-trinsic credibility (plausibility) transferability (context-embeddedness) dependability (stability) confirmability (value expectation, triangulation) publication standards (a priori institutionally bound and structured) positionality/standpoint community as arbiter voice critical subjectivity reciprocity sacredness sharing privilege/power fairness/balance (equity of stakeholder representations) ontological authenticity (knowledge of self revealed) educative authenticity (stakeholder knowledge grows) catalytic authenticity (stakeholder agency develops) tactical authenticity (stakeholder learns self-sufficiency) theoretically structured (postmodern, poststructuralism, new materialisms, posthumanism, postcolonialism) ontologically unrestricted (expansive, posthumanism, postanthropocene) non-replicable (not easily reconstructed) values (non-anthropocentric) educative authenticity (stakeholder knowledge grows, liberationist focused) catalytic authenticity (stakeholder acts autonomously, awakened agency) tactical authenticity (stakeholder self-sufficiency) grant & lincoln editorial journal of urban mathematics education vol. 14, no. 2 7 given qualitative researchers who opt to support stakeholders in these ways, conventional researchers lodge critiques of such choices and frequently label qualitative researchers as “activists” or “moralists” rather than actual researchers, but this is also a place where qualitative inquiry comes closest to action research. some qualitative researchers, especially those taking critical stances, are likely to argue that research itself is a moral activity, and one which endeavors to move communities toward more equitable ends that are more socially just—social justice research. quality as trustworthiness: examples from the jume archives there are many examples of trustworthy qualitative research that show both the methodological and the authentic, ethical, and relational criteria (see table 1, column 3) that characterize or align with socially just research. we found articles from the jume archives2 with examples of trustworthiness; however, in the examples cited, authors rarely made explicit claims of trustworthiness described in table 1. we took this approach because many qualitative researchers omit criteria of trustworthiness, perhaps because they fail to recall them after completing their doctoral studies, because qualitative researchers do not value or discuss trustworthiness as a criterion of quality, or because journal editors and reviewers do not require researchers to make such articulations and claims. i cite my own work as complicit in this failing to make clear our intent to not demonize research that lacks an explicit claim of trustworthiness as a show of transparency. my research report about a group of black boys and their mathematics identity development while learning as a cohort in the algebra project (i.e., grant et al., 2015) has authentic trustworthiness. this research report, which shared findings about six high school boys’ mathematics identity transition from others initially positioning them as “at risk” to the boys demonstrating and the researchers documenting them over time as being confident and productive mathematics learners, suggests both educative and catalytic authenticities. another example of trustworthy qualitative research comes from a study that used a combination of critical race theory and historical critical analyses and revealed how little change has manifested in education for minoritized populations from policy emerging from the equity message of “mathematics for all” (berry et al., 2014). in this example, stakeholders experiencing 2 yvonna and i dedicated most of our time focused on the revisions and expansions of the trustworthiness table. we decided close to the deadline that we should cite examples from the jume archives. so, yvonna asked a graduate research assistant to gather examples of qualitative research published in jume over 10 years. dr. grant randomly selected articles in search of examples, sometimes choosing articles that she was familiar with. for example, dr. grant chose to cite her own research because she was familiar with it and thought it a good example at the time. we were not seeking to cite the “best” articles or to exclude anyone’s work in particular but rather to share an example that might improve clarity of the particular point being made. grant & lincoln editorial journal of urban mathematics education vol. 14, no. 2 8 nuanced understandings are not research participants, as in the case of grant et al. (2015), but the stakeholders are readers who hold positions in educational leadership or as policy makers with potential to effect change. these stakeholders may experience ontological, educative, and ideally catalytic and tactical authenticities in relation to trustworthy research. another study with potential for claims of trustworthiness reported an instance of tactical authenticity experienced by a research participant (leonard & evans, 2018). this stakeholder participant’s involvement in research supported adoption of a culturally sensitive approach to teaching mathematics using guided inquiry instead of direct instructional telling after participating in a community-based immersive learning experience. other examples of research, just to name a few from the jume archives, with potential for instances of trustworthiness claims using authentic, ethical, and relational criteria (see column 3 in table 1) include kurz et al. (2017), mcgee (2013), and ragland & harkness (2014). post-qualitative inquiry: expanded social consciousness readers will notice, however, that we, in dialogue, have added a fourth column to the trustworthiness table, the expansion. some might see this, as we do, as a big deal. we hope they will consider this proposal, build upon this work, and/or critique it to further the metaphysics of this emerging space. this expansion attempts to articulate our nascent understandings from the study and reflective discourse of this project. we refer to the expansion, the last column of the trustworthiness table, as expanded social consciousness paradigms, which includes something st. pierre (2018) calls “post-qualitative inquiry” but also includes postmodernism, poststructuralism, posthumanism, postcolonialism, and new materialism models for inquiry. many of these proposals for an expanded social science can also be aggregated under what might be called postanthropocene, or chthulucene, to borrow a term from haraway (2019), that is, a movement beyond the enlightenment’s concern for the human-as-agent toward consideration of the natural, geo-spatial, and material world. also, these models take into account an ecological realm that situates humans as merely one among many actors and include other formations and life-forms exerting agency on the universe. with many of these models, for example, postcolonialism and poststructuralism, there is a decided turn toward considerations of political and historical power and power relations and of questions of social justice (e.g., gerrard et al., 2017). this social justice focus, for many theorists and proposers of an extended ontological consciousness, extends not just to the marginalized and oppressed groups around the world but also to the biological, geological, and ecological environments that encompass the entire planet, and perhaps beyond into intergalactic spaces. grant & lincoln editorial journal of urban mathematics education vol. 14, no. 2 9 what we have tried to do in expanding the trustworthiness table is to consider the original question that i posed to lincoln, which is, “if you were looking at this table now, how would it be different from 1995’s work on rigor and trustworthiness?” at that point, we began to contemplate the new movements and proposals that added to the richness and complexity of the realm we call qualitative inquiry. we make no claims to this table’s completeness, or even accuracy. what we have tried to do is suggest what previous authors have suggested, and that is what should be in the column itself. we have searched for systematic proposals for quality or ethical proposals, although we have not located formal proposals that lay out a proposed comprehensive metaphysics. thus, what we have provided in the revised and expanded table is a sketch, a kind of liminal, intellectual dotted line that serves as an initial attempt to map an approximation of “quality criteria” for the expanded social consciousness paradigms. of course, ultimately it may be found that our effort to expand the trustworthiness table for the emerging paradigm was insufficient. perhaps multiple columns are needed or the representation itself is wholly inappropriate. in fact, both of us see this fourth column as tentative, incomplete, provisional, troubling, and troubled. it will take many more theoretical explications before any of these proposals become working metaphysics and can be taken into the social landscape for “field trials.” we were, however, charged with thinking about where might qualitative research be headed over the years intervening between the rigor proposal (lincoln, 1995) and now and what would constitute quality, similarly as rigor or trustworthiness previously, in the proposals that enlarged, elaborated, snarled, and made gnarly the terrain of qualitative inquiry. in short, if many of the new proposals beyond constructivist inquiry, broadly, were aimed toward both ecological and human justice, how could we trust the findings sufficiently to act upon them? in light of our thinking about expanding this trustworthiness table regarding parameters of quality and/or rigor, we reflected on where the field had been in the intervening 25 years and what constituted the most promising models. yvonna consulted with another knowledgeable scholar,3 and we settled on the column title, “expanded social consciousness paradigms.” this title fit our thinking and our current understanding about this still emerging research. there are multiple emerging paradigms within this research space; many of them are related and some take very different postures on where inquiry should begin, depending on the positionality of the researcher and the questions she proposes to answer or address (e.g., st. pierre, 2018; 3 we are indebted to gaile cannella for the titling of the column as “expanded social consciousness paradigms.” we had originally labeled this column “post-qualitative criteria” but realized that was entirely too narrow to subsume the many models, shifts, research prototypes, and proposals circulating throughout the inquiry realm. as qualitative researchers ourselves, we were also dissatisfied with the linguistic implication that, somehow, the field had moved on, moved away from qualitative research onto something else, and, consequently, we abandoned this terminology for more specific and clearly descriptive terminology. grant & lincoln editorial journal of urban mathematics education vol. 14, no. 2 10 taylor, 2016). st. pierre (2011) described post qualitative inquiry as “becoming”— one must prepare for this type of inquiry through extensive study of theory, beginning with in-depth and broad reading, thinking as writing, and theorizing followed by experimenting with inquiry without methodology. a primary goal of st. pierre’s vision of post-qualitative inquiry is revealing the unknown while transforming or deconstructing along the way. subtitles used in column 4 of the trustworthiness table depict these ideas: a) transformative/deconstructive criteria and b) reliance on positioned theorizing/deconstructing power structures (or perhaps archaeological uncovering of power structures, had we chosen a shorter metaphoric subtitle). in the third row, we outlined what we believed to be important topics that various models might take up in extended discourses of what might eventuate in the field. we created a new term for this row in the fourth column, a-trinsic. we loosely articulate a-trinsic to contrast it with the extrinsic quality criteria of rationalistic, conventional inquiry and the methodological criteria of constructivist inquiry, as well as the intrinsic authenticity criteria later developed for constructivist inquiry. we term this fourth column “a-trinsic” because quality criteria may derive from many places, including social theory (derrida, 1991), political theory (foucault, 1972/2010), economic theory (see, for instance, the work of deleuze and guattari, 1983, on capitalism and its madnesses), biology, social or natural ecology, linguistics, semiotics, or physics. to add clarity to why the power structure appears with prominence or centrality in the trustworthiness table, consider two examples. a poststructuralist researcher might be searching for a means to deconstruct a series of linguistic and discursive signifiers to comprehend where power structures shaped meanings and observed endless deferrals-as-deflections that served to obscure understanding of, as well as interrupt, extant power structures. on the other hand, a postcolonialist researcher might seek to understand the oppressions or cultural disfigurations remaining long after the colonials had retreated, and in some cases the colonials have taken up residency as claimed natives but are exempt from the treatments because they never went home and retained power. for these proposed inquiries, power structures are still at the center of inquiry but in very different forms for the two researchers. in much the same way, the posthumanist researcher would be framing very different questions from either of the previous two researchers—still about power structure but in a greatly expanded context of biology, ecology, geography, geo-spatiality, and non-human beings. thus, the title of “social consciousness” for a paradigm moves to and through questions of social justice and asks, “social justice for whom? and how?” the criteria are deconstructionist, a la derrida and foucault, and transformative, for example, with feminist critique or the new materialism. when we inspect the criteria themselves, we were not able to extend our investigations deeply but attempted rather to lay out some general themes we were able to discern in the various proposals. our a-trinsic criteria included a) the theoretical grant & lincoln editorial journal of urban mathematics education vol. 14, no. 2 11 structuring of an inquiry; b) the ontology(ies) by which any given model might proceed; c) the replicability or non-replicability of a given study, or, conversely, its dependability, or ability to have its processes and products proceeding therefrom be transparent and traceable even if the study itself is not replicable; d) the values emerging from a worldview that is not human-centric (schulz, 2017); e) the educative authenticity through knowledge and liberation; f) the catalytic authenticity of a model, or the ability to prompt socially just and transformative action on the part of stakeholders, research participants, and beneficiaries of the research, and; g) tactical authenticity, or the ability of the researcher to provide support in “speaking truth to power.” none of these criteria were explicitly mentioned as attesting to rigor in the various models of qualitative research-beyond-the-classical-or-constructivist-paradigm, but together, we have inferred some of them from the reading we have done and from discussions among ourselves and with other qualitative researchers also engaged in understanding these new proposals. so, for instance, we might unpack them in the following way. all of the models subsumed under this set of rubrics is, as far as we can tell, theoretically structured and thus likely arrives with its own ontology(ies), whether postcolonialism, poststructuralism, posthumanism, new materialism, or post-qualitative inquiry. unlike some more classical or constructivist models, which may begin a-theoretically with intentions of potentially creating new theory, many of these models begin and end with a priori theoretical, historical, or hierarchical stances; think foucault and his ideas of policy archaeology, or deleuze and guattari, with their deconstruction of the fault lines of capitalism. other proposals circulating recommend beginning studies with theory in mind beforehand and then thinking your way through writing to new revelations disconnected from the original theory. the idea, however, that these models virtually always begin with some theoretical structuring is a departure from classical ethnography or qualitative research, which frequently, though not always, tries to grasp whether new theory might be created from lived experience. sometimes, qualitative studies are undertaken for the purpose of testing extant theory (e.g., guba & lincoln, 1989; lincoln & guba, 1985). for the foregoing reason, we included “theoretically structured” as one hallmark of quality for these expanded social consciousness paradigms, models or proposals. we also felt these expanded paradigms were “ontologically unrestricted.” that is, we failed to locate any single proposed statement of reality that drew all the models together. thus, the choice of ontology depends on the theoretical stance adopted to guide a given inquiry, whether power relations, deconstruction of human regulation, or postanthropocene/posthumanism. when confronted with the question of reliability, we eventually decided that studies such as these, for the moment, were non-replicable; that is, a second researcher could not conduct them in such a way as to come to the same conclusions. while this is also largely true of new paradigms, such as constructivist inquiry, there grant & lincoln editorial journal of urban mathematics education vol. 14, no. 2 12 are also requirements for such inquiries to be trackable, their methods and methodological choices and decision points to be publicly inspectable, and for analyses to be carried out in systematic and disciplined ways. we found no such requirements within these models and, indeed, in some proposals, found a kind of intellectual abhorrence of rigorous method or publicly inspectable logics leading to conclusions. the labeling of careful and disciplined analyses as “mechanistic” or “mechanical” seems to us to undermine the “science” in social science, but that is a conversation of many weeks and months for another time. when we considered the question of values, we also found multiple perspectives, some of which were definitely anthropocentric and some of which were antianthropocene, or focused on the non-human and its relationship(s) with the human— social, psychic, economic, historic, geographic and place-based, cultural, ecological, mythic (particularly the myth of human mastery, see schulz, 2017). consequently, we concluded that values were embedded in the theoretical positions adopted by the researchers but that many were either liberationist in focus, transformational, or nonanthropocentric. we also examined the potential for some, or all, of these models to provide for educative authenticity, or the ability for stakeholder knowledge to grow in more sophisticated and informed ways and to provide them with the informational tools to begin to re-narrate their own lives toward less oppression. many of the models seek such transformations, although the extent to which the models mandate that new knowledge be shared with stakeholders in such a way as to enable them to begin to reframe their own experience is unclear. some models do, and others are unclear regarding the extent to which stakeholders and research participants become equal partners in the discovery/uncovering of hidden assumptions and social and cultural myths that are formed from colonial experiences that suppress transformation. much of what can be obtained regarding the power of these models to educate stakeholders in new ways of thinking about their own circumstances is sparse and intellectually thin. we likewise looked at the possibility of these models for prompting catalytic authenticity, or the urge to act upon the stakeholders’ own immediate environment or circumstances. while some models do urge increased sophistication and, likewise, deeper understanding of larger geo-political issues related to the distribution of power, other models do not. catalytic authenticity, or quality, is intricately bound up with tactical authenticity, or quality, since the urge to act upon one’s circumstances is intimately bound together with knowing how to act, where to act, and the appropriate means of achieving influence and voice addressed to power. sometimes, the impulse to act is without knowledge of how and where to act for maximum impact; it is at this point that the role of the researcher moves to being a facilitator. she is usually accustomed to the means for making one’s voice heard and is in a position to aid stakeholders in framing grant & lincoln editorial journal of urban mathematics education vol. 14, no. 2 13 their actions and petitions in such a way as to maximize input and impact. this is a role that conventional researchers typically eschew, and the resistance is framed as having the researcher move from a “scientific” role to one of advocacy, which is purported to undermine objectivity. however, objectivity has long been discredited as a mark of qualitative or interpretive inquiry, and so researchers who move from inquirer to teacher are frequently understood and employed in new paradigms, such as constructivist, action, participatory action, and transformational paradigms and/or models. the foregoing are those criteria of quality that we think characterize some, if not all, of the social consciousness and transformation models of research. many of them rely heavily on qualitative methods but frequently mix and match qualitative and quantitative if doing so illuminates some interpretation and extends some understanding on the part of stakeholders and researchers alike. when reflecting back on our conversation as a mathematics educator and qualitative researcher interested in producing high-quality qualitative inquiry, i found the revised and expanded trustworthiness table an informative tool to use for considering new ways to address the question that hovers over most manuscripts submitted for publication: what constitutes this research as trustworthy or of high quality? i envision that my next set of manuscript submissions will include discussions about the intrinsic criteria from column 3. from my perspective, especially after engaging in conversations about them with yvonna, i was reminded about the existence of the authenticities, their importance for equity and valuing others within a socially just environment. these are important ideas worth consideration from the beginning of an inquiry through the end. thus, they are worthy for explicit discussion. i am looking forward toward an expanded social consciousness; delving into the literature piqued my desire to read more deeply about this qualitative moment and dreaming about new possibilities for inquiry. at this point, our conversation must end, or this manuscript will never make it to press. however, we hope this shared conversation addressed, at least in part, the question raised about lincoln’s perceptions about changes to qualitative inquiry and criteria of quality from 1995 to today. through this editorial, we attempted to remind researchers about the value of trustworthiness, shared an updated table that applies to contemporary research, and provided definitions and research examples of the authenticities. we also encourage researchers to make explicit claims and for editors and reviewers to look for authors’ claims about trustworthiness encountered or activated during their research and to work collaboratively to ensure that claims of research quality are clearly articulated in future research reports. grant & lincoln editorial journal of urban mathematics education vol. 14, no. 2 14 references berry, r. q, iii, ellis, m., & hughes, s. 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(2011). post qualitative research: the critique and the coming after. in n. d. denzin & y. s. lincoln (eds.), the sage handbook of qualitative research (pp. 611–626). sage publications. grant & lincoln editorial journal of urban mathematics education vol. 14, no. 2 15 taylor, c. a. (2016). edu-crafting a cacophonous ecology: posthumanist research practices for education. in c. a. taylor & c. christina (eds.), posthuman research practices in education (pp. 5–24). palgrave macmillan. texas a&m university. (n.d.). yvonna lincoln (emeritus). retrieved june 3, 2020, from https://directory.education.tamu.edu/view.epl?nid=ysl toness, a. s. (2002). assessment of participatory rural appraisal: the effects of practicing pra among development institutions and rural communities in paraguay (publication no. 3060912) [doctoral dissertation, texas a&m university]. proquest dissertations and theses global. copyright: © 2021 grant & lincoln. this is an open access article distributed under the terms of a creative commons attribution-noncommercial-sharealike 4.0 international license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. race, identity and learning together: student perspectives on mathematical collaboration journal of urban mathematics education december 2009, vol. 2, no. 2, pp. 18–45 ©jume. http://education.gsu.edu/jume indigo esmonde is an assistant professor in the department of curriculum, teaching, and learning at the ontario institute for studies in education (oise) – university of toronto, 252 bloor st. w., toronto, on, canada, m5s 1v6; email: iesmonde@oise.utoronto.ca. her research interests include equity and identity in mathematics education, sociocultural theories of learning, and teacher education for social justice. kanajana brodie is an undergraduate student in the equity studies program at the university of toronto; email: kanjana.brodie@utoronto.ca. her research interests include minority representation and educational equity. lesley dookie is a recent m.a. graduate and current research assistant in the department of curriculum, teaching, and learning at the ontario institute for studies in education (oise) – university of toronto, 252 bloor st. w., toronto, on, canada, m5s 1v6; email: ldookie@oise.utoronto.ca. her research interests include equity in mathematics education. miwa takeuchi is a ph.d. candidate in the department of curriculum, teaching, and learning, at the ontario institute for studies in education (oise) – university of toronto, 252 bloor st. w., toronto, on, canada, m5s 1v6; email: miwa.takeuchi@utoronto.ca. her research interests include english language learners‘ opportunity to learn in mathematics classroom. social identities and opportunities to learn: student perspectives on group work in an urban mathematics classroom indigo esmonde university of toronto kanjana brodie university of toronto lesley dookie university of toronto miwa takeuchi university of toronto in this article, the authors investigate group work in a heterogeneous urban high school mathematics classroom. two questions are explored: how do students describe cooperative group work in their mathematics class? how do students describe the way their socially constructed identities influence the nature of their group interactions in mathematics classrooms? the authors present a case study of the ways in which race, gender, and other social identities might influence the nature of group work in reform-oriented high school mathematics classrooms. the analysis, based on 14 interviews with high school students, focused on students’ perceptions of group work and their theories about when cooperative groups work well and when they do not. students named interactional style, mathematical understanding, and friendships and relationships as the most influential factors. using an analytic lens informed, in part, by critical race theory, the authors highlight the racialized and gendered nature of these factors. keywords: cooperative learning, gender, identity, mathematics education, race esmonde et al. social identities and mathematics journal of urban mathematics education vol. 2, no. 2 19 and then the next statement was, ―i have been attracted to someone of the same sex,‖ and absolutely everyone in the classroom sat down except for me. and i was standing there and, so uh. it was just horrendous. the next 2 weeks were just really painful to go through, and to have people treating me differently, and girls kind of like, scooting away whenever i sat near them. – willow, a 9th-grade, bisexual, biracial girl backgrounds, if our backgrounds are too different we don’t get along. …i don’t know, [the teacher] always seats me with people she knows i’m gonna get into it with. [who do you get into it with?] i don’t know… most of the white people in the class. – candie, a 10th-grade, lesbian, african american girl n these two quotes, willow and candie tell the interviewer (indigo esmonde) about school experiences in which their socially constructed identities had an influence on their opportunities to learn. in the first quote, willow describes an activity in a social studies class where she outed herself as bisexual, and then went through a very painful few weeks in which other girls acted uncomfortable around her. in the second quote, candie describes her difficulty working with classmates who have a different ―background.‖ when asked to elaborate, candie describes her difficulties with ―most of‖ the white students in her class. in these brief excerpts, the two students describe how some aspects of their social identities influenced classroom collaboration. (willow describes the importance of her gender and sexual orientation, but not race; whereas candie describes the importance of her race, but not gender or sexual orientation.) in many mathematics classrooms, students work together with their peers, in whole-class and small-group discussions. while student interaction has been lauded for its potential to support critical mathematical thinking and independent problem solving, educators have also recognized that these interactions have the potential to reinforce inequity, as highlighted by willow and candie‘s comments. several scholars have written about curricular and pedagogical strategies to enhance cooperative learning, and the issues of equity that might arise in cooperative contexts (see, e.g., bianchini, 1999; cohen & lotan, 1997; webb, nemer, chizhik, & sugrue, 1998). throughout this corpus of research, it has become clear that who students are influences what and how they learn together. in other words, issues of identity are central to the learning that takes place in cooperative i esmonde et al. social identities and mathematics journal of urban mathematics education vol. 2, no. 2 20 group work. nonetheless, there is still little research that explicitly addresses student perspectives on their social identities and why or how they matter. in this article, we build on the research about cooperative learning by documenting what students have to say about group work in mathematics classrooms. two questions guided our inquiry: 1. how do students describe cooperative group work in their mathematics class? 2. how do students describe the way their socially constructed identities influence the nature of their group interactions in mathematics classrooms? identity and mathematics learning we use the phrase socially constructed identities, or sometimes social identities, to refer to social categories—including, but not limited to race, ethnic, or gender categories—that are often imposed on people within a particular context. we use the phrase to refer to identities that are self-imposed as well as those identities that one imposes on others. we are concerned with both because in everyday interaction people attend not only to their own identities but also to the identities of those around them. in this article, our goal is to understand more about which social identities matter in mathematics classrooms, and how they matter. understanding the importance of social identities in structuring opportunities to learn is especially important for mathematics education because the field has only recently begun to take equity as a central focus for research (secada, 1995). a variety of studies have documented the racialized (martin, 2006, 2007, 2009), gendered (mendick, 2005), and classed (lubienski, 2002) nature of mathematics learning in classrooms across the united states. these multiple facets of social identities operate simultaneously to structure people‘s lived experience. therefore, in this article, we take an intersectional approach to analyzing identity (crenshaw, 1991), and consider how people‘s multiple forms of identity, sometimes referred to as a ―nexus of multimembership‖ (wenger, 1998, p. 158), influence the nature of their experiences in school. these social categories are complex, even more so when we consider their intersectional nature. for example, a recent study of racial identity development in high school focused on two prominent categories for african american youth: street savvy and school oriented (nasir, mclaughlin, & jones, 2009). in this study, nasir, mclaughlin, and jones demonstrated the multiple forms of workingclass african american masculinities and femininities that were at play in the school. barnes (2000), in another classroom study, developed an analysis of two types of middle-class masculinities in the mathematics classroom: one dominant and one subordinate. both of these studies highlight the importance of recognizing the multiple ways that even a single social identity category can be instantiated in the classroom. esmonde et al. social identities and mathematics journal of urban mathematics education vol. 2, no. 2 21 therefore, the theoretical framework through which we view identity comes from sociocultural theories of learning in which learners are viewed as people-incontext and learning as shifts in identity within a particular learning context (lave & wenger, 1991; rogoff, 2003; wenger, 1998). as an analytic construct, identity allows the researcher to focus on individual experiences without losing sight of larger social contexts in which identities are constructed and made meaningful. through participation in various kinds of social practices, individuals construct identities and have their identities constructed for them by others (wenger, 1998). nasir and hand (2006) conceptualize identity as when individuals come to participate in a cultural practice, they negotiate an identity that is part what they have come to view as consistent about themselves in their lives, part what they perceive to be available to them in a practice, and part how they are perceived by social others. (p. 467) in this conceptualization, the self, others, and socially organized practices all play a part in shaping practice-linked identities. thus, identities are both enduring and shifting with each new context and experience. the strength of the sociocultural approach is to consider identity as practice—activityor community-based—constantly constructed and reconstructed in interaction. but the strength of sociocultural theory‘s focus on practice-based identities can also be construed as a weakness, given that with a few exceptions (see, e.g., gutierrez & rogoff, 2003; lee, 2007; nasir & saxe, 2003; nasir et al., 2009), the field has not adequately considered the ways social identities like race, gender, and class (or socioeconomic status), interact with and inform the construction of practice-based identities. mainstream sociocultural theories are typically color-blind, and ignore the ways in which social identities are made part of the structure of everyday practices (nasir & hand, 2006). the sociocultural approach has primarily been used to focus on how individuals come to self-identify within a practice, but research has also considered how identities are imposed by others, and how these imposed identities act to structure a practice (rogoff, 2003). in the classroom, how a student identifies others can have consequences for that student‘s own learning, as in the case when a student decides who in the classroom is ―smart‖ and therefore who might provide her or him assistance in her or his own learning. and how others identify a student has consequences as well, because if a student is seen as smart, for example, they are typically offered more opportunities to engage in meaningful academic activity (cohen & lotan, 1997). we therefore turn to other theoretical approaches that inform our sociocultural-based analysis of social identities in schools. we provide three examples of classroom research—informed by three different theoretical approaches—that have attended to the influence of social identities in classroom mathematics learn esmonde et al. social identities and mathematics journal of urban mathematics education vol. 2, no. 2 22 ing. first, we discuss critical race theoretical approaches to study the racial and mathematical identities of african american learners; next, we discuss research on stereotype threat theory with respect to race, gender, and mathematics; and third, we discuss sociological research on expectation states theory, status, and social interaction. first, since ladson-billings and tate‘s (1995) introduction of critical race theory (crt) into education, researchers have increasingly investigated ways in which race structures u.s. schools at the macro policy level and the micro interactional level (see, e.g., two edited volumes on crt and education: dixson & rousseau, 2006b; parker, deyhle, & villenas, 1999). for instance, anderson and powell (2009) used crt to examine the role of race in constructing the historical conditions and the contemporary status of education in particular school districts and schools. whereas, martin (2006) used tenets of crt to explore the racialized nature of mathematics learning for african american people in the united states. during martin‘s interviews with adults, his participants described a number of ways in which white teachers responded to students‘ racial identities by diminishing the nature of the mathematics education they received. along with critical race theorists (see, e.g., dixson & rousseau, 2006b), martin argues that race and racism are pervasive in the united states, and that it is important to challenge claims of objectivity and alleged neutrality—often coded as ―colorblindness‖ in educational settings and in educational research (see also martin, 2009). methodologically, martin uses crt‘s concept counternarrative, which places an emphasis on the voices of people of color to describe the effects of racism in their everyday experiences, as opposed to dominant narratives that deny the importance of race. second, in a psychology-based theoretical approach, steele and colleagues have amassed an impressive body of research documenting the effect of stereotype threat on the performance of people from marginalized groups (for a comprehensive review of stereotype threat theory, see steele, spencer, & aronson, 2002). stereotype threat theory describes the threat, in ―real-time,‖ of one‘s performance being judged based on a stereotype. for example, african americans can face a stereotype threat based on the stereotype that they cannot succeed academically, and women face a stereotype threat based on the stereotype that they are less mathematically capable than men. in some contexts, when (some) members of the marginalized group have reason to believe that they will be judged based on a stereotype, their performance (say, on an academic test) can suffer because of the anxiety of trying to manage the threatening situation. although the majority of stereotype threat research has been conducted using quasi-experimental methods, recently nasir and colleagues have used ethnographic methods to explore the theory in classrooms (nasir et al., 2009). in a similar ethnographic study (one that did not explicitly take stereotype threat as a esmonde et al. social identities and mathematics journal of urban mathematics education vol. 2, no. 2 23 lens but whose findings nevertheless resonate with it), african american students in a school with a well-publicized racialized achievement gap reported that their peers treated them as if they were ―ignorant‖ and expected them to struggle with their schoolwork (rubin, 2003). in this school, even when teachers tried to build community by creating racially diverse collaborative groups, african american students were often isolated from their friends and marginalized in academic work. in diverse classrooms, students may be very aware of the ways that their teachers and peers view them as members of racialized groups, and may react to stereotype threats when they expect that others will treat them negatively because of their racial identities. third, we consider research on expectation states theory, a sociological theory of status that models how individuals construct expectations for one another‘s competence in activity (berger, cohen, & zelditch, 1972). this theory states that in many situations, members of a group will act as if higher status people are more competent than lower status people, regardless of the demands of the activity. status can be conferred based not only on social identity categories like race, gender, and socioeconomic status but also on other hierarchies such as a person‘s rank or credentials (e.g., well-educated people have more status; a manager has more status than the people they supervise). classroom research has demonstrated the importance of status, highlighting how high-status students tend to dominate small-group and whole-class discussions and learn more from these interactions (cohen & lotan, 1997). while cohen, lotan, and colleagues have argued that popularity and perceived ability are the most consequential status characteristics in classrooms, social identities like race, gender, and socioeconomic status are likely to contribute to status hierarchies as well (chizhik, 2001). although the theoretical foundation of expectation states theory asserts that one can simply ―add up‖ status characteristics, we argue, however, from an intersectional approach, that multiple status characteristics can qualitatively influence one another. race is experienced differently depending on one‘s gender and one‘s socioeconomic status, and vice versa. these three approaches to understanding the influence of social identities all point to the importance of not only analyzing how students self-identify in terms of social identities but also how they are identified by others, and how they ident ify others in the group. in other words, one‘s opportunities to learn are not only influenced by one‘s own identities but also by the identities of others. in short, from martin‘s (2006) research and crt, we emphasize the value of listening to narratives and counternarratives as students describe their experiences; from the research on stereotype threat theory, we consider the various stereotypes that might be present in the classroom, whether they are explicitly voiced or not; and from the research on status, we again emphasize the importance of intersectionality be esmonde et al. social identities and mathematics journal of urban mathematics education vol. 2, no. 2 24 cause of the ways multiple status characteristics might interact in classroom interaction. this study considers how students‘ identities (their own identities, imposed identities, and the ways they identify others) are made salient in the narratives they tell about their mathematics classroom. we analyze a set of interviews conducted with high school students about group work in their mathematics classroom, and report on the social categories that students used to describe themselves and their peers. drawing from crt, we highlight the stories shared by students of color and girls, but we also include narratives from students from dominant groups, expecting that students from various social positions will tell different kinds of stories about their collaborative experiences. we discuss how various categories were invoked (or avoided) to explain the success and failure of group work for mathematics learning. through this analysis, we contribute to a broader understanding of how social identities are made salient in mathematics classrooms, specifically as students work together in small cooperative groups. methods the analysis used a case study design (yin, 1989) in an attempt to better understand the nature of cooperative group work in one particular mathematics classroom. we were inspired by willow and candie‘s remarks on their difficulties working with peers because of their social identities, and constructed the present interview study to gain a better sense of the nature of group work in a single mathematics classroom community. (unfortunately, willow had left the school shortly after the interview quoted in the epigraph, in part because of the experiences she described. candie, however, was still a student at the school at the time of the follow-up study but she had completed her required mathematics courses and consequently was not part of the focal classroom.) following yin‘s (1989) recommendations for case study design, we outline the research questions and the related propositions, the unit of analysis, and then the research context, data collected, and strategies for interpreting the data. the research questions were: how do students describe cooperative group work in their mathematics class? how do students describe the way their socially constructed identities influence the nature of their group interactions in mathematics classrooms? we draw on interview data to describe students‘ perspectives on group work in a reform-oriented, urban high school mathematics class, and discuss several ways in which students argue that their socially constructed identities do (or in some cases, do not) influence their group work. the data for this analysis are taken from a set of 14 interviews that were conducted in the winter and spring of 2007. the interviews were part of a larger study examining mathematical thinking and learning at the high school level. the interviews took place at the esmonde et al. social identities and mathematics journal of urban mathematics education vol. 2, no. 2 25 same school, and with many of the classmates of willow and candie, who took part in a related study in 2004–2005 (esmonde, 2009b). although our approach to understanding identity is informed primarily by sociocultural theories, we are also informed by crt‘s emphasis on ―voice‖ in which the voices of people of color (students, parents, teachers, researchers) are highlighted as they tell their own stories (dixson & rousseau, 2006a). often these stories are framed by educational researchers as counternarratives against the dominant racial (and racist) narratives of the educational community. because this study is not limited to a focus on race, but instead discusses social identities more broadly, we chose to interview a variety of students. in our analysis, we will foreground counternarratives against a backdrop of the dominant narratives from our interviews. another reason to interview a broad range of students for this study is because we do not assume that race, gender, or any other a priori set of social categories are the most prominent ways that students conceptualize identity in the classroom. we recognize that most social categories are complex and contested, and that they may be strategically introduced in some contexts, and downplayed in others (gee, 2000; pollock, 2004). in other words, it is of interest to know whether and when students invoke race, gender, and other social categories in their narratives about mathematics classroom experiences. at the outset of the study, we assumed that students would not necessarily openly discuss social identities that are sometimes taboo topics (e.g., race, socioeconomic status), especially if students were from privileged groups, or if they were hesitant to address these topics with the interviewers. therefore, we designed the interview to provide opportunities for students to discuss these issues if they wanted to. by not directly asking about social identities, we could study the ways in which these categories were independently invoked or omitted by various students. we anticipated some explicit, and some covert, discussions of social identities, as well as some avoidance of this topic (pollock, 2004). we employed two levels of analysis for the study. the case focuses on a single classroom because we wanted to capture students‘ practice-linked identities for a single practice-linked community. however, we also expected that students with different social identities would have different perspectives on group work in their classroom; therefore, for our second research question that investigates the influence of social identities, our unit of analysis was the individual student. each student lived at the intersection of multiple forms of identity, and would have a unique perspective on classroom group work that was shaped by these identities. research context the school was situated in a small urban city in northern california, usa, and was a small school program within a larger school. the larger school, bay esmonde et al. social identities and mathematics journal of urban mathematics education vol. 2, no. 2 26 area high school (pseudonym, as are all proper names throughout), was the only high school in the city, and the student body was diverse in terms of race, ethnicity, and socioeconomic status. the small school, media academy, was a program that focused academically on issues of social justice as well as media studies and media production. students in the small school applied to be part of the program, and were admitted, in part, to match the demographics of the broader school community. the school admitted between 60–90 students per year and emphasized community building both within and across grade levels. with respect to social identity categories such as race, gender, socioecono mic status, and sexuality, the media academy included an explicit focus on issues that are often derived from these categories throughout all years of the program. in the 9th grade, this focus manifested particularly in the core classes of social studies and english in which students learned about systems of oppression and privilege. students were encouraged to write about and explore issues of identity in writing and video assignments. in willow‘s quote in the epigraph, she described an activity at the very beginning of her 9th-grade year in her core class in which issues of oppression and identity were explicitly discussed. unfortunately, as willow described, these conversations sometimes created unsafe spaces for some students. additionally, in the same interview, willow described her anger at other white students who did not defend themselves when accused of racism by students of color. despite efforts by teachers to provide an equitable education to all students in the school, the larger school‘s long-standing racial ―achievement gap‖ was not erased in the media academy. in fact, during the year of this study, a group of students and teachers collaborated to educate themselves and the media academy community in general about racialized achievement differences in the school, advocating for more equitable educational opportunities. with respect to mathematics, the media academy had two mathematics teachers in the year of this study. ms. delack, a white woman with a workingclass background, was in her third year of teaching at media academy; she had taught the students of this study for all 3 years (except for three students, one of whom was interviewed, who joined the class in 2007). the three mathematics courses were de-tracked, using the interactive mathematics program (imp) 1 curriculum (i.e., all students experienced somewhat the same ―level‖ of mathematics instruction). students, however, had previously been tracked in middle school; therefore, a given mathematics class might contain students from different grade levels, depending on whether a particular student had taken pre-algebra or algebra i in the 8th grade, or whether she or he had failed a mathematics course since 1 for information about the interactive mathematics program (imp) curriculum, see http://www.mathimp.org/. http://www.mathimp.org/ esmonde et al. social identities and mathematics journal of urban mathematics education vol. 2, no. 2 27 arriving in high school. the class we focus on in the present study was composed of 30 11thand 12th-grade students. ms. delack‘s approach to mathematics teaching relied on students‘ collaborative work. students worked in groups on a daily basis and were responsible for constructing mathematical knowledge together. the imp textbook did not contain the usual series of theorems, worked examples, and definitions. instead, students were presented with lengthy word problems and were expected to work as a small group and then as a whole class to construct methods for solving the problems. definitions, theorems, and generalizations were usually constructed in whole-class discussions, and often represented on collaboratively created posters that would remain displayed around the room. ms. delack employed several techniques to help students work productively in groups. she generally asked them to ask one another questions first, and to only approach her if they could not resolve something on their own. she held what she called ―process quizzes‖ in which groups were responsible for solving problems, while she assessed the quality of their collaboration. she held whole-class discussions in which students and teacher talked about the value of cooperative group work and encouraged all students to stay involved. in the latter half of the year, inspired by research on complex instruction (cohen, lotan, scarloss, & arellano, 1999), ms. delack began assigning roles to students (facilitator, reporter, recorder, process checker) so that each student had a specific task to perform, and groups reflected on their collaborative process every day. interview participants for the purpose of this case study, we invited all students in ms. delack‘s class to participate in the proposed study. out of the 30 students, 14 agreed to participate. the demographics of our interview participants are listed in table 1. all demographic information is derived from student self-identifications on surveys or responses provided during the interviews. the table demonstrates that our interview participants included both young men and young women, both grade levels, and almost all the racial/ethnic groups in the classroom. (the one asian american-identified student in the class did not participate in an interview.) interview protocol and analysis the first author (indigo esmonde) and a colleague from stanford university conducted the interviews; both are white, middle-class women. esmonde (the first author) was well known to the students, having conducted research and volunteering in their classroom for the past 3 years. the other interviewer (kathleen o‘connor) was unknown to the students prior to the study, and visited the classroom several times for the sole purpose of conducting interviews. esmonde et al. social identities and mathematics journal of urban mathematics education vol. 2, no. 2 28 table 1 names and demographics of interview participants 2 name gender race grade namaya w african american 11 tony m latino 12 tariq m latino 11 giulia w white 11 haley w white 11 mike m white 11 noreen w white 11 nuncio m multiracial 11 dustin m white 11 chelsea w white 11 elly w multiracial 12 karmina w latina 12 may w multiracial 12 samantha w multiracial 12 interviews were semi-structured and focused on student beliefs about the efficacy of group work, and were designed to elicit stories from students about group interactions that ―went well‖ and interactions that ―didn‘t go well.‖ after some initial general questions, the interview protocol was subdivided into four major sections. the first section contained questions that asked students to describe an incident in a group that didn‘t go well; the second, to describe an incident in a group that went well; the third, to reflect on how they might design a mathematics classroom community for effective group work; and the fourth, to describe their current group. each major question in the subsections included a number of prompts to elicit detail from students. (see appendix a for the full interview protocol, including prompts.) to address the first research question, all interviews were roughly transcribed and then coded with a series of open codes. all data were coded iteratively, with new codes being added or refined throughout the process. these codes were used to capture the important factors that influenced the outcome of group work, according to students. each of the authors participated in coding. each of the codes was developed collaboratively, and refined until we came to agreement on examples and non-examples for each code. the four basic codes are defined in 2 although the category multiracial masks quite a bit of diversity, the term is used here because it is how students self-identified. esmonde et al. social identities and mathematics journal of urban mathematics education vol. 2, no. 2 29 table 2. (further examples and explanations of these codes are developed in the results section.) table 2 codes for interview data code description interactional style utterances that describe the kinds of actions or ways of being that support or detract from group work mathematical understanding utterances that describe the ways in which the nature of a person or group‘s mathematical understanding supports or detracts from group work friendship and relationships utterances that describe the ways in which friendship, relationships, and feelings of comfort or safety (or lack thereof) support or detract from group work social identities utterances that explicitly make reference to socially constructed identities, and describe how these identities support or detract from group work to address our second research question, we examined the socially constructed identities of each interviewee, and the socially constructed identities of the classmates they mentioned during their interviews. our goal was to identify any patterns about how these identities were used or avoided, and to determine whether and how they influenced the cooperative group work experience. as a part of this analysis, we examined students‘ descriptions of one another to determine whether they matched or challenged stereotypes related to social identities and mathematics learning. our analysis of the second question was guided by both the responses provided during the interviews and the meanings conveyed through the ―subtext‖ of the transcribed interviews (banning, 1999). in other words, during the analysis, we drew on banning‘s analysis of the internal conflicts and contradictions in a feminist course at a public university. banning employed an analysis of the ―text‖ (field-notes, transcribed interviews, etc.) and the subtext (the unspoken assumptions and beliefs) for her study. similarly, in our analysis, we attempted to document the subtext of our transcribed interviews. while students did not always openly discuss socially constructed identities during the interviews, we argue that esmonde et al. social identities and mathematics journal of urban mathematics education vol. 2, no. 2 30 some of their narratives employed covert strategies for discussing race, gender, and other social categories. to explore the subtext of the students‘ transcribed interviews, we first identified (as best we could) the social identities—race, gender, and grade level—of all interview participants. for all participants, we collected narratives from their interviews in which they described their own interactional styles and the interactional styles of specific peers. we identified the social identities of these named students as well. (not all of the students mentioned in this article were participants in the study, so they do not appear in table 1.) from this collection of narratives, we looked for relationships between students‘ identities and the descriptions and stories they told and that were told about them. in our search for these patterns, we were partly guided by familiar stereotypes based on race, gender, and socioeconomic status, and partly guided by any new patterns that might emerge from the descriptions and stories provided during the interviews. this approach is in keeping with grounded theory; in that, we stayed close to the data and allowed the findings to emerge from constant comparison between the interviews and a gradual elaboration of a set of codes (glaser & strauss, 1967). however, we were also guided by crt; therefore, we were expecting that race, gender, and socioeconomic status (among, perhaps, other social identities) would emerge during the interviews but that these might be hidden. as banning (1999) points out, in contexts marked by hierarchies, those with power tend to use ―power-evasive discourse‖(p. 160) and it would be naïve to expect them to explicitly highlight the ways in which they maintain their privilege. we next describe the themes that emerged from this grounded theory approach. results students’ perspectives on successful and unsuccessful groups in the discussion of the findings, we first describe the results of the first research question: how do students describe cooperative group work in their mathematics class? for this investigation, our unit of analysis was the entire class and we did not distinguish between individual students. the four codes (or themes) were developed through repeated analyses of the data to describe factors influencing mathematical group work. we summarize students‘ views on how these four factors—interactional style, mathematical understanding, friendship and relationships, and social identities—influenced success and failure of group work. in presenting the data, we attempt to maintain the complexity and internal tensions that students expressed during their interviews. the analysis for this research question might appear unrelated to our earlier comments on the centrality of social identities in mathematics classrooms. however, we feel it is important to esmonde et al. social identities and mathematics journal of urban mathematics education vol. 2, no. 2 31 describe what students did focus on during their interviews before deconstructing the more covert messages of the interviews in the later analysis. interactional style the most common factor that students mentioned during their interviews was what we called interactional style. this category included descriptions of student personalities, and different ways of interacting in a group. for example, the following two statements were coded for interactional style: i felt like they‘ve been pretty supportive. (noreen) when there‘s people that take, i guess a dominant role in a group? and kinda work through everything themselves and just tell people the answers? like that doesn‘t work for me. just because of the fact that i need to be able to work through it for myself to really get it. i‘m a very experiential learner. (karmina) both statements describe styles of individual group members or the group as a whole. when asked to describe a positive group work experience, all students agreed that ―good‖ group work involved a combination of people who could work well together. students emphasized the importance of sticking together as a group, solving problems together, and not leaving any members behind. there was strong agreement amongst the students in these descriptions of positive group work. despite this near consensus on the characteristics of good group work, there was additional complexity revealed during the interviews. students discussed the constellation of factors that could influence the appropriate interactional style for a group, and in a related note, discussed the tensions inherent in collaborating in school when their primary responsibilities were to themselves as individuals, and not to the group. furthermore, students differed in their opinions about whether there was a single best way to interact, or whether different individuals had different best interactional styles. we discuss each of these findings in turn. according to the students‘ responses, interactional style alone could not account for the success or failure of group work; there was a set of variables at play. during the interviews, it was rare that a student‘s narrative about group work was coded only for interactional style. for example, students recognized that interactional styles were often influenced by the context of the group collaboration—the type of task, their level of mathematical thinking, or their comfort level or friendship with others in the group. students also described a tension between trying to do good group work (i.e., spending as much time as needed so that all group members understood the material) while also trying to complete their schoolwork correctly and quickly. in esmonde et al. social identities and mathematics journal of urban mathematics education vol. 2, no. 2 32 other words, students were torn between their desire to be inclusive and the pressure to succeed academically. for example, some students told us they (or their peers) liked to work quickly so they could complete the homework in class and some students reported that they were reluctant to ask too many questions, because they did not want to ―slow down‖ their peers. noreen, for instance, reported feeling pressured to say she understood explanations from her peers even when she did not in order to allow them to move on. her statement illustrates the dilemma faced by many students in which they had to choose between what they considered good group work, and their personal academic or social goals. an additional complexity relating to students‘ comments on interactional style was found when they acknowledged the diversity of learning needs in the classroom and how it influenced group work. for example, noreen stated, ―there are certain people, with their personalities, [who] are never really going to work really, really well together, and you can‘t really make them.‖ karmina reiterated this point in her earlier quote (in which she identified herself as an ―experiential learner‖), implying that different students might ―need‖ different kinds of interactions to support their learning. thus, what might work for one person might cause difficulties for another. several other students echoed this sentiment. mathematical understanding students often discussed how different levels of mathematical understanding influenced how well a group worked together. for this code, we focused on the instances where students described how their own level of mathematical understanding, as well as their perceptions of the mathematical understanding of other students, affected the group‘s interactions. for example, the following statement by haley is categorized into this code: ―the groups that usually do the best, [are] where someone‘s really good at math and they‘re someone that‘s like broadly liked.‖ (note that haley discusses mathematical understanding in the context of other personality factors, i.e., ―being broadly liked.‖) generally speaking, our interview participants took one of two differing views on the optimal mathematical understanding for a group. some students asserted that groups should consist of a heterogeneous mix of people with different levels of understanding so that those who needed extra help could ask their peers. other students, however, argued that they preferred to work in groups where everyone had similarly high levels of mathematical understanding. all students who advocated this second position characterized themselves as high achievers. giulia stated that she did not like group work because she did not like being put into groups where people were ―behind in math‖ and ―don‘t bother‖ to get caught up. she preferred groups where everyone had ―the same amount of math‖ and successfully finished their work. noreen was another high achiever who felt that some group members with different levels of understanding ―just esmonde et al. social identities and mathematics journal of urban mathematics education vol. 2, no. 2 33 aren‘t going to work.‖ she described the pressure of being in charge, and said she never wanted others to feel like she was ―taking over‖ or being a ―suck up.‖ therefore, she preferred to be in groups where others did not need her help. even among students who said they preferred heterogeneous (ability) groups, we heard many stories about how difficult it was to work in such groups. high achieving students who held such beliefs often admitted to feeling that their groups slowed them down and they were not always keen to stop their work to help out their peers. for example, dustin reported during his interview that he preferred to get his work done as early as possible, rather than help out struggling students. for those who requested help from peers, such behavior was upsetting, as they felt that they were being ignored. samantha stated, ―i‘d ask [a student in the group who was making progress] well, like, do you know what‘s going on and she would be like, ‗oh, and it‘s like this‘ and then kind of just ignore it and i‘d get really frustrated.‖ in contrast to high achievers, students who were positioned as ―slower‖ than the rest of the group sometimes felt they slowed down their group‘s progress. these students reported feeling pressured to say they understood certain concepts when they did not. friendship and relationships with the code friendships and relationships, we focused on the parts of the transcribed interviews where students described whether or not they were friends with group mates, liked someone, had something in common (e.g., common interests), got along with people, had prior knowledge of people, and the importance of building community in the class and in the group. for example, haley said: ―i think that with the right people everyone could be working and learning. but i think that, like, with the wrong people you can really get set back.‖ students agreed that friendships had both positive and negative impacts on group work and explained some associated tensions. while students perceived good relationships as necessary for successful group work, they also reported that friendship could add challenges. for example, tariq claimed, ―some friends will work good together and some friends just won‘t do anything.‖ students said they felt pressured to socialize with friends, felt left out when they were not friends with group members, and also felt that they could not stop friends in their groups from talking with each other and getting distracted. students pointed out that this ―social talk‖ usually did not lead to productive mathematical work. regarding this point, chelsea stated that friendships can be ―a drawback because if we‘re all so close we‘re going to be talking to each other.‖ similarly, dustin said, with friends, ―it‘s a little tempting to like, catch up on stuff you haven‘t caught up on.‖ the negative effects of friendships could also extend to group members who were outside of the circle of friendship. when describing one group in which she esmonde et al. social identities and mathematics journal of urban mathematics education vol. 2, no. 2 34 was with two older girls who were good friends, haley remarked, ―i didn‘t have really like any sort of authority to get them to start working‖ because she wasn‘t friends with them. haley, tony, noreen, and several other students argued that feeling comfortable in a group was an important aspect for the success of group work. when students felt comfortable in their groups they were able to ask questions without embarrassment and this supported their mathematical learning. social identities during the interviews, students rarely discussed socially constructed ident ities such as race, gender, or socioeconomic status. in this section, we report on many of the explicit statements that students made regarding these kinds of ident ities and their influences on group work. in our coding of the interview responses for this category, we searched for any explicit reference to race, gender, socioeconomic status, or other relevant identity markers. in the process of coding, we added one other type of identity that was frequently mentioned: grade level. we also coded statements into this category when students mentioned a value for ―diversity‖ without specifying what type of diversity they were referring to. we felt that the word diversity was often a way of talking about race or culture, without explicitly mentioning this potentially controversial term (pollock, 2004). grade level. many students mentioned grade level during their interviews. their mathematics class included both juniors and seniors, and this fact influenced students‘ narratives about one another. this social identity may have been related to friendship and relationships because students tended to know samegrade peers better. the implications of this social category on classroom hierarchies were complex. on the one hand, we felt that in this high school, as in many others in the united states, the seniors were at the top of a social hierarchy, followed by juniors. a remark from noreen illustrates this hierarchy as she described group dynamics. noreen claimed that kim was in ―an older position‖ because she was a senior in a group of juniors. because kim was older, noreen believed that kim would have felt confident that people were not ―taking over‖ and that she ―had a voice.‖ on the other hand, in this particular class, the students who were seniors had either failed or been tracked into a ―low-level‖ mathematics course in the past. therefore, the juniors in the course were more likely to be positioned as mathematically talented. race, gender, and socioeconomic status. members of marginalized groups most frequently mentioned social categories. that is, girls were more likely to mention gender, and students of color were more likely to mention race. only one student mentioned socioeconomic status (in an oblique way, by referring to students from ―different neighborhoods‖ in a social context where someone‘s socioeconomic status could be predicted based on the neighborhood they lived in). esmonde et al. social identities and mathematics journal of urban mathematics education vol. 2, no. 2 35 when these categories were explicitly raised, students often described valuing diversity—that is, having groups that included students of different racial or ethnic backgrounds, students from different neighborhoods, or half boys and half girls. for example, chelsea, a white girl, described a group that she felt had not worked well because the group contained four ―leader-ish girls.‖ she felt that groups needed people with different personalities, linked personality with gender, and ended by saying that groups worked better when they consisted of half boys and half girls. although many of the students‘ statements superficially applauded diversity, several students described diverse groups that were not functional. specifically, students from marginalized groups told us they preferred groups that were more homogeneous. for example, elly, a multiracial girl of low socioeconomic status told us that she preferred to work with other girls. elly went on to tell us that when she worked in groups she could not help but interpret people‘s individual behaviors through a ―social justice lens.‖ she provided, in some detail, a narrative about being grouped with two white boys and a latina girl in which the two white boys dominated every interaction. she then pointed out that sometimes more homogeneous groups might lessen this type of power struggle. another student, tony, a latino boy, told us that he preferred to work with other people of color, and specifically latino boys, because they socialized outside of school and he knew them fairly well. this comment was interesting, because earlier in the interview tony had said that it did not matter ―who you are.‖ he stated that so long as students stayed focused, group collaboration could go well. we emphasize that in this remark he does not essentialize students of color by saying that there is something about these students that makes collaboration work better. instead, he told us that his social sphere was mainly composed of students of color and that their positive social interactions outside of school supported positive mathematical interactions in class. analysis of subtext of student interviews after we completed the aforementioned analysis on students‘ perspectives of successful and unsuccessful groups, we developed an interest in uncovering some of the more subtle ways in which students‘ socially constructed identities appeared during the interviews. although students rarely discussed the importance of social identities for group work explicitly, upon closer examination of student talk, we anticipated finding implicit messages about race, gender, and social identities. this section focuses on our second research question: how do students describe the way their socially constructed identities influence the nature of their group interactions in mathematics classrooms? for the analysis of the second question, we examined the students‘ interview responses more closely to identify whether and how social identities might have esmonde et al. social identities and mathematics journal of urban mathematics education vol. 2, no. 2 36 been implicated in their descriptions of interactional styles, personalities, and mathematics achievement. by setting individual students as the unit of analysis, we considered how our interview participants described themselves, and how they described their peers. in many cases, we had access to students‘ selfidentifications for race and gender, and after researching and volunteering with these students for 3 years, i (the first author) also had access to the ways in which students were routinely racialized and gendered by others in the classroom. therefore, for each interviewee, we listed all the classmates that they mentioned, the social identities that were associated with each of these classmates, and then we considered how interviewees described their classmates. for example, when chelsea described being irritated with the way haley dominated group work, we noted that both chelsea and haley were white girls. race, gender, and grade level were the most visible categories to us. we were not, however, aware of all students‘ socioeconomic status or other social categories that might have been at play. we recognize that there is danger in imposing social identities on students. we risk simplifying multiple layers of students‘ social identities or erasing diversity within one social category. however, we observed that these identities were routinely imposed on the students every day in the classroom. when we used the construct of social identity for our analysis, we referred to identities that are selfimposed, and also to the identities that one imposes on others. in other words, we were concerned with both self-identification and the identities students imposed on others. when we looked under the surface of students‘ innocuous discussions of interactional style, mathematical understanding, and friendship and relationships (or feelings of comfort), we began to make connections between these themes and students‘ social identities. in particular, we discuss patterns that emerged in three areas: group leadership and other styles of group interaction, the ways in which students handled the tension between getting their own work done and helping others, and student preferences for group mates. our analysis of the interview responses revealed a general tendency for white students to describe themselves as taking on leadership type roles in the group work and for latino/a, african american, and multiracial students to describe themselves as taking on more passive roles. this tendency was the case even for students of color who described themselves as ―dominant‖ or ―leaders‖ in general. more specifically, mike (white boy), haley (white girl), and giulia (white girl) described themselves as leaders throughout their interviews whereas tony (latino boy), karmina (latina girl), and namaya (african american girl) described a general reluctance to take on such a role. when considering the relationship between gender and roles within group work a similar pattern emerged—boys tended to be group leaders. this pattern was particularly evident in groups that consisted of all white students, or all stu esmonde et al. social identities and mathematics journal of urban mathematics education vol. 2, no. 2 37 dents of color (where gender seemed to be the most salient marker of difference between students). as one example, when tony (latino boy) was working with tariq (latino boy), namaya (african american girl), and may (multiracial girl), tony described that he and tariq took on the leadership roles and did most of the work. tony went on to report that he and tariq also did the group presentations. a similar example was noted in the interview with haley, a white girl. when working with a group of girls, haley described herself as being a leader. however, when put in a group that included boys, haley said she took on a more passive role and described a boy who took on a strong leadership role: there was this guy and he was really kind of like the leader and he was really, he got everyone to do their work. …well he just, he dominated every conversation we had, so then when he decided to do math, which he did, he wanted to do his work, he did get good grades, so when he decided to do his math, kind of our group did it. and not to say that i didn‘t have conversations with other people in my group, it‘s just he was like kind of like a leader and kind of, i don‘t know, i don‘t actually like him that much but like it‘s just he is good at getting people to do group work. (haley) as evident in these examples, students‘ social identities—their own, and the identities of others in their groups—became salient and likely influenced the roles they took on during group work. during this second phase of the analysis, we found further examples of the tension that students described between doing good group work (working collaboratively and helping one another) and getting work done. the analysis of the transcribed interviews revealed that students responded to this tension in various ways, which led, in several cases, to the exclusion of some students from the group work. for example, when telling her story of a group that did not work very well together, namaya shared her experience of being excluded that may have contributed to her frustration and lack of participation in the group: i was like, ―what‘s going on?‖and they kept telling me like, ―hold on‖ and then they would just forget that i had even asked them what was going on. i had to beg them to keep me…to tell me what was going on. …i really needed them to keep me included so i could understand, but they didn‘t. (namaya) her group members at the time were chelsea and chris (two white students) and jaime (a latino boy). this tension between getting one‘s own work done and helping the group was reported many times during the interviews. however, students responded to this tension in different ways. karmina, a latina female, tells a story in which she chose to assist a classmate. specifically, she recalled an incident working with dustin (white boy), amelia (white girl), and kayla (african american girl): esmonde et al. social identities and mathematics journal of urban mathematics education vol. 2, no. 2 38 i was in a group with, amelia and dustin, and kayla, i think. um, and kayla had been gone [to a youth leadership program] for a week, so, when she came back, she had a lot of catching up to do. and dustin and amelia are both really dominant people when it comes to, like, the work, um, so, they were kinda speeding through it, not really making sure that, we got it, and i was kinda like catching kayla up. so, it, it was just, really weird cuz then we would get to these things and we‘d be like i don‘t get it and then they‘d get kinda frustrated and not wanna tell us, and just kinda get an attitude. (karmina) while karmina was engaging in what most students would have agreed was good group work, her two peers were rushing to get the work done, thus leaving her behind. good group work, then, does not always benefit all students. her peers in the group, both white students, clearly did not make the same choice. they chose not to slow down to help kayla. we also interviewed dustin, and although he did not specifically discuss this group, he told us that in general, when he was in heterogeneous groups, ―i feel like maybe they‘re not getting it as fast as me, so i feel like i just wanna get it done or go ahead or whatever.‖ these examples illustrate the impact that this tension had on various students and how, as a result, students of color were marginalized and excluded. this exclusion is even more poignant because of the way namaya and karmina described themselves as strong leaders in other groups, consisting primarily of other students of color. we stress, however, that it is important not to see students of color as helpless victims in groups where they do not take on leadership roles. in namaya‘s narrative above, she explained how she repeatedly asked for help from the group, but the group ignored her. she attempted to be an advocate for herself, but the other group members still excluded her from their discussions. as critical race theorists have insisted, namaya‘s story about her participation in this group provides a counternarrative to the dominant stories being told about group work in this classroom. more specifically, these stories contrast with dominant narratives about students who just ―don‘t care‖ and therefore do not voluntarily participate in group work (echoed in giulia‘s previous narrative). far from not caring, namaya‘s counternarrative illustrates that students can be excluded from groups despite their best efforts to engage. that exclusion can lead to frustration and disengagement, which ironically may then further exclude students from productive group work. perhaps as a result of such clashes, we found that when we examined whom students named as good group members we noticed that, by and large, students of color identified other students from marginalized racial groups. for example, tony, a latino boy, explicitly reported his preference for working with latino and african american students. he later explained that he learned better when working with these students because he felt comfortable to ask questions and to answer esmonde et al. social identities and mathematics journal of urban mathematics education vol. 2, no. 2 39 the questions of his peers. when working with these students, he tried to be as ―active‖ as he could in this group. more evidence came during the interview with karmina, a latina girl, when she described working well with her best friend samantha, a multiracial girl. specifically, she recalled that they helped each other work through problems and built understanding together. the fact these students are both seniors is also noteworthy. karmina reported, however, not working well with haley, a white girl, who was also a junior, as haley would rush through work and ―jump in‖ to explain the answers. overall, we found that all of the students of color in our small sample demonstrated a preference for working with other students of color. they were more likely to be friends with one another, more likely to be in the same grade level, and to have similar values or interactional styles. the white students that we interviewed named some white students and some students of color as their most preferred peers to work with. this preference might be because these privileged students did not have to struggle to create opportunities to learn for themselves. this pattern of marginalized students preferring to work with other marginalized students did not hold out for gender, demonstrating that the way gender operated in the classroom was distinctly different from race. most boys and girls displayed a preference for mixed gender groups, although two of the girls (elly and chelsea) mentioned all-girl groups that had worked well. in chelsea‘s story, she described being surprised to find that an all-girl group could work well. discussion in this article, we provided an analysis of the texts and subtexts of a group of high school students‘ narratives about mathematics group work. based on our analysis, we have shown that students experience mathematics classrooms as sites for power struggles that are often related to their social identities, and we have discussed how these power struggles may affect student opportunities to learn. we found that within this classroom, white students, especially boys, despite the best efforts of marginalized students to make their voices heard, often dominated group discussions. the major themes that students described in their narratives—interactional style, mathematical understanding, and friendships and relationships—held covert racialized, gendered, and classed meanings. our analysis of the subtext of the transcribed interviews resonates with other research about the impact of social identity. in california high schools, mathematics achievement and friendship groups tend to be highly racialized and gendered, and, perhaps less visibly, classed (noguera, 2003; olsen, 1997; pollock, 2004; rubin, 2003). as pointed out previously, in this school, students recognized racialized and socioeconomic esmonde et al. social identities and mathematics journal of urban mathematics education vol. 2, no. 2 40 patterns of achievement. students of color were over-represented in lower tracked mathematics classes while white and asian students were under-represented. furthermore, stereotypes about boys‘ superior mathematics skills continued to be a subject of debate in public forums, and in this particular high school, it was well known that middleand upper-class students from the ―hills‖ tended to be higher achievers than their working-class or poor counterparts from the ―flats.‖ thus, when students talked about how mathematics achievement influenced the way groups worked together, and dismissed some students as not caring, or not staying focused, there may have been an unacknowledged racialized, gendered, or classed dimension to these assertions. friendship patterns also tended to be associated with these social identities. although many students had cross-race or cross-gender friendships, and this small school community had done much work to build community across race, gender, and socioeconomic status, it was still the case that friendship groups were relatively homogeneous with respect to race and socioeconomic status. our interview respondents mentioned that they were friends with people they felt ―comfortable‖ with, perhaps indicating some shared experiences, shared language, and shared understandings of the world that can come from sharing similar social identities. this study is particularly important for sociocultural theories because the interviews were grounded in a particular practice—group work in a mathematics classroom—and therefore highlight the interconnections of practice-linked identities with broader social identities. these practice-linked identities as mathematics students were shaped by students‘ social identities (as when tony described his preference for working with other latino/a students), and reciprocally, the social identities were shaped by their mathematics identities (as when some of the ―leader-ish girls‖ decided that this social identity made mathematical collaboration more difficult). our analysis therefore provides one possible methodological and empirical perspective to include these social categories into a sociocultural analysis of learning. the crt perspective proved important to our analysis because of the emphasis on counternarratives. although only a small number of students explicitly discussed social identities during their discussions of group work, we believe it is no coincidence that the students who did so were those who were in traditionally marginalized groups. white students did not discuss race, and boys did not discuss gender, perhaps because their privilege blinded them to these aspects of classroom life. (another explanation, of course, is that they were simply reluctant to discuss these identities during the interviews.) the juxtaposition of these dominant narratives with counternarratives highlights some of the complexity of group work in mathematics classrooms. this complexity was represented many times during the student interviews. students described not only the multitude of factors that influence group work but esmonde et al. social identities and mathematics journal of urban mathematics education vol. 2, no. 2 41 also the real tensions inherent in trying to accomplish the multiple and sometimes competing goals of group work. in telling their narratives, students never relied on just one factor to explain the success or failure of group work. instead, they described how interactional styles, mathematical understanding, friendships and relationships, and (in some cases) social identities all contributed to the process and outcome of group work. group work has been touted as a way to get students from ―different groups‖ to respect one another, but researchers have cautioned that teachers must take an active role in fostering that respect (aronson &bridgeman, 1979; boaler, 2006). in our interviews, boys and girls of color and white girls reported their marginalization in group work in this classroom. students seemed to have mixed feelings about heterogeneous groupings, whether these grouping were heterogeneous based on mathematical understanding, friendship, grade level, or social identities. while students almost universally recommended heterogeneous groups, they also almost universally described problems that arose in such groups. valuing diversity may be a powerful narrative in current high schools, yet these stories provide counternarratives that illustrate that in diverse groups the same power imbalances that characterize social life outside schools might be reinscribed in collaborative groups. these ―heterogeneous groups‖ might actually be supporting privileged groups—especially white boys—rather than creating spaces with rich opportunities to learn for students of color and for girls in general. while our analysis is certainly speculative in some cases, we feel that the stories we were told during the interviews call into question the role of heterogeneous groups in mathematics classrooms. even in a classroom with a teacher who was aware of equity issues, and in a school where students studied issues of oppression and social justice, these systems of oppression were at play and influenced students‘ opportunities to learn mathematics. conclusion as i (the first author) have argued elsewhere (2009a, 2009b), equity issues in mathematics classrooms are complex and simple solutions may not be enough. although in this article we have used student narratives to highlight the ways in which heterogeneous groups may reinforce pre-existing inequities, we do not support racialized or gendered forms of grouping or tracking as solutions to this problem. while in some cases, single-sex or race-based (e.g., afrocentric) schools may help some members of marginalized groups to succeed academically, in most cases the de facto segregation that happens in schools deepens the divide between groups. we urge the mathematics education community to take this issue on di esmonde et al. social identities and mathematics journal of urban mathematics education vol. 2, no. 2 42 rectly, and to consider ways in which teachers in heterogeneous classrooms can support all students‘ opportunities to learn rich mathematics. acknowledgments this article is partially based on a presentation given at the research presession of the annual meeting of the national council of teachers of mathematics, washington, dc, april 2009. this work was supported, in part, by the national science foundation under grant no. sbe-0354453 to the learning in informal and formal environments science of learning center, as well as by the connaught fund at the university of toronto. any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the position, policy, or endorsement of the national science foundation or the connaught fund. the interview protocols were created as part of a collaborative effort frombrigid barron, nicole casillas, leslie herrenkohl, emma mercier, veronique mertl, na'ilah nasir, kathleen o‘connor, roy pea, leah rossman, and kersti tyson. indigo esmonde and kathleeno‘connor conducted the interviews. we would like to thank the students and teacher at bay area high school who generously donated their time to this project. references anderson, c. r., & powell, a. 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(1989).case study research: design and methods (revised ed.). newbury park, ca: sage. esmonde et al. social identities and mathematics journal of urban mathematics education vol. 2, no. 2 45 appendix a interview protocol 1. tell me about yourself (name, grade level, etc.). 2. how often do you work in groups in your math class? how much of each class period is spent working in groups? 3. tell me a story about a time you were in a group that didn‘t work so well. a. who were the characters involved? b. can you describe what the group was supposed to be doing together? what happened? c. how did it work out? d. what do you think the other people were thinking/feeling in this situation? (can probe for a specific person from their story) e. why do you think it didn‘t work so well? (what‘s your theory about why this group didn‘t work so well?) f. if you were to go back and were in this situation again, what do you think you might do differently to make this go better? what might others have done to make this go better? 4. tell me a story about a time you were in a group that worked well together. a. who were the characters involved? b. can you describe what the group was supposed to be doing together? what happened? c. how did it work out? d. what do you think the other people were thinking/feeling in this situation? (can probe for a specific person from their story) e. why do you think it worked so well? (what‘s your theory about why this group worked so well?) 5. imagine you (are a teacher, and you) are going to set up a new collaborative unit for your students. take us through how you would set it up—what would you think about, what would you do to make sure the groups would work well? what resources would you use? when would you use group work? (note: ask each probe question separately after they have had a chance to answer fully.) a. what would the physical setup look like? b. how would you form the groups? c. what kind of feedback would you give the groups? d. how would you assess how the groups are doing? e. what kind of product would you have the groups working on? f. what kinds of relationships would you want people to have, how would you support this? g. what kinds of challenges do you think you would face? how would you address them? 6. tell me about the group you‘re in right now. how is the group working together? a. what is your role within the group? b. what roles do the other group members take on? c. do these roles change for different kinds of activities? microsoft word 381-article text no abstract-1983-1-6-20200603 (galley proof 2).docx journal of urban mathematics education december 2020, vol. 13, no. 2, pp. 26–41 ©jume. https://journals.tdl.org/jume eduardo mosqueda is associate professor in the department of education at the university of california santa cruz, 1156 high street, santa cruz, california, 95064; email: mosqueda@ucsc.edu. his research interests include the mathematics education of emergent bilingual learners and equity in urban school contexts. saúl i. maldonado is an assistant professor in the department of dual language and english learner education at san diego state university, 5500 campanile drive, san diego, california 92182; email: smaldonado@sdsu.edu. maldonado investigates the intersection of language, literacy, k–12 achievement and evaluation. editorial using large-scale datasets to amplify equitable learning in urban mathematics eduardo mosqueda university of california santa cruz saúl i. maldonado san diego state university he united states has administered national mathematics assessments since 1973, and since then researchers have identified a relationship between students’ achievement, racial-ethnic background, and enrollment in advanced mathematics courses (carpenter et al., 1983). patterns of performance on achievement measures such as the national assessment of educational progress (naep) influence policy decisions, and disaggregating data by students’ socioeconomic status, race-ethnicity, and english-language competence remains a federal priority (de brey et al., 2019). although policymakers have used disaggregated student assessment results to justify funding decisions (trujillo, 2016), researchers have argued that the persistent disparities in achievement trends by disaggregated groups reproduces educational inequalities (gutiérrez, 2007; johnson, 2002). disrupting the mathematics achievement disparities of students minoritized by socioeconomic status, race-ethnicity, and english-language competence requires an explicit consideration of schools’ designation as either urban, suburban, or rural. in the united states, schools in metropolitan communities comprised of more than 50,000 persons often serve high percentages of students minoritized by socioeconomic status, race-ethnicity, and english-language competence (lippman et al., 1996). an association exists between mathematics achievement and students’ income, racial-ethnic, and language background; however, student characteristics cannot completely explain differential mathematics achievement, and consideration of urban school contexts is required (capraro et al., 2013). to challenge deficit thinking often associated with urban schools and to provide a tool for researchers studying urban education, milner (2012) provided a threecategory typology for classifying urban schools: urban intensive, urban emergent, and urban characteristic. urban intensive describes schools in large metropolitan cities, urban emergent describes schools in cities with populations under 1 million people, and urban characteristic describes schools not located in cities but facing t mosqueda & maldonado editorial journal of urban mathematics education vol. 13, no. 2 27 contextual challenges similar to those of urban intensive and urban emergent schools (e.g., limited instructional resources, inadequate teacher preparation, and increases in the population of linguistically minoritized (lm) students). milner’s typology of urban schools has recently been amplified to include a) population/location/geography, b) enrollment, c) demographic composition of students, d) resources in schools, e) disparities and educational inequality, and f) social and economic context (welsh & swain, 2020). in addition, welsh and swain (2020) have also found that lms comprise 11% to 15% of students enrolled across all three categories of urban schools. the urban context, therefore, provides important information regarding the learning environment in which students operate. from our perspective, it is critical that researchers distinguish how students’ individual characteristics, schools’ urban contexts, and the provision of advanced mathematics courses influence mathematics achievement trends. the opportunitiesto-learn (otl) conceptual framework informs our research of mathematics achievement in urban schools. researchers have described otl as equitable access to the structural conditions that develop all students’ learning (stevens & grymes, 1993; tate, 2001). we consider otl an appropriate framework for our additive perspective of students in urban schools negotiating mathematics learning opportunities as mechanisms of power (martin et al., 2010). the united states department of education has an agency dedicated to providing education data and research to the public: the institute of education sciences (ies). established in 2002, ies replaced the office of educational research and improvement to address the “mismatch between what education decision makers want from the education research and what the education research community is providing” (whitehurst, 2003, p. 13). ies administers programs such as the what works clearinghouse as well as four centers: a) the national center for education research, b) the national center for education statistics, c) the national center for education evaluation and regional assistance, and d) the national center for special education research. in this article, we present suggestive guidance for accessing and using large-scale datasets from the national center for education statistics (nces) to examine secondary mathematics achievement in u.s. urban schools. our discussion’s focus is on using large-scale datasets produced by nces to examine mathematics achievement, but we intend for this paper to be a guide for researchers in both accessing and using large-scale data in general for the purpose of developing a quantitative research agenda that examines students’ mathematics achievement in urban secondary school contexts in the united states. our purpose is to provide methodological considerations and analytical suggestions for researchers using nces datasets to examine mathematics outcomes of students minoritized by race-ethnicity, socioeconomic status, and english-language background. we are specifically interested in sharing suggestive guidance to equity-informed researchers committed to urban mathematics education issues. in what follows, we provide a mosqueda & maldonado editorial journal of urban mathematics education vol. 13, no. 2 28 description of the nces secondary longitudinal studies and guidance for accessing this data. we then offer methodological considerations for analyses with nces datasets and conclude with analytical suggestions for studies focused on urban school contexts. for clarity, a data source is represented in bold italics, and variables are all upper case. nces secondary longitudinal studies researchers interested in questions about secondary students’ mathematics achievement in urban schools may benefit from accessing data from the following longitudinal surveys: a) national longitudinal study of the high school class of 1972 [nls-72], b) high school and beyond [hs&b], c) national education longitudinal study of 1988 [nels:88], d) educational longitudinal study of 2002 [els:2002], and e) high school longitudinal study of 2009 [hsls:09]. we display a historical overview of the research design of the five longitudinal studies along with information about data collection years and students’ corresponding ages and years in school in figure 1. figure 1. research design for nces secondary longitudinal studies beginning with the nls-72, nces secondary longitudinal studies provide researchers with almost fifty years of quantitative information to analyze students’ mathematics achievement. data collection has concluded for nls-72, hs&b, and nels:88, but data collection is ongoing for els:2002 and hsls:09. the most mosqueda & maldonado editorial journal of urban mathematics education vol. 13, no. 2 29 recent nces secondary dataset, the middle grades longitudinal study of 2017–18 [mgls:2017], is currently collecting mathematics assessment data of students, beginning in the sixth grade and concluding in the eighth grade. accessing and using nces datasets the ies website offers the public free unrestricted access to data, publications, products, and data tools associated with nces. using the online codebook data tool, researchers may download variables of interest in school-level or student-level data as well as create syntax files. ies offers free distance learning dataset training modules online to teach researchers how to acquire, access, and explore nces datasets as well as how to conduct analyses with specific data tools and statistical software. we recommend researchers start with the common modules, such as “analyzing nces complex survey data,” and continue with the 34-minute module “introduction to the nces longitudinal studies: 1972–2020.” after understanding the objectives of the various secondary longitudinal studies, we suggest researchers further familiarize themselves with one of the nces datasets. prior to downloading any public-use data for analysis, we encourage researchers to review the data file documents that describe specific details about each dataset, such as appropriate design weights and information about the suppression of all direct and indirect identifiers that may compromise the confidentiality of participating persons and schools. additionally, we recommend researchers review tables previously created by nces and published online prior to downloading the variable lists for longitudinal studies of interest to identify if research questions will require access to restricted-use schoolor student-level data. researchers from organizations in the united states, such as universities and government agencies, that are interested in analyzing restricted-use data must participate in a comprehensive application process for a license, which includes signed documents, such as notarized affidavits of nondisclosure, as well as the completion of an online training course. students require a faculty advisor to submit the application as the principle project officer, who will work in partnership with both a senior official from the university’s sponsored projects office as well as a systems security officer from the university’s technology department to finalize the application. after reading the restricted-use data procedures manual, researchers submit the license application with signed original documents to the ies data security office. each license application permits up to seven users of the ies data, including students, provided all users access and analyze the data in the same location. if the restricted-use data license is approved, the requested data file(s) are mailed to the license holder as an encrypted cd-rom/dvd for exclusive use on a secured standalone desktop computer that is not connected to a network or modem, and the data must remain in a locked office that is only accessible to persons listed on the license. additionally, ies requires that researchers analyzing mosqueda & maldonado editorial journal of urban mathematics education vol. 13, no. 2 30 restricted-use nces data round unweighted sample size numbers and submit all reports, tables, or presentations for potential disclosure review prior to dissemination. in our research, we have analyzed secondary students’ achievement data from nels:88, els:2002, and hsls:09. when we used hsls:09 data to examine secondary mathematics achievement in u.s. urban schools, the student racial-ethnic background composite variable [x1race] and the mathematics standardized theta score [x1txmtscor] were available on the public-use files, but the student native language background composite variable [x1nativelang] was only accessible on the restricted-use files. using the public-use data, we ran descriptive statistics, such as frequencies, percentages, crosstabs, and correlation tables, to determine if the number of hsls:09 cases would be appropriate for our research questions. methodological considerations complex sampling the nces typically uses complex sample designs in its data collection approach that include the following three strategies: stratification, clustering, and a multi-stage approach. for example, in describing the sampling approach used in hsls:09, ingels and colleagues (2011) note the following: in the base-year survey, students were sampled through a two-stage process. first, stratified random sampling and school recruitment resulted in the identification of 1,889 eligible schools. a total of 944 of these schools participated in the study, resulting in a 55.5 percent (weighted) or 50.0 percent unweighted response rate. in the second stage of sampling, students were randomly sampled from school ninth-grade enrollment lists, with 25,206 eligible selections (or about 27 students per school). (p. v) the nces notes that their use of a complex sampling design is to increase the efficiency in measuring specific subsamples in a population. for example, stratification is used to ensure that different subgroups are adequately represented in the sample. stratification involves dividing a sampling frame into relevant subgroups prior to the sample selection (schneider et al., 2007). although complex sampling strategies are useful to ensure sufficient numbers of underrepresented observations in a sample, such sampling strategies also give more weight to particular observations that are disproportionately included relative to their representation in the overall population. therefore, the weights that are included in each dataset must be included in an analysis to adjust for any oversampling (thomas et al., 2005). the nces uses sampling weights to indicate the relative contribution of each observation in order to produce adequate population-level estimates. for example, if a student in a dataset is assigned a weight of 1050, this means that the student represents 1,050 students in the population who have the characteristics used in the sampling design, such as racial-ethnic mosqueda & maldonado editorial journal of urban mathematics education vol. 13, no. 2 31 background and grade level. the nces provides the sampling design information needed to use complex variance estimation software to compute estimates of variance that reflect the complex sample design of the data collection process. if this information is not used, or if used incorrectly, the results of hypothesis testing, or the p values, will also be incorrect. sample weights are used to account for sample selection processes, meaning they adjust for the fact that not all units had an equal probability of selection into the sample. sample weights can be adjusted for the fact that nonresponse may be greater among certain subgroups of the population. this adjustment is important because when there are differential patterns of nonresponse the data can be biased, or not representative of either the population or subgroups of interest. if all sources of nonresponse bias are accurately captured in the nonresponse weight adjustments, the end result produces estimates that represent the target population. data clustering clustering of observations results from the selection of groups of units, such as first selecting schools then selecting students. if clusters of students are internally homogeneous, meaning students within schools are more similar than students across schools, then the estimates of the overall variance on measures will be lower than is the case if a simple random sampling strategy was employed (muthén & satorra, 1995; thomas et al., 2005). multi-stage cluster sampling can also be used to facilitate multilevel analysis of the relationships between distinct levels of data. multilevel modeling is a variance estimation technique that is appropriate for complex sample designs, as it addresses the clustered samples into the analytical models (muthén & satorra, 1995; raudenbush & bryk, 2002). for each participant in the sample, the total score on a dependent variable is decomposed into an individual (or withingroup) component and a between-group component. the decomposition of variables from the sample data into their component parts can be used to compute a withingroup covariance matrix (i.e., the covariance matrix of the individual deviations from the group means) and a between-group covariance matrix (i.e., the covariance matrix of the disaggregated group means), and the variation at each level can then be explained simultaneously with sets of predictors at each level of the data structure (raudenbush & bryk, 2002; thomas & heck, 2001). there are multiple statistical software packages, such as hlm, mlwinn, spss, stata, and r, to name a few, that can take advantage of the nesting of students (level 1) within schools (level 2), which will adjust for the variance estimation appropriate for complex designs used by nces. mosqueda & maldonado editorial journal of urban mathematics education vol. 13, no. 2 32 causal inference the american educational research association has availed a report that provides researchers (and funding agencies) with a set of guidelines for evaluating various methods and analytic approaches for estimating causal effects from large-scale observational datasets that can inform policy (schneider et al., 2007). these guidelines focus on the following analytic approaches for estimating causal effects: fixedeffects models, instrumental variables, propensity scores, and regression discontinuity (e.g., schneider et al., 2007). all of these methods provide useful strategies for eliminating bias in observational designs, such as the secondary longitudinal studies produced by nces. in research designs that have a clear treatment variable, one popular method for addressing selection bias in such observational datasets is propensity score matching (psm). although participants in a specific treatment activity are not randomly assigned to specific interventions in observational data sets, students may differ systematically in such assignments. this results in selection bias in the estimation of the treatment effects. psm analysis can be used to correct for such selectivity bias (rosenbaum & rubin, 1984; singer & willett, 2003). the benefit of psm is that it produces a conditional probability, or propensity, of being in the treatment or control group based on a set of observed variables (rosenbaum & rubin, 1984). an additional benefit to psm is that it can be utilized to match students based on key observable characteristics in order to minimize selection bias in the estimated treatment effects and simultaneously support causal inferences based on the results for participants in the treatment relative to those in the comparison group. an important limitation of using propensity scores to address selection bias in observational datasets is that psm only accounts for observed covariates and does not account for unobserved characteristics of participants (murnane & willett, 2011; rubin, 1997). despite the method’s limitations, we believe psm improves researchers’ ability to more effectively compare treatment and comparison group students in large-scale dataset observational studies. analytical suggestions from our published studies analyzing multiple levels of data simultaneously there are multiple measures that reflect the urban school context at the individual level and at the school level that influence mathematics achievement, often times with negative effects on the overall outcomes. in other words, characteristics of minoritized students, including race/ethnicity, language status, and socioeconomic status (ses), can all depress achievement performance on standardized tests. for instance, an nces study shows that in 2015 the average mathematics score on the mosqueda & maldonado editorial journal of urban mathematics education vol. 13, no. 2 33 naep assessment for white 12th-grade students was 22 points higher than the test performance of their latinx peers and 30 points higher than their black peers (mcfarland et al., 2019). this naep study also showed that the mean mathematics score was 115 for 12th-grade english language learner (ell) students, reflecting a negative difference of 38 points, on average, relative to the mean score of 153 for their nonell counterparts. in addition, schools with higher concentrations of low-ses students had lower aggregated test scores in mathematics. the naep study showed that, on average, the aggregated mathematics scores for 12th-grade students in high-poverty schools was 129, a mean score that is markedly lower than the average scores for 12th-grade students in mid-high-poverty schools (145), mid-low-poverty schools (154), and low-poverty schools (164) (mcfarland et al., 2019). the disparities highlighted in this naep report across race and ethnicity, student family incomes of varying levels, and school poverty levels and between ells and non-ells suggest that urban mathematics education analyses must account for income at the individual level and also at the school level by including measures of low-income student concentration. the inclusion of socioeconomic measures at the individual level in addition to the socioeconomic concentration and makeup of a school’s student population, which is also an important marker of a school’s urbanicity, is consistent with extant research that has found ses exerted the largest influence on academic achievement rather than schools’ racial-ethnic composition (rumberger & palardy, 2005). other research by orfield and lee (2007) has focused on the connection between urban school contexts and academic achievement as influenced by within-school factors, such as students’ racial-ethnic segregation and family income. these results showed that students who attended schools in which the mean of students’ ses was high received increased academic benefits, while students in schools where the mean ses was low showed lower achievement performance outcomes. the effects of the concentration of ells in specific schools has been identified as an additional factor that negatively moderates achievement. gándara and orfield (2010) researched how disproportionate segregation of latinx ells negatively impacted achievement in addition to negatively influencing students’ social and emotional development. another study on the segregation of latinx ells showed their concentration in schools was associated with lower achievement outcomes in mean scores and was negatively influenced in terms of content knowledge, english-language fluency, academic-language fluency, and literacy development (gifford & valdés, 2006). results from these studies show the importance of including race/ethnicity, language status, and ses school-level variables in analyses of achievement in urban schools. most nces datasets include multiple urbanicity-related measures for studies to account for individual-background and school-context variation that may negatively influence academic performance outcomes. these predictors may also help mosqueda & maldonado editorial journal of urban mathematics education vol. 13, no. 2 34 account for selectivity bias in the sample. such measures include individual-level race/ethnicity, gender, and ses. at the school level, we suggest researchers use aggregate urbanicity measures, which include the following: whether the school is public or private, the percentage of low-income students within each school, the number of students that qualify for free or reduced lunch (used as a proxy for poverty), the percentage of ells, and the percentage of teachers that are not fully credentialed. in addition, the nces datasets provide a robust measure of family ses that includes income, parents’ occupational status, and educational attainment. for these reasons, we recommend the use of the ses indicator over the unidimensional family income variable in studies of urban schools. in a study titled “systematized discrimination: the relationship between students’ linguistic minority status, race-ethnicity, opportunities to learn, and college preparatory mathematics,” mosqueda and colleagues (under review) investigated differential patterns of opportunities to learn among secondary school students minoritized by race/ethnicity, ses, and lm status in urban schools. the researchers accounted for individual level covariates that included ses, gender, race-ethnicity, lm status, and each student’s self-reported level of english-language proficiency (elp). in addition, the following urban school context measures were integrated into the analysis: school type indicator (1 = public and 0 = catholic or other private), urban designation (0 = suburban or rural and 1 = urban), schoolses (percent of 12th grade students eligible for free or reduced lunch), percent lep (percent of lm students in 12th grade), percent college track (percent of 12th grade students in the college/academic track), and percent credentialed teachers (percent of 12th grade credentialed teachers). each of these measures was highlighted in the literature reviewed related to the mathematics achievement of ethnically and linguistically minoritized students in urban schools. variability in english-language proficiency our work has examined the mathematics achievement patterns of lm students in urban schools. we have primarily focused on two indicators of linguistically minoritized status: whether students are native english speakers (coded as non-ell or as linguistic minority status) and, when available, we have concurrently examined the effect of students’ self-reported level of elp. mosqueda (2010) as well as mosqueda and maldonado (2013) used els: 2002 data to examine the relationship between mathematics achievement and academic track placement or course-taking patterns of latinx students in 12th grade, respectively. in mosqueda (2010), the results showed that the effect of academic track placement and mathematics achievement differed as a function of both lm status and each lm student’s degree of elp, relative to non-lm students. the findings suggested that latinx ells with lower elp benefitted less from high-track placement than both latinx ells with high degrees of elp and their non-ell peers. mosqueda and maldonado (2013) found that mosqueda & maldonado editorial journal of urban mathematics education vol. 13, no. 2 35 lm status and elp were strongly related to mathematics course-taking, and students who took higher level mathematics courses benefitted more than those in lower level math courses. the findings from both studies suggest that the variability in the range of elp from a low to a high degree is a better indicator of the effects of language than a binary indicator of lm status. perhaps because lm students are included with students of emergent levels of proficiency and others within this same reference category, lm students may have elp that is near english-proficient and can potentially compare to the english proficiency of a native english speaker. there are multiple language background measures in nces datasets that can inform studies of mathematics in urban schools (mosqueda & maldonado, 2013). for example, lm status can be coded from the variable f1stlang, a survey question that asked students, “is english your native language (the first language you learned to speak when you were a child)?” (ingels et al., 2004). additionally, we have used students’ self-reported elp measure in both nels:88 and els:2002. in these datasets, four ordinal dimensions included how well students “understand spoken english,” “speak english,” “read english,” and “write english” and comprise elp. for each of these dimensions of english proficiency, students provided one of following ordinal responses: “very well,” “well,” “not well,” or “not at all” (ingels et al., 2004). in order to only account for the variation in elp of lm students, we have differentiated among the level of english proficiency of lm students, using the crossproduct of lmij * engprofij. the variable engprofij was a weighted composite resulting from principal components analysis of the four dimensions of elp in the survey. we also acknowledged a limitation of using such self-reported measures; however, we note that the reliability of similar measures has been established in large-scale studies of immigrant students (portes & rumbaut, 2001). because there are no measures of elp at a national level, this method is one of the most useful measures of elp available. although nels:88 and els:2002 datasets both include an elp indicator for lm students, hsls:09 does not include a measure of elp. maldonado and colleagues (under review) used psm to examine the relationship between latinx students’ english-language and immigrant background, teachers’ mathematical proficiency practices, and urban school contexts in hsls:09 data. researchers also used three restricted-use variables to determine students’ english-language background: nativelang (indicates the language the student first learned to speak), duallang (indicates if student is multilingual), and ebl (indicates if student is currently an emergent bilingual ell and enrolled in an ell program). findings from the quasi-experimental multi-level linear regression models showed that the relationship between teachers’ mathematical practices, urban school contexts, and latinxs’ mathematics achievement was not influenced by english-language background. without measures that account for the variability of students’ elp, researchers may neglect an important dimension in urban students’ mathematics achievement. mosqueda & maldonado editorial journal of urban mathematics education vol. 13, no. 2 36 immigrant generational status when nces studies like hsls:09 do not include elp measures for lm students, researchers may consider analyzing the interrelated nature of elp and immigrant generational status. we have concurrently analyzed the mediating effects of english-language background and immigrant generational status on students’ mathematics achievement using both els:2002 and hsls:09 data. when we analyzed els:2002 data (a study that includes elp variability for lms), the effect of immigrant generational status on mathematics achievement was not statistically significant (mosqueda & maldonado, 2013). when we analyzed hsls:09 data (a study that does not include elp variability for lms), the effect of immigrant generational status on mathematics achievement was statistically significant (maldonado et al., under review). although closer examinations of the degree of confoundedness between immigrant generational status and english-language background were important, our results imply that immigrant generational status information is also an important factor to consider when analyzing urban students’ mathematics achievement, particularly for nces studies that do not include elp information. maldonado and colleagues (under review) used three restricted-use hsls:09 variables to determine students’ immigrant background: scountry (country in which student was born), p1country (country in which parent1 was born), and p2country (country in which parent2 was born). using scountry, p1country, and p2country data, we created three dichotomous immigrant generational status variables: first, second, and third. first indicates students who were born in a country other than the united states and had at least one parent who was born in a country other than the united states. second indicates students who were born in the united states and had at least one parent who was born in a country other than the united states. third indicates cases wherein students and both parents were born in the united states. our coding scheme for students’ immigrant generational status was consistent in our analyses of both hsls:09 data and els:2002 data. another reason to consider the inclusion of immigrant generational status information in studies of urban school students’ mathematics achievement is the potential effect of unobserved values connected to interrelated student background characteristics of race-ethnicity, english-language proficiency, and immigrant generational status. prior research has found that academic achievement and immigrant aspirational optimism provide explanations for how the second generation outperforms first and third generation immigrant students (kao & tienda, 1995; portes & rumbaut, 2001). results from these studies show the importance of including individual-level variables of language background and immigrant generational background in analyses of achievement in urban schools. mosqueda & maldonado editorial journal of urban mathematics education vol. 13, no. 2 37 inadequately defined indicators there will be cases in which specific variables of interest to a study may not be adequately defined and so are unable to provide useful results to an analysis. for example, when analyzing els: 2002 data, we were interested in a variable that would gauge whether or not the presence of a mathematics teacher with specialized limited english proficiency training was associated with higher latinx ells’ mathematics test scores. our analysis revealed that the presence of teachers with specialized limited english proficiency training did not have a statistically significant effect on the assessment outcomes of latinx ells (mosqueda, 2011). however, this finding was not conclusive, because the variable for the language training of teachers was inadequately defined. in the els:2002 survey, the associated question specifically asked whether “teachers had at least 8 hours of specialized training over the last 3 years” in working with ells. for those teachers reporting having had such training, some may have attended a single-day (eight-hour) workshop for training on teaching ells over the last three-year period. alternatively, it could mean that a teacher earned a graduate degree in second-language instruction. given the wide range of this measure, more work is needed in future nces studies to create more reliable measures of specialized professional development for teachers. our more recent studies attempt to go beyond existing information about mathematics teachers’ certification, credentialing, years of experience, and participation in professional development. for example, we are now analyzing teachers’ specific pedagogical practices that are consistent with the strands of mathematical proficiency (maldonado et al., under review). illuminating inequity to improve urban mathematics achievement informed by otl, this article offers analytical guidance for education researchers interested in examining the structural conditions that influence mathematics learning and achievement in urban schools. specifically, we argue for researchers to consider how the intersectional interplay of students’ ses, race-ethnicity, and englishlanguage background and urban school contexts influences mathematics achievement. our purpose in this article was to provide researchers examples of prior analyses to inform methodological considerations when using nces datasets to analyze secondary students’ mathematics achievement in urban schools. our rationale for our recommendations is to emphasize the importance of capturing the effects of specific characteristics in urban school contexts that influence students’ mathematics achievement. second, we consider it important to inform researchers that studies that include school-level percentages of students minoritized by ses, race-ethnicity, and english-language background in statistical models but omit examinations of curricular resources or pedagogical practices may reproduce a mosqueda & maldonado editorial journal of urban mathematics education vol. 13, no. 2 38 deficit discourse of urban communities (capraro et al., 2013; moses & cobb, 2001). policy and practice inequities in urban schools are complex, and we recognize that students’ participation in advanced mathematics courses is only one variable in a multitude of social, cultural, and political factors that influence mathematics teaching and learning. we consider our research aligned with studies that promote equitable access and meaningful participation in mathematics learning as a civil right (moses & cobb, 2001; perry et al., 2003). we are hopeful that researchers interested in students’ mathematics achievement in urban schools will consider accessing and using nces studies to deepen our collective understanding beyond ies summary statistics featured in reports such as naep, timms, and tables from the condition of education. we believe that studies of large-scale data that are attentive to the methodological considerations of complex sampling, data clustering, and causal inference will contribute nuanced perspectives of promising policy and practice directions for improving urban students’ mathematics achievement. additionally, we believe this article details specific data sources and student, school, and community variables that are important to consider when analyzing achievement in urban schools. although we have exclusively analyzed standardized mathematics achievement scale scores, nces datasets offer multiple measures of students’ achievement, including grade point averages and transcripts with course-taking information. we recommend researchers consider the appropriateness of all outcome variables in relation to research questions as well as methodological fit and analytical design. for example, the advantage of using a standardized mathematics achievement scale score may offer readers useful analytical interpretations due to item response theory methods, but the disadvantage may be that test scores in one content area may not be as comprehensive of a students’ academic profile, relative to a composite grade point average across multiple content areas (e.g., english-language arts, science, history/social studies). moreover, it is important to consider the significance of language in mathematics teaching and learning, specifically for ells (celedón-pattichis, 2008; martiniello, 2008; turner & celedón-pattichis, 2011). considering that mathematics assessments are primarily administered in english in the united states, researchers are responsible for addressing the confoundedness of assessments simultaneously evaluating students’ comprehension of mathematical concepts and skills and students’ competence in english-language fluency and literacy (american education research association et al., 2014). references american educational research association, american psychological association, national council on measurement in education, & joint committee on standards for educational and psychological testing 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(re)defining urban education: a conceptual review and empirical exploration of the definition of urban education. educational researcher, 49(2), 90– 100. https://doi.org/10.3102%2f0013189x20902822 whitehurst, g. j. (2003). the institute of education sciences: new wine, new bottles (ed478983). eric. https://files.eric.ed.gov/fulltext/ed478983.pdf copyright: © 2020 mosqueda & maldonado. this is an open access article distributed under the terms of a creative commons attribution-noncommercial-sharealike 4.0 international license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. journal of urban mathematics education july 2015, vol. 8, no. 1, pp. 10–13 ©jume. http://education.gsu.edu/jume david w. stinson is an associate professor of mathematics education in the department of middle and secondary education in the college of education and human development, at georgia state university, p.o. box 3978, atlanta, ga, 30303; e-mail: dstinson@gsu.edu. his research interests include exploring socio-cultural, -historical, and -political aspects of mathematics and mathematics teaching and learning from a critical postmodern theoretical (and methodological) perspective. he is a co-founder and current editor-in-chief of the journal of urban mathematics education. editorial reviewing for jume: advancing the field of urban mathematics education david w. stinson georgia state university eer review—i think all of us in the academy have a love–hate relationship with this facet of our multifaceted academic lives. most of us, specifically those who have earned, sought, or will seek tenure, have experienced a full range of emotions with respect to the peer-review process: joy, sadness, pride, disappointment, confusion, success, defeat, and so on. i certainly have experienced this smorgasbord, if you will, of emotions throughout my 15 years in the academy (which includes my 4 years as a doctoral student). during the past decade and a half, i have been a receiver, writer, requester, and assessor of peer review. within each of these roles, although i might not always want to admit it, i have learned something invaluable about the process: when written with both evaluative and educative purposes in mind, peer review, more times than not, just makes us and, perhaps more importantly, our research smarter.1 silver (2003), in his journal for research in mathematics education (jrme) editorial “reflections on reviews and reviewers,” made a distinction between the evaluative and educative purposes of reviews. the evaluative purpose, as discussed by silver, is rather self-evident: researchers with particular expertise identify and discuss what they see as the strengths and weaknesses of a submitted manuscript and make a recommendation regarding its publication (i.e., they evaluate the publication worthiness of a manuscript). the educative purpose is not always as self-evident: researchers with particular expertise provide comments and suggestions about theoretical and methodological choices of a submitted manuscript and often attempt to extend the findings and implications. they do all of this with the intention of assisting the author in revising the manuscript or in making decisions about future projects (i.e., they educate the author about other possibilities). 1 one should not infer from this statement that i accept and participate in the peer-review process—a more than 1,000-year-old idea (spier, 2002) argued to be “a flawed process at the heart of science” (r. smith, 2006)—without critique. the extensive scholarship of foucault on discourse and discursive practices, surveillance and discipline, and power/knowledge critically interrogates the entire enterprise called science, including (directly and indirectly) the peer-review process (see, e.g., 1969/1972, 1966/1994, 1975/1995). p http://education.gsu.edu/jume mailto:dstinson@gsu.edu stinson editorial journal of urban mathematics education vol. 8, no. 1 11 pushing further into this distinction between evaluative and educative purposes, smith (2004) suggested that reviewers approach the task as one of mentoring authors by writing “reviews that teach” (p. 292). such reviews, he claimed, “clearly articulate the main reasons why authors’ current arguments are inadequate and also present strategies they might pursue in correcting those deficiencies” (p. 293, emphasis in original). borrowing three practices from good classroom teaching, smith offered a frame of sorts of how to write reviews that teach. the reviewer needs to (a) “understand what the author thinks and, to the greatest extent possible, why he or she thinks it” (p. 293); (b) identify not only particular errors in the manuscript (or project), but also the general research misconceptions that are evident; and (c) model clear and sound arguments throughout the review, especially those that conclude with an unfavorable recommendation. writing reviews that teach evidently are more time consuming. nonetheless, smith’s proposal for reviews that teach was grounded in his conviction that (a) it is what [the field of mathematics education] produces that counts, and (b) the field does not principally advance when we make individual contributions to the literature. rather, it advances when the average quality of published research rises. reviews that teach are an important influence on average quality because they supplement, in a very costeffective way, the professional education of researchers. (p. 294) in repositioning smith’s (2004) argument in the context of urban mathematics education, we have: it is what the field of urban mathematics education produces that counts. the field does not principally advance with individual contributions to the literature but rather when the average quality of published research rises. reviews that teach are an important influence on average quality because they supplement, in a very cost-effective way, the professional education of urban researchers. but before discussing what it might mean to advance the field of urban mathematics education, we must first ask, “is it even a field?” well, let us see. it has it own urban education special issue (tate, 1996). it has its own journal of urban mathematics education (matthews, 2008). it has its own handbook of urban education chapter (martin & larnell, 2013). it has its own encyclopedia of mathematics education entry (stinson, 2014). it has its own (developing) theoretical framework (larnell & bullock, 2015). and google and google scholar searches of the phrase return 5,610 and 365 hits, respectively. so, let us dispense with arguing whether or not urban mathematics education constitutes a unique disciplinary field, and just agree that it does.2 2 let us also dispense here with the challenges of “defining” or describing urban mathematics education. the complexities of that task have been critically discussed at length in martin and larnell (2013). stinson editorial journal of urban mathematics education vol. 8, no. 1 12 so then, given that urban mathematics education is its own unique disciplinary field, how might it be advanced and who is responsible for that advancement? over the past eight years, one group that has been instrumental in advancing the field is comprised of the scholars, researchers, practitioners, and graduate students who have offered their time and expertise to review manuscripts for jume. without authors and reviewers the journal would not exist. there have been approximately 230 double-blind reviews written by over 100 unique reviewers.3 we celebrate those reviewers who have written reviews that teach. these reviewers, who understand that reviewing is indeed scholarly work (heid & zbiek, 2009), have assisted authors in getting smarter about their work and, in turn, they have made the research and scholarship available in the disciplinary field, well, just smarter. recently, i completed a 3-year term as a member of the jrme editorial panel. during my time on the panel, i learned much about the historical beginnings and inner workings of what is certainly the most established and arguably the most respected journal in mathematics education. 4 i also learned much from the extraordinary leadership of cynthia langrall, jrme editor during my tenure on the panel. through the next several months, i plan to use my newly acquired insights about knowledge production and dissemination and work with members of the jume editorial team to think and rethink the inner workings of jume, including the peerreview process. we will keep readers, reviewers, and authors up to date as changes are implemented that will hopefully continue to advance the field of urban mathematics education in more ethically and just ways. as jume matures, going through growing pains along the way, we hope to forever get closer to our mission: to foster a transformative global academic space in mathematics that embraces critical research, emancipatory pedagogy, and scholarship of engagement in urban communities. in the mean time, the jume editorial team invites you to become part of a unique group that is advancing the field through writing reviews that teach. if you have not reviewed for jume in the past, “make this your year to review” (langrall, 2015, p. 2)—you can sign up here. and if you have reviewed in the past, please 3every other year the jume editorial team acknowledges the significant and time consuming contributions of our reviewers, please see january 2008–december 2009, january 2010–december 2011, and january 2012–december 2013. 4 one should not infer from this statement that i uncritically position jrme as the “gold standard” in mathematics education knowledge production and dissemination. to do so, would be wrong. the genesis of jume was in reaction to the very fact that the mainstream or, more aptly, the “white-stream” journals (gutiérrez, 2011) of mathematics education research are all but void of the kind of research that is published in jume (see matthews, 2008; stinson, 2010). http://ed-osprey.gsu.edu/ojs/index.php/jume/user/register http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/55/36 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/157/98 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/225/146 stinson editorial journal of urban mathematics education vol. 8, no. 1 13 make sure that your profile information, including your areas of interest, is up to date and complete. —we look forward to receiving your reviews that teach! references foucault, m. 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(2015). toward a geo-spatial framework for urban mathematics education scholarship. in s. mukhopadhyay & b. greer (eds.), proceedings of the eighth international mathematics education and society conference (vol. 3, pp. 712–722). portland, or: ooligan press. martin, d. b., & larnell, g. (2013). urban mathematics education. in r. milner & k. lomotey (eds.), handbook of urban education (pp. 373–393). london, united kingdom: routledge. matthews, l. e. (2008). illuminating urban excellence: a movement of change within mathematics education [editorial]. journal of urban mathematics education, 1(1), 1–4. retrieved from http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/20/9 silver, e. a. (2003). reflections on reviews and reviewers [editorial]. journal for research in mathematics education, 34(5), 370–372. spier, r. (2002). the history of the peer-review process. trends in biotechnology, 20(8), 357– 358. smith, j. p., iii. (2004). reviews that teach. journal for research in mathematics education, 35(4), 292–296. smith, r. (2006). peer review: a flawed process at the heart of science and journals. journal of the royal society of medicine, 99(4), 178–182. stinson, d. w. (2010). how is it that one particular statement appeared rather than another?: opening a different space for different statements about urban mathematics education [editorial]. journal of urban mathematics education, 3(2), 1–11. retrieved from http://edosprey.gsu.edu/ojs/index.php/jume/article/view/116/69 stinson, d. w. (2014). urban mathematics education. in s. lerman (ed.), encyclopedia of mathematics education (pp. 631–632). dordrecht, the netherlands: springer. tate, w. f. (ed.) (1996). urban schools and mathematics reform: implementing new standards [special issue], urban education, 30(4). http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/20/9 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/116/69 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/116/69 microsoft word reflecting back to forge the path forward (proof 1).docx journal of urban mathematics education may 2022, vol. 15, no. 1, pp. 1–8 ©jume. https://journals.tdl.org/jume journal of urban mathematics education vol. 15, no. 1 1 editorial reflecting back to forge the path forward editorial team robert m. capraro melva grant jamaal young tarcia hubert michael s. rugh mary margaret capraro marlon james jemimah young alesia mickle moldavan miriam sanders chance lewis eduardo mosqueda ali bicer susan ophelia cannon jonas l. chang hank you to all our reviewers, editorial board members, authors, and those who chose the journal of urban mathematics education (jume) as their outlet of choice. in collaboration with the editorial team, we are releasing critical data concerning the performance of our journal for the 2021 calendar year. as such, the editorial team seeks to uphold our goal of transparency through analysis of both our shortcomings and achievements. in the annual state of jume report, we provide a discussion of significant issues to the health and success of the journal, such as review and acceptance rate, time to publication, author demographics, and lessons learned along the way. we believe that providing transparency to our readers will support our greater goal and mission of fostering a transformative global academic space for critical research and scholarship in urban mathematics. the editorial team has worked to bring timely issues to press as quickly as possible without jeopardizing the review process. however, the review process has been tough at times. given the very difficult year with the added challenges of the ongoing global pandemic, whose name shall never be spoken, reviewers exceeded expectations. our typical time to send manuscripts to reviewers was two work days, and our average for days to decision was 31. unfortunately, some of the variation around those numbers has been less than laudable. the range for time to send manuscripts to reviewers was the same day to seven days, and the range for days to decision was three days to 124 days. we could never have imagined the difficulties we would face t editorial team editorial journal of urban mathematics education vol. 15, no. 1 2 in moving manuscripts quickly through the process. although these timeliness indicators are certainly not the best case, they are clear benchmarks for moving forward. when reflecting back from last year, we have improved on nearly all the metrics, with the exception of the extended range on the time to decision criterion. there were three areas through which we worked to improve time to reviewers and time to decision. first, we made efforts to expand our reviewer pool. we broadened and deepened our reviewer pool by adding 51 new reviewer accounts in the 2021 calendar year. one goal held by our team is to ensure every manuscript receives an excellent, positive, and productive review. in response to that goal, we started a mentoring program for reviewers. our editors still tended to rely on very specific reviewers who excelled at providing a caring and respectful review, however. they depended on these reviewers even when their recommendation was to decline the submission. we are pleased that through our mentoring program, we provided examples of helpful reviews to members of our community through two zoom mentoring meetings for current and potential reviewers. this one initiative led to an increase in reviewers and better, more caring reviews. this in turn enabled each member of the jume team to feel accountable and empowered to make their own informed decisions and to move quickly and decisively for every submission on which they are the action editor. this is evidenced in the reduced average time to decision. we are building a track record of collaboration and trust, and this benefits our community in reducing all the time metrics. submissions have risen, and we now have a publication backlog. once a manuscript is accepted, we are currently running about six months to publication. our goal for 2023 is that jume will no longer be constrained by using traditional publishing norms and move to a publish-when-ready model. two key features of jume’s intended implementation of that model are that there will still be two editorials published per year and opportunities for guest editors to lead special issues that will be published as stand-alone issues. however, all other manuscripts will be “published when ready” without time constraints or delays. we believe that this change will positively impact many of jume’s metrics. marketing jume articles and authors continues to be of paramount importance. we have taken additional steps to ensure jume authors receive broad recognition and marketing that helps their work be found and cited. we have added a new weekly read feature (see figure 1) and a most-read metric to our site. figure 1 shows that the referenced article has been read or downloaded 180 times, and the graph indicates the frequency of access by day for the date range provided. we have also added orcid as a new service to the journal. orcid is receiving broad acceptance, and its adoption is becoming more commonplace among program officers as part of their due diligence. the power of orcid allows reviewers to receive credit for completing reviews and provides easier indexing of author contributions. if you do not already have your orcid number, please consider all the benefits having one affords. editorial team editorial journal of urban mathematics education vol. 15, no. 1 3 figure 1. sample graph of readership of an article between december 2021 and january 2022 our acceptance rate for 2021 was ~19% (see figure 2). we completed our first year of being a scopus-rated journal, and we present our first metrics, nearly one year ahead of schedule (see figure 3). although our sjr is modest, we are mighty and this score will increase over time. we believe that as more readers recognize the prominence of the emerging scholars and the quality of the work being published in jume that more researchers will cite the work and the work will be foundational. we will continue to carefully scrutinize how the journal fits in the urban mathematics landscape, and we encourage all readers to be sure to appropriately cite jume when and where possible. figure 2. acceptance rate by year editorial team editorial journal of urban mathematics education vol. 15, no. 1 4 figure 3. scopus rating for jume’s first year of being indexed as a result of participating with a group of editors committed to increasing opportunities and reforming journal practices, the jume team undertook a reflective look at our historical and current methods last year. this required us to gather preliminary data concerning the demographics of our authors, and in doing so we used the word “appear” to carefully categorize authors by our interpretation or familiarity with them. however, this is by no means how any one author may identify and was a precarious practice, but it afforded us the opportunity to begin following our plan to collect and publish author demographic information in our end-of-year review. this year, we have authors’ self-reports of identification. over the past year, the jume team piloted voluntary submission of author demographic information. if authors were published more than one time in the journal, they were asked to complete one survey response per publication. emails were sent to authors published in jume with invitations to complete a qualtrics survey. in this survey, we requested information on race, ethnicity, and gender as well as rank or graduate student status and institution/employer for when the article was published. “i prefer not to answer” was an option for nearly every item to respect the authors’ comfort level in disclosing the requested information. for authors published from 2008 to 2019, we collected 38 survey responses out of the 166 requests sent (response rate of 23%). additionally, out of the 21 email requests sent to authors published from 2020 to 2021, 11 survey responses were collected (response rate of 52%). the responses provide insight into the characteristics of research scholars published in jume. moving forward, we hope to increase the survey response rate to provide a more accurate representation of the published authors. to accomplish our goals of becoming a disruptor in mathematics education and reducing bias in publication, we will provide transparency as we reflect and refine our practices. additionally, we will discuss survey responses in relation to the authors’ levels of power and privilege at the time of publication in jume, as this may be one of many reasons why authors have decided not to respond to the survey (e.g., lack of perceived power, threats of identification, and concerns about how data will be used). for this year, however, we only want to report the information without editorial team editorial journal of urban mathematics education vol. 15, no. 1 5 making broad conclusions about what it might mean and rather allow the reader to simply absorb the numbers. we encourage readers nonetheless to contemplate how they can engage with the data and in turn how their contributions to the model can reinvigorate jume. table 1 author demographics comparison note. *the response rate was 23% for 2008–2019 and 53% for 2020–2021. categories with too few responses were not recorded to retain author anonymity. we are concerned about the low response rate of the survey, because any conclusions based on these data may not adequately represent jume’s authors in general. what we know is that the survey results (see table 1) provide insight into a limited subset of authors published in jume, describing those authors’ authentic identities. historically (2008–2019), 16% of responding authors identify as black and 8% identify as latin or hispanic. the representation of white authors (53%) may be due to a greater sense of security in answering the question(s). the higher response rate for authors published between 2020–2021 could be attributed to the more recent time of publication and engagement with the journal. nonetheless, there is still room for improvement in terms of response rate. again, the survey results only reflect a subset of the authors published in jume. approximately 36% of the authors identify as asian, while approximately 45% identify as white. moreover, categories with too few responses were not reported to preserve author anonymity because of the identifiability of the data due to the small sample size. it is difficult to think how the results may be different if we had received 100% complete data. despite the difficulties we have faced in acquiring a complete data set, we will persist in attempting to secure as much information as possible. we will make that information readily available and ensure that readers feel secure in knowing that the jume team is cognizant of the struggles of our diverse community, including threats, description 2020–21 2008–2019 black 16%* latin or hispanic 8%* asian 36%* 8%* multi-racial 11%* white 45%* 53%* editorial team editorial journal of urban mathematics education vol. 15, no. 1 6 acts of prejudice, and discrimination. the editorial team views our practice of transparency as a strength of the journal and our editorial team’s commitment and dedication to encouraging a transformative global space in mathematics teaching, mathematics learning, and mathematics culture. the qualtrics survey also contained an item for information regarding the authors’ gender identities. the choices available were genderqueer, man, transgender, trans man, trans woman, woman, not listed (with an option to add a descriptor), and “i prefer not to answer.” respondents who selected “not listed” did not add a descriptor. additional descriptors would be useful for us to make more comprehensive subsequent surveys. according to the data from authors published between 2008– 2019, authors mostly identified as women, and 13% either chose not to answer or said their gender identity was not listed. similar to the authors from 2008–2019, almost half of the authors from 2020–2021 reported identifying as a woman (see figures 4 and 5). however, a larger percentage of those authors chose not to disclose information regarding gender identity. the percentage of authors identifying as a man decreased by over ten percent from the 2008–2019 data to the 2020–2021 data. figure 4. author gender identity 2008–2019 figure 5. author gender identity 2020–2021 editorial team editorial journal of urban mathematics education vol. 15, no. 1 7 to provide a more holistic representation of the professional identities of authors, we included a survey item regarding institution/employer for when the author’s article was published in jume. using the survey responses and the carnegie classification of institutions of higher education, we examined authors’ university employment affiliations (see table 2). we found a higher concentration of authors publishing from r1 institutions than from r2 institutions from 2008–2019, whereas, per author responses, there was equal representation of authors from both r1 and r2 institutions who published from 2020–2021. additionally, published jume authors represented institutions outside of the united states as well as historically black colleges and universities and public school districts. although the representation of such affiliations is marginal, authors from these institutions provide critical perspectives and scholarship. table 2 author professional identities note. *the response rate was 23% for 2008–2019 and 53% for 2020–2021. additionally, we solicited information regarding authors’ position types and funding when their article was published in jume. over half of the authors published from 2008–2019 (63%) held tenure track positions, with the majority classification of assistant professor. similarly, 54% of the authors published in 2020–2021 held tenure-track positions. furthermore, in both samples, less than 20% of the authors were graduate students. moreover, less than a quarter of authors in both samples reported receiving national or international funding as a pi or co-pi. this supports our description 2020–21 2008–2019 affiliated with r1 institution 27%* 47%* affiliated with r2 institution 27%* 18%* affiliated with public institution 55%* 79% * affiliated with hbcu 3%* affiliated with institutions outside of the u.s. 9%* 8%* employed by public school districts 9%* 5%* editorial team editorial journal of urban mathematics education vol. 15, no. 1 8 belief that jume is a viable outlet for junior faculty members to break ground on their research agenda as well as for senior scholars to make high-quality, meaningful contributions to the field. aggie stem at texas a&m university has housed jume since 2019. over the past three years, under the leadership of dr. robert m. capraro, the jume team has worked to meet our goals to establish a permanent home for the journal in tdl, to expand the editorial team, to improve metrics, and to obtain a scopus ranking. we posted a call to fill a jume editor position eight months ago, and we were extremely pleased to receive three applications. each of the applicants received guidance on addressing missing or unaddressed points in their application. finally, we are pleased to announce that dr. jamaal young has been elected as editor-in-chief for jume with an appointment of 2023–2027. the current editorial team is working with dr. jamaal young on a transition plan to a new editorial team, and we are excited see the journal continue to grow and serve as a space for exemplary scholarship under his leadership. copyright: © 2022 capraro, capraro, lewis, grant, james, mosqueda, young, young, bicer, hubert, moldavan, cannon, rugh, sanders, & chang. this is an open access article distributed under the terms of a creative commons attribution-noncommercial-sharealike 4.0 international license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. microsoft word 4 final valero b vol 11 no 1&2.doc journal of urban mathematics education december 2018, vol. 11, no. 1&2, pp. 103–117 ©jume. http://education.gsu.edu/jume paola valero is professor in the department of mathematics and science education, faculty of science, stockholm university, svante arrheniusväg 20, se-106 91 stockholm, sweden; email: paola.valero@mnd.su.se. her research explores the school mathematics curriculum as a field where power relations produce certain subjectivities and generate inclusion/exclusion of different types of students. human capitals: school mathematics and the making of the homus oeconomicus paola valero stockholm university, sweden he title of my contribution to this special 10th year anniversary issue contains the terms human capitals—and there is not a spelling mistake—school mathematics, making, and homus oeconomicus. apparently, the terms do not say much about mathematics and its learning and teaching, but more about the economy. i argue here, however, that it is of great importance for mathematics educators to understand the conditions in which the practices of teaching and learning, as well as the contents of school mathematics, acquire a particular meaning while directing new generations into becoming certain types of people. now more than ever before, the connection between mathematics education and the functioning of free market, financial capitalist economy is explicit and has a direct effect of power on the types of children that we produce in education, as well as in processes of social and economic inclusion and exclusion. therefore, research in mathematics education is no longer a matter of studying the best way to steer teachers, learners, and topics to achieve the best possible mathematical results, as if such orchestration were neutral and detached from the overall political and economic configuration of current societies. what is it, then? from critical mathematics education… in mathematics education research, the political perspectives have been in the air for a while (valero, 2018b). there has been a group of researchers from different countries, adopting diverse but connected theoretical perspectives, working on the political dimensions of mathematics education (e.g., gutiérrez, 2013; jablonka, wagner, & walshaw, 2013; valero, 2004). among these, critical mathematics education is recognized as a way to engage with questions such as what is mathematics in relation to society, what does mathematics do as part of the school curriculum, and what are the potentials of mathematics education to produce or challenge inequalities in society and among students. a number of researchers, ole skovsmose, eric gutstein, and marilyn frankenstein, among others, are key references in this area. when i started my research path many years ago in colombia and when i moved to denmark to do my ph.d. under the mentoring of ole skovsmose, my concerns and research aligned with this trend. indeed, the paper that was published in the first issue of jume was a piece t valero making human capitals journal of urban mathematics education vol. 11, no. 1&2 104 co-authored among helle alrø, pedro paulo scandiuzzi, ole skovsmose, and me. we examined the stories of a group of brazilian students about their experience of mathematics education in secondary school using the notion of foreground (skovsmose, 1994). this notion refers to the “interpretations of life-opportunities in relation to what appears to be acceptable and available within the given sociopolitical context” (skovsmose, scandiuzzi, valero, & alrø, 2008, p. 38). foreground, we thought, proposed a shift of interpretation of the students’ intentions to engage with mathematics from paying attention to their past or background, toward focusing on their future possibilities or foreground. in other words, the notion of foreground helped in seeing that engagement in mathematics did not simply depend on what students “lack” given their family or socio-economic background which, more often than not for students in disadvantage, is conceived as determining their failure. students’ intentionality may well be constructed with respect to what students see as future possibilities for themselves, within the conditions in which they live. furthermore, the experience of students living in disadvantaged conditions invited us to propose the notion of borderland position, “a relational space where individuals meet their social environment and come to terms with the multiple choices that cultural and economic diversity make available to them” (p. 35). this notion helped us with thinking that the experience of living in disadvantage always makes visible in tangible ways the life opportunities of others in privilege and, at the same time, those opportunities that may or may not belong to oneself. a borderland position “makes evident the harshness of social division, stratification, and stigmatization” (p. 56). in this constant confrontation with what is or is not possible, students’ intentions to engage in and with mathematics in school may blur. the meaning of mathematics and its function in making a transition toward a different form of life may also blur. the dreams for a future in a borderland position are fragile. as i read it now, this piece is representative of critical mathematics education—at least as proposed by skovsmose in his multiple writings. first, there is the concern for how mathematics education entangles with issues of democracy and inequality. second, there is the idea that mathematics education is critical because it can both empower or oppress. it can go in “both directions” (skovsmose & valero, 2001). third, there is an assumption on the individual as a social being with intention who can make choices of engaging (or not) in the activity of learning mathematics. fourth, there is the assumption that there is hope and that seeing reasons to decide to engage in mathematics learning may indeed lead to a brighter future. this assumption is visible, for example, in the discussion on the obscurity of mathematics (skovsmose et al., 2008, pp. 53–54): education is recognized to be important to ensure a change in life opportunities; however, the tradition of mathematics education that students in a borderland position experience does not make it possible for them to recognize how the content of mathematics contributes to that. such formulation has the implicit idea that connecting the meaning of the mathematical content valero making human capitals journal of urban mathematics education vol. 11, no. 1&2 105 with possibilities in the future is important in moving toward the opportunities that adequate mathematical learning opens for people in disadvantaged positions. in other words, there is the underlying idea that learning mathematics represents an empowerment for the future. in the years to follow, i started problematizing such ideas as my investigations of foreground turned into investigations of students’ identity (e.g., ander-sson, valero, & meaney, 2015; stentoft & valero, 2009). a discursive approach to identity opened up the possibility of developing sharper analytical lenses to understand the relation between the individual and the socio-cultural-political context, which during those years became one of the central concerns for mathematics education researchers who were building social, cultural, and political frameworks in mathematics education (e.g., radford, 2008; sfard & prusak, 2006). moving toward discourse also offered better analytical tools to think about how power unfolds in and through the practices of mathematics education. it was also a way of starting a problematization that would move critical mathematics education from critical theory and marxist views of power, toward the terrain of critical post-modern theories (stentoft & valero, 2010). from this point of view, the assumptions present in skovsmose and colleagues (2008) were revisited. later in this contribution, i will return to this point. …to the cultural politics of mathematics education in the last 10 years, i have moved away from critical mathematics education and started exploring what i call the cultural politics of mathematics education, a term that navigates in some recent research (e.g., craig, 2018; diaz, 2017) with a particular interest in examining the wide network of mathematics education practices and its cultural and political significance for the constitution of notions, in time and space, of the modern subject. as i see it, the study of the cultural politics of mathematics education is a displacement—or even better, an expansion—of mathematics education research from the demand of the field to study the mathematical specificity of relationships of teaching and learning, toward the terrain of curriculum studies (e.g., appelbaum & stathopoulou, 2016) with the intention of tracing the specific role that mathematics education has played in the making of modern and contemporary culture and society. what does it mean to say that one researches or understands mathematics education in relation to cultural politics? think of picasso’s famous painting guernica. picasso was painting at the time of the spanish civil war, before the outbreak of the second world war. he painted a scene of the tragic attack by a german air bomber on civilian population in a little town in northern spain. he was irreverent in his painting. he broke with the ideas of how the world should be represented. the first time one sees picasso’s guernica, one thinks: what is that? is this a per valero making human capitals journal of urban mathematics education vol. 11, no. 1&2 106 son? is this a horse? or are they mixed together? what is in and what is out? the spectator has to think, observe, and react because there is no way of simply watching the painting without having to strange what has become familiar to the spectator. picasso himself said about his work: you have to wake people up. to revolutionize their way of identifying things. you’ve got to create images they won’t accept. make them foam at the mouth. force them to understand that they’re living in a pretty queer world. a world that’s not reassuring. a world that’s not what they think it is. (as cited in malraux, 1976, p. 110) picasso made it possible for people to see the world in different ways. but this was not easy because at the time he—and other modernist painters—started developing his techniques, he struggled with norms and rules of what it meant to do “proper” painting. my point is that as part of culture, we are always in struggle and negotiation of values, ways of seeing and ways of understanding the world. and all of us do this. we do that in a classroom teaching mathematics; in conversation day after day the explicit and implicit definitions of that is valid in a particular classroom emerge. but also, when we are in society, we negotiate all the time what is right or wrong and what we take as true, as much as we negotiate what we take as mathematics education and why. and such struggle and negotiation are political because the privileging of certain values and ways of seeing the world—mathematics and mathematics education included—over others is connected to how power creates orders and classifications that, through knowledge, make us who we are and how we engage in the world. the last decade has witnessed a growing interest for how mathematics education is related to equity, access, and inclusion. numerous journal issues, special volumes, and overviews of literature have compiled research produced around the world with a socio-political interest (e.g., forgasz & rivera, 2012; valero & meaney, 2014). even conservative agendas about what should count as mathematics in schools are launched and promoted as the solution to the issues of differential performance by different types of students. nowadays even the necessity of steering education to finetune results in international comparative studies of mathematical achievement are justified as the means to promote equity. it has become natural for us to think that mathematics is crucial for equity, for the well-being of nations, communities, and of the individual. the statement that learning mathematics is empowerment for the future is a naturalized truth. everybody repeats it; nobody seems to question it. and yet, what is known is that the more society desires this to be the case, the more differentiation, classification, and segregation happens with and through achievement in mathematics. isn’t this weird? isn’t this strange? don’t we feel like cutting the idea in pieces to reconfigure it in a different fashion in an attempt to make people “foam at their mouth,” and open the uncomfortable and not reassuring task of inviting us to see that the world is not what we think it is—as pi valero making human capitals journal of urban mathematics education vol. 11, no. 1&2 107 casso once said? mathematics education has increasingly become a very important part of the school curriculum, and it plays a very important role in current societies—and we all mathematics educators would like to think that is the case. despite its recognized importance, there is a constant struggle to define what are valid and what are illegitimate meanings and forms of mathematics education, at any level of the work that we do, in the classroom, with our colleagues, with colleagues from other subject areas, and also in a variety of sites in society. neither the practices nor the meanings of mathematics education are defined once and for all. power struggles about what counts as valid mathematics are being played out constantly, and many times the cultural significance of mathematics education and its meanings are not exclusively dependent on teachers and classrooms. therefore, i have taken a broader perspective, to research within the broader network of practices of mathematics education, how in western societies through history, there have been negotiations and struggles that constitute the meaning, significance, and objects/subjects involved in the practices of mathematics education. the cultural politics of mathematics education as an area of research is an attempt to understand the constitution of the practices of mathematics education as part of a larger cultural space where the meanings of mathematics in relation to education are constantly negotiated. when working in that landscape there are sources of inspiration. i feel particularly inspired by the work of philosophers who have tried to think cultural politics for different cultural objects and practices in modern societies. i am very fond of the work of michel foucault among others. the intellectual exercise consists in studying his work to “think with him” in mathematics education. he almost never mentioned the word mathematics and certainly not mathematics education. foucault was not interested in the problems that i am interested in. but i am interested in some of his problems, namely how people become subjects as an effect of power, and how knowledge in modern times functions as part of the technologies of governing and power. what i do as a researcher is play with some of the notions and forms of analyzing that foucault proposed to rethink how mathematics education, as part of the school curriculum in particular time and space, frames the directions in which one becomes a kind of subject in relation to the knowledge of mathematics as it is configured in pedagogical practices within the school as a social institution. the work of michel foucault has prolifically nurtured critical studies of schooling and the curriculum (ball, 2017). in mathematics education it has been appropriated to think about the constitution of learners and teachers as subjects within the web of power of the institutional discourses of mathematics education practices (e.g., stinson, 2006; walshaw, 2014). such studies have provided insightful interpretations of students’ and teachers’ identity formation in terms of their process of subjectivity in the practices of mathematics education. although important in a quest for understanding the socio-political constitution of mathematics valero making human capitals journal of urban mathematics education vol. 11, no. 1&2 108 education practices, many of these earlier studies still focus on some of the traditionally defined actors and elements of the classic didactic triad of mathematics education research. more recent studies inspired by foucault deploy analytical strategies to explore the functioning of mathematics education discourses and their effects of truth in generating ways of thinking about mathematics education and its participants (e.g., llewellyn, 2018). mathematics education drawing on foucauldian tools, provide a critical stance toward how power, knowledge, and subjectivity connect in the multiple practices of mathematics education (kollosche, 2016). the research questions in studies that displace the focus of teaching and learning build on the assumption that mathematics and mathematics education are political because the historical constitution of the knowledge and associated practices both have emerged and make part of the classifications and organizations that regulate social life and, within them, notions of who people are and should be. this means that as much as mathematics education is thought to empower and make possible a better future, at the same time, the same practices create differences among people which are meant to classify, rank, and thus include and exclude. it is for this reason that different types of interrogation push the limits of mathematics education research to locate its understanding and study in the realm of the cultural and political history of schooling and education. in other words, researching the cultural politics of mathematics education allows revealing the way in which mathematics education generates concepts, distinctions, and categories that regulate the possibilities of thinking and being in/with mathematics as a privileged area. through these analytical moves it becomes evident how mathematics education and power are connected in the school curriculum. in the following section, after having clarified the types of analytic stances that i take, i go back to my initial concern: how are current forms of mathematics education creating the classifications and orderings that govern the fabrication of the homus oeconomicus? problematizing the desire of more mathematics achievement in the 1980s, there started to appear in mathematics education research clear statements about mathematics being cultural and mathematics education being political. in national policy documents regulating school mathematics the association between mathematics education and democracy became a strong way of justifying good reasons for expanding the desire of access to learn mathematics (skovsmose & valero, 2008). it has also been argued that access to learning mathematics is a “human right in itself” given that mathematics is seen as a cultural product of purposeful human activity (vithal & volmink, 2005, p. 3). nowadays such types of statements have become frequent, and the idea that it is desirable that all should be offered the opportunity at one point in life to learn mathematics because “studying mathematics will bring associated benefits—personal, social, and political—for all” valero making human capitals journal of urban mathematics education vol. 11, no. 1&2 109 (clements, keitel, bishop, kilpatrick, & leung, 2013, p. 8). more recently, policies in various countries tend toward over privileging mathematics and science at the expense of other school subjects and considering the latter to be irrelevant to current social and productive needs. just think of targeted, large investments in education in all countries to “strengthen mathematics”—such as the swedish matematiklyftet [mathematics lift] since 2014, and u.s. president obama’s investing in america’s future: preparing students with stem skills in 2016. even the controversial open move of the japanese minister of education in 2015 to cut humanities and social sciences at universities to open space to “useful education” for the future, or the danish downsizing of humanity university studies for privileging “education that leads to secure employment” in 2015 resonates with the logic that mathematics and science are core subjects to protect and promote. they are fundamental for global economic competitiveness, and this has become an important justification for mathematics education practices, for reforms, for research, for teachers and even for politicians to increase the number of mathematics lessons per week in many countries. these statements of course please mathematics educators and are part of the perceived status and relevance of our work. however, a closer, critical examination to their emergence as part of the cultural politics of schooling and the constitution of its social epistemologies invites to their problematization. that is, to de-familiarize them by asking how they became truths that are part of current commonsense notions of people participating in mathematics education. the question then is not why those statements emerge, as if one could find the chain of causes in history that result in these statements. rather the issue is to delve into how the statements were articulated and became possible, plausible formulations about the role of school mathematics in society. answering such questions requires a type of genealogical study, a “history of the present” (foucault, 1975/1991, 31) that traces the lines of how becoming mathematically competent is effected in multiple games and through the workings of technologies that render the subjects governable. such a study is a huge enterprise for the scope of this essay. some work already published has advanced in that direction, providing a grounding in time and space of how such statements have formed. diaz (2017) investigates the way in which the emphasis on the adequate teaching and learning of the equal sign in current reforms in mathematics in the united states link with broader meanings in society about equality. mathematics education reform and research that supports reform build on the assumption that “with knowledge of the equal sign, children will have a better understanding of equality, greater access to ‘higher’ levels of mathematics, more academic opportunities, and an overall increase in economic and social standing” (p. 36). notwithstanding the apparent goodness of this stated purpose, the pedagogies of mathematics in the curriculum operate classifications and differentiations of those children who learned the right equality and those who fail to do so. thus, discourses around valero making human capitals journal of urban mathematics education vol. 11, no. 1&2 110 the neutrality and goodness of learning “the equal sign” are neither neutral nor natural but render children objects of the calculations of power. kollosche (2014) argues that mathematics is a form of knowledge that has, from its very beginning, served the interest of power. challenging skovsmose’s (2005) formulation that mathematics education is critical because it has no essence and therefore can serve the purposes of oppression as well as of empowerment, kollosche asserts that mathematics as a form of knowledge is imbricated in fine technologies of power. particularly, he examines how logic and calculation, as part of mathematics, emerged in concrete time and space configurations of practice and related to power. through the incorporation of logics, “mathematics represents a form of thinking and speaking which provides powerful techniques for the government of others” (p. 1067). calculation is one of the core skills of a numerate person; however, calculation has historically been connected not only with commerce but also with the formation of bureaucracy. calculation was central in the creation of an objectivizing epistemology that was central for modern forms of government. school mathematics has developed side by side with the consolidation of bureaucracy and, thus, kollosche argues, it “can be considered an institution which (alongside other functions) identifies and trains a calculatory-bureaucratic elite and teaches the rest to subordinate to the calculatory-bureaucratic administration of our society” (p. 1070). in other words, mathematics cannot go in “both directions.” mathematics and its entering into school has historically created inclusions and exclusions. my intention here is not to say that people’s mathematical competence and achievement are not important. rather, i want to contend that the idea of people’s mathematical competence being an important constitutive element of citizenship is historically contingent and does not depend on the intrinsic characteristics of mathematics, but on how mathematics and mathematics education operate as effective technologies of governing for effecting contemporary forms of subjectivity. in previous writings, i have unfolded the elements of a genealogical study of the arguments for making mathematics for all a necessary part of human capital development (valero, 2017). there are a series of points that i would like to highlight here. first, a historization of the idea that mathematics and mathematics education are connected to the economic and social growth of a nation shows that it is important to place such an idea in the context of the consolidation of nation states and in the attempt to build and expand a national, politically regulated provision of education. in this context, (mathematics) education is a matter of the creation of a political body called “the nation” and of a political individual called “the citizen” (tröhler, 2016). these are however notions that change in time and space. school pedagogy and school knowledge are central in how the political fabrication of the citizen in the practices of education happen because these are conscious attempts to direct the conduct of the school participants to become particular type of desired beings. in valero making human capitals journal of urban mathematics education vol. 11, no. 1&2 111 this sense, the school curriculum encapsulates the political aspirations for who the citizen and the nation should be. second, the fundamental question of a curriculum is “what is the type of person that should result out of the curriculum” and “what kind of knowledge and moral attributes contribute to make such desired type of person.” in other words, a curriculum is not just a structured set of lists of school-adapted disciplinary content in different subjects. it is an articulated political device to steer and govern the population through the creation of desired subjectivities. from this perspective, the historical tension has been that up to the beginning of the 20th century, it was generally considered that the study of the humanities mainly was what provided the knowledge and moral qualities needed to become a virtuous citizen. what happened from the end of the 19th to the middle of the 20th century was that the growing natural sciences, technical knowledge, and mathematics—connected to the industrialization and the technical transformation of production and society—changed from being perceived as technical knowledge of limited value for the citizen to be, little by little, considered as a more valuable knowledge. it was only after the second world war and the period of extreme technological optimism that mathematics— and to certain extent science—started to be considered central elements of the formation of the modern citizen. it was at this time that clearly different discourses articulated to argue for the necessity of mathematics “for all.” from this point on, the traditional accounts provided in the historiography of mathematics education to justify the importance of mathematics as an area of the curriculum is known to mathematics educators (e.g., karp & schubring, 2014). third, it is also important to take into consideration the wider institutional articulation of the desire of mathematics for all—which is more often than not, the untold history of mathematics education. the boost for mathematical qualifications of the population is often portrayed as a natural development of the needs of the population to cope with the demands of a changing world. for example, mathematicians organized in the international commission for mathematical instruction (icmi) started to articulate the contribution of mathematics to the construction of an organized, structural, and systemic new world. for that, mathematics had to be promoted (kurepa, 1955). from another perspective, the boost of mathematics education has been a clear political strategy to direct the population toward the adoption of forms of thinking and being that resonate more with a scientific rationality (popkewitz, 2004). that mathematics education inserts in children the qualities of rationality, objectivity, universality is, of course, an evident and desired effect of power in the making of subjectivity. mathematics education operates as a technology of government to conduct the self. this move is explored in, for example, andrade-molina and valero (2017). they have shown how the training in school geometry is not really about visualization, but about the making of a scientific self that can see with and through the eyes of reason. this type of configuration is relat valero making human capitals journal of urban mathematics education vol. 11, no. 1&2 112 ed to the idea mentioned above that mathematics is meant to empower due to its intrinsic characteristics as the form of thinking that epitomizes a rational logic. fourth, the boost of mathematics education as a part of a political dispositive in education has also a strong link to economy. this means that from the 1960s, there has been a growing connection between the economic steering of society and mathematics competence and achievement. the efforts of mathematicians found resonance with the interest of the new organization for european economic cooperation (oeec), the antecessor of the organization for economic cooperation and development (oecd), in linking education to notions of economic growth through theories of human capital (tröhler, 2015, pp. 751–752). economists at the university of chicago in the 1960s proposed to think about capital not only in terms of materials, assets, or production but also in the great potential in humans. human capital is the “stock of skills and knowledge accumulated by workers through education, on-the-job training, and self-improvement” (mcfadden, 2008, p. 380). this capital is a source to generate value and wealth to individuals, organizations, and nations. this form of “embodied capital” in people (becker, 1993) became an important construct with relation to education: the input of education and provision of qualifications then can be connected to explanations of sustained economic growth in nations and increase in individual income. in this way, the investments in mass education and skill improvement entered economic models as an important variable to steer in order to generate value in a time of rapid advancement. oecd and united nations educational, scientific and cultural organization (unesco) became important institutions to collaborate with international organizations of mathematics education such as the icmi and the commission internationale pour l’étude et l’ amélioration de l’enseignement des mathématiques (cieaem) (furinghetti, 2008; furinghetti & giacardi, 2010). indeed, oeec was the economic supporter and organizer of the royaumont seminar in 1959 with the intention of discussing the fundamentals for reforming the contents and pedagogies of mathematics, vis-à-vis societal and economic needs to increase citizens’ mathematical knowledge and “appreciation for the numerical point of view” (fehr, bunt, & organisation européenne de coopération économique [oece], 1961, p. 11; author’s translation). if human capital, education and mathematics education were important factors for technological development and economic growth, it became clear that these factors had to be activated, monitored, and controlled. the history of the connection between different types of policies with a clear economic interest and initiatives to make mathematics education a key area of school curricula is fascinating, and still unexplored. this study is however important in understanding the articulations that have built the strong narrative of salvation connected to mathematics, and at the same time making mathematical achievement an element of inclusion and exclusion. valero making human capitals journal of urban mathematics education vol. 11, no. 1&2 113 making human capitals we are now in a particular kind of society, some have called it a post-political society (wilson & swyngedouw, 2014). this phrasing refers to the tendency of the state to not necessarily defend the public interest as a way of governing, but to defend the advancement of private interests and the interest of the capital. the idea of politics is concerned with what is public and common and that has to be defended and administered in the government and the political organization of the state. now we are in times where the logic of the political is surpassed and sometimes even overruled by the protection and advancements of private interests. some other scholars have also called this time, the time of the financial neoliberalism, understood as a form of rationality that reconfigures all spheres of human life in economic terms. brown (2015) explores how such reconfiguration has had an effect on subjectivity in transforming the very same notion of the human to become a value for the capital. she provocatively asserts that, in current times, one could use the term “human capitals” instead of simply human beings (pp. 32–37). she highlights the idea that it is natural to think that almost any kind of action can be seen as an investment for an expected profit. and this change is being operated by the way in which in many institutions of public and private life, the economic logic renders humans a factor in a calculation of optimization of value. in mathematics education, pais (2014) has argued how mathematics education research has been disavowing the function of mathematics education in the capitalist configuration of society. he notes that mathematics achievement has become an important way of granting economic value to the individual, and that efforts of mathematics education to combat injustice through the betterment of mathematics teaching and learning do reinforce the market value of mathematical qualifications given that, in a capitalist system, the failure of the many is a precondition for the success of the few. nowadays, the statements that argue for the prominence of mathematics clearly link mathematical competence with entrepreneurship and economic competitiveness. if one examines the wordings of national policies (e.g., doğan & haser, 2014), international policies, teacher education programs (e.g., montecino, 2018), education programs (e.g., de toledo e toledo, knijnik, & valero, 2018) and even textbooks (da silva & valero, 2018), mathematics education is being presented as a central value for the individual. the mathematical value of the individual becomes an exchange value and it is a measurement of the person’s human capital which, aggregated, is a measurement of the human capital of a system or of an economy. this exchange value results in the fact that mathematics is an object of desire to sell and trade with. nowadays, we might have a polarizing competition among each other of who goes to the most prestigious school of mathematics because that will secure a good job and, thus, a bright future. and i would dare to say that it is possi valero making human capitals journal of urban mathematics education vol. 11, no. 1&2 114 ble to hear in many of the discourses of mathematics education a similar kind of reasoning because it is most appealing. the constant failure of many to learn and like mathematics in each country and among countries has been a justification for the need of more research, particularly the one that wants to reform and fix “the problems of practice.” now the evidence for the need of research “that matters for practice” is no longer provided by small projects and results in national testing. it is minutely produced at large scale, so that there could be no doubt that mathematics achievement and competence matter for economy and for democracy. again, i am not saying that these truths are bad or good. my point is to highlight that they constitute a particular way of thinking about the value of mathematics now and why is it that we are educating people mathematically. the whole issue is that when we naturalize the reasoning, we classify people with respect to how well they performed in tests, how well they sorted out a problem, and all these differentiations generate combined mechanisms of inclusion and exclusion. we create a group of people to whom we, in our practice, are giving and granting more value, while some are devalued. a similar logic applies for us as university professors or as teachers; we also have an exchange value as better or worse mathematical educators, more recognized because we get more grants, or for teachers who can produce better results of students. of course, there are differences in the institutions that we work for, but the mechanisms seem to be similar. this distinction is the point of brown (2015) when she analyzes how the neoliberal logic permeates and reconfigures different types of institutions. the result is that neoliberalism is not a distant economic policy, but it ends up being the very same organizing principle of our own conduct and our own sense of being. we take these effects for granted; and we even come to believe that they are necessary. it is in this type of free market, financially capitalist societies, in this kind of economic organization, that we are performing mathematics education; and mathematics education, more than ever, is being politically governed by the logic that was exemplified previously. i think that we run a serious risk of reducing the meaning of mathematics education to education for the qualification of a submissive workforce. in such context, we end up educating not a human being but a homus oeconomicus. this person does not need to be a thinking person, or a rational being, but an economic exchange being. the studies on the cultural politics of mathematical education pose the question of which are the ethical and political commitment of mathematics educators in such political and economic rationality. in other words: in which directions are we governing and being governed with and through the teaching and learning of mathematics in the school curriculum? what we are doing is not innocent: we are always directing people, guiding toward a particular direction, so: which one is that? in the current configuration, i find it unacceptable that we operate on the assumption that we are only doing good for the empowerment and the future of students if they just valero making human capitals journal of urban mathematics education vol. 11, no. 1&2 115 know and come to master more mathematics, even with a hint of critique. i hope that by sharing some of the work on the cultural politics of mathematics education one can come to think politically seriously about what we are doing and about how it could be otherwise. acknowledgments this paper is based on the opening lecture at the 4th national forum of mathematics curricula in são paulo, brazil, august 3–5, 2017. some of these ideas have been published in portuguese (valero, 2018a). i would like to thank kenneth jørgensen, vanessa neto, and marcio a. silva for comments to and discussion of these ideas. references andersson, a., valero, p., & meaney, t. 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(2014). the post-political and its discontents: spaces of depoliticisation, spectres of radical politics. edinburgh, united kingdom: edinburgh university press. microsoft word when one lacks will they denounce editorial introduction jume (proof 2).docx journal of urban mathematics education july 2020, vol. 13, no. 1b (special issue), pp. 1–4 ©jume. https://journals.tdl.org/jume robert m. capraro is professor of mathematics education and co-director of the aggie stem center at texas a&m university, department of teaching, learning and culture, 4232 tamu, college station, tx 77843-4232; email: rcapraro@tamu.edu. his research interests are centered on stem educational research initiatives, urban mathematics achievement and representational models, and quantitative methods. jonas l. chang is editorial assistant at the aggie stem center at texas a&m university, department of teaching, learning and culture, 4232 tamu, college station, tx 77843-4232; email: jchang1066@tamu.edu. his research interests include communication strategies in stem education and the role of language and word choice in stem learning. editorial when one lacks will, they denounce… robert m. capraro texas a&m university aggie stem jonas l. chang texas a&m university aggie stem o much needs to change, and so many who have so much do so little that it makes us wonder why. recently, walmart; the american college of cardiology, the association of black cardiologists, and the american heart association (american college of cardiology, 2020; walter, 2020); the memphis interfaith coalition for action and hope; the president of the church of jesus christ of latter-day saints (mccombs, 2020); columbia sportswear and nike (bjorke, 2020); and sports legends alike have pronounced their positionality to denounce racism and violence. we are not sure, but it is quite likely that those police officers who were involved in mr. george floyd’s death would denounce racism and violence. some police chiefs have made the news denouncing racism and violence, though each of those who did so was unarmed and among peaceful protestors. so many voices are being raised to call for change and address the need for systemic racism and violence to end. these announcements are comforting, for sure, but we as american consumers understand that business is business and that nike and many other companies denouncing racism and violence make their money on the “dollar-acracy” of black super athletes; in this case, positionality is good for business and words are cheap. isn’t it time for action now? i want to push that further actually. perhaps it is time for a little less business positioning and a lot more action. the facts are facts concerning the events that spurned this national conversation. well, the media and the public relations moguls seem to have a specific idea of what those facts are, and perhaps too many people believe those “facts.” there were news stories indicating that mr. floyd was a criminal, a spurious designation that was intentionally misleading and not supported by the events that transpired. yes, there have been many arguments centered on this “fact” and a multitude of others, but i think there are several facts we can all assess and agree upon: mr. floyd was suspected of a crime, mr. floyd remained on the scene long enough for police to respond, and mr. floyd cooperated with police by allowing himself to be arrested s capraro & chang editorial journal of urban mathematics education vol. 13, no. 1b (special issue) 2 and handcuffed. furthermore, no charges had been filed, so mr. floyd was not yet accused, and, of course, he was not found guilty in a court of law by a jury of his peers. therefore, he was not a criminal. now he is denied the opportunity to face his accusers and was denied his right to a speedy trial. he was accused, tried, and sentenced on the street in public view while all too many people watched. it was a public execution. "life, liberty and the pursuit of happiness." these are guaranteed to all americans, well, as long as you are not black. right? the united states declaration of independence contains these three "unalienable rights." within the declaration of independence, it says that these rights have been given to all humans by their creator and that our governments (local, state, and national) were created to protect them. is the government protecting these rights for everyone? how can students learn, how can our society progress, how can families be secure when there is no law that demands that those in police custody remain safe and at least in the same life state as when they are arrested? when will police training be focused on preserving the life of those in their custody? well, custody is an interesting word. police use it as a term for when someone has been arrested. the technical definition of custody is the protective care or guardianship of someone or something. was mr. floyd in custody? nope! well, i have heard that mr. floyd was under arrest and had not yet moved to custody before his death; that’s another “fact” floating around. the definition of arrest, interestingly, is to seize (someone) by legal authority and take them into custody. yes, an arrest is the seizing of someone or something, and that act places the person or thing in the custody of the duly authorized officers performing the arrest. mr. floyd was in the custody of the arresting officers. so, how should someone treat someone or something once they are in their custody? this is a tough question, because what keeps coming up is the concept of ordinary care. ordinary care necessitates the preservation of the status of the person or object in police custody to all reasonable but not extraordinary measures. was there an effort on the officers’ part to exercise ordinary care? watch the video of mr. floyd’s arrest and his treatment while in police custody—see what you think. them ask yourself how would you feel if mr. floyd was your father, child, brother, cousin, or nephew? what is our responsibility though in the face of this horrific act? i think to answer this question, we first must recognize who we are and what power we have. we are teachers, researchers, friends, and colleagues. in short, we are influencers, ones who have privileged status and positions. but how are we going to use this privilege? do we simply submit our words of support with all the other people and organizations who have done so? that doesn’t seem enough. it has never seemed enough. we must use our individual and collective privilege to act and spur change. if you are not using it now, you don't deserve it. give it back. capraro & chang editorial journal of urban mathematics education vol. 13, no. 1b (special issue) 3 we need to go beyond words and help educate those who are doing the most to better society in the present by peacefully demonstrating and thereby placing themselves at risk of covid-19 infection. we must help them develop messaging that provokes change and not just dissention. this is one of those pivotal moments where one must act to avoid being the disruption and become a disruptor. productively using frustration and anger there have been many trying times in u.s. history, each punctuated by precipitating events that stemmed from long-standing social ills, many of which are still unresolved. perhaps none are more pervasive and insidious than white privilege and systemic, pernicious, and well-entrenched racism. recently, that racism came to the forefront with mr. floyd’s execution. unfortunately, these current trying times seem to be playing out just as others have. while i applaud high-profile people and organizations taking on aspects of the big issue, i am concerned that the big issue is not at the forefront of the conversation. how will we as educators use these events to educate children, to reach out to our peers, colleagues, and friends, and, finally, how will mathematics education change as a result of these conversations? i think the current needs of society require us as mathematics educators to not only do what we can to support the current peaceful protests and calls for reform, but to also reflect on our own practices and field and make necessary changes. a number of researchers have claimed that mathematics success is an equalizer and that those who are successful in mathematics are among a privileged class. when and how are we going to use mathematics to privilege young black and brown children? how will we educate new teachers to assume the mantle of leadership in the classroom and foster the mathematical success of black and brown children? as editor-in-chief, i challenge you to think deeply about these events, consider responses to the questions i raise, and use your time, talents, and research prowess to answer these questions in order to make a change today for the world of tomorrow that we must build together. our team has secured permission to reprint two amazing publications that were clearly ahead of their time and speak to our current situation. a special thank you to mr. george f. johnson, president & publisher of information age publishing inc., and dr. chance lewis, professor and founder of the international conference on urban education, for granting permission to reprint these two articles. i hope you find them timely and inspirational. capraro & chang editorial journal of urban mathematics education vol. 13, no. 1b (special issue) 4 references american college of cardiology. (2020, june 1). abc, acc and aha denounce racism and violence plaguing communities. https://www.acc.org/latest-in-cardiology/articles/2020/06/ 01/09/25/abc-acc-and-aha-denounce-racism-and-violence-plaguing-communities bjorke, c. (2020, june 1). portland businesses and leaders add to calls denouncing racism and violence. portland business journal. https://www.bizjournals.com/portland/news/2020/06/01/social-roundup.html mccombs, b. (2020, june 1). mormon president denounces racism, escalating violence. standardexaminer. https://www.standard.net/news/state/mormon-president-denounces-racism-escalating-violence/article_e1822c30-38e8-5c21-a1c8-186d95795503.html walter, m. (2020, june 1). in wake of george floyd’s death, cardiovascular groups denounce ‘incidents of racism and violence.’ cardiovascularbusiness. https://www.cardiovascularbusiness.com/topics/healthcare-economics/george-floyd-cardiovascular-denounce-racism-violence copyright: © 2020 capraro & chang. this is an open access article distributed under the terms of a creative commons attribution-noncommercial-sharealike 4.0 international license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. microsoft word 7 final reviewer acknowledgment vol 11 no 1&2.doc journal of urban mathematics education december 2018, vol. 11, no. 1&2, pp. 175–180 ©jume. http://education.gsu.edu/jume reviewer acknowledgment 2008–2018* it really goes without saying, nevertheless, a peer-reviewed journal is not possible without reviewers. over the past decade, as a way to honor our reviewers, we have listed reviewers’ names and affiliations at the end of every other volume. below are those lists as originally available. rather than compile one master list, here we provide each of the five lists as a way to recognize those who contributed their expertise time, after time, after time. whether contributing one, two, three, or multiple reviews, the jume editorial team members throughout the past decade are forever grateful for the vital contribution and service to the success of jume and to the larger (urban) mathematics education community that each of you provided. thank you! sincerely, the jume editorial teams 2008–2018 january 2008–december 2009 reda abuelwan, sultan qaboos university dan battey, arizona state university joanne becker, san jose state university clare bell, university of missouri robert berry, university of virginia tonya bartell, university of delaware angela brown, piedmont college evelyn brown david brown, texas a&m university, commerce joan bruner-timmons, miami-dade county public schools lecretia buckley, jackson state university leonides bulalayao gustavo bermúdez canzani theodore chao, university of texas, austin carrie chiappetta, stamford public schools karen cicmanec, morgan state university marta civil, university of arizona shelly jones, central connecticut state university karen king, new york university richard kitchen, university of new mexico steven kramer, baltimore freedom academy brian lawler, california state university, san marcos della leavitt jacqueline leonard, temple university dorothy lewis-grace, dekalb county public schools sarah lubienski, university of illinois, urban champaign danny martin, university of illinois, chicago donna mccaw, western illinois university jennifer mccray, erikson institute allison mcculloch, north carolina state * if you were a reviewer for a manuscript during january 2008–december 2018 and your name and affiliation is not listed, please contact david stinson at dstinson@gsu.edu. it is our sincere intent to recognize all those who have generously given of their expertise and time to the success of jume. reviewer acknowledgement january 2008–december 2018 journal of urban mathematics education vol. 11, no. 1&2 176 raeshaun costley ubiratan d'ambrosio ella-mae daniel, florida a&m university julius davis, university of maryland donna dodson marilyn evans, national council of teachers of mathematics cassie freeman, university of chicago joseph furner, florida atlantic university mary foote, queens college, city university of new york imani goffney, university of michigan lidia gonzalez, york college, city university of new york susan gregson, university of illinois, urbana-champaign barbro grevholm, university of agder victoria hand, university of colorado, boulder shandy hauk, university of northern colorado daphne heywood, university of toronto crystal hill, indian university purdue university indianapolis leslie hooks, fort worth independent school district tisha hyman christopher jett, georgia state university alanna johnson jason johnson, middle tennessee state university martin johnson, university of maryland university regina mistretta, st. john’s university kamau mposi mayen nelson, houston independent school district madeline ortiz-rodriguez, interamerican university of puerto rico pamela paek, national center for the improvement of educational assessment angiline powell, university of memphis arthur powell, rutgers, the state university of new jersey malik richardson, charlotte mecklenburg schools fatimah saleh, universiti sains malaysia walter secada, university of miami stanley shaheed, dekalb county public schools joi spencer, university of san diego megan staples, university of connecticut william tate, washington university james telese, university of texas, brownsville la mont terry, occidental college carmen thomas-browne, art institute of pittsburgh thomas thrasher david wagner, university of new brunswick erica walker, teachers college, columbia hersh waxman, texas a&m university desha williams, kennesaw state university maya wolf january 2010–december 2011 noor aishikin adam joshua oluwatoyin adeleke, institute of education shuhua an, california state university lorraine baron tonya bartell, university of delaware dan battey, rutgers university clare bell, university of missouri cigdem haser crystal hill roberta hunter, massey university mine isiksal, middle east technical university andy isom, center for literacy laura jacobsen, radford university martin johnson, university of maryland reviewer acknowledgement january 2008–december 2018 journal of urban mathematics education vol. 11, no. 1&2 177 robert berry, university of virginia denise brewley, georgia gwinnett college angela brown, piedmont college david brown, texas a&m university, commerce lecretia buckley, jackson state university stephanie byrd, clayton county schools patricia campbell, university of maryland theodore peck-li chao, university of texas, austin carrie lynn chiappetta, stamford public schools haiwen chu, graduate center, cuny karen cicmanec, morgan state university marta civil, university of arizona nicholas cluster, university of georgia lesa covington clarkson, university of minnesota cynthia cromer jaime curts, university of texas, pan american ubiratan d'ambrosio julius davis, morgan state university sandy dawson irene duranczyk, university of minnesota indigo esmonde, university of toronto gheorghita faitar mary foote, queens college, cuny cassie freeman, university of chicago joseph furner, florida atlantic university imani goffney, university of michigan lidia gonzalez, york college, cuny susan gregson, university of illinois, urbana-champaign rochelle gutierrez, university of illinois, urbana-champaign eric gutstein, university of illinois at chicago victoria hand, university of colorado, boulder deborah harmon, eastern michigan university shelly jones, central connecticut state university joyce king, georgia state university richard kitchen, university of new mexico courtney koestler, university of arizona della leavitt, rutgers university shonda lemons-smith, georgia state university jacqueline leonard, temple university julie livingood danny martin, university of illinois, chicago donna mccaw, western illinois university jennifer mccray, erikson institute eduardo mosqueda, university of california, santa cruz nirmala naresh ellen pechman, emp consulting gerard petty, henry county public schools arthur powell, rutgers university laurie rubel, brooklyn college, cuny walter secada, university of miami megan staples, university of connecticut william tate, washington university in st. louis dante abdul-lateef tawfeeq, adelphi university la mont terry, occidental college lanette waddell, vanderbilt university anita wager, university of wisconsin, madison dorothy white, university of georgia kimberly white-fredette, griffin regional educational service agency candace williams, dekalb county public schools desha williams, kennesaw state university curt wolfe, mt. carmel christian school jamaal rashad young, texas a&m university reviewer acknowledgement january 2008–december 2018 journal of urban mathematics education vol. 11, no. 1&2 178 january 2012–december 2013 nathan alexander, teachers college, columbia university dan battey, rutgers university joanne becker, san jose state university robert berry, university of virginia lecretia buckley, jackson state university carrie chiappetta, stamford public schools haiwen chu, wested teresa dunleavy, university of san diego indigo esmonde, university of toronto gheorghita faitar, d’youville college lidia gonzalez, york college, cuny jessica hale, georgia state university jacqueline hennings, griffin regional educational service agency jennifer jones, rutgers university brian lawler, california state university, san marcos maxine mckinney de royston, university of california berkley james telese, university of texas, brownsville la mont terry, occidental college anita wager, university of wisconsin, madison morgin jones williams, georgia state university candace williams, dekalb public schools desha williams, kennesaw state university khoon wong, national institute of education, singapore special issue guest editors and open peer reviewers volume 5, number 1 spring/summer 2012 proceedings of the 2010 philadelphia and 2011 atlanta benjamin banneker association conferences – beyond the numbers erika bullock, georgia state university nathan alexander, teachers college, columbia maisie gholson, university of illinois, chicago volume 6, number 1 spring/summer 2013 privilege and oppression in the mathematics preparation of teacher educators (prompte) david stinson, georgia state university joi spencer, university of san diego january 2014–december 2015 nathan alexander, san francisco state university dan battey, rutgers university joanne becker, san jose state university clare bell, university of missouri–kansas city robert berry, university of virginia jennifer jones, rutgers university rick kitchen, university of denver brian lawler, california state university, san marcos percival matthews, university of wisconsin–madison reviewer acknowledgement january 2008–december 2018 journal of urban mathematics education vol. 11, no. 1&2 179 angela brown, piedmont college joan bruner-timmons, miami-dade county public schools lecretia buckley, jackson state university patricia campbell, university of maryland college park susan cannon, georgia state university robert capraro, texas a&m university carrie chiappetta, stamford public schools ervin china, georgia state university marta civil, university of arizona teresa dunleavy, vanderbilt university indigo esmonde, university of toronto gheorghita faitar, d’youville college mary foote, queens college, cuny susan gregson, university of cincinnati jessica hale, georgia state university victoria hand, university of colorado boulder shandy hauk, wested crystal hill, indiana university-purdue university indianapolis keith howard, chapman university maxine mckinney de royston, university of pittsburgh eduardo mosqueda, university of california, santa cruz angiline powell , university of memphis mary raygoza, university of california, los angeles laurie rubel, brooklyn college, cuny james telese, university of texas at brownsville luz valoyes chávez, university of missouri anita wager, university of wisconsin– madison erica walker, teachers college columbia university candace williams, dekalb county school district desha williams, kennesaw state university morgin jones williams, georgia state university khoon wong, national institute of education, nanyang technological university, singapore maria zavala, san francisco state university january 2016–december 2017 glenda anthony, massey university lorraine baron tonya bartell, michigan state university dan battey, rutgers university nermin bayazit, fitchburg state university john bragelman, university of illinois at chicago susan cannon, georgia state university robert capraro, texas a&m university theodore chao, the ohio state university ervin china, georgia state university marta civil, university of arizona stephanie cross, georgia state university corey drake, michigan state university anthony fernandes, university of north carolina charlotte danny martin, university of illinois at chicago jasmine mathis, georgia state university percival matthews, university of wisconsinmadison maxine mckinney de royston, university of wisconsin-madison alesia mickle moldavan, georgia state university eduardo mosqueda, university of california, santa cruz sarah oppland-cordell, northern illinois university alexandre pais, manchester metropolitan university reviewer acknowledgement january 2008–december 2017 journal of urban mathematics education vol. 11, no. 1&2 180 mary foote, queens college, cuny toya jones frank, george mason university maisie gholson, university of michigan susan gregson, university of cincinnati eric gutstein, university of illinois at chicago jessica hale, georgia state university victoria hand, university of colorado shandy hauk, wested crystal hill, indian university-purdue university indianapolis keith howard, chapman university signe kastberg, purdue university rick kitchen, university of wyoming gregory larnell, university of illinois at chicago brian lawler, kennesaw state university jacqueline leonard, university of wyoming angela lopez pedrana, university of houston downtown elijah porter, georgia state university arthur powell, rutgers university-newark mary raygoza, saint mary’s college laurie rubel, brooklyn college, cuny tesha sengupta-irving, vanderbilt university james telese, university of texas at brownsville luz valoyes-chávez, university of chile, santiago anita wager, vanderbilt university erica walker, teachers college columbia university craig willey, indiana university-purdue university indianapolis morgin jones williams, university of south carolina beaufort maria zavala, san francisco state university january 2018–december 2018 special issue open peer reviewers volume 11, number 1&2 fall/winter 2018 a decade of critical mathematics education knowledge dissemination erika bullock, university of wisconsin–madison susan cannon, georgia state university christopher jett, university of west georgia alesia mickle moldavan, fordham university david stinson, georgia state university microsoft word final nzuki vol 3 no 2.doc journal of urban mathematics education december 2010, vol. 3, no. 2, pp. 77–115 ©jume. http://education.gsu.edu/jume francis m. nzuki is an assistant professor of developmental mathematics in the school of general studies, at the richard stockton college of new jersey, p.o. box 195, pomona, nj 08240; email: francis.nzuki@stockton.edu. his research interests include issues of equity in mathematics education with a focus on students’ identity construction and use of technology in the mathematics classroom. exploring the nexus of african american students’ identity and mathematics achievement francis m. nzuki the richard stockton college of new jersey in this article, the author explores five african american students’ racial, mathematical, and technological identity construction and how these identities shape each other and the sense of agency exhibited in the process. data collection for the study included classroom observations and interviews, including a task-based interview. the stories told by the participants, their solutions for the mathematical tasks, and their participation in the figured world of mathematical learning illuminated their sense of identity and agency. an analysis of the data revealed that the participants’ positioning and authoring of their identities were influenced by how they negotiated and interpreted the constraints and affordances in the figured worlds in which they participated. it is through this process of negotiation and interpretation that the participants exhibited a sense of agency, or lack thereof, which, in turn, shaped their opportunities to participate in mathematics and hence the authoring of their mathematical identities. keywords: african american education, agency, equity, graphing calculators, identity, mathematics achievement ecently, mathematics education researchers have begun to utilize the notion of identity to examine issues of equity (see, e.g., gutstein, 2003; martin, 2000, 2006a, 2006b; nasir & hand, 2006; stinson, 2010a). from this perspective, researchers consider the social and cultural features of the figured worlds (holland, lachicotte, skinner, & cain, 1998) in which students participate in mathematics and how these figured worlds shape how they position themselves as learners and doers of mathematics and how they are positioned by others. this perspective also considers the concept of agency—the understanding that individuals have the capacity to author their identities by resisting and/or reacting against the structural and cultural forces that might shape their identities. holland et al. (1998) define figured worlds as socially and culturally constructed realms “of interpretation in which particular characters and actors are recognized, significance is assigned to certain acts, and particular outcomes are valued over others” (p. 52). within holland et al.’s figured worlds the concept r nzuki exploring identity and achievement journal of urban mathematics education vol. 3, no. 2 78 space of authoring refers to the responses that people give with human agency and with improvisation, and the concept positional identity refers to the ways in which people understand and enact their positions in the worlds in which they live, implying that identities are developed in and through practice (boaler & greeno, 2000). boaler and greeno (2000) claim that the mathematics learning environment is a particular social setting where teachers and students construct interpretations of the actions that take place in it; thus, the mathematics classroom could be considered as a figured world or a community of practice (wenger, 1998). several studies have documented underachievement and limited persistence of african american1 students in the figured world of the mathematics classroom (martin, 2000, 2003; oakes, 1985; tate, 1997b). however, drawing on critical race perspectives, including critical race theory (tate, 1997a), a few scholars have begun to focus on the salience of race and identity in regards to african american students’ mathematics learning and participation (see, e.g., gutstein, 2003; martin, 2006a, 2006b; nasir & hand, 2006; stinson, 2008), uncovering the structural and cultural factors that perpetuate the racism embedded within american social structures and practices. these scholars are interested in understanding the relationship between the ways that african american students, who come to learning contexts with their experiences as african americans, think about themselves as african americans and their conceptions of themselves as learners and doers of mathematics. in other words, the focus is on the dialectic relationship between racial and mathematical identities. moreover, in discussing equity issues in mathematics, the use of technology in classrooms has been recommended as useful in helping engage low-ses2 and minority students in learning that encourages them to use complex thinking skills in mathematics (hennessy & dunham, 2002). the national council of teachers of mathematics (nctm) (2000) states, “technological tools and environments can give all students opportunities to explore complex problems and mathematical ideas” (p. 13, emphasis added). nctm also posits that technology can attract students who disengage from non-technological approaches to mathematics, and that all students should have opportunities to use technology in appropriate ways that 1 the terms african american and black are used throughout to refer to a person of african ancestral origins who self-identifies or is identified by others as having the cultural identity of the united states. 2 according to national assessment of educational progress (naep), individual social economic status (ses) is defined by a student’s participation in the federal free and reduce-priced lunch program, while a school’s ses is defined by the percentage of students enrolled in the federal free and reduce-priced lunch program. thus, schools with a high (or low) percentage of students participating in the federal free and reduce-priced lunch program are classified as either low or high ses schools. nzuki exploring identity and achievement journal of urban mathematics education vol. 3, no. 2 79 will afford them access to interesting and important mathematical ideas. as such, through the opportunities afforded to them to interact with technology when learning mathematics, it is worth examining the relationship between the technological identity that low-ses and racial minority students develop and their developed identity in mathematics. the study reported in this article, derived from my dissertation (nzuki, 2008), builds on the emerging research about identity and the mathematical learning of historically marginalized students. in particular, i explore african american students’ identity construction—racial, mathematical, and technological—and how the nexus of these identities affects the students’ mathematics participation and achievement. two questions guided the study: 1. what are african american students’ perceptions of their mathematical experiences in the figured worlds of mathematics education in which they participate as learners? 2. how do african american students position and author their identities—racial, mathematical, and technological—within the figured worlds of mathematics education in which they participate as learners? here, i take a sociocultural perspective and view identity to be the basis “from which people create new activities and new ways of being” (holland et al., 1998, p.5), as well as the means through which individuals assert themselves, care about the conditions of their lives, and attempt to direct their own behavior. in other words, i view identity as the means through which individuals enact agency. it is a dynamic concept, one that is constructed by individuals as they actively participate in cultural activities, and one that both shapes and is shaped by the social context. in this study, racial identity refers to the ways in which individuals perceive themselves in relation to their group, while mathematical identity considers the perceptions of individuals regarding their abilities to participate and perform effectively in mathematical contexts. and technological identity pertains to the ways that individuals appropriate and interact with technology. (here, i restrict technology to the use of graphing calculators in the mathematics classroom.) theoretical framework in the study, i draw on the following sociocultural theoretical perspectives: (a) holland et al.’s (1998) framework of figured worlds, positioning, and authoring; and (b) goos, galbraith, renshaw, and geiger’s (2003) metaphors of technology as master, technology as servant, technology as partner, and technology as extension of self, that describe the varying degrees of sophistication with which students and teachers interact with technology. additionally, in theorizing nzuki exploring identity and achievement journal of urban mathematics education vol. 3, no. 2 80 about how race and racism operate to affect african american students’ mathematics learning and participation, i draw on critical race theory (crt). sociocultural theory the sociocultural perspective that i draw upon is holland et al.’s (1998) identity and agency framework that locates identity and agency as aspects of participation in particular communities of practice. holland et al. discuss social systems in terms of figured worlds, positioning, and authoring. cultural artifacts play an important role in figured worlds because they can serve as pivots (vygotsky, 1978), which shift the frame of activity and provide the means by which figured worlds are “evoked, collectively developed, individually learned, and made socially and personally powerful” (holland et al., 1998, p. 61). through engagement with artifacts, learners enact their identities and agencies through processes of positioning and authoring within the figured worlds. additionally, this enactment is patterned and governed by not only the learners’ negotiation of classroom norms with the teacher and among themselves but also by their adaptations to the constraints and affordances of the figured worlds in which they participate (boaler & greeno, 2000). this sociocultural perspective, therefore, places emphasis on the socially and culturally situated nature of mathematical activity, where the classroom, as a community of practice, supports a culture of sense making in which meanings are shared among students and the teacher. from this perspective, learning entails the collective process of enculturation into the practices of mathematical communities (galbraith, goos, reinshaw, & geiger, 1999) where students interact among themselves, with the teacher, the mathematics tasks, and classroom artifacts within the social context of the classroom. these interactions are patterned and governed by social expectations, conventions, norms, habits, and rituals (galbraith et al., 1999; goos, galbraith, renshaw, & geiger, 2000; warschauer, knobel & stone, 2004). an essential aspect of sociocultural theory is that learning is mediated by cultural tools and is fundamentally transformed in the process. the graphing calculator technology is an example of how such tools transform mathematical tasks. learning, thus, is a process of appropriating the cultural tools (e.g., graphing calculators) recognized by a community of practice, and participation in such classroom communities requires learners to acquire new forms of reasoning and action that is beyond their established capabilities (galbraith et al., 1999; goos et al., 2000). sociocultural theory and technology to describe the varying degrees of sophistication with which students and teachers work and interact with technology and the ways in which technology can nzuki exploring identity and achievement journal of urban mathematics education vol. 3, no. 2 81 mediate learning, goos et al. (2003) draw from a sociocultural perspective of learning to theorize four metaphors of technology usage: technology as master, technology as servant, technology as partner, and technology as an extension of self. the lowest level is that of technology as master. here students have limited operational skills, and the complexity of usage confines their activity to the few operations over which they have competence. if students have insufficient mathematical understanding, they blindly accept the output produced irrespective of its accuracy or worth. in the next level, technology as servant, the user is in control and applies the technology as a fast and reliable mechanical aid to replace mental, or pen-and-paper computations, but the technology is not used in creative ways to change the nature of the mathematical tasks. at the third level, technology as partner, technology is seen as a companion with which to explore rather than just a tool for producing results. aside from being in control, the user not only appreciates that the outcome has to be judged against mathematical criteria other than just the technology-produced response but also recognizes that there needs to be a balance between the authorities of the technology and mathematics. technology as an extension of self is the highest level of functioning. here, the technology provides as extension of students’ mathematical abilities and becomes an integral part of their mathematical repertoire, something that shares and supports their mathematical argumentation. while these modes of interaction are hierarchical in that they depict an increasing level of technology use that students and teachers attain, goos et al. (2003) contend that these modes are not necessarily related to the level of mathematics taught or the sophistication of the available technology, and once a user has shown that he or she can work at a higher level, it does not mean that he or she will do so on all tasks. rather, these modes describe an expansion of the technological repertoire, which gives the user a wider range of modes of operation available to engage with a particular mathematical task. inequities that arise from differential access and use of educational technology in mathematics for racial minority students and low-ses students are usually considered in terms of (a) physical access, the physical presence of the technology, and (b) experiential access, how the technology is used, by whom and for what mathematical tasks (gorski, 2005; warschauer et al., 2004). research shows that, even when they had the physical access to technology, many racial minority students and low-ses students were more likely to use technology for drill-andpractice activities that involve lower thinking skills (hennessy & dunham, 2002; national center for educational statistics, 2002). thus, the dynamics of technology access and use often end up reflecting, recycling, and strengthening the already existing inequities in mathematics education related to race and ses. moreover, dunham and dick’s (1994) and penglase and arnold’s (1996) allegation of graphing calculators’ relative physical access when compared to other nzuki exploring identity and achievement journal of urban mathematics education vol. 3, no. 2 82 forms of technology, because of price, portability, and ease of use, still holds true today. thus, inequities in the use of graphing calculators are more likely to arise because of experiential access—the extent of instruction incorporating graphing calculators received by students, as well as the opportunity that they have to make use of the graphing calculator’s mathematical functions, which is what the nctm (2000) envisions as the appropriate use of technology that can promote equity in mathematics classrooms. furthermore, the consensus of research reviews is that students who use graphing calculators display better understanding of function and graph concepts, enhance their problem-solving skills, and score higher on achievement tests for algebra and calculus (see, e.g., adams, 1997; graham & thomas, 2000; hollar & norwood, 1999; schwarz & hershkowitz, 1999; thompson & senk, 2001). the use of graphing calculators has also been shown to improve students’ attitude towards mathematics (ellington, 2003). the aforementioned modes of interacting with technology can provide the lens through which to examine technological inequities by looking at the various ways in which students’ appropriate technology to engage with mathematical tasks. as students gain more control in the use of technology, the technology becomes a means to equip them with skills and strategies to solve mathematical tasks by engaging them in active and meaningful learning that stimulates their creativity and critical thinking, thereby, in turn, increasing their level of understanding and participation. critical race theory (crt) to understand the role of race and racism in the academic learning and participation experiences of african american students, it is important to consider crt as a framework for understanding the social inequalities arising through race and racism. scholars of color who were working in academic legal circles initially developed crt; it grew out of their dissatisfaction with the slow rate of racial reform since the growth of the civil rights movement (ladson-billings, 1998). grounded in the recognition that african americans have a unique history of oppression and discrimination in the united states, including slavery, crt posits that this distinct historical background contributes to a racialized minority experience and cultural identity for african americans (tate, 1997a). the lives of african americans in the united states continue to be impacted by this history of race and racism. for this reason, crt advocates scholarly discourse that raises race consciousness, rather than masking racial identity through colorblindness or race neutrality (ladson-billings & tate, 1995; solórzano & yosso, 2002; tate, 1997a). additionally, critical race theorists espouse the idea that race is socially constructed to mean that race is neither biologically determined nor fixed. instead, race is ever evolving as a function of social, political, legal, and economic pres nzuki exploring identity and achievement journal of urban mathematics education vol. 3, no. 2 83 sures (delgado & stefancic, 2001). this thinking implies that race categories are created by the dominant society in order to manipulate these categories to its own advantage. in educational research, crt has been used to expose racism within existing educational practices and policies. while some researchers have drawn from crt to explore the experiences of people of color as students and faculty in secondary or higher education (bernal, 2002; solórzano & yosso, 2002), others have used crt to critique certain legal cases in education (villenas, deyhle, & parker, 1999). additionally, crt has been used to examine practices for preparing teachers to teach culturally diverse students (ladson-billings, 1999). crt often relies on counter-storytelling, which is a method of telling a story with the objective of casting doubt on the validity of accepted premises or myths, in particular, the premises or myths held by the majority (delagafo & stefanic, 2001; solórzano & yosso, 2002). in drawing from crt’s style of storytelling, i used mcadams and bowman’s (2001) definition of a life story. they define a life story as an internalized narrative that depicts an individual’s life in time and consists of the reconstructed past, perceived present, and anticipated future. further, life stories reflect an individual’s understanding of self by both the individual himself or herself and also through the wide variety of cultural influences within which the individual’s life is situated. therefore, the life stories that the five participants of this study told through interviews entailed their own narratives of their mathematical experiences. this process allowed the participants to locate their stories within a context, which provided a fuller understanding of how they perceived themselves as learners and as doers of mathematics. methods this research was a case study of an intermediate algebra iii mathematics classroom at graham high school (pseudonym, as are all proper names used in this study), serving a culturally diverse student population with the majority being african american. of the 30 students enrolled in the class, 22 were juniors (11th graders) and 8 were seniors (12th graders) who were taking their last mathematics course before they graduated from high school. the school was chosen because of its demographic factors. first, it was a low-ses school based on the percentage of students eligible for free and reduce-priced lunches. second, although there were relatively more black students in the school, there was a fair share of other racial groups, particularly white students, at the school. this factor allowed for the examination of african american students’ schooling experiences as they interacted with students from other “races.” i chose the intermediate algebra iii class, one of the “lower-track” mathematics courses at graham high school, because (a) i wanted a classroom where there were several african american students, most of whom had been relegated nzuki exploring identity and achievement journal of urban mathematics education vol. 3, no. 2 84 to the lower-track courses, and (b) i wanted a classroom where students used the graphing calculator on a frequent basis. generally, teachers of the lower-level mathematics courses at graham used graphing calculators less frequently than those of higher-level courses. this was particularly true among the lower-tracked courses. thus, on this basis, of the lower-tracked mathematics courses, the intermediate algebra iii class students were more likely to employ the use of graphing calculator. additionally, by studying only one class, i wanted to examine how success and failure could exist among african american students with similar experiences within the figured worlds and learning communities in which they participated. these equivalent experiences include similar ses backgrounds, similar mathematical experiences, and, most notably, the same teacher. data collection and participant selection data were collected through both quantitative and qualitative means. i employed a survey instrument, as well as classroom observations, and three interviews with each of the five student participants and the classroom teacher. data collection took place in two stages. in stage 1, i administered a survey instrument (see appendix a) to all students in the classroom. the survey had two distinct parts. the first part consisted of five-point, likert-scale statements based on surveys developed by fleener (1995) and the fennema-sherman mathematics attitudes scales (fennema & sherman, 1976). the second consisted of open-ended questions, which asked the students to provide their names, descriptions of their race and/or ethnicity and their experiences with and use of graphing calculators. the survey instrument intended to (a) collect demographic information on the students (i.e., race, ethnicity, gender, etc.), (b) provide insights into students’ perceptions towards graphing calculators and mathematics, (c) assist in preparing interview questions, and (d) facilitate the selection of the five participants for the study. from the survey instrument, 12 out of 30 students self-reported their race to be african american. of these 12, i examined each student’s responses to the items on the survey instrument. after reversing the responses of the negative statements so that the (strongly) disagree responses were reported as (strongly) agree and vice versa, i determined my initial pool of high achievers to be those students who agreed or strongly agreed with at least half of the items. from this pool, i chose three “high achievers” (two male students and one female student) who had the highest strongly agree/agree responses. i followed the same procedure to determine the “low achievers,” except here the focus was on those with the highest strongly disagree/disagree responses. in the end, i chose three low achievers (two female students and one male student). in the next step of choosing the participants, i consulted with the teacher to get his opinion of my list of high achievers and low achievers. in doing so, i wanted to determine if the teacher nzuki exploring identity and achievement journal of urban mathematics education vol. 3, no. 2 85 agreed with the students’ self-reporting and responses to the items on the survey instrument and if these responses corresponded with their overall performance in the class. i also wanted to get the teacher’s assistance in determining the regular attendees. based on the teacher’s assessment of the students’ performance and participation in class, the teacher disagreed with only one male student’s responses, leading me to classify him as a high achiever. in addition, according to the teacher, one of the male students i had classified as a low achiever was not a regular attendee. after my discussion with the teacher, i narrowed my participants to two high achievers (caleb and karen) and two low achievers (annabel and danielle). i decided to retain the male student, whose responses were challenged by the teacher, without initially classifying him as either a low or a higher achiever, because i wanted to (a) find out why he self-reported as a high achiever when and if indeed he was not, and (b) have a fairly balanced number of female and male participants. with extended time in the classroom and from my observations and interviews, i came to classify this student (amos) as a low achiever. by focusing on these five participants, i was able to examine in depth the issues of identity and agency among african american students and how they impacted the students’ mathematical learning. summary information about the five participants is provided in table 1 (for a detailed description of the participants see nzuki, 2008). in the next stage of data collection, i conducted classroom observations for two to three classroom periods per week from september 2007 to december 2007, for a total of 20 classroom observations. for these classroom observations, guided by a sociocultural framework, i observed the classroom dynamics in terms of classroom interactions between teacher and students, among students themselves, and among teacher, students, and technology (as previously noted, in this case, graphing calculators). i also explored the classroom discourse and the instructional methods employed by the teacher including how the graphing calculator was used to enhance teaching and learning strategies. during each classroom observation, i took field notes to record details of classroom tasks, teacher actions, and student actions involving graphing calculator use. i also documented the important visual or physical components of the classroom interactions. to capture accurately the teacher and student actions during the classroom interactions, all the classroom observations were audio taped. to gain a deeper understanding of students’ sense making of identities and agencies within the figured worlds of mathematical learning, i conducted three interviews with each of the participants. i conducted the first semi-structured interview at the beginning of the study. given that the figured worlds in which students participate shape their beliefs, values, and understandings that they develop, the prompts for the semi-structured interview included questions pertaining nzuki exploring identity and achievement journal of urban mathematics education vol. 3, no. 2 86 to those beliefs and understandings related to their sense of identity and agency (holland et al., 1998). table 1 description of participants name family background occupational aspirations year course grade annabel 18-years old, lived with mother and two brothers, no family member had graduated from high school, father jailed because of drug-related charges elementary school teacher 12 47 amos 17-years old, 16 brothers and 4 sisters, lived with dad and a few siblings, other siblings either in college or working professional football player, musician, computer engineer 12 50 l ow a ch ie ve rs danielle 17-years old, one brother and one sister, lived with aunt because her mother could not care for them, did not know where her father lived pediatrician 12 56 caleb 17-years old, six sisters and two brothers, lived with mother and stepfather, biological father died when he was 8-years old, working class computer programmer 11 80 h ig h a ch ie ve rs karen 16-years old, six sisters and four brothers, godmother to her friend and schoolmate’s baby, lived with her mother and one of her sisters, two of her sisters and mother are in nursing profession licensed practical nurse 11 88 the second interview occurred toward the middle of the study, at which time i asked the students to share their reflections and reactions to some of the mathematics lessons covered since the beginning of the study. in particular, i asked them about (a) the lessons they particularly liked or disliked, (b) the effect of graphing calculators on their enjoyment, skills, and understanding, (c) features of the graphing calculator used in lessons and their relative importance, (d) students’ use of graphing calculators in the observed lessons, and (e) students’ perceptions of the role of graphing calculators in mathematical learning. moreover, i nzuki exploring identity and achievement journal of urban mathematics education vol. 3, no. 2 87 investigated the students’ perceptions of the opportunities offered by the classroom interactions between the students and the teacher, among themselves and with graphing calculators. the central focus was how these interactions allowed students to engage constructively and critically with mathematical ideas within the specific classroom contexts. the last interview, which was conducted near the end of the study, was a task-based interview. i engaged the students in problem-solving tasks that were similar to those they had worked in class during the course of this study. i was interested in assessing the students’ understanding of the mathematical content and the extent to which the graphing calculator appeared to contribute (or not) to their understanding, and the choices of strategic purposes of calculator use favored by the students. some of the requirements of the tasks presented to the participants included: determining the roots and the axis of symmetry of a quadratic function, solving an algebraic linear equation with one unknown variable, finding the output of a quadratic function given a specific input, solving questions of a quadratic model describing the height of a baseball in the air after it is hit and of an exponential model depicting the elimination of caffeine from the body at a given rate. to capture how students used the graphing calculator, i videotaped them as they solved the tasks. i also asked the students to explain their thinking and justify their strategies. in addition, i made notes about the strategies they used and collected all the written work that they produced as they solved the problems. data analysis i used the qualitative data from the classroom observations, student interviews, and field notes to address the research questions. i analyzed these data using a grounded theory approach (strauss & corbin, 1998). i began my analysis by conducting a general read-through of the data, paying attention to the data as a whole, and making analytic memos about the insights, patterns, possible themes, and categories emerging from the data. i then explored these categories in a holistic manner to answer the research questions, grouped them into meaning categories, and compared them repeatedly to identify any possible links (strauss & corbin, 1998). in coding the data, i used a combination of pre-established codes, drawn from my theoretical frameworks, and open codes, which emerged from my initial review of the data. during my analysis, i focused on the participants’ conceptions of mathematics and how they related mathematics to themselves and to their lives to investigate their mathematical experiences within the figured worlds of mathematics education in which they participated as learners. moreover, to gain an understanding of how the participants positioned and authored their identities, i analyzed all data from each of the participants, including interviews and classroom interactions. in my analysis, i examined what participation in mathematics “looked like” nzuki exploring identity and achievement journal of urban mathematics education vol. 3, no. 2 88 across time and how it might fit into each participant’s construction of identity and his or her sense of agency exhibited in the process. in analyzing students’ interaction with and use of the graphing calculator in the classroom and as they solved the mathematical tasks presented, i drew upon the framework described by goos et al. (2003). research site and context the school. graham high school was one of the four public high schools in a mid-sized, urban school district in the northeastern part of the united states. it was located on the west side of the city. from the state school report card, graham high school serves approximately 1000 students in grade 9–12, most of them from low-income families. the percentage of students eligible for free and reduce-priced lunch was 64%, 70%, and 68%, in the 2003/04, 2004/05, and 2005/06 academic school years, respectively. during the 2004/05 and 2005/06 academic school years, african american students had the highest enrollment compared to other racial groups (see table 2). additionally, student stability in the 2005/06 academic year was 47%, which was significantly lower compared to that of 75% and 86% in the 2003/04 and 2004/05 academic years, respectively. tracking and ability grouping of students in mathematics courses did not begin when students entered graham high school; it started in eighth grade. based on their teachers’ assessment of their achievement level, test scores and the final course grade from seventh grade, students were placed in either a “regular” mathematics course (lower track) or an integrated mathematics 1 course (upper track). middle school teachers recommended which students should be placed in which track when they began the 9th grade at graham high school; teachers recommended either a math 1 course (upper track) or math 1a (lower track). those students who were successful in the math 1 course and continued to be successful in the upper-track courses move to math 2, then math 3, and then take pre-calculus, calculus, and/or statistics courses before they graduated high school. these students took the state’s regent exams, math a and math b, after the completion of math 2 and math 3 courses, respectively. however, if an individual student did not perform “well” in any of the upper-track courses, he or she was relegated to the lower-track courses. the lower-track courses were slower in pace compared to the upper-track courses in the sense that it took a longer time period to cover the same material. thus, students were exposed to a different type of instruction and curriculum based on their track placement (oakes, 1985). students who were placed in math 1a (lower track) in 9th grade, if successful continued to the next lower track, math 2a. those students who did not pass these courses after the first attempt were required to repeat the courses. as such, a student could have been in the 10th grade, taking math 1a; or could have been in the 11th grade, taking math nzuki exploring identity and achievement journal of urban mathematics education vol. 3, no. 2 89 2a; and so on. from math 2a, the next track in the sequence was usually math 3a, which was the intermediate algebra iii course that i chose for this study. table 2 demographic factors of graham high school source: state school report card the teacher. at the time of the study, mr. samson was in his second year of teaching at graham high school. a white man in his late 20s, mr. samson held a b.s. in mathematics from a public university in the northeastern part of united states and a m.s. in mathematics education from a private university in the northeastern part of united states. mr. samson completed his student teaching and one of his field placements at graham high school. thus, in total, he had spent at most three years in the school during and after the completion of his master’s degree program. similar to most teachers in u.s. schools, mr. samson claimed that he held all of his students to the same “standard” regardless of their cultural background or ethnicity (williams & land, 2006). in other words, similar to many professed “race neutral” teachers, he did not attribute differences in academic performance between lowand high-achieving students to their racialized identities and, in turn, to their racialized learning experiences, but rather to their low-ses backgrounds. he often associated the low-ses backgrounds of his low-achieving students with the likelihood of having “dysfunctional” families that “caused” students’ problems such as skipping school to care for siblings, failing to do 3 student stability is the percentage of students in the highest grade in a school who were also enrolled in that school at any time during the previous school year. for example, if school a, which serves grades 9–12, has 100 students enrolled in grade 12 this year, and 94 of those 100 students were also enrolled in school a last year, the stability rate for the school is 94%. 4 the source of this data, which is the only public data available, does not explain this drop. it also does not explain why the percentage of the students with limited english proficiency is zero in 2003/04 and 2004/05. 2003/04 2004/05 2005/06 eligible for free lunch 54% 60% 59% student stability3 75% 86% 47%4 limited english proficient 0% 0% 8% racial/ethnic origin american or alaskan native 2% 2% 2% black or african american 38% 41% 41% hispanic or latina/o 18% 19% 20% white 39% 34% 32% asian or native hawaiian/other pacific islander 4% 5% 4% nzuki exploring identity and achievement journal of urban mathematics education vol. 3, no. 2 90 homework, or coming to school unprepared and/or hungry. furthermore, mr. samson supported the educational policy of academic tracking, particularly in mathematics, because the slower, lower-level tracks provided the students “more time to understand the material, maybe they can do more stuff so that they don’t have to do [homework]…maybe [the course] can be [taught] at a slower level for them to make them understand the material.” in the end, mr. samson’s short-term goal was to ensure that his students were excited about mathematics. but by his own account, he did not envision his students proceeding into mathematics related fields in the future. although mr. samson’s claims to race neutral standards and efforts to have students excited about mathematics taken collectively were well intended, by not recognizing the effects of the persistence and permanence of race and racism in the educational experiences of racial minority students, he, similar to too many u.s. teachers, was subscribing to a color-blind ideology that overlooks the salience of race and racism in the daily lives and educational experiences of racial minority students (ladson-billings & tate, 1995; solórzano & yosso, 2002; tate, 1997a; williams & land, 2006). under the pretense of race neutral policies, this color-blind ideology masks an underlying reality of racialized educational practices and policies such as tracking and ability grouping, low teacher perceptions and expectations, and high-stakes standardized testing that legitimize the placement of racial minority students in a subordinate position (williams & land, 2006). mr. samson’s support and justification of some of these practices not only failed to acknowledge institutional racism but also concealed “dysconscious racism”—an “uncritical habit of mind that justifies inequity and exploitation by accepting the existing order of things as given” (king, 1991, p. 135). moreover, the color-blindness ideology afforded mr. samson (and other white teachers) “a safe space” (williams & land, 2006, p. 581) that allowed him to not confront race-issues in regard to the academic experiences and performance of racial minority students. mr. samson instead pointed to students’ socioeconomic backgrounds as an explanation for the low academic achievement of many of his racial minority students. mr. samson’s sentiments are in line with other teachers (of any “race”) who subscribed to this color-blind ideology, using ses as a bad proxy for race. such teachers most often (un)consciously fail to conceptualize race as a social construction and mistakenly believe that racism is no longer an issue in the post civil rights era (ladson-billings & tate, 1995; williams & land, 2006). the endurance of this color-blind ideology among u.s. teachers—as suggested in my crt analysis of mr. samson—demonstrates the crucial need for teachers to be provided multiple opportunities to learn about and understand the complexities of the racialized identities and, in turn, racialized learning experiences of racial minority students in their teacher preparation and professional development programs (king, 1991; ladson-billings, 1999). nzuki exploring identity and achievement journal of urban mathematics education vol. 3, no. 2 91 the classroom. in his mathematics classroom, mr. samson usually began class by giving students some warm-up questions that were a review of the previous day’s lesson. as the students worked on the questions, mr. samson walked around the classroom to monitor the progress of the students and to collect their homework, which students generally placed in the upper left hand corner of their desks. in nearly each class period, mr. samson reprimanded somebody for failure to bring his or her homework. mr. samson employed whole-class instruction techniques for the most part and made use of the various resources in the classroom to present the material to the students. these resources include transparencies, overhead projector, chalkboard, and the smartboard interactive whiteboard. for example, he used the overhead projector to display the image of the graphing calculator whenever he wanted to integrate the use of graphing calculator. mr. samson often demonstrated the algorithm to solve a mathematical task and expected that the students would be able to follow the same sequence of steps when given a similar task to solve. mr. samson also expected the students to ask questions regarding a specific step in the problem-solving process at any time during the class period. it is also noteworthy that, although the students did not have a permanent pre-assigned seating arrangement and could choose their seat in the class, the students had their favored sitting positions. there were those who preferred to sit near the front, as was the case for caleb, and those who preferred to sit at the back, as amos did. one of the observations i made over time was that the students’ seating positions affected their participation in the classroom. more often than not, the students who sat in the front were more active in the classroom than those who sat at the back of the classroom who were relatively less engaged. there was a classroom set of ti-83 graphing calculators that students were free to use whenever they wanted to. mr. samson’s perception of the use of the graphing calculator technology of his students whom he felt did not have “decent math skills” influenced how he integrated the graphing calculator technology in the classroom. he believed the graphing calculator served as an impetus to the students’ mathematical problem-solving efforts. in this regard, the most exploited aspect of the graphing calculator technology was the facilitation of the multiple representations of mathematical tasks—algebraic, graphical, and tabular. the graphing calculator amplified the students’ speed and accuracy of problemsolving strategies like graphing and reviewing a table of values. thus, for the most part of his instruction, mr. samson and the students interacted with the technology as a servant. additionally, students occasionally exhibited a level of subservience to the graphing calculator technology through their failure to connect their mathematical knowledge with their graphing calculator knowledge in solving mathematical tasks. for example, students sometimes specified unreasonable nzuki exploring identity and achievement journal of urban mathematics education vol. 3, no. 2 92 values for the window range and, as a result, scaling interfered with their efforts to solve problems. findings the goal of this study was to investigate african american students’ racial, mathematical, and technological identities construction. the following discussion focuses on these identities and how they coalesce to influence the participants’ mathematical learning. racial identity all of the participants—the low achievers (annabel, amos, and danielle) and the higher achievers (caleb and karen)—were fairly consistent in indicating that being an african american was important to their self-definition, and they felt good about being african americans. for example, annabel talked about how proud she was to be an “african american…people of african descent,” while, when asked to describe herself, karen said, “i am black and am 16.” additionally, all the participants, associated with the collective term of race with a perception of “we,” meaning that they felt a strong and positive attachment to being african americans. as such, a collective identity played a part in their perception of self. moreover, the participants were aware of the societal constraints and challenges that affected the academic participation of many african american youth including being “downed” or devalued by the society. as an example, amos felt that “we are downed…they are just like we can’t do nothing…like one time i came here because i wanted to try to go to [name of university] and there were all white people and they looked at me like…they didn’t say anything but the way they looked at me…it was a mean look…i felt funny.” caleb argued: a lot of us…because of where we come from we do not have the type of…like the mental stability to be able to uphold…like i can’t go to college…i can’t do this or that because we are not in the right kind of environment around us to be able to do that…you see what i mean…i am lucky to have a mom that makes me want to do that, and she raised me to be the best i can be. other constraints they mentioned include (a) the lack of role models and encouraging family and community members who would make students appreciate the importance of academic (and mathematics) success, (b) the media that portrayed african americans as being involved predominantly in sports, music, violent and crime-related activities, (c) the lack of resources, and (d) the low expectation that society had about urban public schools and about the success of african american students. peer pressure had much to do with the courses afri nzuki exploring identity and achievement journal of urban mathematics education vol. 3, no. 2 93 can american students took, their attitudes towards school, and their academic performance. caleb also talked about peer pressure that increased with school level. he shared: i think it harder for us to do better in school because we have…i mean it depends on what kind of school you go to…and what race is dominant…i mean there is more peer pressure...you know what i mean? and you kinda like start falling in and start falling out of the good crowd into the bad crowd…like…you know what i mean? …and the bad crowd is really the guys who do not do good in school, don’t come to school, stuff like that…i mean the older you get as an african american, the harder it gets…unless you have education. thus, all the participants not only had strong and positive affiliation to their race but also they were cognizant of the obstacles and the social devaluation prevalent in society that face african americans. differences emerged in the ways in which the participants interpreted and negotiated these obstacles and the sense of agency they exhibited in the process. this difference, in turn, affected how they positioned themselves and the kinds of mathematical identities they authored. for example, annabel appeared to blame her african american status for her tarnished identity in mathematics. she seemed to associate the barriers and social devaluation that african americans face in the society with her poor performance in mathematics. thus, instead of exhibiting a sense of agency to overcome these barriers, annabel appears to have cast herself as a victim who was trapped in a society that looked down upon african americans. danielle was disinterested and unfocused on racial issues in the school setting. i did not take this to mean that she had adopted what fordham (1988, 1996) describes as a raceless persona—the notion that, for african american students to be successful they must distance themselves from african american cultural attributes. indeed, danielle saw herself as, and said that she was proud to be, an african american. her responses to questions pertaining to her perceptions of the impact of race on her academic experiences were, however, noncommittal. for example, in responding to a direct question about race and her academic experiences, she said, “honestly, i can’t even answer that question because i don’t remember anyone saying or doing something racial and if they did it was a joke…i take everything for a joke.” because danielle, in general, was aware of the racial bias in society against african americans, i took her response to mean that she was not interested in interrogating the salience of race in her academic pursuit. indeed, her focus seemed to be to “just finish high school and graduate.” this focus appears to coincide with her overall air of disinterest towards schooling and particularly towards taking mathematics courses. amos perceived his intellectual capability to perform well in the course was questioned by some white students and was unhappy about this experience. he nzuki exploring identity and achievement journal of urban mathematics education vol. 3, no. 2 94 remarked, “when i was going to this class there was a bunch of people, some white students were telling me i have to be smart to be doing mathematics…to be in the class…lots of stereotypes…it’s like you have to prove yourself.” these are some of the constraints that can cause african american students to feel ambivalent about their success in school and even doubt their own abilities. as an african american who faced these constraints, amos did not respond to them as a passive victim; rather, he reacted through resistance and criticism. he resisted by being argumentative and disruptive in class, which sometimes took away time from covering new material. his failure to do homework, which was another form of resistance, sometimes led to heated verbal exchanges between amos and mr. samson. additionally, amos blamed and criticized his teachers for their reactions towards him although he admitted his anger and attitude, which annoyed the teachers, played a part in these reactions. amos’s racial identity, which was influenced by the social devaluing of african americans, corresponds to the oppositional student described by fordham and ogbu (1986). fordham and ogbu offered an explanation of the school-resistance phenomenon as it relates to african american students in terms of the sense of collective oppositional identity that blacks develop in response to racial stigmatization by whites. like the low achievers, the high achievers, caleb and karen, were aware of the societal constraints and challenges that operated inequitably to affect the mathematical participation of many african american students negatively. in response to the social devaluation of african americans by society karen remarked: being an african american…like people do down you but if you put your effort, you don’t have to worry about what people say, that is how i feel it…like i don’t care…i am equal to everybody else, i can do the same thing everybody else can do…i don’t see what i cannot do because of my skin color…like i can’t learn mathematics, i can’t do this or i can’t do that…i don’t see nothing like that…i just think an african american can do whatever you want to do…like white people, they can down you a lot but you just stick around and do what you want to do to make it in life. caleb shared karen’s sentiments by saying, “i don’t care what people say,” and “there are certain things you have to prove people wrong and show them that you can do it.” he also attributed his success to his mother who kept pushing him to be the best he could be. thus, the awareness of these societal constraints and challenges shaped the high achievers’ authoring of positions for themselves as learners of mathematics and as african americans. unlike the low achievers, they perceived these constraints as a source of motivation and that as african americans they had to work twice as hard to overcome them, thereby demonstrating their sense of agency. their racial identity shaped their resourcefulness and resiliency in the sense that nzuki exploring identity and achievement journal of urban mathematics education vol. 3, no. 2 95 they did not see race as a limiting factor but a resource that empowered them. they had a firm belief that in spite of the obstacles and challenges that negatively affected the mathematical participation of many african american students, they had the capacity to overcome them and be successful in mathematics. mathematical identity martin (2000) defines mathematical identity as being shaped by students’ “beliefs about (a) their ability to perform in mathematical contexts, (b) the instrumental importance of mathematical knowledge, (c) the constraints and opportunities in mathematical contexts, and (d) the resulting motivations and strategies used to obtain mathematics knowledge” (p. 19). in examining students’ mathematical identity, it is imperative to draw from their past and perceived present experiences and to link these experiences to their anticipated future experiences. the students’ stories assist in this regard. by linking past, present, and future experiences, one can examine the students’ trajectory of mathematical experiences given that it is within this trajectory that students’ identities are developed and refined as a result of the cumulative effect of their mathematical experiences. to illustrate, take the case of two of the participants—annabel, a low achiever, and caleb, a high achiever. annabel’s mathematical identity could be traced back to her earliest unpleasant mathematical experiences. she claimed that “the first time i did division and i did a horrible, horrible job because i did not know what i was doing…i felt sad…and i lost the interest right there.” annabel remarked that she had since hated mathematics all her life because she “couldn’t learn it…they don’t stick.” annabel created a portrait of herself as a mathematics victim and blamed her failure on various constraints in the figured worlds of the mathematics classroom and school. she complained about her lower grade teachers who passed her to the next grade “even though they knew i didn’t know mathematics,” instead of giving her the help she needed. over and over again she said that she hated mathematics, and she didn’t know whether it “is because i am black american or not.” saying that she had failed mathematics all her life and that she did not have the essential basic skills which left her feeling helpless, she argued that she needed “an assessment of disability but they won’t do it.” she claimed that she never had a high point in mathematics and positioned herself as “just getting by” because “i have to take the course in order to graduate.” caleb’s trajectory, however, tells a different story. recalling his earliest memories of mathematics, caleb claimed that he was good at mathematics at first, but when it got to multiplication and division he started falling behind. he attributed this falling behind to his failure to work hard or pay attention. other constraints included the difficulty of the subject and also his lack of realization of the importance of mathematics. with time, however, there was a shift in his trajectory of participation in mathematics that was influenced by the positive identity that nzuki exploring identity and achievement journal of urban mathematics education vol. 3, no. 2 96 caleb developed upon realizing the importance of mathematics for college admission. this realization caused him to be more actively involved in the learning process by paying more attention, writing extra notes, and asking questions in class. such behaviors demonstrate his sense of agency through improvisation and innovation: i started paying more attention…i started writing stuff down and extra side notes like...instead of just the notes he [mr. samson] writes on the board…i write my own so that i may have my own way of remembering because sometimes when i don’t write my notes i don’t remember. additionally, linking past and present experiences to future experiences provides a lens through which one can examine students’ perceptions of who they are and who they would like to become—their occupational aspirations—and how this coincides with their mathematical identities. caleb’s future aspirations were “going to college and becoming a computer programmer” and as a consequence he saw himself being “big on mathematics because my major is technology, and technology and mathematics go together.” thus, caleb believed that, to be successful in his future career, he had to be good at mathematics; that is, have a positive mathematical identity. on her part, annabel also contemplated going to college and “major in education but i am not gonna teach like high school or middle school…’cause i don’t think i can teach mathematics beyond elementary level…so i will teach the elementary kids and i am gonna minor in african american studies.” as such, annabel’s perceptions towards mathematics seemed to have created tensions and limitations to her aspiration of becoming a teacher because she did not see herself capable of teaching beyond elementary school level mathematics. karen, a high achiever, aspired to join the nursing profession and believed that mathematics was useful in careers like nursing because “it helped in measurements.” she further believed that mathematics would help her gain admission to college. indeed, compared to low achievers, high achievers understood the role that mathematics would play in fostering or hindering their prospects of gaining entry to college. amos, a low achiever, believed that his football skills would earn him a place in college. his future aspirations were working in the music industry, becoming a professional football player, and becoming a computer engineer. however, unlike caleb, amos did not see the connection between being good at mathematics and becoming a computer engineer, because “you know how certain people grow up knowing things…i think i just grew up with computers,” implying that his knowledge about computers was innate, and he did not have to go through any formal training. as i got to know more about amos, i interpreted his behavior and actions to be that of a student who occasionally positioned himself as cocky and conceited. nzuki exploring identity and achievement journal of urban mathematics education vol. 3, no. 2 97 in other words, it was hard to talk him out of something. this positioning might have prompted the occasional arguments he had with his teachers who determined if and when he was wrong and at times ended up sending him out of the of the classroom. moreover, i found the way amos presented his abilities and achievements to be sometimes misleading and pretentious. this overstating might explain his self-reporting on the statements of the survey instrument, which had led me to initially classify him as a high achiever. when i asked about his experiences of the current mathematics course, amos, who was retaking the course after failing the first time, responded: i took the course last year and i failed the test—that is why they wanted me to take it again…i get everything in this class…this class is so easy. everything i can do because i have learned it before. i know all he [mr. samson] knows. amos’s overall performance and participation in the course, however, told a different story. indeed, while repeating a course may place a student at a somewhat more advantageous position to understand the topics than a student taking the course for the first time, the student will still have to put it in effort and work hard to author his or her positioning as a capable learner and doer of mathematics. all these attributes, which culminated in amos authoring a somewhat distorted mathematical identity, affected his participation in mathematics. danielle, another low achiever, perceived mathematics to be useful in her future aspirations of becoming a pediatrician because it helped in measurements of the “right amount of scoops and right amount of diets.” however, she did not quite appreciate the role that mathematics would play in facilitating the process of gaining entry to college and studying to eventually become a pediatrician. she claimed that she did not see herself taking any mathematics courses in the future. she described her mathematical experiences as “a pain in the neck, and every day is a low point for me” and that she took mathematics courses not because she wanted to but because she had to. no matter how hard she tried, she claimed she “still don’t get it…my brain cells are gone…i have probably like three brain cells working and they ain’t working hard enough…mathematics is just…i don’t care for mathematics…mathematics don’t care for me.” these comments seem to reveal that during her years of schooling she had developed a disinterested mathematical identity. further, her disinterest in the subject had reached to the point where she did not feel she had the agency to author a positive identity and position herself as a capable learner and doer of mathematics. closely related to the students’ future aspirations were their valuations of mathematics or their perceptions of the utility of mathematics. their perceptions appear to have been limited to experiences and functions to do with measuring, numeracy, and calculations of mundane tasks particularly pertaining to handling nzuki exploring identity and achievement journal of urban mathematics education vol. 3, no. 2 98 money and balancing a checkbook. karen’s statement captures this perception best: when you think about saving money you need to know how to add this, to multiply this, subtract that…and like when you go to the store, you need to know how much money to spent…you know…be able to add stuff up…they try to cheat you sometimes…so you need to know mathematics so they won’t. the students’ perceptions of the utility of mathematics not only reveal that mathematics had little purpose outside of school but also expose the disconnect between their in-school and out-of-school mathematical experiences. this disconnect could also explain the gap between the students’ perceptions of future aspirations and the importance of mathematics in helping them achieve those goals and aspirations. further, although all participants mentioned that they thought mathematics was important because it was a requirement for high school graduation, only the high achievers saw mathematics as important for success with future college coursework. to this end, therefore, there appears to be a connection between students’ perceptions of their future aspirations and the value and utility of mathematics on one hand, and the kinds of mathematical identities they develop on the other hand. when students see the usefulness and value of mathematics and its role in their future aspirations, they seem to develop more positive identities in mathematics and vice versa. it is also imperative to examine how students’ identities are shaped by both their negotiation of classroom norms with the teacher and among themselves, and also by their adaptation to the constraints and affordances of the figured worlds in which they participate (boaler & greeno, 2000). equally important is the individual agency exhibited by students in the process. one of the norms that emerged in mr. samson’s classroom was the expectation and obligation for students and the teacher to ask and answer questions. this study showed that this norm both constrained and afforded students’ opportunities to participate in mathematics. for example, asking questions was one of the aspects of caleb’s participation that made him visible in the classroom. this agentive role, which positioned him as a questioner, ensured that he was able to (a) seek clarification on concepts that he did not understand, and (b) pose challenging questions for the rest of the students to think about. this classroom norm of questioning afforded karen the opportunity to be an active participant in the figured world of the mathematics classroom where she positioned herself as an explainer. whenever the teacher asked a question, karen participated by both raising her hand and calling out answers. in some cases, the teacher would call upon a student to work a problem on the board, which is something that karen loved to do. her seating position in the front row of the class was beneficial because it helped the teacher to not only see her hand first whenever he nzuki exploring identity and achievement journal of urban mathematics education vol. 3, no. 2 99 asked a question but also to easily hear her responses. as for annabel, because of her lack of confidence in mathematics, she believed that asking questions in class would waste time for other students. as such, the figured world of the mathematics classroom where she had to interact with other students created a constraint for annabel that prompted her to avoid participating in the classroom. some of the constraints in the classroom included disruptions from other students talking in class. high achievers exhibited a sense of agency to overcome the temptation of giving in to pressures from their peers and misbehaving in class. in contrast, low achievers were more prone to being distracted and losing focus. the many students in the classrooms also constrained the opportunity to participate for low achievers because they felt that they had to compete for the teacher’s attention with the other students. high achievers, caleb and karen, exhibited a sense of agency to overcome this constraint by sitting in the front where the teacher could easily spot them. the perception that some students, particularly the low achievers, lacked the skills and understandings necessary to perform mathematical tasks also created a constraint that limited their participation. we see, therefore, that students’ effort to enact agency within the figured world in which they participated created spaces that were inherently full of tension that holland et al. (1998) referred to as contested spaces or spaces of struggle. these contested spaces help to illuminate the interplay between students’ participation in figured worlds and the sense of identity and agency they exhibit in the process. moreover, how students positioned themselves and authored their mathematical identities was influenced by how they negotiated the classroom norms and the constraints and affordances in the figured world of mathematics learning in which they participated. it is through these negotiations that they exhibited a sense of agency that afforded or constrained their opportunities to learn and participate in mathematics. technological identity to describe students’ technological identities, i engaged the participants in problem-solving tasks that were similar to those they had worked in class during the course of this study and examined the varying degrees of sophistication with which students interacted with and appropriated the use of the graphing calculator to solve the mathematical tasks. in doing so, i drew from goos et al. (2003) and their framework that theorizes four metaphors of technology usage: technology as master, technology as servant, technology as partner, and technology as an extension of self. the lowest level of interacting with technology is that of technology as master, while technology as an extension of self is the highest level of functioning. the graphing calculator amplified the participants’ problem-solving strategies and scaffolded them through the tasks where they lacked the algebraic facil nzuki exploring identity and achievement journal of urban mathematics education vol. 3, no. 2 100 ity. by influencing the participants’ capabilities to solve mathematical tasks, the graphing calculator had an effect on how the participants positioned themselves and authored their identities as learners and doers of mathematics. however, the extent to which this authoring happened was contingent upon how the participants positioned themselves in using the technology—the modes with which they interacted with graphing calculators. for example, karen, a high achiever, always took time to go through the tasks to make sure that she fully understood their requirements. she showed her sense of control of technology in knowing when and how to use the graphing calculator. for instance, in determining the maximum height of a baseball in the air after it is hit, karen knew it was important to have a good view of the graph of the quadratic model. before going to the window function to adjust the window to a reasonable domain and range, karen looked at the table function to get a sense of how big or small the values were. this sense making also enabled her to get a rough idea of what the maximum value was. by doing so, she blended her mathematical knowledge (of domain and range) and her graphing calculator (of adjusting the window). she then used the maximum function of the graphing calculator to find the maximum value and correctly interpreted this output to be the maximum height of the baseball in the air. consequently, karen’s ability to blend her mathematical knowledge and her graphing calculator knowledge to interpret the output of the technology meant that the use of graphing calculator did not result in the replacement of her mathematical skills but rather formed an extension of those skills and her abilities to solve the mathematical tasks. karen’s process represents the highest functioning of technology as an extension of self, which shaped her positioning as a capable doer of mathematics. nonetheless, in a few instances, caleb, another high achiever, failed to blend his mathematical and graphing calculator knowledge, which hampered the effectiveness of the graphing calculator use. based on when and how he appropriated the use of the graphing calculator, however, caleb exhibited a sense of autonomy and control of the technology that could be enhanced by more guidance and support from the teacher and enabled him to achieve the highest level of functioning—technology as an extension of self. among the low achievers, annabel and danielle, there was the tendency to immediately reach out for the graphing calculator even before, in my estimation, they had thoroughly understood the task. this tendency was particularly evident for tasks where the algebraic equation was given because they could easily enter the equation into the y=editor of the graphing calculator. while they may have used the graphing calculator as a starting point with the goal of getting oriented to the task, failure to take time to reflect on the information relevant to the mathematical task at hand affected how they appropriated the use of graphing calculator. when confronted with tasks where they had to find the algebraic equation, nzuki exploring identity and achievement journal of urban mathematics education vol. 3, no. 2 101 they were unable to determine the equation and were, thus, unable to solve the task altogether. the lack of strategic use of the graphing calculator, however, was not the only issue that affected how the participants, particularly the low achievers, interacted with technology and hence the kinds of identities they exhibited in the process. another important issue was how students blended their mathematical and graphing calculator knowledge in solving the problems. for example, in some cases, the low achievers were unable to juxtapose the mathematical knowledge (of domain and range) and the graphing calculator knowledge (of adjusting the viewing window). this failure restricted them to lower levels of interacting with technology. in addition, given that a student’s ability to use technology influences how he or she interacts with technology, amos (another low achiever) resisted the use of graphing calculator by saying: “the graphing calculator has many steps…you have to press the 2nd [key] and do this and that…you are not going to be able to remember all of that—it’s just that it is too many steps to do it…i don’t like using it…i will never have a calculator in the class unless somebody gives it to me.” as degennaro and brown (2009) point out, having the knowledge and seeing the need to use technology is an issue of identity concern. this resistance impacted amos’s knowledge base concerning using the graphing calculator technology and shaped his potential interaction with the technology. it is also important to point out how the figured world shaped the participants’ interaction with graphing calculators. in this regard, the teacher’s role in integrating technology into classrooms is pivotal given that it is the teacher who guides instruction and shapes the instructional context in which the technology is used. this guiding and shaping implies that the decisions the teacher makes pertaining to the use of the graphing calculator directly affect students’ technological identities as they interact with the technology to perform different tasks. first, by his own account, mr. samson did not think that, “a lot of those kids are going to go into mathematics fields outside of high school…i think they are so phobic of mathematics and everything in mathematics scares them.” mr. samson, in addition, believed that the graphing calculator was most beneficial to students who “have decent mathematics skills.” second, mr. samson admitted that he had not spent as much time in class teaching them how to use graphing calculators because he had expected them to have had acquired the graphing calculator knowledge already. unfortunately, most of these students were products of previous lower-tracked mathematics courses where the appropriate use of graphing calculators was limited. mr. samson, however, remarked that in the future he would make a concerted effort to ensure that his students had an adequate knowledge of using the technology. third, because most of the students did not own their own graphing calculators, mr. samson did not assign homework that required the use of graphing calculators. mr. samson found this lack of ownership problematic be nzuki exploring identity and achievement journal of urban mathematics education vol. 3, no. 2 102 cause students did not get the opportunity to hone their skills in using graphing calculators to solve mathematical tasks. discussion this study contributes to knowledge on issues of african american students’ identity construction—racial, mathematical, and technological—and how these identities shape each other, and the sense of agency exhibited in the process. in doing so, i injected the role of race and racism and their relationships to mathematics learning and participation by exploring how the overlapping learning communities including school, peer, family, community, and societal forces shape students’ construction of themselves as learners of mathematics (martin, 2000). this study also brought out the individual agency that students enact within the figured worlds or learning communities in which they participate as learners of mathematics. the analysis demonstrated that students’ efforts to enact agency created contested spaces or spaces of struggle (holland et al., 1998), which were inherently full of tension. additionally, how students positioned themselves and authored their mathematical identities were influenced by how they negotiated the classroom norms and the constraints and affordances in the figured world of the mathematics learning in which they participated. it is through these negotiations that they exhibited a sense of agency that afforded or constrained their opportunities to learn and participate in mathematics. to some extent, this study offers an explanation as to why, in spite of all the participants facing similar constraints as african americans who came from similar ses backgrounds and attended the same class, taught by the same teacher, there were both high and low achievers. as such, this study’s focus on students’ identities revealed how both success and failure can exist within the same figured worlds and learning communities of the students. this study also challenges fordham and ogbu’s “burden of acting white” theory (1986; see also fordham 2008; ogbu, 2004). this theory suggests that long periods of oppression and discrimination by the dominant society have led african americans to develop responses and behaviors that emphasized their distrust of and opposition to the dominant society and its institutions, including schools. a significant limitation of this theory is that the notions of agency and resistance are not thoroughly interrogated, if at all. as foley (1991) noted: this perspective sees african americans as “discouraged and trapped in the racist myths of the dominant society. they are unable to see that they can both be successful and black” (p. 77). by so doing, this theory disregards the ability and facility of disenfranchised groups to respond to the oppression and the struggles in their lives with a sense of agency and positive resistance. in other words, this theoretical perspec nzuki exploring identity and achievement journal of urban mathematics education vol. 3, no. 2 103 tive “underestimates the capacity of ethnic resistance movements to empower individuals” (foley, p. 78). in his critique of fordham and ogbu’s (1986) theory, stinson (2010b) claims that it offers an insufficient justification as well as an oversimplified explanation for the black–white achievement gap. his critique emanates from a participative inquiry study (stinson, 2004) in which he examined the influence of socio-cultural and -historical discourses on the agency of four academically and mathematically successful african american male students. stinson defined agency as “the participants’ ability to negotiate—that is, to accommodate, reconfigure, or resist—the available sociocultural discourses that surround male african americans in their pursuits of success” (p. 3) the participants were requested to respond to theoretical perspectives that discussed the schooling experiences of african american children, including fordham and ogbu’s burden of acting white theory. stinson (2010b) explains that throughout the participants’ “individual and collective counter-stories of success there were instances where they managed the burden of acting white by accommodating, reconfiguring, or resisting the discourse” (p. 15). further, the participants’ counter-stories demonstrated their “complex, nuanced, and multilayered schooling experiences” (p. 18). stinson argues that these experiences provided a context within which the participants negotiated, with a sense of agency, the discourses that unjustly shifted the responsibility for african american students’ underachievement away from factors like the structure of u.s. public schooling and onto the shoulders of the students themselves. stinson contends that it is this contextualization that negates the oversimplification posed by fordham and ogbu’s (1986) theory. additionally, according to fordham and ogbu’s burden of acting white theory (1986), the dysfunctional aspects of african american students that undermine their academic success are adaptations to the hostile sociocultural discourses that they face and not inherent cultural traits as posited by the deficit theory. however, like the participants in stinson’s (2010b) study, the participants in this study, particularly the high achievers, did not respond to the unfavorable discourses by adapting to them but by negotiating them within the figured worlds in which they participated. thus, the factors that enhance or constrain meaningful academic and mathematics participation of african american students should be contextually traced in how they negotiate and interpret the socio-cultural and socio-historical forces that shape their schooling experiences (stinson, 2006, 2008, 2010a). in the context of learning mathematics, foley’s (1991) and stinson’s (2010b) remarks are in alignment with the recent emphasis on student identity and agency by other researchers who examine issues of equity in mathematics education (e.g., gutstein, 2003; martin, 2000, 2006a, 2006b; nasir & hand, 2006). nzuki exploring identity and achievement journal of urban mathematics education vol. 3, no. 2 104 these researchers use the notion of agency to examine the importance of historically underachieving students having a robust and empowering identity in relation to mathematics. additionally, this perspective serves to challenge the stereotypical and essentialist perspective that often comes as an entailment of classification—the notion that members of a particular ethnic, racial, socioeconomic, gender, or other group, by virtue of their membership in that group, all share important attributes and therefore can be treated in ways appropriate for members of that group. as martin (2000) and nasir (2002) contend, this shift in perspective affords students the sense of agency to act and change their experiences in spite of the constraints and obstacles that they face. from this viewpoint, learning is therefore associated with the practices in which students engage, and as such the learning communities within which they participate shape the identities and agencies that students develop. given that these learning communities influence students’ mathematical background, and experiences that they come with in the figured world of the mathematics classroom, they ultimately shape the students’ perceptions of their future aspirations, the value and utility of mathematics, and their ability to do mathematics, all of which are connected to their mathematical identities. indeed, when students see the usefulness and value of mathematics and its role in their future aspirations, they seem to develop more positive identities in mathematics and vice versa. focusing on students’ development and enactment of their identities within the mathematics classroom offers mathematics educators an opportunity to gain an understanding of the contested spaces within which students negotiate the classroom norms and the constraints and affordances in the figured world of the mathematics learning in which they participate. given that it is through these negotiations that students exhibit a sense of agency which affords or constrains their opportunities to learn and participate in mathematics, mathematics educators, including teachers and administrators, can begin to look into factors that affect, in a positive or negative way and at a more personal level, students’ participation in mathematics. one of the important aspects of this study regards listening to the voices of the students in examining their identities. the counter-stories told by the students brought to the fore the confluence of the forces that filter into the figured world of the mathematics classroom, affecting their perspectives of themselves, their abilities in mathematics, and their future goals and aspirations. to ensure that these students have an opportunity to learn, special consideration must be given, at a more individual level, to the factors that affect their identities. mathematics educators and researchers, therefore, need to develop and identify strategies to help students with negative mathematical identities. students like annabel, who had a disabled mathematical identity, need numerous opportunities to experience success in mathematics. these successes can nzuki exploring identity and achievement journal of urban mathematics education vol. 3, no. 2 105 be achieved through recognizing and rewarding small accomplishments and diversifying assessment methods to allow students to demonstrate multiple capabilities and intelligences of participating in mathematics. according to wenger (1998), learning is a process of identity formation in which learners locate themselves within particular communities in a process of belonging and ultimately knowing. thus, if students like annabel and other low achievers are given opportunities to be successful in learning mathematics, these experiences can relate and become part of their trajectories. to increase the participation of these students, and especially students like danielle who are disinterested in mathematics, teachers must ensure that the figured worlds of mathematics classrooms are learning environments that motivate, encourage, challenge, and empower students. one way of creating such environments is by introducing instructional activities that relate mathematics to students’ life experiences and communities. low achievers like amos, with distorted perceptions of mathematics, can benefit from an awareness of themselves as students and as future adults. these types of students need to know that their future success and life chances depend on their current actions and academic performance. moreover, the societal and school discourses that have a disproportionately negative affect on the academic and mathematical participation of many african american students point to the racialized schooling experiences of these students (martin, 2006a). in other words, being an african american within the figured world of mathematics learning influences how one is socially constructed and framed as a learner and doer of mathematics (martin, 2006, 2009). for example, research has shown that teachers’ preconceived notions about african american students may guide differential expectations for, and interactions with, these students and impact their academic achievement (see, e.g., berry, 2008; pringle, lyons, & booker, 2010). in his study, consisting of eight african american middle school boys who had experienced success in mathematics, berry (2008) reported that teachers who, unjustly, had the propensity to focus on his participants’ real or perceived behavioral problems instead of their academic work had discouraged most of them from taking advanced mathematics courses. in addition, amos’s discipline issues and frequent brushes with authority figures point to a form of institutional racism that contributes to the marginalization of african american students through the construction of black identity as an oppositional social identity that is in need of discipline, punishment, and control (ferguson, 2000). there is consensus among most educators and scholars about the importance of having safe and orderly environments for meaningful teaching and learning to take place. consequently, in a society where the construction of race shapes the interpretations of behavior, trying to keep an orderly learning environment can interfere with the learning opportunities of certain groups of students, particularly african american male students. berry (2008) argues that the nzuki exploring identity and achievement journal of urban mathematics education vol. 3, no. 2 106 unfair judgment of african american students’ academic ability based on their behavior undermines their success because “their behavior could be in reaction to symptoms of structural issues, such as teachers not being multi-culturally competent” (p. 479). as such, to assist in deconstructing this perception and possibly other misconceptions about african americans, teachers need to be prepared to teach different racial/ethnic or cultural groups. providing teachers and other school personnel with opportunities to become familiar with african american history and culture through staff development sessions or by attending classes that encourage positive counseling methods to enrich the lives of their student populations would assist in this deconstruction. moreover, the notion that students do not have control of their academic destiny, and that no matter how hard they try failure is inevitable, is yet another ideology of intellectual helplessness, exemplified by low achievers like annabel, that teachers and school personnel in general must be prepared to counter. according to perry, steele, and hilliard (2003) school personnel should make efforts to organize sessions that are inspirational and motivational and also attempt to create various environments within which students can construct positive academic identities. one way of organizing such sessions would be through inviting prominent and successful african americans, such as mathematicians, lawyers, scientists, professors, to schools as guest speakers to speak to the students as role models and instill in them a sense of optimism—that despite the constraints, they too can be successful. indeed, a study conducted by zirkel (2002) revealed that having the same race-and-gender-matched role models were “significantly and consistently predictive of a greater achievement concerns” (p. 371) on the part of minority students. having matched role models can provide concrete information to minority students in regard to what is possible for them as members of specific social groups. zirkel noted: similar others in desirable positions may enable young people to construct their own images of themselves in similar contexts, helping them to generate not only the thought “if he (or she) can do that, maybe i can too,” but also “if he (or she) can do that, maybe people like me can do any number of different things.” (p. 359) thus, having the same race-and-gender-matched role models can impact the development of identities of minority students and their academic achievement. in addition, the high achievers in this study reported that one of the factors that influenced their academic achievement was their parental support, expectations, and involvement. this is consistent with other research findings (e.g., berry, 2008; ladson-billings, 1995), which indicate that parental involvement in terms of advocating for, helping, motivating, and encouraging their children at home and school contributes to improved academic performance, behavior, and self-esteem of minority and low-ses students. nzuki exploring identity and achievement journal of urban mathematics education vol. 3, no. 2 107 moreover, the role of peers in the socialization process of african american youth and in shaping their racial identities and academic achievement is paramount. fordham and ogbu (1986) attributed the underachievement of african american students to oppositional behavior that involved peers in school settings. this opposition was brought about by the notion that because black students viewed academic success as the domain of whites, this was seen as fundamentally in opposition to black culture and established achievement norm of peer groups. as caleb, one of the high achievers, argued, an african american student has to exhibit individual agency that defies the negative influences of peer group pressure in order to be academically and mathematically successful. berry (2008) reported that it is this sense of agency and resiliency to overcome peer pressure that influenced his mathematically successful participants to construct alternative identities. berry further argued that in doing so the participants did not develop alternative identities that chose white culture over african american culture; rather, they used racial identities and parental discussions of racialized experiences to promote achievement. three primary components influenced these alternative identities: (a) co-curricular and special academic program identity (b), religious identity, and (c) athletic identity. (pp. 482–483) research has also shown that the pervasive notion regarding negative peergroup influences among minority students does not always hold true. indeed, the role of supportive and high-achieving peer groups has been reported as instrumental in promoting the mathematical achievement of minority students (walker, 2006). in her study involving high-achieving minority students at one of the highpoverty schools in new york city, walker found that collaboration and peer support played a critical role in supporting the participants’ mathematical knowledge. walker noted that the participants helped each other in several ways and on several mediums, with academic work (tests, homework), with advice about problem solving and course taking, and with encouragement. also they helped each other in multiple settings; in class, out of school, on the phone, and in the cafeteria (as well as other ‘school spaces’). (p. 68) it was through the intellectual collaboration and interaction with their peers that walker’s participants were afforded the opportunity to co-construct their individual and collective identities in relation to mathematics. research reviews also reveal that educators and school personnel should capitalize on cultural learning styles and culturally relevant curricula because students bring different cultural patterns to the classroom through language use, problem-solving techniques, and interactional styles. they also bring different prior experiences and frames of reference for imagining concrete applications of abstract ideas. school administrators should provide instruction that supports nzuki exploring identity and achievement journal of urban mathematics education vol. 3, no. 2 108 these varied cultural styles and experiences using culturally relevant materials (delpit, 1995; ladson-billings, 1995; reyes & stanic, 1988). to ensure that classroom-learning experiences are meaningful and relevant to african american students, instructional strategies should be employed that allow them to see the connections between what they learn in the classroom and the mathematics that occurs in their day-to-day lives. that is to say, their appreciation of the value and importance of mathematics needs to be expanded to go beyond numeracy and calculations of mundane tasks particularly limited to money. this expansion is essential in fostering the development of identities related to mathematics. moreover, one way of ensuring that african american students effectively develop technological identities is to give them the opportunity to create with and learn through technology (degennaro & brown, 2009). this goal can be accomplished by using graphing calculators to model and interpret real-world situations using data collected from sources and activities that connect to the students’ everyday experiences (gutstein, 2003). to achieve the nctm’s (2000) goal of ensuring that access to and use of technology in the mathematical learning process of students does not result in another dimension of inequity, it is also important to look into factors that relegate the students’ use of graphing calculator technology into lower levels of functioning. for one, failure to plan and implement a strategy that appropriates when and how to use the graphing calculator so that its use is not always the starting point has the danger of technology taking the place of, instead of extending, students’ conceptual understanding, computational fluency, or mathematical problemsolving skills. there is no doubt that technology has the potential to positively impact students’ mathematical identities. however, this depends on how technology is appropriated in the mathematical learning of students. in the lowest level of interaction, technology as master, students’ lack of mathematical understanding and subservience to technology influences them to blindly accept the output generated by the technology irrespective of its accuracy or worth. for students to develop the capacity or sense of agency to question the output of the graphing calculator, they need to have a mathematical knowledge base. through guidance and support, teachers can provide students with opportunities to blend their mathematical and technological knowledge base, given that this will allow them to interpret the graphing calculator output and judge it against the pertinent mathematical criteria. in other words, as demonstrated in this study, the blending of the mathematical and graphing calculator knowledge bases is important in facilitating the co-construction of african american students’ technological and mathematical identities. low-achieving students can learn a few lessons from the high-achieving students. for example, they can learn that they have the capacity to author positive mathematical identities by responding to the constraints and affordances of nzuki exploring identity and achievement journal of urban mathematics education vol. 3, no. 2 109 the figured worlds in which they participate as learners and doers of mathematics with a sense of agency. indeed, while the high achievers negotiated the unfavorable discourses of the figured worlds with a sense of agency, the low achievers appeared to adapt to these discourses with a sense of helplessness, constraining their opportunities to participate in mathematics. regarding racial identities, the low-achieving students can learn that being in touch with one’s own racial heritage should not lead to opposition but can and should be resourceful and powerful, as was the case for karen and caleb. the high achievers demonstrated that increasing knowledge about their heritage and being aware of the societal challenges and barriers that negatively affect the academic and mathematical participation of african american students can make students better prepared to face these obstacles. to these students, racial identity was a positive and motivating source that made them work even harder. as such, these students showed that being an african american and being successful in mathematics are not mutually exclusive occurrences. when interacting with technology to solve mathematical tasks the low-achieving students can learn from the high-achieving students the necessity to exhibit a sense of control of the technology—understanding and knowing how and when to use the tool. this sense of control, however, requires that students first take the time to carefully read the mathematical tasks presented and make sure that they fully understand the requirements of the tasks. it is this sense of control when using technology that often affords students the ability to mutually reinforce their mathematical knowledge and their graphing calculator knowledge and thereby interact with technology at higher levels of functioning. conclusion this study showed that both success and failure do exist within the same figured worlds and learning communities of students. it is the individual agency that students exhibit as they participate in the figured worlds of mathematical learning that determines the kinds of identities they author. thus, in our efforts to improve the mathematical learning and achievement of african american students, and indeed all other historically marginalized groups, it is important to inculcate into the students this sense of individual agency—the recognition that an individual has the capacity to act and make choices. it is the understanding that, in spite of the odds and the seemingly insurmountable obstacles that one may face, people have the capability to overcome these constraints if they make the appropriate choices of who they want to be. to ensure that african american students have opportunities to learn mathematics, our challenge as mathematics educators is to find ways of providing students with opportunities to develop the sense of agency that will ensure that they author positive mathematical identities. african american students need to nzuki exploring identity and achievement journal of urban mathematics education vol. 3, no. 2 110 understand that it is not contradictory to be an african american and be successful in mathematics. african american students, and indeed all students, i believe, can navigate their way through the vast array of obstacles that they may be confronted with and become successful in mathematics. acknowledgments the author would like to thank his colleagues pam cross and frank cerreto, the anonymous reviewers, and jume editors for their critiques, comments, and suggestions on earlier drafts of this article. references adams, t. l. 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(2002). is there a place for me? role models and academic identity among white students and students of color. teacher’s college record, 104, 357–376. nzuki exploring identity and achievement journal of urban mathematics education vol. 3, no. 2 114 appendix a survey instrument part i (cont. on next page) key: sa: strongly agree a: agree n: neutral d: disagree sd: strongly disagree statement sa a n d sd 1 i enjoy learning mathematics. 2 graphing calculators make mathematics fun. 3 doing mathematics is doing something, which i think i just can’t do. 4 i do not prefer working problems with a graphing calculator. 5 i am good at mathematics. 6 i feel that the use of graphing calculators has caused a decline in my basic arithmetic facts. 7 if i had my choice this would be my last mathematics course. 8 mathematics is useful for solving everyday problems. 9 graphing calculators motivate me to do mathematics. 10 i am looking forward to taking more mathematics courses. 11 no matter how hard i try, i am not the type to do well in mathematics. 12 i am able to do more interesting mathematics with graphing calculators. 13 i feel confident in solving problems in mathematics. 14 i find mathematics to be very boring and dull. 15 i try math harder when i have a graphing calculator. 16 i have a lot of self-confidence when it comes to mathematics. nzuki exploring identity and achievement journal of urban mathematics education vol. 3, no. 2 115 17 i understand mathematics better if i solve problems using paper and pencil. 18 i will use mathematics in many ways as an adult. 19 learning mathematics involves mostly memorizing. 20 using graphing calculators makes me a better problem solver in mathematics. 21 i see mathematics as a subject i will rarely use in daily life as an adult. 22 a graphing calculator enables me to solve problems i could not solve before. 23 learning mathematics mostly involves exploring problems to discover patterns and make generalizations. 24 i rely on my graphing calculator too much when solving problems. 25 it is important to know mathematics in order to get a good job. part ii please fill in the following information: 1. your name_____________________________________ 2. gender: (check one) male_________ female________ 3. how would you describe your race/ethnicity? (you can check more than one)  african american  hispanic/latina/o  american indian/alaskan native  asian/pacific islander  white  other 4. experience with and use of graphing calculators (check one)  i have the ability to work with graphing calculators yes______no_____  i use the graphing calculator to do my homework yes______no_____  i have my own graphing calculator which i use at home yes______no_____  my teacher allows the students to use graphing calculators whenever they feel like it yes______no_____  there are enough graphing calculators in my classroom for all the students yes______no_____ journal of urban mathematics education december 2016, vol. 9, no. 2, pp. 29–32 ©jume. http://education.gsu.edu/jume _______________________ *editor’s note: in the spring/summer 2015 issue (vol. 8, no. 1) jume published, as a commentary, dr. danny bernard martin’s invited plenary address delivered at the nctm research conference april 2015 in boston, massachusetts (martin, 2015). in the fall/winter 2015 issue (vol. 8, no. 2), jume published a response commentary, authored by drs. diane j. briars, matt larson, marilyn e. strutchens, and david barnes (briars et al., 2015). the response commentary here continues this important discussion; we invited others to keep things going while they are still stirring (see “contributing a commentary to jume: keeping things going while they are still stirring”). bryan meyer is a mathematics teacher on special assignment for the escondido union high school district, 302 north midway drive, escondido, ca, 92027-2741; email: meyer.bryan@gmail.com. his interests include students’ mathematical identity and authority, the transformation of inequitable and unjust systems in mathematics education, and the function of education in society. response commentary* a critical dialogue: continuing the conversation about “the collective black and principles to actions” bryan meyer escondido union high school district n a previous volume (vol. 8, no. 1) of the journal of urban mathematics education (jume), professor danny bernard martin, in his commentary “the collective black and principles to actions” (martin, 2015), provided a thoughtful and direct critique of the national council for teachers of mathematics (nctm) and its latest policy document principles to actions: ensuring mathematical success for all (nctm, 2014). in the following issue of jume (vol. 8, no. 2), members of the nctm leadership responded with an open call for engagement with the issues raised by professor martin (see briars, larson, strutchens, & barnes, 2015). as a teacher on special assignment for mathematics in a high-poverty school district, as a critical educator, and as a human being, the issues raised in professor martin’s commentary are of both personal and professional importance to me. i have struggled in thinking through how i might engage in contributing to this ongoing dialogue: what is the purpose of my response? how might i express my views and concerns while also acknowledging that i don’t have answers to his hard questions? what is my role as a white, male educator in getting involved (or not) in the remaking of mathematics and mathematics education, which professor martin describes as “white institutional spaces” (martin, 2015, p. 20)? still, i am compelled to respond, even if imperfectly. there are reasons to appreciate the contents of principles to actions. as noted by martin (2015), it outlines what many would recognize as aspects of good teaching practices. the document has been helpful in my work with teachers and it is useful to have an organization like nctm speak openly about the need for instructional i http://education.gsu.edu/jume http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/308/193 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/308/193 mailto:meyer.bryan@gmail.com meyer response commentary journal of urban mathematics education vol. 9, no. 2 30 change. yet, as martin points out, these practices are not new and there has been little progress toward erasing inequitable outcomes in the context of these (and previous) recommendations. it raises questions for me: why is that? why does nctm not openly acknowledge that? how might the framing and policies of mathematics education actually be causing these problems? most of my concern is with respect to what nctm is silent about and/or about the unspoken messages of their current framing and policies. some of those concerns are:  “equity” is framed almost exclusively in the dominant terms of access and achievement. as gutiérrez (2009) has written, “access and achievement can be thought of as the dominant axis, preparing students to participate economically and privileging a status quo” (p. 6). the dominant axis measures how well students can “play the game called mathematics” (p. 6). by contrast, her critical axis attends to issues of identity and power, and “ensures that students’ frames of reference and resources are acknowledged in ways that help build critical citizens so that they may change the game” (p. 6). i know that nctm is aware of these critical perspectives, as the research is included in the reference section of principles to actions. why don’t these issues feature more prominently in the framing of equity and in the recommendations for practice?  in the opening of principles to actions, there is a section titled “progress and challenge.” in it, progress and challenges are referenced in terms of dominant measures—naep scores, pisa scores, act/sat scores, ap course taking patterns, “college and career readiness,” and “readiness for college mathematics” (nctm, 2014). how we choose to report progress and challenge reveals something about our educational goals, yet those goals are not clearly stated. broader, or different, educational aims— perhaps including aims such as self-actualization, critical consciousness, intellectual autonomy, strengthening community, anti-bias education—would likely lead to a different set of measures. what does nctm take to be the goals of education?  nctm’s participation in the “mathematics for all” (or, “with math i can”) narrative continues to perpetuate the notion that dominant mathematics is synonymous with power and intelligence; and, by contrast, that those not having been “bestowed this gift” through a dominant mathematics education are mathematically lacking and/or powerless (lawler, 2005; martin, 2003). it is worth asking: what or whose mathematics are we talking about? what or whose interests does it serve? an alternative framing might recognize that “all people are mathematical.” such a position would refute meyer response commentary journal of urban mathematics education vol. 9, no. 2 31 the ontological status of mathematics; make explicit attempts to include multiple ways of being mathematically smart; recognize the mathematical activity of various individuals and (sub-)cultures; and resist measuring people against a predetermined, dominant mathematics.  in addition to its positive impacts, mathematics can and has caused harm and oppression both in education and in society (d’ambrosio, 1990; skovsmose, 1994). educationally, mathematics has played a role as a gatekeeper, resulted in intellectual trauma, and been used as a tool for the preservation of white privilege (e.g., through the justification of tracking). in society, mathematics has been used to further capitalistic priorities, support the development of harmful machinery and weaponry, and preserve or exacerbate systemic racism. why is nctm virtually silent about the role of mathematics as an instrument of oppression? what policies and practices might be recommended if they wholeheartedly supported a mathematics education that sought to improve the world?  what is crowded out of schools by a bloated eurocentric (joseph, 1997) mathematics curriculum? what questions and inquiries are “sanctioned” by the current standards and purposes of mathematics education? what opportunities are there for all students to see themselves in their educational experiences through mathematics? these are not issues i have seen nctm take up in principles to actions, or otherwise.  situating school (mathematics) in our historical and cultural context might be helpful in understanding the educational experiences of students who are marginalized by society. schooling and education operate in a racist, classist, patriarchal society. at least, we must acknowledge how education functions in that societal context and, at best, we must wrestle with how education actually functions to replicate that social order. how might nctm recognize the ways that racism manifests itself structurally in schools and interpersonally in classrooms? what recommendations, practices, or policies might help combat the racist, sexist, and classist (and others) issues? my hope in writing this response commentary is to keep the critical dialogue going and to provide a practitioner’s perspective and support to the issues raised by professor martin. these are not merely theoretical issues. there are implications for the ways these policies, framings, and professional recommendations impact our work with children. as practitioners, we have been charged with moving from “principles to actions,” but what are we to do if the institutional principles guiding the project of mathematics education result in harmful educational practices or lead to an education that preserves the status quo? we are forced to carve out a creative space meyer response commentary journal of urban mathematics education vol. 9, no. 2 32 for ethical practice when our institutional equity slogans are misaligned with the educational hopes of marginalized children and families. i am abhorred and increasingly frustrated with the way this misalignment often results in locating the problem with the individual student or, worse, with the generalized racial or socioeconomic group we take that student to identify with or belong to. the aforementioned bulleted points are by no means meant as an exhaustive list, but reflecting on them has assisted me to think through some of the ways nctm is currently silent about significant issues as well as how their current framing of issues might be problematic. i realize that the questions raised here are not new (see, e.g., freire, 1970; martin, 2003) and i am disheartened that these critical issues have yet to be taken up with serious consideration. i am hopeful that now is the time. finally, i wonder: is nctm only willing to produce recommendations that are palatable to dominant (white) interests in mathematics education? i think this is professor martin’s central critique. i am thankful for professor martin’s courageous voice in continuing to raise these important issues, for nctm’s opening themselves to critical analysis, and for jume’s hosting of this important dialogue. i hope that nctm will embrace these and other issues with genuine interest and action and that all of us as educators will critically interrogate our own complicity. references briars, d., larson, m., strutchens, m. e., & barnes, d. (2015). a call for mathematics education colleagues and stakeholders to collaboratively engage with nctm: in response to martin’s commentary. journal of urban mathematics education, 8(2), 23–26. retrieved from http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/292/178 d’ambrosio, u. (1990). the role of mathematics education in building a democratic and just society. for the learning of mathematics, 10(3), 20−23. freire, p. (1970). pedagogy of the oppressed. new york, ny: herder & herder. gutiérrez, r. (2009). framing equity: helping students “play the game” and “change the game.” teaching for excellence and equity in mathematics, 1(1), 4–8. joseph, g. g. (1997). foundations of eurocentrism in mathematics. in a. p. powell & m. frankenstein (eds.), ethnomathematics: challenging eurocentrism in mathematics education (pp. 61–81). new york, ny: state university of new york press. lawler, b. r. (2005). persistent iniquities: a twenty-year perspective on “race, sex, socioeconomic status, and mathematics.” the mathematics educator, monograph 1, 29–46. martin, d. b. (2003). hidden assumptions and unaddressed questions in the ‘mathematics for all’ rhetoric. the mathematics educator, 13(2), 7–21. martin, d. b. (2015). the collective black and principles to actions. journal of urban mathematics education, 8(1), 17–23. retrieved from http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/270/169 national council of teachers of mathematics. (2014). principles to actions: ensuring mathematical success for all. reston, va: national council of teachers of mathematics. skovsmose, o. (1994). towards a philosophy of critical mathematics education. dordrecht, the netherlands: kluwer. http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/292/178 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/270/169 journal of urban mathematics education december 2015, vol. 8, no. 2, pp. 44–52 ©jume. http://education.gsu.edu/jume matthew g. caputo is a teacher in the department of mathematics at thurgood marshall high school in the fort bend independent school district, 1220 buffalo run, missouri city, tx 77489; email: matthew.caputo@fortbendisd.com. his research interests include classroom technology, reading and writing in the mathematics curriculum, and use of projects as a learning tool. public stories of mathematics educators practices and benefits of reading in the mathematics curriculum matthew g. caputo thurgood marshall high school i think that the article shows that some people think that math can be changed by laws, when it actually cannot. – ninth-grade algebra i student, reading assignment 1 the differing opinions of mathematics and mathematics education tell me that the world isn’t perfect. everybody has their own opinions, and that can or cannot be changed. – ninth-grade algebra i student, reading assignment 1 hile these quotes may lack complexity, they are actual written responses by ninth-grade algebra i students in reaction to a reading assignment based on the article “alabama’s slice of pi,” written by mark boslough (2015). as part of the reading assignment, the students were asked to summarize and critique the fictitious article in their own words. while the ideas represented in these quotes are somewhat contradictory, they illustrate how these students read and analyzed the story with respect to current trends in mathematics and mathematics education. in the past several years, especially with the adoption of the common core state standards, emphasis has been placed on connectivity among different academic subjects as well as between those subjects and the world. simultaneously, there has been an increasing trend toward examination questions that are wordier and require more thinking and unpacking of the concept (mctighe & wiggins, 2014). the purpose of this public story is to elaborate on some classroom techniques that might assist students in becoming comfortable and more successful with interpreting and answering text-based mathematics questions. this past school year was the first year that i had taught ninth-grade algebra i in the past several years. also, it was the first year since reading and writing across the curriculum has received increased national attention (sultan & artzt, 2010). early in the year, i noticed that several students, from all ranges of abilities, were struggling with word problems. while all students were having difficulty with understanding and unpacking word problems, it was especially evident among the students classified as english language learners (ell), roughly 5% of my class w http://education.gsu.edu/jume mailto:matthew.caputo@fortbendisd.com caputo public stories journal of urban mathematics education vol. 8, no. 2 45 enrollment. obviously, this struggle was not something i considered to be news, but i took the opportunity to employ some new techniques in the classroom. current research shows that reading and writing in the content areas leads to a higher level of understanding in the content, more use of content-based vocabulary, and greater ease in unpacking the content from questions that involve lengthy reading passages (tompkins, 2013). in addition, reading and writing in mathematics has often coincided with increased problem-solving skills, creativity in solutions, and in-depth explanations of concepts (sobecki & mercer, 2014). early in the semester, i noticed a number of students in my algebra i class struggling with an examination question involving the classification of numbers that was presented in such a way that many were confused. the question read: a set of numbers is said to be closed under a certain operation if, when you perform the operation on any two numbers in the set, the result is also a number in the set. is the set of irrational numbers closed under addition? explain. the four choices gave “yes” and “no” answers followed by explanations to support the claim. the correct answer: “no, the set of irrational numbers is not closed under addition. for example, the sum of is not an irrational number” (burger et al., 2007, chapter 1 examination question 16). the concept of closure was new for many of my algebra i students, and that is why it was difficult for them to understand, especially when they were first introduced to it during a test. while the concept of classifying numbers was one that many of the students understood thoroughly, they had difficulty extracting the necessary information from the question. seeing a term to which they had not been previously introduced confused some; others were simply intimidated by the length and wordiness of the question. when i originally taught number classification, i compared the idea to classifying living things according to their taxonomy. i felt this method was particularly relevant given that many of the students were also taking biology. after introducing the topic, i asked the usual barrage of questions, such as:  list all categories the number would fit into.  list all the categories that ½ would fit into.  if a number was classified as rational, must it also be an integer?  if a number was classified as an integer, must it also be rational?  is it possible that an integer is rational? considering that i felt my students understood the nomenclature of numbers well, and considering that the question on closure was the only one that stood out as in caputo public stories journal of urban mathematics education vol. 8, no. 2 46 correct by a significant amount, i decided to review it with the entire class in the following session. we analyzed the question together carefully, and i showed them an example using integers with addition, and then integers with division to show them the idea of a set with closure and then one without closure. next, we looked at the choices, one by one, and most of the students identified the correct choice. afterward, i spoke with one of my students who did not answer the question correctly. this question was, in fact, the only one on the test that he answered incorrectly. i asked him to explain why he thought he made the mistake when, obviously, he understood the concept being tested. his explanation was simple and predictable, “all the reading made the question confusing.” further discussion with several students revealed that they were not comfortable with reading for content. many of them felt intimidated by questions that were given in paragraphs containing words they had not seen before. almost all the ell students tried to guess at these questions. i remembered my own experiences in education when textbooks transitioned from nothing more than bound sets of examples to be completed for homework to actual paragraphs and chapters that needed to be read, understood, and then applied. as with most students making such a transition, i floundered when that happened, and i was watching my own students do the same. it has been shown that a growing number of students are discouraged by the increased rigor they experience when transitioning from elementary school to secondary school. this increase in rigor can lead to a change in student attitudes toward the subject and, more generally, their academic future, which results in lower grades, higher drop-out rates, and decreased interest (brunner, 2013). i did not feel that the “sink or swim” approach that i experienced would be successful with all of my students, so i decided to try a different method. as i do often in the classroom, i tried to make it something a little more fun and a lot more unexpected. i assigned the students a passage to read and respond to (see figure 1); it was a fictional article titled “alabama’s slice of pi” (boslough, 2015). the passage was written in the style of an actual journal article. it described how, for religious reasons, the legislature of alabama voted to redefine pi from an irrational number with an approximate value of 3.14 to the exact value of 3, referencing a passage from the old testament. i advised my students that it was a fictitious article and asked them to read the explanation as part of their assignment. in response, the students gave varied summaries and reactions to the article. many of the students did not find the article funny or entertaining, and several thought the references to religion in an article about mathematics were not appropriate. however, regardless of whether they understood the author’s statement or received the message that was covertly delivered, they all read the article and gave some sort of analysis. caputo public stories journal of urban mathematics education vol. 8, no. 2 47 figure 1. instructions for reading assignment 1. i did not make an effort to align the reading assignment with the content we covered at the time because the purpose of this assignment aimed at providing the students with an opportunity to practice their skills at extracting subject-related content from a work of writing. the reading assignment was meant to be concurrent to the curriculum as a means to exercise their skills at unpacking a wordy question. when selecting the writing, my objective was to find a genre the students would enjoy and a content level that was not beyond their ability to interpret. i felt that “alabama’s slice of pi” satisfied the indicated criteria. i eagerly waited until their next chapter examination. when those results came in, i was pleasantly surprised. the number of students receiving grades above 80% increased from 15% to 25% of the class population. on average, the ell students showed a 5% improvement over their previous test scores. what was even more impressive was that in performing the item analysis of these results, i noticed no particular question stood out as missed by a majority of students. in reviewing caputo public stories journal of urban mathematics education vol. 8, no. 2 48 their work and speaking to them individually, i understood a great many of the mistakes to be simple errors rather than conceptual misunderstandings. as with the previous examination, there were several wordy and lengthy questions. in the administration of this test, however, the students appeared more comfortable with reading the question and interpreting what was actually being asked. it would appear that the students were beginning to understand how to unpack a word problem to see the concept that was actually at the heart of the question. despite these improvements, a number of students mentioned to me that they did not like the story or the assignment. they felt that the author’s attempts at humor were unsuccessful, and that the story was not applicable to their lives or interests. when the assignment asked them to describe how the supposed legislation to truncate the value of pi to an integer value related to the fields of mathematics or mathematics education, it appeared that the students lacked enough background information to make such an assessment and, from their point of view, it became an exercise in tedium. from day one, i tried to create an atmosphere where the students felt comfortable discussing these issues with me, so these reactions were not unexpected. i felt that this time was crucial in their work with literacy in mathematics, and i did not want to discourage any of the students by forcing them to read an article that they did not enjoy. i found myself with the profound task of trying to find another reading assignment for the next marking period that both would be interesting to the students and would address the appropriate mathematical content. based on what i observed about student interests during the following weeks, i selected an old newspaper article as their next assignment (see figure 2). “mars probe lost due to simple math error,” written for the los angeles times by robert lee hotz (1999), was a nonfiction newspaper article about how a mistake in converting standard units to metric units caused nasa to lose millions of dollars in a space probe explosion, thankfully with no loss of life. in addition to the benefit of working within students’ reading abilities, i decided this reading assignment would further reinforce the need for accuracy in calculations, something that the students had insisted was not that important in previous conversations. additionally, this topic was more interesting for the students, especially given that part of the assignment was to describe a similar experience they had in their lives. again, the number of students that obtained a grade above 80% on the next chapter examination increased to 40% of the class. the actual written responses improved with every attempt as well. while grading, the summaries of the articles became of decreasing interest and the student analyses became much more amusing. when asked to describe their own experiences with calculation errors, students provided some interesting stories about purchasing shoes in foreign countries, talking to relatives overseas about their height caputo public stories journal of urban mathematics education vol. 8, no. 2 49 and weight, and problems concerning culinary mishaps as a result of not monitoring time or temperature closely. aside from my students having further honed their reading skills in mathematics, they learned more about how the accuracy of their calculations could make a difference in real-life situations. figure 2. instructions for reading assignment 2. bringing this personal aspect into the assignment allowed the students the opportunity to share cultural experiences in ways that previously had not been available. many students who struggled in class and felt disconnected from the main cultural dynamic of the school began to feel that they could speak openly about their family’s heritage even if it was only limited to mistakes they made cooking empanadas or trying to account for time zone changes when calling relatives in turkey. in addition, the students enjoyed this article more than the previous one. they felt it was far more applicable to their lives, and they enjoyed sharing their embarrassing stories about how they struggled in converting units or miscalculated measurements at some point in their lives. seeing that everyone—including nasa engineers—can make mistakes made these teenagers feel less self-conscious about the mistakes they would make. caputo public stories journal of urban mathematics education vol. 8, no. 2 50 i decided to up the ante, so to speak, a little bit more in the following marking period with reading assignment 3 (see figure 3). they had to read “the secret number” by igor teper (2000), a fictitious story that involved a mathematician who discovered another integer, called bleem, between 3 and 4, resulting in his being ostracized by the mathematics community and eventually committed to an asylum. after many failed attempts to prove the existence of this integer to his therapist, the main character escapes toward the end of the story. i decided to make the assignment a little more interesting than the previous two. along with asking the students to describe a mathematics problem or assignment they struggled with and how they resolved it, i also asked them to research a persecuted mathematician or scientist from history and to describe why that person was shunned for her or his beliefs. figure 3. instructions for reading assignment 3. considering that the students had grown accustomed to this sort of assignment by this point, the results were nothing short of stellar. i was impressed by the caputo public stories journal of urban mathematics education vol. 8, no. 2 51 research component of this assignment as well as the personal anecdote and the description of the story plot. descriptions of galileo and archimedes were accompanying my students’ stories of their own experiences struggling to explain a problem. the students enjoyed both the story and the opportunity to share how frustrating a difficult task could be. again, giving the students the chance to describe the story in terms of their own lives gave the assignment a more personal feel that they appeared to enjoy. while the research component frustrated some of the students, they enjoyed learning more about people from the history of mathematics and the sciences that they had heard about in their classes. they found it remarkable that even brilliant people were not always praised for their insights. still, their examination grades continued to increase. in the last mock state of texas assessments of academic readiness (staar) examination that was given to my algebra i students, the passing rate increased from 96% to 98% and the number of commended students (those earning scores over 80%) increased from 30% to 40%. in may of that year, the students took the staar. we reviewed earnestly, as did everyone else, and i spent weeks pacing my classroom like an expectant father waiting for the results. finally, in the last week of the school year, we received our results. an impressive 99% of my algebra i students passed the assessment and 40% of the students received commendations for their scores of 80% or above. needless to say, there was a communal sigh of relief in my classroom when those results were posted. the issue of students being unable to unpack the mathematics from a lengthy word problem concerned me in the beginning of the school year. creating assignments in which the students read, analyzed, and wrote about published works in mathematics provided them with practice of this skill. with high-stakes examinations moving toward an increased amount of reading and interpretation, i felt it was important to expose my students to reading and writing in the discipline early in the school year. giving them these opportunities to practice extracting the mathematical content from a work of writing helped them to interpret the paragraph-sized questions they encountered on the algebra i state assessment. while this was not the only factor their success can be attributed to, i feel this approach is worth further scrutiny and replication. references boslough, m. (2015). alabama’s slice of pi. retrieved from http://www.snopes.com/religion/pi.asp brunner, j. (2013). academic rigor: the core of the core. retrieved from http://www.nassp.org/tabid/3788/default.aspx?topic=academic_rigor_the_core_of_the_core burger, e., chard, d., hall, e., kennedy, p., leinwand, s., renfro, f., seymore, d., & waits, b. (2007). algebra i. orlando, fl: holt, rinehart and winston. http://www.snopes.com/religion/pi.asp http://www.nassp.org/tabid/3788/default.aspx?topic=academic_rigor_the_core_of_the_core caputo public stories journal of urban mathematics education vol. 8, no. 2 52 hotz, r. l. (1999, october 1). mars probe lost due to simple math error. los angeles times. retrieved from http://articles.latimes.com/1999/oct/01/news/mn-17288 mctighe, j., & wiggins, g. (2014). from common core standards to curriculum: five big ideas. retrieved from http://www.hopefoundation.org/from-common-core-standards-to-curriculumfive-big-ideas/ sobecki, d., & mercer, b. (2014). pathways to math literacy. new york, ny: mcgraw hill. sultan, a., & artzt, a. (2010). the mathematics that every secondary school math teacher needs to know. london, united kingdom: routledge. teper, i. (2000). the secret number. retrieved from http://www.strangehorizons.com/2000/20001120/secret_number.shtml tompkins, g. (2013). literacy for the 21st century: a balanced approach (6th ed.). new york, ny: pearson. http://articles.latimes.com/1999/oct/01/news/mn-17288 http://www.hopefoundation.org/from-common-core-standards-to-curriculum-five-big-ideas/ http://www.hopefoundation.org/from-common-core-standards-to-curriculum-five-big-ideas/ http://www.strangehorizons.com/2000/20001120/secret_number.shtml microsoft word 405-article text no abstract-2001-1-6-20200615 (proof 1).docx journal of urban mathematics education july 2020, vol. 13, no. 1b (special issue), pp. 12–37 ©jume. https://journals.tdl.org/jume jacqueline leonard is professor of mathematics education in the school of teacher education, university of wyoming, 1000 e. university avenue, department 3374, laramie, wy 82071; email: jleona12@uwyo.edu. her research interests include computational thinking, self-efficacy in stem education, culturally specific pedagogy, and teaching mathematics for social justice. erica n. walker is professor of mathematics education at teachers college, columbia university, 525 w. 120th street, new york, ny 10027; email: ewalker@tc.columbia.edu. her research focuses on the social and cultural factors that facilitate mathematics engagement, learning, and performance, especially for underserved students. mathematics literacy, identity resilience, and opportunity sixty years since brown v. board: counternarratives of a five generation family1 jacqueline leonard university of wyoming victoria r. bloom university of the sciences in philadelphia erica n. walker columbia university nicole m. joseph vanderbilt university in this chapter, the authors use black feminist thought (bft) to examine the mathematics education and the educational attainment of african american females in a matrilineal line that spans five generations. a cross analysis of school experiences, from a maternal great-great-grandmother to her great-great-granddaughter, reveal a portrait of segregation, desegregation, and resegregation. the impact of these educational contexts on the mathematics literacy and mathematics identity of four african american women and the hope and promise of a young girl in the class of 2026 are also presented. from sharecropper schools in mississippi to prestigious universities in the eastern united states, the challenges and successes of one family’s struggle to obtain mathematics literacy and the american dream are discussed through the historical lens of brown v. board of education. using this historical context, the specific experiences of these five family members encourage a dialogue about a larger narrative—the mathematics attainment of all black children. keywords: black feminist thought, counternarratives, mathematics identity, segregation 1 this article was first published by information age publishing as a book chapter: leonard, j., walker, e. n., cloud, v. r., & joseph, n. m. (2017). mathematics literacy, identity resilience, and opportunity sixty years since brown v. board: counternarratives of a five-generation family. in j. ballenger, b. polnick, & b. irby (eds.), women of color in stem: navigating the workforce (pp. 79–107). information age publishing. permission to reprint was given to the journal of urban mathematics education by george johnson on june 15, 2020. leonard et al. mathematics literacy, identity resilience, & opportunity victoria bloom is a licensed physical therapy and certified wound specialist and serves as adjunct faculty at the university of jamestown in north dakota and the university of the sciences in philadelphia, 600 s. 43rd street, philadelphia, pa 19104; email: v.cloud@usciences.edu. her research focus is physical therapy, skin and wound care, and rehabilitation techniques. nicole m. joseph is an assistant professor of mathematics education in the department of teaching and learning at vanderbilt university, 230 appleton place, nashville, tn 37203; email: nicole.m.joseph@vanderbilt.edu. her research focuses on the role of race, gender, and other socially constructed identities and structural systems of oppression in shaping black women and girls’ mathematics identity development. journal of urban mathematics education vol. 13, no. 1b (special issue) 13 he purpose of this qualitative study, which uses case study design, is to present and examine the mathematical experiences and educational attainment of five african american females. they represent five generations in one african american family whose roots began in rural mississippi in the 1860s—nearly one hundred years before the supreme court ruled in brown v. board of education (1954) that the racial segregation of schools is unconstitutional. while the experiences of the female members in this family are not meant to be generalized to all black families, their struggle to obtain mathematics literacy can be used as a theme to understand the black struggle for quality education preand post-brown v. board of education. this study begins with a great-great-grandmother (now deceased) born in washington county, mississippi, in the 1910s during the black nadir and ends with her six-year-old great-great-granddaughter, who began first grade in delaware county, pennsylvania, in the fall of 2014—sixty years post-brown v. board of education. descriptive content analysis of their mathematical experiences provides a sociohistorical (i.e., changes in society over time) account of five african american females’ mathematics literacy and mathematics identity. additionally, we describe the impact their mathematics education had on their career trajectories and everyday lives. these narratives are told through the historical lens of brown v. board of education. review of the literature the bodies of literature that support this study are mathematics literacy, mathematics identity, and the intersection of race and gender in mathematics education. terry (2011) described literacy broadly as not only the ability to read and write but also its importance for understanding literacy as the means to liberation and freedom. this view is grounded in frankstein’s (1990) notion of critical mathematics literacy and gutstein’s (2006) notion of reading and writing the world with mathematics. mathematics literacy mathematics literacy is using mathematics as a cognitive enterprise to communicate mathematically with others (national council of teachers of mathematics, 2000), engage in society as an informed citizen (moses & cobb, 2001), and position oneself as a doer of mathematics for empowerment (leonard, 2009). thus, t leonard et al. mathematics literacy, identity resilience, & opportunity journal of urban mathematics education vol. 13, no. 1b (special issue) 14 mathematics literacy is racialized and situated “within the larger contexts of african american, political, socioeconomic, and educational struggle” (martin, 2006, p. 197) and the struggle for civil rights (moses & cobb, 2001). mathematics identity we examine mathematics identity among these five multigenerational females. martin (2006) defined mathematics identity as “the dispositions and deeply held beliefs that individuals develop, within their overall self-concept, about their ability to participate and perform effectively in mathematical contexts and to use mathematics to change the conditions of their lives” (p. 206). martin (2000) suggested that students with a well-developed mathematics identity are successful doers of mathematics. students with well-developed mathematical identities have the following characteristics: (a) believe in their ability to perform in mathematical contexts, (b) understand the instrumental importance of mathematics, (c) realize there are constraints and opportunities in mathematical contexts, and (d) exhibit motivation and engagement in strategies to obtain mathematics knowledge (clark et al., 2009). the authors believe that individual and collective experiences should be studied and examined within the contexts of mathematics literacy, mathematics identity, and cultural practices as black women. this study adds to the literature on black feminism (collins, 2009), black resilience in mathematics (mcgee, 2013), and black self-determination (dixson, 2011), offering a multi-generational analysis seldom seen in mathematics education research (gholson, 2013). the intersection of race and gender in this study, we not only examine the individual beliefs and mathematical understandings of five related but distinct females in a matrilineal line but also how their identities are interwoven and intersect with race and gender. specifically, we show how their mathematics identities are shaped and influenced by their mathematics attainment and positioning in the family across different generations. using the lens of brown v. board of education, we present the counternarratives of five african american females, their struggle for mathematics literacy, and how it shaped or limited their access to higher education; science, technology, engineering, and mathematics (stem) education; and stem-related careers. this study stands in contrast to the literature base on african american mathematics education in general, which is often related to gap-gazing (gutierrez, 2008; lubienski, 2008) and cultural deficit theory (martin, 2006; mcgee & pearman, 2014; terry & howard, 2013) rather than mathematics literacy, identity, resilience, and agency (martin 2000, 2006; walker, 2012, 2014). yet, narratives about mathematics achievement and attainment among african american women and girls are sparse in the literature (lim, 2008; lubienski leonard et al. mathematics literacy, identity resilience, & opportunity journal of urban mathematics education vol. 13, no. 1b (special issue) 15 & bowen, 2000). in the small number of studies that compare mathematics achievement among male and female students, lim (2008) claimed: the majority of previous studies of gender issues in mathematics education have focused on the differences between boys’ and girls’ motivational constructs, performance levels, or learning styles while ignoring the dynamic socio-cultural context of their mathematics learning in and out of schools. (p. 308) thus, focusing on the intersection of race, gender, and mathematics attainment, this study adds to the research on both gender studies and mathematics education. theoretical framework the theoretical framework that undergirds this study is black feminist thought (bft), which collins (2009) describes as an epistemology used to validate black women’s knowledge and experiences. the core themes of bft are “work, family, sexual politics, motherhood, and political activism” within the u.s. context of racial and gender oppression (collins, 2009, p. 269). the principles of this epistemology rely on two types of knowing that derive from black women’s experience— knowledge and wisdom (collins, 2009). in this way, black women from all walks of life participate in a type of knowledge that is based on collective experiences that emerge from similar forms of oppression. domestics had to learn how to function in two worlds—one where they were responsible for rearing white children, while simultaneously being viewed as inferior, and the other where they raised their children to resist such definitions and to strive for something better. in the narratives to be described, a great-great-grandmother’s school and work experiences in rural mississippi and later in an urban city in missouri shaped and informed the educational trajectory of her granddaughter in urban st. louis during the 1960s and 1970s. thus, mother wit is valued alongside institutional knowledge, providing voice and legitimacy to four generations of women. black feminist thought uses dialogue to assess knowledge claims, promoting an ethic of care that is characterized by “personal expressiveness, emotions, and empathy…central to the knowledge validation process” and the ethic of personal accountability (collins, 2009, pp. 281–282). black women validate each other’s experiences through dialogue and storytelling that has roots in “african-based oral traditions and in african-american culture” (collins, 2009, p. 279). bft is used as a framework to discuss how mathematics literacy can be used to empower black women to challenge oppression and the status quo. we acknowledge that bft as a framework has limitations. as a social theory, it lies at the intersection of “race, class, gender, sexuality, ethnicity, nation, and religion” and can only partially tell the entire story (collins, 2009, p. 12). the hope of leonard et al. mathematics literacy, identity resilience, & opportunity journal of urban mathematics education vol. 13, no. 1b (special issue) 16 brown v. board of education sixty years ago was that the united states would turn the corner on race and racism to provide all students with equal access and educational opportunity. the brown v. board of education decision took a first step to eradicate the separate but equal doctrine that perpetuated jim crow for a half-century and reversed black advancements in politics and education acquired during reconstruction (patterson et al., 2008; rucker & jubilee, 2007). however, the voices of black women and their educational experiences, particularly in mathematics, are often missing from the extant literature (walker, 2014). thus, bft is a powerful analytical and theoretical tool to examine the ways in which seemingly neutral educational policies and practices reproduced gendered and racialized mathematics education in african american communities. these policies are evident in the counternarratives of a five-generation family. from a sharecropper school system in mississippi with burgeoning class sizes and high dropout rates to a suburban school district, the females in this family experienced segregation, desegregation, or resegregation. such educational contexts had and continue to have an impact on their mathematics education and educational attainment in general. research questions the research questions that guide this study on five african american females in a matrilineal line are as follows: 1. how do the counternarratives of a five-generation matrilineal line of african american females compare and contrast in terms of their mathematics literacy, mathematics identity, and resilience? 2. what role did mathematics literacy and social agency play in terms of educational opportunities and career paths? 3. how do their mathematics education and educational attainment illustrate a broader social and political context of race, class, and gender? to answer these research questions, we engaged in descriptive content analysis of the counternarratives, document analysis of historical records and artifacts, and comparative analysis to find themes and patterns from multiple data sources. methodology we used the counternarrative or counter-storytelling approach to examine the mathematics literacy, identity, and resilience of a five-generation matrilineal line. solorzano and yasso (2002) present three general forms of counter-storytelling: (a) personal stories or narratives told in first-person, (b) other people’s stories or narratives told in third-person, and (c) composite stories or narratives constructed through various forms of data, historical records, or archives. in this paper, we employ all leonard et al. mathematics literacy, identity resilience, & opportunity journal of urban mathematics education vol. 13, no. 1b (special issue) 17 three of the aforementioned methods of counter-storytelling to examine historical data, school records, and test data that span 70 years of schooling from the rural south to the urban east coast. these sources also include data obtained from interviews and questionnaires as well as oral history (i.e., recollections and oral traditions told by elders to members of successive generations). two authors of this paper are members of the five-generational family reported in this study. to ensure validity and reliability, an unrelated third-party analyzed the qualitative data to find emergent themes. data analysis and data sources while counternarratives are used to uncover the use of simple language and thick descriptions, we also examined anecdotal records and artifacts to understand the varied mathematical experiences of each family member by using descriptive content analysis (neuendorf, 2002). the data sources consist of artifacts and documents that span more than 70 years. these data include census records, report cards, sat and gre scores, interviews, and questionnaires collected at different time periods. additionally, oral stories are used to describe the mathematical experiences and educational attainment of four adult women and one child who represent five generations in one family. these stories reveal details about these black females’ mathematical literacy, identity, resilience, and educational opportunity that can be used to tell a larger narrative about the education of african american children. figure 1. cross maternal family tree leonard et al. mathematics literacy, identity resilience, & opportunity journal of urban mathematics education vol. 13, no. 1b (special issue) 18 participants and settings the participants in the study are members of the cross family (all names are pseudonyms (see figure 1 for family tree and birth order). results of recent dna testing revealed the ancestry of this family is 76–80% yoruba (west african tribe near nigeria) and 20–24% european. tracing the maternal side of the family tree through census records, the authors found that seven generations of the cross family resided in oktibbeha county, mississippi, beginning in 1860. the counternarratives we present the individual counternarratives of five females in a matrilineal line to illuminate their early life, school experiences, mathematics literacy, mathematics identity, and resilience within historical and educational contexts. each counternarrative is unique, exploring the individual and social agency that influenced each family member’s life. the counternarratives begin with lou ellen, the great-great-grandmother. living into her nineties, she had a strong influence on the family for nearly 75 years. lou ellen’s story: “it was hard for me because i did not have the learning.” early life. lou ellen was born in washington county, the heart of the mississippi delta, during the 1910s. the family moved during her early childhood years, and she was reared in oktibbeha county. lou ellen’s responses to a questionnaire administered by her great-granddaughter, rita, in 1997 as part of a college course reveal many nuances about her school and adult life. i was born in leland, mississippi. i grew up in starkville, mississippi, on my grandmother’s farm. i loved that farm very much. i have a cousin still on that farm now, but it much different. [my grandparents] had nine children—four boys and five girls. my grandparents owned their own land…raise the children and take care of the land. [my father] worked on a dairy farm; he helped to milk the cows. he got paid for working. we raised all our food. [mother] did not work; she took care of the house and children. my mother died when i was ten years old. i lived with my oldest sister after she married. historical and educational contexts. lou ellen went to school in the 1920s. during this period, the agrarian south was dotted with farms and sharecroppers. however, teaching was a stable, high-status profession for blacks during this period. ladson-billings (2005) noted there were about 66,000 black teachers in the united states in 1910. unable to teach in the north, many of these teachers taught in the south (tillman, 2004). leonard et al. mathematics literacy, identity resilience, & opportunity journal of urban mathematics education vol. 13, no. 1b (special issue) 19 school experiences. lou ellen often told stories of how she and her sisters had to walk 10 miles to school. the stories revealed the nature of the one-room schoolhouse where reading, spelling, and figuring were the lessons of the day. given the demands of farm life in rural mississippi, it was common for black students to work on the farm instead of attending high school, which was not compulsory. although blacks represented 29% of all high school students, only 5% of black children were enrolled in high school in the south during the 1910s (anderson, 1988; rucker & jubilee, 2007). i was not able to go to high school. my father could not send us to high school. it was three of us. he could not pay for us to go…. we did not have a lot of money but that was no problem. we had food and shelter, and we was [sic] happy then…. lou ellen’s statement implied the family could not afford for her to attend high school. if she was old enough to work in the fields or perform day work, the family needed her to bring in additional income. although she did not complete high school, she had aspirations to further her education. i wanted to go to college and learn a traid [sic] to work with older peoples [sic] and to help peoples [sic] that can’t help themselves. mathematics literacy and mathematics identity. although there were no copies of school records, lou ellen often told her family that she completed school as far as the eighth grade. while her literacy in mathematics was unknown, she read the bible on a daily basis and was able to fix anything that was broken. she often said she was good at figuring, which was evident by her ability to manage and save money. life choices and career trajectory. as a young adult in the 1940s, lou ellen made a life-changing decision. she left her family in mississippi and relocated to st. louis, missouri, during the great migration. there were relatives in st. louis that helped her to get settled. she joined the african methodist episcopal church, where she was a member for 65 years before her death in 2010. i been without a job, without food, and [did not] know where i would get my next meal or get money for rent, but the lord made a way for me to get a job. it was hard to get a job without training or education. it was hard for me because i did not have the learning. so i wint [sic] back to school. my first job was day work. lou ellen lived and worked in st. louis as a baker, cook, and dietician until she was 70 years old. because of her self-determination to leave mississippi and start a new life, lou ellen influenced her daughter, bernice, to leave mississippi as well. thus, her decision to relocate led to opportunities that changed the career trajectories of both women. she shared her pride as a black elder and offered advice to her progeny. leonard et al. mathematics literacy, identity resilience, & opportunity journal of urban mathematics education vol. 13, no. 1b (special issue) 20 it is nice to be black at any age. it does not [matter] what color you are. age does not change anything. i am proud to be black. i am proud [to] be what god intended for me to be. glad i am black. stay in school and get [some] learning. do not smoke or drink. don’t let men take advanted [sic] of you. don’t let your relationships into fear [sic] with your education or with religion. bernice’s story: “i wanted to finish high school, but i needed a job.” early life. bernice was born in oktibbeha county, mississippi, in the late 1930s. she was the only child living in the household. she appeared to live an uncomplicated life with her mother and stepfather prior to matriculating in school. when she was seven years old, her life became complicated after her mother, lou ellen, left for st. louis. because lou ellen was uncertain that she could find a job right away, she left bernice with an aunt in mississippi who could provide a home and a nurturing environment. historical and educational contexts. there were two educational philosophies that were prevalent during the 1940s—classical (liberal arts) and industrial (technical) education influenced by du bois and carver, respectively (russell, 2014). during this period, many black students learned applied mathematics, which allowed them to learn a vocational trade. however, other black students, influenced by du bois’ philosophy, studied algebra and plane geometry in high school (russell, 2014). interestingly, the high school mathematics curriculum in oktibbeha county, mississippi, was predominantly influenced by du bois since tenth through twelfth-grade students took algebra, geometry, and trigonometry in high school. school experiences. school records revealed that her mother, lou ellen, registered bernice for school at the age of five. education in the south often took place in segregated schoolhouses like the one shown in figure 2. the photograph shows bernice smiling as she stood among 50 members of the third-grade class at the oktibbeha county training school. figure 2. schoolhouse in oktibbeha county, mississippi leonard et al. mathematics literacy, identity resilience, & opportunity journal of urban mathematics education vol. 13, no. 1b (special issue) 21 mathematics literacy and mathematics identity. examination of report cards revealed that bernice earned b grades in first-grade arithmetic and as and bs in second grade. her grades dropped to cs in grades three through five. by sixth grade, her arithmetic grade slipped to a d. however, attendance records also showed 17 absences during her sixth-grade year. these absences no doubt had an impact on her performance. when asked about these absences, bernice replied: when i was old enough, i had to stay home to watch auntie’s younger children while she went to work. i also stayed with another aunt sometimes who lived further away from the school. if the weather was bad, i couldn’t go. my other aunt lived more than 10 miles from the school. i had to leave at 6 a.m. to walk to school to get there before 8:30 am. because of d grades in arithmetic and history, bernice had to attend summer school to be promoted to the seventh grade. however, her grades in mathematics showed little improvement. when asked about her grades, bernice stated: math gets harder as you go along, and you need someone to help you. auntie had an eighth-grade education and a family of her own, and i didn’t have anyone to help me with my homework. in high school, bernice was enrolled in a course that used a book called math at work. george washington carver and the current thinking of the day may have influenced such a text. while her behavior was good, and her grades in english and reading were passing, she reluctantly claimed, “i was not good in math.” she earned a d grade in general math in ninth grade. in tenth grade, she was enrolled in algebra but later withdrew from the course (see table 1). when queried about this, bernice said: in grade 10 you had algebra, geometry in grade 11, and trigonometry in grade 12. there was [sic] four years of math in high school. it was difficult for me because i did not understand the concepts. there were 40 or 50 students in my elementary classes. in high school, students came from the surrounding areas and there could be 90 in one class. mom and auntie dropped out of school to work on the farm. i was the first one to go to high school, but i dropped out in the tenth grade. leonard et al. mathematics literacy, identity resilience, & opportunity journal of urban mathematics education vol. 13, no. 1b (special issue) 22 table 1 generational mathematics achievement: high school courses and sat scores *sat scores scaled 200–800 **percentile rank is for all students only ***enrollment, graduation, and minority data from 2012 ****sat recentered in 1995 life choices and career trajectory. as an adult, bernice did not lose sight of her desire to finish her high school education. she enrolled at a community college in st. louis during the 1970s and completed studies to earn a ged and later an associate’s degree. this self-determination opened the door of opportunity. bernice was able to get a better job where she became a union worker at a local bakery chain. her wages doubled, enabling her to become a homeowner and to move from workingclass to lower-middle-class status. this decision impacted her life and the lives of her mother and children, who were able to enjoy the comforts of homeownership. family member high school type 8th grade course 9th grade course 10th grade course 11th grade course 12th grade course sat* math score bernice oktibbeha county training school, starkville, ms enrollment: na minority: 100% general math – d mathematics for work – d algebra – w withdrawn withdrawn na belinda public high st. louis, missouri enrollment: 2,500 graduation rate: 50% minority 99% general math – a algebra i – a na geometry – w algebra ii/trig – a 510 (81st percentile**) rita public high school howard county, maryland ***enrollment: 1,155 graduation rate: 90% minority: 66% pre algebra – b algebra i – a geometry/trig – a algebra ii honors – a calculus, ap – a, score 3 560**** (72nd percentile) leonard et al. mathematics literacy, identity resilience, & opportunity journal of urban mathematics education vol. 13, no. 1b (special issue) 23 belinda’s story: “i decided to go back to college.” early years. belinda was born in missouri in the late 1950s. she was bernice’s older daughter and one of three siblings born in st. louis. she responded to the same questionnaire, which was administered by her daughter, rita, in 1997. belinda’s responses revealed details about her life and upbringing during the sixties and seventies. i remember looking out the window of my neighborhood when i was four years old and liking what i saw. streets were lined with brick apartments and bungalow houses dotted with trees, flowers, and green grass. my neighborhood consisted of corner stores and confectionaries owned by blacks and jews. there was a grocery store, library, and several clothing and shoe stores a few blocks away. historical and educational contexts. two major events defined the 1950s—the u.s. supreme court decision on may 17, 1954, known as brown v. board of education, and the launch of sputnik by the russians on october 4, 1957. as a result of sputnik, children in the 1950s and 1960s were introduced to “new math” (berry et al., 2013). a chapter on sets was common in every textbook as new math became synonymous with set theory (berry et al., 2013; raimi, 1995). despite the millions of dollars spent to design the curriculum, the new math program was a failure (raimi, 1995). mathematics achievement scores dropped during this period, revealing students, in general, were worse off than they were before new math was introduced. school experiences. de facto segregation existed in the st. louis public schools when belinda began matriculating in 1963, despite the passage of brown v. board of education almost ten years earlier. belinda attended a k–8 neighborhood school for nine years. the majority of her teachers were black, and some lived in the community. discipline was strict, and parents unequivocally supported the teachers. my neighborhood [school] was all black…. [my mother] was a hard worker. her unhappiness inspired me to do better. i valued education as the key to success and worked hard in school. i valued family. mathematics literacy and mathematics identity. belinda disliked mathematics in elementary school because teachers tended to focus on rote memorization and computation, if they taught mathematics at all. while she did not remember what mathematics curriculum was used, she recalled studying set theory (i.e., new math) in seventh and eighth grade using old books that were passed down from white schools. belinda’s eighth-grade teacher was an african american woman who liked teaching math. she spent a great deal of time teaching the class how to solve word problems, fractions, decimals, and percent. working with rational numbers (i.e., leonard et al. mathematics literacy, identity resilience, & opportunity journal of urban mathematics education vol. 13, no. 1b (special issue) 24 fractions) was belinda’s earliest recollection of enjoying mathematics, and she realized that she was good at it. the eighth-grade teacher recommended that belinda take algebra in ninth grade at the local high school. to her surprise, she enjoyed solving quadratic equations and using the foil method to distribute binomials. her grade in algebra i was an a, reflecting the time and effort she put into the course. unlike the schools in mississippi in the 1950s, four years of mathematics were not mandatory in the st. louis public schools in the 1970s. therefore, belinda appealed to the high school counselor to skip geometry and take algebra ii/trig instead. permission was granted, and she earned an a in algebra ii/trig as well. in her senior year, belinda’s score on the sat was higher than most students who attended her predominantly black high school (see table 1). encouraged by her counselor, teachers, and college-bound peers, belinda decided to attend college in new england. my biggest challenge was staying focused on completing my education. i dropped out of college to get married. family and belonging were important to me. because i could type, i took jobs as an office clerk. it was hard to emerge from the domesticated role of being a clerical worker and mother. i decided to go back to college. life choices and career trajectory. belinda’s decision to return to college was one of self-determination. she enrolled in the teacher education program at a private urban university in the midwest during the early 1980s. she graduated with a b+ average and a teaching credential in general science and began teaching science at a middle school in a nearby suburb. after relocating to texas with her family, belinda obtained a master’s degree in mathematical sciences and taught middle grades mathematics for eight years. ten years later, she obtained a ph.d. in mathematics education in the late 1990s from a research university in the eastern united states. belinda eventually became a tenured professor. her life choices and career trajectory firmly planted the cross women in the middle-class. the civil rights movement gave me a militant attitude, but i believe i am tolerant of others. the women’s movement gave me strength to break out of stereotypical jobs for women and to pursue mathematics education and ordained ministry. rita’s story: “i was accepted to an ivy league university.” early years. rita, belinda’s older daughter, was born in the late 1970s on a military base in the south. three years later, the family relocated back to missouri after her father was honorably discharged. rita started kindergarten at the age of four because she had a fall birthday. she was most often the youngest student in her class. rita initially attended elementary school at a private christian school in missouri and later two public elementary schools in texas when the family relocated. the racial leonard et al. mathematics literacy, identity resilience, & opportunity journal of urban mathematics education vol. 13, no. 1b (special issue) 25 makeup of the private school in missouri was predominantly white, whereas the public schools in texas consisted mainly of african american and latinx students. during her adolescent years in texas, rita attended a small, private, predominantly white episcopalian school for seventh and eighth grade. historical and educational contexts. on the educational front, the national commission on excellence in education presented a report entitled a nation at risk: the imperative for educational reform (1983). during this time, u.s. students were compared with their international counterparts and found to be lagging, particularly in mathematics and science. in response to public criticism of the report, many states began requiring algebra i as a graduation requirement. from 1982 to 1992, student enrollment in algebra i and advanced mathematics courses increased dramatically: algebra i from 65% to 89%, algebra ii from 35% to 62%, and calculus from 5% to 11% (raizen et al., 1997). mathematics literacy and mathematics identity. in elementary school, rita was part of the talented and gifted program, where she enjoyed solving logic problems. she learned early on that she was good in mathematics and enjoyed word problems. throughout high school, rita was placed in honors or advanced placement mathematics courses. responding to a protocol in 2013 provided by her mother, belinda, she shared her mathematics experiences: in my math classes in high school, i was usually the only black person in class. there was a 99 club where the students’ average grades were 99 or higher for each report card, and there were at least 20–30 students in that club, if not more. i was a member of the 99 club as well. in the middle of 11th grade, rita moved to the east coast to finish high school, as her mother was pursuing a ph.d. the high school demographics were primarily mixed. african american and white students attended high school together as a result of a planned community that prided itself on developing neighborhoods to house both middle-class and affluent families. rita remembered her algebra ii and calculus teacher was a mentor and role model. he was very encouraging and advised me while applying to college. he was very excited when he found out i was accepted to an ivy league university and encouraged me to attend that school over others. after graduating from the ivy league university, rita was admitted to a master’s degree program in allied health. note the growth in quantitative reasoning on her gre scores as shown in table 2 compared to her sat scores in table 1. after obtaining the advanced degree, rita began a career in allied health that has spanned 14 years. she credits her positive experience in mathematics as a factor in acquiring leonard et al. mathematics literacy, identity resilience, & opportunity journal of urban mathematics education vol. 13, no. 1b (special issue) 26 higher education, obtaining a stable job in healthcare, and maintaining the middleclass lifestyle she enjoyed as a child. table 2 generational mathematics achievement: college mathematics courses and gre scores *stlcc data from 2007 **slu & penn data from 2011 ***gre scores scaled 200–800 zoe’s story: “every number plus zero is the same number.” early years. zoe was born in a research hospital in pennsylvania in the late 2000s. she is rita’s only daughter. zoe has one brother who is 18 months older than she. prior to enrolling in a neighborhood public school, zoe and her brother attended daycare less than a mile from their home. at the daycare, zoe learned to recognize the alphabet, counting numbers, the calendar, shapes, and colors. she also enjoyed playing with dolls and playing house. family member college information college statistics college mathematics courses gre*** scores bernice st. louis community college st. louis, missouri public two-year college gender: 66.8% f; 33.2% m black: 45.7% asian: 0.04% hispanic: 0.02% white: 37.0% other: 0.05% *total enrollment: 7,232 general math – c na belinda boston university, boston, massachusetts private st. louis university, st. louis, missouri private jesuit gender: 57.6% f; 42.2%m black: 8.3% asian: 8.2% hispanic: 0.04% white: 72.3% two or more races: 0.04% **ft undergraduate: 7,716 college/algebra trig – b calculus – a calculus ii/analytic geometry – b+ trigonometry a 570 (53rd percentile) rita university of pennsylvania philadelphia, pennsylvania private ivy league gender: 51.3% f; 48.7% m black: 7.1% native american: 0.3% asian: 18.6% hispanic: 8.1% white 46.2% two or more races: 2.2% international: 10.9% **ft undergraduate: 10,324 calculus i – c calculus ii – b intro to stats i – b+ intro to stats ii – c+ 660 (81st percentile) leonard et al. mathematics literacy, identity resilience, & opportunity journal of urban mathematics education vol. 13, no. 1b (special issue) 27 historical and educational contexts. notwithstanding, one of the most important legal cases in recent history was parents involved in community schools v. seattle public school district (dixson, 2011). the crux of this case was deciding whether a school district could remedy racial segregation voluntarily. race was considered, among several factors, to determine admission to highly demanding seattle high schools. parents filed a lawsuit, contending that admission based on race was unconstitutional (dixson, 2011). the supreme court ruled in favor of the parents, stating that the way race was used to admit students was unconstitutional and violated the 14th amendment. some legal scholars saw this case as overturning brown v. board of education, offering no remedy for school resegregation (dixson, 2011) and leaving black parents with few choices other than charter and neighborhood schools (leonard et al., 2013; morris, 2004). school experiences. zoe currently attends a public school in pennsylvania, where she began kindergarten in the fall of 2013 with a great deal of exposure to technology. her kindergarten teacher reported that zoe knew how to count up to 40 and was an active listener in class. according to child trends, a research center based in maryland, 4 million children (52% white; 23% hispanic/latino; 16% black/african american; 5% asian/pacific islander; 2% multiple races; and 1% american indian/alaska native) started kindergarten in the fall of 2013 as the class of 2026 (samuels, 2013). this class has been described as happy, resilient, and eager to learn. mathematics literacy and mathematics identity. during kindergarten, zoe enjoyed counting various objects, such as light displays, during the christmas season as her mother drove the car. she often shouted out math problems, such as “2 + 2 is 4 and 3 + 3 is 6.” when her mother asked her how she knew, zoe said, “i already know that.” she frequently played games on her mother’s iphone and ipad (see figure 3). zoe also had a leap pad that she used to play math games. her grandmother, who is a mathematics educator, heard her count backwards from twelve during the christmas holidays: “12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0.” surprised to hear zoe state the number zero, her grandmother wondered how deep zoe’s mathematical understanding was at the age of five. figure 3. zoe and friend using ipad mini leonard et al. mathematics literacy, identity resilience, & opportunity journal of urban mathematics education vol. 13, no. 1b (special issue) 28 a few days later, while riding in the car, zoe asked, “what’s zero plus 3?” her older brother responded the answer was three. the next week zoe asked the same question again. her grandmother explained that any number plus zero was that number and drilled both children by asking: “what’s 4 + 0? 10 + 0? 25 + 0? 200 + 0?” they responded in unison: 4, 10, 25, and 200. several days later, her grandmother asked, “what’s 0 + 300?” zoe responded 300. then her grandmother asked, “why?” zoe responded, “zero plus 300 is 300 because 300 plus zero is that number.” a few minutes later, she told her brother, “every number plus zero is the same number.” thus, not only was zoe able to recognize zero as a number, but she was also able to generalize the zero property of addition, regardless of the order of the addends. as a kindergartener, zoe’s brilliance in mathematics suggests that black children are capable of learning much more mathematics and that their mathematics identity can be developed as young children (leonard et al., 2013). zoe also modeled her teacher at home by posting grids on the wall and counting down to a reward: “when we get to the number 17, we will have a dance party!” her mother encouraged math concepts by dividing hot dogs and pancakes into parts or cutting a pizza into slices. she often asked zoe, “would you like your pizza cut into quarters or halves?” after zoe responded, her mother told her, “you know math!” in kindergarten, zoe aspired to become either a teacher or a boss. data analysis the counternarratives describing the mathematics experiences of five generations of cross females (lou ellen, bernice, belinda, rita, and zoe) in many ways reflect the 20th century history of school mathematics in the united states. as described earlier in this paper, many contemporary narratives of the history of mathematics education emphasize key historical landmarks, such as the sputnik launch in 1957, the “new math” of the 1960s, and a nation at risk in 1983, as well as their impact on school curriculum, instruction, and assessment. however, these counternarratives also reveal a simultaneously occurring story: that of how one particular set of american citizens—black americans—were finally gaining legal access to rights as citizens. the brown v. board of education decision in 1954 and the civil rights movement of the 1960s, along with increased attention to standards and equity as described in a nation at risk, had profound implications for education generally and mathematics specifically. thus, to consider the mathematical lives of these five african american females, and african americans in general, without attending to the broader historical, educational, and social contexts would be ahistorical and incomplete. when comparing and contrasting the counternarratives, two themes emerge: mother wit and like a boss. we draw upon bft as a theoretical framework to analyze these themes within a sociohistorical context (i.e., changes in society over time). leonard et al. mathematics literacy, identity resilience, & opportunity journal of urban mathematics education vol. 13, no. 1b (special issue) 29 “mother wit”: the drive for education within constrained opportunity we see this broader historical context writ large in lou ellen and bernice’s stories. the south, as others have described (anderson, 1988; morris & monroe, 2009), holds an important place in education for african americans. here is where most african americans were enslaved and, thus, barred from educational opportunity. following the end of slavery, a sharecropping system was developed, which kept blacks in poverty. the education provided for blacks in mississippi and other southern states during this period was minimal, inadequate, and relegated african americans to second-class citizenship designed to maintain the status quo (rucker & jubilee, 2007). despite significant obstacles to black education in the south, black parents valued education (mizell, 2010). moreover, blacks supported community schools, which were largely built, maintained, and supported by all-black communities (anderson, 1988). the quest for education was also at times hampered by the financial realities of living in the agrarian south. children were needed for labor—on the farm or at home—and schooling was often interrupted. in the case of lou ellen, families simply could not forego the financial benefit of additional laborers working the farm. thus, for this generation of young people, going to school would have decreased family income significantly. further, lou ellen and bernice’s description of family life resonates with the broad and extended family networks of many african americans in the south. for a variety of reasons, children lived not only with their parents but also with grandparents, aunts, uncles, or older siblings. at times, schooling was interrupted due to family obligations—bernice was entreated to stay home and look after younger cousins because her aunt had to go to work. demands of family and work also prevented bernice from getting help with her math homework. the limited education of her elders adversely affected bernice’s educational trajectory. as was discovered in a study of african american mathematicians who described cases of older generations of family members being born ‘too early’ to benefit from advances in civil rights (walker, 2014), lou ellen’s desire to go to high school and college was thwarted by her circumstances. there is evidence that lou ellen exhibited talent (“able to fix anything that was broken”) that was unrealized within a traditional educational context, in part because she was black and arguably because she was a woman. in black mathematicians’ descriptions of family members who exhibited mathematical talent but were undereducated, the men often worked at skilled and technical but menial labor. women, however, were in most cases limited to the domestic sphere whether they worked in their homes or were domestics in the homes of others. lou ellen’s “cautionary tales” to her own descendants—about getting an education and not allowing relationships with men to interfere with education or progress in leonard et al. mathematics literacy, identity resilience, & opportunity journal of urban mathematics education vol. 13, no. 1b (special issue) 30 life—are deeply centered in a black feminist ideology of self-determination. the emphasis on acquiring education to support oneself, help extended family members, and not ‘fall,’ while prominent among african american women, did not have the same urgency among white women of a similar era (giddings, 1984; jones, 1985). the tension between vocational and classical education in terms of what was “most appropriate” for african americans was evident in bernice’s experience— although classes like algebra, geometry, and trigonometry were offered to 10th through 12th graders, a high school course that bernice took in the 9th grade used a book called math at work. bernice felt her lack of progress in mathematics limited her opportunities for employment. therefore, she exhorted her progeny to do better and learn more than she had. although she had not benefitted from direct intervention on the part of her mother or aunt, she was determined to help her children go even further in school and to be successful. in the urbanized midwest, there were varied opportunities for employment—better paying than the farm and domestic jobs of the south—and more opportunities for young people to be exposed to educational and social experiences. bernice’s quest for education continued when she earned her ged and was able to secure better employment, benefitting not just herself, but her elders and descendants. like “a boss”: honoring and redefining “women’s work” in mathematics and beyond the self-determination ethic inspired the mathematical lives of bernice’s descendants, belinda and rita. belinda, who was born just after the brown v. board of education decision, experienced mathematics at a time when educational opportunity was increasing for blacks and the traditional curriculum in mathematics was changing. unlike her mother, who took standard courses in arithmetic and algebra, belinda recalled studying set theory in addition to traditional mathematics topics in her neighborhood school. for many schools in the south, the brown v. board of education decision did not ensure speedy desegregation. likewise, belinda’s school had a predominantly black teaching force, with teachers and their families living and working in the same community. she attended a progressive high school, where she was able to take advanced mathematics courses up to pre-calculus if she chose to do so. belinda’s daughter, rita, like many black children of the 1970s and 1980s attending integrated schools, still found herself one of only a few black students in advanced mathematics classes. the narrative about advanced mathematics classes “being the most segregated places in american society” (stiff & harvey, 1988, p. 190) is a familiar one to high-achieving mathematics students, educators, and researchers (walker, 2006). disturbingly, there is significant evidence that even when black students are ‘qualified’ for these courses, they are shunted into lower-level courses leonard et al. mathematics literacy, identity resilience, & opportunity journal of urban mathematics education vol. 13, no. 1b (special issue) 31 (oakes, 1995), or they may self-select lower-level courses themselves to avoid the isolation they experience in advanced courses (walker, 2006). like high-achieving students describing family members whose own experience in school was not indicative of their mathematics potential (walker, 2006), belinda saw her mother’s life as a cautionary tale: working hard at unsatisfying labor because of limited education and lack of success in school. knowing the narrative of her mother’s interrupted education, she knew that she had to do well in school to avoid having similar experiences. belinda advocated to take advanced mathematics courses and defined her path through the high school mathematics curriculum. she graduated from high school and attended college but interrupted her college education to get married. as belinda described it, the pull of family life was strong. nevertheless, she continued her education in mathematics and mathematics education, becoming a teacher and later a professor. her daughter, rita, attended an ivy league university, and although she is not in a mathematics career, she described mathematics as being an important means to securing her university education and well-paying career. rita’s daughter, zoe, is a beneficiary of her grandmother’s mathematical expertise and her mother’s encouragement of mathematics discussion and learning. in addition, she undoubtedly benefits from the ease of access to technology, which supports mathematics learning through games and apps. what zoe sees in her mother and grandmother are women who are invested in and committed to her education and who have time to talk to her about math. despite zoe’s elders, bernice and lou ellen, not having this luxury of time to devote to their descendants’ education, the lessons they imparted—directly and indirectly—about determining one’s future are evident. that zoe wants to “be a boss” is a testament to the sacrifices of her elders and her freedom to declare herself as such without censure. discussion the multiple and interlocking layers of race, gender, and mathematics among these five generations of females are of interest from a historical perspective, to be sure, but also have significant implications for how we consider these constructs in current educational settings. much of the literature on race and mathematics focuses on racialized hierarchies of performance with little attention to the role of gender and cultural contexts. much of the literature on gender and mathematics omits discussion of race and focuses on (white) women’s attitudes towards and performance in mathematics. however, more recent literature (e.g., walker, 2001; riegle-crumb, 2006) and cross-sectional analyses of national datasets show that the picture is considerably more complex. for example, african american girls have very positive attitudes towards mathematics, higher than those of their white and latina counterparts, and are as likely as white boys to persist in advanced mathematics classes in high school. leonard et al. mathematics literacy, identity resilience, & opportunity journal of urban mathematics education vol. 13, no. 1b (special issue) 32 the societal meme that women are not good in mathematics or are not confident in mathematics largely does not apply to black women. yet, given the counternarratives of these five females, mathematics has multiple meanings. despite their interest in school, educational opportunity for lou ellen and bernice was limited, and their mathematical trajectories were cut short. we hear about math enjoyment from belinda and zoe, while rita sees it as more of a driver to facilitate access to college and career. for belinda, who saw the struggles of her mother, and undoubtedly heard the stories from her grandmother, school was a way to ensure social mobility and success, as it was for many african americans in the united states. however, without the trio programs instituted by the johnson administration, educational attainment and economic success may have been limited during the 1970s. further, in a time of significant economic stress, it is important to consider that some of the same choices that lou ellen and bernice had to make are those that resonate with young women of color today. given that girls are still more likely than their brothers to be called upon to provide domestic care for elderly relatives and younger siblings, the impact of these experiences on their schooling might be considerable. how do we ensure that close familial ties and a desire to support their families do not conflict with girls’ mathematics success? finally, and rather disturbingly, there is substantial evidence that opportunities to learn mathematics are more limited in rural educational settings. various reform movements in mathematics and school organization may not necessarily be aligned—as one example, course offerings in small high schools (a popular school organization reform) in mathematics are often limited. despite their best efforts to stay in school and commit to their education, a ceiling effect in terms of coursework can affect black girls’ college and career chances. while the last century has brought great progress, in many ways, these cautionary tales abound. the mathematical experiences of five generations of black females show the strides blacks have made in education and professional settings. however, there is still much to be done to ensure that children of color have the opportunities to determine their paths. conclusions this study examined the mathematical experiences and educational attainment of five african american females in a matrilineal line that spanned five generations. their mathematics literacy and mathematics identity developed differently due to various educational opportunities based on sociohistorical contexts. the elder women, who grew up prior to brown v. board of education, had family responsibilities that hindered their educational attainment. nevertheless, the elders seized leonard et al. mathematics literacy, identity resilience, & opportunity journal of urban mathematics education vol. 13, no. 1b (special issue) 33 opportunities to advance their education by enrolling in training programs or returning to college. they also encouraged the women in younger generations to stay in school and avoid the distractions that might derail their education. while effort, resilience, and perseverance were important factors in actualizing successful careers, school segregation and integration had an impact on belinda and rita’s academic preparation, respectively. segregation is hegemonic and disadvantages students of color. yet, caring teachers pushed african american students and influenced students like belinda to succeed. in contrast, attending integrated schools provided rita with broader opportunities to learn than her mother, grandmother, and great-grandmother. nevertheless, being the only african american in advanced mathematics courses was intimidating and isolating for rita. yet, these experiences prepared her to succeed at an ivy league institution, whereas her mother dropped out but later returned to college. the counternarrative of this family is not complete because the educational journey continues for young zoe. at the present time, zoe attends a predominantly african american school in a suburban school district near a large urban city. her future is dependent upon the strength of family support and learning opportunities provided by the school district or her mother’s ability to relocate to a better district in the future. while the success of previous generations of women provides no guarantee, the family legacy of educational attainment is a strong predictor of zoe’s future success. what remains to be seen is whether the nation turns the corner in the 21st century to deeply invest in the mathematics education of african american females. recommendations this study reveals the challenges that african american females face in mathematics classrooms in public school systems in the united states. the literature reveals they are underserved when it comes to opportunities to learn advanced mathematics (leonard, 2009; walker, 2012, 2014). low enrollment in advanced courses, lack of rigorous coursework, poor perceptions of ability, and feelings of isolation hinder the success of african american women and girls in mathematics. given that most careers require more mathematical knowledge than in the past, three recommendations emerge from this study. the recommendations are to (a) continue to advance efforts to remedy segregation in public schools (dixson, 2011), (b) broaden opportunities for african american girls to take advanced mathematics and science courses, and (c) support longitudinal studies on the stem education of african american girls. while we acknowledge educational progress for african american children since the passage of brown v. board of education, evidence suggests that public schools remain highly segregated along racial and economic lines (berry et al., 2013; leonard et al. mathematics literacy, identity resilience, & opportunity journal of urban mathematics education vol. 13, no. 1b (special issue) 34 dixson, 2011; leonard et al., 2013). as evident in this study, middle-class, african american families often live in large urban cities or first-tier suburbs where schools serve predominantly black and brown children. african americans continue to be underserved by receiving a different kind of mathematics education than their white counterparts (berry et al., 2013; martin, 2006). until academic success cannot be correlated with one’s zip code, the intent of brown v. board of education remains unrealized. broadening opportunities for african american children in general and african american girls in particular requires broader access and creative innovation. to promote student success in stem, some schools and districts are implementing cuttingedge programs like robotics and game design as early as third grade in afterschool clubs (leonard et al., 2016; repenning et al., 2010). however, encouraging girls to participate in these activities during afterschool clubs has been challenging (leonard et al., 2016). what has been successful in some school districts is offering robotics and game design classes during the regular school day. in these settings, female and minority students have reported high self-efficacy on video gaming and use of computers when a combination of robotics and game design is offered (leonard et al., 2016). connecting robotics and game design to problem solving and mathematics may help to increase the need for learning advanced mathematics, particular among young african american women and girls who are growing up in a gaming culture. for students like zoe, who are intrigued with ipads and tablets, opportunities to learn mathematics and computational thinking through robotics and game design hold promise. computation thinking is a problem-solving process that promotes algorithmic thinking (wing, 2006). in fall 2015, zoe’s grandmother, belinda, facilitated a study on computational thinking in zoe’s school district. this study will unpack teacher learning and facilitation of robotics and game design as well as student learning in stem and development of 21st century skills (see newton et al., 2020). the impact on african american students and girls in these schools has yet to be realized. finally, we recommend conducting longitudinal studies to examine the relationship between african american girls’ exposure to innovative stem curriculum and role models and their persistence in advanced mathematics courses and academic success. such studies are needed to understand issues related to recruiting and retaining african american women as stem majors and professionals. from euphemia lofton haynes (first african american woman to receive a ph.d. in mathematics in 1943) to tasha inniss, sherry scott, and kimberly weems 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(2006). computational thinking. communications of the acm, 49(3), 33–35. https://doi.org/10.1145/1118178.1118215 copyright: © 2020 leonard, walker, bloom, & joseph. this is an open access article distributed under the terms of a creative commons attribution-noncommercial-sharealike 4.0 international license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. journal of urban mathematics education december 2018, vol. 11, no. 1&2, pp. 69–77 ©jume. http://education.gsu.edu/jume julius davis is an associate professor in the department of teaching, learning, and professional development in the college of education at bowie state university, 14000 jericho park road, bowie, md 20715; email: jldavis@bowiestate.edu. his research and scholarly interest focus on black male students and teachers, critical race theory, culturally relevant pedagogy, and social justice in mathematics education. redefining black students’ success and high achievement in mathematics education: toward a liberatory paradigm julius davis bowie state university i always take two sets of notes, one to ace the test and one set i call the truth, and when i find historical contradictions i used the first set as proof– proof that black youths’ mind are being polluted, convoluted, diluted, not culturally rooted. … they’ll have you believe other wise their history is built on high-rise lies the pyramids were completed before greece or rome were conceptualized, then they’ll claim the egyptians race was a mystery you tell them to read herodotus book ii of the histories it cannot be any clearer. … black students, always take two sets of notes. – asante (2008, p. 191) n the opening poem m. k. asante, jr. speaks of two sets of notes (partially excerpted above). these words eloquently capture the dichotomy of the racial reality black students face inside and outside of mathematics spaces. though mathematics is typically considered completely objective, race-neutral, and culture-free, martin (2008) asserts that black students often learn in white institutional space: one can also understand mathematics education as white institutional space by considering who is allowed to speak on issues of teaching, learning, curriculum, and assessment and who dominates positions of power in research and policy contexts. in each instance, white scholars disproportionately fill these roles, an important signifier of white institutional space. (p. 390) white teachers dominate the field of mathematics, another signifier of a white institutional space. in addition to being taught by white teachers, black students learn that white men created mathematics, and the purpose for learning mathematics is to get a high-paying job in white institutions. these assertions make up the crux of the euroi http://education.gsu.edu/jume mailto:jldavis@bowiestate.edu davis toward a liberatory paradigm journal of urban mathematics education vol. 11, no. 1&2 70 centric paradigm that pervades school mathematics and only serve white interests. these assertions do not lead to liberatory outcomes for black people. in a liberatory paradigm, black students must be taught to think communally and must be part of the process of developing institutions and systems to support the economic, political, social, and cultural advancements of their people. thus, as asante explains, black students must take two sets of notes: one to pass the test and another to learn how to use mathematics to support liberation for themselves and their people. there has been a movement in mathematics education to achieve liberatory outcomes for black students. over the last 10 years, there has been a paradigm shift in mathematics education towards a focus on black student success and high achievement (berry, 2005, 2008; jett, 2010, 2011; moody, 2003, 2004; terry & mcgee, 2012) and providing them with a liberatory mathematics education (martin, 2010; martin & mcgee, 2009). the paradigm shift has been led by critical black scholars who seek to challenge the deficit discourse, inadequate conceptualizations of race and racism, privileged perspectives of mathematics, substandard instruction, and mistreatment of black students in mathematics education research, policy, classrooms, and society (anderson, 1990; martin, 2008, 2009; martin, gholson & leonard, 2010; tate, 1993). the current definitions of success and high achievement for black students are based only on static data (e.g., test performance, grade point averages). the test scores and grades of white students are usually situated as the standard for how black student success and high achievement are judged. this standard promotes an individualistic focus that mirrors a eurocentric paradigm, conforms to the standard of whiteness, and creates inadequate definitions and conceptualizations for a liberatory paradigm (anderson, 1990; harris, 1993; martin, 2009). the standard of whiteness is a form of property that dictates acceptable norms, behaviors, cultural practices, status, reputation, achievements, and performance in mathematics spaces and society. it also includes the exclusion of black community. a liberatory paradigm is responsive to the distinct historical and contemporary needs of the collective black community in mathematics education and society at large. as the field of mathematics education moves toward a liberatory paradigm, definitions of success and high achievement must be reconceptualized to support liberatory outcomes. given the united states’ racist history and culture, the collective black community is affected and has specific needs that deserve specific considerations that do not always align with those of the white community. therefore, a eurocentric paradigm will not and cannot serve black people in the same way. for example, there was a period in history where it was illegal for black people to receive an education. they were banned from learning reading and mathematics skills, but many of them clandestinely learned anyway, even in the face of death, because they knew the knowledge would not only benefit them, but also their family, community, and generations to come. black people have always had to think and move with the davis toward a liberatory paradigm journal of urban mathematics education vol. 11, no. 1&2 71 community in mind. true liberation cannot be achieved through individual pursuits. in this essay, i describe and critique the eurocentric paradigm that pervades mathematics education. i explain how the paradigm situates black student success and high achievement to be aligned with the interests and standard of whiteness. i then describe key elements of the paradigm shift in mathematics education focused on successful and high achieving black students and liberation. i redefine success and high achievement in mathematics by aligning it with a liberatory paradigm. for liberatory outcomes to be achieved, the expectations, preparation, and definitions of success and high achievement must be connected to the larger goal of black liberation in mathematics education and society. i ground this liberatory paradigm in a diasporic view of black history in mathematics, culture, values, and interests. the eurocentric paradigm in mathematics education the definitions used to describe success and high achievement for black students in mathematics education are based on a eurocentric paradigm (anderson, 1990; bishop, 1990; martin, 2008) and standard of whiteness (harris, 1993). the paradigm rests on the foundation that europeans created mathematics and possess superior intellectual ability. this paradigm is based on european history, culture, interests (e.g., economics, military), and ways of knowing. it is also based on individualism, capitalism, european/white superiority, and the ranking and sorting of racial groups. there are cultural, political, and economic ramifications for the use of a eurocentric paradigm with black students (apple, 1993). this paradigm misinforms black students about their people’s place in the history of mathematics and seeks to destroy their racial and mathematics identities. the eurocentric paradigm teaches successful and high-achieving black students to align themselves with european interests (e.g., capitalism, warfare), which are causing black people harm the world over. the mathematics curriculum used to determine success and high achievement for black students is mainly based on the contributions of white men from europe and north america. anderson (1990) notes: the dominant curriculum in use today throughout the united states is explicit in asserting that mathematics originated among men in greece and was further developed by european men and their north american descendants. … from generation to generation for centuries this type of eurocentric “scholarship” has been reproduced in the objective and subjective pursuits of justifying racism. (pp. 349–350) as asante’s (2008) poem suggests, the curriculum used in most schools does not teach black students about the mathematics contributions of their ancestors and elders. it represents the implied, understood power of europeans and whites: “the decision to define some groups’ knowledge as the most legitimate, as official davis toward a liberatory paradigm journal of urban mathematics education vol. 11, no. 1&2 72 knowledge, while other groups’ knowledge hardly sees the light of day, says something extremely important about who has power in society” (apple, 1993, p. 222). this (mis)representation has been the status quo for hundreds of years, and at no point has there been a national effort to incorporate the vital contributions of black people from africa, north america, and across the world. the grading and scoring system used in schools is based on a eurocentric curriculum, individualism, and white superiority. it quantifies black students’ knowledge of eurocentric ideas and gives no weight to their knowledge of the contributions of their own people. the grading and scoring system creates a ranking and sorting system that promotes individualism and white superiority that runs counter to the goals and objectives of black liberation. by ranking and sorting the students based on the regurgitation and synthesis of racially biased information, american schools implicitly support the notion of white intellectual superiority. martin (2009) describes this ranking system as the racial hierarchy of mathematical ability, which positions black students at the bottom and white students at the top. the racial hierarchy also applies to black students’ participation and achievement in higher level mathematics courses, in which they are underrepresented. this system supports the notion that few black students possess mathematical ability worthy of participation in higher level courses. in essence, the system determines black students’ academic worth based on a white standard. this imposed hierarchy of mathematical ability inhibits liberatory outcomes for black students. the paradigm shift in mathematics education there have been two major paradigm shifts in mathematics education that seek to challenge deficit views of black students and achieve liberatory outcomes. the focus on successful and high-achieving black students is one of the longest standing and most developed paradigm shifts. this body of research has given insight into the experiences of successful and high-achieving black students at the k–12 and collegiate levels (berry, 2008; jett, 2011; terry & mcgee, 2012; thompson & davis, 2013; thompson & lewis, 2005). success and high achievement most often have been defined based on grade point average, standardized test scores, and advanced placement course participation. this narrow definition is aligned with the eurocentric paradigm and standard of whiteness. there must be holistic definitions that support a liberatory paradigm. researchers have found that successful and high-achieving black students have had (a) early exposure to mathematics; (b) family support and advocacy (e.g., parents, aunts, uncles, guardians); (c) participation in college level mathematics courses and program; (d) teacher and peer support; (e) involvement in extracurricular activities (e.g., math programs, sports); and (f) strong spiritual beliefs (berry, 2008; ellington & frederick, 2010; jett, 2010, 2011; noble, 2011; stinson, 2008; terry & mcgee, davis toward a liberatory paradigm journal of urban mathematics education vol. 11, no. 1&2 73 2012). an important finding emerging from this research is that black adults and peers contribute to and support black students in achieving at high levels in mathematics. this finding illustrates the importance of relationships and community to black students and how they help them to perform at much higher levels. this support also speaks to the need for successful and high-achieving black students to give back to their community as a means of ensuring that others are supported as well. the liberatory paradigm in mathematics education has primarily focused on pedagogy and research related to black students. martin and mcgee (2009) argue, “any relevant framing of mathematics education for african americans must address both the historical oppression that they face and the social realities that they continue to face in contemporary times” (p. 210). in martin’s (2010) edited book mathematics teaching, learning, and liberation in the lives of black children, he assembled black mathematics education scholars and others committed to providing black students with a meaningful mathematics education to “change the direction of research on black children and mathematics” (p.vi). these researchers have described black students’ experiences, socialization, learning, and identity development through the lens of liberation. however, there has been limited focus on the expectations of black students in a liberatory paradigm in mathematics education. this focus is important given that black students have been instrumental in advancing the black struggle for liberation inside and outside of mathematics spaces. as the field moves toward a liberatory paradigm, the expectations for successful black students must be defined. historically, black people who have had intellectual, material, economic, and political resources have used them to advance the collective agenda and interests of their people. this collective effort also highlights the distinct cultural importance of the community over the individual. in a liberatory paradigm, successful and highachieving black students must make a similar commitment to use their resources for the collective upliftment of black people in mathematics education and beyond. a liberatory paradigm of mathematics education for black students a liberatory paradigm requires black students to first develop a liberatory mindset anchored in their history, culture, and interests. central to this paradigm is the understanding that black students have a responsibility to use their mathematical knowledge to help others and their community become self-sufficient. ture and hamilton (2011) assert: black people must redefine themselves, and only they can do that. throughout this country, vast segments of the black communities are beginning to recognize the need to reclaim their history, their culture; to create their own sense of community and togetherness. (p. 37) davis toward a liberatory paradigm journal of urban mathematics education vol. 11, no. 1&2 74 black people must redefine themselves and their ideas of success and high achievement to align with efforts toward total liberation. a focus on liberation in mathematics education is necessarily in opposition to the eurocentric paradigm as well as eurocentric interests and outcomes, such as growing their workforce and supporting international capitalism and warfare. the success of black students in a liberatory paradigm includes elements inside and outside of mathematics education. black students must be taught their role in deconstructing and dismantling the eurocentric paradigm. black students must operate from the premise that mathematics is not purely an objective, neutral, culture-free subject matter or space (ernest, 1991; ernest, sriraman, & ernest, 2016). it is undergirded by the eurocentric paradigm and, therefore, skewed toward white stand-ards and achievement. a liberatory paradigm must inform black students that the way mathematics is currently taught serves white interests and disadvantages black students. this paradigm must include a re-education of the history of mathematics, specifically highlighting the contributions of blacks from africa and throughout the diaspora. moreover, black students must know that white adults and children are not the standards of achievement, success, or participation. they must also be taught that earning high course grades and test scores, while admirable, is not true success if it is based solely on eurocentric mathematical knowledge, individualism, and the standard of whiteness. true success must benefit the entire black community and cannot be achieved if black students have no knowledge of their ancestors’ contributions to the field of mathematics. they must know that their people made important contributions and used mathematics to build great civilizations, specifically the first highly technological civilization known to humankind on the continent of africa. there is evidence that arithmetic, algebra, geometry, trigonometry, and other advanced forms of mathematics originated in egypt with black people (anderson, 1990; bishop, 1990; browder, 1992; jackson, 1970; van sertima, 1991). there is also evidence that many prominent white male greek scholars (e.g., pythagoras, euclid) studied mathematics from black egyptians (anderson, 1990; bishop, 1990; browder, 1992; jackson, 1970; van sertima, 1991). this inclusive history is why black students must take two sets of notes, to identify historical contradictions and provide proof for themselves that they come from great mathematical thinkers. black students must know they are descendants of mathematically competent people who created great civilizations on the continents of africa, the americas, and around the world. over time, this understanding can transform their thinking regarding their own ability to achieve. the history of black people’s contributions to mathematics and how they used it provides a clear direction and purpose for what successful black students must do with their mathematical knowledge. the liberatory paradigm for black student suc davis toward a liberatory paradigm journal of urban mathematics education vol. 11, no. 1&2 75 cess does not focus solely on individual achievement but instead includes components of giving back, sharing and creating for the whole black community. mathematics should be used to help others and build black communities all over the world. black student success should be about the collective mathematical achievement of the black community (thompson & davis, 2013). therefore, successful students must help others acquire mathematical knowledge, especially those experiencing challenges. no black student has achieved success alone. there has always been family and community members who have made sacrifices and offered support in becoming mathematically successful (hrabowski, maton, greene, & greif, 1998). successful black students must do their part to ensure that others in their community are mathematically proficient. this requirement reinforces the commitment to the collective achievement of the black community. in the eurocentric paradigm, there is no focus on using mathematics for the benefit of the black community or liberatory purposes. it has an individualistic aim, to benefit and promote european history, culture, and interests. although this focus does not align with black cultural norms, there are still black students who have to navigate white institutional spaces to become successful in mathematics. however, they are taught to advance white global interests, not black liberation. these “successful” black students are taught to use their mathematical knowledge to go to college and get high paying jobs, in companies owned mostly by white men (kroll & dolan, 2018). this focus diverts black intellectual resources away from causes and efforts that support liberation. it also supports the advancement of technology that can be used to further international capitalism and warfare. this career track is perfectly aligned with the goals and objectives of the larger societal eurocentric paradigm. liberated successful black students know they should not acquire mathematical knowledge solely to get a high paying job. the current and historical position of their people makes that an unwise choice because they see the benefit of a cooperative economics agenda versus an individual and capitalistic agenda. black students should use their mathematical knowledge to support cooperative economics and collective work and responsibility, which can benefit them, their family, and black communities around the world. perhaps the biggest difference between a eurocentric paradigm and a liberatory paradigm is that the former can produce successful black students in mathematics who go on to be good employees, but the latter produces black students who are agents of change and purveyors of community improvement and institution building. closing words the reality of learning mathematics in white institutional spaces requires that black students take two sets of notes. the first must be used to pass the test; the sec davis toward a liberatory paradigm journal of urban mathematics education vol. 11, no. 1&2 76 ond must be used to deconstruct and challenge the faulty mathematical knowledge that has been presented as objective, true, and culture-free. the second set of notes self-empowers black students by informing them of their people’s contributions to mathematics and the world. it encourages them to use their knowledge to produce liberatory outcomes for their people. the second set of notes is crucial for the development of a liberatory paradigm to situate black students’ success and high achievement in mathematics education. until all black people are liberated, black students must be provided the knowledge, skills, and dispositions to navigate and achieve in white spaces, all while contributing to the liberation of their people. the eurocentric paradigm does not focus on the collective achievement of black students, so it must be abandoned. the focus on liberation for black people in society has always been about the collective, not the individual. the same must be true for mathematics education. references anderson, s. e. (1990). worldmath curriculum: fighting eurocentrism in mathematics. the journal of negro education, 59(3), 348–359. apple, m. w. (1993). the politics of official knowledge: does a national curriculum make sense? teachers college record, 95(2), 222–241. asante, m. k. (2008). it’s bigger than hip hop: the rise of the post-hip-hop generation. new york, ny: st. martin’s press. berry, r. q., iii. (2005). voices of success: descriptive portraits of two successful african american male middle school mathematics students. journal of african american studies, 8(4), 46–62. berry, r. q., iii. (2008). access to upper-level mathematics: the stories of successful african american middle school boys. journal for research in mathematics education, 39(5), 464–488. bishop, a. j. (1990). western mathematics: the secret weapon of cultural imperialism. race & class, 32(2), 51–65. browder, a. t. (1992). nile valley contributions to civilization: exploding the myths (vol. 1). washington, dc: institute of karmic guidance. ellington, r. m., & frederick, r. (2010). black high achieving undergraduate mathematics majors discuss success and persistence in mathematics. negro educational review, 61(1–4), 61–84. ernest, p. (1991). philosophy of mathematics education. new york, ny: routledge. ernest, p., sriraman, b., & ernest, n. (eds.). (2016). critical mathematics education: theory, praxis and reality. charlotte, nc: information age. harris, c. i. (1993). whiteness as property. harvard law review, 106(8), 1707–1791. hrabowski, f. a., iii, maton, k. i., greene, m. l., & greif, g. l. (1998). beating the odds: raising academically successful african american males. new york, ny: oxford university press. jackson, j. g. (1970). introduction to african civilization. charleston, sc: the citadel press. jett, c. c. (2010). “many are called, but few are chosen”: the role of spirituality and religion in the educational outcomes of “chosen” african american male mathematics majors. the journal of negro education, 79(3), 324–334. jett, c. c. (2011). “i once was lost, but now am found”: the mathematics journey of an african american male mathematics doctoral student. journal of black studies, 42(7), 1125–1147. davis toward a liberatory paradigm journal of urban mathematics education vol. 11, no. 1&2 77 kroll, l., & dolan, k. a. (2018, october). forbes 400 2018: a new number one and a recordbreaking year for america’s richest people. forbes. retrieved from: https://www.forbes.com/forbes-400/#17359e367e2f martin, d. b. (2008). e(race)ing race from a national conversation on mathematics teaching and learning: the national mathematics advisory panel as white institutional space. the mathematics enthusiast, 5(2), 387–398. martin, d. b. (2009). researching race in mathematics education. teachers college record, 111(2), 295–338. martin, d. b. (ed.). (2010). mathematics teaching, learning, and liberation in the lives of black children. new york, ny: routledge. martin, d. b., gholson, m. l., & leonard, j. (2010). mathematics as gatekeeper: power and privilege in the production of knowledge. journal of urban mathematics education, 3(2), 12–24. retrieved from http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/95/57 martin, d. b., & mcgee, e. (2009). mathematics literacy for liberation: reframing mathematics education for african american children (pp. 207–238). in b. greer, s. mukhopadhyay, a. b. powell (eds.), culturally responsive mathematics education. new york, ny: routledge. moody, v. r. (2003). the ins and outs of succeeding in mathematics: african american students’ notions and perceptions. multicultural perspectives, 5(1), 33–37. moody, v. r. (2004). sociocultural orientations and the mathematical success of african american students. the journal of educational research, 97(3), 135–146. noble, r. (2011). mathematics self-efficacy and african american male students: an examination of models of success. journal of african american males in education, 2(2), 187–213. stinson, d. w. (2008). negotiating sociocultural discourses: the counter-storytelling of academically (and mathematically) successful african american male students. american educational research journal, 45(4), 975–1010. tate, w. f. (1993). advocacy versus economics: a critical race analysis of the proposed national assessment in mathematics. thresholds in education, 19(1&2), 16–22. terry, c. l., sr., & mcgee, e. o. (2012). “i’ve come too far, i’ve worked too hard”: reinforcement of support structures among black male mathematics students. journal of mathematics education at teachers college, 3(2), 73–84. thompson, l., & davis, j. (2013). the meaning high-achieving african-american males in an urban high school ascribe to mathematics. the urban review, 45(4), 490–517. thompson, l. r., & lewis, b. f. (2005). shooting for the stars: a case study of the mathematics achievement and career attainment of an african american male high school student. the high school journal, 88(4), 6–18. ture, k., & hamilton, c. v. (2011). black power: politics of liberation in america. new york, ny: vintage books. van sertima, i. (1991). blacks in science: ancient and modern. piscataway, nj: transaction. https://www.forbes.com/forbes-400/#17359e367e2f http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/95/57 journal of urban mathematics education july 2016, vol. 9, no. 1, pp. 1–6 ©jume. http://education.gsu.edu/jume david w. stinson is an associate professor of mathematics education in the department of middle and secondary education in the college of education and human development, at georgia state university, p.o. box 3978, atlanta, ga, 30303; e-mail: dstinson@gsu.edu. his research interests include exploring socio-cultural, -historical, and -political aspects of mathematics and mathematics teaching and learning from a critical postmodern theoretical (and methodological) perspective. he is a co-founder and current editor in chief of the journal of urban mathematics education. editorial contributing a commentary to jume: keeping things going while they are still stirring david w. stinson georgia state university s jume continues to grow in both years and influence, i have had an increasing number of mathematics educators (i.e., scholars, researchers, teacher educators, and/or classroom teachers) inquire about contributing a manuscript to the commentary or response commentary sections of the journal. some of the usual questions: can anyone submit a manuscript for consideration to these sections? or are manuscripts by invitation? are manuscripts peer reviewed? what is the turnaround time? what issues or topics might be included? how long is a typical manuscript? what is the purpose (or purposes) of a jume commentary? here, i aim to answer these and other questions (see highlighted hyperlinks throughout for additional information). i begin by responding to the last question, given that the subtitle of this editorial—a paraphrase of sojourner truth’s words spoken at the first annual meeting of the american equal rights association in 1867—reflects the purpose of a jume commentary. her extended remarks certainly convey, i believe, the purpose: “so i am for keeping the thing going while things are stirring; because if we wait till it is still, it will take a great while to get it going again” (truth, 1867, p. 20). that is to say, the purpose of the commentary section—and its companion, the response commentary section1—is to keep conversations about critical issues going in constructive directions, forever bringing those critical issues into the center.2 when stirring, not only do things keep going but also those things on the margins are brought to the center. a perusal of the titles of jume commentaries over the past 9 years provides a listing of sorts of some of the critical issues that need to be contin 1 manuscripts submitted to the response commentary section should be in direct response to commentaries published in jume, either in the current issue or past issues. these response commentaries can provide a different viewpoint, extend the conversation, or take the conversation in a new direction. 2 when bringing critical issues to the center, the aim is not to somehow normalize such issues but rather to include them as central components of productive discussions and actions. a http://education.gsu.edu/jume mailto:dstinson@gsu.edu stinson editorial journal of urban mathematics education vol. 9, no. 1 2 uously engaged and brought to the center (see appendix a). because many, if not most, of these issues can be perceived as troubling and uncomfortable topics for “polite conversation,” too often the way they are discussed or “managed” in the larger mathematics education community is through journal special editions; set-aside meetings, conferences, workshops, or courses; or themed edited volumes, to name just a few. in other words, rarely are these issues integrated throughout the day-to-day discussions and activities of the vast majority of mathematics educators. but here at jume these too-uncomfortable-for-polite-conversation issues are the very ones that are openly integrated and, most importantly, interrogated throughout the online pages of every jume edition. in many ways, the commentary (or commentaries) of each published edition sets the stage, so to speak, to remind our readers about the journal’s mission: “to foster a transformative global academic space in mathematics that embraces critical research, emancipatory pedagogy, and scholarship of engagement in urban communities.” can anyone submit a manuscript for consideration to these sections? or are manuscripts by invitation? the responses to these two questions: yes and yes. submitted manuscripts to the commentary and response commentary sections are both unsolicited and solicited.3 but it is crucial to note, whether unsolicited or solicited, manuscripts must be scholarly essays solidly grounded in the literature. manuscripts are not to be confused with blog postings, letters to the editor, or op-eds. similar to these writing spaces, authors are encouraged to submit manuscripts in first-person narratives but these narratives must be grounded in the science of the author(s), the science of others, or, preferably, both. in fact, when manuscripts are solicited, we (the editorial team) request that the author(s) cite heavily her or his own work and the work of others so that the reference list might become an educative resource for our readers. most often, solicited authors are noted senior scholars who have an extensive and established body of research and scholarship that reflects the mission of jume. nonetheless, throughout the past 9 years, commentaries have been authored and co-authored not only by senior scholars but also by mid-career folks, freshly minted phds, and doctoral students. are manuscripts peer reviewed? what is the turnaround time? yes, all manuscripts submitted to both the commentary and the response commentary sections are open peer reviewed4 by the editor and members of the editorial team (and, at times, other senior members of the larger mathematics education community). ini 3 solicited manuscripts might also include revised versions of delivered talks (see, e.g., leonard, 2012; martin, 2015; nasir, 2016); the aim here is to bring unpublished talks to the larger mathematics education community. 4 submissions to the pubic stories of mathematics educators and the book review sections are also open peer reviewed by the editor and members of the editorial team. stinson editorial journal of urban mathematics education vol. 9, no. 1 3 tially, manuscripts were sent out for double-blind peer review. the process was changed to open peer review so that solicited authors might receive reviews in a timely manner. given that the journal is published only twice a year, it was important that time from initial solicitation to published commentary be no more than six months. in most cases, for both unsolicited and solicited submitted manuscripts, authors receive reviews within eight to ten weeks, with time from initial submission to publication being around six to eight months. (see peer review process for additional information.) what issues or topics might be included? how long is a typical manuscript? the issues or topics of submitted manuscripts vary widely. to provide an idea of what might be addressed in a jume commentary, i borrow partially from kilpatrick (2007) when he provided a list on what topics might be included in a manuscript submitted to the research commentary section of the journal for research in mathematics education. i modify and extend his list here to focus explicitly on an urban mathematics education context:  commentaries on research within an urban context;  discussions of the connections between research, policy, and/or practice within an urban context;  scholarly analyses of policy trends related to urban mathematics education (e.g., research funding, national policies);  scholarly essays on sociopolitical issues that relate to urban mathematics education;  commentaries on the relationship between research and evaluation within an urban context; and  scholarly debates among proponents of different viewpoints on issues that relate to urban mathematics education. this list is certainly not exhaustive, but does provide an idea of the different possible directions a jume commentary might take (for additional guidance, see appendix a). the length of manuscripts typically range from 1,500–4,500 words, inclusive of references, appendices, footnotes, figures, and tables. (see section policies and author guidelines for additional information about submitting a manuscript to jume.) with the aim of keeping things going while they are sill stirring, we look forward to receiving your submission to the commentary or response commentary sections. if you have additional questions, please email me at dstinson@gsu.edu. references kilpatrick, j. (2007). a commentary on research commentaries [editorial]. journal for research in mathematics education, 38(2), 106–107. http://ed-osprey.gsu.edu/ojs/index.php/jume/about/editorialpolicies#peerreviewprocess http://ed-osprey.gsu.edu/ojs/index.php/jume/about/editorialpolicies#sectionpolicies http://ed-osprey.gsu.edu/ojs/index.php/jume/about/editorialpolicies#sectionpolicies http://ed-osprey.gsu.edu/ojs/index.php/jume/about/submissions#authorguidelines mailto:dstinson@gsu.edu stinson editorial journal of urban mathematics education vol. 9, no. 1 4 leonard, j. (2012). er’body talkin’ ‘bout social justice ain’t goin’ there. journal of urban mathematics education, 5(2), 18–27. retrieved from http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/192/117 martin, d. b. (2015). the collective black and principles to actions. journal of urban mathematics education, 8(1), 17–23. retrieved from http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/270/169 nasir, n. s. (2016). why mathematics educators should care about race and culture? journal of urban mathematics education, 9(1), 6–17. retrieved from http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/298/192 truth, s. (1867). speech, proceedings of the first anniversary of the american equal rights association, held at the church of the puritans, new york, may 9 and 10, 1867 (pp. 20–21). new york, ny: r. j. johnston, printer. http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/192/117 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/298/192 stinson editorial journal of urban mathematics education vol. 9, no. 1 5 appendix a commentaries and response commentaries by title and author (2008–2016) note: scroll over titles and click; all are hyperlinked.  putting the “urban” in mathematics education scholarship william f. tate – washington university in st. louis  the common core state standards initiative: a critical response eric (rico) gutstein – university of illinois at chicago  mathematics as gatekeeper: power and privilege in the production of knowledge danny bernard martin, maisie l. gholson – university of illinois at chicago jacqueline leonard – university of colorado denver “both and”—equity and mathematics: a response to martin, gholson, and leonard jere confrey – north carolina state university engaging students in meaningful mathematics learning: different perspectives, complementary goals michael t. battista – the ohio state university  changing students’ lives through the de-tracking of urban mathematics classrooms jo boaler – stanford university  positive possibilities of rethinking (urban) mathematics education within a postmodern frame margaret walshaw – massey university  neoliberal urbanism, race, and equity in mathematics education pauline lipman – university of illinois at chicago  erbody talkin bout social justice aint goin there jacqueline leonard – university of wyoming  why (urban) mathematics teachers need political knowledge rochelle gutiérrez – university of illinois at urbana-champaign  place matters: mathematics education reform in urban schools celia rousseau anderson – university of memphis  why should mathematics educators learn from and about latina/o students’ in-school and out-of-school experiences? marta civil – the university of arizona http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/19/2 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/88/43 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/95/57 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/95/57 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/108/53 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/108/53 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/115/58 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/115/58 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/138/85 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/138/85 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/141/89 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/141/89 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/191/116 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/192/117 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/223/148 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/231/150 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/251/159 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/251/159 stinson editorial journal of urban mathematics education vol. 9, no. 1 6  the collective black and principles to actions danny bernard martin – university of illinois at chicago call for mathematics education colleagues and stakeholders to collaboratively engage with nctm: in response to martin’s commentary diane j. briars – nctm president matt larson – nctm president-elect marilyn e. strutchens – nctm board of directors david barnes – nctm associate executive director, research, learning and development  mathematics and social justice: a symbiotic pedagogy gareth bond, egan j. chernoff – university of saskatchewan, canada  from implicit to explicit: articulating equitable learning trajectories based instruction marrielle myers – kennesaw state university paola sztajn – north carolina state university p. holt wilson – university of north carolina at greensboro cyndi edgington – north carolina state university  why should mathematics educators care about race and culture? na’ilah suad nasir – university of california, berkeley http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/270/169 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/292/178 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/292/178 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/256/170 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/288/177 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/288/177 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/298/192 journal of urban mathematics education december 2015, vol. 8, no. 2, pp. 1–10 ©jume. http://education.gsu.edu/jume david w. stinson is an associate professor of mathematics education in the department of middle and secondary education in the college of education and human development, at georgia state university, p.o. box 3978, atlanta, ga, 30303; e-mail: dstinson@gsu.edu. his research interests include exploring socio-cultural, -historical, and -political aspects of mathematics and mathematics teaching and learning from a critical postmodern theoretical (and methodological) perspective. he is a co-founder and current editor-in-chief of the journal of urban mathematics education. editorial the journal handbook of research on urban mathematics teaching and learning: a resource guide for the every student succeeds act of 2015 david w. stinson georgia state university s a critical1 mathematics educator, it is difficult not to be pessimistic about the every student succeeds act of 2015 (essa), signed into law by president barak obama on december 10th. the essa, similar to it predecessors, has an admirably worded purpose statement: “to provide all children significant opportunity to receive a fair, equitable, and high-quality education, and to close educational achievement gaps” (essa, 2015, sec. 1001). but after more than a decade of suffering through federal legislation that left far too many children behind and yielded far too many losers in the race to the top, i have become increasingly doubtful that any organization, including the federal government, has “the will” (hilliard, 1991, p. 31)2 to facilitate “the kind of violent reform necessary to change the conditions of african american, latin@, indigenous, and poor students [i.e., the collective black3] in mathematics education” (martin, 2015, p. 22). nevertheless, it is being 1 by critical, i mean in the critical theoretical sense. bronner (2011), in providing a definition of sorts of critical theory, writes: critical theory refuses to identify freedom with any institutional arrangement or fixed system of thought. it questions the hidden assumptions and purposes of competing theories and existing forms of practice. … critical theory insists that thought must respond to the new problems and the new possibilities for liberation that arise from changing historical circumstances. interdisciplinary and uniquely experimental in character, deeply skeptical of tradition and all absolute claims, critical theory…[is] concerned not merely with how things [are] but how they might be and should be. (pp. 1–2) 2 in his article titled “do we have the will to educate all children?” hilliard (1991) writes: if our destination is excellence on a massive scale, not only must we change from the slow lane into the fast lane; we literally must change highways. perhaps we need to abandon the highways altogether to take flight, because the highest goals that we can imagine are well within reach for those who have the will to excellence. (p. 36, emphasis in original) 3 martin (2015), attributing the term to eduardo bonilla-silva, named this group of currently and historically underserved students the collective black. a http://education.gsu.edu/jume mailto:dstinson@gsu.edu https://www.congress.gov/114/bills/s1177/bills-114s1177enr.pdf stinson editorial journal of urban mathematics education vol. 8, no. 2 2 critical that makes me optimistic as well, albeit a “non-stupid optimism” (mcwilliam, 2005, p. 1).4 it is this forever oscillating between pessimism and optimism that drives me and many other critical educators to do the work that we do. for the past 8 years, exemplars of this crucially needed work—completed by a particular group of (largely) critical mathematics educators—are found within the online pages of the journal of urban mathematics education (jume). the readers, editors, reviewers, and authors of jume (a collective group that numbers more than 1,000 strong) have brought to life over 1,700 pages of scholarly editorials, commentaries, response commentaries, public stories, research articles, and book reviews. this group of educators includes those who have spent decades working to provide all children significant opportunity to receive a fair, equitable, and highquality education (many with a specific focus on the collective black), as well as those who are just beginning their careers as critical mathematics classroom teachers, teacher educators, and/or education researchers. the purpose behind the creation of jume was and continues to be to create a movement of change in mathematics education (matthews, 2008). over the past 8 years, jume has offered different statements—that is, different knowledges (cf. foucault, 1969/1972)—about “urban” mathematics education and, in turn, different statements about urban children and urban schools (stinson, 2010). to date, web views of jume content have exceeded 140,000 views, and google scholar citations have exceeded 400, with google and google scholar web searches returning over 2,300 and 340 hits, respectively. four years ago, based on the power, in the foucauldian sense (see, e.g., foucault, 1980), of the academic edited handbook to produce and reproduce knowledge in both social science research, in general (e.g., denzin & lincoln, 1994, 2000, 2005, 2011), and mathematics education research, in particular (e.g., grouws, 1992; lester, 2007), i suggested that jume be envisioned “as a both–and rather than an either–or research and pedagogical resource” (stinson, 2011, p. 3). that is, jume can function as both a peer-reviewed journal and an academic edited handbook on urban mathematics education. i then proceeded to provide the table of contents, if you will, of the first edition of the handbook of research on urban mathematics teaching and learning. here, i offer an expanded version of that table of contents, including the research and scholarship published in jume over the past 4 years (see appendix a). 5 4 mcwilliam (2005) argues that teachers who maintain their passion for teaching even after seeing endless rounds of ideas and polices come through do not indulge in mindless optimism but rather a nonstupid optimism. 5 see also two jume special issues: the benjamin banneker association and national science foundation (bba-nsf) special issue (bullock, alexander, & gholson, 2012) and the privilege and oppression in the mathematics preparation of teacher educators (prompte) special issue (stinson & spencer, 2013), as well as the editorials, public stories, and book reviews published in nearly every issue. http://www.scholar.google.com/citations?user=dyb3gm0aaaaj&h1=en http://www.scholar.google.com/citations?user=dyb3gm0aaaaj&h1=en http://www.google.com/search?num=20&site=source=hp&q=%22journal+of+urban+mathematics+education%22&oq=%22journal+of+urban+mathematics+education%22gs_1=hp.3..0i22i30.3768.3768.0.6502.2.2.0.0.0.0.488.639.0j1j4-1.2.0....01c.2.49.hp..1.1.150.0nh1o_atnmo0 http://scholar.google.com/scholar?q=%22journal+of+urban+mathematics+education%22&btng=&hl=en&as_sdt=0%2c11 stinson editorial journal of urban mathematics education vol. 8, no. 2 3 i also suggest here an expanded use for jume beyond its use as a research and/or pedagogical resource. i suggest that jume be used as an easily accessible resource guide to assist those mathematics education leaders and policy makers who will be busy in the coming months and years translating essa into policies and practices intended to ensure that every “urban student” succeeds in mathematics. this time around, however, i hope that members of the larger mathematics education community will neither allow politics to take the place of scientific inquiry (boaler, 2008) nor erase “race” from a national conversation on mathematics teaching and learning (martin, 2008), among other policy missteps and omissions of the past.6 as the single largest and most up-to-date collection of theoretical and empirical social science on urban mathematics teaching and learning, i hope those members of the mathematics education community who will be charged (both directly and indirectly) to translate essa will turn to jume often as they consider bullock’s (2015) most recent direct and timely question: – “do all lives matter in mathematics education?” references bronner, s. e. (2011). critical theory: a very short introduction. oxford, united kingdom: oxford university press. bullock, e. c., alexander, n. n, & gholson, m. l. (eds.). (2012). proceedings of the 2010 philadelphia and 2011 atlanta benjamin banneker association conferences – beyond the numbers [special issue]. journal of urban mathematics education, 5(2). retrieved from http://ed-osprey.gsu.edu/ojs/index.php/jume/issue/view/10 bullock, e. c. (2015, november 18). do all lives matter in mathematics education? invited speaker to the lappan-phillips-fitzgerald mathematics education colloquium series, program of mathematics education at michigan state university, east lansing, mi. boaler, j. (2008). when politics took the place of inquiry: a response to the national mathematics advisory panel’s review of instructional practices [special issue]. educational researcher, 37(9), 588–594. denzin, n. k., & lincoln, y. s. (1994). handbook of qualitative research. thousand oaks, ca: sage. denzin, n. k., & lincoln, y. s. (2000). handbook of qualitative research (2nd ed.). thousand oaks, ca: sage. denzin, n. k., & lincoln, y. s. (2005). the sage handbook of qualitative research (3rd ed.). thousand oaks, ca: sage. denzin, n. k., & lincoln, y. s. (2011). the sage handbook of qualitative research (4th ed.). thousand oaks, ca: sage. 6 for instance, although it is stated that the views expressed in foundations for success: the final report of the national mathematics advisory panel [nmap, 2008] “do not necessarily represent the positions and polices of the [u.s.] department of education” (p. ii), both the panel and the resulting report were commissioned under the no child left behind act of 2001. the panel was charged “with the responsibilities of relying upon the ‘best available scientific evidence’ and recommending ways ‘… to foster greater knowledge of and improved performance in mathematics among american students’” (p. xiii). for critiques of the final report, see kelly (2008) and sriraman (2008). http://ed-osprey.gsu.edu/ojs/index.php/jume/issue/view/10 stinson editorial journal of urban mathematics education vol. 8, no. 2 4 every student succeeds act of 2015, pub. l. no. 114-95. foucault, m. (1972). the archaeology of knowledge (a. m. sheridan smith, trans.). new york, ny: pantheon. (original work published 1969) foucault, m. (1980). power/knowledge: selected interviews and other writings, 1972–1977 (c. gordon, ed.; c. gordon, l. marshall, j. mepham, & k. soper, trans.). new york, ny: pantheon. grouws, d. a. (ed.). (1992). handbook of research on mathematics teaching and learning. new york, ny: macmillan. hilliard, a. g., iii. (1991). do we have the will to educate all children? educational leadership, 49(1), 31–36. kelly, a. e. (ed.). (2008). reflections on the national mathematics advisory panel final report [special issue]. educational researcher, 37(9). lester, f. k. (ed.). (2007). second handbook of research on mathematics teaching and learning. charlotte, nc: information age. matthews, l. e. (2008). illuminating urban excellence: a movement of change within mathematics education. journal of urban mathematics education, 1(1), 1–4. retrieved from http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/20/9 martin, d. b. (2008). e(race)ing race from a national conversation on mathematics teaching and learning: the national mathematics advisory panel as white institutional space. the montana mathematics enthusiast, 5(2-3), 387–398. martin, d. b. (2015). the collective black and principles to actions. journal of urban mathematics education, 8(1), 17–23. retrieved from http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/270/169 mcwilliam, e. (2005). schooling the yuk/wow generation. apc monographs, 17, 1–10. national mathematics advisory panel. (2008). foundations for success: the final report of the national mathematics advisory panel. washington, dc: u.s. department of education. no child left behind act 2001, pub. l. no. 107-110, 115 stat. 1425 (2002). sriraman, b. (ed.). (2008). critical notice on the national mathematics advisory panel report [special section]. montana mathematics enthusiast, 5(2-3). retrieved from http://www.math.umt.edu/tmme/vol5no2and3/ stinson, d. w. (2010). how is it that one particular statement appeared rather than another?: opening a different space for different statements about urban mathematics education. journal of urban mathematics education, 3(1), 1–11. retrieved from http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/116/69 stinson, d. w. (2011). both the journal and handbook of research on urban mathematics teaching and learning. journal of urban mathematics education, 4(2), 1–6. retrieved from http://edosprey.gsu.edu/ojs/index.php/jume/article/view/156/96 stinson, d. w., & spencer, j. a. (eds.). (2013). privilege and oppression in the mathematics preparation of teacher educators [special issue]. journal of urban mathematics education, 6(1). retrieved from http://ed-osprey.gsu.edu/ojs/index.php/jume/issue/view/12 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/20/9 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/270/169 http://www.math.umt.edu/tmme/vol5no2and3/ http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/116/69 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/156/96 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/156/96 http://ed-osprey.gsu.edu/ojs/index.php/jume/issue/view/12 stinson editorial journal of urban mathematics education vol. 8, no. 2 5 appendix a note: scroll over titles and click, all “chapters” are hyperlinked. handbook of research on urban mathematics teaching and learning (expanded edition) table of contents part i: issues 1. putting the “urban” in mathematics education scholarship william f. tate – washington university in st. louis 2. the common core state standards initiative: a critical response eric (rico) gutstein – university of illinois at chicago 3. mathematics as gatekeeper: power and privilege in the production of knowledge danny bernard martin, maisie l. gholson – university of illinois at chicago jacqueline leonard – university of colorado denver 3.1 “both and”—equity and mathematics: a response to martin, gholson, and leonard jere confrey – north carolina state university 3.1 engaging students in meaningful mathematics learning: different perspectives, complementary goals michael t. battista – the ohio state university 4. changing students’ lives through the de-tracking of urban mathematics classrooms jo boaler – stanford university 5. positive possibilities of rethinking (urban) mathematics education within a postmodern frame margaret walshaw – massey university 6. neoliberal urbanism, race, and equity in mathematics education pauline lipman – university of illinois at chicago 7. erbody talkin bout social justice aint goin there jacqueline leonard – university of wyoming 8. why (urban) mathematics teachers need political knowledge rochelle gutiérrez – university of illinois at urbana-champaign http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/19/2 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/88/43 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/95/57 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/95/57 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/108/53 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/108/53 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/115/58 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/115/58 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/138/85 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/138/85 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/141/89 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/141/89 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/191/116 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/192/117 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/223/148 stinson editorial journal of urban mathematics education vol. 8, no. 2 6 9. place matters: mathematics education reform in urban schools celia rousseau anderson – university of memphis 10. why should mathematics educators learn from and about latina/o students’ in-school and out-of-school experiences? marta civil – the university of arizona 11. the collective black and principles to actions danny bernard martin – university of illinois at chicago 11.1 call for mathematics education colleagues and stakeholders to collaboratively engage with nctm: in response to martin’s commentary diane j. briars – nctm president matt larson – nctm president-elect marilyn e. strutchens – nctm board of directors david barnes – nctm associate executive director, research, learning and development 12. mathematics and social justice: a symbiotic pedagogy gareth bond, egan j. chernoff – university of saskatchewan, canada 13. from implicit to explicit: articulating equitable learning trajectories based instruction marrielle myers – kennesaw state university paola sztajn – north carolina state university p. holt wilson – university of north carolina at greensboro cyndi edgington – north carolina state university part ii: theoretical perspectives 14. a metropolitan perspective on mathematics education: lessons learned from a “rural” school district celia rousseau anderson, angiline powell – university of memphis 15. mathematical counterstory and african american male students: urban mathematics education from a critical race theory perspective clarence l. terry, sr. – occidental college 16. caring, race, culture, and power: a research synthesis toward supporting mathematics teachers in caring with awareness tonya gau bartell – university of delaware 17. ethnomodeling as a research theoretical framework on ethnomathematics and mathematical modeling milton rosa, daniel clark orey – universidade federal de ouro preto, brazil http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/231/150 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/251/159 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/251/159 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/270/169 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/292/178 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/292/178 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/256/170 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/280/177 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/280/177 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/34/12 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/34/12 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/98/87 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/98/87 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/128/84 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/128/84 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/195/143 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/195/143 stinson editorial journal of urban mathematics education vol. 8, no. 2 7 part iii: teachers and teaching 18. comparing teachers’ conceptions of mathematics education and student diversity at highly effective and typical elementary schools richard s. kitchen – university of new mexico francine cabral roy – university of rhode island okhee lee, walter g. secada – university of miami 19. preservice teachers’ changing conceptions about teaching mathematics in urban elementary classrooms mindy kalchman – depaul university 20. evolution of (urban) mathematics teachers’ identity mary q. foote – queens college, cuny beverly s. smith, laura m. gillert – the city college of new york, cuny 21. when am i going to learn to be a mathematics teacher? a case study of a novice new york city teaching fellow michael meagher – brooklyn college, cuny andrew brantlinger – university of maryland, college park 22. success made probable: creating equitable mathematical experiences through project-based learning dionne i. cross – indiana university bloomington rick a. hudson – university of southern indiana olufunke adefope – georgia southern university mi yeon lee, lauren rapacki, arnulfo perez – indiana university bloomington 23. regarding the mathematics education of english learners: clustering the conceptions of preservice teachers laura mcleman – university of michigan flint anthony fernandes – university of north carolina charlotte michelle mcnulty – university of michigan flint 24. k–8 teachers’ concerns about teaching latino/a students cynthia oropesa anhalt – the university of arizona maría elena rodríguez pérez – universidad de guadalajara 25. affinity through mathematical activity: cultivating democratic learning communities tesha sengupta-irving – university of california, irvine 26. delegating mathematical authority as a means to strive toward equity teresa k. dunleavy – vanderbilt university 27. “i just wouldn’t want to get as deep into it”: preservice teachers’ beliefs about the role of controversial topics in mathematics education ksenija simic-muller – pacific lutheran university anthony fernandes – university of north carolina at charlotte mathew d. felton-koestler – ohio university http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/24/14 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/24/14 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/97/79 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/97/79 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/109/93 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/120/99 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/120/99 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/129/121 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/129/121 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/170/123 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/170/123 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/158/141 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/208/161 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/208/161 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/242/172 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/259/181 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/259/181 stinson editorial journal of urban mathematics education vol. 8, no. 2 8 part iv: teacher education 28. teaching mathematics for social justice: reflections on a community of practice for urban high school mathematics teachers lidia gonzalez – york college, cuny 29. math links: building learning communities in urban settings jacqueline leonard – temple university brian r. evans – pace university 30. learning to teach mathematics in urban high schools: untangling the threads of interwoven narratives haiwen chu – graduate center of city university of new york laurie h. rubel – brooklyn college, cuny 31. the mathematics learning discourse project: fostering higher order thinking and academic language in urban mathematics classrooms megan e. staples, mary p. truxaw – university of connecticut 32. collaborative evaluative inquiry: a model for improving mathematics instruction in urban elementary schools iman c. chahine – georgia state university lesa m. covington clarkson – university of minnesota 33. k–2 teachers’ attempts to connect out-of-school experiences to in-school mathematics learning allison w. mcculloch, patricia l. marshal – north carolina state university 34. “estoy acostumbrada hablar ingéls”: latin@ pre-service teachers’ struggles to use spanish in a bilingual afterschool mathematics program eugenia vomvoridi-ivanović – university of south florida 35. recruiting secondary mathematics teachers: characteristics that add up for african american students tamra c. ragland – hamilton county educational service center shelley sheats harkness – university of cincinnati part v: student learning and identity 36. social identities and opportunities to learn: student perspectives on group work in an urban mathematics classroom indigo esmonde, kanjana brodie, lesley dookie, miwa takeuchi – university of toronto 37. exploring the nexus of african american students’ identity and mathematics achievement francis m. nzuki – the richard stockton college of new jersey http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/32/13 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/32/13 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/5/5 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/50/60 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/50/60 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/74/49 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/74/49 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/44/38 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/44/38 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/94/92 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/94/92 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/172/122 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/172/122 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/221/163 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/221/163 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/46/35 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/46/35 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/45/68 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/45/68 stinson editorial journal of urban mathematics education vol. 8, no. 2 9 38. how do we learn? african american elementary students learning reform mathematics in urban classrooms lanette r. waddell – vanderbilt university 39. (in)equitable schooling and mathematics of marginalized students: through the voices of urban latinas/os maura varely gutiérrez – elsie whitlow stokes community freedom public charter school craig willey – indiana university purdue university-indianapolis lena l. khisty – university of illinois at chicago 40. high-achieving black students, biculturalism, and out-of-school stem learning experiences: exploring some unintended consequences ebony o. mcgee – vanderbilt university 41. urban latina/o undergraduate students’ negotiations of identities and participation in an emerging scholars calculus i workshop sarah oppland-cordell – northeaster illinois university 42. latina/o youth’s perspectives on race, language, and learning mathematics maria del rosario zavala – san francisco state university 43. latinas and problem solving: what they say and what they do paula guerra, woong lim – kennesaw state university 44. black male students and the algebra project: mathematics identity as participation melva r. grant, helen crompton, deana j. ford – old dominion university part vi: policy 45. racism, assessment, and instructional practices: implications for mathematics teachers of african american students julius davis – morgan state university danny bernard martin – university of illinois at chicago 46. practices worthy of attention: a search for existence proofs of promising practitioner work in secondary mathematics pamela l. paek – university of texas at austin 47. an examination of mathematics achievement and growth in a midwestern urban school district: implications for teachers and administrators robert m. capraro, jamaal rashad young, chance w. lewis, zeyner ebrar yetkiner, melanie n. woods – texas a&m university 48. compounding inequalities: english proficiency and tracking and their relation to mathematics performance among latina/o secondary school youth eduardo mosqueda – university of california, santa cruz http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/62/72 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/62/72 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/112/91 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/112/91 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/178/142 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/178/142 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/213/151 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/213/151 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/188/152 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/198/162 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/284/182 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/284/182 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/14/8 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/14/8 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/6/1 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/6/1 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/33/20 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/33/20 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/47/48 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/47/48 stinson editorial journal of urban mathematics education vol. 8, no. 2 10 49. success after failure: academic effects and psychological implications of early universal algebra policies keith e. howard – chapman university martin romero – santa ana college allison scott – level playing field institute derrick saddler – university of south florida part vii: international perspectives 50. learning mathematics in a borderland position: students’ foregrounds and intentionality in a brazilian favela ole skovsmose – aalborg university pedro paulo scandiuzzi – university são paulo states paola valero – aalborg university helle alrø – aalborg university bergen university college 51. transforming mathematical discourse: a daunting task for south africa’s townships roland g. pourdavood – cleveland state university nicole carignan – university of quebec at montreal lonnie c. king – nelson mandela metropolitan university 52. forging mathematical relationships in inquiry-based classrooms with pasifika students roberta hunter, glenda anthony – massey university 53. mathematics as (multi)cultural practice: irish lessons from the polish weekend school stephen o’brien, fiachra long – university college cork 54. reflecting heritage cultures in mathematics learning: the views of teachers and students robin averill – victoria university of wellington 55. financial literacy with families: opportunity and hope lorraine m. baron – university of hawaiʻi at mānoa http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/248/171 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/248/171 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/4/4 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/4/4 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/18/15 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/18/15 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/100/81 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/100/81 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/147/124 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/147/124 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/166/125 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/166/125 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/258/173 journal of urban mathematics education december 2016, vol. 9, no. 2, pp. 5–10 ©jume. http://education.gsu.edu/jume erica n. walker is professor of mathematics education at teachers college, columbia university, 525 west 120th street, new york city, ny 10027; email: ewalker@tc.edu. her research focuses on the social and cultural factors that facilitate mathematics engagement, learning, and performance, especially for underserved students. commentary the importance of communities for mathematics learning and socialization1 erica n. walker teachers college, columbia university espite myths to the contrary, students attending urban schools are interested in learning and in mathematics, and they have communities and networks that are committed to their education and mathematics development. too often, particularly for black and latina/o students, these networks and communities have gone unnoticed and unacknowledged—and, in fact, are often disregarded and denigrated as not essential to students’ academic success. in much of my work and research with schools and students, i have placed at the forefront students’ voices and experiences, because these are so often missing from research discussions about teaching and learning. while it is important to consider the role of school-based learning communities (namely, the valuable relationships and interactions between students and teachers, primarily, but also extending to counselors and administrators) in students’ learning, it is also important to value—and to further capitalize upon—the rich learning communities that young people may have outside of school. one story from a research project exploring the formative, educational, and professional experiences of black mathematicians (walker, 2014) demonstrates the power of an extended learning community. nathaniel long (a pseudonym) grew up in pittsburgh, and lived on a street where there were close familial and intergenerational ties: i grew up in a—well, call it little italy—mostly italian and some irish and german, but all catholic [neighborhood]. and then there were a smattering of black families. my mother actually grew up [on the block] the generation before, so they all knew each other. it was very close-knit … it was a pretty bright group of kids. their mothers 1 this commentary draws extensively from my teachers college press book building mathematics learning communities: improving outcomes in urban high schools (walker, 2012a). my thanks to lesley bartlett for directing me to deborah brandt’s work. editor’s note: for a review of professor walker’s 2012 book see “keeping the ‘welcome sign’ lit: a review of building mathematics learning communities: improving outcomes in urban high schools.” d http://education.gsu.edu/jume mailto:ewalker@tc.edu http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/299/194 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/299/194 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/299/194 walker commentary journal of urban mathematics education vol. 9, no. 2 6 had grown up with my mother on that street and they were all college educated…. they’re still living there. but there was a sort of intellectual atmosphere…. one of the kids, henry, was about five years older than me and he would play ball with us younger kids. i would play chess with him, and he started giving me these little puzzles. he was very interested in mathematics. he ended up majoring in mathematics and became a secondary school math teacher. i was the oldest kid in that group—this group of kids that were really second generation on that street. so i was the one that was older and perhaps more interested in mathematics than the other kids. (pp. 39–40) many other mathematicians in that study reported that there were numerous people in their homes and neighborhoods who supported their mathematics learning and socialization beyond school walls. from their earliest mathematical memories engaging in family traditions and being exposed to mathematics concepts by family members and other adults; to teachers going above and beyond formal classroom parameters to introduce mathematicians to advanced material; to their participation in summer, after-school, or saturday mathematics enrichment programs, mathematicians described significant exposure to mathematics during childhood and adolescence. noteworthy is that the educational attainment of the family members (immediate and extended) who exposed them to these mathematics concepts ranged from little formal schooling to phd completion. are teachers aware of students’ potentially rich mathematical lives outside of school? are they aware of the various networks and support structures for mathematics learning that students may be bringing with them to school? in my research with young people in urban high schools, they reported similarly rich out-of-school mathematics experiences and networks supporting their school mathematics learning (walker, 2012a, 2012b). furthermore, high school students and mathematicians alike, when reflecting on their adolescent experiences in school, reveal that some of the most pivotal moments for their mathematics interest and learning occurred when they were exposed to mathematics that was different from the typical textbook or worksheet exercise. some of these moments were based on a brief conversation with a teacher outside of class about a mathematics concept, or a novel problem that incorporated mathematics content already “covered.” other moments were critical to young people’s developing an understanding of mathematics as a wide-ranging discipline that could be appreciated for both its beauty and its usefulness. one unfortunate outcome of multiple cycles of reform in mathematics education has been a narrowing effect on conceptions of mathematics. in an effort to improve student performance, too often teachers, schools, and districts—particularly, urban school districts—revert to assessments that primarily measure basic skills and elementary concepts and instruction that focuses on “recipes” for solving particular types of problems. students come away from these experiences thinking that mathematics is a series of exercises without meaning, that mathematics is a walker commentary journal of urban mathematics education vol. 9, no. 2 7 solitary activity, and that mathematics is solely about formulas and procedures with little use or relevance to their lives. the broad mathematics opportunities afforded by learning communities extend beyond peers’ focus on school mathematics (class problem sets, test preparation, and homework). such learning communities can be composed of solely similarly aged classmates, or can include youth and adults across generations. they can focus on what is traditionally understood as “school” mathematics or they can focus on exploring mathematical topics not usually covered in school, the kind of mathematics that is often engaging and interesting for students and draws on their creativity in ways that school mathematics may not. furthermore, the communities that support mathematics learning and socialization can be informal and inadvertent, or formal and intentional. young people’s reports about learning communities reveal some important characteristics. the communities are marked by informed participants and/or observers who have high expectations for young people’s learning of rigorous mathematical content, who often provide support or “push” for young people to persist in mathematics, and who engage in traditions that foster the development of effective mathematics learning behaviors. they also help to establish a sense of belonging to a broader mathematics community. as one mathematician reported about a committed teacher who extended the school day to teach her and her classmates advanced mathematics when the school district administration would not permit the offering of advanced courses: “when mr. holly said i could do math— that was it! i could do math.” we can draw from these characteristics to think about actions in classrooms that support the development of strong learning communities. for example, within the classroom, teachers should be aware that their instructional behaviors signal their interest in students’ mathematics learning, and that students interpret these behaviors as indicators of teachers’ perceptions and expectations (or lack thereof) for student success. students speak very compellingly, honestly, and knowledgeably about the work of teachers. they are not expecting friendships but do expect (and respond well) to teachers who exhibit care about and interest in them and their learning and success. teachers must be aware that their practices with students (for example, assigning low-level mathematics work repeatedly, not exposing students to engaging problems) may reflect their stereotypical notions of students’ interest, competence, and potential, and that these practices can impede student progress and have cumulative and long-ranging impact on students’ life outcomes. teachers should also be aware of the social aspects of learning, in particular, how students’ peer groups support mathematics learning. they should be aware that students’ peer group support for academic and mathematics behaviors is complex and nuanced—and that high achieving and so called low-achieving peers can support each other. more broadly, teachers should be aware that there are extensive walker commentary journal of urban mathematics education vol. 9, no. 2 8 support networks for students’ mathematics learning that may incorporate extended family members, peers, adults with little formal education beyond high school, and previous teachers. one way to capitalize on students’ informal learning communities within and beyond school is to develop peer tutoring programs for mathematics. in a study of one such peer tutoring program (walker, 2012a), the work of the peer tutors revealed that students drew upon pedagogical strategies and interconnected content to help their struggling peers. to their peers, these explanations were novel and provid-ed a new way of thinking about mathematics problems, even those problems that were not particularly complex. indeed, there have been numerous studies that show the effectiveness of peer academic support for the learning of mathematics in college and university settings. the best of these programs incorporate several key concepts that bear repeating: first, they assume students are excellent and competent—they are not remedial programs. second, they incorporate academically supportive peer groups that foster learning and engagement. third, they are sustainable—most often sustained by committed leadership, but also by participants who carry out the mission of the program within the program’s confines and beyond them. these programs and experiences are both formal and informal and occupy spaces that cross important boundaries—they encompass activities within and outside of classrooms, within formal educational institutions and within neighborhoods, among novices and experts, and among peers and mentors. this work supports the notion of repositioning students in the classroom as worthy contributors to the development of mathematics knowledge and understanding. in particular, promoting discussion and group problem solving in the classroom increases students’ agency and active interest in their own mathematics learning, and ensures the development of metacognitive and problem-solving skills on which students can rely when they are participating in lifelong mathematics endeavors. there is a rich and vibrant research landscape relating to mathematics learning communities—such research contributes to our understanding of mathematics identity and socialization (e.g., boaler, william, & zevenbergen, 2000; martin, 2000) as well as the roles of power and agency in mathematics classrooms (wagner, 2007) and the relationship of the quality of discourse (white, 2003) and student–student and/or student–teacher interactions to mathematics outcomes (hand, 2010). i would argue, however, that with the prominent and largely negative popular culture and media discourse about mathematics in the united states there is considerable value in exploring how these constructs operate in out-of-school settings (e.g., nasir, 2000) as well as in intergenerational contexts that broaden exposure to rigorous mathematics. how do young people describe their mathematics learning communities outside of school? what kinds of learning experiences for mathematics do these communities provide for young people, and how can we craft more of them? how should they be crafted? how do young people make sense of the connections between their out of walker commentary journal of urban mathematics education vol. 9, no. 2 9 school mathematics learning experiences (especially those not focused on school mathematics activities) and their in-school mathematics learning? what types of interactions around mathematics are the best drivers for mathematics interest and engagement over time? for example, one mathematician recounted a vivid story about his grandfather using a porch to explore questions of infinity with him at an early age; the mathematician now shares this story with his students in his college classes to help develop their understanding of limit (walker, 2014). we can draw from the extensive literature on literacy practices (within and beyond schools) to develop a broader theoretical understanding of mathematics learning and socialization fostered by students’ communities, whether those communities are informal or formal, intentional or serendipitous. burns (2015) recently noted that elementary teachers have an extensive and multi-dimensional repertoire for reading instruction, but are apprehensive when it comes to designing mathematics instruction. they have less experience crafting learning experiences that elicit student imagination and creativity in mathematics than in reading and writing. furthermore, there are many examples of spaces (book clubs, spoken word and essay contests, libraries) in which young people can engage in out-of-school literacy practices involving multiple modalities of expression (e.g., vasudevan, 2009), but decidedly fewer analogous examples of extracurricular mathematics activities (for the most part, these are limited to mathematics circles, clubs, and teams). indeed, i have argued elsewhere (walker, 2012a, 2012b) that rich and varied mathematical spaces may be key to the development of mathematical identity as well as to the dissemination of mathematical knowledge. finally, brandt (1998) suggests that “sponsors of literacy” are key agents who “enable, support, teach, model, as well as recruit, regulate, suppress, or withhold literacy—and gain advantage by it in some way” (p. 2). it is useful to consider who sponsors of mathematical learning are, what they do, and how “sponsorship” can enable rich practices of learning communities or impede them. such a research agenda can inform how we make mathematics learning experiences as diverse, rewarding, engaging, transformative, and common-place as we do experiences related to multiple forms of literacy. it has the potential to not only enhance and improve mathematics teaching and learning, but also develop cadres of students with strong and positive mathematics identities, who are excited about mathematics, see themselves and are seen as talented, knowledgeable doers and users of mathematics, and are leaders of robust learning communities. references boaler, j., william, d., & zevenbergen, r. (2000). the construction of identity in secondary mathematics education. paper presented at the second international mathematics education and society conference. montechoro, algarve, portugal. brandt, d. (1998). sponsors of literacy. college composition and communication, 49(2), 165–185. walker commentary journal of urban mathematics education vol. 9, no. 2 10 burns, m. (2015, april 1). what reading instruction can teach us about math instruction. education week, 34 (26), 32. hand, v. (2010). the co-construction of opposition in a low-track mathematics classroom. american education research journal, 47(1), 97–132. martin, d. b. (2000). mathematics success and failure among african-american youth: the roles of sociohistorical context, community forces, school influence, and individual agency. mahwah, nj: erlbaum. nasir, n. s. (2000). points ain’t everything: emergent goals and percent understandings in the play of basketball among african american students. anthropology and education quarterly, 31(3), 283–305. vasudevan, l. (2009). performing new geographies of literacy teaching and learning. english education, 41(4), 356–374. wagner, d. (2007). students’ critical awareness of voice and agency in mathematics classroom discourse. mathematical thinking and learning, 9(1), 31–50. walker, e. n. (2012a). building mathematics learning communities: improving outcomes in urban high schools. new york, ny: teachers college press. walker, e. n. (2012b). cultivating mathematics identities in and out of school and in between. journal of urban mathematics education, 5(1), 66–83. retrieved from http://edosprey.gsu.edu/ojs/index.php/jume/article/view/173/100 walker, e. n. (2014). beyond banneker: black mathematicians and the paths to the excellence. albany, ny: state university of new york press. white, d. y. (2003). promoting productive mathematical classroom discourse with diverse students. journal of mathematical behavior, 22(1), 37–53. http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/173/100 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/173/100 microsoft word author approved mendez perez et al vol 16 no1 (1).docx journal of urban mathematics education june 2023, vol. 16, no. 1, pp. 96–106 ©jume. https://journals.tdl.org/jume karina méndez pérez is a graduate student studying stem education at the university of texas at austin, email: kmendezp@utexas.edu. she primarily wants to learn how asset-oriented pedagogies draw upon multilingual students’ multiple ways of understanding and communicating to create equitable science learning experiences. alexandra r. aguilar is a graduate student studying stem education at the university of texas at austin, email a.aguilar@utexas.edu. she primarily wants to learn how students, particularly students of color, have resisted the dominance of school mathematics both within and outside the classroom. amy rae johnson is a doctoral student studying stem education at the university of texas at austin, email arjohnson@utexas.edu. she primarily wants to learn how prospective and practicing teachers reflect on their lived mathematic experiences to support elementary students with building their mathematics identity. chandel burgess is a doctoral student studying stem education at the university of texas at austin, email: chandelburgess@utexas.edu. she primarily wants to learn about black women’s experiences in postsecondary stem spaces and to identify ways to improve the quality of these experiences. carlos nicolas gómez marchant is an assistant professor of stem education at the university of texas at austin, email: nico.gomez@utexas.edu. he primarily wants to learn about the mathematical brilliance and experiences of latine learners in navigating predominantly white spaces. critical reads centering humanity within ethnographic research: a book review of black boys’ lived and everyday experiences in stem our research collective explores latine learner’s experiences with mathematics. therefore, we must consider possible methods to de-settle the white gaze surveilling and erasing latine learners in k-12 schools, as well as the white ideologies in educational research. in this book review, we discuss kimi wilson’s black boys’ lived and everyday experiences in stem (2021) and explore his use of ethnographic research to tell the story of his boys (carter, malik, darius, and thomas). wilson highlights how he disrupts the norms of educational ethnography through his research and posits the need to amplify black voices and experiences in stem education. he challenges the reader to push against white ideologies and reconsider the deficit narratives surrounding black boys. by reflecting on wilson’s work and our own, we consider two points of reflection: centering humanity and emotionality, and the importance of place. we explore how wilson addresses these two points through his stories of his boys and how our research collective considers these ideas in our work with latine learners in mathematics. as educators, educational researchers, and policy makers, we must reflect, acknowledge, and create transformative actions centered around humanity and emotionality, as well as the importance of place, to ensure equitable learning spaces for black and latine learners. keywords: stem education; black boys; latine learners; educational ethnography karina méndez pérez the university of texas at austin alexandra aguilar the university of texas at austin carlos nicolas gomez marchant the university of texas at austin amy rae johnson the university of texas at austin chandel f. burgess the university of texas at austin pérez et al. critical reads 97 latine learners have resisted and persevered through a legacy of racial inequities (e.g., san miguel, 1989; yosso, 2006). the mi lengua, mi raza, mis matematicas collective (mi3) recognizes the role mathematics education plays in the legacy of maintaining white institutional spaces (moore, 2008) and perpetuating white supremacy. since mi3 explores the intersection of mathematics education, race, and language, we are committed to resisting, interrogating, and disrupting white supremacy for latine learners. our individual backgrounds, social identities, and experiences are combined and weaved through our work and research processes. those collaborating in mi3 work continuously reflect on ways to center and amplify latine learners’ voices and experiences. as mi3 pursues this work, it is imperative to continue learning not only from colleagues working directly with latine populations, but also from co-conspirators who challenge whiteness, while amplifying the voices of other historically and contemporarily excluded and exploited racial groups. consequently, a sect of mi3 chose to read black boys’ lived and everyday experiences in stem by kimi wilson (2021) to determine what it means to conduct ethnographic research focused on latine learners pushing back against the dominant white gaze in mathematics (see jones, 2022). we share our collective’s reflections here to create space for conversation and transparency into the messiness of learning in this community, like other groups before (see hand et al., 2021; jones et al., 2017; translanguaging study group, 2020). wilson’s (2021) book challenges its readers to decenter white ideologies in research through his investigation, which centers both the black experiences of his boys (carter, darius, thomas, and malik) and his own blackness when navigating white spaces. those seeking transformation and disruption of the education system will be challenged to (re)think research norms with learners of color. according to wilson (2021), the way one conducts ethnographic work must be attached to the humanity of the individual. when done correctly, ethnography is humanity, uninterrupted–the unknown, the good, the bad, and the ugly, remembering that it all is subjective. while i would somehow love to romanticize this notion — ethnography is not void of one’s own experience. you bring your whole self to this work, and there is no possible way to detach me from this process. (wilson, 2021, p. 54) wilson’s (2021) perspectives of ethnography aligns with solórzano and yosso’s (2001) claim that researchers’ narratives and relationships with oppressive systems are data and should be considered as part of the analysis and research project. our collective’s individual indigenous, latine, black, and pérez et al. critical reads 98 white1 identities are each part of the humanity we bring to our work with latine learners. like wilson, mi3 members strive to be transparent about their racialized selves and the intersections with other related oppressive systems (e.g., linguicism, ableism, heteronormativity). our individual identities were important when reflecting on the central themes of wilson’s book. while reading through the book and meeting as a collective, we shared stories and testimonios about our own experiences with school and mathematics, the sanctuaries and spaces protecting our identities from assimilation, and the way our emotions and humanity have been erased in academic spaces. our positionalities were laid bare for the group to learn and grow from. we weave our social identities together to serve as a reminder of the problematic nature of viewing these social categories as mutually exclusive. for example, it is important to understand the nature of overlapping oppressive systems of one’s individual blackness and latinidad and one’s afro-latinidad identity (see comas-díaz, 2021). in this review, the authors reflect on wilson (2021) and the way he challenges our perspective of ethnographic research in mathematics education. after providing a summary of his book, the authors follow up with a discussion of two reflection points: 1) centering our humanity and emotionality in mi3’s work, and 2) the importance of place. the review concludes by discussing how these points help mi3 be unapologetic and unafraid. summary black boys’ lived and everyday experiences in stem (2021) is broken into nine sections, including an interlude halfway through titled “sanctuary.” each section describes a different facet of wilson’s journey as an educational ethnographer, while weaving in the voices of his family and community members. wilson illustrates his humanizing journey by sharing his own lived experiences and those of four high school learners who wilson refers to as “my boys,” establishing the importance of the relationships he has cultivated. wilson shares his ideas to develop a new and hopeful future for educational ethnography, especially in regards to black learners. 1 like dumas, we see no need to capitalize white: “white is not capitalized in my work because it is nothing but a social construct, and does not describe a group with a sense of common experiences or kinship outside of acts of colonization and terror…. thus, although european or french are rightly capitalized, i see no reason to capitalize white” (2016, p. 13). pérez et al. critical reads 99 in this educational ethnography, it was my priority to honor my boys, the school site, their families, and redwood…after all, i want to see a world that not only values equity, diversity, and inclusion through lip service but one that centers and prioritizes the needs of black children because i believe in the statement: “when black children win, we all win.” (wilson, 2021, p. 21–22) throughout the book, wilson weaves stories of his experiences as a child growing up in los angeles2, a classroom teacher, a graduate student, a mathematics teacher, educator, and researcher, and a role model for black students. recognizing his positionality, wilson’s goal as a scholar is “to be an example that liberates black boys from feeling that they have to withdraw from south central los angeles to be successful” (wilson, 2021, p. xiii). through his reflections, wilson provides an opportunity to make connections between the importance of his positionality and the way it influences his support of carter, malik, darius, and thomas. wilson helps his readers further understand the importance of going beyond reflecting on their experiences to acknowledging how these experiences shape the transformative actions necessary to support black learners’ stem identities. wilson (2021) challenges traditional ways of conducting ethnographic research and encourages others to do the same by writing, “a prescribed set of rules and regulations have gotten us nowhere — not even close to where we need to be, our final destination” (p. 39). maintaining the status quo of teaching, learning, and generating educational ethnography will continue to hurt and oppress black learners. a transformative action, wilson emphasizes that ensuring school environments are safe for black learners is vital by demonstrating the time and space he dedicates to getting to know his boys, both in regards to their lives outside of school spaces and their identifies as stem learners. wilson stresses that educators, researchers, and policy makers must do better in their mission to support historically and contemporarily excluded racialized learners: “i write because i want black boys situated at the forefront. i’m tired of my boys being featured as a sideshow in ethnographic museums” (wilson, 2021, p. xiv). he recognizes that ethnographic work goes far beyond a 2 while south los angeles is considered an urban area, wilson (2021) does not describe the context of his childhood or work in this way. according to larnell and bullock (2018), “urban” is a problematic term conflated with race and poverty. wilson challenges these “overly simplified renderings” (larnell & bullock, 2018, p. 44) in the way he describes the city and its influence on him. therefore, we also stand in defiance against such terminology used to describe place. pérez et al. critical reads 100 research paper or grant. he exemplifies the necessary use of time, resources, and passion to improve opportunities for black students. wilson (2021) ends his story with a call to action for his readers: “i hope my words have caused you to look at black bodies’ dry bones around this country. i vow to breathe life into them by doing the work. i challenge you to do the same” (p. 102). in short, he reflects on the importance of centering humanity, emotionality, and place in his work, while inviting others to consider how they work with and support communities of color. centering humanity and emotionality by further developing the relationships between his boys, wilson was privy to their stories, emotions, and family and community circumstances. their vulnerability challenged him to reconsider the purpose of educational ethnography in his work. humility taught me that educational ethnography is about serving others. as researchers, teachers, administrators, and policy makers, this requires giving up privilege, status, and preconceived notions. exploring sacred life histories forced me to place my plans on the back burner – abandoning my schedule, altering how i gathered information pertinent to school’s inner workings. scared stories gathered through impromptu walks. (wilson, 2021, p. 42) humanizing and centering his boys’ voices and lived experiences in his work allowed wilson to reflect upon his role as an educational ethnographer and the need for flexibility when developing partnerships with schools and communities. by immersing himself into the school and communities, he was able to recognize the role of emotions in educational ethnography. i have to be honest, navigating the messiness of schooling as a researcher caused a range of emotions that quite honestly could have jeopardized the project. i used these moments of uncertainty to apply the challenges i faced as a small microcosm of what my black parents and children must go through daily. (wilson, 2021, p. 50) instead of excluding his feelings from his work, wilson used emotions to gain insight on what his boys were experiencing in mathematics and scientific spaces. for instance, wilson described the fascination carter had with science as a means to make things and support his family and community. however, the traditional call and response environment fostered in his science classroom resulted in carter questioning his own scientific knowledge and ability and abandoning his dream of becoming a scientist: “i have given up on my dream of becoming a scientist because i don’t think it will happen” (wilson, 2021, p. 61). understanding what his pérez et al. critical reads 101 boys were going through emotionally as they navigated stem spaces enabled wilson to learn about the complexities of k-12 schooling, specifically teaching and learning mathematics and science, experienced by black children. our reflection on humanity and emotionality reflecting on wilson’s work with his boys, mi3 considered the crucial role humanity and emotionality plays in our work to understand latine learners’ experiences with mathematics. through our different projects, we have the privilege of building relationships with latine learners, their families, and their communities. we have learned about the explicit and implicit ways latine learners’ identities, languaging practices, and funds of knowledge (moll et al., 1992) have (or have not) been acknowledged and leveraged in mathematics. latine learners’ emotions and stories of mathematics classrooms must not be collapsed into one unified monolith. similar to our work with latine learners, wilson’s work acknowledged carter, malik, darius, and thomas’s bravery in sharing their emotionality, and demonstrated the importance of listening to singular voices in educational ethnography to understand the black male experience in stem. however, wilson does not explicitly address the need for educational ethnographers to recognize the power dynamics between researcher(s) and participant(s). as relationships are built, learners of color and researchers can share their vulnerabilities, lived experiences, and knowledge, yet what is shared by learners of color is commonly seen as data sources (de lissovoy et al., 2013). elevating the contributions of learners of color as valuable insights can create spaces to collectively heal from traumatic experiences in stem, especially in mathematics (kokka, 2019). as mi3 engages in our work with latine learners, we work to recognize the crucial role emotionality plays in our methodological choices. moreover, we attempt to make space for our own emotionality and humanity as we navigate white academic spaces to create pathways for amplifying the voices and experiences of latine learners. in the future, we will look at the work of matias (2016) and zembylas (2006) to help us further conceptualize emotionality in our work. the importance of place throughout the book, wilson highlights his intention to navigate the physical space of the research, from the micro (e.g., sanctuary space) to the macro (e.g., south central la). however, his intentional moves with space did not always guarantee mitigation of harm, as wilson shared in an anecdote of his prior teaching years, where he altered his classroom to bring a ‘beach day’ to his students, not realizing they “...had never left the community and had no concept of what a beach was” (wilson, 2021, p. 4). despite having great intentions, wilson created a place that harmed his students, who did not have beach experiences of pérez et al. critical reads 102 their own that they could use to develop the classroom into a space that was authentically theirs. the students’ experiential knowledge was not prioritized or made accessible in his classroom, because it could only be framed through wilson’s beach experiences. this experience taught wilson how easy it was to choose his own understanding over his students. by sharing this story early on, wilson provided an initial opportunity for readers to reflect on the importance of place. stories of redwood and the people in it highlight the urgency of wilson’s work by showcasing moments of vulnerability between him, his boys, and school staff. wilson cultivated a comforting sanctuary space in one of the school’s classrooms to interview and connect with his boys. however, there were many other instances throughout his journey wherein wilson had to adapt to spaces his boys navigated. we chose to highlight one of these stories that we learned a great deal from regarding how wilson faced challenges in settings with variables that were out of his control. in one such instance, wilson administered a fitness test to one of his boys, malik, at a running track on campus. upon seeing kids from a rival housing project on the track, malik requested to take the test another time, exposing wilson to his fear. through wilson’s account, we are able to visualize the area surrounding the running track as not only a physical location, but also a place inhabited and complicated by the interactions between each person in and outside of the school. malik experienced the running track as tangible danger and panicked because of the other boys inhabiting that space, indicating that there was a part of the history of the community previously unknown to wilson. at that moment, malik became wilson’s cultural broker (aikenhead, 1997). malik felt comfortable sharing his feelings of fear of the other kids on the track and his concern that this fear would diminish his teacher’s perception of his masculinity within the school environment. in response, wilson permitted malik to take the test another time and covered for him with his teacher. wilson’s connection to malik allowed him to experience another dimension of the track, one rich with malik’s lived experience with its actors. by exiting the space, wilson demonstrated to malik that, over all else, he prioritized malik’s safety. our reflection on place place is as much a collaborator in ethnographic work as the learners inhabiting it (see emerson et al., 2011; soja, 2010). in our work, the physical spaces of latine learners’ communities and schools are sources of experiential knowledge, emotional connection, and understanding (see rodríguez, 2020). no one from mi3 is native to this neighborhood. consequently, we have to create connections with the community through hosting local events and participating in existing ones. unfortunately, a few years of participation is not equivalent to a lived childhood in the community. as ethnographers of latine learners, we have the incredible opportunity to learn from place as an important aspect of our collaborators’ pérez et al. critical reads 103 identities. wilson taught our collective to reflect on how we can approach each space we encounter with a willingness to engage with our own and our collaborators’ emotions, and to respect and prioritize these emotions as valuable knowledge worth reporting. we must be careful to not perpetuate assimilation and erasure (see jones, 2022). in addition to our differences from wilson’s experiential knowledge of his research site, our collective is working with a very different population of students. latine elementary students in central texas and black high school boys from southern california have distinct spatial experiences and knowledge. wilson’s advice, while valuable, cannot simply be copied and pasted onto our context. therefore, as a research collective, we interpreted wilson’s teachings as an encouragement to leverage our emotions and lived experiences (see solórzano & yosso, 2001). many in mi3 experienced school as latine learners, and some as texas learners. we drew upon these experiences in our discussions of how to begin approaching the community. we will continue to listen to our emotions when we feel uneasy or hesitant in preliminary planning. we have learned from wilson to respect the caution of our team members. he shared the potential harm that mistakes regarding place can have on our participants. conclusion a humanistic approach and consideration of the multidimensionality of black boys is pivotal in discussions of lived experiences. wilson unapologetically shared his frustration with the inequities that black learners face in k-12 educational settings, specifically within mathematics classrooms. he highlights these inequities by providing himself and the reader with contextual understanding of his boys’ mathematical experiences. the emotion and realness of his approach to this ethnographic study emphasizes the experiences of black learners. reflecting on these experiences allows the reader to grapple with their potential role in maintaining and perpetuating these educational inequities. the reader may feel uncomfortable during their reflection, which wilson uses as a tool to portray the urgency for black learners, who are often failed by school systems. to productively create equitable learning spaces for black and latine learners, considerable reflection, acknowledgement, and transformative actions must be taken to determine how we as educators, researchers, and policy makers will (1) center humanity and emotionality in our work, and (2) emphasize the importance of place. wilson provides readers with an opportunity to journey through the lives of his boys and understand their distinctive aspirations, personalities, and struggles that force them outside of the stereotypical box often used to limit their capabilities. wilson guides readers through the evolution of his relationship with his boys and explains how building a sanctuary or navigating the spaces his boys know better can allow researchers to be collaborators and meet their needs, while pérez et al. critical reads 104 maintaining space for the boys to set their boundaries. though our work varies from wilson’s in regards to age, location, and racialized identities of our collaborators, we strive to (1) create sanctuaries for the latine learners we work with and identify the sanctuaries they may already have, and (2) develop meaningful relationships with students, their communities, and their families as we navigate their environment. ethnographic studies should be messy and genuine and should embrace the gravity of the work necessary to give justice to the voices of black and latine learners. approaches to investigating with and for black and latine learners should work to distinguish them as complex beings with varying and endless talents. similar to wilson, we strive to frame the latine comunidad’s narratives as those of a community that has persevered and resisted erasure and assimilation, rather than relying on the common deficit narrative of latine learners (see jones, 2022). wilson’s work encourages uncomfortable conversations and self-reflection. it can be difficult and intimidating to share our thoughts on the racial inequities in our daily environments. those who do not identify as black or latine may find it harder to empathize because issues of inequity may be irrelevant or challenging to relate to their lived experiences. therefore, it is imperative that we break down the boundaries prohibiting us from these meaningful conversations and work toward a shared goal of uplifting black and latine learners. wilson emphasizes the shared need to create spaces and moments that allow marginalized students to see and hear themselves in stem contexts. moreover, educators and researchers need to question (1) how we have worked with black and latine children in the past, and (2) how we can work to improve their stem opportunities and learning environments. consequently, individual projects have emerged that work closer with latine caregivers (see gutiérrez et al., 2022), provide an oral history of prospective teachers of color (see johnson et al., 2023), and delve deeper into the discourses of whiteness (see gómez marchant et al., 2023). our collective is committed to greater community involvement, allowing time for conversations and activities to build relationships and to strengthen our understanding of the communities we engage with. this includes creating, maintaining, and reinforcing new and established sanctuaries for latine learners in schools and the greater community. references aikenhead, g. s. 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(2005). discursive practices, genealogies, and emotional rules: a poststructuralist view on emotion and identity in teaching. teaching and teacher education, 21(8), 935–948. https://doi.org/10.1016/j.tate.2005.06.005 microsoft word the end or the beginning (proof 1).docx journal of urban mathematics education may 2021, vol. 14, no. 1 (special issue), pp. 1–11 ©jume. https://journals.tdl.org/jume journal of urban mathematics education vol. 14, no. 1 (special issue) 1 editorial the end or beginning? either way, the credits are not rolling yet! editorial team robert m. capraro chance lewis eduardo mosqueda tarcia hubert hyunkyung kwon mary margaret capraro melva grant jamaal young alesia mickle moldavan michael s. rugh jacqueline leonard marlon james ali bicer susan ophelia cannon jonas l. chang jume quick facts new leadership term began april 15, 2019 first volume delivered may, 2020 current acceptance rate: ~11% now a scopus indexed journal (as of 2020) average time to initial assignment: ~3 days average time to decision: ~36 days average time to publication: ~8 months double-blind peer review: yes number of reviewers assigned: 2–3, plus an editorial board member editorial team editorial journal of urban mathematics education vol. 14, no. 1 (special issue) 2 hank you to all our reviewers, editorial board members, authors, and those who chose the journal of urban mathematics education (jume) as their outlet of choice this past year. jume has had many recent successes, and we in the editorial team plan to release the salient performance data for the journal. for jume to advance its mission, we believe that accountability and transparency are essential. to this end, our readers will from now on receive an annual progress report about jume in our first issue of each year. the editorial team has worked to bring timely issues to press as quickly as possible without jeopardizing the review process. however, the review process has been tough at times. given the very difficult year amidst the dual pandemics of racism and covid-19, reviewers exceeded expectations. our typical time to send manuscripts to reviewers was three days, and our average time to decision was 36 days. unfortunately, some of the variation around those numbers has been less than laudable. the range for time to send to reviewers was 0–9 days, and the range for time to decision was 12 days to 101 days. we could never have imagined the difficulties we would face in moving manuscripts quickly through the review process. while these timeliness indicators are certainly not the best case, they are clear benchmarks for moving forward. we are starting here and hope that next year we are able to report improvements. in order to improve time to reviewers and time to decision, we have three focuses. first, we will work to expand our reviewer pool so that we can select from a broader population of committed reviewers and burden the few much less. second, we will seek to start a mentoring program for reviewers. a goal held by our team is to ensure every manuscript receives an excellent, positive, and productive review. therefore, the team tended to rely on a few reviewers who excelled at providing a caring and respectful review even when their recommendation was to decline the manuscript. if we can get this mentality to spread, it has the potential to change the education publishing landscape as a whole. our third focus is to ensure that each member of the jume team feels accountable and empowered to make their own decisions and to move quickly and decisively for every submission on which they are the action editor. the number of submissions is already on the rise, but this will be a double-edged sword. time to publication is slowly increasing, as is our backlog. we are concerned that the granularity of this first reflection will not be sustainable. the relatively modest number of submissions currently allows for a great deal of detail with regard to the important metrics in our jume quick facts table, but as submissions increase, we are sure a much less fine-grained analysis will result and other issues related to circulation growth will creep into the process. for the next two years, we are committed to meeting our goals of publishing two issues per year with 2–3 research manuscripts in each issue. jume will publish special issues as they align to the interests of our readership and mission of the journal. we will not substitute special issues for a regular issue unless that special issue t editorial team editorial journal of urban mathematics education vol. 14, no. 1 (special issue) 3 is of such importance that making it a regular issue makes a statement. when we assumed the mantle of leadership, there was at least one failed search for an editorin-chief and the journal was closed to submissions for a year. therefore, when we began our term, the manuscript flow was zero, as was the backlog. we have worked to build both. we are pleased to say that manuscript flow is steadily growing; a graph at this point would be gratuitous and the growth unsustainable. but if we just compare the first month of last year to the first month of 2021, we are on track to show an increase. if we were ceos tasked with predicting first quarter numbers, we would expect a 10% increase for the quarter and a 20% gain year over year in manuscript flow. we are currently building a backlog, and we are actively working to manage that backlog to be no more than one issue. what this means is that no author should wait for more than one issue after entering the post-production phase for the publication of their manuscript. we are pleased that jume is both free and open access, and our goal is at some point to move to publishing when ready and moving away from issues. a backlog is important to this endeavor and to producing a quality journal; it means we are able to select the highest quality manuscripts while providing time to nurture authors through the revise and resubmit phase. when we attached the targets to our chests, and unfortunately to our backs as well, one tenet on which we stood was that everyone who can make a decision understands and is committed to nurturing every manuscript on which there is a revise and resubmit. it is a fact that even when all the best things happen some authors decide, for what are often deep personal reasons, to not resubmit. we stand ready to help. in our first year of leadership, we sought to bring relevant editorials to the field and allow our readers and authors to determine the research direction for the journal. our planned range for 2020 was 4–6 research articles, and there were five research articles published. we also published editorials that we believe help readers to decide into which sections of the journal to submit their work. those editorials also help to provide clarity about what is expected in those sections and how to best situate their work to fit into a section that best fits an author’s research agenda and scholarly mission. additionally, we provided guidance in some editorials about our expectations for quantitative rigor and have the same planned for qualitative methods. our international research section is receiving a large number of submissions, and there is a great deal of interest in that section. our publication range for 2021 and 2022 for research articles will continue to be 4–6. we will work to move to a publish-when-ready format and move past any arbitrary limit on the number of research articles we publish in each issue. however, citations to our articles are essential to our metrics both for those published in jume and in articles submitted to other journals as well. in doing so, we will continue to get the message out about the impactful research published under the jume masthead recently and historically. editorial team editorial journal of urban mathematics education vol. 14, no. 1 (special issue) 4 the path to acclaim starts with ensuring that authors receive recognition and reputational prestige from their association with the journal. to accomplish this, we have retroactively applied digital object identifiers to every article and enabled credit for reviewers through the orcid system. this allows the proliferation of third-party indexers, speedier distribution, and access through the digital cloud. we continue to be indexed in the directory of open access journals and to comply with their very high standards. now every article has a graph indicating its online, direct access readership (see figure 1). we also have a new interactive artificial intelligence agent that examines the keywords for all the articles published in jume, compiles the most common, indexes them, and creates an interactive word jumble. that jumble allows a reader to click a keyword, after which the agent retrieves all the articles that used that keyword (see figure 2). this same system, through orcid, is linked to google scholar and automatically updates an author’s google scholar profile and directs readers to their work published in jume directly without the need to have library or university credentials when accessing articles through our other third-party indexes. now this is what we mean by being truly open access. figure 1. sample graph of direct access for an article editorial team editorial journal of urban mathematics education vol. 14, no. 1 (special issue) 5 figure 2. example of interactive keywords our success this past year has been remarkable. our acceptance rate for 2020 was ~11% (see figure 3). when comparing the rate to historical data, the trend is declining. however, we do not have information to make a fair comparison based on flow per year. several updates to the open journal system software prevents this metric. despite this, we believe that moving forward we will be able to report the flow rate and acceptance rate over time and to be able to chart this information. while flow was sufficient to exceed all our first year’s goals and to achieve a comparable acceptance rate to high-quality journals, we hope that eventually moving to a publishwhen-ready format will both increase the acceptance rate while improving the timeliness for publishing manuscripts. the end result will be that some years we may publish more articles and in other years less. we are in our first year of being a scopus-rated journal, and we look forward to our first metrics being posted. it will be important to carefully scrutinize how the journal fits in the urban mathematics landscape and to be sure that we are citing jume appropriately and working to make sure that jume articles get the best possible publicity. we have additionally reestablished the journal’s facebook account. we have also installed an orcid plugin to ensure that an orcid link is listed for every author. these changes allow each author to have the necessary tools to get the word out about their publications and to help generate citations of those works. editorial team editorial journal of urban mathematics education vol. 14, no. 1 (special issue) 6 figure 3. acceptance rate by year we have discussed our successes of this past year, but let us now look forward to the year that is to come. we at jume have long sought to be disruptors, both in mathematics education and at national funding agencies more broadly. this role requires foresight, transparency, and a proactive mindset, which we seek to bring to a growing conversation regarding discrimination and prejudice in the publication process. jume is not without fault, but one of our goals is to make our weaknesses as well known as our strengths, to look within and without, and to seek assistance in remediating where we fall short. we are excited to join our peer editors in mathematics education and urban education should they decide to follow suit, and we hope to issue unified and univocal stances to reduce discrimination and prejudice in publication, funding, and access to publication outlets. every endeavor, however, must have a beginning, and these often are composed of small steps. discrimination and prejudice in the publication process is complex, and clear data is a necessity for understanding the myriad issues involved, particularly if we, as a field, wish to understand whether major problems arise in the role of the reviewer, as is so often cited to be the case, or in the role of the editor. to this end, we at jume join the american psychological association (2021) in the mission “to better understand the demographics of our participants, and to identify and improve any gaps in representation across our network of authors, reviewers, and editorial team editorial journal of urban mathematics education vol. 14, no. 1 (special issue) 7 editors” (para. 1). as such, jume will soon initiate a plan to collect demographic information for published articles inclusive of all authors and reviewers and to publish this data in each year-end review, for authors who wish to disclose such information. in discussing this decision, it was suggested that we have all submissions include demographic information; however, there was a concern that authors might be worried about how such information would be used and whether or not requesting demographic information at the submission stage would influence the publication process. instead, we are going to pilot the voluntary submission of demographic information of published authors, including race and ethnicity, gender, sexual orientation, disability/ability status, rank or graduate student status, institution one graduated from, and terminal degree graduation year as well as if authors have ever received international funding or funding from the author’s home nation’s national funding agencies as a pi or co-pi. we believe this last category of information is important to collect because funding is nearly essential for most of us in the academy to do our work. without that demographic variable, it is impossible to know if any group is underrepresented because they are omitted from the funding stream. the decision to collect this voluntary information is not without cause. the jume editorial team understands that there is a problem in the editorial process; we know that there are too few faculty of color and too much service work required of these few individuals who are often disproportionately taxed with cultureand equity-related tasks. how do we know this? we know this because we performed a careful review of our own practices and because we have looked at the demographic information currently available for our past publications, which is presented in table 1. we feel it is necessary here to state that this information has gaps, as it was our goal to avoid misrepresenting any scholars while compiling the data presented in this table. to this end, we only included those scholars who the members of the editorial team are familiar with and are certain of how they would identify themselves. we did not categorize any scholars whose self-identity we were not certain of. we were confident in doing so due to the many years of experience collectively represented by the members of the jume editorial team and the resulting familiarity that these members have with most major and upcoming scholars in the field of urban mathematics education. this method also resulted in a minimal loss of data, with each column of the table losing less than four percent of its original pool of scholars. yes, there are many flaws to this approach to data collection, but some practice toward wokeness is better than none, even if a few steps are made in error. again, major systemic changes must begin with small steps, and we will reiterate that the demographic information collection policy of jume is changing in order to fill gaps and minimize inaccuracies in the available data. the information presented now, however, we believe to be satisfactory to discuss larger historical trends in jume’s own publication process and to justify the need for collecting more accurate demographic data. editorial team editorial journal of urban mathematics education vol. 14, no. 1 (special issue) 8 table 1 demographics of jume authors and reviewers demographic authors 2008–2020* reviewers who completed a review 2016–2020 reviewers who declined to conduct a review 2016–2020 reviewers who accepted to conduct a review and later remitted 2016–2020 asian 5.3% <1% <1% <1% black (non-hispanic) 41.0% 5.0% 79.0% 9.0% latin or hispanic descent or origin 36.0% 11.0% 21.0% 9.0% white (non-hispanic) 14.0% 84.0% <1% 82.0% note. all numbers in this table are estimates based on editorial team members’ familiarity with the author or reviewer. *this includes both primary authors and co-authors. the demographic survey will not be limited to the broad categories presented in table 1. authors and reviewers will be able to provide their own demographic information with more fine-grained insights into their classifications or place of origin. for example, categorizing the initial data presented in table 1 was complicated by the act of describing certain geographic regions as subsets of larger ones (e.g., should “asian” be inclusive of everyone with roots going back to the continent of asia despite the many different contexts present within the landmass?). with the implementation of our new demographic survey, jume’s authors and scholars will be able to more accurately self-identify their ethnicities and heritages (e.g., “south asian” or “east asian” rather than “asian.”). our desire is that collecting such demographic information will capture nuances of identity with great accuracy and that doing so will promote discussion of urban mathematics within an international framework and additionally enable and empower discussion of how each author’s mathematical identity is shaped by unique mathematics education experiences. another goal of the editorial team is to better understand the representation of authors of all gender identities and sexual orientations in urban mathematics education and to recognize their contributions. it is for this reason that the voluntary demographics survey will allow contributors to indicate their gender identity and sexual orientation. additionally, we recognize the unique experiences of lgbtiq+ students, teachers, and mathematics education scholars and seek to make the journal a space where conversations and studies rooted in such experiences can be shared. editorial team editorial journal of urban mathematics education vol. 14, no. 1 (special issue) 9 our mission to strengthen the inclusive nature of jume’s scholarly community will be further supported by the voluntary demographics survey allowing contributors to indicate and specify their disability/ability status. scholars in the field of urban mathematics education have long studied how best to support the mathematics achievement of exceptional learners, and such learners who enter the academy must be able to share their unique research perspectives, developed from their own mathematical identities and experiences, with their peers. we will strive for jume to be a space where such scholars know their work fits and will be fairly represented. as we implement these changes, we recognize that our path is not without faults, but it is important to keep in mind that most journals are reluctant to provide this level of transparency. as such, there is no long-standing model for transparent editorial practices for us to follow, though we find the recent guidance of the american psychological association and the implementation of its own demographics survey a good place from which to start. the best we can do for jume, for the field and community at large, and for urban mathematics students is to provide further transparency of our own editorial practices. for these reasons, we believe that collecting demographic information for authors and reviewers is the correct course of action to take in order to improve the representation of all scholars in jume. with all this being said, the editorial team agrees that without there ever having been a previous discussion about demographic characteristics or opportunity with the journal, it appears that the people doing the work in urban settings are represented well within our authorship pool. we feel this is a strength of the journal and of the team historically. we have discussed our desire and reasoning for collecting new data with our demographics survey, but what do we have to announce regarding our current data in the first annual progress report for jume? in answering this question, let us first ask one that is essential for discerning an equitable editorial process: “who is doing the reviewing?” often, reviewers act as the gatekeepers to publication. additionally, it is often the reviewers who tend to favor their own paradigms and topics and who approach reviewing myopically. at least, this tends to be the scapegoat for when editors and editorial teams are questioned about publication decisions. so what are the characteristics of a jume reviewer? to answer this, we reviewed the last four years of reviewer information. we were limited to four years because software updates made reviewer information older than this unreliable. additionally, we do not have sufficient information on reviewers for manuscripts that were not published. when considering the last four years in which manuscripts were published, of those reviewers who completed a review, the ethnic/racial breakdown appears to be the following: less than 1% asian, 5% black, 11% latin or hispanic, and 84% white (see table 1). we were also interested in knowing more about reviewers who either decline or remit a review. of the 62% of those who declined to conduct a review, less than 1% appear to be asian or white, 79% black, and 21% latin or hispanic. for reviewers who accepted to conduct a review and who remitted that review, 82% editorial team editorial journal of urban mathematics education vol. 14, no. 1 (special issue) 10 appear to be white, 9% black, 9% hispanic or latin, and less than 1% asian. what this means is that when an author of color submits a manuscript, they have an ~80% chance to have white reviewers. on the surface, these trends may appear problematic, but with context, they are signs of the health and utility of jume. considering that the national center for education statistics confirms that 77.5% of full and associate professors are white americans (mcfarland et al., 2018), we fully expect that the vast majority of our reviewers would reflect these demographics. in fact, given our focus on the education of youth from disenfranchised communities and schools, it is inspiring to see that so many of our senior white professors are supporting, strengthening, and facilitating the publication of research by jume’s diverse authors. more than eight in 10 authors for jume are faculty of color publishing work concerning mathematics education for an equally diverse population of students in urban schools. moreover, our data also note a relatively low rate of reviews by faculty of color (16%); specifically, only 5% of reviewers are african american. yet, within an appropriate context, these data should be expected. only 5.5% of full and associate professors are african american in the united states (mcfarland et al., 2018). on the other hand, latin and hispanic scholars only account for 4% of full and associate professors in the united states (mcfarland et al., 2018) but represent roughly 11% of jume’s reviewers, which we recognize and applaud. jume also fully recognizes that faculty of color are faced with a “minority tax” (baez, 2000; trejo, 2020), asked to perform diversity work and support underrepresented students in addition to their academic and administrative duties. black and brown mathematics education faculty are often too overburdened to be able to accept every extra task asked of them. it is also a reality that black and brown faculty are trying to navigate the tenure and promotion process in a typically white institution, devised by white faculty, and governed by white administrators. therefore, many faculty of color find themselves making choices between their personal missions and those for which they receive credit toward promotion, tenure, and recognition. given all of this, jume will continue to focus our support for equitable access to a research publication that advances minority scholarship and research to improve mathematics education in urban schools and classrooms. we hope that the steps being taken in 2021 will advance this goal and that the jume editorial team’s decisions allocate the effort necessary to raise jume to the prominence of other major journals both within and beyond our field. we expect that being included in scopus, working on our author metrics, adopting orcid identification, and getting the journal’s social media accounts active will have a positive impact that will allow faculty of color to more easily prioritize jume for their service to the field and to become active reviewers. these steps already appear to be bearing fruit. we are pleased that every one of our inaugural editorial board members decided to extend their service. we were also able to add one new international board member who is interested in our collective work. editorial team editorial journal of urban mathematics education vol. 14, no. 1 (special issue) 11 there are some other prejudicial considerations in the publication community, far too many to detail here, but they range from the notion that great mathematics education research comes from only a small number of universities or from those graduates who come from them to the idea that urban mathematics education is a niche and not a mainstream educationally scientific area of interest. we have not completed our review of institutions represented in our author and reviewer pools. too many years, as well as faculty tendency to move or change universities, make this work arduous to complete with a high level of dependability. therefore, we intend to collect these data moving forward so that we will have accurate information to determine if any institutional bias exists, and we will report this information in the 2021 jume report that will be published in the first issue of 2022. we would like to conclude this introduction to this issue of jume by thanking the many wonderful scholars who will join the journal in bettering the editorial process and strengthening the voices and representation of all urban mathematics education scholars, especially those in the most vulnerable positions within and outside of the academy. we believe that jume is well positioned to make a meaningful contribution to our field and to provide a high-quality outlet suitable for graduate students and junior faculty looking to break ground on their research agenda and establish their reputation as well as for senior scholars looking to make powerful statements about the teaching and learning of mathematics in urban contexts, at home, across the street, or across the globe. while this editorial summarizes our first year of publication as the new editorial team, it also welcomes in the new year with all the hopes and dreams it brings, but make no mistake, this editorial team is still working hard, reevaluating our procedures, and establishing new benchmarks by which to make thoughtful decisions moving forward. references american psychological association. (2021). apa journals demographics survey 2021. baez, b. (2000). race-related service and faculty of color: conceptualizing critical agency in academe. higher education, 39(3), 363–391. mcfarland, j. hussar, b. wang, x. zhang, j., wang, k. rathbun, a., barmer, a., forrest cataldi, e., & bullock mann, f. (2018). the condition of education 2018 (nces 2018-144). u.s. department of education, institute for education sciences, national center for education statistics. https://nces.ed.gov/pubs2018/2018144.pdf trejo, j. (2020). the burden of service for faculty of color to achieve diversity and inclusion: the minority tax. molecular biology of the cell, 31(25), 2752–2754. https://doi.org/10.1091/mbc.e20-08-0567 copyright: © 2021 capraro, capraro, leonard, lewis, grant, james, mosqueda, young, bicer, hubert, moldavan, cannon, kwon, rugh, & chang. this is an open access article distributed under the terms of a creative commons attribution-noncommercial-sharealike 4.0 international license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. microsoft word 410-article text no abstract-2520-1-6-20211020 (proof 1).docx journal of urban mathematics education november 2021, vol. 14, no. 1b (special issue), pp. 6–20 ©jume. https://journals.tdl.org/jume dan battey is a professor of mathematics education in the department of learning & teaching in the graduate school of education at rutgers graduate school of education, 10 seminary place, 232, new brunswick, nj 08901; email: dan.battey@gse.rutgers.edu. his research focuses on understanding mathematics education as a racialized and gendered space through researching classroom spaces. monique a. coleman is an adjunct instructor at san francisco state university in the program in visual impairments, p.o box 4214, highland park, nj 08904; email: moniquecoleman@sfsu.edu. an educator-activist, she is a doctoral candidate at rutgers researching the intersections of race, class, and disability in education.=. antiracist work in mathematics classrooms: the case of policing dan battey rutgers, the state university of new jersey monique a. coleman rutgers, the state university of new jersey uprisings against police violence have placed the institution of policing front and center in conversations about societal change. in our work with activists, we have engaged in collecting and analyzing public records data to inform community organizing for change. in this editorial we discuss how to obtain, interpret, and analyze public records on policing as a way to support educators and youth to investigate policing in their own communities. specifically, we discuss how to embed this work in youth participatory action research as a way to respond to youth concerns about their community in a way that leads to action. briefly, we illustrate how this occurred in one classroom in an urban school where students made meaningful connections between their analyses of local police data and their personal experiences with racially disparate policing, which led some students to become change agents in their communities. this work demonstrates the simultaneous cultivation of youth engagement with mathematics and activism. keywords: mathematics, policing, public records, racism, ypar battey & coleman antiracist work in mathematics classrooms journal of urban mathematics education vol. 14, no. 1b (special issue) 7 ver the last year, uprisings demanding justice for george floyd, breonna taylor, and ahmaud arbery have put a focus on the need for change in policing, whether reforming, defunding, or abolishing the institution. as the world isolated in their homes in the midst of a global pandemic, people could not look away from these incidents—which lit a fire more broadly than we have seen before. these protests were spurred by activists such as darnella frazier, a teenager, who recorded the murder of george floyd, posted footage, and used her voice to challenge officers. these actions further raised the question of what can be done before incidents occur and, specifically, what mathematics teachers can do to address such events in the classroom. in response to these events, many teachers looked for antiracist resources. however, it is not clear whether lists and resources will turn into the concrete antiracist action and policy change that is needed. we often focus in mathematics classrooms on what we teach when how we teach may be even more important. certainly, when lower track classrooms teach rote mathematics, the “how” keeps students from accessing more complex mathematical ideas, but it goes further than that as well. many times, the mathematics we teach is meant to stay in the classroom rather than be useful to students in their current and future lives. we can teach unit rates for the costs of widgets or we can apply unit rates to something meaningful in the world. at a time when society is rethinking the role of police, this seems a critical moment to use police data to engage students in a way that will not only help them learn the mathematics but, more importantly, make plain ways in which they can use mathematics for the rest of their lives. in this editorial, we share work we have been engaged in with community activists for the last four years focused on addressing policing practices and immigration policy at the local level. one of the key tools that we have been using is collecting public records to track policing practices and examine racial profiling incidents, examine the collaboration of prisons with immigration and customs enforcement, and to monitor systems in an effort to change town, county, and state policy and hold each of these entities accountable. here, we specifically focus on obtaining and analyzing documents around policing. first, we provide some background on state and federal laws that allow for access to public records. then, we discuss how we used our state’s public records laws to track and examine local policing practices. we end with a discussion of researching these records with students in mathematics classrooms. public records laws in new jersey, the law allowing us to access records is the open public records act (opra), but each state has a specific public records policy. two other examples are california’s public records act and texas’ public information act. a number o battey & coleman antiracist work in mathematics classrooms journal of urban mathematics education vol. 14, no. 1b (special issue) 8 of resources as well as specific state laws can be accessed through the national freedom of information coalition (www.nfoic.org). at the federal level, the law is called the freedom of information act (foia). foia was enacted in 1966 and has gone through many revisions, but it allows the public to access records of the executive branch. extending from foia, state policies govern the type of records that can be accessed at the local, county, and state levels, but they vary extensively from state to state. for example, police use of force reports are considered public records in many states, but not in north carolina. internal affairs reports, key in documenting police disciplinary action, are public records in alabama, georgia, and arizona, but not in california and new jersey. additional examples of public records are court cases, voting records, city council meeting minutes, municipal budgets, government correspondence (e.g., emails), government contracts, and administrative policies and procedures. as noted, in this article we focus on obtaining and analyzing records focused on policing given our work and recent police murders. analyzing police data with youth central in our efforts to enact change in policing policy and practices is ibram x. kendi’s notion of racist and antiracist policies. “a racist policy is any measure that produces or sustains racial inequity between racial groups” (kendi, 2019, p. 18). what this means is that we can evaluate policies by the racial disparities that they produce or, in the case of antiracist policies, the racial equity they produce. simply put, any policy that increases or sustains racial disparities in policing is racist. we have found this definitional space to support our work with students in examining policing. although the tendency is to see racial disparities or police violence as an issue in urban or predominantly black communities, from 2013–2019 police killed more people in suburban and rural communities, and this trend has been increasing (sinyangwe, 2020). racial disparities and violence in policing then are issues everywhere. policies can be examined for their impact in terms of racial inequities, even as predominantly white communities police their boundaries to restrict the movement and actions of their urban neighbors (boyles, 2015; meehan & ponder, 2002). although our work has been focused on monitoring municipal agencies and police departments to build a critical mass of community members pressing for change to racist policies, these same data have been used in policy and non-profit work to address the problem nationally. specifically, the open policing project at stanford (openpolicing.stanford.edu) and mapping police violence (mapping policeviolence.org) have tracked racial disparities in traffic stops and police violence, respectively. both websites have datasets that can be downloaded and analyzed, though your specific community may not be represented. collecting and analyzing data can also be used to provide arguments for how to reimagine public safety. battey & coleman antiracist work in mathematics classrooms journal of urban mathematics education vol. 14, no. 1b (special issue) 9 having students access and analyze data offers a critical lesson in exercising their rights and can lead to social justice actions, such as gutstein (2006) discusses in detail. in the case of our work, starting in the context of local policing provided a connection to the data that heightened student motivation and engagement. community activists that we work with assisted a number of high school students who expressed an interest in learning more about racist policing in their town and getting involved in the organized response to this problem. in a multi-year analysis of their small town’s municipal budget, a group of students and adult activists discovered that a disproportionate amount of funds were allocated to public safety, four times more than the community services budget. the analysis also revealed that an outsized share of the public safety budget was apportioned for the police department. in the interest of increasing awareness of this data in their community, the students created bar and pie charts highlighting the magnitude of the budget disparities and displayed postersize models of these visual representations during local black lives matter and reimagining policing protests. this analysis informed community arguments around reallocating funding away from the police in order to provide more community services, aligning with broader calls to defund or abolish the police. however, in order to support students in making data-based arguments, they need to be able to access government records. specifically, we focus on supporting teachers and students in accessing police data that is not already available via a website or an existing report. first, we discuss how to access public records as well as some issues in analyzing and framing the data. obtaining police records one of the biggest barriers to accessing data is sometimes figuring out the right language to use. public record requests must ask for government documents, not general information. we have had support from the american civil liberties union– new jersey (aclu–nj) and specifically opra lawyer c.j. griffin of pashman stein walder hayden, who serves on the aclu–nj board. community organizations working on similar issues or your own state aclu chapter might be a support if you have difficulty accessing public data. a useful strategy is to start small, as you can always request more data. around policing, we have focused our energy on tracking traffic stops and use of force. here are two requests we have submitted: pursuant to opra and the common law right of access, i seek all use of force reports from january 1, 2016 to october 23, 2017. pursuant to opra and the common law right of access, i seek the monthly traffic stop race/gender report for the months of january 2016 october 2017 to include the gender and race/ethnicity of persons stopped, persons ticketed, and persons given warnings. battey & coleman antiracist work in mathematics classrooms journal of urban mathematics education vol. 14, no. 1b (special issue) 10 it may be necessary to make multiple requests before you get the data you want, but these laws are in place for citizens and reporters to investigate government institutions. however, if you ask for too much data, government agencies may start to say they cannot meet the request or charge fees. they are allowed to charge fees for work time with large requests and for copying physical materials and dvds (in the case of video footage). however, if you request smaller amounts multiple times, it is possible to avoid charges, and if you ask for electronic copies, there are no fees for the physical materials. therefore, appending “you can send all documents in pdf form to [insert your email address]” at the end of any request will result in quicker responses and lower to no fees. depending on the public records law governing your institutions, they may not aggregate data by race or gender. therefore, obtaining data on individual traffic stops for a month and going individually through them could be necessary to examine disparities. this is also something to urge policy change with the state attorneys general so all state and local agencies aggregate data to look at disparities based on race and gender. requesting records for individual incidents another reason to request information about individual incidents is if there was a specific interaction with the police that you want more information on. for example, in contrast to traffic stops, which can be initiated by license plate readers, instances of use of force are based on police decisions alone. such instances also raise questions of training and implementation of de-escalation practices, and collecting a year’s worth of data can spur investigation of suspicious individual incidents. in these cases, opra requests specific to an incident rather than overview data can be helpful. the following two requests illustrate this type of incident specificity. pursuant to the opra, i seek dash camera footage and body camera footage of the use of force incident that occurred on [insert date] involving [insert name] and officers [insert officer names]. please note that the supreme court said in north jersey media group v township of lyndhurst that dash and body cam videos of the use of force should be available to the public under the common law. pursuant to the opra and the common law right of access, i seek the arrest report, pedestrian stop report, traffic stop report, police log, use of force reports, and any other reports the police department has on file for the [insert date] incident involving [insert name]. video footage and arrest reports can raise serious inconsistencies about the story police tell to the public (morrison, 2017). therefore, they can serve as a check for the stories police tell about incidents or reveal blatant lies in reports (for additional request examples, see hunsdon & battey, n.d.). battey & coleman antiracist work in mathematics classrooms journal of urban mathematics education vol. 14, no. 1b (special issue) 11 data interpretations and limitations in addition to policy limiting available data, traffic stops may be initiated by license plate readers, so police departments can say they are not responsible for racial disparities in traffic stops. however, tickets are generated by the police themselves. the benefit of looking at traffic stops is that they are not community initiated and make it more difficult for police departments and municipalities to shift blame to the community. a common practice in many communities is to have elected officials and police departments blame racism in the community for calling the police (e.g., amy cooper & the central park birdwatching incident; see sacchetti et al., 2020). although this is a real issue, it is also a way to deflect responsibility from the police. however, even these incidents can raise problems in the ways that dispatchers handle calls in the community. in new jersey, a law was recently enacted to consider a racially based false 911 call as a bias crime (state of new jersey, 2020). washington and new york have similar laws, and california and michigan are considering them. monitoring the number and types of community calls as well as how dispatchers and the police respond to such calls can be an important effort. for example, we uncovered an incident where a homeowner called 911 because a black man was on a cell phone near a parked car in front of their house. the police department responded by sending not one, not two, but four police cars. we collected the dispatch call and the multiple incident reports for each police officer that arrived on the scene. therefore, even community racism can show how dispatchers and police departments respond to incidents where no crime is under way. it is also important to note that data obtained from police records is always incomplete. race data classification is usually attributed from the visual observation by the police officer. therefore, the dependability of race classification data is questionable. the racial lens of police can diminish the appearance of racial disparities by underreporting incidents with black and brown individuals. in addition to racial information, police reports may be missing other data, like why someone was arrested or the types of force used. even dashcam videos have limitations, as they may not show the incident because of the direction of the car. we have also had occasions when police sit down with community members and show partial or edited video. it is possible that video is edited even when obtaining it through public records. however, even with these limitations, it has not compromised the data we have obtained, across multiple municipalities and the county (or national datasets mentioned earlier), from revealing significant racial disparities in policing. battey & coleman antiracist work in mathematics classrooms journal of urban mathematics education vol. 14, no. 1b (special issue) 12 analyzing the data for action both of the requests above dealing with traffic stops and use of force reports were fulfilled. although sophisticated analysis is possible, dividing the number of traffic stops by the local population gives you a comparable racial rate (see figure 1 for one police department’s statistics). usually this is then multiplied by 1,000 or 100,000 to give the rate “per 1,000 persons.” in terms of mathematics lessons, this can be used to learn about rates as well as learn about demographics of the students’ community versus neighboring communities and to encourage them to consider policies that segregate within or across communities. population data can be accessed through the u.s. census bureau or american community survey (www.census.gov or censusreporter.org). figure 1. traffic stop rate by race figure 1 shows that black drivers were stopped at over 3.5 times the rate by population of white drivers in this community over a two-year period. the police department in this example pushed back to say that the town population was not a fair representation because there was a busy road going through town. this argument would require that the neighboring communities had a higher percentage of black drivers in their populations. however, neither neighboring town was predominantly black. for educational purposes, students can explore the implications of using different population statistics in the analysis. clearly, the traffic stop rate for black drivers is over three times higher than for white drivers. battey & coleman antiracist work in mathematics classrooms journal of urban mathematics education vol. 14, no. 1b (special issue) 13 we mentioned earlier that just looking at stops can allow police departments to push back due to license plate readers. this raised the need in the community to look at the citation data in comparison to the earlier stop data to speak to such concerns (see figure 2). figure 2. traffic citation rate by race once again, the data show black drivers being cited at over three times the rate of white drivers. additionally, asian drivers were cited about 2.5 times more than white drivers. therefore, the patterns shift somewhat from traffic stops, because the citation rate for white drivers is lower in comparison to other racial/ethnic groups. however, the disparities remain from the earlier analysis. analyzing this data with students can help them realize disparities in policing for different communities. it also places calculating unit rates within an everyday context rather than those from textbooks. therefore, traffic stop data provide an important context to develop mathematics skills while doing it in a context that raises issues of social justice. another consideration for data to analyze is police use of force. traffic stops and citations can be initiated by license plate readers and radar outside the decisionmaking purview of police officers. however, police officers decide whom to use force on, not community members calling the police or automatic license plate readers. the decision to use force is made by individual or groups of officers. when looking at the same community that we performed the traffic stop analysis on for cases of police use of force, we found that 64% of use of force incidents were with people of color and 58% of them were specifically with black individuals despite the battey & coleman antiracist work in mathematics classrooms journal of urban mathematics education vol. 14, no. 1b (special issue) 14 community being 64% white and 6% black. police departments and municipalities have less room for argument on these incidents except to try to consider them individual cases. however, similar results across months and years can speak volumes about patterns of use of force. mental health crises can play a role in use of force data, which can raise the need to reconceptualize public safety and to consider whether police are sufficiently trained and should be the ones handling mental health incidents (westervelt, 2020). these data and analyses are drawn from public records in a suburban, predominantly white community. it is a small town that borders a city with a major university, and it also, according to our data and analysis, is one that polices black drivers heavily as they live in and pass through the community. as sam sinyangwe (2020) has stated, racial disparities in policing are increasingly becoming a suburban issue as the communities become more diverse and police the borders of their urban neighbors. this means that although recent coverage highlights killings in urban communities, this is work predominantly white communities also need to take on, as their police fine, incarcerate, and kill black people that pass through. antiracism in urban mathematics classrooms mathematics activities involving analysis of policing data can also be implemented in the context of youth participatory action research (ypar), a civic education framework that engages youth in research projects that tackle complex and challenging social issues (mirra et al., 2015). under adult guidance and research training, ypar student groups identify an issue of concern that impacts their lives and/or the lives of others in their school or larger community and conduct research that seeks solutions to the problem. ypar research culminates in a student presentation of their findings, proposed solutions, and action (cammarota & fine, 2008). with social justice and student empowerment as central tenets, ypar is an effective means of empowering historically marginalized student voices (zaal & terry, 2013). schools across the country have implemented ypar projects that engage youth in knowledge production on critical social issues such as school violence, health promotion, experiences of new immigrants, restorative justice, and educational inequalities (rubin et al., 2016; zaal & terry, 2013). rooted in local contexts and driven by student inquiry, ypar projects are not guided by scripted lesson plans and curriculum guides. rather, the research process and lesson plans unfold in an organic manner, facilitated by skilled adults who assist students in using tools and resources for data collection, analysis, and reporting. mathematics settings have employed ypar projects for examinations of issues including the prison industrial complex (terry, 2011), racial and class-based educational battey & coleman antiracist work in mathematics classrooms journal of urban mathematics education vol. 14, no. 1b (special issue) 15 inequalities (yang, 2009), racism in housing data (gutstein, 2003), and school food injustice (raygoza, 2016). these real-world applications of mathematics understanding in the service of social justice placed marginalized youth in urban spaces at the center of data gathering, analysis, and reporting thereby reinforcing deeper mathematical learning, cultivating increased awareness of the utility of quantitative data, and creating opportunities for students to contribute to important reform discussions in their schools and communities (gutstein, 2003; raygoza, 2016; terry, 2011; yang, 2009). whether in the context of a ypar project or in a mathematics lesson that simply analyzes data, examining policing practices through data analysis allows students to examine policies for racist practices. analyzing racial disparities in policing data can prove particularly engaging for youth in urban school settings who develop an understanding of some of the statistics behind the racial injustices that they may see and hear about from people directly impacted by biased policing in their homes and communities. empowered by their research and related findings on racial disparities in policing, students can become civic actors in discussions of reimagining public safety in their own neighborhoods. given the existence of racially disparate policing across different community settings—urban, suburban, and rural—the practice of social justice mathematics education has relevance for students in many k–12 school environments. there are a number of ways for students to mathematically analyze police data. it can be helpful to start with a question; for example, “what do we want to know about traffic stops by police in our community?” at the upper elementary level, issues of fairness are of concern to many students. students may want to know how traffic stops differ by race, and they can look at total counts of traffic stop data for one month and compute percentages to compare stops with black or white drivers with all stops. it is in comparing the data using counts and percentages for different racial groups that issues of fairness are raised for discussion for students. younger learners can also compare the percentage of stops and population using charts and graphs. at the middle school level, students can extend this by examining proportions, dividing the stop percentages by the percentage of the population. analyzing the data this way moves students into considering proportions (stops to population) and comparing proportional relationships across races. additionally, as the rates of stops or citations are compared to the population, students are computing unit rates (per person or 1,000 persons), which they can represent graphically as well. lastly, students can consider different ways to represent population and the implications for how arguments are constructed. for example, what happens if they use county data or include the populations from neighboring communities that may drive through their community? exploring the different ways to battey & coleman antiracist work in mathematics classrooms journal of urban mathematics education vol. 14, no. 1b (special issue) 16 represent populations can engage students in considering different methods of calculating results, the arguments the different considerations support, and opportunites to critique how data is organized and presented in the media. at the high school level, students can extend analyses of proportions and unit rates to develop models for examining trends across years of police data. driven by a question of whether traffic stops are increasing or decreasing inequities, they could gather multiple years of traffic stop data and develop an equation that shows the relationship of stops over time for each racial group. this traffic stop data can also be represented graphically, and students can interpret each other’s graphs. furthermore, students can evaluate police reports or watchdog reports to interpret the data and various graphical representations used. as we have shown, there are ways to engage students at all levels in k–12 education in critically analyzing policing. ypar in one mathematics classroom we have found that it is important to start with the questions asked by students and to examine decisions about how to analyze and represent data. based on our work collecting and analyzing police records, a college student from a local community activist group worked with an urban high school class to share policing data for that community. during the initial discussion, a number of questions were raised about the data and possible interpretations, which prompted the high school students to further examine the policing data. students asked about the citation rates by police officer to determine if some officers had higher than average rates of citing black, latinx, and asian drivers. one student wondered if black and latinx teens walking, riding their bikes, or hanging out in town were more likely than their white counterparts to be stopped and questioned by the police. students also made connections to police interactions with young people. specifically, one high school student shared an experience of being harassed in an interaction with the police. the discussion resulted in the students deciding that they wanted to examine the demographic data on youth involved in suspicious person incidents in the community. learning about one peer’s experiences and the strategies for gathering the data prompted using mathematics to examine concerns about policing in the students’ community. in order to investigate the suspicious person incidents in the community, the high school students had to learn about police data and how to request, organize, and analyze it. however, students first needed to decide on a research question that could be answered with the data available. students had to consider various issues like what data to use, how much data to collect, and over what period of time to collect the data. accessing and analyzing the police records presented students with the opportunity to review raw data and determine ways to reduce it to interpretable battey & coleman antiracist work in mathematics classrooms journal of urban mathematics education vol. 14, no. 1b (special issue) 17 conclusions. when the data came in, they decided how to organize it in a format to do analysis of counts, rates, and proportions. they decided to do this in google sheets, which allowed the students to work collaboratively. students explored the meaning of simply analyzing counts without consideration of the population of the surrounding community. this raised the need for proportions and relating the number of incidents to the population to draw a comparison across various racial groups. across this work, students considered additional variables (e.g., location of stop, time of day), disaggregated analyses, and used and applied measures of center and spread to determine suitable comparison groups and techniques to know when their results diverge from the norm. still, any analysis leaves a number of interpretations of racial disparities. however, in centering conversation on kendi’s (2019) definition, the existence of racial disparities themselves speaks to racist policy. the issue then becomes which policy and how it is producing such disparities. this also shifts the conversation away from blaming certain groups to looking for structural reasons for racial inequities. as students engage in analyzing police data, other questions are sure to arise. in this case, students became interested in the role of police in the schools. they were generally aware of the popular term “school-to-prison pipeline,” which refers to the pathway from school suspensions and punitive disciplinary practices to criminalization and incarceration in the juvenile justice system that disproportionately impacts black, latinx, and indigenous students of color (morris, 2012). this phenomenon is closely tied to the use of school resource officers (sros) “that either are or act like police officers” (tyler & trinkner, 2017, p. 163). although the proliferation of sros has been widespread across the nation’s schools, this crime-control approach to discipline is more common in schools that predominantly serve black and latinx students (tyler & trinkner, 2017). although a nearby urban school with a majority black and latinx student population employed an sro, the students that we worked with were in a school district that did not employ an sro. however, their general concern about police in schools resulted in students commenting that they regularly see a youth police officer in their own school, even though they are not designated as an sro. the presence of the officer in their school raised several questions about the presence of the youth police officer, as he had regular access to the school. is the youth police officer used for security at all school events? is an officer/s present for some sporting competitions more or less than others? how much of the school district’s budget is allotted for law enforcement? are there racial disparities in the number of discipline incidents that require police involvement? therefore, the ypar work with these youth raised more questions about their community and additional research to pursue. pursuing answers to these and related questions involves engaging with school officials to ensure that they will provide access to the data or requesting public records through opra if battey & coleman antiracist work in mathematics classrooms journal of urban mathematics education vol. 14, no. 1b (special issue) 18 they are unwilling to share it. this activity can also highlight the importance of youth voices in determining what they need to feel safe in their buildings. ultimately, ypar is about producing knowledge and informed action. as youth begin to identify racial inequities, which point to racist policies—those that are producing racial disparities—they can shift to advocating for antiracist policies— those that produce racial equity. in the case of this high school, the action that resulted came in the form of speaking at board of education and city council meetings, organizing direct actions, and partnering with community organizations. the public officials and community groups stated that they benefitted from the analysis that students conducted. ultimately, the youth police officer was limited to having access to the school for specific incidents through alliances students built with parents and members of the school board. though there are clear benefits to developing mathematical understanding through analyzing public records, we believe that the more important contribution is the action in the service of creating a more just society that can result. concluding remarks doing ypar based on requests for public data can be complex and tedious but equally rewarding to students and necessary work in marginalized communities. not only is it detailed work, but it is also work that goes against the socialization of many community members who have been conditioned to trust the police. having conversations with those who do not bring systemic lenses to understand policing often lead to discussions focusing on individual police officers (e.g., “bad apples”). a historical analysis of the police reveals just how policing has been an arm of the state to control black and brown people in an effort to keep white people feeling safe (muhammad, 2010). challenging white institutional spaces and anti-blackness (martin, 2019) is long-term discouraging work. police departments will sometimes note limitations of their own data, questioning the validity of it, which raises the question, “why not collect better data?” city councils often frame issues as “good” and “bad” cops rather than looking at issues systemically (stanley, 2015). in urban centers, data show widespread use of force in policing (mapping police violence, n.d.). engaging youth in these communities, who are themselves harassed or subjects of force by police, can make them agents of change. this work needs to be done by concerned residents forming multi-generational coalitions that nurture and encourage new generations of activists. young people in predominantly black and brown communities who have accumulated, lived experiences with overpolicing and racial profiling can enrich and deepen their understanding of longstanding racial discrimination by engaging policing data analyses. in the process, youth researchers are exposed to ways they can analyze, interpret, and represent data to inform direct actions around policing in their communities. student perspectives are battey & coleman antiracist work in mathematics classrooms journal of urban mathematics education vol. 14, no. 1b (special issue) 19 then at the center of community education on policing, leading to activism and calls for change. by making the mathematics relevant to students’ lived experiences and analyzing their own community’s data, students are encouraged to continue taking mathematics and using it for their own purposes. the goal is not to make all students mathematicians, but it is to give them access to consciously use mathematics in their lives as a form of socially transformative participation. data alone cannot solve our national problem around police violence, but it can show where specific issues lie within police data, raise the need for specific practices or policy changes, and bring to light blatant racial profiling incidents to build community support for change. as students uncover these blatant illegal and unethical policies and practices and the mathematics underlying their cases, they can be empowered for a lifetime of engaged mathematics activism. in particular, supporting them in accessing public records will make them better equipped to find problems, present their case using mathematics, counter alternative arguments, and enact change. pushing for that change through analysis shared with the community and done in individual classrooms is antiracist work. in turn, this empowers students to engage in action to push for antiracist policies and practices that address police violence in their communities. acknowledgements the authors sincerely thank amiri tulloch for his work in the school and jessica hunsdon for feedback on a previous version of the paper. references boyles, a. s. (2015). race, place, and suburban policing. university of california press. cammarota, j., & fine, m. (eds.). (2008). revolutionizing education: youth participatory action research in motion. routledge. gutstein, e. (2003). teaching and learning mathematics for social justice in an urban, latino school. journal for research in mathematics education, 34(1), 37–73. gutstein, e. (2006). reading and writing the world with mathematics: toward a pedagogy for social justice. routledge. hunsdon, j., & battey, d. (n.d.). transparency and accountability in public safety/policing: open public records act (opra) requests. https://bit.ly/oprarequestsforpoliceaccountability kendi, i. x. (2019). how to be an antiracist. one world. mapping police violence. (n.d.). police accountability tool. https://mappingpoliceviolence.org/cities martin, d. b. (2019). equity, inclusion, and antiblackness in mathematics education. race ethnicity and education, 22(4), 459–478. doi.org/10.1080/13613324.2019.1592833 meehan, a. j., ponder, m. c. (2002). race and place: the ecology of racial profiling african american motorists. justice quarterly, 19(3), 399–430. mirra, n., garcia, a., & morrell, e. (2015). doing youth participatory action research: transforming inquiry with researchers, educators, and students. routledge. battey & coleman antiracist work in mathematics classrooms journal of urban mathematics education vol. 14, no. 1b (special issue) 20 morris, m. w. (2012). race, gender, and the school-to-prison pipeline: expanding our discussion to include black girls. african american policy forum. http://schottfoundation.org/file/1765/download?token=xt7cgqpi morrison, c. m. (2017). body camera obscura: the semiotics of police video. american criminal law review, 54(3), 791–841. muhammad, k. g. (2010). the condemnation of blackness: race, crime, and the making of modern urban america. harvard university press. raygoza, m. c. (2016). striving toward transformational resistance: youth participatory action research in the mathematics classroom. journal of urban mathematics education, 9(2), 122– 152. https://doi.org/10.21423/jume-v9i2a286 rubin, b. c., abu el-haj, t. r., graham, e., & clay, k. (2016). confronting the urban civic opportunity gap: integrating youth participatory action research into teacher education. journal of teacher education, 67(5), 424–436. sacchetti, m., jacobs, s., & hauslohner, a. (2020, may 27). public outrage, legislation follow calls to police about black people. the washington post. https://www.washingtonpost.com/national/public-outrage-legislation-follow-white-womans-call-to-police-about-black-man-incentral-park/2020/05/27/94b219a6-a049-11ea-9590-1858a893bd59_story.html sinyangwe, s. (2020, june 1). police are killing fewer people in big cities, but more in suburban and rural america. fivethirtyeight. https://fivethirtyeight.com/features/police-are-killing-fewerpeople-in-big-cities-but-more-in-suburban-and-rural-america/ stanley, j. (2015, march 19). we need to move beyond the frame of the “bad apple cop.” american civil liberties union. https://www.aclu.org/blog/national-security/we-need-move-beyondframe-bad-apple-cop state of new jersey. (2020, august 31). governor murphy signs legislation criminalizing a false 91-1 call based on race or protected class [press release]. https://nj.gov/governor/news/news/562020/approved/20200831b.shtml terry, c. l., sr. (2011). mathematical counterstory and african american male students: urban mathematics education from a critical race theory perspective. journal of urban mathematics education, 4(1), 23–49. https://doi.org/10.21423/jume-v4i1a98 tyler, t. r., & trinkner, r. (2017) why children follow rules: legal socialization and the development of legitimacy. oxford university press. westervelt, e. (2020, september 18). mental health and police violence: how crisis intervention teams are failing. national public radio. https://www.npr.org/2020/09/18/913229469/mentalhealth-and-police-violence-how-crisis-intervention-teams-are-failing yang, k. w. (2009). mathematics, critical literacy, and youth participatory action research. new directions for youth development, 123, 99–118. zaal, m., & terry, j. (2013). knowing what i can do and who i can be: youth identify transformational benefits of participatory action research. journal of ethnographic & qualitative research, 8(1), 42–55. copyright: © 2021 battey & coleman. this is an open access article distributed under the terms of a creative commons attribution-noncommercial-sharealike 4.0 international license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. microsoft word 5 final young & young vol 9 no1.doc journal of urban mathematics education july 2016, vol. 9, no. 1, pp. 79–93 ©jume. http://education.gsu.edu/jume jamaal rashad young is an assistant professor at the university of north texas, 1155 union circle #310740, denton, texas 76203; email: jamaal.young@unt.edu. his research interests include critical race theory for black students in mathematics, multicultural stem projectbased learning, literature synthesis, and meta-analysis methodology. jemimah lea young is an assistant professor at the university of north texas, 1155 union circle #310740, denton, texas 76203; email: jemimah.young@unt.edu. her research interests include the academic achievement of black girls, achievement gap research, in addition to culturally responsive pedagogy, particularly in stem. young, black, and anxious: describing the black student mathematics anxiety research using confidence intervals jamaal rashad young university of north texas jemimah lea young university of north texas in this article, the authors provide a single group summary using the mathematics anxiety rating scale (mars) to characterize and delineate the measurement of mathematics anxiety (ma) reported among black students. two research questions are explored: (a) what are the characteristics of studies administering the mars and its derivatives to representative populations of black students? (b) what is the 95% ci for the reported ma of black students in the mars literature? a literature search yielded 21 studies after inclusion criteria were applied. analyses suggest that black participants and their scores are not well represented in the current ma research using the most popular instrument the mars. based on available mean point estimate data, the reported ma of black students can best be described as consistent across measurements, and population parameter estimates are between 200 and 220 on the mars scale. moreover, although substantial research in the area of ma exists, much work is needed to fully comprehend the nuances of black ma and its influence on achievement in mathematics. keywords: african american students, black students, confidence intervals, mathematics anxiety athematics literacy affords individuals the opportunity to participate fully in the community of practice. theorists describe general identity development as the process of acquiring membership in a community of practice (nasir, 2002; wenger, 1998). martin (2007) suggests that mathematics literacy in the black student population is linked to identity construction at the intersection of their racial identity (black) and mathematics identity (becoming a doer of mathematics). thus, mathematics identity development for black students is the acquisition of membership as a “doer” of mathematics while negotiating elements of their racial identity to gain full participation in the community. subsequently, full participation in the mathematics community of learners is essential to the development of a mathematm young & young black mathematics anxiety journal of urban mathematics education vol. 9, no. 1 80 ics identity. mathematics anxiety (ma) can serve as a major impediment to the student’s participation in mathematics consistently observed in avoidance patterns. many anxious students avoid mathematics tasks and courses to prevent exacerbating feelings of anxiety. the mounting interest in leveraging the constructs of identity development and community of practice in mathematics education research is predicated on the opportunities these constructs provide for analyzing and explaining affective aspects of mathematics learning, such as motivation, engagement, interest, anxiety, and participation (cobb & hodge, 2010). research has yet to fully examine the possible differential effect of ma on underrepresented pre-k–12 students. specifically, given the long-standing “achievement gap” between black and white students, it is imperative that the possible differential effect of ma on black students1 is addressed (national center for education statistics, 2009). historically, compared with their peers, children of color educated in urban schools disproportionately struggle with mathematics (elias, white, & stepney, 2014; kellow & jones, 2005, 2008). given this trend, students and educators alike are under unprecedented pressure to achieve state proficiency standards and to close gaps in student achievement (yeop kim, zabel, stiefel, & schwartz, 2006). robust solutions, however, remain elusive. the perpetual mathematics achievement gap trends are extremely detrimental to the career aspirations and future success of black students; therefore, if the effects of ma can be addressed in this population the ramifications would be substantial. several educators have suggested that researchers forego the inclination to “gap-gaze” and shift their attention toward studies of between group variance and toward identifying the mathematics strength of students of color rather than continuous investigations of underachievement (e.g., gutiérrez, 2008; martin, gholson, & leonard, 2010; stinson, 2013). these ideas support the need for more single group summaries that provide pertinent theoretical and practical knowledge to enhance instructional praxis. prior research synthesis and meta-analysis prior research suggests that non-statistically significant interaction effects exist between ma, mathematics achievement, and race and ethnicity (e.g., hembree, 1990; ma, 1999). these results suggest that ma does not differentiate among racial and ethnic groups. for example, in the foundational work “a meta-analysis of the relationship between anxiety toward mathematics and achievement in mathematics,” ma (1999) concluded that the relationships between ma and heterogene 1 throughout this article, when referencing and reporting aggregated data on specific racial, cultural, and ethnic groups (e.g., black students), we acknowledge the significant within group variation embedded (and made invisible) in such data, specifically in regards to academic achievement and performance. we also acknowledge the dangers of somehow representing such groups as monolithic across a number of demographic characteristics, which they are not. young & young black mathematics anxiety journal of urban mathematics education vol. 9, no. 1 81 ous and homogeneous populations were essentially the same. yet, black students were missing from the 27 studies included in the analysis. heterogeneous samples included latina/o, thai, native american, australian, and lebanese students, but black students were not identified in the sample pool. in an analogous work, hembree (1990) concluded that there was a nonstatistically significant effect of ma between black and white college students. however, of the 151 studies in the analysis only three effect sizes were based on representative numbers of black students, all of which were extracted from studies conducted in post-secondary settings. generally speaking, large meta-analyses have consistently concluded that ma does not have a differentially larger effect on diverse students of color. the majority of the studies included in these analyses, however, consisted of large homogeneous populations of white students compared to small heterogeneous populations of underrepresented groups of racially and ethnically diverse students. given the underrepresentation of black students in mathematics research literature in general, more inclusive research reporting and data presentations are necessary to better ascertain the magnitude and prevalence of ma in the black student population. mathematics anxiety rating scale the mathematics anxiety rating scale (mars) is one of the most popular and widely used instruments to assess ma. the original mars, developed by richardson and suinn (1972), was a 98-item instrument that assessed levels of anxiety related to mathematics analogous tasks. given the substantial amount of time necessary to complete the original instruments, recent derivatives of the mars have reduced items substantially (alexander & martray, 1989; suinn & winston, 2003). these derivatives of the mars have emerged to assess specific populations of students and to reduce the administration time, resulting from the popularity and persistence of the mars in mathematics education over the years. despite these changes, mars remains one of the most popular ma instruments due to its historically consistent validity and reliability across administrations (plake & parker, 1982; suinn & edwards, 1982; suinn, taylor, & edwards, 1988). notwithstanding its strong history as a robust ma instrument, assessments with diverse populations of students are consistently less prevalent and the scores are less reliable. in a reliability generalization study, capraro, capraro, and henson (2001) concluded that across all the studies sampled, the mars produced scores with high reliabilities, but results from 11 studies conducted with heterogeneous populations that used the mars had a negative correlation to cronbach’s alpha reliabilities. this negative correlation is relevant because the mars is one of the most used measures of ma, thus reliability of scores must remain stable across racially heterogeneous populations. this instability indicates that ma research involving stu young & young black mathematics anxiety journal of urban mathematics education vol. 9, no. 1 82 dents of color may have lower reliability estimates, but without direct inspection it is impossible to draw conclusive conclusions. analyzing black student ma as an isolated variable supports the ability to aggregate studies for meta-analytic thinking and can help to contextualize the mathematics performance of black students as we move away from between-group investigations. in “characterizing the mathematics anxiety literature using confidence intervals as a literature review mechanism,” zientek, yetkiner, and thompson (2010) posit that confidence intervals (cis) can serve as an efficient mechanism for characterizing scores on the mars. the present study seeks to use cis to summarize the measurement of ma using the mars with a black student population. problem statement and research questions the prevalence of ma and the magnitude of effects of ma on black student mathematics achievement have important ramifications for subsequent student success. for example, many underrepresented students of color experience anxiety that is detrimental to their mathematics performance (adler, 2007). these episodes can foster lower grade point averages and poor performance on college entrance exams, limiting access to many colleges and universities. ma significantly influences cognitive functions and test performance; however, its influence on black pre-k–12 students, albeit researched, has yet to be coherently synthesized. this phenomenon creates limitations that complicate or eliminate the application of traditional meta-analysis methods. first, pertinent participant descriptive data, such as race or ethnicity is generally absent from many studies. second, studies that do present racial frequency counts often do not disaggregate the ma results by race or ethnicity. third, when disaggregated sample descriptive data are present, the necessary statistical data necessary to calculate effect sizes are absent. despite these analytical challenges, a structured synthesis of the influence of ma on black student achievement has explicatory significance. specifically, the use of meta-analytic thinking can help provide credence to more extensive studies of this phenomenon. meta-analytic thinking systematically allows researchers to benchmark their results by comparing them to prior results from analogous studies. thus, researchers need to explicitly design and place studies in the context of the effects of prior literature (henson, 2006). this shift in empirical thinking promotes the empirical replication of results and supports meta-analytic thinking. one analytic medium for the comparison of effect sizes is the ci. according to thompson (2002), cis for effect sizes are exceptionally valuable because they facilitate both meta-analytic thinking and the elucidation of intervals via comparisons with the effect intervals reported in analogous prior studies. furthermore, cumming and finch (2001) suggest four reasons to use cis: young & young black mathematics anxiety journal of urban mathematics education vol. 9, no. 1 83 • cis provide point and interval information that is accessible and comprehensible, which supports substantive understanding and interpretation. • there is a direct link between cis and null hypothesis statistical significance testing (nhsst). • cis support meta-analytic thinking focused on estimation. • cis communicate information about a study’s precision. in addition, sample size is a reasonable consideration when applying metaanalytic thinking to compare and evaluate, for example, technology professional development in urban mathematics classrooms. the application of meta-analytic thinking through cis provides a lens to compare effects across large and small samples. along with strong evidence of effect, cis also provide two other advantages. first, when sample sizes are considerably small, nhsst may not capitulate statistically significant results. unfortunately, the conclusion typically associated with non-statistically significant results is that the effect is not real (cumming & finch, 2005). cis, however, allow researchers to place results in a broader context to establish practical and clinical significance. second, because all cis report both (a) point estimates and (b) characterize how much confidence can be vested in a given point estimate (zientek, yetkiner, & thompson, 2010), comparing point and interval estimates to other studies examines precision and quality of the results of a particular study across other studies. thus, the purpose of this study was to conduct a single group summary of studies using the mars to characterize and delineate the measurement of reported ma within the black student population. two research questions guided the inquiry: 1. what are the characteristics of studies administering the mars and its derivatives to representative populations of black students? 2. what is the 95% ci for the reported ma of black students in the mars literature? methods to investigate the aforementioned research questions, we conducted a literature search using the following key terms: african american/black, mathematics anxiety, mathematics anxiety rating scale (mars), and math anxiety rating scale (mars). we use operational definitions to capture essence of the constructs under investigation and to help guide our consistent identification of pertinent studies. ma was operationalized as an anxious state in response to mathematicsrelated situations measured by self-reported surveys, physical reactions, or observations. but for the purpose of this study, we included only studies that measured ma using young & young black mathematics anxiety journal of urban mathematics education vol. 9, no. 1 84 one of the mars instruments. black students were operationalized as nonimmigrant pre-k–12 and post-secondary students racially self-reported as black. using these operational definitions as an initial guide, we used a three-step approach to search for studies that used the mars to measure black student ma. we applied a broad search of several educational research databases using the key-topic descriptors (previously noted). the specific databases searched were (a) educational resources information center (eric), (b) academic search complete (asc), (c) psycinfo, and (d) educational research compete (erc). next, using the same key descriptors, we searched meta-analyses and systematic reviews published after 1990 to locate prior systematic reviews as a means to augment the initial pool of studies. we manually searched the reference list from the foundational works of ma (1999), hembree (1990), and ma and kishor (1997) to retrieve pertinent studies. five additional studies were located using this procedure. using the aforementioned search procedures, we located a total of 224 studies. study abstracts were screened initially and promising studies were read and evaluated with respect to the objective of this study. a study was included in this synthesis if it included (a) pre-k–12 or post-secondary black participants, (b) use of the mars or one of its derivatives, and (c) descriptive statistics. based on these criteria, we identified 21 individual studies and 24 individual mean point estimates of ma. the complete inclusion and exclusion procedures are presented in figure 1. we coded each study for pertinent independent variables that captured design features as well as other pertinent characteristics. design features included: grade, composition of black students, sample size, mars instruments used to measure anxiety, and cronbach’s alpha. to summarize the measurement of reported ma in the black student population, we created a 95% ci from the baseline mean point estimates of homogenous samples of black participants. all point estimates presented in the study were collected before participants received any treatment or intervention, and this represent the reported ma of black students before manipulation. to maintain the fidelity of the cis, racially heterogeneous populations had to disaggregate black student data to be included in the ci calculations. a 95% ci was chosen by convention, a 90% or any other level would be equally valid, but the 95% ci is a more strict measure (zientek et al., 2010). cis were selected because they provide point estimates for population parameters, as well as a measure of precision of these estimates that can be compared across administrations (cumming & finch, 2001). the point estimates are sample statistics, two of the most commonly used of which are means and effect sizes (zientek et al., 2010). in this study, we extracted mean mars scores across the administrations as the point estimate to summarize the mean scores. to calculate the mean point estimates we included only studies that disaggregated and presented mars scores for black students. because the 98-item mars and mathematics anxiety rating scale for adolescents (mars-a) represent the young & young black mathematics anxiety journal of urban mathematics education vol. 9, no. 1 85 oldest, most established, and most psychometrically similar instruments in present across our selected timeframe, we decided to include only these instruments in the ci calculations and presentation. from the six studies that present mean mars scores for black students, seven mean point estimates were extracted. two were calculated using the mars and five were calculated used the mars-a. mean mars scores, standard deviations, and sample sizes were then used to calculate the 95% cis. figure 1. literature search flow diagram for mars. we used the microsoft excel confidence macro and the stock option to plot the intervals. the mars and mars-a are five point likert scaled instruments with 98 individual items. the ma scores for each instrument are determined by the calculated sum of the individual items, thus the scores range from 98 to 490. due to the relative underrepresentation of the mars in the sample of scores, coupled with inconsistency in measurement specificity and parity between adolescent samples and the mars-a instruments we combined all the 95% ci onto a single summary (see figure 2). mars-a point estimates were indicated by black squares and diamonds used to identify point estimates of original mars. using the procedures and eric (n = 76), asc (n = 24), pyscinfo (n = 89), erc (n = 30) records identified through database searching (n = 219) sc re en in g e lig ib ili ty additional records identified through foundational works (n = 5) in cl ud ed records after duplicates removed (n = 190) records screened (n = 190) id en tif ic at io n records excluded (n = 93) full-text articles assessed for eligibility (n = 97) full-text articles excluded (n = 76) studies included in synthesis (n = 21) young & young black mathematics anxiety journal of urban mathematics education vol. 9, no. 1 86 guidelines presented in prior studies (e.g., cumming, 2007, 2009; zientek et al., 2010), we systematically identified a feasible estimate for the range of mars scores represented in this single group summary. based on visual inspection of the overlap and lack thereof between studies a reasonable inclusion ranged for the population parameter estimate of ma scores was identified. results the characteristics of the studies in this review are presented in table 1. it includes citation information, publication source, sample size, number of black participants, and whether or not the data were disaggregated to present the mean point estimates of reported ma for black participants. table 1 characteristics of mars studies citation year source grade n nb bmars glover 1994 d middle 67 67 yes johnson 1997 d high 123 24 no kazelskis et al. 2000 a college 321 70 no cox 2001 d college 88 5 no hopko et al. 2002 a college 42 2 no sloan et al. 2002 a college 72 1 no hopko 2003 a college 815 32 no hopko et al. 2003 a college 64 2 no husni 2006 d college 62 62 yes miqdadi 2006 d high 168 47 no baloglu & zelhart 2007 a college 559 81 no solazzo 2007 d college 131 2 no gleason 2007 a college 261 13 no reed 2008 d college 84 6 no sprybook 2008 d college 28 3 no brocato 2009 d college 29 29 yes bryant 2009 d college 132 39 no jones 2009 d high 67 67 yes grassl 2010 d college 351 55 no meritt 2011 d middle 105 105 yes steiner & ashcraft 2012 a college 369 32 yes note: d = dissertation; a = article; b = black young & young black mathematics anxiety journal of urban mathematics education vol. 9, no. 1 87 for this pool of studies, the year of publication ranged from 1994 to 2012, with a median year of 2006. the reported representation of black participants ranged from 0.6% to 100%, with a mean represented of 40.7% based on the ratio of the number of black participants to non-black participants. dissertations were the most prevalent publication source in the pool of studies and represented approximately 62% of the included studies. articles represented only 8 of the 21 studies or approximately 38%, and college students were the population of interest in the majority of the studies. mars scores were not presented for black students in 15 of the 21 studies, or approximately 71% of the studies. mars scores for black students were extracted from five dissertations, including homogenous populations of black students and one article with heterogeneous participants. reliability scores were only presented in five of the studies reviewed in this synthesis. the reported reliabilities, however, were relatively high and ranged from .84 to .96. of the four available mars instruments, the revised mathematics rating scale (rmars) was administered most often, followed by the mars-a. a full description of each mars instrument and its administration frequency is presented in table 2. table 2 description of mars survey instruments administrations description of mars survey items freq. mathematics anxiety rating scale (mars) • measures students’ anxious reaction when they do mathematics in ordinary life and academic situations (richardson & suinn, 1972) 98 3 mathematics anxiety rating scale for adolescents (mars-a) • a revised form of the mars that involves changes in wording and substitutions appropriate for adolescents (suinn & edwards, 1982) 98 5 revised mathematics anxiety rating scale (rmars) • revised from mars and shorten to measure three factors: math test anxiety, numerical anxiety, and math course anxiety (alexander & martray, 1989) 25 9 mathematics anxiety rating scale brief version (mars-brief) • a shorter revised version of the original mars that is comparable in construct measurement (suinn & winston, 2003) 30 3 figure 2 presents the 95% cis for black participant performance on the mars. the cis allow the amount of error to be quantified from sample to sample for comparison. the level of precision in each point estimate controls the error associated with each point estimate. the width of the ci represents the precision associated with each point estimate; specifically, the smaller the width of the ci, the more precise the measurement and the wider the ci the less precise the measure young & young black mathematics anxiety journal of urban mathematics education vol. 9, no. 1 88 ment (cumming & finch, 2005). furthermore, the width of the ci is directly related to the standard deviation and inversely related to sample size, two major components of measurement precision in statistics. appropriately, if the variability (standard deviation) is small then the point estimate is more precise and likewise a larger sample size is more representative, which increases the precision of the point estimate. all of the 95% cis presented in figure 2 are relatively wide, indicating a lack of precision in the measurement that could be attributed to large standard errors of consistently smaller and less representative sample sizes of black students. figure 2. 95% ci for black ma measured on the mars. based in the data presented in figure 2, the mars scores for black students can be best described as relatively similar in magnitude and precision. all of the confidence bands are relativity wide and all of the bands overlap at least partially with the other studies. these visuals indicate that across the observed studies there are not statistically significant differences between the mean point estimates based on the degree of overlap between the studies (cumming, 2009). it is also worth noting that the mars and mars-a scores substantially overlap, indicating that the scores are fairly similar, which is expected given the similar scaling and similar participant characteristics. the position of the mean point estimates and the overlap between studies suggest that the mean point estimates for the black population parameter is somewhere between 200 and 220. a more conservative estimate is presented given the limited number of studies and the aggregation of the mars and mars-a scores. an appropriate interpretation of these intervals is necessary. a young & young black mathematics anxiety journal of urban mathematics education vol. 9, no. 1 89 95% ci does not indicate that a point estimate correctly represents the population parameter with 95% certainty, but rather that if an infinite number of cis are constructed, than one can be 95% certain that the population parameter is present. discussion these analyses suggest that black participants and their scores are not well represented in the current ma research utilizing the most popular instrument the mars. although the mars was developed in the early 1970s, the first instance of an administration that reported the inclusion of black participants was not until the mid 1990s. this time lapse is approximately 20 years after the development and validation of the instrument. in addition, the majority of the studies presented in the current synthesis did not include reliability scores for researcher interpretation. this review located 21 studies that included black participants, but only 6 studies included mean mars scores for black students most of which were dissertation studies. furthermore, the mars scores for black students were disaggregated and presented in only one study with heterogeneous participants, despite representative samples of black students in many of the studies. this lack of data disaggregation inhibits meta-analytic thinking as it pertains to the ma challenges of black students. for instance, if a researcher seeks to conduct a power analysis before soliciting participants for a ma intervention in an urban school district serving a large black population, it is nearly impossible to accurately generate a representative estimate of the number of black students necessary to have adequate power against type i error. one could argue that the researcher could just use the information from the general population, but given the scarcity of resources and opportunities to affect change in urban schools this is not a process that is worth leaving to chance. finally, based on normative ma measurements the data analyzed here suggest that ma may be slightly higher in the black population, despite previous research that suggest otherwise. according to suinn and edwards (1982) normative scores for ma should range from 197.6 to 204.7. the differences between these ranges of scores are slightly higher for the black student data analyzed. however, access to mars point estimates from representative homogeneous samples of black participants remains a measurement limitation. conclusion ma is thought to influence learning and mastery of mathematics from an early age, but its precise developmental origins is unknown (rubinsten & tannock, 2010). given that ma is one of the many elements associated with the affective domain of mathematics learning and has a well-documented detrimental effect on young & young black mathematics anxiety journal of urban mathematics education vol. 9, no. 1 90 performance (ashcraft, 2002; hadley & dorward, 2011; harari, vukovic, & bailey, 2013), it is imperative that we find solutions to reduce its effects. thus, researchers, teachers, and parents must approach ma in black children armed with accurate and representative estimates of this construct. the results of this study suggest that, although substantial research in the area of ma exists, much more work is needed to fully comprehend the nuances of black ma and its influence on achievement in mathematics. more non-comparative or within-group analysis is necessary to develop more representative estimates of ma in black children. much of the available research uses between-group designs involving black and white students or male and female students. one major limitation of racialor gender-comparative designs, however, is that when group differences are found, investigators can only speculate about the causes of those differences (dotterer, lowe, & mchale, 2014). these activities perpetuate the trend of gap gazing and fail to yield information that is practically significant for classroom use. in addition to the need for more within-group analysis, researchers must begin to disaggregate student data by race and gender and to report individual group descriptive statistics (capraro, young, lewis, yetkiner, & woods, 2009). the lack of reporting of sufficient statistics to calculate effect sizes for all students is a major impediment to generalizable practices for students of color, and it impedes the application of metaanalytic thinking for the purposes of supporting the mathematics teaching and learning of large populations of students in urban schools. in order to move away from the era of gap gazing, it is imperative that we as researchers and educators provide representative reports on all of the students in the participant pool. in the end, the consistent disaggregation of mathematics data by race remains elusive, and subsequently, so do 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(2003). the mathematics anxiety rating scale, a brief version: psychometric data. psychological reports, 92(1), 167–173. thompson, b. (2002). what the future quantitative social science research could look like: confidence intervals for effect sizes. educational researcher, 31(25), 25–32. wenger, e. (1998). communities of practice: learning, meaning, and identity. cambridge, united kingdom: cambridge university press. yeop kim, d., zabel, j., stiefel, l., & schwartz, a. e. (2006). school efficiency and student subgroups: is a good school good for everyone? peabody journal of education, 81(4), 95–117. zientek, l. r., yetkiner, z. e., & thompson, b. (2010). characterizing the mathematics anxiety literature using confidence intervals as a literature review mechanism. the journal of educational research, 103(6), 424–438. journal of urban mathematics education july 2015, vol. 8, no. 1, pp. 1–9 ©jume. http://education.gsu.edu/jume robert q. berry, iii is an associate professor at the university of virginia in the curry school of education, 405 emmet street, charlottesville, va, 22904; e-mail: robertberry@virginia.edu. his scholarship focuses on equity issues in mathematics education, qualitative metasynthesis as a methodological approach for evidence-based practices, and mathematics instructional quality. mark w. ellis is a professor of secondary education at california state university, fullerton, 2600 e. nutwood ave., suite 600, fullerton, ca, 92831; e-mail: mellis@fullerton.edu. a national board certified teacher of mathematics, his scholarship focuses on equity issues in mathematics education and professional development for teachers of mathematics aimed at creating learning environments that promote success for all students. crystal h. morton is an associate professor of mathematics education in the school of education at indiana university–purdue university, 902 west new york street, indianapolis, in, 46202; e-mail: cranhill@iupui.edu. her scholarship focuses on gaining a greater understanding of african american students’ mathematics knowledge and learning through rigorous and relevant curricular innovations. jan a. yow is an associate professor of mathematics education in the college of education at university of south carolina, 820 s. main street, columbia, sc, 29208; e-mail: jyow@sc.edu. as a national board certified teacher, her scholarship focuses on the development of mathematics teacher leaders who can provide high quality mathematics instruction for all students. in memoriam “i am a teacher. that’s what i’ve done almost all my life. i teach.” dr. carol e. malloy (june 6, 1943–january 17, 2015) robert q. berry, iii university of virginia mark w. ellis california state university, fullerton crystal h. morton indiana university–purdue university jan a. yow university of south carolina hese words, handwritten by a middle-school student, sat framed on dr. carol e. malloy’s desk until her retirement from the university of north carolina at chapel hill (unc) in 2009. the words reminded her of how many students feel and perceive mathematics. dr. malloy stated, “all students have the ability to learn if given the opportunity. …that student did not have the opportunity” (hobbs, 2009, ¶3). opportunities to learn, access, and equity are themes found throughout t http://education.gsu.edu/jume mailto:robertberry@virginia.edu mailto:mellis@fullerton.edu mailto:cranhill@iupui.edu mailto:jyow@sc.edu berry et al. in memoriam journal of urban mathematics education vol. 8, no. 1 2 dr. malloy’s work. in an interview upon her retirement, she reflected on the field of mathematics education, stating: the issues related to opportunity have come to the surface and are being discussed, though we haven’t quite figured out how to make universal changes in opportunity and access. the mathematics education community is broader and more diverse now, which i think is extremely important. (hobbs, 2009, ¶10) dr. malloy’s contribution is one of the primary reasons why “the mathematics education community is broader and more diverse.” dr. carol e. malloy was a mathematics teacher, teacher educator, and mentor whose career spanned more than four decades. reflecting on her work gives us the opportunity not only to speak to the tremendous impact dr. malloy has had on the field, but also to share some personal memories of the impact she had on those of us in the field who were privileged to spend time with her as mentees, colleagues, teachers, and friends. dr. malloy’s vision, courage, and commitment made issues of equity in mathematics education visible to and relevant for all. while there is still much work to be done, her work has established that equity is not peripheral to efforts to improve mathematics teaching and learning but rather at the heart of this work. it has been said that dr. malloy communicated to her students the importance of not trying to be “the best” but of doing “their best” (m. malloy, personal communication, january 24, 2015). it is with this admonition in mind that we strive for our best. for dr. malloy, striving to be our best meant having a plan of action and being thoughtful and purposeful in our actions. her son, michael malloy, described a central lesson she taught us all by quoting her, “if you do not have a plan to succeed, you have a plan to fail” (hobbs, 2015, ¶6). dr. malloy’s career and accomplishments represent a model plan for success in mathematics education. teaching dr. malloy was first and foremost a teacher of mathematics and was a model for not only how to scaffold deep learning but also how to engage in lifelong learning. she taught more than twenty years as a middle and high school mathematics teacher in four urban school districts in pennsylvania, florida, and wisconsin. it was in her work as a teacher that she established herself as a champion for the brilliance and resiliency of all learners—especially african american learners—to understand and excel in mathematics. she marked her seventh year of teaching high school as the year she became a “real teacher”: i started going to see parents in their homes, writing notes to them and calling when their children did well and also when i needed help with different issues with students. berry et al. in memoriam journal of urban mathematics education vol. 8, no. 1 3 …i engaged in extra-curricular activities with the students, played on the teachers’ basketball team and attended many after-school activities. it changed the whole way i taught. that was the year i became a real teacher. my students and i became a part of the same learning community. (hobbs, 2009, ¶12) as a teacher, dr. malloy prioritized building conceptual understanding by drawing on her knowledge of mathematics and knowledge of students as thinkers. in all of her work, she aimed to demonstrate both the beauty of mathematics and the students’ ability to make sense of mathematics. for example, in her article “perimeter and area through the van hiele model” (malloy, 1999b) she used what appeared a simple geometry task to illustrate significant pedagogical issues related to allowing all students access to meaningful mathematics learning. dr. malloy’s teachings demonstrated that while “teachers motivate and facilitate learning, they must recognize and use students’ characteristics and behaviors of resiliency to encourage intrinsic motivation and to help students become more responsible about learning mathematics” (malloy & malloy, 1998, p. 314). she believed that promoting resilient processes in students is critical to the success of students in learning mathematics. dr. malloy’s influence as a teacher was not limited to her time as a high school teacher. her focus on mathematics was supported in her work for mcgrawhill education in which she authored middle grades and high school mathematics textbooks. in 1997, unc recognized her excellence in teaching by awarding her the favorite faculty award. scholarship dr. malloy’s scholarship and teaching on access and equity is well noted in mathematics education and school reform. her dissertation research is an example of how she integrated mathematics and issues of equity. in this work, she examined the problem-solving characteristics, strategy selection and use, and verification actions of 24 african american eighth-grade students. she stated her dissertation was “motivated by the lack of empirical research available about how african american students solve mathematics problems and by the uneven achievement reports for these students” (malloy, 1995, p. iii). dr. malloy’s dissertation was significant at the time because its focus on african american students as learners of mathematics was unprecedented. this work helped pave the way for researchers with similar interests to position african american learners as the focal point of study rather than taking on an achievement lens focusing on between-group gaps and comparisons. her body of work communicates the implicit message that african american students are worth studying in their own right and comparisons to other groups of learners are not always necessary or instructive when it comes to berry et al. in memoriam journal of urban mathematics education vol. 8, no. 1 4 understanding how to promote success for those served poorly by traditional schooling practices. understanding african american students’ mathematical learning was a central theme through much of dr. malloy’s work. she brought light to the fact that “mathematics educators have little knowledge of how african american students perceive themselves as mathematics students, how they approach mathematics, or the role of culture in their perception and mathematics performance” (malloy, 1997, p. 23). dr. malloy’s pioneering work precipitated a still-growing knowledge base on african american learners of mathematics. during the almost twenty years since her dissertation, there has been a significant increase in research documenting and examining the experiences of african american students in mathematics. much of the increase in the knowledge base is built on the foundations set forth by dr. malloy, including the work of scholars whose dissertations she chaired (berry, 2003; eatmon, 2007; hill, 2008; noble iii, 2009), but her influence is not limited to her own students. mathematics education scholars far and wide have looked to dr. malloy and her work as seminal to the study of african american children in mathematics. one can argue that dr. malloy’s work has inspired and informed a new generation of researchers who have increased the knowledge base about african american learners of mathematics and who reject theories and discourses that suggest african american learners are deficient or inferior to other learners. a review of dr. malloy’s scholarship provides the mathematics education community with a framework for how to include african american students (see, e.g., malloy, 1997; 1999a; 2000; 2004; 2008a; 2008b). she suggested: (a) providing teachers with training to develop positive student-teacher interactions; (b) facilitating positive peer interaction in multiracial settings that promote communication; (c) mentoring of students and social support systems; (d) providing additional learning opportunities through co-curricular activities; (e) collaborating with community-based agencies; (f) offering career exploration, appropriate course selections, and preparation for postsecondary schooling; and (g) providing students with access to high quality mathematics teaching, curricula materials, and opportunities to learn (malloy, 1997). she stated that these “recommendations can be implemented for all students; however, they are particularly important to the mathematically underserved and underrepresented african american student populations” (p. 23). service dr. malloy was a valued, long-term servant-leader in the mathematics education community. she served on the board of directors for the national council of teachers of mathematics (nctm) and as president of the benjamin banneker association (bba). additionally, she served on the writing team of berry et al. in memoriam journal of urban mathematics education vol. 8, no. 1 5 nctm’s (2000) principles and standards for school mathematics and as lead of the revision to the standards for teachers of mathematics for the national board for professional teaching standards (nbpts). she was lead editor for the nctm book series mathematics for every student: responding to diversity (2009) and many others. among dr. malloy’s numerous service awards she received the bba distinguished member award in 2003, the west chester university (pa) distinguished alumna award in 2004, the first annual unc-chapel hill school of education black alumni impact award in 2010, the bba lifetime achievement award in 2013, the nctm lifetime achievement award in 2013, and the unc school of education distinguished leadership award in 2014. as important as service at the national level was to dr. malloy, she also recognized the opportunity and responsibility she had as a faculty member to mentor students at all levels. unc recognized dr. malloy’s outstanding mentoring by awarding her the unc faculty mentoring award in 2009. each of us (i.e., the authors of this memorial tribute) benefitted from her willingness to take an interest in not only our academic growth but also our personal growth and well-being. we each have memories of times dr. malloy took notice of a change in demeanor and made time to listen to what might be going on in our lives. for her, mentoring meant personal engagement with the whole person. in an interview upon her retirement in 2009, dr. malloy glanced across a multitude of photos of former students that fill her office and stated: “this is my life. i’m so fortunate to have had these relationships. i look at these kids and think to myself, ‘oh, my goodness. they are a legacy that would make anyone extremely proud. they’re wonderful!’” (hobbs, 2009, ¶19). lessons learned the field of mathematics education is better because of dr. malloy’s tireless dedication to ensure it would be—not for professional gain, but because of a personal love for people and a drive to do what is right for the students who most need access to opportunities and support to succeed with mathematics. those of us who humbly follow in her legacy must keep in mind that still too many students “don’t like anything that deals with math.” we must therefore act upon our responsibility to further efforts to expand quality mathematics education for all students. it is our intent and our hope that each of us do right by our mentor and remain focused on what matters most in our work as mathematics educators— understanding and respecting the students we serve and those who are served by the teachers with whom we have the privilege to interact. for more than four decades, the mathematics education community has felt dr. malloy’s strong presence as a role model and an exemplar of the power of sincerity, grace, persistence, and action. she leaves a legacy of excellence. included berry et al. in memoriam journal of urban mathematics education vol. 8, no. 1 6 in the appendices are abbreviated examples of this legacy with a listing of some of her publications (appendix a) and doctoral dissertations she chaired (appendix b). references berry, r. q., iii. (2003). voices of african-american male students: a portrait of successful middle school mathematics students (doctoral dissertation). retrieved from proquest dissertations & theses full text: the humanities and social sciences collection. (order no. 3086494). eatmon, d. (2007). understanding the mathematics success of african-american students at a residential high school (doctoral dissertation). retrieved from proquest dissertations & theses full text: the humanities and social sciences collection. (order no. 3272782). hill, c. a. (2008). making the invisible visible: an examination of african american students' strategy use during mathematical problem solving (doctoral dissertation). retrieved from proquest dissertations & theses full text: the humanities and social sciences collection. (order no. 3304277). hobbs, m. (2009, april 7). mathematics educator carol malloy to retire this summer. retrieved from http://soe.unc.edu/news_events/faculty_news/2009/090417_malloy.php hobbs, m. (2015, january 27). carol malloy remembered as accomplished teacher, beloved mentor. retrieved from http://soe.unc.edu/news_events/faculty_news/2015/012015-malloy-obit.php malloy, c. e. (1995). african-american eighth grade students' mathematics problem solving: characteristics, strategies, and success (doctoral dissertation). university of north carolina at chapel hill, chapel hill, nc. malloy, c. e. (1997). including african american students in the mathematics community. in j. trentacosta & m. kenney (eds.), multicultural and gender equity in the mathematics classroom (pp. 23–33). reston, va: national council of teachers of mathematics. malloy, c. e. (1999a). developing mathematical reasoning in the middle grades: recognizing diversity. in l. stiff & f. curcio (eds.), developing mathematical reasoning in grades k–12 (pp. 13–21). reston, va: national council of teachers of mathematics. malloy, c. e. (1999b). perimeter and area through the van hiele model. mathematics teaching in the middle school, 5(2), 87–90. malloy, c. e. (2000). the kids got it and the teachers smiled: a charter fulfills its vision. the high school journal, 83(4), 19–26. malloy, c. e. (2004). equity in mathematics education is about access. in r. rubenstein & g. bright (eds.), perspectives on the teaching of mathematics (pp. 1–14). reston, va: national council of teachers of mathematics. malloy, c. e. (2008a). looking throughout the world for democratic access to mathematics. in. l. d. english (ed.), handbook of international research in mathematics education (2nd ed., pp. 20–31). new york, ny: routledge. malloy, c. e. (2008b, march). a historical perspective on the preparation of mathematics teachers in the areas of student diversity and the education of disadvantaged students. paper presented at symposium on the occasion of the 100th anniversary of icmi: rome, italy. retrieved from https://www.unige.ch/math/ensmath/rome2008/wg2/papers/malloy.pdf malloy, c. e. (ed.). (2009). mathematics for every student: responding to diversity. reston, va: national council of teachers of mathematics. malloy, c. e., & malloy, w. w. (1998). resiliency and algebra i: a promising non-traditional approach to teaching low-achieving students. the clearing house, 71(5), 314–317. national council of teachers of mathematics. (2000). principles and standards for school mathematics. reston, va: national council of teachers of mathematics. noble, r., iii. (2009). the impact of self-efficacy on the mathematics achievement of african american males in postsecondary education (doctoral dissertation). retrieved from proquest dissertations & theses full text: the humanities and social sciences collection. (order no. 3352993) http://soe.unc.edu/news_events/faculty_news/2009/090417_malloy.php http://soe.unc.edu/news_events/faculty_news/2015/012015-malloy-obit.php https://www.unige.ch/math/ensmath/rome2008/wg2/papers/malloy.pdf berry et al. in memoriam journal of urban mathematics education vol. 8, no. 1 7 appendix a bibliography for dr. carol e. malloy (listed chronologically) malloy, c. e. (1995). african-american eighth grade students’ mathematics problem solving: characteristics, strategies, and success (doctoral dissertation, university of north carolina at chapel hill). malloy, c. e. (1997). including african american students in the mathematics community. in j. trentacosta & m. kenney (eds.), multicultural and gender equity in the mathematics classroom (pp. 23–33). reston, va: national council of teachers of mathematics. malloy, c. e. (1997). mathematics projects promote students’ algebraic thinking. mathematics teaching in the middle school, 2(4), 282–288. malloy, c. e., & brader-araje, l. (eds.). (1998). challenges in the mathematics education of african american children: proceedings of the benjamin banneker association leadership conference. reston, va: national council of teachers of mathematics. malloy, c. e., & jones, m. g. (1998). an investigation of african american students’ mathematical problem solving. journal for research in mathematics education, 29(2), 143–163. malloy, c. e., & malloy, w. w. (1998). issues of culture in mathematics teaching and learning. the urban review, 30(3), 245–257. malloy, c. e., & malloy, w. w. (1998). resiliency and algebra i: a promising non-traditional approach to teaching low-achieving students. the clearing house, 71(5), 314–317. malloy, c. e. (1999). book review of standing outside on the inside: black adolescents and the construction of academic identity. multicultural perspectives, 1(1), 43. malloy, c. e. (1999). developing mathematical reasoning in the middle grades: recognizing diversity. in l. stiff & f. curcio (eds.), developing mathematical reasoning in grades k–12 (pp. 13–21). reston, va: national council of teachers of mathematics. malloy, c. e. (1999). perimeter and area through the van hiele model. mathematics teaching in the middle school, 5(2), 87–90. pugalee, d. k., & malloy, c. e. (1999). teachers’ action in community problem solving. mathematics teaching in the middle school, 4(5), 296–300. malloy, c. e. (2000). a new look at geometry taught in the middle grades. new england mathematics journal, 32(2), 78–87. malloy, c. e. (2000). the kids got it and the teachers smiled: a charter fulfills its vision. the high school journal, 83(4), 19–26. malloy, c. e., & guild, d. b. (2000). problem solving in the middle grades. mathematics teaching in the middle school, 6(2), 105–108. noblit, g. w., malloy, w. w., & malloy, c. e. (2001). the kids got smarter: case studies of successful comer schools. understanding education and policy. cresskill, nj: hampton press. bright, g., jordan, p., malloy, c., & watanabe, t. (2002). navigating through measurement in grades 6–8. reston, va: national council of teachers of mathematics. malloy, c. (2002). democratic access to mathematics through democratic education: an introduction. in l. d. english (ed.), handbook of international research in mathematics education (pp. 17– 26). mahwah, nj: erlbaum. malloy, c. e., & jones, m. g. (2002). an investigation of african american students’ mathematical problem solving. in j. sowder & b. schapplle (eds.), lessons learned from research (pp. 191– 196). reston, va: national council of teachers of mathematics. pugalee, d. k., frykholm, j., johnson, a., slovin, h., malloy, c., & preston, r. (2002). navigating through geometry in grades 6–8. reston, va: national council of teachers of mathematics. berry et al. in memoriam journal of urban mathematics education vol. 8, no. 1 8 malloy, c. e. (2003). teaching and learning geometry through student ownership. new england mathematics journal, 35(2), 16–27. malloy, c. e. (2004). equity in mathematics education is about access. in r. rubenstein & g. bright (eds.), perspectives on the teaching of mathematics (pp. 1–14). reston, va: national council of teachers of mathematics. ellis, m., & malloy, c. e. (2007). preparing teachers for democratic mathematics education. in d. pugalee, a. rogerson, & a. schinck (eds.), proceedings of the 9th international conference: mathematics education in a global community (pp. 160–164). charlotte, nc: mathematics in the 21st century. retrieved from http://math.unipa.it/~grim/21_project/21_charlotte_ellis%20and%20malloypaperedit.pdf ellis, m. w., malloy, c. e., meece, j. l., & sylvester, p. r. (2007). convergence of observer ratings and student perceptions of reform practices in sixth-grade mathematics classrooms. learning environments research, 10(1), 1–15. malloy, c. e. (2008). looking throughout the world for democratic access to mathematics. in. l. d. english (ed.), handbook of international research in mathematics education (2nd ed., pp. 20– 31). new york, ny: routledge. malloy, c. e. (2008, march). a historical perspective on the preparation of mathematics teachers in the areas of student diversity and the education of disadvantaged students. paper presented at symposium on the occasion of the 100th anniversary of icmi: rome, italy. retrieved from https://www.unige.ch/math/ensmath/rome2008/wg2/papers/malloy.pdf malloy, w. w., malloy, c. e., & noblit, g. w. (2008). bringing systemic reform to life: school district reform and comer schools. cresskill, nj: hampton press. malloy, c. e. (2009). instructional strategies and dispositions of teachers who help african american students gain conceptual understanding. in d. b. martin (ed.), mathematics teaching, learning, and liberation in the lives of black children (pp. 88–122). new york, ny: routledge. malloy, c. e. (ed.). (2009). mathematics for every student: responding to diversity. reston, va: national council of teachers of mathematics. malloy, c. e., & noble, r., iii. (2009). the education of african american children in charter schools. in l. c. tillman (ed.), the sage handbook of african american education (pp. 367–382). thousand oaks, ca: sage. strutchens, m., bay-williams, j., civil, m., chval, k., malloy, c. e., white, d. y., & berry, r. q., iii. (2012). foregrounding equity in mathematics teacher education. journal of mathematics teacher education, 15(1), 1–7. http://math.unipa.it/~grim/21_project/21_charlotte_ellis%20and%20malloypaperedit.pdf https://www.unige.ch/math/ensmath/rome2008/wg2/papers/malloy.pdf berry et al. in memoriam journal of urban mathematics education vol. 8, no. 1 9 appendix b dissertations chaired by dr. carol e. malloy (listed chronologically) sliva, j. a. (1998). factors that relate to middle grades mathematics teachers’ attitudes toward mainstreaming (doctoral dissertation). retrieved from proquest dissertations & theses full text: the humanities and social sciences collection. (order no. 9840991). jeffries, m. j. (2000). in their own words: african-american educators’ perceptions of the sustaining characteristics associated with segregated schools (doctoral dissertation). retrieved from proquest dissertations & theses full text: the humanities and social sciences collection. (order no. 9968609) berry, r. q., iii. (2003). voices of african-american male students: a portrait of successful middle school mathematics students (doctoral dissertation). retrieved from proquest dissertations & theses full text: the humanities and social sciences collection. (order no. 3086494). edmunds, j. a. (2004). “just good teaching”: viewing effective teachers’ use of technology with lowperforming students through multiple lenses (doctoral dissertation). retrieved from proquest dissertations & theses full text: the humanities and social sciences collection. (order no. 3129703). bryan, w. r. (2005). effects of a study abroad experience on student views of whiteness (doctoral dissertation). retrieved from proquest dissertations & theses full text: the humanities and social sciences collection. (order no. 3170405). ellis, m. w. (2005). school mathematics practices and the games of truth that are school mathematics (doctoral dissertation). retrieved from proquest dissertations & theses full text: the humanities and social sciences collection. (order no. 3190242). eatmon, d. (2007). understanding the mathematics success of african-american students at a residential high school (doctoral dissertation). retrieved from proquest dissertations & theses full text: the humanities and social sciences collection. (order no. 3272782). gould, t. o. (2007). a longitudinal analysis of the effects of collective bargaining on interstate teacher salary differences from 1960 to 2000 (doctoral dissertation). retrieved from proquest dissertations & theses full text: the humanities and social sciences collection. (order no. 3272552). yow, j. a. (2007). “visible but not noisy”: a continuum of secondary mathematics teachers’ thinking about teacher leadership (doctoral dissertation). retrieved from proquest dissertations & theses full text: the humanities and social sciences collection. (order no. 3257555). hill, c. a. (2008). making the invisible visible: an examination of african american students' strategy use during mathematical problem solving (doctoral dissertation). retrieved from proquest dissertations & theses full text: the humanities and social sciences collection. (order no. 3304277). gordon, e. m. (2009). mathematically successful latina and latino students: stressors and supports (doctoral dissertation). retrieved from proquest dissertations & theses full text: the humanities and social sciences collection. (order no. 3366335). joyner, r. l. (2009). adkin high school and the relationships of segregated education (doctoral dissertation). retrieved from proquest dissertations & theses full text: the humanities and social sciences collection. (order no. 3352892). mcglone, c. w. (2009). a case study of pre-service teachers experiences in a reform geometry course (doctoral dissertation). retrieved from proquest dissertations & theses full text: the humanities and social sciences collection. (order no. 3352934). noble, r., iii. (2009). the impact of self-efficacy on the mathematics achievement of african american males in postsecondary education (doctoral dissertation). retrieved from proquest dissertations & theses full text: the humanities and social sciences collection. (order no. 3352993). microsoft word 413-article text no abstract-2250-1-6-20210130 (proof 1).docx journal of urban mathematics education may 2021, vol. 14, no. 1 (special issue), pp. 45–70 ©jume. https://journals.tdl.org/jume gregory a. downing is an assistant professor of stem education in the department of curriculum and instruction within the school of education at north carolina central university, 700 cecil street, #2090, durham, nc 27707; gdowning@nccu.edu. his research explores equity and diversity issues within stem education, specifically how current teaching and learning practices within the k–16 system (dis/en)able students of color and other marginalized students to/from entering stem careers. whitney n. mccoy is a postdoctoral research associate in stem education on the making engineering real national science foundation grant in the department of curriculum, instruction & special education within the school of education and human development at the university of virginia, 405 emmet st. charlottesville, va 22904; wnm3mx@virginia.edu. her research explores identity development for black girls in educational settings, with particular focus on critical race theory, racial identity development, self-efficacy, and stem education exploring mathematics of the sociopolitical through culturally relevant pedagogy in a college algebra course at a historically black college/university gregory a. downing north carolina central university whitney n. mccoy university of virginia in collegiate mathematics, college algebra continues to be a barrier to graduation for students (specifically non-science, mathematics, engineering, and science majors). each year, nearly half of enrolled students struggle to “pass” this course with a grade of c or better (herriott, 2006). using innovative constructed lessons geared towards african american students, this research study was designed to investigate the effects of a sequence of such lessons grounded in the principles of culturally relevant pedagogy on students enrolled in an introductory college algebra course at a historically black college/university. using critical race theory as a lens, along with culturally relevant pedagogy, this study explored students’ abilities to apply mathematics to address contentious and present-day sociopolitical problems through eight in-depth semi-structure student interviews. further, findings also suggest the need for collegiate mathematics instruction to have more emphasis on cultural components to build students’ sociopolitical consciousnesses, because this is integral in helping students be able to think critically and use mathematics in their everyday lives. students in this experimental course were able to discuss difficult issues, such as the pervasiveness of racism in america (decuir & dixson, 2004) and the importance of cultural identity for african american students (martin, 2009). keywords: college mathematics, critical race theory, culturally relevant pedagogy, mathematics education downing & mccoy exploring mathematics of the sociopolitical through culturally relevant pedagogy journal of urban mathematics education vol. 14, no. 1 (special issue) 46 ationally, nearly 50% of students who are unsuccessful withdraw from servicelevel college algebra courses each year (herriott, 2006). with the growth of diverse students in america and research that is often focused on gap gazing, which gutiérrez (2008) defines as “document[ing] disparities in achievement between middle-class white students and students who are black, latina/latino, first nations, english language learners, or working class” (p. 357), we must research and invest in pedagogical approaches that are specifically geared towards students of color who stand to benefit most. racially minoritized students face additional barriers to achieving in mathematics courses due to deficit perspectives and perceptions on perceived indications of what (and how) they can or cannot learn (harper, 2010). racially minoritized students are often regulated and oppressed under the guise of teachers “doing what is best for their students.” these views also help teachers rationalize that “certain students” are not capable of the critical thinking needed to engage in higherorder tasks; therefore, they must be explicitly told the information they need to learn. under this mindset, the only way to “teach” students is through direct instruction (i.e., lecture), a method that is still the predominant mode of instruction in mathematics; yet, the failure rates for students who receive primarily direct instruction are 55% higher than students who are engaged in a more active learning approach (freeman et al., 2014). deficit perspectives, traditional instruction, and low expectations are factors that help influence whether students take more advanced coursework in high school. when students are not advised to take more advanced courses, they are inadequately prepared for college. this often results in students being placed in college algebra, where failure rates are extremely high. there is a high need to help minoritized students in these service-level mathematics courses to improve outcomes for students (boyce & o’halloran, 2020; president's council of advisors on science and technology, 2012). using culturally based pedagogical frameworks has been shown to improve and combat negative learning experiences for students of color (gutstein et al., 1997; ladson-billings, 1995a, 2014). to explore the effects of such pedagogy (in hopes to improve the outcomes for these minoritized students), this study utilized culturally relevant pedagogy (crp) within a college algebra mathematics course taught at a historically black college/university (hbcu). literature review underrepresented minority groups, including african americans, hispanics and latinos(as/x), and others, make up about 30% the population in the united states (u.s. census bureau, 2017). there has been some research conducted (and currently being conducted) on how best to improve aspects of the education system to impact diverse groups of students and their teachers (ladson-billings, 1995a, 1995b; martin n downing & mccoy exploring mathematics of the sociopolitical through culturally relevant pedagogy journal of urban mathematics education vol. 14, no. 1 (special issue) 47 et al., 2017). however, much of this research has been conducted in grades k–12, with less focus at the college level. powell and frankenstein (1997) curated the anthology ethnomathematics: challenging eurocentrism in mathematics education in order to challenge the following statement on the notion of white dominance in the field of mathematics: in the eurocentric account, europe (and “europeanized” areas like the u.s.a.) has always been and currently is the superior center from which knowledge, creativity, technology, culture, and so forth flow to the inferior periphery, the so-called underdeveloped countries. (p. 1) this hegemonically developed notion of how mathematics should be learned, taught, and who created mathematics is problematic on multiple levels. thus, researchers are working to fight to end these and other similar notions that mathematics is naturally, and by definition, “neutral” with regards to culture and the teaching of it (nasir et al., 2008). mathematics is not culturally neutral. thus, it is worthwhile investigating how an approach to teaching mathematics that embraces the culture of students affects students’ engagement with and performance in mathematics. social justice pedagogy is an equity-oriented practice that is conceptualized by what is and what is not fair and just with regards to the relationships between the individual and society (baily & katradis, 2016). this is measured by the explicit and implicit ideas of wealth distribution, opportunity, and social privileges based on one’s social status in society. social justice is often referred to as something researchers and educators move towards, never quite reaching it, for “[i]t is in seeking to understand the ways in which we simultaneously accept/unaccept the other that we move closer to becoming agents of social justice” (aguilar et al., 2016, p. 252). practitioners and researchers interested in social justice have found ways to address inequitable learning opportunities. some have done so by using crp, which integrates the cultural identity of the students involved and academic rigor of the content being taught (ladson-billings, 1995a). theoretical framework by utilizing the tenant of sociopolitical consciousness (crp) and challenging the notions of colorblindness, dominance, and the hegemonic nature of mathematics (critical race theory [crt]), we are able to address anti-racism and social justice issues (sjp) through mathematics education. thus, researchers are moving past sociocultural views in attempts to "espouse sociopolitical concepts and theories, highlighting identity and power at play” (gutiérrez, 2013, p. 37). culturally relevant pedagogy. crp has teaching mathematics for social justice at its heart. it looks beyond notions of functional mathematical literacy and concerns downing & mccoy exploring mathematics of the sociopolitical through culturally relevant pedagogy journal of urban mathematics education vol. 14, no. 1 (special issue) 48 itself with gaining mathematical knowledge and skills that are necessary for participating in society as it is now and pushes towards critical mathematical literacy. this emphasizes the development of knowledge, practices, and discourses for transformative purposes (tan et al., 2012). these transformative purposes come from freire (1968/1970), who reminds us that education should provide opportunities to understand, challenge, and re-create preconceived understandings of the self and the world. with crp, the goals are clear: to afford students who have been historically marginalized the opportunity to achieve. the effect of teaching with students’ culture has been shown to have a substantial increase in self-confidence and self-efficacy (enyedy & mukhopadhyay, 2007; hubert, 2014), effectively replacing feelings of failure and alienation that is all too common with the subject of mathematics and students of color (tate, 1995). crp is founded upon three tenets: academic achievement, cultural competence, and sociopolitical consciousness. academic achievement concerns itself with student learning, focusing on what “students actually know and are able to do as a result of pedagogical interactions with skilled teachers” (ladson-billings, 2006, p. 34). cultural competence involves aiding and empowering students to recognize and honor their own cultural beliefs and practices while also acquiring access to wider cultures. sociopolitical consciousness focuses on students becoming more conscious and aware of sociopolitical issues not only on a national or a global level, but perhaps even more so on a local level. the focus of this research paper and subsequent literature review will be on the third tenet (sociopolitical consciousness). sociopolitical consciousness. possibly the least addressed of the tenets of culturally relevant pedagogy is sociopolitical (or critical) consciousness. this section will show how researchers attempt to unveil the policies and practices that are happening in the world outside (and sometimes inside) the classroom. william tate (1995) stated that this type of pedagogy is beneficial because it enables students to pose their own questions as they relate to their communities and how they feel they are negatively impacted. therefore, students are in charge of their own learning, and crp allows students “to see the world from the perspective of others” (tate, 1995, p. 170). students can incorporate the problems facing many african american communities to make mathematical learning more relevant to themselves as shown in this dimension. power through the sociopolitical. gutiérrez (2013) charges us to make transparent the realities that are plaguing us so that we can empower our students to rise beyond the current dynamics that are at play in mathematics education and in our society. teachers must continually be educated and attend professional development so that they can see their mathematics classrooms as a smaller part within a large social and political history. they need support to challenge discourses that seek to instill inequality though the use of high-stakes standardized test scores as the only downing & mccoy exploring mathematics of the sociopolitical through culturally relevant pedagogy journal of urban mathematics education vol. 14, no. 1 (special issue) 49 measure for learning. this involves developing more than just the “[pedagogical content] knowledge of mathematics, pedagogy, and learners, but also the political knowledge and experiences necessary to negotiate the system and develop working networks with other educators who share their emancipatory visions" (gutiérrez, 2013, p. 62). during a series of professional development sessions in the bahamas, teachers wanted to enhance their abilities to "(a) foster critical mathematical and critical consciousness, (b) build on informal mathematical and cultural knowledge; and (c) utilize empowerment orientations toward students’ culture" (matthews, 2003, p. 61). the institute hoped to marry crp with a heavier emphasis of social justice pedagogy (gutstein et al., 1997) to reflect on the use of mathematics in schools and the world, explore enhancing algebraic mathematics instruction, and encourage using students' cultures as a tool in mathematics class. some teachers were more successful than others. the teachers who were successful relied on the relationships built between their students and themselves. teachers also built on students’ mathematical and cultural knowledge simultaneously through rich mathematics explorations. through these explorations, students’ critical-thinking capabilities were expanded though mathematics, and connections to the government and other societal entities were investigated as context within their mathematics courses (matthews, 2003). social justice pedagogy in action some teachers have found success in infusing project-based curriculum with crp (gutstein, 2003; gutstein et al., 1997; westheimer & kahne, 1998). lynn (1999) interviewed african american teachers of predominantly african american students about their successful practices with this demographic. these teachers were sought out due to their liberative-styled instruction. with most being elementary teachers, they commented on the pervasiveness of racism in america (decuir & dixson, 2004), the importance of cultural identity for african american students, and the intersection of class and race (crenshaw, 1989, 1991). aligning with crp, this style of teaching (liberatory), according to lynn (1999), consists of the following: a) teaching children about the importance of african culture [cultural competence], b) encouraging and supporting dialogue in the classroom [constructivist methods], c) engaging in daily self-affirmation exercises with students [cultural competence], and d) actively and consistently resisting and challenging authorities who advocate practices that are hegemonic and counter-emancipatory [sociopolitical consciousness]. (p. 619) downing & mccoy exploring mathematics of the sociopolitical through culturally relevant pedagogy journal of urban mathematics education vol. 14, no. 1 (special issue) 50 teachers can choose to see the connections between mathematics teaching and social justice in the form of activism (gutstein et al., 1997). this study uncovered that teaching for social justice and using crp is not removed from teaching standards. in fact, they overlap. both social justice and teaching standards understand and advocate for teachers using students' cultural knowledge as platforms for instructional activities and require a place for critical thinking. there is an important distinction that exists between thinking critically in mathematics (necessary for mathematical aptitude) and viewing knowledge critically in general (necessary for effective social change; gutstein, 2003; gutstein et al., 1997). teachers should continue to make connections between families in order to create and foster a sense of community in which students will see their culture in their work in order to build "a curriculum of empowerment and as a way to promote cultural excellence" (gutstein et al., 1997, p. 733). crt: challenging colorblind mathematics perspectives increasing mathematical knowledge and ability is certainly a goal of this body of research, and so is affording historically marginalized students the opportunity to succeed. if students see no links between the subject they are trying to learn and their lives and future goals, then they will not respond to it. sticking to the status quo and continuing not to teach mathematics with a cultural perspective will continue to exclude people from cultures from outside the dominant majority because they just simply will not be interested in it (nasir et al., 2008). as herzig (2005) states, “some individuals may reject mathematics not out of a sense of choice but because they feel that mathematics has rejected them” (p. 253). thus, the transition to the inclusion of these practices can be enhanced through listening and learning from students and other adults who are invested in making the students’ learning environments productive and conducive to developing students’ identities of themselves and the world. the driving force behind much of crp literature in this literature review is summed up by martin (1997): “[e]xposing the links between mathematics and social [awareness] should not be seen as a threat to ‘[academic] mathematics’ but rather as a threat to the groups that reap without scrutiny the greatest material and ideological benefits from an allegedly value-free mathematics” (p. 169). offering students proven and valuable methods to a new view on mathematics should not come as a threat to anyone, for enriching the mathematical educational experience for students of varying backgrounds that do not reflect that of what is shown on posters and in textbooks through the use of culturally relevant teaching has positive effects on those students (ascher, 1991). colorblind ideology allows people to look beyond race and reinforce the mindset that everyone is the same, thereby invalidating racial identity and cultural experiences. as a challenge to colorblind ideology, crt allows us to utilize its downing & mccoy exploring mathematics of the sociopolitical through culturally relevant pedagogy journal of urban mathematics education vol. 14, no. 1 (special issue) 51 undergirding tenets to critique the colorblind perspectives when teaching mathematics at the collegiate level. crt addresses race, racism, and power structures in the united states. for this study in particular, the permanence of racism and whiteness as property are the focus. crt posits two notions: (a) the permanence of racism, which states that race is permanent and a constant that controls social, political, and economic mobility in our society, and (b) whiteness as property, which states that “the law’s construction of whiteness defined and affirmed critical aspects of identity; of privilege; and of property,” making it easy to exclude african americans (harris, 1993, p. 1725; see also delgado and stefancic, 2017). these factors allow for limited opportunities and numerous educational injustices, especially in mathematics. mathematics as an educational content area is not racial, but historically the instruction of mathematics is embedded in racism (joseph et al., 2017). crt allows us to challenge conversations regarding mathematics, and african american students are centered around the underperformance and access these students have. in particular, they have disparate access to advanced courses and certified teachers at early ages, which disenfranchises these students mathematically, thus maintaining a culture of white supremacy (bullock, 2017). in collegiate settings, professors are not typically hired based on their pedagogical skill sets but rather on their research ability, thus contributing to poor achievement, poor instruction, and negative ideals towards mathematics amongst african american students and privileging the white experience (joseph et al., 2017). additionally, through mathematical knowledge (specifically statistics), we can see how racism contributes to inequity amongst african americans in our society. crt posits that the idea of property allows americans to gain power; in education, intellectual property such as access to certain technologies, certified teachers, and advanced courses are in fact limited resources that every student does not have access to (bullock, 2017). quality education is given to those who have access and is situated around specific nuances of housing and socioeconomic status (bullock, 2017; joseph et al., 2017). for example, diversity in teaching mathematics may differ at a private school with board certified teachers compared to a public school where new teachers are hired to instruct mathematics (bullock, 2017). in collegiate settings, some of these same disparities exist when it comes to class size, access to teacher assistants or tutors, and other resources that can assist with achievement. deep engagement in culturally relevant mathematic pedagogy not only allows students to develop social awareness and find value in mathematics, it also challenges the current narrative that mathematics is white property and provides a way for african american students to connect their own experiences to instruction. downing & mccoy exploring mathematics of the sociopolitical through culturally relevant pedagogy journal of urban mathematics education vol. 14, no. 1 (special issue) 52 methods the purpose of this study was to investigate the effects of a series of lessons grounded in crp on students in a college algebra course. to this end, the research question that was explored was the following: how are college algebra students able to critique discourses of power using mathematics as a tool? study context this study took place at a large hbcu located in the southeastern united states during the fall 2018 semester. students were enrolled in one of two sections of college algebra i at bernard st. stephen state university (bsssu [pseudonym]). nestled amongst an urban black neighborhood that is recognized by the national register of historic places, bsssu resides in the middle of an urban community with deep roots focused on african american heritage, history, development, and educational advancement that contributed to the growth of the hbcu. course context this three-credit hour course, college algebra and trigonometry i, has been historically taught using traditional approaches and is vastly procedural. topics include linear, quadratic, higher order, exponential, and logistic functions. most students will end up taking only one additional mathematics course to satisfy general education requirements at bsssu, either college algebra and trigonometry ii (for majors that are more science, technology, engineering, and mathematics-focused) or elementary statistics (non-science, technology, engineering, and mathematics focused majors). there is a non-credit hour course that students can place into (introductory college algebra), but most students begin in college algebra and trigonometry i. participants participants included 25 students—overwhelmingly, many of the students were african american (96%) and female (76%). a total of eight students agreed to participate in semi-structured student interviews (table 1). the student sample that was selected to participate self-identified as either all african american (six women and one man) or of mixed heritage (one woman, half-african american and half-white). all of these students were first-year students, and one was a non-traditional student (who was not admitted to bsssu directly from a high school). the age of these participants was between 18 and 19, except for the non-traditional student, who was 36 at the time of the study. downing & mccoy exploring mathematics of the sociopolitical through culturally relevant pedagogy journal of urban mathematics education vol. 14, no. 1 (special issue) 53 table 1 interview participant information name (pseudonym) race gender age year major final grade tiffany black female 18 first year nursing a asia-la'rae black female 18 first year nursing a pamela black female 18 first year elementary education a matthew black male 18 first year business administration a ahshante black female 18 first year social work b lindsey black/white female 18 first year nursing b jamie black female 18 first year business administration b kanisha black female 36 junior business administration c intervention four lessons were designed and developed to be grounded in crp and for the specific population of students—primarily traditional college students attending an hbcu in the southeastern region of the united states (see table 2). these lessons were vetted for their alignment with crp using matthews, jones, and parker’s (2013) culturally relevant cognitively demanding task rubric. these lessons were implemented during the final three (of five) units in the college algebra course at bsssu. beginning with a problem statement that would guide the focus of the lessons, students had various opportunities to explore the statement through mathematical exploration using data sets, mathematical manipulations, and information from the internet. students were also encouraged to draw from their own prior knowledge and past experiences of various topics to help make meaning. downing & mccoy exploring mathematics of the sociopolitical through culturally relevant pedagogy journal of urban mathematics education vol. 14, no. 1 (special issue) 54 table 2 table of lessons mathematical content guiding question/ problem statement linear functions what does incarceration look like in this county and the us? does race play a role? quadratic functions how are sexually transmitted diseases and cuffing season related to each other? exponential functions how do poor people bank? what does college loans/debt look like at hbcus? exponential and logarithmic functions how are demographics in the us changing? data collection and analysis semi-structured interviews were a crucial data collection tool in this study to explore students’ views and perceptions gleaned from participants in the course. this semistructured interview used broad, open-ended questions that allowed for flexibility to let the respondents open up about what they were thinking yet also allowed consistency amongst participants (merriam & tisdell, 2016). these interviews were separated into two main sections. the first part contained questions related to the course and how students thought and reacted to the content, more specifically, the delivery and interactions with and about the content. the second part of the interview was focused on students’ abilities to think about real-world situations and come up with opinions. the goal with these situations were to see how students could use mathematics to help them conceptualize the issue and support their argumentation with the mathematics students were describing. of these three scenarios, two were focused on the sociopolitical construct of crp. these scenarios were surrounding minimum wage and police shootings and killings of unarmed black americans. the third scenario was around planning for social events. this scenario was used to further investigate students’ mathematical thinking through a culturally relevant situation. voice recordings of the interviews were transcribed verbatim, with indications of any elongated pauses or breaks in communication documented along with turns, which were marked by changes with the speaker at each timestamp (ochs, 1979). transcripts were shared with participants to make sure that what was recorded downing & mccoy exploring mathematics of the sociopolitical through culturally relevant pedagogy journal of urban mathematics education vol. 14, no. 1 (special issue) 55 aligned with what the participants tried to portray. atlas.ti 8 was used to analyze the interview data. data-driven coding (decuir-gunby et al., 2011) took place through line-byline analysis of transcripts. statements were coded through the use of open coding in order to conceptualize what was said by respondents (strauss & corbin, 1990). after open coding, axial coding was conducted to collapse codes into larger categories through constantly comparing codes with each other and developing categories from pools of concepts discovered in the data. constant memoing took place to capture the researcher’s thoughts during this process (charmaz, 2006). findings when pedagogy has a goal of building students’ sociopolitical consciousness, it is charged with equipping students with the knowledge to view social and/or political actions that are happening in their local communities, nationally, and globally through a critical lens (ladson-billings, 1995b, p. 476). the experiences, as reflected by students in this course, revolved around themes related to recognizing (that a problem exists), caring (empathizing with groups of minoritized people), critiquing (established systems and structures), and acting (towards finding solutions). these themes were observed in both parts of the interview. part one included questions about the course, and part two included questions surrounding scenarios involving a topic/issue and the use of mathematics to help explain that scenario. interview part one: questions about the course recognizing. several of the students who were interviewed expressed a feeling of shock when they were asked what they thought about the lessons. the lesson that really shocked the students with some of the raw facts and figures was the lesson on incarceration and linear functions because they were able to see and recognize the pervasiveness of racism that exists in our society. tiffany said, “the incarceration one is like… wow! is that many people in jail!?? like, black people at that! it’s very informational, all that. i didn’t know that before.” connecting what was learned in this class with other things, matthew had the following to say about the topics in the course. matthew: the changing populations in race in america and then the incarceration [were the lessons that addressed my culture in this class]. today, a lot of people see african american, you automatically think bad sometimes. we talked about in these lessons, well… people made comments about how they always going to jail and stuff; drugs, you see a lot of people getting killed, police brutality, and stuff like that. i think we’re already at a disadvantage, so for us to fix that, it would take us all to come together as one and everything. but as to what was talked about in the class, i do believe the black racial population and incarceration were very important things that are important [to know] today for us. downing & mccoy exploring mathematics of the sociopolitical through culturally relevant pedagogy journal of urban mathematics education vol. 14, no. 1 (special issue) 56 matthew acknowledged that some of the topics that were discussed in class involved issues that are salient to him and his cultural community (african american). ahshante also agreed with this sentiment when she was responding to the question of how discussing these topics in a mathematics class made her feel. ahshante: well, we basically grown, so it's not like it was too much because it's something that needed to be talked about and still needs to be talked about and pushed for. if you don't talk about it… you don't want to make a mistake, or you want to be aware. you want to be cautious. so, i think all of them was good. like it is not just about one thing. it's about everything. all the topics should come with this education, and school. that's what we are now. we need to learn about this stuff and a class seems like a good place for it—at least the way we did it, mixed in with the math. discussing these topics for these students was something they felt was needed, because these topics reflected what was currently going on in society and thus were affecting their day-to-day lives. the culturally relevant topics influenced not only their lives, but also people who looked like them and were a part of their same culture. caring. when learning about these sociopolitical topics throughout this course, students often expressed varying degrees of empathy for the populations of people that were being discussed. with this, students began to use the mathematical lessons to create their own intellectual property. after the course was over, and throughout the lessons themselves, students thought heavily about the issues that were being discussed with the mathematics. jamie: [these lessons] made people in the class think not just about like what the equations was or whatever math we were doing but how it could affect us. so, they basically depict what can happen in the future if this continue to happen—happen to us and other people. feelings of wanting to prevent and/or help alleviate some of the issues that were talked about (specifically incarceration and banking choices by poor people) came up in several of the interviews. take lindsey’s response when she was asked about the role culture should play within a mathematics course. she stated the following thoughts deeply rooted in her background and her feelings about incarceration and how it has played a role in her life. lindsey: so, when i'm doing math, now i'm thinking of these real-world situations— like this is actually happening, you know. how can i help prevent that? like, think about the incarceration [lesson]. i knew a lot of people that have gotten locked up for no reason… well, to me for no reason. i know a lot of people that do dirty things and don’t get locked up, and i feel like it’s just because of the color of their skin. yeah, i take the incarceration thing a little far. i just feel really bad for them because race has a lot more to do with them being locked up than their actually petty crime. downing & mccoy exploring mathematics of the sociopolitical through culturally relevant pedagogy journal of urban mathematics education vol. 14, no. 1 (special issue) 57 the empathy that lindsey exhibited was for people of color who themselves were in the incarceration system. she, and others interviewed, really connected with people in the situations that were brought up during these class discussions during the crp lessons. like other students, lindsey was able to see how mathematics was connected to racism within her own community, kanisha took a different approach with the empathy she was exhibiting. it was not directed towards a particular group of people like some of the other statements were above. she wanted people, specifically white americans, to show empathy towards african americans. kanisha took note of how whiteness as property guided what statistics are shared in america. she believes that crp conducted at predominantly white institutions could be a potential conduit to perform such a task to illicit such a response. kanisha: a lot of times, even in the predominantly white schools, a lot of times they show; rather, they wouldn't show about statistics about the white becoming smaller in number [exponential lesson about rising population of minorities in america]. they wouldn't show this. but i think like in their life in general, a lot of white kids don't get the information that they need, so they're only stuck with what they know, but this is reality, you know. this is our reality; the incarceration is reality. i mean i think it would give them a better outlook on us as black people too. there could actually be something wrong with a system that is incarcerating us more often than them, even though they more than outnumber us in overall population. critiquing: incarceration. throughout all the interviews, the most comments were on the crp lesson centered around incarceration. perhaps it was because this was the first lesson that was presented to the students or it was an issue that students were invested in. kanisha had the following to say as it relates to incarceration. kanisha: as black adults and young adults, you know, we need to know certain things especially about incarceration and its role in life… like this is our life! pretty much. that's what they’re doing is putting it as a part of our life. it’s all racism. incarceration is part of my culture too, but not my culture per se, like not directly towards me, but it is a part of black culture. like i said it's how they see us. it's our life. it's worse the prisons are filled with us because of the crime rate or the so-called crime rate that they put us in. yeah, that is definitely a part of our culture. sad but true. show them the graphs and the numbers and they'll rethink somethings—probably not though. using that lesson as a launching pad for her thoughts, kanisha began to think about what students did in class and wondered if looking at the statistics, making this her own intellectual property, would lead to a change in the system of incarceration, crime rates, and arrests made by law enforcement. downing & mccoy exploring mathematics of the sociopolitical through culturally relevant pedagogy journal of urban mathematics education vol. 14, no. 1 (special issue) 58 health the topic of health really got jamie talking and deeply reflecting on her own personal home life with her family and issues dealing with health and insurance. she began her statements discussing the sexually transmitted disease contraction lesson (in relationship to quadratic equations), and this discussion progressed into one about the overall well-being and health of african americans. this then evolved into a discussion about what she believed to be the true heart of the matter: expensive health insurance premiums, connected to the pervasiveness of racism in medicine. jamie: i guess the std patterns. african americans have a hard time telling their doctors that they have something that they can't get rid of. so that's the same thing with just regular illnesses in general, like diabetes, high cholesterol. because we are, we're not too known about going to the hospital and looking after our health because that's [not] the main priority. but we as african americans need to keep ourselves in check whenever it comes to these types of things. a lot of times, we weren’t taught these things. this leads to an even bigger issue of what i would really want to talk about is good health care for just for all different types of races. it's too expensive, and a lot of people can't afford it. if we got data on these types of diseases and compared when people actually come in to get help based on whether they had insurance or not, and plotted it on graphs and showed them the equations, i would hope that would change some things. these discussions held with jamie and kanisha were extremely enlightening in what students were already passionate about, or had previously thought about these issues, and how they connected what was done in these classes on these “experimental” lesson days. acting. discussions from reflecting upon the mathematics that students learned (the content) led them to focus on the issues (the context) and how it affected them. these reflections were often quite deep, and students had been constantly thinking about these sociopolitical contexts since speaking about them during the specific lesson classroom period. lindsey: a lot of my family has been locked up, and i can, i see the difference. like something that the people that raised me [african americans] would do that would get them locked up, but something that if my mom's side of the family [caucasian] did, it probably would get them a slap on the wrist. i like the fact that i can see it from both sides and perspectives, but it's just messed up, and i wish i could just do something about it. kanisha echoed this feeling of “wanting to do something” about the reality of the situation as well. she also wants other african american students to feel the same way because of the reality of this pervasive issue in society. downing & mccoy exploring mathematics of the sociopolitical through culturally relevant pedagogy journal of urban mathematics education vol. 14, no. 1 (special issue) 59 kanisha: the only thing that really made me feel some type of way is, i believe, would have been that incarceration one. just to know like this is real. like just introducing it to the black students because it deals with black people. it makes them, or it should make them, even look like for their future like, “i don't want to be a statistic. i don't want this to be me,” you know. the total prison population went down a little in that time period, but we're still the majority. but it's like: okay, as a black student what can i do? what can i do so i'm not that statistic? what can i do to help my community so we can bring down the number? this idea about finding the next steps or “taking action” is something that was explored in the final part of the interviews with students. they were asked to reflect upon several situations and indicate how they could potentially use mathematics to explore the situation, if they thought it could be explored with mathematics, what mathematics was being used, and how it was being used. interview part two in the final portion of the interviews, students were given three scenarios (two sociopolitical and one culturally relevant) where they were asked first for their opinion about the topic, then second if mathematics could be used to help either explain the situation or bolster their stance on the topic being asked about. the topics included whether the federal minimum wage should be increased from $7.25 to $15, social gathering decisions (attending parties), and the killing of unarmed black americans by police officers. in these discussions, students were able to identify how racism existed based on mathematical situations and utilized the knowledge they gained in class to share how mathematical concepts influenced the decisions they made, as well as cultural and community issues. the analysis will not primarily look at the specific responses that students gave regarding their opinions on the topics or the specific mathematics that students were using; rather, the analysis is focused on how students were using the mathematics that they were employing during their commentary. it is important to note that students were not asked to do any mathematical calculations during this portion of the interview. they were asked if mathematics could be used and to explain how they would use it if they had access to information they thought was necessary to fully explain their thinking and reasoning. minimum wage the first topic students were asked to respond to dealt with the sociopolitical topic surrounding minimum wage. students were prompted with the following statement: “there are some jobs in the university, especially those for students, where some people are making minimum wage at $7.25 per hour. people have been talking downing & mccoy exploring mathematics of the sociopolitical through culturally relevant pedagogy journal of urban mathematics education vol. 14, no. 1 (special issue) 60 about raising the minimum wage to either $10 per hour or even $15 per hour.” after the prompt, students were asked if they believed that the minimum wage should be increased and why they thought this way, followed by a second question in which they were asked the following: “in order to support their argument and position, in what ways could mathematics be used, if at all.” students used mathematics in three main ways to describe their thoughts about minimum wage to justify that a problem actually exists, to compare and contrast situations between groups of people, and to explain how a change in wages affects an entire system. recognizing: justifying a problem exists. the first theme that showed up during analysis was one where students used mathematics to justify that a problem existed by describing and modeling a person’s situation. several students described how they would use iterative calculations, linear equations, and ratios to show how by not increasing minimum wage, people would not have their basic needs met. asia-la’rae believed that the minimum wage should be increased because she felt that as an 18-year-old, single, non-parent college student it is impossible to make a living on $7.25 per hour even with the few bills she has to pay every month. she found it highly improbable that people with families and children could manage to make that living situation work. when asked how she could use math to help explain the reasoning behind her opinions, asia-la’rae described calculating a living wage dependent upon the number of children a person has and comparing someone’s current finances over a month and looking at that to show that a person making minimum wage cannot make enough money to meet their basic needs. students used mathematics to justify that a problem existed and proposed that raising the minimum wage would help alleviate that problem (although asia-la’rae would love to see that number be even higher). while doing the crp lessons, students had to work with data that they used to help explain the existence of a problem just as these students were doing with this sociopolitical topic of setting a minimum wage. caring: comparison of situations. students also gave responses using mathematics that allowed them to compare groups of people based on differing situations. students reflected on their own situation, where they found themselves earning minimum wage, and compared this with the potential to earn above minimum wage. for this use of mathematics, students proposed using equations, such as profit functions, to show comparisons. tiffany believed that the minimum wage should be increased because college students had limited options to work beyond the normal hours of operation on bsssu’s campus and due to the rising costs in tuition and fees. she felt that if students are trying to be responsible and contribute to their own educational expenses, the only way for that to happen is for them to be able to make more money in the downing & mccoy exploring mathematics of the sociopolitical through culturally relevant pedagogy journal of urban mathematics education vol. 14, no. 1 (special issue) 61 time that they are able to. she stated the following as her explanation of her use of mathematics in helping her craft an argument for her supposition. tiffany: so, definitely [i need] information from the internet and like make a graph of like the people that are making minimum wage, like by the end of the year how much are they doing. and they also do like their living expenses their car expenses like every bill and stuff that they had to pay. and then the people that's making above minimum wage, how much they're also making their expenses, and everything like in the same situation like saying living expenses cost and everything is the difference in making $7.25 and making above $7.25. she believed that showing the difference in the yearly net gain for people making minimum wage and those making above minimum wage would itself justify increasing the minimum wage from $7.25 to $15 (or “somewhere around that number”). pamela also used a similar argument; however, her belief was that the minimum wage should not be increased, because, as she puts it, “a mcdonald’s worker should not get paid the same as someone with a college degree.” she proposed showing this difference in yearly net gain or savings to high school students as a warning to show them that getting a college degree would afford you a lifestyle that you would probably want to live. looking at profit functions was a key part of the experimental lesson based on health and flu shot maker/distributor companies. these students particularly used the principles of that lesson and their knowledge of how “life works” to formulate an argument—to different audiences—of the projected outcome of different groups of people based on hourly wages. critiquing: explaining how a change would affect a larger system. finally, students crafted arguments for and against raising the minimum wage using mathematics to justify their responses that revolved around explaining how such a large increase in minimum wage would affect not only the people receiving said increase, but also everyone else working within this environment/system. generally, the students that used this argumentation disagreed with most students in that increasing minimum wage from $7.25 to $15 should be done. both matthew and lindsey used mathematical principles of optimization through inequalities and graphs of these functions and systems of equations to demonstrate and explain their reasoning. matthew believed that in order to accommodate such an increase in the minimum wage, schools and (other work places) would have to lay off people because they would not be financially able to support giving raises to a vast number of people. he stated the following in support of his position on the topic. downing & mccoy exploring mathematics of the sociopolitical through culturally relevant pedagogy journal of urban mathematics education vol. 14, no. 1 (special issue) 62 matthew: i don’t know how much money the school has to give out , but from what i would think, i would say no because if you do that, you wouldn’t be able to hire a lot of people. some people’s hours may get cut short. maybe that’s when it could be shown using a graph; it'll probably be quadratic-looking because the higher the salary, the less people could be hired. also, look at if you do raise it, how much money is left for the school to use for different things for stuff that we need, like athletics and stuff like that. you would see a lot of the important stuff is going down and then you just put two and two together and then you’ll see that you need to keep minimum wage what it is. lindsey (much like pamela) stated that people working at fast food restaurants should not make more money than emergency medical technicians, whom she felt/knew that some of them did not even make between $10 and $15 per hour. she did not go as far as to say these people should not get raises at all but believed it should not be higher than $10 per hour. lindsey: you would have to see how much you have to distribute out to the people that you're paying and raise it out to a certain extent to where everyone gets paid the same. so, like where you're not breaking the budget you have to pay everyone. so, you would have to incorporate how many employees you have, how many hours they work, and then figure out okay if we pay them $8 an hour, we pay everyone on campus as jobs like this $8 an hour, and they work this amount we have enough money to pay all of them instead of like, oh, we're just gonna give you $10 an hour and then not have enough money to pay everybody. similar to the previous theme of caring: comparison of situations, students who used this theme of explaining the ramifications of such an increase in minimum wage used the reasoning used during the health lesson and combined it with the discussions that were had throughout that lesson to help them craft an argument to support their position on minimum wage. planning the second topic students were asked to respond to was less sociopolitical in nature but related to the culture of living on an hbcu college campus, and one that the researchers initially thought students would say did not involve mathematics at all. this topic was surrounding the decisions people make to attend or not to attend a homecoming party (or some other social gathering) at a college campus. students were prompted with the following statement: “remember the class after halloween, people went to a lot of different homecoming parties the night before. is there mathematics involved in deciding which party to go to?” responses given by students were interesting in that no one said “no;” rather, they explained what application of mathematics they use or what they would use in such situations. responses revolved around the concepts of modeling fun and modeling profit. downing & mccoy exploring mathematics of the sociopolitical through culturally relevant pedagogy journal of urban mathematics education vol. 14, no. 1 (special issue) 63 recognizing: modeling fun. most students gave responses that involved the reason for wanting to go to a party to begin was first, and foremost, enjoying themselves and having a good time mingling with their peers. lindsey embodied the responses from most students, except she hit on every point that was discussed in all others’ responses. she was extremely keen on this question due to her high involvement and participation in the crp lesson delivered on the class period after halloween (which also coincided with homecoming). she stated the following in response to this question. lindsey: there are many equations involved! the first involves the population of the party and the amount of space available. if there's way too many people, it’s going to get shut down. i went to a party halloween night, and we had to choose between the q-party and a block party. the q-party was down the road, and we knew everyone was going to go to the q-party, because it’s a q-party… who’s not going to go?!? so, everyone's gonna be there, so you gotta think about this space cuz it was at a house, so the space around the house and all the people inside the house and then eventually the party was gonna get shut down cuz every q-party gets shut down. … another equation is the type of fun can we have. like we already knew there's gonna be smoking, underage drinking—it's halloween and homecoming. it’s like, are we gonna be able to get out in time? are we gonna get caught up? that’s another equation. how fast we can fun if something crazy pop off, like a fight or gun shots—and remember this is depending on the first equation with space and people. this may be an exponential function because we’d probably be able to run fast at first, then slow up as we got tired. [laughter]. um, ok… then we got to think about the block party, which we had to think about the time walking there, which was 25 minutes, which was another factor. so, it was 25 minutes to walk there, and not that many people know about it, so hopefully it's gonna be laid back, you know we can just chill, listen to music, do what we do. so, we decided to go to the block party cuz even though it was a farther distance, we figured since it was a farther distance everyone's gonna be a little bit too messed up to walk that far, so we decided to go to that one for the factors of less people times a farther distance. the block party seems quadratic with all the two factors multiplying that gave us a better answer in the long run since we didn’t have to think about the other stuff that a q-party brings with it. this quote was extracted from lindsey after she originally did not know how to respond—which was frequent for her. she would often have long pauses of silence, followed by blocks of fruitful statements after simply needing a rephrasing of the question and some time to think. lindsey brought together many of the experimental lessons that were discussed throughout the course, including linear functions, quadratic functions, and exponential functions. to say it like pamela phrased it, lindsey thought about this topic using equations, “but typically most people do this without thinking about the specific math or equations. people just make decisions without thinking its math, but it can be if you want it to be. almost anything can be!” recognizing: modeling profit. jamie thought through this scenario as if she were the one throwing the actual party and not going to one (counter to the intended downing & mccoy exploring mathematics of the sociopolitical through culturally relevant pedagogy journal of urban mathematics education vol. 14, no. 1 (special issue) 64 perception of the scenario). using the quadratic profit-revenue function discussed during the crp lesson on health, she explained how several factors would determine the constraints of a certain party they would try to throw and its chances of being successful. jamie: yeah [party planning involves math] because it depends like how many tickets you sell. so, like say if there is one party everybody talking about, “oh, we gonna go to this, everybody is going to go to this.” but after presales you raise the price at the door. so, like if you start with a $5 presale, you start with $10 until 10:00 or 11. then the price will go up more around midnight. you can only raise the price so much to make sure you get the most number of people possible because people won’t pay but so much, and all this depends on the size of the house the party is in. this discussion focused on making money from throwing a party with variables about the total number of tickets, space, and advertising. other students also made responses for this situation of throwing a party. responses in this theme used mathematics and described it using concepts that were experienced in the crp lessons, which helped students make a more compelling argument to support the points they were trying to make. deaths of unarmed black americans by police the final topic discussed at the end of the interview was focused on issues surrounding the killing of unarmed black americans by police officers in the united states. students were prompted with the following statement: “as i am sure you are familiar with news stories about unarmed black americans being killed by police officers in america, what are your thoughts about this?” after students gave their thoughts on the topic at hand, they were then asked, “who, if anyone, has the power to affect change or impact such shootings?” the final question was, “how, if possible, can you use mathematics to address this situation to help you enhance your arguments?” responses on these final questions revolved around major themes as to how students were using the mathematics to show the severity of the problem (using statistics) and then projecting what would happen if the pattern continued. critiquing: describing the severity of the situation. the way that students began using mathematics to help their arguments on why the shooting and subsequent killings of unarmed black americans was problematic (which everyone agreed was a major issue that makes them feel an array of emotions) was to show how this was an issue. all students began down this path of discussion and described how they needed to gather the data and statistics that has been collected. lindsey had the following to say. downing & mccoy exploring mathematics of the sociopolitical through culturally relevant pedagogy journal of urban mathematics education vol. 14, no. 1 (special issue) 65 lindsey: i feel like, using statistics and showing like stuff has happened in the past and like the ratio black men being—i’m trying to remember because i just did a project on this [for another class]. ummm… i think black men are ten or fifteen times more likely to be shot and killed by an officer than white men. they’re (i think) one in sixteen are more likely to be abused by officers. so, i think just using statistics and showing them the numbers and that, this is real. even though, you know, “one of your homeboys just got shot,” yeah. it’s more than just your homeboy. there’s people out here everywhere getting shot and killed by police for no reason, so i feel like showing the statistics to both sides [black people and police officers]. i feel that it would help both sides. she then goes on to discuss her personal feelings and emotions and how practical steps could be taken with this information that involves both african americans and police officers. focusing on the statistics is the way lindsey saw how mathematics could be used. others also began with this usage of the subject and took it a step further by explaining how if the problem persists, then they could project these effects using mathematical representations of the data. acting: projecting and exploring trends. several of the interview participants commented on how they would use the statistics gathered to display the data visually in hopes of compelling change. matthew: umm… i would make a graph showing the past few years. first, i would put how many people who are like, do an estimate of how many black people died within the last, say, 10 years, and then continue that equation to project in the future. i would show this to people, police, to show that if they don't change, these numbers would probably continue to increase. it'll probably be linear because i feel the number goes up more and more each year—not a lot, like exponents, but consistently. using principles explored in class, students were able to verbally describe what they wanted to use and how to use it, which comes to mathematical exploration. for these students, the predictive modeling was important, and they felt it would be the primary reason for catalyzing change—to stop the continued killing of unarmed black men by police officers. students really mirrored their arguments in the same way that the crp lesson on incarceration was presented and explored. first they explored data that had been collected on the issue, then they used linear equations and functions to explain what is currently happening with this data set. finally, by extrapolating responsibly and relatively close in the future to show that if all other factors held constant, they could know what to expect. perhaps the most sociopolitical topic discussed during the crp lessons, this topic was one that students stated they enjoyed because of how it really allowed them to see how mathematics could be useful and helpful in the “real-world.” downing & mccoy exploring mathematics of the sociopolitical through culturally relevant pedagogy journal of urban mathematics education vol. 14, no. 1 (special issue) 66 discussion students were able to use the critical-thinking and examination skills that were practiced and utilized in these exploratory lessons in the contexts of other sociopolitical and cultural situations that had not been directly explored in the course. mathematics and emotions were utilized in a variety of ways to advocate for a certain position. the complex nature of these positions and the mathematics used are advantageous for students who most likely will not take another mathematics course; however, they will be able to utilize these skills in their various fields of study and as a part of becoming productive members of society. recognizing, caring, critiquing, and acting although higher education is a very diverse setting, the teaching practices within it, especially in mathematics, are still very limited and are embedded in white supremacy, thus making it difficult for black students to establish mathematics identities (joseph et al., 2017; larnell, 2016). thus, through a crt lens, the findings in this study challenged the pervasiveness of racism that exists within mathematics instruction, while the intervention itself allowed students to engage in a critical examination of racism in america while they built their intellectual property mathematically. through this approach, students not only became more knowledgeable about the content matter, they became community advocates of issues that they are faced with daily. the themes that transpired during student interviews embodied one of the following: recognizing that a problem existed, caring, critiquing, and acting. students described how they felt after the lessons, often stating that they either did not know about certain issues or were just shocked to see the numbers in relation to them. once they expressed or emphasized the situation, some empathized with the groups of people that were being discussed. after caring about and empathizing with various people for myriad reasons, students moved to a more critical approach with their statements. students began to critique various institutional systems, including incarceration, education/college loan providers, and healthcare—all topics that had culturally relevant lessons built around them. some students wanted to seek to change the various systems they had just critiqued. these discussions led to the scenarios (two sociopolitical and one culturally specific) where students voiced their opinions and concerns. again, students used these same themes of recognizing, caring, and critiquing to explain how they would potentially act on these issues. students were able to use the experimental lessons they had been a part of in various ways. one way was by using the mathematics that was learned. another way was using the structure of the experimental lessons themselves to explain what they would do. the ability to utilize mathematics in such a way that students find it useful and see its applicability in their day-to-day lives is a strong benefit for the advocation of downing & mccoy exploring mathematics of the sociopolitical through culturally relevant pedagogy journal of urban mathematics education vol. 14, no. 1 (special issue) 67 such critical mathematics pedagogy (tate, 1995). it is important to mention again that most of these students will not take another mathematics class, and that is something most of them are all too excited about. students were able to discuss difficult issues, such as the pervasiveness of racism in america (decuir & dixson, 2004), the importance of cultural identity for african american students (martin, 2009), and the intersection of class and race (crenshaw, 1989, 1991) through the use of mathematics. this occurred during the classroom activities themselves but also during the interviews when they were asked to take stances or discuss their opinions on various topics. evidence from this study shows that students were able to examine inequalities in various areas of life and expound upon their critiques of various issues in their local, state, and global community—such as the goal for this type of pedagogy (gutstein, 2003; ladson-billings, 1995b; milner, 2017). implications for future research and practice teacher training programs must lead the charge in developing the next generation of culturally relevant pedagogues. this study has helped lay some groundwork at the collegiate level in showing that greater positive student outcomes were held at this level of education. this call for teacher educators extends to graduate programs in mathematics as well as mathematics education programs. to better serve and meet students where they are to get them where we want them to be, using the culture of students was shown to be a great start, as it allowed them to take ownership of their mathematics education and transform it into their own intellectual property. the education of students who take service-level mathematics courses like college algebra need this pedagogy to get them more interested and have a better outlook about mathematics and its utility in the real world. for teachers who are already in the classroom, teacher educator programs should provide professional development on crp. showing in-service teachers the foundations and how to use students’ cultures in non-trivial and innovative ways to enhance their teaching practices will benefit already good teaching practices and make them better. conclusion historically, students taking college algebra and other introductory mathematics courses have struggled to “pass” these courses (herriott, 2006). when looking at what is taking place in collegiate mathematics classrooms, culture and the interests of students continue to lack a place within the day-to-day activities of classes, even if the courses are designed to be student centered and rely less on teacher-as-lecturer. in this study, it was shown that by incorporating these cultural facets of students’ downing & mccoy exploring mathematics of the sociopolitical through culturally relevant pedagogy 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(1998). education for action: preparing youth for participatory democracy. in w. ayers, j. a. hunt, & t. quinn (eds.), teaching for social justice (pp. 1–20). the new press. copyright: © 2021 downing & mccoy. this is an open access article distributed under the terms of a creative commons attribution-noncommercial-sharealike 4.0 international license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. microsoft word +benoit+salopek+vol+16+no+1.docx journal of urban mathematics education june 2023, vol. 16, no. 1, pp. 40–71 ©jume. https://journals.tdl.org/jume gregory benoit is a lecturer in the mathematics education department as well as the assistant director of the earl center for learning and innovation at boston university, 2 silber way, boston ma 02215; gbenoit1@bu.edu. his research focuses on designing and understanding expansive mathematical spaces that nurture strong positive mathematics identities. this includes but not limited to: critical media literacy, gamification/gamebased learning in mathematics, culturally responsive mathematics instruction and educational simulation design. gábor salopek is a mathematics education researcher and graduate of teachers college of columbia university, 525 w 120th st, new york, ny 10027; salopek@tc.edu. his research explores identity development in mathematics education, with particular focus on the impacts of mathematics portrayal in social media and technology on academia. what do you meme? an investigation of social media and mathematics identity gregory benoit boston university gábor salopek columbia university, teachers college mathematical spaces extend far beyond the classroom and physical environments into virtual spaces. today’s students have more to consider than just their face-to-face experiences with mathematics inside or outside the classroom; they have the online perspectives of others to consider as well. to gain critical insight, we conducted this research with semistructured focus groups using an interactive mathematics internet meme activity. using positioning theory, this article highlights students’ stances and three storylines as conceptual tools for a better understanding of their offand online mathematics identities. results show that the two spaces are not mutually exclusive and that students are succumbing and adhering to a larger hegemonic construction of mathematics found in the online communities with various points of tension found. keywords: social media, internet memes, mathematics identity, sociotechnological as a former high school mathematics teacher, i remember finishing up a lesson on factoring polynomials (i.e., completing the square), and one of my freshman students (at the time), pedro, waited until class was over to hand me a folded-up piece of paper. immediately, pedro jovially headed out of the classroom and yelled to me, “told you so.” as i unfolded the paper, i noticed the mathematics internet meme in figure 1, smiled, laughed, and thought, “that’s just pedro being pedro.” but as the year went on, i noticed a change in him. anytime pedro struggled in mathematics, he would say, “i’m never going to use this anyway.” and i thought back to the mathematics internet meme. it appears pedro’s new demeanor aligned with the internet meme’s message. i questioned whether pedro’s experience was unique. subsequently, i began to consider my former and present students’ mathematics beliefs and how societal representations of mathematics have influenced or are still influencing them. benoit & salopek an investigation of social media and mathematics identity journal of urban mathematics education vol. 16, no. 1 41 figure 1. pedro’s mathematics internet meme the narrative story above illustrates the current critical period we find ourselves in with regard to mathematics education, where we must acknowledge and examine how the social milieu interacts and reinforces other dimensions of mathematical teaching and learning (stephan et al., 2015). what was once a field that focused traditionally on classroom activities, sequencing, and execution of lessons has been pushed to consider the many social perspectives that influence it (gutiérrez, 2000; jackson, 2009; lerman, 2000). although mathematics education researchers have considered the socialization process that occurs inside and outside of schools to be influential to a student’s mathematics identity (anderson, 2007; boaler, 2000; boaler & greeno, 2000; hodge & cobb, 2019; nasir & mckinney de royston, 2013), the online socialization process that also perpetuates ideologies of mathematics has yet to be studied (benoit, 2018). the various social media platforms accessible to k–12 (primary and secondary) students today are full of mathematics messages contributing to young people’s mathematics identity, especially when they are developing ideas about themselves and their relationship with mathematics (benoit, 2018; salopek, 2018). martin (2012) observed that social media such as youtube and facebook are not only responsible for exporting and importing culture, ideology, protest, and revolution, but also for exposing the human condition and helping black children to contextualize their lives vis-à-vis the conditions in which other children live and learn. (p. 51) as students negotiate their way through the world, they make sense of who mathematicians are and who can or cannot be mathematicians. the perceptions and images students process about mathematics or mathematicians ultimately affect their beliefs and attitudes and play a decisive role in mediating whether young people (dis)engage with the mathematics field (ernest, 2008). young people today live in a world that is vastly different than that of their parents’ adolescence. we must acknowledge that today’s generation lives in an accelerated, interconnected, and sociotechnological world—where part of a young person’s everyday routine consists of spending large amounts of time online engaging in virtual spaces. previous generations were influenced by popular media such as newspapers, magazines, television, movies, and radio. today’s generation of students has an assortment of media sources, including social media platforms such as twitter, facebook, and instagram. with the introduction of social media platforms, students can connect and form online communities with users they feel a stronger sense of affinity to regardless of their geographical locations (chen et al., 2014). benoit & salopek an investigation of social media and mathematics identity journal of urban mathematics education vol. 16, no. 1 42 marshall and sensoy (2011) argued that social media offers snapshots that document life, and its popularity continues to grow with many people having a social media account. social media offers today’s youth a “portal for entertainment and communication and [has] grown exponentially in recent years” (o'keeffe & clarke-pearson, 2011, p. 1). this virtual space is an expressive medium (radovanovic & ragnedda, 2012) that widens accessibility to most and allows users to share at their discretion. social media integration allows students to access the uncensored thoughts and feelings of other users on mathematics, not just their own. also, as students engage and spend time on social media platforms, it becomes virtually impossible to “protect” them from unsympathetic messages about mathematics that can strongly influence their perceptions of mathematics or mathematicians. consequently, today’s students have more to consider than just their face-to-face experiences with mathematics inside or outside the classroom; they also have the online perspectives of others. it is also important to recognize the relationship between social media and mathematics identity as sophisticated work, even if issues of sociotechnological perspectives may not “easily translate into large-scale policy recommendations” (gutiérrez, 2013, p. 1). however, the goal of mathematics education research should reflect not only the diversity and complexity of the individuals they are trying to serve but also the social context they are doing it in. just as the social turn in mathematics education research sought to examine the emergence of “theories that see meaning, thinking, and reasoning as products of social activity” (lerman, 2000, p. 23), the sociotechnological lens seeks to do the same in virtual spaces. issues framed around the sociotechnological realm are powerful cornerstones that may lead to helping students negotiate their continuously developing mathematics identities in a more productive and positive light. for this reason, they deserve an increase of attention in the mathematics research field. this research investigates the results of a study with 31 high school students as they explored and discussed a set of curated mathematics internet memes that sought to capture the negative, neutral, and positive depictions of mathematics that can be found on the web. further focus group interviews with the students about their social media responses (smrs) provided evidence of how such memes could impact their developing identities in this digital age. this article starts with an overview of relevant literature on the portrayal of mathematics in the media. it then presents a discussion on the theoretical perspectives of social media and internet memes, as well as a framework to conceptualize students’ smrs. literature review influence of mathematics media portrayals over the past decades, researchers have focused on the representations of mathematics and mathematicians in the media from a mathematics education context, mainly exploring how popular cultural images influence students’ relationship with the subject (appelbaum, 1995; epstein et al., 2010; mendick, epstein, et al., 2008; mendick, moreau, et al., 2008; moreau et al., 2010; picker & berry, 2000). medick and colleagues collected over 500 questionnaires, facilitated numerous focus groups, and found that images of mathematics and mathematicians in popular culture are “simultaneously invisible and ubiquitous” (mendick, moreau, et al., 2008, p. ii). although some students could not reference any mathematics examples in popular culture, most drew on common mathematics stereotypes popularized by the media. interestingly, although students were aware of the conventionalized notions of mathematics in their answers, it did not prevent or alter their responses. epstein et al. (2010) stated that students benoit & salopek an investigation of social media and mathematics identity journal of urban mathematics education vol. 16, no. 1 43 knew they were drawing on stereotyped images—which did not stop them from finding these the first that came to their heads. indeed, it would be surprising if these were not the first images that came to them, given their pervasiveness in popular culture. (pp. 54–55) the authors posited that students mainly drew on stereotypical imagery because of their pervasiveness in popular culture and the lack of alternative images available. lisa darragh (2018) looked at popular young adult fiction books as a conduit to better understand the relationship between mathematics and the societal perceptions of mathematics. in particular, she paid close attention to the storylines about school mathematics and mathematics teachers embedded in each fictional book. darragh’s (2018) final analysis included 59 books listed on the teen’s top ten list from the american library association (ala). she partitioned books into four categories—positive, negative, neutral, or mixed—based on how they depicted mathematics and thematically coded excerpts regarding school mathematics or mathematics teachers. findings from her study indicated that over 50% of the sample depicted a negative mathematics storyline. in fact, mathematics was characterized as “nightmarish, inherently difficult, and something to be avoided” (darragh, 2018, p. 197). although it was obligatory, mathematics was often seen as unimportant in these popular texts. mathematics class was generally seen as useless and something to endure rather than productive and fruitful. additionally, mathematics teachers had various negative storylines attached to them as well. darragh (2018) stated mathematics teachers were commonly depicted as boring and crazy, and they were overall “positioned as villains” (p. 197). these studies examined popular culture artifacts coupled with student discourse, only to conclude that stereotypes of mathematics and mathematicians (or the people who do well in mathematics) are salient in their respective popular culture domains. today, these messages have transcended former mediums of popular culture and appear on social media platforms through internet memes. as in previous studies, there is potential for these artifacts to influence students’ mathematics identities. although social media was once considered a trivial pastime, researchers recognize it as a point of concern in academic conversations (appelbaum, 1995; knobel & lankshear, 2007; milner, 2012; schifman, 2013; wiggins, 2019; yus, 2018;). social media and internet memes researchers have found that defining popular culture is an extremely difficult proposition due to its trendy nature; it can become dated as quickly as it is produced, rendering its definition susceptible to constant revision (fiske, 2017; marshall & sensoy, 2011). the task of understanding popular culture becomes even more difficult because today’s generation of students will have a different perspective on it than the previous one. fishwick (2002) suggested that, in defining popular culture, the operative term is new: new age, new generation, and new definition. for decades, institutions such as broadcast television, radio, recorded music, and film have been increasingly responsible for creating and distributing popular culture (hrynyshyn, 2017). however, today’s mass media is being displaced by social media and digital culture. although interactions are still “face-to-face,” technology is used to facilitate communication user to user. digital culture helps to describe the way technology shapes the communication mechanisms, interactions, and behavioral patterns we exude toward one another on and off social media platforms (wiggins, 2019). benoit & salopek an investigation of social media and mathematics identity journal of urban mathematics education vol. 16, no. 1 44 social media has manifested a great deal; it is made up of several networking sites, including facebook, twitter, tiktok, and instagram, connecting the world to ideas, beliefs, and different perspectives. social media harnesses today’s participatory culture by creating networks and connecting peers, as well as empowering users to share their ideas and exchange messages and news items, including photos and videos. social media provides an outlet for dynamic interaction and cultural production (wiggins, 2019). unlike previous technologies, such as television and film, social media provides its users with unique access to public discourse, cultivating a wide variety of societal perceptions on various content in one central location (hrynyshyn, 2017). previously, popular culture was based on the concept of single institutions disseminating content to the masses, but social media allows users, who were once merely consumers, to produce user-generated content (hrynyshyn, 2017; schifman, 2013). in this way, social media users have agency that they never had before; they have the capacity to constitute or reorient discourse, which can be both liberating and constraining (wiggins, 2019). during this 21stcentury digital era, social media has not only impacted our social, cultural, and political worlds by circumventing traditional pathways into public discourse but also decentralized communication practice. by eliminating some communication barriers and restructuring modernday communication practices, social media has helped give a voice to more individuals, including students, so they can participate in a democratic fashion.1 internet memes although slightly different than its original conception,2 internet users defined a meme as a virtual phenomenon. wiggins (2019) defined an internet meme as a “remixed, iterated message that can be rapidly diffused by members of participatory digital culture for the purpose of satire, parody, critique, or other discursive activity” (p. 11). an internet meme acts as a vehicle carrying ideas, practices, culture, or symbols from person to person in various forms, including social media (e.g., facebook, instagram, twitter). internet memes can take on various forms and formats, such as image macro memes—a line of text on top of the meme, another one at the bottom, and one picture in the middle; gifs—compressed data image files that support both animated and static images; and so much more. although internet memes are often regarded as humorous discourse, to simply regard them in this manner is merely surface-level understanding (benoit, 2018; schifman, 2013; wiggins, 2019; yus, 2018). in this digital age, internet memes offer a cultural analysis of public discourse and popular culture (johnson, 2007). within the participatory power of social media, internet memes that are shared on a micro basis have the 1 although digital culture allows certain accessibility to public discourse, it is not without its complications. a common misconception could be that digital culture and social media act as social equalizers providing an equal “playing field” for public discourse. although digital culture is participatory (content is user generated) and helps provide accessibility to public discourse, notions of power such as access and impact are still not equally distributed. consider the 21st-century job title influencer. the financial negotiation of an influencer is to use their social capital of amassing an established large following for persuasion of products, goods, and services, and, in return, they receive a monetary contribution. in this case, an influencer serves as a liaison from corporation to consumer and is compensated for it. other social media users simply operate on a whole different scale; they do not have the mass followings and have a different definition and objective of participation. although this is just one of many lenses, it helps illuminate that the shareability of digital items and discourse is not equally distributed and too can be informed by capitalism. 2 the term meme was first introduced by the biologist richard dawkins in his book the selfish gene (1989), where he defined memes as “an idea, behavior, or style that spreads from person to person within a culture” (p. 192). benoit & salopek an investigation of social media and mathematics identity journal of urban mathematics education vol. 16, no. 1 45 potential to scale up and spread to countless individuals within minutes, having a macro impact. the decentralization of user-generated content posits new arguments, visually influencing discourse and social norms (baym & burnett, 2009; boyd, 2008; jenkins, 2006; o'reilly, 2007). the discursive power of internet memes the discursive power of internet memes is best rooted in their ideology, semiotics, and intertextuality (wiggins, 2019). the internet meme–creation process is not passive. a great deal of contemplation goes into the organization of images and chosen words (benoit, 2018; schifman, 2013; wiggins, 2019; yus, 2018). the author of an internet meme makes intentional choices throughout the whole process, from conception to completion, that reflect the internet meme’s purpose. intertextuality also plays a critical role; it assists in shaping the meaning of the internet meme (wiggins, 2019; yus, 2018). internet memes do not exist in a vacuum; they rely on a shared cultural and semiotic understanding for collective interpretation. they are entangled in their sociocultural environments and cannot exist outside of the events and practices in which they appear (schifman, 2013). they are experienced as encoded information that demands shared knowledge to properly understand and can therefore create barriers of discursive specificity for those unable to access the internet meme’s meaning. in this way, intertextuality and relationality become “purposeful, unavoidable, and ubiquitous” (wiggins, 2019, p. 35). further, in foucault’s (1989) writing on discursive power, he explained the duality of the social relations embedded in discourse as consisting of “practices that systematically form the object of which they speak” (p. 49). a parallel from this idea of discourse to internet memes is that meaning is created and negotiated socially and not in terms of the actual physical world. framework for positioning of internet memes identity and positions are multifaceted researchers have categorized identities in a variety of ways. what was once considered relatively stable and generalizable and that had the connotation of an essentialized self (stinson, 2013) is now referred to as increasingly fluid (chronaki, 2011; solomon et al., 2011) and as an always-developing process (e.g., black et al., 2015, nasir, 2002) of multiple identities forming in different moments of time (esmonde, 2009; wood, 2013). identities can be conscious or subconscious (bishop, 2012), independent (based on self-perception; davies & harré, 2001), or interdependent (based on affiliation with a group [institution or affinity]; gee, 2001). identities can be formed by material, relational, and ideational resources (nasir & cooks, 2009) and impacted by characteristics such as race, gender, and disciplinary performance (varelas et al., 2012). identities can zoom in and look at specific moments or expand over periods of time (darragh, 2018), and they depend on various factors and life experiences such as culture, race, gender, and so on (gee, 2001; wenger, 1998). using these ideas as a basis, we use bishop’s (2012) definition of identity, “a dynamic view of self, negotiated in a specific social context and informed by past history, events, personal narratives, experiences, routines, and ways of participating” (p. 38). closely related to identities are positions that are either self-perceived or offered by others. davies and harré (2001) explained that “positions are identified in part by extracting the autobiographical aspects of a conversation in which it becomes possible to find out how each conversant conceives of themselves and of the other participants by seeing what position they take up” (p. 264). positions are dynamic; they are negotiated in the moment and can be accepted benoit & salopek an investigation of social media and mathematics identity journal of urban mathematics education vol. 16, no. 1 46 or refused (suh et al., 2013). positioning is not unidirectional but relational depending on wider contexts of history and life experiences (gee, 2001; wenger, 1998) and only if one accepts that positioning (davies & harré, 2001). harré et al. (2009) explained that individuals “use words (and discourse of all types) to locate themselves and others” (p. 3). three constructs are key to understanding how students engage in positioning within digital culture: stance, smrs, and storylines. we define and describe each briefly below and exemplify them with a short analysis of pedro’s mathematics internet meme discussed in the opening vignette. stance an internet meme’s stance serves to make connections between the information conveyed by internet memes and subtler categories of identity. stance signifies the ways in which authors and users position themselves in relation to the internet meme, its text, its linguistic codes, the addressees, and other online participants that engage (schifman, 2013). for example, let us examine the meme pedro handed to his teacher (figure 1). the internet meme’s content, namely the ideas and ideologies communicated, relate to the caption, which reads, “i'm still waiting for the day that i will actually use 𝑥𝑦 + (4 ∗ 20) > 𝑥 − 5𝑦[2 + 9 − 7] in real life.” the caption questions the purpose of mathematics and exemplifies its uselessness. this is further supported by the caption that includes an invalid inequality that cannot be understood. another thing to note is that the image, which depicts an individual with a straight face, is often used to convey mild irritation (emojipedia, 2020). overall, the internet meme conveys the ideological message that mathematics is not relevant, which describes where pedro wants to locate himself. like identity and positions, stances are not concrete but are rather fluid (chronaki, 2011; solomon et al., 2011) and can be in an always-developing process (e.g., black et al., 2015; nasir et al., 2013). as artifacts, internet memes are created with both cultural and social attributes and possess the power to reconstitute our social system (wiggins, 2019). social media responses: the acts of positioning relevant for this analysis are smrs, which are the communicative actions individuals can perform on social media platforms such as facebook, twitter, and instagram. smrs include but are not limited to the following: liking,3 commenting,4 tagging,5 sharing,6 and creating7 (muntinga et al., 2011). although many smrs are available today, at the time of data collection in 2017, only the “like” button was available. like in-person conversations, online users can use smrs as positioning tools that negotiate and locate themselves and others. smrs enable social media users to openly accept or reject social media content, all while opening the door for follow-up communication. consider pedro’s earlier action in a social media context. he not only shared an internet meme but also tagged his teacher in the meme. sharing places the user at the controls as they select, confirm, and broadcast content for their friends and others to see. presumably, when users engage with internet memes on social media platforms, they “decide to imitate a certain position 3 liking is a quick-and-easy way to express your support of a certain content. 4 commenting is a response to a post on social media. 5 tagging a user ties that particular user to an internet meme. if users are friends on social media, they do not have control of the internet meme or piece of content they are tethered to. 6 sharing is when a user broadcasts web content on a social network to their connections (their friends, followers, associated groups, etc.). 7 creating and posting: authoring content on social media. benoit & salopek an investigation of social media and mathematics identity journal of urban mathematics education vol. 16, no. 1 47 that they find appealing” (schifman, 2013, p. 367) and that aligns with the discourse they are choosing to replicate or engage in. tagging explicitly directs the social media content to specific people; it is akin to saying someone’s name in a conversation. multiple people can be tagged on a social media post, drawing them to regard the post (in this context, the meme) and creating a positioning opportunity for them to accept or deny it. further, if the teacher liked the meme, it would signify support for the meme’s content, or the teacher could openly comment in a variety of ways, including rejecting the meme’s ideology. in other words, smrs can communicate textual, verbal, and visual cues that illustrate a position (wiggins, 2019). these, too, are not arbitrary actions; users make intentional decisions to perpetuate the discourse they are choosing through social media for future deliberation. we are not claiming individuals’ smrs to be synonymous with their identity and that like identity is complex to understand. we are saying it is important to examine what students are choosing to perpetuate and why they choose to do that. storyline: through such positionings, students can construct storylines through their smrs regarding mathematics and their mathematics identities. storylines are the broader, culturally shared narratives intertwined in the social interaction (herbel-eisenmann et al., 2015). within conversations, several storylines can exist simultaneously, all drawn on and from the participants’ cultural, historical, and political backgrounds, and that help to define conventions for interactions online (herbel-eisenmann et al., 2015). participants can interact differently as storylines can operate on different scales; therefore, it is crucial to listen to what students are saying and how they are saying it. again, let us consider pedro’s meme. the more global storyline of mathematics, as a difficult subject that few students do well (walker, 2012), can make it seem useless. this storyline is fostered through acts in socially recognized ways such as an overemphasis on algorithms, correctness, and speed, as opposed to deep conceptual thinking (sherin & jacobs, 2011). we understand both students’ stances and storylines to be situated in a larger social and cultural context in which mathematical discontent is not only widely accepted but to be expected and encouraged in some cases. it appears to be quite normal behavior and socially acceptable to hear or say, “i am bad at mathematics” or “i am not a mathematics person.” storylines can illuminate the point of view for students, influenced by moral or personal traits (harré & van langenhove, 1998). storylines help to frame positionings as they “draw on [the] knowledge of cultural structures and the positions that are recognizably allocated to people within those structures” (ritchie, 2002, p. 27). research questions the overarching research questions guiding this study are the following: 1. what mathematics internet meme stances can be identified in students’ social media responses (smrs)? 2. what mathematics internet meme storylines are evident in students’ collective social media responses (smrs)? methodology participants the researchers recruited 31 students between the ages of 13 and 18 in new york city via a short promotional youtube clip and/or informational session at their respective schools. benoit & salopek an investigation of social media and mathematics identity journal of urban mathematics education vol. 16, no. 1 48 twenty students were from public schools (westpine high school,8 silvercliff academy, and moorhall), five were from a private school (fairbourne prep), and six were from a charter school (marblepond charter). the researchers organized focus groups for participating students at the respective schools outside instructional time that lasted 60 min. students received food for their participation. fifty-two percent (16) self-identified as males and 48% (15) self-identified as females. forty-five percent (14) were in 9th grade, 29% (9) were in 10th grade, 19% (6) were in 11th grade, and 6% (2) were seniors. participants represented diverse racial and cultural communities: 48% (15) identified as african american, 26% (8) identified as hispanic, 10% (3) identified as asian, 6% (2) identified as caucasian and “mixed,” and 3% (1) identified as indigenous american. research activities the goal of this research was to gain insight into students’ smrs regarding their mathematics identities. to accomplish this, we used a mixed-methods design that included an individual internet meme activity (ima) and small focus group–based follow-up discussions about the ima. in this section, we describe the methods and materials associated with each. individual internet meme activity to acquire the math memes for the individual ima, the researchers used a google search, typing “math memes” into the search bar.9 researchers gathered a sample of 100 mathematicsrelated memes from the search. a “convenient jury”10 of 10 mathematics experts (i.e., teachers, doctoral students, and mathematicians) were asked to examine the sample of memes and categorize them as positive, negative, or neutral. positive internet memes were defined as memes that emphasized good and laudable characteristics about mathematics or when the character(s) was portrayed as loving the subject; negative internet memes emphasized bad and negative characteristics about mathematics or when the character(s) was portrayed as hating the subject or finding it unreasonably difficult; and neutral internet memes referred to mathematics but did not impose any feeling (positive or negative) about the subject (e.g., an internet meme that illustrated a mathematics problem) or explained a description of a mathematics experience that did not incite negative or positive reactions. an 80% agreement (8 out of 10 experts) threshold was needed to obtain each internet meme’s code. the jury selected a subset of nine memes. (see table 1 for a display of the nine memes chosen for the study and their expert evaluations.) to start, each participant was asked to identify their grade level, age, gender, and ethnicity and to rate themselves as a mathematics student (self perceived math ability [spma]) on a scale from 1 to 10, with 1 being the lowest and 10 being the highest. later, when examining student responses, researchers partitioned the scale into the following categories: excellent (10), good (7–9), average (4–6), bad (2–3), and poor (1). so, in total, 3% of my sample identified as a 8 student pseudonyms were intentionally chosen to match their focus group; for example, students in westpine high school’s focus group will all start with a “w.” 9 a preliminary finding was that of the 13 predetermined categories, three (23%) had labels associated with women (e.g., lady, blonde, women) and two (15%) were focused on confusion (e.g., confused, confused math). these filters provide insight on the abundance of math internet memes created having these tags or associations; hence, there is a wide variety of internet memes that center on women and confusion. 10 the jury assembled in 2017 consisted of eight males. seven of whom were white, which had implications on how internet memes were eventually coded (e.g., internet meme 7 was coded as positive but might be perceived differently, as it can be considered biased toward females). benoit & salopek an investigation of social media and mathematics identity journal of urban mathematics education vol. 16, no. 1 49 bad mathematics student, 16% identified as an average student, 77% identified as a good mathematics student, and 1 student identified as an excellent student. on average, students rated themselves at a 7.69, with a standard deviation of 1.5. approximately 80% of my sample identified as having above-average mathematics skills. students’ self-assessment is included in appendix a. next, everyone was presented with nine 5x7 index cards showing each of the internet memes depicted in table 1, all at once in random order. participants were asked to place the nine memes into one of three predetermined groups (positive, neutral, or negative) according to how they perceived the message or determined the relationship. students were instructed to give a short description of why they made the selections. students were in control and were strongly encouraged to express whatever mathematical idea(s) they wanted. as part of the research design, participants were asked to select one internet meme they would post on their social media account, simulating students’ “posting” regiment. a blank copy of the complete ima protocol is included in appendix b. table 1. memes used in mathematics internet meme activity mathematics internet meme expert jury’s judgments mathematics internet meme expert jury’s judgments meme a. “homer simpson” negative meme f. “chicks dig math help” positive meme b. “polar bears” positive meme g. “it gets complicated” neutral benoit & salopek an investigation of social media and mathematics identity journal of urban mathematics education vol. 16, no. 1 50 meme c. “me neither” negative meme h. “that’d be great” neutral meme d. “nice try, math” negative meme i. “dance lessons” positive meme e. “live dangerously” neutral focus group follow-up discussion after students completed the ima, they came together as a focus group to share their selection choices and were asked how they would comment (if at all) to the other internet memes if they were to come across them virtually. furthermore, participants were asked an open-ended set of questions designed to probe their exposure to mathematics internet memes, their smrs, and their mathematics identities. although the focus groups varied according to students’ interests and responses, they all had the same overarching structure: discussions began with a set of planned questions, which were followed up with impromptu questions to stimulate further discussion and additional probes as needed (lindlof & taylor, 2002). a copy of the protocol is included in appendix c. at times, the researchers found it helpful to repeat a question and the students’ verbal responses to ensure their own comprehension. all focus groups were audiotaped and later transcribed. in addition, the researchers took extensive notes on phrases, sentences, actions, concepts, body language, opinions, and quotes as the discussions unfolded. data collection and analysis analysis consisted of five semistructured focus groups of 31 student participants across new york city. the researchers used a grounded theory approach first, in which codes and theoretical themes could emerge inductively from the data and be used as lenses (creswell, 2012). as such, transcription data were organized in several ways (i.e., questions and schools), several different lenses (i.e., frequently repeated ideas, reference to the literature, and shocking or surprising remarks), and several different variables (i.e., grade, gender, and students’ benoit & salopek an investigation of social media and mathematics identity journal of urban mathematics education vol. 16, no. 1 51 perceived mathematical ability [spma]). the researchers then used indexing (miles & huberman, 1994) as a process of defining codes according to research questions and the theoretical framework, cross-referencing and pairing them into an axial coding system. later, a hierarchical tree diagram was created to interpret the connectivity between current codes. the researchers then used selective coding to help build and illustrate a story to connection between codes into overarching themes (see table 2 as an example of the ability affects susceptibility theme). codes were cross-referenced from all five focus groups and then combined into three salient storylines: vindicate or villainize mathematics; ability affects susceptibility; and influence or isolate, that is the question? these will be discussed later in the article. table 2. data analysis framework question westpine high school silvercliff academy marblepond charter fairbourne prep moorhall theme do you think math internet memes are perceived the same way? for example, if we were to all look at an internet meme, would we have the same understanding ? willow: because, for some people, math is easy, and for other people, math is hard; like, they would have a different perception of them [negative mathemati cs memes]. samuel: yeah, it’s true ‘cause, as long as you understand math, then you’re going to probably relate to at least one of these. sharon: it probably has nothing to do with ethnicity or race. joshua: or gender. alfred: depends on someone’s personalities and how they perceive things. daniel: yeah, because it depends on if you like math or you don't. alfred: if someone was really good at math, then felicia: it depends because, when you grow older, you also have a bigger mindset of knowledge then a younger child may have, depending on what they learn from other people. falyn: i think it’s also michelle: ‘cause some people are good at math, and some people are bad . . . some cannot relate, and some of them can. mike: someone can take it as an insult . . . ‘cause, you know, people can get bullied ability affects susceptibility : people who are good at math are not affected by an internet meme’s message. different abilities will interpret things differently. benoit & salopek an investigation of social media and mathematics identity journal of urban mathematics education vol. 16, no. 1 52 samuel: it also depends on your math skills, like, even if you’re good at math. they will probably have a deep debate about these kinds of memes. experience that affects your interpretation of things. for being smart . . . it’s like saying, “oh, he’s smart . . . let’s trouble him.” limitations the researchers were cognizant of the various degrees of bias used throughout, including how the memes were collected and selected, the questions that were asked by the researchers, the themes and codes that were created, and the messages that were highlighted. this study was facilitated in 2017, and the individual ima sheet only listed male and female as genders. although there was space to write in “nonbinary gender,” the fact that we did not have it as a predetermined choice could have pigeonholed students to make a selection that may not have fully represented how they identify. however, we do refrain from presenting data from a gendered perspective. furthermore, the main purpose of this study was to gain insight on how students interact with mathematics messages on social media, and the convenient jury ultimately selected the memes students eventually engaged with. as stated previously, the convenient jury was not a representative subset of the diversity contained in the mathematics education community and not well positioned to recognize the problematic nature of some of the meme selections (i.e., that many feature white males and that at least one is offensive to those who identify as a woman). however, we do believe that what students claimed as their “why” speaks to a larger context that extends beyond the selection of memes and the ima, and we focus our analysis on that. findings in this section, we present findings from the ima and the focus group discussions. first, we present numerical data regarding the mathematics internet meme that students selected to share via social media. then we present an overview, based in thematic analysis, of students’ reasons for choosing the meme they did and how they would expect to position themselves with respect to that meme. finally, we present three salient storylines that emerged during our close analyses of the focus group discussions: vindicate or villainize mathematics; ability affects susceptibility; and influence or isolate, that is the question? students’ mathematics internet meme selections students were asked to choose a single internet meme from among eight options (see table 3) to post and share (e.g., students' smrs), providing insights into their affiliation with mathematics messages on social media. table 3 shows findings from the ima task.11 table 3. mathematics internet meme activity cards 11 although 31 students participated, five did not identify the mathematics meme they would post, decreasing the sample from 31 to 26 for the ima. benoit & salopek an investigation of social media and mathematics identity journal of urban mathematics education vol. 16, no. 1 53 mathematics internet meme percentage of students’ selection (n = 26) meme a. “homer simpson” 27% (7) meme b. “polar bears” 15% (4) meme c. “me neither” 15% (4) meme d. “nice try, math” 12% (3) meme e. “live dangerously” 12% (3) meme f. ‘chicks dig math help” 8% (2) meme g. “it gets complicated” 8% (2) meme h. “that’d be great” 4% (1) meme i. “dance lessons” 0% (0) memes a (homer simpson), b (polar bears), and c (me neither) were the three that were most often selected by students to post on their social media accounts. (see table 2 for images of the memes used in the ima.) twenty-seven percent (7 of 26 students) chose homer simpson, 15% (4 of 26 students) chose polar bears, and 15% (4 of 26 students) chose me neither. memes a (homer simpson) and c (me neither) were judged by the convenient jury as negatively depicting mathematics, and meme b (polar bears) was judged by the jury as positively depicting mathematics. meme a, homer simpson, is titled with the phrase, “me in math class” and shows a picture of homer simpson, a relatively dimwitted character on the simpsons tv series, asking the question, “can you repeat the part of the stuff when you said all about the things?” internet meme c (me neither) shows a man laughing, sarcastically asking the question, “do you know that awesome feeling when you finally understand math? . . . me neither.” both of these internet memes convey negative ideologies about mathematics (i.e., “math is confusing” and “math is hard”). although both emote a humorous tone, their stances reflect a negative one regarding mathematics identity. meme a, homer simpson, asks viewers to locate themselves within the meme (i.e., “me in math class”) and to consent to its message about being confused. senders of meme c, me neither, directly engage in the social community by asking a question and then positioning themselves as not knowing mathematics (i.e., “me neither”). meme b (polar bears) is titled with the phrase “when you understand something in math” and shows two polar bears dancing. although the convenient jury judged it as positive, its stance can be perceived in multiple ways. one is the right to feel a sense of accomplishment when solving a mathematics problem. a second unrelated message is that the feeling of understanding is so infrequent in mathematics that, when it happens, it is a cause for celebration: a mixed message at best. perhaps tellingly, no student chose internet meme i (dance lessons), which illustrates a connection between mathematical functions and dance moves and was judged positively by the jury. benoit & salopek an investigation of social media and mathematics identity journal of urban mathematics education vol. 16, no. 1 54 overall, 54% of all student participants selected a negative internet meme to share, more than double the selection rate of positive (23%, n = 26) or neutral internet memes (23%, n = 26). perhaps most telling is that approximately 80% of the students self-identified as average or above in their mathematical ability, and yet most memes they selected illustrated negative positions with regard to mathematics. perhaps not surprisingly, student selection choices are consonant with the negative and widespread story of mathematics that is portrayed in popular culture, as discussed above in prior research (darragh, 2018; mendick, epstein, et al., 2008; mendick, moreau, et al., 2008). overview of student positioning through social media responses (smrs) as they selected their internet memes, students also chose other smrs (e.g., “liking,” “tagging,” “commenting”) that they would tag to their meme and that described a similar routine. first, they would assess a meme for its relatability, which often included laughter, some type of phatic communication like emojis, and tagging other friends who might also relate to the meme’s message. for example, steven and scott, 9th graders from silvercliff academy, felt that internet memes often reflect true circumstances: steven: like, sometimes, you realize how this can relate to you in a certain way. scott: if they’re more relatable, it means more entertainment . . . funnier. willow, a senior from westpine high school, agreed, acclaiming, “well, it’s funny because they agree with it; that’s why they relate to it and laugh.” other focus groups shared similar sentiments. students elaborated on the theme of relatability, suggesting the kinds of comments they would expect others to post or that they themselves might post. for example, steven, from silvercliff academy, stated, “[homer simpson] would get facts, likes, and laughing emojis.” samuel added, “all you would see under [me neither] is facts11 and the skull face.”12 also, sean, a 9th grader from silvercliff academy, said, “[if i posted meme e (it gets complicated)] there would be laughing emojis, and it would get all likes.” melissa, a 9th grader from moorhall, indicated she would expect to see comments such as, “oh my god, this is so true [referring to polar bears].” mary, also a 9th grader from moorhall, indicated that she would put “the tears of joy” emoji under memes a (homer simpson), d (nice try, math), e (live dangerously) and g (it gets complicated). in addition, students explained they would tag—create a direct link to—their friends to the original meme. unlike sharing, tagging calls specific users’ attention to the post. sharon, from silvercliff academy, specifically voiced, “i’ll tag a friend under it [internet meme c, me neither] and be, like, ‘this is so true.’” additionally, mike from moorhall ecstatically chose meme c and stated, “i’m tagging, like, 30 friends.” francine from fairbourne prep said she would laugh at homer simpson because, “that’s how i feel” and would tag a friend. in short, the students’ choices and discussions of smrs (e.g., sharing or “liking” negative internet memes, “tagging,” putting a “joking” comment beneath a negative post) portray a clear stance toward a socialization that reflects a public degrading of mathematics. 11 to add further clarity to the students’ dialogue, “facts” implies “factual” and is generally used to concur a proposition. 12 here, “skull face” is being used to represent “died” laughing. benoit & salopek an investigation of social media and mathematics identity journal of urban mathematics education vol. 16, no. 1 55 interestingly, students rarely decided to share positive internet memes that depicted uplifting messages about mathematics. in addition, students did not express “liking,” “tagging,” or “commenting” on positive mathematics internet memes, reflecting their disinterest in sharing positive math messages. our analysis indicated that positive mathematics internet memes did not generate the same desire; only one (polar bear) garnered students’ attention. these findings support previous work, as discussed earlier, and suggest the dominance and prevalence of negative images and messages about mathematics. although students may not associate their smrs as mathematical in nature per se, their choices and discussions of smrs call attention to, reinforce, preserve, and amplifiy negative messages about mathematics in virtual spaces. in the next section, we present findings with respect to three storylines that emerged from close analysis of the focus group discussions. these storylines were present across discussion groups, and, in our analysis, they both reflect and perpetuate negative rhetoric and public images about mathematics. storylines in students’ discourse after examining all transcripts and facilitating the coding process described above, three centralized storylines emerged: vindicate or villainize mathematics; ability affects susceptibility; and influence or isolate, that is the question? each theme will be presented in a subsequent section. villainize > vindicate mathematics: as described above, an initial analysis of students’ smrs demonstrated several negative tagging practices toward mathematics, illustrating an active culture among young people of socially accepted contempt for mathematics. closer examination of students’ responses in the focus groups revealed that although students rated themselves highly as mathematics students, they were more comfortable in adhering to a socialization that demeans it. students did not feel compelled to defend or endorse it, positioning themselves as spectators with no vested interest in how mathematics is discussed, or they portrayed a theme we call “vindicate or villainize.” this theme was present in the responses of 24 students. we present the statements of six, whose responses are particularly revealing. willow, an african american senior from westpine high school’s focus group, expressed that, on social media, “most of the time, with stuff like that [negative mathematics memes], everyone agrees. nobody will disagree; everyone will just agree.” samuel from silvercliff academy similarly added, “when everyone is making fun of math, you’re also going to want to make fun of math. when everyone’s saying math is hard, even if you’re good at it, you’re probably going to say it’s hard.” both willow and samuel’s comments are extremely powerful statements that call attention to the established culture of social media regarding mathematics and mathematics internet memes. these remarks indicate that negative mathematics internet memes and their associated messages are not only met with acceptance; they are also not confronted or challenged. these negative social media practices help illuminate a seemingly valid truth of the assimilative culture online (wiggin, 2019), one that legitimizes a negative affinity with mathematics and adds to a negative narrative. perhaps even more telling are the contradictory stances students took on the vindication of mathematics on social media. on one side, willow, who self-identified as an average mathematics student (spma 7), expressed, “i would not like or comment [on a negative mathematics internet meme], but i would share it.” interestingly, as a student who sees herself as benoit & salopek an investigation of social media and mathematics identity journal of urban mathematics education vol. 16, no. 1 56 a good math student, willow did not feel obligated or positioned with the need to defend it. moreover, her actions to perpetuate internet memes that display contempt for mathematics contradict her own personal identity as a skillful mathematics student. this reflexive action provides insight on the possible internal tension social media users engage in. as a student with supposedly high reverence for mathematics, willow can disregard and distance her own personal performance and opinions from her smr, highlighting a deeper socialization and affinity toward the popular stance and collective identity than her individual one. alfred and antwon, two african american students from marblepond charter school’s focus group, offered a different perspective. when probing further about their smrs, the boys revealed the following: alfred: if someone was really good at math, then they will probably have a deep debate about these kinds of memes [negative mathematics internet memes]. antwon: yeah. like, people who leave school, who really like math and they actually do it for fun, they might see this [negative mathematics internet meme] and be serious. it is clear from the onset that both alfred and antwon do not feel obligated to protect or defend mathematics. although both students have high self-perceived mathematical ability (both self-identified as a 9), they do not perceive people teasing mathematics or those who do well at mathematics as a problem. additionally, they separate themselves from other “good” mathematics students (e.g., they actually do it for fun) and create a storyline where it is the responsibility of the other “good” mathematics students to endorse/defend mathematics on social media. alfred’s justification emphasizes really good as if there is this abstract threshold of ability needed to act. antwon, on the other hand, distinguishes between ability and affection for his defense. ability affects susceptibility: as students described the villainization of mathematics online, their comments revealed the belief that negative messages from internet memes are not uniformly received by all. further, their comments suggested that mathematical ability may dictate the perceptions and susceptibility level of an internet meme’s message. students’ commentaries point to lower mathematically performing students being more susceptible to negative mathematics internet memes, as it may more closely relate to their experiences, and higher mathematically performing students being less susceptible, as it may relate less (or not at all) to their experiences. as a result, we call this emergent theme ability affects susceptibility. this theme was salient across various focus groups, as it was present in 15 student responses. we include five such student remarks. for example, when asked why people would perceive internet memes differently, mike from moorhall clarified that, “[for] some people, math is easy, and for other people, math is hard; they would have different perceptions of them [internet memes].” here, mike is making the claim that individual mathematics experience and ability potentially influence susceptibility. during another focus group, francine from fairbourne prep added to these sentiments as she described her sister’s mathematical ability and her beliefs of what she is now afforded. she explained, “no, my sister likes math, and she’s good at it, so i don’t think it’s [negative benoit & salopek an investigation of social media and mathematics identity journal of urban mathematics education vol. 16, no. 1 57 mathematics internet meme] going to affect her like that.” here, francine is building on previously mentioned claims and explicitly drawing a link from high mathematical ability to susceptibility; in particular, strong mathematical abilities are less susceptible to negative mathematics messages because they are not relatable to her. francine expressed that, due to her sister’s high ability, she is shielded from negative mathematics messages. these students articulated that those with positive mathematics experiences would not relate to negative internet memes. this limited view sees mathematics experiences, whether positive or negative, as mutually exclusive and having nothing to do with one another. they take mathematics experiences and abilities to be static, as they create concreate labels as “good” and “bad.” the overall implication here is that they perceive mathematics as one overall mathematics experience instead of collective combinations of multiple experiences. it neglects the tendency to see mathematics experiences on a continuum, conjoin past mathematics experiences with the present, and allow shifts and growth to happen. to further add to this storyline, michelle from moorhall stated, “some people are good at math, and some people are bad . . . some cannot relate, and some of them can.” additionally, antwon from marblepond charter school stated, “it depends on who you [are] showing it to; probably someone who struggles with it [mathematics] might think something different.” francine’s and antwon’s comments help to elucidate the storyline as they explicitly mentioned mathematical ability as the deciding factor of susceptibility. although francine’s statement regarded the lens of a strong math student and antwon’s regarded that of a struggling math student, both ultimately positioned those with a higher mathematical ability to be impervious to negative mathematics messages. the students believe that “good” mathematics students have direct in-person examples to counteract the negative messaging they are receiving, whereas other students may not have those experiences and are vulnerable to the meme’s message. influence or isolate, that is the question?: although not as significant as the shakespearean work of hamlet, contemplating life and death, this storyline was named after the philosophical ways in which students attended to the specific question, “would they share these [negative] internet memes with their younger sibling?” witnessing students construct complex arguments to explain the advantages and disadvantages of their choice, while having to address rebuttals and counterexamples, was truly philosophy in motion. this line of questioning was asked to every student and was important because it removed participants from the focal point, which emphasized their honest perceptions of social media influence and delivered an unpredicted storyline. below we include the comments of 10 students. influence. as predicted, most participants affirmed that they would not want their younger sibling(s) to view negative mathematics internet memes because of their subliminal influence. antwon from marblepond charter admittedly stated, “if they see a whole lot of [negative] things about math, they might not see math as the best, or maybe not interesting or popular to be good at.” antwon’s response aligns with an earlier comment about the prevalence of negative mathematics messaging impacting students’ views about mathematics. further, scott from silvercliff academy declared that he would not expose his younger sister to negative internet memes because “my sister, just like samuel’s sister, dislikes math, but i want her to like math . . . for me, i say that the subject is very important . . . you have to learn it [math], and without math, you can’t do anything.” scott’s reactions convey his fear. he is convinced that sharing negative internet memes would only confirm and normalize his sister’s conceptions benoit & salopek an investigation of social media and mathematics identity journal of urban mathematics education vol. 16, no. 1 58 about mathematics. also, as a brother, he would not want to see her lose tenacity or to stop trying. alternatively, mary from moorhall shared that although her sibling was doing well in math, she would not share negative internet memes with her. she stated, “they might even be, like, ‘oh, i’m the only person who could understand this stuff; maybe i’m weird.’” here, mary is expressing her fears that her sister would be outcast among other people for not agreeing with the internet meme’s message, and she is alluding to the social pressures that exist for k–12 students. willow and wakanda from westpine high school respectively asserted that negative internet memes would influence their sibling’s conception about mathematics as well. willow: no, because i feel like, in these [internet] memes, they are more negative than positive or neutral. so, it’s like i want them to see the better side of math, not people complaining about math. wakanda: i wouldn’t want my younger siblings to see that math is really complicated or confusing. i want them to excel, to do better and be strong, so that the positive things can come to them. intriguingly, the duo opted to only portray one side of mathematics to their siblings, a side that consists of rudimental and effortless mathematics. one concern is that this decision appears to be an overcorrection (e.g., students see all negative messaging about mathematics, so let us instead show all positive messaging), and eventually this conception of mathematics can be troublesome when mathematics becomes onerous. felicia and fae from fairbourne prep affirmed that they too would not share negative internet memes with their younger siblings. they expressed the desire for their siblings to encounter mathematics in an unbiased manner and define it through their experiences, not others’ perceptions. felicia added that she wanted her younger sibling “to have an expectation of what math is going to be like. even though it is going to be challenging, they will figure it out eventually.” fae correspondingly added, i do have a little brother, and when he comes home and shows me all this stuff, he says math is going to be so hard. and i want him to be able to not think like that; i want him to think he can be able to do it. it is important to note that students are not misleading their siblings or falsifying the mathematics experience. in felica’s case, she even acknowledges that math will be difficult, setting the expectation for her sibling, but is overall optimistic that her sibling’s perseverance will hold true. isolate. unexpectedly, 23% (7 out of 31) of participants took a different approach and agreed that exposing their younger siblings to negative mathematics internet memes would be a benefit. participants’ responses mainly emphasized that the sharing of memes is a method to alleviate feelings of isolation and become socially accepted. for example, sharon from silvercliff academy mentioned that although negative memes are dissatisfactory, they help attract people to a shared experience or view, which ironically is a sign of relief for not being the only one. she stated, “it’s like, sometimes, when you think you’re the only one who experiences that, but, like, it’s actually not only you.” michelle from moorhall similarly stated that she would show negative mathematics internet memes to her sister because she is “bad” at mathematics and the memes would help her feel “normal.” she added, “it will make her feel better about herself benoit & salopek an investigation of social media and mathematics identity journal of urban mathematics education vol. 16, no. 1 59 because then she knows she’s not the only one out there that’s bad at math.” in each of these statements, participants express seclusion as the root of their actions because they view negative internet memes as a sign of support. they stated that sharing allows their siblings to release therapeutic feelings, express virtual empathy, and validate social acceptance. although their actions may seem outwardly negative, their rationale provides insight into the socioemotional and sociocultural lenses of mathematics online. in the same vein, samuel from silvercliff academy added, “yeah, ‘cause it’s not like my little sister is dumb. she obviously understands she also hates math. she’s past the point in kindergarten where you’re counting, you know.” although the rationale of samuel’s statement is not grounded in the same theme of isolation like the others, it still draws on the perspective of being part of a community. here, it seems samuel’s actions are a sign of confirmation. samuel implies that his younger sibling’s identity is fixed and concrete; therefore, negative internet memes would not change her or reorient her perceptions toward mathematics. samuel’s statement also eludes to the fact that his sister’s dislike for mathematics is a result of its level of difficulty; once it became challenging, her affection changed. conclusion this study serves as a catalyst, signifying the conceptualizations of power and identity that are rooted in social media. through the course of focus group interviews and the ima, we were able to deduce students’ smrs (e.g., liking, tagging, commenting, creating, sharing), which highlighted their stance toward mathematics messages. students’ smrs to internet memes reinforced negative messages about mathematics. furthermore, students’ comments inspired three storylines: vindicate or villainize mathematics; ability affects susceptibility; and influence or isolate, that is the question? vindicate or villainize mathematics given the participatory nature of digital culture and its invisible but rigid enculturation process, social media becomes saturated with negative mathematics messages, with little to no resistance. in other words, social media users understand the memetic social system, and with no oppositions, they are motivated to produce and reproduce negative mathematics internet memes. this potentially has the power to skew the perceptions of other users who engage with internet memes, permitting the perception that it is acceptable to publicly belittle mathematics, a continuous harmful cycle where mathematics is left defenseless. one deep concern is that social media is an open medium that young people can access and that it can be extremely difficult to form a positive mathematics identity when negative internet memes and pervasive thoughts of mathematics are constantly portrayed, as this collective identity may be difficult to resist. ability affects susceptibility this storyline explicated students’ perspective of mathematics experiences as a static or singular experience, instead of dynamic and having the ability to change from moment to moment. high mathematically performing students are perceived as individuals who have never, or will never, face difficulties with mathematics, making negative internet memes unrelatable to them. lower mathematically performing students are seen to have never, or will never, be able to succeed in mathematics, making them always susceptible to negative internet meme messages. influence or isolate, that is the question? benoit & salopek an investigation of social media and mathematics identity journal of urban mathematics education vol. 16, no. 1 60 a majority of students opted not to display negative internet memes to their siblings for fear it may impact their perceptions about mathematics. students wanted their siblings’ feelings toward mathematics to be defined by them and not by another’s perspective. although the reality is that students cannot shield their siblings from negative mathematics messages forever, some students suggested intentionally displaying positive messages to counteract the negative ones. surprisingly, some students saw sentimental benefits in displaying negative internet memes to their siblings, as they may signify there are others who dislike mathematics. students’ discussions about negative internet memes could become a sign of relief (i.e., “not being the only one”). collectively, students’ testimonies illustrate mathematics internet memes as influential to one’s mathematics identity. lessons learned though the study was facilitated 5 years ago, there are some pressing lessons that are applicable for today. we watched as the pandemic shut down the world and people gravitated to online spaces for human interaction and connection. and although these virtual spaces extend far beyond the classroom and are often regarded as fun and unacademic, they are inherently full of mathematics messages. we offer students’ stances and storylines as conceptual tools leading to a different perspective, as well as an insight into possible tension points of student identities and their smrs. in this discussion, we share some lessons on the intersection of students’ mathematics identities (offand online). throughout the stances and storylines displayed above, we noticed that students who identify themselves as being good at mathematics not only chose to post and redistribute negative mathematics internet memes but also “liked” and left supportive comments on other negative internet memes, adding to their notoriety. although we expected students who think highly of themselves in mathematics to choose to share positively messaged mathematics memes, they did not. instead, they chose to perpetuate negatively messaged ones, even if they disagreed with their individual identity. a lesson learned is that students’ responses do not necessarily mirror their own mathematics identity. they are not just considering their own experiences with mathematics; it is deeper than that. these results lend themselves in support of students’ succumbing and adhering to a larger hegemonic construction of mathematics and embracing the online community’s perspectives over their own. when looking at willow’s and samuel’s comments above, they are highlighting the deeply rooted culture surrounding mathematics in the sociotechnological space and the pressure to follow it. part of the online socialization process is learning to read the social cues and reacting to them accordingly (boyd, 2008; buckingham, 2008). as students join online communities, the already-established social norms about mathematics undergird their interactions way before their accounts are created. looking at other users’ profiles provides critical clues of what is socially acceptable and what is not (boyd, 2008). it is apparent that they are following suit with the established culture and how everyone else is talking about mathematics. as online spaces grew, we watched as students took time to carefully curate their online personalities. also, as we stand back to examine their design choices, they call attention to possible tensions between their digital bodies and their offline selves. when alfred and antwon, who both think highly of themselves in mathematics, were reluctant to post positive mathematics internet memes or defend mathematics online, what were they saying and why? offline, in person, both students were happy to proclaim their mathematical brilliance, but online they faltered, each declaring someone else should vindicate mathematics. in addition, mary’s benoit & salopek an investigation of social media and mathematics identity journal of urban mathematics education vol. 16, no. 1 61 comments above indicate she does not want her sibling to be ridiculed for being a good mathematics student. instead, she wants her sibling to “play the game.” consequently, she presents her sibling with online mathematics discourse to become familiar and camouflage into the charade. another lesson learned is that writing oneself online can be a difficult process, especially when it is in direct opposition with the dominant sociotechnological perspectives (boyd, 2008). for a lot of students, being mathematical is synonymous with being a “nerd,” and, in some cases, students may think being a “math person” means subverting parts of their identity (boaler & greeno, 2000). although students theoretically have the potential to (re)create themselves online, the social structures that regulate popularity offline are present online as many students’ online networks primarily consist of their localized friends (buckingham, 2008). thus, the looming pressure for validation present during offline face-to-face conversations may also exist in online communication. understandably, students are cognizant of what they post, knowing that there may be some face-to-face interactions following. although these two spaces (offand online) may be presented as unique, they are not mutually exclusive. students’ comments and actions illustrate a duality that tethers their onand offline mathematics identities that may not be as simple to align. offline, the structures of schools and classmates provide a sense of community (walker, 2012), but online it is unclear what those supportive structures are, as their actions are individual experiences that cause students to learn primarily by doing. therefore, one thought is to create a formal space (in person or virtual) for young people to deliberate and debrief their thoughts, curiosities, and sentiments about mathematics and/or mathematics messages. these discussions can offer students a space to openly reflect on their past and present experiences, including experiences that might support mathematics stereotypes, in hopes of dispelling misconceptions and possibly helping students reimagine and reconstruct their experiences moving forward. references anderson, r. 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(2018). identity-related issues in meme communication. internet pragmatics, 1, 113– 133. benoit & salopek an investigation of social media and mathematics identity journal of urban mathematics education vol. 16, no. 1 66 appendices appendix a: focus group demographics focus group data table school/ type/ location students grade self-perceived math ability (spma) (1–10) w es tp in e h ig h sc ho ol pu bl ic / b ro nx , n y w1-wilma 12th 8 w2-willow 12th 6 w3-wilbur 11th 7 w4-wilson 11th 5 w5-wakanda 11th 7 w6-william 11th 7 si lv er cl if f a ca de m y pu bl ic / b ro ok ly n, n y s1-sharon 9th 8 s2-sean 9th 8 s3-susan 9th 6 s4-samuel 9th 8 s5-scott 9th 9 s6-steven 9th 8 m ar bl ep on d c ha rt er c ha rt er / m an ha tta n. n y ma1-adam 10th 8 ma2-alex 10th 10 ma3-anthony 10th 8 ma4-antwon 10th 9 ma5-akil 10th 9 ma6-alfred 11th 9 fa ir b ou rn e pr ep pr i va te / m an ha tta n . n y f1-felicia 10th 6 benoit & salopek an investigation of social media and mathematics identity journal of urban mathematics education vol. 16, no. 1 67 f2-francesca 10th 8 f3-falyn 11th 3 f4-francine 10th 9 f5-fae 10th 9 m oo rh al l pu bl ic / q ue en s, n y m1-mike 9th 5 m2-maureen 9th 8 m3-michelle 9th 9 m4-melissa 9th 8 m5-mark 9th 8 m6-matt 9th 9 m7-mary 9th 8 benoit & salopek an investigation of social media and mathematics identity journal of urban mathematics education vol. 16, no. 1 68 appendix b internet meme activity (ima) protocol benoit & salopek an investigation of social media and mathematics identity journal of urban mathematics education vol. 16, no. 1 69 appendix c student focus group questions name of school: day: date: time: script for students: my name is _______. we are researching math in popular culture. the focus group will be in three parts. we will ask you a couple of questions about math in popular culture as a group, give you some memes to sort, and then return as a whole group for more discussion. your answers will help us determine if there is a link between math education and popular culture, so we need you to be open and honest. this focus group could take anywhere from 45 min to 1 hr. your participation is totally voluntary, and refusal to participate will not result in any consequences. should you need to stop the interview at any point, we can reschedule it for another time. everything you say will be confidential; this is a safe space, so we encourage you to tell the truth. any questions, comments, or concerns about what we just talked about? what is the best method to follow-up with you should we need further clarification on anything you said? group (part i) 1) while on the internet, do you go on any social media websites? which ones, how many times? 2) what images do you see now? do any of them pertain to math? 3) what are the people doing when math is being portrayed? 4) how do the people look? (jeans/clothes/braids, etc.) show them an unrelated meme à funny meme has nothing to do with math 5) what do you see in it? describe it. do you see this as positive, negative, or neutral? 6) have you seen this one before? 7) in your own words, what’s a meme? what do you think its general purpose is? do you think all memes have the same purpose? do you think there are messages in them? 8) do you think memes are influential in any way? benoit & salopek an investigation of social media and mathematics identity journal of urban mathematics education vol. 16, no. 1 70 9) if you had to pass a message, would you use a meme? do you think they are influential ways to pass messages? 10) have you seen any memes relating to math? (if yes, describe them.) explain. individual (part ii) show them math memes on the table {read} direction: in front of you are nine different mathematics memes. your job is to place each meme into one of the three categories (p/neu./neg). place the number located on the back of the meme in the appropriate category on your student answer sheet. there is no right or wrong answer. there may not be an even number of each category present; as a matter of fact, one or two categories may not be used. it is up to you to place the memes where you feel they should go. you are in complete control! so, when you are done, write a description of why all of those memes are in their category (i.e., why are the ones you’ve selected positive?). afterward, write down the number specifying which memes you would post. also, if you come across an interesting one and you want to discuss it further, place the number in the discussion category, and then we will discuss the most frequently chosen meme as a group. if you come across one that you absolutely agree with or absolutely disagree with, write the number in the “agree with” box or the “disagree with” box. also, use the tape recorder and explain why you disagree or agree with it. group (part iii) 11) have you seen any of these memes before or memes like these? please describe them. 12) which meme would you post? have you shared or posted any memes like this? if so, where? why do you think you did that? what were the comments like? *student may choose a number to discuss* 13) do you feel like there are messages behind any of them? what’s the message behind it? how do you know? benoit & salopek an investigation of social media and mathematics identity journal of urban mathematics education vol. 16, no. 1 71 14) if you had a younger sibling, would you let them read or see these memes? why do you think? 15) was it the picture that decided the message or the words (caption)? 16) do you think these memes are influential in any way? 17) do you think everyone interprets the meme in the same way? do you think the message changes depending on gender, race, or age? 18) if students consistently see these memes, do you think it will influence the way they think about mathematics? microsoft word 418-article text no abstract-2230-1-18-20201224 (proof 1).docx journal of urban mathematics education may 2021, vol. 14, no. 1, pp. 71–95 ©jume. https://journals.tdl.org/jume cathery yeh is an assistant professor at chapman university in the attallah college of educational studies, 1 university dr., orange, ca 92866; email: yeh@chapman.edu. her research focuses on critical mathematics education, humanizing practices, ethnic studies, and social justice teaching and organizing. ricardo martinez is assistant professor in mathematics education in the department of teaching, learning and teacher education at the university of nebraska–lincoln, 1430 vine st., lincoln, ne 68505; email: rmartinez21@unl.edu. his research centers on paradigms of critical youth studies in mathematics education. sarah rezvi is a doctoral student at the university of illinois at chicago in the college of education, 1040 w. harrison st., chicago, il 60607; email: rezvi@uic.edu. her research interests explore the intersections of pre-service and early career mathematics teacher preparation in the united states, critical race feminism, and identity construction. shraddha shirude is a mathematics teacher at garfield high school in the seattle public school district, 400 23rd ave., seattle, wa 98122; email: ssshirude@seattleschools.org. her pedagogy uses an ethnic studies approach to dismantle white supremacy norms in the mathematics classroom. radical love as praxis: ethnic studies and teaching mathematics for collective liberation cathery yeh chapman university sarah rezvi university of illinois at chicago ricardo martinez university of nebraska–lincoln shraddha shirude seattle public schools ethnic studies is a growing movement for curricular and pedagogical practices that reclaim marginalized voices and histories and create spaces of healing for students of color; however, its application to mathematics education has been limited. in this essay, we provide a framework of five ethea of ethnic studies for mathematics education: identity, narratives, and agency; power and oppression; community and solidarity; resistance and liberation; and intersectionality and multiplicity. we describe key concepts and examples of the ethos of ethnic studies. keywords: ethnic studies, love, mathematics for liberation, transformative resistance yeh, martinez, rezvi, & shirude radical love as praxis journal of urban mathematics education vol. 14, no. 1 72 i see your lesson plan book scrawled in red, black or purple whichever color best fits your mental state fatigued eyes from too-blue screens the deluge of emails` confusion and frustration schedules and systems built upon the brokenness of the world the failure to recognize each other’s humanity somehow, you must navigate all of this somehow, you must subvert all of this always, there is lack always, there is more always, there is comparison always, the message give, give, give until you snap in two give, give, give until you collapse under this weight pause. observe. breathe. reflect. and remember that in times of tumult full of lashing winds the trees that protect each other that share the trees that dance and weave that bend in the eye of the storm are the ones that do not break our interconnected roots whisper to us softly i am because we are we are because i am a gentle reminder a firm demand do not forget to take care of you do not forget to care for each other this is how we win ~ —poem by sara rezvi, “on the eve of this school year | a love letter to radical math teachers” yeh, martinez, rezvi, & shirude radical love as praxis journal of urban mathematics education vol. 14, no. 1 73 e write this out of love! our love for students. our love for teachers. our love for our communities. we recognize the epistemological force of love and how it is entangled in how we make sense of the multiple and often contradictory spaces we traverse as scholars of color, as children of immigrants, and as mathematics educators (anzaldúa, 1987; darder, 2017; hooks, 2001). once more, we write this out of love. we write this looking into the mirrors of our souls. we look into each other’s reflections and see collective grief, collective mourning, collective pain. we see the calls for teachers to sacrifice their bodies, their beings, their lives. we bear witness to the pain of families still reeling from mass death, loss, pain, and poverty (wrigleyfield, 2020). we recognize the gross inhumanity and violence of these calls and the similarities between and within mathematics education (capraro & chang, 2020; leonard, 2020). liberation in mathematics education cannot be conceived without a commitment to our humanity and to love as a grounding force (darder, 2017; freire, 1998). the orientation to love proposed here is not a romanticized or liberal notion of love. our definition of radical love draws on orientations from activist and critical scholarship (boggs; 2012; davis, 1990; freire, 1968/1970; 1998). paulo freire, a critical humanist, stipulates both education and love as political projects and called for an “armed love”—the fighting love of those convinced of the right and the duty to fight—with a revolutionary, radical essence, conceiving love as an act of freedom that becomes the pretext for other actions towards emancipation (freire, 1998). radical love offers us possibilities to understand our relationships with each other and oppressors in ways that challenge complacency. angela davis (1990) reminds us that radical means “grasping things at the root.” therefore, to ground ourselves in the tradition of radical love, we argue that mathematics education must adopt a radical form of resistance that is interdependent, collective, healing, and nurturing for our collective spirits. this is especially important as we, mathematics educators and teachers, navigate complex terrain and search beyond privileged spaces to locate knowledge from communities that disrupt colonial systems of knowledge making. white settler colonialism has been a major force that not only defines but also sets the conditions for mathematics teaching and learning as it appears today (bullock, 2017; larnell et al., 2017; martin et al., 2010). school mathematics in the united states is tied to histories of racial apartheid, which shape both curriculum and schooling structures and are still perpetuated today through the seemingly benign practices of tracking and ability grouping as well as teacher expectations and curricular differentiation (louie, 2017; martin et al., 2010). perhaps the most oppressive aspect of a colonizing education is the way in which it functions to deculturalize by “destroying a people’s cultural identities (cultural genocide)” (spring, 2016, p. 8). we join a growing group of mathematics educators, researchers, and activists to contend that mathematics is political (e.g., bullock, 2017; goffney et al., 2018; gutiérrez, 2013; leonard, 2020). mathematics has been used as a weapon to legitimize capitalist w yeh, martinez, rezvi, & shirude radical love as praxis journal of urban mathematics education vol. 14, no. 1 74 interests, produce stratified achievement, and position children and families of color at the bottoms of social strata (martin, 2013; yeh, 2018). as dr. gloria ladsonbillings (2017) reminds us, “school is often a place of trauma” for many students of color (p. 5). our affirmation of the problems in mathematics education is not unique or new. these issues have existed since the inception of the united states. our intervention in this paper uses an ethnic studies framework to deliberately center radical love, hope, and liberation within mathematics education. we are cautious here not to propagate myopic and pervasive understandings that sever mathematics education from the histories of colonialism. we choose to not subscribe to over deterministic renderings of schools as either mere instruments of social and economic domination or as sites of liberation and possibilities (freire, 1998). as educators committed to a more humanizing pedagogy, we see education as a site of social reproduction and as a potential site for transformation (freire, 1968/1970, 1998; hooks, 1994). schools can be a place in which students’ ideas and identities are honored and leveraged, and mathematics education can help bring equality and justice to an unjust world (freire, 1968/1970; gutstein & peterson, 2013; leonard, 2020). as educators of color whose work in mathematics education centers on antiracist and decolonizing frameworks, we acknowledge the many privileges that we have and our roles as both the oppressor and the oppressed (freire, 1968/1970). we offer this writing as a collective—of classroom teachers, mathematic educators, and community activists—in which our commitments to ethnic studies transpire in both institutional and community spaces through community-based education, youth participatory action research, and community organizing. while the ethnic studies movement is growing in both college and k–12 schools, its application to mathematics education has been limited (sleeter & zavala, 2020). here, we humbly share how ethnic studies gave us the language and concepts to put our dreams of humanizing and rigorous academic experiences for youths of color into action. what follows are ethea developed through sessions of dialogue and shared reflection on our praxis and then an excerpt of our dialogue. this dialogue models ethnic studies as a process of communal learning towards collective consciousness and action. it is only with others are we able to identify structural and internalized oppression and to dismantle these structures in mathematics education by naming them, questioning their existence, and then envisioning and working towards alternatives (boggs, 2012; freire, 1968/1970, 1998). what follows is a conceptualization of ethnic studies mathematics by epistemically situating ethnic studies in the socio-cultural-historical-mathematical world (frankenstein, 1983; gutstein, 2008). yeh, martinez, rezvi, & shirude radical love as praxis journal of urban mathematics education vol. 14, no. 1 75 there is emotion in mathematics: ethos and epistemology we are the body of light that will inspire an entire lifeline you are the key to the clouds that block the moon as long as you believe in you (kumari, 2016, stanza 3). in this section, we discuss the ethos of ethnic studies mathematics, along with the epistemological posture of our life-work presented throughout this paper. when we say life-work, we honor the power with(in) (ayala et al., 2018) each of us, along with our inherent connections to others through acts of spiritual activism (anzaldúa, 2009). spiritual activism represents non-binary modes of thinking and being when engaging in acts of social and individual transformation, which arise from gloria e. anzaldúa’s own lived experiences (keating, 2008). we center our identities as mathematics educators across multiple privileged and oppressed lived experiences to challenge the current state of mathematics education by providing a humanizing alternative. only those subject to oppression can fully grasp the injustice in education (freire, 1968/1970), and it is only an epistemic unraveling of injustice that can lead to the resistance (medina, 2013) necessary to humanize mathematics. ethnic studies creates a nexus for such hope and love in the teaching, learning, and living of/with mathematics. while ethnic studies is a contested and emergent field, we define it broadly as a movement for curricular and pedagogical projects that reclaim marginalized voices and histories and create spaces of healing (dingle & yeh, in press; martinez, 2020; rangnath et al., in press). both of these two processes are tied to social actions that challenge and transform oppressive systems and cultures of domination. the goal of the ethnic studies movement is to rehumanize experiences of communities of color, challenge problematic eurocentric narratives, and build community solidarity across categories of difference, such as race, class, gender, and sexuality (ferguson, 2012; omatsu, 2003). ethnic studies can be framed as anti-racist in the sense that it attempts to unpack, challenge, and eradicate racism as it takes place in our schools and in the broader society, but ethnic studies strives to do more than this. ethnic studies can also be framed as part of a broader process of decolonization or “delinking that leads to de-colonial epistemic shifts that brings to the foreground other epistemologies, other principles of knowledge, and understanding” (sleeter & zavala, 2020, p. 24). building on our own experiences with ethnic studies projects (dingle & yeh, in press; rangnath et al., in press) as well as the ethnic studies literature (e.g., banks, 2008; cammarota, 2016; cuauhtin, 2019; sleeter & zavala, 2020; tintiangcocubales et al., 2015), we developed the framework of five ethea of ethnic studies for mathematics education: identity, narratives, and agency; power and oppression; yeh, martinez, rezvi, & shirude radical love as praxis journal of urban mathematics education vol. 14, no. 1 76 community and solidarity; resistance and liberation; and intersectionality and multiplicity. ethos of ethnic studies: interconnectedness figure 1. ethea of ethnic studies mathematics ethnic studies mathematics seeks to humanize the classroom for students of color, who have a history of experiencing dehumanization through silencing of their identities, perspectives, and intellectual abilities. humanization is a collective endeavor linked by the dialectical of human-to-human and human-to-self communication (ayala, et.al., 2018). communication is an act both in the political power of education and in its ability to provide self-reflection. ethnic studies is a bridge that cultivates multiple ethos of love (rangnath et al., in press). in figure 1, the different ethea lie centered on the perimeter of the circle that divides ethnic studies mathematics from community-family, meaning the ethea inhabit space in both. the perimeter acts as a link between ethea, signifying that these ethea are interconnected and mutually dependent. additionally, the ethea are informed by the historical-culturalworld that the community-family is situated in, hence the arrow from the historical yeh, martinez, rezvi, & shirude radical love as praxis journal of urban mathematics education vol. 14, no. 1 77 cultural-world manifests in families and communities to inform and develop the ethea. the ethea are placed on the same circle to emphasize that they are interconnected, non-linear, and non-hierarchical. for example, the ethos of power and oppression is directly connected to the ethos of identity, agency, and narratives however, the individual ethea do represent specific bodies of history/research, thus they are discussed as independent at times. the arrows between each ethos and the center circle is where/how ethnic studies makes education an intimate experience for learners and where ethnic studies informs the learner while simultaneously allowing the learner to contribute to what is ethnics studies. ethnic studies mathematics is a process that connects mathematics learning to the community and to the real world while acknowledging the situated, embodied, and emotional aspects of learning (goffney et al, 2018; lakoff & núñez, 2000; lipka et al., 2005). hence, a one-size-fits-all approach to ethnic studies can be toxic by ignoring the uniqueness of individual learners, families, and communities. the center circle of figure 1 highlights the mathematics that is already inside of every learner (goffney et al, 2018; lakoff & núñez, 2000; lipka et al., 2005). the false dichotomy between self and mathematics created by eurocentric narratives ignores and perpetuates racism, heterosexism, ableism, capitalism and subjectivism (bullock, 2017; gutiérrez, 2013; madden et al., 2019; yeh et al., 2020; yeh & rubel, 2020). the fact that people have been socialized into believing that they are either “math people” or not is why the divide exists, but note that the line is dotted because mathematics naturally seeps into and builds from a person's identity, and all people contribute to mathematical knowledge. the dotted line represent the resistance of marginalized people to eurocentric-only mathematics and the hope they harbor to one day fully erase the lines that divide us. ethnic studies mathematics is a call to reclaim the inner mathematics already present in all of us. the inward and outward arrows in figure 1 highlight the constant push and pull that is needed to heal mathematics and the self. the ethea of ethnic studies are rooted in the interconnectedness between individuals and their experiences with mathematics, which centers the focus of ethnic studies on the connection between history, culture, family, and youth development as an external and internal process of mathematical becoming (freire, 1974). below is a description of each of the five ethos. ethos of identity, narratives and agency the ethos of identities, narratives, and agency acknowledges identity formulation as not independent but in communion with others, with our ancestral experiences, and with the world around us. while mathematics education often positions identity as an individual trait or as situated within classroom interactions, our ancestors’ experiences also shape our identity (banks, 1999; leonard, 2020). leonard (2020) specifically calls attention, in this era of anti-blackness and strengthened yeh, martinez, rezvi, & shirude radical love as praxis journal of urban mathematics education vol. 14, no. 1 78 white nationalism, of the importance of “discover[ing] one’s roots” (p. 10). knowing the counterstories of our ancestors—the contributions and resilience of people of color in the face of hardship and violence—reaffirm both our cultural and american identity. counternarratives build from the belief that communities of color hold within them deep resources and ways of knowing and being that are particularly important in disrupting colonial ideologies pervasive in mathematics (anzaldúa, 1987; freire & macedo, 1987; leonard, 2020). banks (2004) explains, “the knowledge that emanates from marginalized epistemological communities often contends existing political, economic, and educational practices and calls for fundamental change and reform. it often reveals the inconsistency between the democratic ideals within a society and its social arrangements and educational practices” (p. 237). yet, much of these rich and collective ways of knowing and being have not yet been made visible in mathematics education. ethnic studies requires the rewriting of mathematics curriculum from the perspective of people of color, grounding pedagogies and curriculum in counternarratives that offer historical accounts, interpretations, and cultural practices that question dominant narratives. d’ambrosio’s (1985) work in ethnomathematics and the mathematics in cultural context project, in which mathematics curriculum was developed in collaboration with yup’ik elders (kisker et al., 2012; lipka et al. 2005), highlight the importance of reclaiming identities as central to students’ personal and academic development. as au, brown, and calderon (2016) state, “as peoples of color—what we must know to survive, to understand who (and where) we are, to imagine freer and more joyful futurities—we demand curricula that honor the knowledge production of our ancestors; engage the yearnings of our children, families, and communities; and interrogate the enduring tradition of white supremacist subjugation and misrepresentation” (p. 151). ethos of power and oppression equity-based reforms in mathematics education have focused on access and inclusion to acquiring knowledge specific to the discipline with minimal attention to the historical, social, cultural, and political situatedness of mathematics education; thus, these initiatives have perpetuated existing racial, gendered, linguistic, and ableist hierarchies (bullock, 2017; louie, 2017; yeh et al., 2020). freire and macedo (1987) remind us that “the intellectual activity of those without power is always characterized as non-mathematical” (p. 122). what we argue here is the importance of examining and abolishing ideological systems of oppression in mathematics education. mathematical “smartness” is an ideological system that perpetuates whiteness and serves as a tool for stratification (bullock, 2017; leonardo & broderick 2011; yeh et al., 2020). ability is commodified and seen as property; success of mathematics is measured in terms of achievement of numerical target, and such data are used to construct and constitute some students as “mathematically smart” while yeh, martinez, rezvi, & shirude radical love as praxis journal of urban mathematics education vol. 14, no. 1 79 simultaneously constructing and constituting other students as “not-so-smart,” deficient, or learning dis/abled (leonardo & broderick, 2011; yeh et al., 2020). the construct of mathematical “smartness” intersects both race and ability. mathematics education has a legacy and history of systemic epistemicide of peoples of color—the erasure of knowledge systems, including languages, experiences, and interpretations of the world, and ways of coming to know and understand—through exclusion within schooling from its functions, curriculum, and pedagogy (louie, 2017; martin, 2013; martin et al., 2010). despite 40+ years of research on culturally responsive pedagogies and multicultural education, the mathematics learning experiences of students of color still mirror u.s. politics of xenophobia and assimilation. students of color educated in western schooling practices learn to perform mathematics using algorithms that are not their own, in a language different from their native tongue, and solving mathematics problems irrelevant to their interests and experiences. challenging oppression in mathematics education requires us to shift from the single conceptual standpoint to recentering nondominant perspectives and knowledge traditions. freire (1968/1970) posits that the oppressed have insights into the nature of oppressions that are necessarily hidden from the oppressors. it is the oppressed who must lead the struggle for liberation on their own behalf and for the oppressors, not only because of their insights but because any attempts for the oppressors to lead the struggle is a reenactment of the power relations that exist under conditions of exploitation. it is in the moment that the oppressed (student, educators, and communities of color) take charge of the struggle for liberation that they evidence to themselves and to the oppressors a clear understanding of their own humanity as agents in history. ethos of community and solidarity the third ethos is about community and solidarity. the central purpose of ethnic studies is rooted in the third world liberation front movement (ferguson, 2012; omatsu, 2003). ethnic studies emerged out of students’ need to belong, to see themselves in the curriculum, and to be respected as human beings. ethnic studies mathematics makes explicit the centrality of community as a site for learning, organizing, and activism (civil, 2007; cuauhtin, 2019; sleeter & zavala, 2020; tintiangcocubales et al., 2015). it is important to stress, however, that community contexts differ across geographic locations, which is to say, there is no single way to approach mathematics teaching and learning from and with youth of color. this is where ethnic studies begins, through community responsive methodologies and pedagogies (civil, 2007; sleeter & zavala, 2020; tintiangco-cubales et al., 2015). mathematics education that centers on equity for students of color must place them out front and create methodologies and pedagogies that are responsive to the specific needs, strengths, and contexts that communities possess. ethnic studies draws strength from community and from the differential spaces of struggle and solidarity. learning is a process yeh, martinez, rezvi, & shirude radical love as praxis journal of urban mathematics education vol. 14, no. 1 80 that is lived, engraved in students’ bodies and memories, not just what is experienced cognitively (anzaldúa, 1987; collins, 2000; hooks, 2001; lakoff & núñez, 2000). students come to the mathematics classroom with not just an identity but a community of people who have helped influence, impact, and shape that identity. connecting to students’ communities is a critical aspect of countering the master narrative of who and what mathematics is. ethos of resistance and liberation local and community-based knowledge cannot be used solely as a resource to critically read and write the world through mathematics (freire, 1968/1970; gutstein & peterson, 2013); ethnic studies projects must also lead to resistance and liberation (sleeter & zavala, 2020; tintiangco-cubales et al., 2015;). the ethnic studies program in arizona consisted of a youth participatory action research program called the social justice education project (sleeter & zavala, 2020). this provided students the opportunity to engage in transformational resistance (solorzano & bernal, 2001), which requires a critique of social oppression and a motivation driven by social justice (cammarota, 2016; gutstein & peterson, 2013; san pedro, 2018). ethnic studies mathematics is a revolutionary praxis of mathematical spiritual wisdom, which consists of the cultural, historical, spiritual, and logical forms of mathematics that help us connect to other individuals and additional forms of mathematics that are collectively created (martinez et al., 2021; shirude, in press). a true revolutionary praxis (freire, 1968/1970) where action and reflection harmonize to create the sound needed for social transformation is a necessity to reach resistance and liberation. mathematical reflection requires mathematical spiritual wisdom (martinez et al., 2021; shirude, in press). reflection alone cannot lead to liberation, and we cannot forget the need for mathematical action in the classroom. mathematical action must connect mathematics learned in the classroom to the community and greater society (martinez et al., 2021; yeh & otis, 2019). ethnic studies mathematics is a revolutionary mathematical praxis that balances mathematical action and mathematical reflection towards transforming mathematics education. rahman (1991) was speaking of participatory action research (par) in saying, “an immediate objective of par is to return to the people the legitimacy of the knowledge they are capable of producing through their own verification systems, as fully scientific, and the right to use this knowledge” (p.15), which we can extend here to say ethnic studies mathematics has the same goal. as we return the memory of already being mathematicians to the people, the people can then liberate us all as they liberate themselves. ethos of intersectionality and multiplicity rather than conceptualizing identities as fixed and essentialized, ethnic studies probes intersectionality and multiplicity as a fluid, temporal spectrum. it recognizes yeh, martinez, rezvi, & shirude radical love as praxis journal of urban mathematics education vol. 14, no. 1 81 the need to attend to multiple ways of being and knowing, especially from gendered contexts in which mathematics has been socially constructed with(in) (gholson & martin, 2019; walkerdine, 1998; yeh & rubel, 2020). although ethnic studies centers its analysis on race and racism, an intersectional approach also attends to multiple identities, as all of us embody multiple identities. teachers and researchers working from an ethnic studies framework attend to students’ multiple social identities and their positions within intersecting relations of power in mathematics. crenshaw (2017) defines intersectionality as a vehicle to represent the practices and embodied knowledges that, historically, have characterized the lived experiences of her women-of-color foremothers, who simultaneously navigated the complexities of gender, race, language, and other identity politics (collins & bilge, 2016). structural intersectionality “refers to the connectedness of systems and structures in society and how those systems affect individuals and groups differently” (few-demo, 2014, p. 181). crenshaw (1989) explained, for example, black women’s marginalization is “greater than the sum of racism and sexism” (p. 140). furthermore, her racialized, classed, and gendered being is constructed “by and through race, and that the production of [black] women and other stable gender categories require[s] violence” (haley, 2016, p. 8). in schooling, violence is enacted through institutionalized policies, such as school disciplinary practices that offer brutality, suspension, expulsion, and detention. a department of education study from 2011–2012 revealed that black girls were six times more likely to be suspended than their white female counterparts (lyfe, 2012). we demand an end to these practices and the carceral mechanisms currently operating in k–12 institutions that disproportionately impact black girls particularly and students of color more broadly. an ethnic studies framework attending to the ethos of intersectionality and multiplicity recognizes, affirms, interrogates, and then disrupts hegemonic gendered violence and trauma experienced in mathematics classrooms. the notion of multiplicity not only connects to oppression but also empowerment. borderlands consciousness (anzaldúa, 1987) refers to identities of peoples of color as produced in multiple and often contradictory physical, social, and political spaces—spaces that are inhabited by both dominant and nondominant forms of feeling, seeing, and doing. as such, our worldviews as students and educators of color include homegrown ways of knowing as well as western epistemes. these approaches empower educators and students of color in that they recognize the advantage of having insights into the experiences of the oppressed and the oppressors and may reveal the cracks that support social change (anzaldúa, 1987; collins, 2000; davis, 1990; hooks, 1989). as educators and researchers, our work should attend to and leverage the multiple ways of knowing through an intersectional lens. the flexibility that an ethnic studies framework offers acts as a site of disruption and counterweight to how oppressive power morphs and diffuses into mathematics classroom spaces. yeh, martinez, rezvi, & shirude radical love as praxis journal of urban mathematics education vol. 14, no. 1 82 based on the ethea provided in this framework, the design of an ethnic studies mathematics curriculum needs to a) center on the perspectives of people of color, b) critically identify and challenge how racism and colonialism functions in mathematics education, c) attend to students’ multiple identities and their positions within intersecting relations of power, d) engage in community-based pedagogies and experiences that bridge mathematics classroom to community and social movements, and e) respect all students’ curiosity, thinking, and intellectualism. table 1 ethos of ethnic studies mathematics ethos description identities, narratives & agency identity, narratives, and agency in mathematics, as defined by ethnic studies, are the ways in which we view and learn about ourselves as mathematical beings that go beyond the dominant narrative. this requires an intentional grounding of pedagogies and curriculum on counter-narratives that offer historical accounts, interpretations, and cultural practices of communities of color. as mathematical beings, our humanity is tied to our ancestors, to each other, and to our relationship with the natural world defined by mathematics. power & oppression power and oppression in mathematics, as defined by ethnic studies, is the acknowledgement of the coercive powers that have historically been used to silence and disrupt the liberation of peoples of color. dismantling hegemonic structures requires naming them, questioning their existence, and then envisioning and working toward alternatives. the internalization of our true mathematical identities as a tool against coercion and disenfranchisement is our power. community & solidarity community and solidarity in mathematics, as defined by ethnic studies, see mathematics as integral to activist movements for social justice. ethnic studies is a process that connects learning to the community and to the real world, acknowledging the situated, embodied, and collective nature of learning and change. mathematics learning is not just experienced cognitively; it is a process that is lived, engraved in students’ bodies and memories, and shaped by our histories, ancestors, and communities. resistance & liberation resistance and liberation in mathematics, as defined by ethnic studies, is the determination to resist & disrupt oppression. ethnic studies mathematics projects engage critical consciousness that moves toward praxis—students take action at the individual and/or community level to create change in the world using mathematics. peoples of color find empowerment in the internalization of their own mathematical identities. intersectionality & multiplicity intersectionality and multiplicity in mathematics, as defined by ethnic studies, explicitly examines tensions at the intersections of identities and attends to broader sociohistorical discourses that are at work. rather than conceptualizing identities as fixed and essentialized, identity is seen as a fluid, temporal spectrum shaped by sociohistorical context. yeh, martinez, rezvi, & shirude radical love as praxis journal of urban mathematics education vol. 14, no. 1 83 dialogue: ethnic studies, mathematics, and healing the framework of the five ethea of ethnic studies mathematics shared above developed through sessions of dialogue, storytelling, and shared reflection on our praxis as a collective of classroom teachers, mathematic educators, and community activists. we continue the traditions of our ancestral heritage by storytelling here. we invoke the cultural memories of how we process, of how we understand, of how we listen and honor each other's brilliance through the form of dialogue. the excerpt included was selected to highlight our own meaning-making and application of ethnic studies as a humanizing process for us, our students, and the readers. we began meaning-making when yeh asked the question, “how can the application of ethnic studies in mathematics serve as a process of healing?” ethnic studies is not curriculum; it is a pedagogy, a process that is lived. what follows is our conversation, a living manuscript (shor & freire, 1987) of knowledge construction. shirude: my instinct is to go back to what i had said earlier: using ethnic studies within mathematics changes the narrative of what mathematics is, what it has been, and what it can be. mathematics is used all day long by every human being all the time, and we just don't acknowledge these doers as mathematicians. mathematics has actually come from the same communities who have historically been told it was an accident. the things we hear, read, watch in books or movies, would have you believing that these mathematical wonders were accidents (dewey, 1986), that the people who created them didn't really recognize that they were doing mathematics. but, like, how would you know? why do you get to decide it was an accident? did you ask them? i can’t believe that it was an accident, given how accurate the mathematics are, but that's the narrative that's been told (zinn, 1980/2015). an easy example to contextualize this is to look at our students’ thinking in the classroom. something that's commonly used with mostly k–8 educators, but i wish k–12 teachers used, is number talks. a simple explanation of what a number talk is you put up a relatively simple arithmetic problem, for example 7 * 8, then you ask students to figure out multiple different ways that answer could be found. finally, you ask students to share their way, and we write down all of the different ways. the actual process has more steps; i’m just giving a quick brief on it. by not just valuing but encouraging multiple ways of solving this problem, we begin to start changing the narrative of what is correct in math class (anderson, 2007). it’s small and it's not ethnic studies per se, but it's kind of leaning into what we're talking about here because it yeh, martinez, rezvi, & shirude radical love as praxis journal of urban mathematics education vol. 14, no. 1 84 changes the narrative that, well, 7 * 8 is 56, that's it. it starts to break into “how are you thinking? why do you think that way?,” and it starts to lean into ethnic studies. the way that i leaned into ethnic studies here was just talking about numbers. and that math talk allows us to look at math history: how did different peoples learn and understand the idea of multiplication. it is often taught as though its inception was a linear process from addition straight to multiplication. perhaps there was more to it than that. maybe a student can understand multiplication far easier than they can understand addition or repeated addition. maybe patterns that showed the commutative property were observed and then connected with patterns regarding addition that were later observed. when you consider both of those, you start to break down the reality that the way we have often taught math isn't the only way that things have been learned or discovered or created. this becomes a healing process for children because it's freeing (fasheh, 1997); it literally frees their hearts and minds and souls from the idea that the way that they think is right or wrong. rather, it encourages the idea that they are mathematical beings and the way that they think and feel and learn is mathematical. rezvi: i have to wholeheartedly agree with what shraddha is saying. mathematics is a human construct, a human invention, and thus a way of knowing that belongs to all people. we know that mathematics has powerful origins, for example, from the islamic world. the word “algebra” itself is arabic and roughly translates to “balance and completion,” which resonates for me as words of healing in and of itself. both this word and the word “algorithm” come from one of the most brilliant mathematicians of the islamic world, al-khwarizimi. we also know that the oldest library in the world was founded by fatima al-fihri, a muslim woman in morocco, in 859 a.d. (siddiqi, 2018). we don't typically learn this kind of history in our schooling—that there was this great love across cultures and generations for learning and a profound respect for knowledge. we are forcefully prescribed a very specific lens to look through and then internalize. we typically learn about fermat, euler, newton, leibniz, plato, as if these were the sole owners and creators of mathematics. their contributions were valuable, but they weren’t the only inventors of their findings. yeh, martinez, rezvi, & shirude radical love as praxis journal of urban mathematics education vol. 14, no. 1 85 i am thinking about rudine sims bishop’s work in literacy and her call for books to function as windows, sliding glass doors, and even mirrors that reflect the reader’s own experience. i want mathematics, just like literacy, to reflect and honor its multicultural heritage of pathfinders and voyagers, of weavers and artists, of predictors rather than the truncated version we are required to teach in schools. it reminds me of the words of ezra hyland, “humans don’t make our stories, but it’s our stories that make us human [paraphrasing amiri baraka]… it’s not until we learn the stories of each other that we are able to embrace our true humanity… when i know the story of my people and my culture, that’s when i become human to myself” (jacobs, 2016, 18:00). what are the mathematical stories we need for our own healing, both individually and collectively, to become fully mathematically human? healing can happen when we can know our authentic selves reflected in multiple modalities, not as a side story, nor as a brief mention or a historical footnote, but rather a multidimensional narrative that travels through space and time to center these understandings. i think about the fact that i come from two people, who came from two people before them, who came from two people before them, and so on and so forth. if i use the powers of exponentiation to analyze this and simply go back sixteen generations, approximately 32,768 of my ancestors needed to breathe, love, and live in order for me to be here, reflecting in this space with you today. when we know our full stories, our intellectual origins, our interdependent connections upon which these mutually shared knowledges were built, we engage in healing the epistemological ruptures caused by the toxic diet of western mathematical consumption. i am ending my thoughts here on the brilliance of adrienne maree brown’s thoughts on healing behavior: “we are all the protagonists of what might be called the great turning, the change, the new economy, the new world. and i think it is healing behavior, to look at something so broken and see the possibility and wholeness in it. that’s how i work as a healer: when a body is between my hands, i let wholeness pour through. we are all healers too—we are creating possibilities, because we are seeing a future full of wholeness.” (brown, 2017, p. 14) yeh, martinez, rezvi, & shirude radical love as praxis journal of urban mathematics education vol. 14, no. 1 86 incorporating an ethnic studies framework in our teaching praxis offers a potential space for healing the brokenness currently standardized in mathematics education across the united states. this is particularly true, i feel, for children navigating kindergarten through 12th grade. i am stating explicitly that this is harmful to all children because white children grow up thinking that only western mathematics has value and children of color grow up believing the same. it is the same narrow narrative and must change if we, too, are america. yeh: this is so powerful! this makes me think of a quote from bell hooks that defines for me ethnic studies. as bell hooks (1989) reminds us, the journey of naming and reclaiming who we are is a space of healing and radical opening: “we are transformed individually, collectively, as we make radical creative space which affirms and sustains our subjectivity, which gives us a new location for which to articulate our sense of the world” (p. 23). what you're describing here goes beyond curriculum, beyond the replacing of eurocentric curriculum with one that is “non-eurocentric,” beyond ethnomathematics (d’ambrosio, 1985), or of studying mathematics in student communities. it’s about reorganizing knowledge, curriculum, and processes around questions that are central to the well-being of communities of color. as an immigrant and a person of color in the u.s., i never felt a sense of belonging because i was always invisible—the stories, the contributions, ways of thinking and being of folx of color have been erased. ethnic studies is healing because for too many of our students of color, it has been cultural erasure. valenzuela (1999) describes this process as subtractive schooling. in the mathematics classroom, erasure occurs through denial of our history. the vast majority of discussions on the origins of mathematics and science include only the greeks, romans, and other western populations. yet, africa is home to the world’s earliest form of mathematical thinking and the first known use of measuring and calculation, confirming the continent as the birthplace of both basic and advanced mathematics. sharing the true roots of mathematics history should be part of the mathematics curriculum. culture also goes beyond race/ethnicity but applies to multiple social identities. my oldest child is multiply marginalized (annamma et al., 2013) as a dis/abled young woman of color. when we think of mathematics as being only one way, following a set of yeh, martinez, rezvi, & shirude radical love as praxis journal of urban mathematics education vol. 14, no. 1 87 rules, only expressed verbally or in written form, and when she never sees herself in the context or the text of the problems posed, it is an act of violence to her. it silences her, and it makes her feel as she shouldn't and doesn’t belong in our math classrooms. by honoring her ways of problem solving, her choice of modalities in expression and engagement and of mathematics as embodied (lakoff & núñez, 2000), this is also ethnic studies mathematics. marginality is often described as imposed by oppressive structures; however, marginality is much more than a site of deprivation... it is also the site of radical possibility, a space of resistance (hooks, 1994). those are the margins; we fight whiteness and oppression daily, and our ancestors and our youth have fought this for so long. they hold the key to changing our current state of oppression in math education. they've been disrupting, organizing, and creating change. let’s look to them for guidance. martinez: one word that comes to mind is multiplicity. multiplicity as it relates to identity, fragmentation, and healing that requires us to embrace the complexities of identity in being okay that we cannot assume a specific order to identity development. as such, i want to make sure to honor a lot of women of color whose theorizing around multiple identities has led to ways of resistance, transformation, and healing (torre & ayala, 2009). for me history is important, and i was listening to the new nas album, and he states “the stupidest part of africa produced blacks that started algebra—proof, facts—imagine if you knew that as a child” (2020, 0:43). even though i do not like saying anyone or thing is stupid, i found it powerful, as it relates to reclaiming your history and the role history has in making an individual's multiple futures. if we dive deeper, math is already a part of our ancestors, along with the intergenerational knowledge being described with cathery’s story. i like what was mentioned earlier about ethnic studies, that it's not just ethnomathematics, it is a reflection upon who is centered. note ethnomathematics emerged in the academy, and, yes, d’ambrosio considered himself a freirean and he did amazing work, but it’s not truly centered with the people because we cannot ignore the privilege that comes with being in the academy. we begin to differ in the multiplicity that each of us bring. i am glad we have different experiences from teaching teachers in the university, to our community work, and especially in shraddha being a current ethnic studies high school teacher, which ties back to the multiplicity yeh, martinez, rezvi, & shirude radical love as praxis journal of urban mathematics education vol. 14, no. 1 88 being multiple modes to connect with others in developing our collective identities. cathery, your story reminded me of a counseling session where i was asked what do i think i deserve and responded with tears. it took years to realize that i do deserve to be happy in this world, and i feel ethnic studies can provide an opportunity for such growth in others. people deserve to know that they come from mathematics (freire et al., 1997). one final thing, ethnic studies is a connection to the indigenous history, to the indigenous people across the world. this allows us to spiritually heal in reconnecting to our ancestors and their spirituality (anzaldúa, 1987). the cosmology of many indigenous people did not even have a direct word for mathematics because mathematics was part of their everyday being (león-portilla, 1963/2012). mathematics is another energy that can connect us to other people and to the natural world. no person can be complete in terms of their identity development without acknowledging that mathematics is already a part of them and thus contributed to their identity development. shirude: first, thank you, ricardo, for sharing that really deep personal story about your life. it inspired me to think about the ways mathematics has impacted my life. i remember my k–8 experience: i love mathematics! it is so fun, it is so easy, it just makes sense. i don’t need to try hard at any of it. i’m in a multi-age classroom, grades 1–3, and when second grade comes around, the first graders show up, and i'm like, “let me show you how it's done here.” i guess it was another reason i decided i wanted to be a math teacher. so much of it was just, you know, i had only really experienced two years of school, and i still had a big love for math and specifically teaching it. my grandfather was a math teacher in india, and your last statement reminded me of him because of the idea of mathematical energy. my grandpa didn't say a whole lot, he was a very thoughtful man. i was in second grade, and i taught him how to do a sudoku puzzle. he had never done one before; it wasn't something he knew about, and i taught him how to do it! and literally, every day since i was in second grade until he died when i was a senior in college, he did a sudoku puzzle, every single day for 15 years. i remember this and think, “i had that level of impact on his life, every single morning, simply by teaching him when i was 8 years old yeh, martinez, rezvi, & shirude radical love as praxis journal of urban mathematics education vol. 14, no. 1 89 how to do a sudoku puzzle.” like, wow! it's that mathematical energy you're talking about: teaching him, nothing crazy, simply how to do a puzzle. but, it changed his daily life for the rest of his life, and it inspired me as an eight-year-old. if i can teach this math teacher something about math, then i can change the world. i'm going to be a math teacher! i literally changed the daily existence of my grandfather’s life as an 8-year-old by teaching him something so simple. it is that mathematical energy you talked about; no one would think that i had necessarily changed anything at the time, but my grandfather’s energy changed. i felt it. and what i had felt was that exact phrase that you use: mathematical energy. that just speaks to me so much… i was so inspired as a kid, filled with mathematical energy. then i hit high school, and that passion just shattered. i was like, “i'm not a mathematician. i can't do this.” i struggled so much! but i also just loved it so much that i kept pushing through. then i got to college. i was studying to become a teacher. i was told by the middle school math teacher that i was working with when i was 18 years old (they wouldn’t put me at the high school because i look too young or something), she straight up told me, “you should really just consider switching to be an elementary school teacher. no one's going to respect you here, you look like one of the kids.” i don’t know why she didn’t respect me as a math teacher, she didn’t really get to know me. but, ultimately, i didn’t argue or say anything against what she said to me, probably because i was too afraid or i believed i was too young or too quiet to say anything. i don't even remember all the reasons anymore. but, really, it was her mathematical energy that just demoralized, dehumanized, me and made me agree with her. it was an act of violence. it was her saying without saying it: “you are not a mathematician, the way you do things is not real or true or good enough for math education. you think puzzles are fun? go teach the kindergarteners; they'll love that.” as though, if they're learning different ways to live life, they're not learning. all of that comes from ways of knowing. ways of knowing are so deeply rooted to energy, somehow, that we see humanity and the ways in which we learn and live through that lens (cruz, 2001). mathematical energy and ways of knowing—i love those phrases forever, and i think i said them a hundred times in the sentence. yeh, martinez, rezvi, & shirude radical love as praxis journal of urban mathematics education vol. 14, no. 1 90 martinez: i feel like you just described what ethnic studies is. it cultivates and takes the beauty that is mathematics and refines it into love. love amongst the people. your story about math made me feel a deep sense of love, and thank you for sharing that. conclusion our paper structure and the discussion above pivots from what counts and what is traditionally considered excellence in educational research. this is an intentional act to challenge the essentialization, individualism, and competition pervasive in the field of mathematics education. the dialogue serves as an example of ethnic studies mathematics in practice, emphasizing its collective process. knowledge is co-constructed in dialogue and through story-sharing; thus, acknowledging the centrality of context, communities, histories, relationships, interactions, and connections in learning. it is only with others are we able to identify structural and internalized oppression and to dismantle these structures in mathematics education by naming them, questioning their existence, and then envisioning and working towards alternatives (boggs, 2012; freire, 1968/1970, 1998). ethnic studies mathematics is grounded in radical love—love for self, our ancestors, and the communities we serve. the process of healing begins as we center on the stories (and mathematics) of black, indigenous, and people of color forgotten and forcibly erased (banks, 2008; dingle & yeh, in press; rangnath et al., in press; rezvi et al., 2020), creating a transformative space in which mathematics education can be humanizing by providing a panorama of what is and has always been mathematics. in this paper, five ethea—identities, narratives, and agency; power and oppression; community and solidarity; resistance and liberation; and intersectionality and multiplicity—are shared. birthed from our conversations and the sharing of our stories as mathematics teachers, mathematics educators, and community organizers, the process of developing the ethea of ethnic studies mathematics contributed to our own healing. we ask our readers to consider the same. how does the doing, teaching, learning, and studying of mathematics lead to healing for you, for students, and for our communities? bell hooks says, “i came to theory because i was hurting... to grasp what was happening around and within me. i saw in theory a location for healing” (1994, p. 59). we share our stories as sites of healing, but they are also sites of theory and contribution. our educational journeys and commitments have important implications for mathematics education, which from its origin has never been neutral but instead has been invested in maintaining particular perspectives and realities while at the same time rendering invisible others (patel, 2016). there is wisdom in our stories and the stories of our students, teachers, and teacher educators of color whose voices are too often silenced in our field. how we communicate our experiences, listen to yeh, martinez, rezvi, & shirude radical love as praxis journal of urban mathematics education vol. 14, no. 1 91 them, make meaning of them, study them, and share them allows for mathematics education as a field to advance, interrupt, and create alternatives outside of this deliberate design of domination. ethnic studies mathematics is an epistemology and a pedagogy grounded in radical love. in other words, it is a commitment to the life work for individual and social transformation. we ask the reader to take time to focus on radical love (davis, 1990) in the mathematical spaces you occupy and consider how to challenge and transform what is considered “correct” math or “valid” research and envision healing. we close this piece in meditation and blessing. we are speaking to the silenced you we who keep us safe may the past inform the present may the present transform the future may your ancestors guide you you who come from big shoulders you with the wind whistling in the ribcage of your heart may there be patience, may there be guidance from above and below— i wish we were not being sacrificed this way i wish our society loved us what would the world be like? i wish people would love math like they love their mother like they love their children as a source of strength, wisdom, and courage. do we have to love math? is it not enough to love humanity? a first attempt at an answer our humanity is inextricably tied to our mathematics like moon and sun our orbits always connected with one another may this dance grow ever brighter. — sara rezvi & shraddha shirude, “a prayer for healing and rupture” yeh, martinez, rezvi, & shirude radical love as praxis journal of urban mathematics education vol. 14, no. 1 92 references anderson, r. 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(original work published 1980) copyright: © 2021 yeh, martinez, rezvi, & shirude. this is an open access article distributed under the terms of a creative commons attribution-noncommercial-sharealike 4.0 international license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. microsoft word 2 final nasir vol 9 no 1.doc journal of urban mathematics education july 2016, vol. 9, no. 1, pp. 7–18 ©jume. http://education.gsu.edu/jume na’ilah suad nasir is a professor of education and african american studies at the university of california, berkeley, 660 barrows hall, mc 2572, uc berkeley, berkeley, ca, 947202572; email: nailahs@berkeley.edu. she studies race, education, and learning, and focuses on equity in mathematics education. commentary why should mathematics educators care about race and culture?1 na’ilah suad nasir university of california, berkeley athematics learning has traditionally been thought of as unrelated to matters of culture. it is also typically viewed as outside of the realm of race, aside from concerns about the mathematics “achievement gap.” i argue, as do many others in emerging scholarship on race and mathematics learning (e.g., martin, 2003, 2013; nasir & shah, 2011; shah, 2013), that this long-held perception of the separation of mathematics learning from issues of race and culture is a fallacy. to be effective, mathematics educators and the field of mathematics education in general must be centrally concerned with these issues. race and culture are not only core to the learning process but also they are central forces that organize our society and determine access to high-quality mathematics instruction. in this commentary, i build the case for why the aforementioned statement is of critical importance. i begin by considering why the field of mathematics education (most often) does not attend to issues of race and culture. i then offer three critical reasons why mathematics educators should consider race and culture. as i do so, i describe findings from several studies. specifically, i focus on two lines of inquiry: (a) a set of studies on students’ understandings about and management of racial stereotypes; and (b) a case study of one high school mathematics department, referred to as “railside high,” that successfully enacted an equity pedagogy in mathematics. i close with a few thoughts about why these issues are so critical in this historical moment in our schools and in our nation. why mathematics education does not attend to race and culture there is a long history in mathematics education of seeing race and culture as outside of the purview of the discipline of mathematics, and the myth that race and culture do not matter for mathematics teaching and learning continues to perpetuate. this position is often justified by a set of assumptions that are blatantly false, 1 this commentary is a revised and abridged version of the plenary address delivered at the 35th annual meeting of the north american chapter of the international group for the psychology of mathematics education, chicago, il; november 15, 2013. m nasir commentary journal of urban mathematics education vol. 9, no. 1 8 but which we as a society are deeply invested. they include ideas about the nature of mathematics thinking and learning, such as that mathematics is racially and culturally neutral, and that we should focus on developing “universal” approaches to instruction, rather than culturally specific ones. contemporary theories of learning, particularly theories from sociocultural or ecological perspectives, however, do not support these assumptions (cole, 1996; gutiérrez & rogoff, 2003; howard, 2010; lee, 2007; nasir, 2002; rogoff, 2003; saxe, 1999, 2002; wenger, 1998). instead, these theories view learning as an inherently cultural endeavor that occurs in the context of cultural practices, which happen in relation to cultural artifacts and social interactions carried out to achieve socially and culturally defined goals. not only is learning inherently cultural in nature, but schools, themselves, are cultural institutions. schools are culturally lived and experienced; in that, they are culturally organized, guided by norms, conventions, artifacts, and involve social interaction. they are also potential spaces of empowerment, marginalization, and identity building (i.e., spaces where cultural and identity trajectories are offered and taken up). thus, schooling institutions house complex ecologies and sets of cultural practices and agentic actors. another reason mathematics education does not attend to issues of race and culture is that we live in a society that holds problematic and unrealistic ideas about race (bonilla-silva, 2010; lopez, 2013). these ideas include the belief that when we focus on race, we call attention to it, and thus exacerbating (or creating) race as an issue in schools. this belief is oftentimes connected to the notion that we are in a “post-racial” era, and that racism is a thing of the past, or the notion that when we account for race it is akin to providing “special treatment” (e.g., affirmative action), and that people should be able to achieve on their own merit. although these prevailing “common-sense” notions are powerful, scholarship on race and racism do not support them. sociologists suggest that the very purpose of race as a construct is to provide differential access to resources for the intention of maintaining the social order (massey, 2007; omi & winant, 1994). social stratification, or the process by which differential access is provided to different social groups, involves two mechanisms: allocating people to social categories, and institutionalizing practices that allocate resources unequally between the groups (massey, 2007). key in this process is creating narratives (stereotypes) about social groups (including racial and ethnic groups) to justify the unequal allocation of resources. obviously, most teachers and administrators are not, and do not, consider themselves to be racist. most are well intentioned, and want the best for all students. if that is true, then how do practices of racism persist in school? one of the ways bonilla-silva (2010) argues that racism persists is through how we frame our conversations and ideas about race. he contends, “a new powerful ideology has emerged to defend the contemporary racial order: the ideology of color-blind rac nasir commentary journal of urban mathematics education vol. 9, no. 1 9 ism” (p. 25). color-blind racism refers to the core ideas deeply held in our society that appear not to be about race but in fact are, which, in turn, perpetuate racial ideology. the central component of any racial ideology is its frames for interpreting information. according to bonilla-silva (2010), there are four key frames that allow racism to be perpetuated without anyone appearing racist. the first is naturalization, which refers to situations when people make sense of racial phenomena as natural occurrences. an example is assuming that neighborhood segregation reflects the preferences of racial (or ethnic) groups to live near people like them, rather than reflecting a set of institutionalized practices of real estate or lending practices (e.g., redlining). the second is cultural racism, which refers to the use of culturally based arguments to explain the social standing of minorities. for example, the idea that blacks underachieve in school because they hold a cultural belief that school is not important. the third frame is minimization of racism; people assume that discrimination is no longer a central factor, and thus minimize the proposed effects of race in their explanations or understandings. and the final frame of colorblind racism is abstract liberalism. this frame is a bit more complicated. bonillasilva writes: by framing race-related issues in the language of liberalism, whites can appear to be “reasonable” or even “moral” while opposing almost all practical approaches to deal with de facto racial inequality. for instance, the principle of equal opportunity, central to the agenda of the civil rights movement and whose extension to people of color was vehemently opposed by most whites, is invoked by whites today to oppose affirmative action policies because they supposedly represent the “preferential treatment” of certain groups. the claim necessitates ignoring the fact that people of color are severely underrepresented in most good jobs, schools, and universities, and hence, it is an abstract utilization of the idea of “equal opportunity.” (p. 28, emphasis in original) thus, abstract liberalism refers to the set of ideas about meritocracy and equality in the abstract, and thus, does not consider how these ideals play out in a highly racially stratified society. taken together, these four frames function to allow racism to be perpetuated, without ever acknowledging its presence. throughout this remainder of this commentary, i return to bonilla-silva’s frames as i elaborate why it is of critical importance for mathematics educators to care about race and culture. three reasons why mathematics educators should care about race and culture bonilla-silva’s (2010) work underscores the idea that race is operating even when we do not (or choose not to) see it. this color-blindness (and cultural blindness) is certainly true in mathematics teaching and learning. i argue, however, that nasir commentary journal of urban mathematics education vol. 9, no. 1 10 there are (at least) three key reasons why mathematics educators should attend to issues of race and culture. they are: 1. our society is racially stratified and students experience access to highquality mathematics instruction by virtue of race. 2. racial stereotyping influences access to mathematical identities for students, and thus disrupts mathematics learning. 3. high-quality mathematics instruction (potentially) disrupts unequal access to mathematics learning for students from marginalized groups. i discuss each of these reasons in turn, considering examples and data that support each assertion. reason #1: our society is racially stratified and students experience access to highquality mathematics instruction by virtue of race. we know from research that in nearly every single life outcome, people from marginalized groups (african americans, latinas/os, native americans, and asian pacific islanders), as well as poor people, have less desirable outcomes (nasir, scott, trujillo, & hernández, 2016). our society in the united states is highly racially stratified (carter, 2012) and this stratification plays out in specific ways in the educational arena (carter & welner, 2013; kozol, 2005; orfield, 2001; reardon, 2011). with respect to mathematics teaching and learning in particular, we see consistent and long-standing disparities by race in not only mathematics achievement but also in opportunities to learn on a wide range of dimensions. test scores and course completion outcomes continue to show disparities by race and income level. analysis of the 2005 national assessment of educational progress (naep) scores in mathematics show that 39% of white eighth graders are proficient in mathematics, versus 9% of black and 13% of latina/o. by the end of high school, black and latina/o students’ overall mathematics scores are not significantly different from those of white eighth graders (haycock, 2001; lubienski, 2002). with respect to course completion, 49% of latina/o and 47% of black students have taken algebra or pre-algebra, compared to 68% of white students, and as of 2000, 13% of students from poor families were “proficient” or “advanced,” compared to 38% of students from non-poor families (flores, 2007). while white students are overrepresented in “gifted,” “honors,” and “advanced placement” programs, black and latina/o students are severely underrepresented (darling-hammond, 2010; oakes, 2005; tyson, 2006). the situation for the english language learner (ell) is similarly challenging. ells are also frequently blocked from higher-level tracks because of their english skills (darlinghammond, 2010). perhaps even more alarming is that though gaps between white, black, and latina/o students narrowed through the 1970s and 1980s, they have widened again in nasir commentary journal of urban mathematics education vol. 9, no. 1 11 recent decades (darling-hammond, 2010). this widening may be due, in part, to the increasing emphasis in schools on standardization and high-stakes accountability (e.g., no child left behind act of 2001), which exacerbated inequity by driving the schools that are least successful to focus on basic skills as a means of test preparation (abandoning a focus on critical thinking and problem solving). these schools also often pushed out students who are struggling and in need of support (haney, 2000; mcneil, 2000; mintrop, 2003; pedulla et al., 2003). these disparities in achievement gaps are often rooted in severe inequities in opportunities to learn mathematics. students from marginalized groups are less likely to have well-trained teachers, and have less access to other resources as well, such as material supplies and technology (darling-hammond, 2010). in addition, disciplinary systems in schools operate in ways that disproportionately penalize latino and african american male students (gregory, skiba, & noguera, 2010; monroe, 2005; nasir, ross, mckinney de royston, givens, & bryant, 2013; noguera, 2003; skiba, michael, nardo, & peterson, 2002), which too often result in more out-of-class time (i.e., lessened opportunity to learn). perhaps the most critical access issue is that of access to high-quality mathematics instruction. students from marginalized groups not only attend schools with fewer qualified teachers (darling-hammond, 2010) but also have less access to college preparatory pathways, and are more likely to be enrolled in a district that employs instructional practices that center on preparation for standardized tests (davis & martin, 2008). students who are english learners are often incorrectly placed in lower-track courses, as counselors and teachers may not realize that they had already studied the material in their home country (gutiérrez, 2002). despite these documented inequities, as a society, we maintain the narrative that all students have an equal chance to learn. our general unwillingness to acknowledge the reality of stratification in access to opportunities to learn is related to the frame of minimization of racism. in other words, while we note the achievement gaps, we rarely acknowledge the extent of racial difference in educational access. our unwillingness to center these opportunity gaps is also related to the abstract liberalism frame; in that, we ascribe to the idea that our job is to provide “equal opportunity” through sameness, even while students face very different levels of multiple kinds of challenges by virtue of race and class in school and in society. this idea of equal opportunity through sameness leaves us unable to redress the myriad inequalities students and communities are forever facing. reason #2: racial stereotyping influences access to mathematical identities for students, and thus disrupts mathematics learning. the second reason mathematics educators should care about race and culture is that students experience mathematics classrooms as racialized spaces, where black and latina/o students are subject to negative stereotypes about their ability to do nasir commentary journal of urban mathematics education vol. 9, no. 1 12 mathematics (martin, 2013; shah, 2013). in my own work, i have come to think about these stereotypes as “racial storylines” that are, themselves, cultural artifacts, as they are the way that we collectively make sense of (and reproduce) achievement patterns (nasir, 2011; nasir, snyder, shah, & ross, 2013). the extent to which students saw mathematics achievement as racialized and were aware of societal storylines about who could be good at mathematics was the subject of a study that my colleagues and i carried out with upper elementary and middle school students (nasir, o’conner, wischnia, & mckinney de royston, forthcoming). the research team surveyed over 150 fourth through seventh graders, and interviewed and observed 12 case study students to explore (a) the extent to which students understood that racial stereotypes about mathematics achievement exists, and (b) the extent to which they believed such stereotypes. the results were sobering. the students we surveyed overwhelmingly reported that they were aware of racial stereotypes which purported that asian students were the smartest at mathematics, followed by white students, followed by latina/o students, and lastly by black students. what is perhaps even more alarming is that this awareness intensified by the time students were in middle school. the pattern was the same for students’ reporting that they believed the stereotypes. perhaps even more striking is that when the research team looked at these data by race, we found that african american and latina/o students were more likely to be aware of negative racial stereotypes about school than asian and white students. they, however, were also less likely to say that they themselves believed such stereotypes. this reported awareness seemed to leave these students in a bit of a quandary. they were both highly aware that others thought that people like them were not good at school, yet they did not believe it. thus, these middle school students had to cognitively and emotionally process the difference between how others thought about their school and mathematics ability, and how they thought about their ability. observations of case study students revealed that students took different approaches to the interpretation and management of identity and stereotypes, with different consequences on student achievement and engagement in class. these included: (a) students who were simply unaware of racial stereotypes about school and did quite well academically; (b) students who took up the negative stereotypes and found their academic achievement negatively impacted; (c) students who endorsed the stereotypes about their own group in the abstract, but who saw themselves as the exception (these students had mixed academic outcomes); and (d) students who overtly resisted the stereotypes and had strong academic outcomes. these findings make salient the deep ways that students are impacted by racial stereotypes, and the power of negative stereotypes in their lives. bonilla-silva’s (2010) frame of cultural racism may explain why while students articulate the power of these stereotypes in their academic lives, our schools nasir commentary journal of urban mathematics education vol. 9, no. 1 13 rarely attend to supporting students in managing the negative racial stereotypes about schools that are thrust upon them. invoking the cultural racism frame would imply that our society may make sense of racial differences in achievement outcomes as a product of different sets of values about school, and thus not take seriously the impact of the stereotypes on children’s lives in school. reason #3: high-quality mathematics instruction (potentially) disrupts unequal access to mathematics learning for students from marginalized groups. the third reason mathematics educators should care about issues of race and culture is that the way we teach mathematics has incredible power to disrupt the troubling opportunity gaps and the negative processes of stereotyping and racialization that have been discussed so far. this power to disrupt is profoundly illustrated by the work of teachers at a school called “railside high school” in the research literature (see, e.g., nasir, cabana, shreve, woodbury, & louie, 2013). the mathematics department at railside has been touted nationally not only for being successful in achieving strong learning outcomes in mathematics with a diverse student population but also for developing an extraordinary teacher professional community in the mathematics department that embraced mathematically rigorous reformminded curriculum and institutional and instructional practices (boaler, 2006, 2008; boaler & staples, 2008; hand, 2003; horn, 2005, 2007, 2012; horn & little, 2010; jilk, 2007; little, 2002; little & horn, 2007; nasir et al., 2013). with respect to learning outcomes in mathematics, boaler’s (boaler & staples, 2008) team conducted a 4-year study that involved over 400 students and found that when compared to two other comparison schools, railside students were slightly more likely to score “basic” or better on the california standardized test (49% vs. 41%), take advanced mathematics courses (e.g., calculus and pre-calculus) as seniors (41% vs. 27%), report that they “like math” (74% vs. 54%), and demonstrate interest in mathematics-related careers (39% vs. 5%). these results are particularly striking given that railside is an urban, comprehensive high school in northern california with a student population (at the time of the study) that was 54% latina/o, 21% black, 17% asian, 30% qualified for free or reduced-price meals, and 25% ells (i.e., unfortunately, too many believe such demographics prevent school success.) one of the hallmarks of the mathematics instruction is that the department ran completely de-tracked classes; utilized a multi-ability, project-based curriculum; and supported the success of all learners to engage in complex mathematics. the work of using the power of high-quality mathematics instruction to disrupt not only stereotypical narratives and achievement outcomes but also unequal access to mathematics learning at railside was undergirded by a set of practices as well as a key set of beliefs and values. practices. the railside approach toward teaching and learning included institutional practices at the level of the department and school as well as instructional nasir commentary journal of urban mathematics education vol. 9, no. 1 14 practices at the level of the classroom (nasir et al., 2013). the approach was developed over a 20-year period of working together as a department. core elements included, first and foremost, a commitment to equity. teachers in the railside mathematics department collectively decided that one of the most important goals they had was to serve all of their students well, and to teach high-quality mathematics to all of their students, including those from marginalized groups. equity, then, was viewed as the opportunity for all students to have access to high-quality mathematics teaching. this commitment to equity was reflected in the ways that mathematics teaching and learning were structured at the institutional level at railside. these institutional practices included block scheduling, so that mathematics courses ran for 90 minutes daily and were a semester long, which allowed for in class time to do extended projects and made it possible for students to take more mathematics courses during high school. another institutional practice was the building and maintenance of the teacher professional community. this professional community was centered on discussing problems of practice together, and providing a supportive environment to help one another maintain challenging instructional practices by working together. with respect to instructional practices, mathematics teaching at railside was guided by a commitment to complex instruction (cohen & lotan, 1997) and group work. complex instruction is a pedagogical approach that emphasizes teaching in a way that attends to power and status issues in a classroom, and has the goal of equalizing status among students by providing mathematical problems that require deep thought and collaborative problem solving—what cohen and lotan call “group-worthy” tasks. group-worthy tasks have to meet several design criteria: focus on core mathematical ideas, offer multiple solution paths or entail multiple representations, and require (more times than not) the collective resources of the group. thus, group-worthy problems lend themselves to opening up the opportunity for complex mathematical thinking for all students. the railside approach to mathematics teaching also involved key practices, such as utilizing multiple representations (numerical, linear, graphical, etc.), focusing on big mathematical ideas (rather than procedures), asking students to justify and explain their work, and having students present their thinking and problem solving in front of the class. this focus on presenting student work had the goal of making students’ thinking public and valued through presentations at the overhead. beliefs and values. these institutional and instructional practices were undergirded by a set of beliefs and values that were held and reinforced within the teacher professional community. these beliefs included: (a) acknowledging both teachers and students as learners, (b) working from strengths and making space for vulnerability, (c) redefining “smart,” (d) rethinking what it means to do mathematics in school, and (e) valuing relationships. taken together these beliefs convey a key set of interrelated values and assumptions. the acknowledgment that all teachers and nasir commentary journal of urban mathematics education vol. 9, no. 1 15 students are learners meant that teachers viewed themselves as learners alongside their students, and saw the work of learning to teach and of getting to know their students as a part of what it meant to teach at railside. this belief connected to the second belief: the goal in the classroom was to emphasize students’ strengths, while creating a classroom space where students (and the teacher) could be vulnerable. the vulnerability was made possible by redefining what it meant to be smart, as well as what it meant to do mathematics—both involved working through difficult problems collaboratively and putting forth effort when things seem difficult. all of these processes were undergirded by a value on relationships—between teachers and students and among students. building trusting and caring relationships with students was important so that they were willing and able to take the significant emotional risks the department was asking them to take in the classroom. interestingly, the disruption of racialized patterns of engagement and achievement at railside was accomplished through increased access to instruction that fostered deep mathematical thinking, and not through explicit discussion about race. this differs from approaches that other efforts have taken (see nasir, ross, mckinney de royston, givens, & bryant, 2013, and givens, nasir, ross, & mckinney de royston, 2016, for an example of explicitly racialized approaches). one way to think about this is that at railside, the negative stereotypes were disrupted and reframed by the outcome of having a large number of potentially stereotyped (i.e., african american and latina/o) students being successful in mathematics and developing strong identities as mathematics learners. closing thoughts in this commentary, i have made the argument that taking up issues of race and culture is imperative for mathematics educators and the mathematics education community. it is imperative because racial and cultural dynamics operate even when we are not paying attention to them, and are especially dangerous when they go unnoticed and unacknowledged. specific to mathematics education, the racialized and cultured nature of schools and of the teaching and learning of mathematics have continued to exert an influence on learning and achievement outcomes, outcomes that get identified as issues of individual achievement or individual difference. this influence is exerted both through the unequal access to high-quality mathematics instruction, and the existence of persistent racial stereotypes about mathematics “ability.” perhaps never has the imperative to make explicit the racialized and cultured nature of mathematics learning been more relevant than in today’s political and educational climate, where racial disparities in policing and in access to higher education make the daily news. to not discuss or address issues of race, culture, and inequality is to accept the current patterns of inequality and marginali nasir commentary journal of urban mathematics education vol. 9, no. 1 16 zation. as robert moses (moses & cobb, 2001) contends, access to high-quality mathematics instruction is one of the key civil rights issues of our time. references boaler, j. 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(1998). communities of practice: learning, meaning, and identity. cambridge, ma. cambridge university press. microsoft word 386ontology to ontologies (proof 2).docx journal of urban mathematics education december 2021, vol. 14, no. 2, pp. 42–70 ©jume. https://journals.tdl.org/jume brian tweed is a senior lecturer in te kura o te mātauranga/the institute of education at massey university, private bag 11-222, palmerston north, new zealand; email: b.tweed@massey.ac.nz. his research interests focus on the intersections of the ethics, sociology, and philosophy of education. he has a special focus on interactions between legitimation regimes, societal/social conditions, and the nature of different pedagogies and how non-māori teachers and researchers engage with māori knowledge and practices. indigenous struggle with mathematics education in the new zealand context: from neo-liberal ontology to indigenous, urban ontologies? brian tweed te kūnenga ki pūrehuroa, aotearoa/massey university, new zealand in this article, the learning of conventional curriculum mathematics in one indigenous māori school in aotearoa/new zealand is conceptualized as a site of ontological struggle. the major finding of a research project which analyzed extensive ethnographic data gathered in partnership with this school identified an ontological disjunction between curriculum mathematics education and the ethos of the school. this disjunction can be related to the complex and emergent phenomenon in aotearoa/new zealand of indigenous schools in neoliberal, capitalist urban conditions. centering the ontological commitments of the māori school challenges the ontological hegemony of curriculum mathematics education and points to a consideration of the possibilities of forms of contemporary mathematical education based on indigenous ontologies. embedded in the discussion is a consideration of the ethical position of non-indigenous researchers working in indigenous contexts. keywords: critical culturally sustaining/revitalizing pedagogy, indigenous educational sovereignty, indigenous māori mathematics education, urban indigeneity research ethics tweed indigenous struggle with mathematics education journal of urban mathematics education vol. 14, no. 2 43 he focus of this article is on the ontological work that the presence of mathematics education based on a mandated national curriculum does in an indigenous māori school. the discussion is necessarily intimately connected with the aotearoa/new zealand context, which will be briefly outlined in the next sections. aotearoa (the long white cloud) is the original, indigenous name for the land now governed by the new zealand state. to be clear from the outset, this article is not about the nature of disciplinary mathematics knowledge, indigenous mathematical practices (ethnomathematics), or pedagogical practices designed to improve indigenous student achievement in the terms of the new zealand national curriculum. to paraphrase foucault (dreyfus & rabinow, 1983, p. 187), the focus is on what the “doing of curriculum mathematics education does” when located in an indigenous, māori school, itself embedded in a long-term emancipatory social movement complicated by living in neo-liberal urban conditions. this focus prompts an investigation into the ontological commitments of both the māori school and the curriculum mathematics education as constructed through the social practices of both. urbanization and urban māori new zealand society is now a thoroughly neo-liberalized, capitalist society that has established a neo-liberal ontology as the basis of almost all social and economic life (fisher, 2013; harvey, 2005; kelsey, 2015). national education systems have also been transformed as part of the establishment of a neoliberal, capitalist hegemony (ward, 2012). urbanization is understood as intimately and inextricably connected with capitalism and, in the early 21st century, its neo-liberal governance formations (rossi, 2017). in new zealand, colonization can be viewed as both a settlercolonization project and a capitalist project that has successfully created dependency on market transactions for survival as well as a predominantly urban-dwelling indigenous population (poata-smith, 2015). māori have to a large extent migrated to urban centers, a process that intensified after world war 2 (peters & andersen, 2013). now, over 80% of māori live in towns and cities, with one quarter of all māori living in auckland, new zealand’s largest city, and one sixth living in australia, with the greatest concentration in sydney (haami, 2018). today, in terms of location at least, māori can be described as an urban people (kukutai, 2013). in this migration to urban living, the nature of māori identity has become contested (see hokowhitu, 2013; kukutai, 2007). whilst genealogical connections to ancestral, pre-colonization communities and homelands remain central to being māori, urban living has resulted in an emergent range of māori identities and adapted social practices (kukutai & webber, 2017). generations of māori have grown up in urban centers and identify primarily with these locations as urban māori, who are far from t tweed indigenous struggle with mathematics education journal of urban mathematics education vol. 14, no. 2 44 a homogeneous group (ryks et al., 2016). in some ways, urbanization has blurred the traditional cultural and spatial separation between pākehā (european new zealander) and māori, but even so, māori communities remain distinctive and centered on māori values and practices. these practices have been adapted to urban conditions and maintain commitments to māori ontologies, values, and epistemologies in novel and creative ways (gagné, 2013; metge, 1995). for example, māori have created successful māori businesses and business models (mika et al., 2019), founded health providers (durie, 2011), influenced the criminal justice system (tauri, 1998), and established early childhood centers, schools, and universities (g. h. smith, 2000). māori continue to resist neo-colonization and the toxic aspects of urbanization whilst taking advantage of the beneficial aspects by reorganizing the resources available to maintain māori social practices, community integrity, and social structures (keiha & moon, 2008). in addition, māori are developing sufficient strength politically and economically to renegotiate relationships with the state and the multicultural nature of general society (gagné, 2016). this also positions māori to be able to engage on different ontological and epistemological terrain with the challenges that face humanity in the 21st century. in relation to this article, this includes forging new relations with all existing knowledge domains, regardless of cultural location, and, in particular, the discipline of mathematics. the perspective adopted here is that māori schools are integral to this urban experience and are another example of how māori have adapted to urban, neo-liberal conditions in ways that maintain maori language and cultural reference points. the school that partnered in the research reported in this article was based on māori language and culture but also dealt with events and attributes related to the school being situated within urban conditions. for the purposes of this article, māori schools are understood in this context: they are part of a wide-ranging, multidimensional, indigenous response to urbanization and the neo-colonial, neo-liberal subtext as a fundamental part of a māori emancipatory movement. non-indigenous researchers and indigenous contexts linda tuhiwai smith explains that there has been a history of researcher abuse of indigenous populations, which rightly sounds a warning and creates caution about the engagement of non-indigenous researchers in indigenous contexts (l. t. smith, 2013). in aotearoa/new zealand, the broad research orientation known as kaupapa māori theory has established itself at the center of research ethics and excellence with, by, and for māori (bevan-brown, 1998; pihama et al., 2015; g. h. smith, 2009, 2011, 2012; l. t. smith, 2005, 2011). in this regard, two factors need to be considered. the first is the non-indigenous ethnicity of the researcher. the second is the influence of non-indigenous theory in the construction of analytical frameworks. tweed indigenous struggle with mathematics education journal of urban mathematics education vol. 14, no. 2 45 who is this researcher? the researcher was born and brought up in england and migrated to new zealand in the late 1980s. strictly speaking, because the author is not māori, the research described here cannot be described as kaupapa māori. the researcher, however, has been closely associated with māori schools and communities for over 20 years, has taught, as a fluent speaker of te reo māori (the māori language), in māori schools and universities, and has created professional learning opportunities for māori teachers in mathematics and science (te maro et al., 2008). moreover, the researcher continues in and is committed to these relationships with indigenous communities built up over many years (tweed, 2019). by marriage, the researcher is also connected to the iwi (tribe) ngāti porou. the long association and commitment to te ao māori (the māori world) by the researcher through work, family, and social life places this nonindigenous researcher appropriately for research in the māori communities of which the researcher is already an accepted member. the researcher also has a longstanding relationship with the māori school that partnered in this research. this relationship was established well before the research was conducted and continues in the present well after it has been completed. non-indigenous theory and engagement in indigenous research contexts? whilst acknowledging the power and relevance of theoretical frameworks grown from entirely within māori/indigenous cultural spaces and communities, it is suggested here that this does not mean that research employing non-indigenous, eurocentric frameworks has nothing meaningful to contribute to indigenous research. the use of theoretical frameworks created by european researchers/thinkers is not sufficient by itself to assert that the research is automatically eurocentric in its entirety. eurocentrism also entails that european/north american (western) interests and outcomes as well as conceptual understandings are centered. research using theory produced by european thinkers to critique this eurocentrism counters oppressive systems that prevent the emergence of spaces in which indigenous culture, language, and worldview are centralized. this signposts a direction towards an ethical position for non-indigenous researchers working in indigenous contexts. the exclusionary stance by some scholars that dogmatically rejects research by non-indigenous researchers and/or using non-indigenous theoretical frameworks is problematical and ultimately unproductive. several prominent māori scholars have commented on this issue. te kawehou hoskins (2012) explains that māori essentialism, which lies behind such dogmatic exclusions, has been a necessary political stance over the last 40 years in order to resist european-based cultural, material, and political dominance in new zealand. she also contends that such a stance is unsustainable because it reifies both māori and non-māori identities and thereby cannot recognize the heterogeneity tweed indigenous struggle with mathematics education journal of urban mathematics education vol. 14, no. 2 46 of both māori and non-māori (the researcher and the participants in this research, for example, do not match these reified identities). instead, hoskins argues for an open, relational stance that accepts the potential and risks for new knowledge generation of engaging ethically with the non-indigenous other and non-indigenous knowledge frameworks. in a similar vein, stewart (2018) argues for the deliberate and critical working of the indigenous-settler relationship. she refers to dualities between social reality and discourses that pervade māori lives in all spheres of life, creating “hyphens” that she contends should be productive relationships rather than antagonistic and exclusionary. stewart comments that critical māori scholarship is better off working with hyphens rather than against them. this can also be expressed as seeing hyphens, such as the māori-pākeha [europeans settler] hyphen, not as symbols of either-or exhausting battles, but of both-and enriching collaborations. (stewart, 2018, p. 773) the research detailed in this article could be viewed in these terms as a critical and productive working of the māori-pākeha hyphen, which explores the tensions created by a duality established between the discourses of curriculum mathematics education and social realities of māori. for te ahukaramū charles royal, who has written extensively about māori knowledge (mātauranga māori) and epistemology, it is understood that …movement ‘towards a new indigenous epistemology,’ at least in aotearoa/new zealand, will involve the development of a way of studying the nature of knowledge (and its attendant questions) that finds inspiration both in conventional western epistemology and in mātauranga māori. (royal, 2009, p. 119) royal contends that contemporary mātauranga māori can benefit from and inform other knowledge domains and proposes a new indigenous epistemology based on contemporary forms of māori concepts and practices, which has the potential to engage with any other domain of knowledge from a māori ontological position. although this is not an argument for the involvement of non-indigenous researchers, it does carry with it an understanding that non-indigenous people may be involved in a contextualized manner, that is, appropriate to the context of research. here it is understood that in some contexts, non-indigenous involvement is appropriate with sufficient ethical and cultural understandings, whilst in other contexts it is entirely inappropriate regardless of the competencies of the researcher. moreover, it is for indigenous communities themselves to decide on the appropriacy of such involvement in any given context. in this regard, it should be noted that the research outlined in this article is not a purely indigenous one; it involves the interaction between a western knowledge system (curriculum mathematics education) and an indigenous school in urban tweed indigenous struggle with mathematics education journal of urban mathematics education vol. 14, no. 2 47 conditions. there is an entanglement here that sees history as simultaneously both māori history and pākehā (european settler) history, with multiple and varied interactions, overlaps, and ambiguities. as jones (2007) points out, there is also the possibility that multiple distinct and irreconcilable events occur simultaneously, resulting in an interminable tension” between them, which may be the case in this research. penetito (2010) contends that māori education is in fact circumscribed by government, majority culture concerns, policies, and agency. māori schools are still charged with producing the same outcomes as any other school in addition to their efforts towards emancipation. in this context, both indigenous and non-indigenous theoretical frameworks have a place. te maro (2019), for example, uses both foucault and marx to reveal how curriculum mathematics education confines time, space, and subjectivity to the instrumental domains of the economy. she argues that indigenous frameworks must drive development of indigenous communities from the inside, but western theory can be powerful in the analysis, and therefore minimization of harm, of systems that impinge upon/oppress māori from the outside. thus, it is considered here that the use of non-indigenous frameworks in indigenous contexts is challenging but, with care, critical engagement, and cognizance of the history of the māori/pākehā relationship, potentially productive. this care and critical engagement with theory involves the researcher as bricoleur, carefully attending to how concepts from many disciplinary domains are reinterpreted and relocated in the research context to create robust and rigorous research methodologies (kincheloe, 2001, 2005). done well, the use of non-indigenous theory can support the purposes of māori communities and maintain their centrality whilst generating useful knowledge of interest in other contexts. kaupapa māori research prioritizes the usefulness and ownership of the research but maintains an open stance about the methods used (moewaka-barnes, 2015). the māori school involved in this research certainly owns this research and continues to use the findings to develop its own approaches to engaging with mathematics. having argued for the conditional legitimacy of non-indigenous researcher involvement in indigenous-oriented research (conditional upon context), more remains to be unpacked about the nature and ethics of this involvement. the “doing” of research always comes under an ethical umbrella, the importance of which is amplified when non-indigenous researchers engage in indigenous-oriented research. here, i will draw briefly on the work of biesta (2015) and yukich and hoskins (2011), who are inspired by the philosophical work of levinas’s concept of ethical responsibility for the other (see levinas, 1961/1969, 1974/1998). biesta (2015) critiques what he describes as the dominant discourse of learning on the grounds that it positions teachers and students (and people in general) in a particular relationship with the world. in this relationship, we are meant to grasp and know the world in its entirety and to continue learning until this is achieved. the world, which includes other people, is, in this view, a thing that is at our disposal, tweed indigenous struggle with mathematics education journal of urban mathematics education vol. 14, no. 2 48 subject to and an object of our knowing. this relationship is a form of possession. instead, biesta argues for an education that allows us to be confronted by the world and for us to be taught by it. in this relationship, the world is not positioned as a thing to be known but as an entity with its own purposes and actions that in certain aspects must always remain unknowable to us. yukich and hoskins (2011) take up this levinasian perspective in the context of pākehā school principals who have developed competencies in engaging with māori communities. the central theme here is that these principals developed practices of simultaneous knowing and not-knowing with respect to māori. over time, they developed understandings of language and practices and were able to participate competently in them. at the same time, this knowing automatically highlighted what they, as pākehā, do not yet know and can never know. the position suggested here is an ethical one. whilst learning from māori and developing cultural competencies and responsivity as non-māori is pursued, this must not be a pursuit of ownership, possession, and control. it must rather be a pursuit of self-knowledge precisely so that an ethical care for māori, as the other, can exist, which reminds “…dominant groups such as pākehā to resist the will to mastery, the desire to wholly know and ‘see’ others… to protect the radical alterity and the cultural difference of others” (yukich & hoskins, 2011, p. 63). taking up this same ethical position in indigenous research, demands that nonindigenous researchers, even as we gain valuable insights and generate new knowledge and understanding, acknowledge the historical relationship that has existed between western research and indigenous communities, a relationship which l. t. smith (2013) describes as a claim to ownership of indigenous ways of knowing and being and a rejection of the people who created them. this entails the understanding that the knowledge generated by research is not knowledge of the other but knowledge for the other, spoken inevitably from a position of power but spoken in such a way that attempts to simultaneously challenge this power; it is spoken by the non-indigenous researcher without expectation of reciprocity, without any expectation that it will be implemented. resisting the will to know here means that the nonindigenous researcher offers their work not as a claim to truth but as a tentative possibility, a suggestion of possible meaning always prefaced by statements of fallibility and subjectivity. indigenous people and communities may accept or reject such work on their own sovereign terms (which always remain partially unknowable to the nonindigenous researcher). research becomes a mixed practice of tentative knowing, speaking, and deliberate ignorance. tweed indigenous struggle with mathematics education journal of urban mathematics education vol. 14, no. 2 49 research methodology the overall aim of the research was to infer ontological commitments from social practices operating in mathematics classes and in the general operation of school life (the ethos). since the researcher already had an established relationship with the partner school in the research, an ethnographic methodology was used. the researcher spent a period of 12 months collecting data from classroom video recordings, individual interviews, focus groups, and field notes. analysis of data employed concepts taken from legitimation code theory (lct; maton, 2014), recontextualized to suit the indigenous context of the school. the main concept employed from lct was that of specialization, which refers to the ways in which social practices specialize a person’s subjectivity. lct informs the analysis by alerting us to the nature of social life as involving (amongst other things) a specialization of the relation to knowledge and a specialization of the relation between people. thus, analysis of data involves asking the following question: what kind of epistemic relation (how knowledge is understood and is to be acquired) and what kind of social relation (how people live with each other) are legitimized in the social field? lct further recognizes two forms of each relation. the epistemic relation is considered as ontic or discursive, indicating a direct relation to the object of study (ontic) or the study of an established discourse about it (discursive). the social relation is considered as subjective or interactive, indicating social practices based on belonging to a certain group (subjective) or based on ways of interacting (interactive). these concepts come together by viewing social fields as having characteristic configurations of how knowledge and knowers are structured, which in turn specialize subjectivity in particular ways (maton, 2006). the specialization of subjectivity is considered to be equivalent to an ontological commitment to a certain kind of person constructed through participation in the practices of that field. findings and discussion throughout analysis of empirical data, these specialization concepts were recontextualized and sharpened for the indigenous context of the school. the ontic epistemic relation was reinterpreted to correspond to direct experience of students and teachers in real event (a participatory epistemology) as a preferred form of knowledge generation and learning for students. the discursive epistemic relation came to mean an engagement in a formalized knowledge discourse about an object of study rather than a direct engagement with that object of study. similarly, for the social relation, a subjective social relation came to mean the possession of genealogical links to tribal ancestors (of any iwi/tribe), and an interactive social relation became a focus on tweed indigenous struggle with mathematics education journal of urban mathematics education vol. 14, no. 2 50 explicit, institutional/disciplinary specific rules of behavior/interaction between people, regardless of genealogy. this led to the major finding of the research: the overall specialization of the whole school ethos was based on an ontic epistemic relation and a subjective social relation, but the specializations in mathematics classes were based on a discursive epistemic relation and an interactive social relation. in other words, the school ethos was about an engagement with the world as it currently exists from a position of indigenous inclusion in ways (both formal and situational) dictated by the nature of this engagement. curriculum mathematics education was about mastering the existing discourse of curriculum mathematics from the position of a generic learner. the following sections present some examples of empirical data, with interpretations in italics relating them to specializations of the school ethos and three mathematics classrooms. quotations appear in english, but the original data was entirely in te reo māori (māori language). this presented the methodological problem of how to maximize the fidelity of the english translations. this was accomplished by an iterative process of collaborative translation and checking with participants (who are all bilingual speakers of māori and english) until participants confirmed that the english version conveyed the intended meanings. the school ethos the school is small, with around 120 students and six full-time teachers. it is located in the center of a medium-sized town and has the full range of students from age five to 18. students come from a wide variety of socio-economic backgrounds, with all of them being māori and being able to function in a cultural context in which māori language and cultural practices are centralized. the school is a second home for students, teachers, and their families. it is common for teachers, students, and visitors to sleep overnight in the school. there is a supply of mattresses and bedding for that purpose. teachers and students have a responsibility to look after the school as if it were their home. the caring of the grounds and buildings of the school is direct; responsibility for its upkeep rests with everyone who lives in the school. interpretation: although the school is small, it caters for a full range of students from all socio-economic groups in the town. being māori and forming a community based on māori cultural protocols and norms (tikanga) is a way in which the school responds to urban conditions and the wealth disparities that have been exacerbated under neo-liberal reforms over the last 40 years. this has a clear association with a subjective social relation. tweed indigenous struggle with mathematics education journal of urban mathematics education vol. 14, no. 2 51 the school has a particular orientation towards students being kaitiaki (custodians) of their lands and natural resources, being upholders of the māori language, and being exemplary representatives of their iwi (tribe). quoting a kaumatua (an elder, koro1), one teacher, te mana, made this statement: koro says that we must learn all of the real (māori) names of places and the histories of them in our iwi (tribal) area. if we don’t, we will be just like the pākehā (european new zealander), who are only visitors to our lands and will soon be gone. the school operates on the basis of tikanga (culturally correct actions), which are not written down as a set of policy statements. tikanga are culturally correct actions learned through participation in real cultural activities. several comments from various teachers during family and staff meeting reiterate the prioritization of language and tikanga. the prioritization is certainly related to both a need to protect the māori language and tikanga māori as well as a response to perceived oppression. two such examples are given here: what’s important is that our own knowledge is fed to our children… we have been oppressed for long enough by those external systems that tell us what knowledge we should be teaching and how we should organize ourselves what’s important is not math, it is the māori language... mathematics is not endangered; it can look after itself and will be there when we are ready for it... if we don’t speak māori, it will die. 1 all names are pseudonyms. interpretation: the social relation includes a relation between people and the land. being māori and having an ancestral connection with the land on which you stand gives a more nuanced understanding of what a subjective social relation means in this context. interpretation: the strong desire to base all aspects of school life on māori philosophy is clearly expressed here. tikanga relates to an underlying māori worldview (ontology) that recognizes language as a fundamental aspect of how māori relate to the world and think of themselves in it. the frustration with external systems, including curriculum mathematics education, is also clearly expressed. the desire to relate to the world not through external knowledge systems but through an autonomous indigenous ontology characterizes a strong ontic epistemic relation in this context. the recognition that mathematics will look after itself recognizes its discursive nature, but māori language and indigenous knowledge have a high degree of ontic urgency about them—the ontic imperative exists for them to survive. tweed indigenous struggle with mathematics education journal of urban mathematics education vol. 14, no. 2 52 a general feature of the school is a prioritization of naturally occurring, experiential learning. priority is always given to real cultural events, such as pōwhiri (welcome ceremonies) and tangihanga (funerals). often the whole school will travel large distances in order to attend such events. the school is a genuine cultural institution in its own right that engages in actual cultural life. it is already embedded in the wider cultural constellation of the iwi and in this sense is much more than just a school. celebration of times of the māori calendar are real participations. for example, the celebration of matariki in june or july (the beginning of the indigenous new year in the southern hemisphere winter) is not a school learning experience but a genuine celebration of the changeover of a natural cycle in aotearoa/new zealand—it is a cultural connection with the land, a recognition of change and the continuing presence of ancestors. one teacher commented on it in this way: participating in matariki reestablishes a spiritual link with ancestors. in the early morning, it is still dark, its cold... the connection is easier to make... somehow nanny is close, and i remember her and mihi [greet] to her for all that she has done for us... she is still doing it right now actually. the relationship with the new zealand national curriculum is ambivalent. one teacher commented as follows: the [national qualifications] are done as quickly as possible so that we can get on with what really counts... developing students as māori people located in a māori reality. officially mandated knowledge is seen as an imposition in some respects, but this is completely consistent with the school’s basis on tikanga and real learning through real cultural participation; time spent on curriculum learning is time taken from the main purposes of the school. students must have qualifications in order to gain access to university or other tertiary training or employment, but the real job of the school is to grow māori speaking and thinking people. interpretation: there is a strong sense of the immediacy and authenticity of this event. teachers and students experience a closeness with ancestors at this time. this can be related to a strong ontic epistemic relation; teachers and students participate in this event and experience it in all aspects of their being—intellectually, emotionally, physically, and spiritually. the learning is both collective and personal, but it is not reproducible in precisely the same way. this represents another important difference from a curriculum mathematics discourse, which to a large extent is about fungibility; students are expected to pass through the levels of the curriculum and emerge at the end in possession of similar knowledge, skills, and dispositions. from a curriculum mathematics viewpoint, students are effectively interchangeable, a situation which is consistent with an economic demand for cohorts of students who have been equipped with necessary knowledge and skills for employability. tweed indigenous struggle with mathematics education journal of urban mathematics education vol. 14, no. 2 53 assessing students against external measures based on the age of students, such as the levels in the new zealand national curriculum, is rejected. the principal of the school expressed it in this way: it feels like a real intrusion, an intrusion of judgmental thinking into our family where the development of our tamariki-mokopuna (children and grandchildren) as people is the most important thing. despite a strong tendency for integrated and thematic approaches throughout the school, mathematics remains as the most defined standalone learning area. all teachers have dedicated times when mathematics learning happens in isolation. there are also dedicated assessments of students. the principal also commented on a mismatch between family interests and the school ethos in relation to mathematics: even when we report how well children are doing in terms of growing and learning in their māori identities, like growth of manaakitanga (caring), tautoko (support), and mātauranga māori (māori knowledge), they still want to know where their child is in mathematics. i think even if we reported about integrated mathematics, they would still want to know about basic facts and levels... and it’s all pressure from outside. it’s not our kaupapa (reason for being/purpose), it doesn’t belong to us. if the compliance thing wasn’t there, mathematics could be very different. interpretation: this illustrates the mixed and somewhat confused discursive nature of societal attitudes to knowledge even in māori communities. urban conditions make the necessity to gain employment paramount, and competency in mathematics is seen as fundamental to this economic imperative. both families, teachers, and students carry these discourses about the universality of mathematics and its importance to “everything” into the school. resisting this pressure from outside takes a lot of energy. the desire for a different form of mathematics education is very apparent and represents a manifestation of the clash between ontic and discursive epistemic relations. interpretation: this highlights the disjuncture between the ethos of the school and mathematics classes, which remain outside the integrating themes uniting the rest of learning in the school. this is a clash between the ontic epistemic relation of the school and the discursive epistemic relation of the mathematics classes. the rejection of assessment against the levels of the national curriculum also emphasizes the ontic epistemic relation; for this school, students (and people in general) develop knowledge at their own pace through their own experiences over time and in real events. this cannot be hurried or planned ahead in a step-by-step hierarchical structure. tweed indigenous struggle with mathematics education journal of urban mathematics education vol. 14, no. 2 54 the principal expresses a very strong feeling of frustration with the need to comply with external regulations, especially the national curriculum. the thinking is certainly present that the school should create its own curriculum. coupled with a rejection of a curriculum that is designed in levels that are used to measure students’ learning, there is a desire to ground any new curriculum in the local iwi context. one teacher put it like this: i went through an english-medium school, and the teachers knew exactly what we were going to do each lesson... but that destroyed my creativity... it was like i couldn’t have my own thoughts... that’s why i think it’s better for us to create our own curriculum, so that the people who are based here, who live here, decide what’s in it and how its organized... how it can be planned so that the students are alive in it. the school as an institution prioritizes the personal development of each unique child located in a māori reality. the collection of unique and very different people who make up the school community is bound together by a common understanding of tikanga, māori language, and iwi identity. tikanga provides a central common grounding that is the basis on which individual development is made. the following two comments, the first by the principal and the second by a senior teacher, express this aspect of the school: tikanga is about meeting each person’s physical and spiritual needs... that’s really what maslow’s hierarchy of needs is about as well... when a person’s needs are met, they have everything they need to become who they are, and they will be able to do anything they want. rather than preparing our children for university and conventional careers, we should be preparing them to exist in the world as unique people defined in their own ways... it’s not about economics even, or about preparing students so that they can bring skills back for the iwi, like becoming doctors or accountants... it’s about students being their own unique selves in the world. interpretation: again, the tension and the sense of constraint and oppression through external regulation is clearly expressed. importantly, the deeper impact of this on indigenous people’s thought processes and identity is also expressed. the response to this is to create a curriculum based not only on indigenous worldviews but also, specifically, driven by the lived experiences of māori people in this locality. in other words, a curriculum based on an ontic epistemic relation and a subjective social relation is desired. tweed indigenous struggle with mathematics education journal of urban mathematics education vol. 14, no. 2 55 teachers and students responded in a variety of ways in mathematics lessons. the following sections focus on three mathematics classes and the teachers te ara, te mana, and te ao. classroom 1, teacher: te ara te ara organized very structured lessons that adhered very closely to official curriculum resources and engaged students in learning the standard procedures, strategies, and algorithms required to solve problems. in many ways, te ara embodies a strong version of the standard curriculum mathematics discourse. te ara expresses a high level of anxiety about mathematics achievement data. major concerns center on the lack of adequate achievement data and a lack of consistency between different teachers. te ara comments: the school hasn’t given me any information about the children i am teaching. i have no knowledge or where the children in the class are at in mathematics. i think things would be so much easier for everyone if we all follow the same journey and use the same assessments. we will all know then where we are in the curriculum and what we should be doing next. for te ara, curriculum mathematics knowledge is essential for survival in the modern world and is a fundamental aspect of the happiness of the child. if the school does not prioritize mathematics in favor of developing children in a holistic sense, problems will arise. this is associated with the ubiquitous nature of mathematics as an inherent part of everything, especially the well-being of children. mathematics is important because it is all around us… and i have seen a lot of children who lack confidence and have felt they are dumb and have hidden and shied away from mathematics because they don’t want to other people to see that. outside the gate, there are numbers all around us. children have to be competent to deal with the numbers they will meet in their lives. there is nothing wrong with building the child’s wairua (spirit), but the reality is that mathematics is in everything… this building has to do with numbers… the school runs on mathematics really, i think. interpretation: these quotes all circulate around an emerging conception of education in this māori school based on being māori and engaging in/with the contemporary world as unique beings. this integration of collectivity (being māori, being iwi) and individuality (flourishing as a unique being embedded in the world) is expressed in the more formal terms of this research as the overall legitimation of people based on a subjective social relation and an ontic epistemic relation. tweed indigenous struggle with mathematics education journal of urban mathematics education vol. 14, no. 2 56 for te ara, mathematics has its own protocols and culture that must be learned if students are to find a place in the economic system outside of a māori context and, more importantly, be able to understand the world, which is inherently mathematical. the children at my last school were happy in mathematics because... in a psychological sense... they were strengthening their minds, and this strengthens their spirit… because they have overcome a challenge, and this is a good thing no matter what the challenge; to overcome it strengthens the spirit. mathematics is construed as having fundamental connections to the material world and human life. mathematics is necessary for understanding and being able to participate in the modern world. te ara expands the importance of mathematics, which is the same as english-medium curriculum mathematics, from being about numbers to being about “the ability of people to speak and to understand the world.” it is a form of knowledge that underpins the nature of being human itself. students in te ara’s class are for the most part happy because they can follow the procedures laid out for them to follow. te ara has a very definite set of behavioral rules. students are strongly controlled to adopt a quiet, thoughtful approach and pay careful attention to correct procedures and language use. a common practice is for individual students to stand up in front the class and explain how they solved a problem using correct mathematical terms and symbols. students occasionally experienced distress when they were unable to follow a procedure. when this happened, te ara momentarily relaxed the usual behavioral rules and defaulted to ways of speaking and interacting that were aligned with the school ethos; students’ well-being and personhood were prioritized. interpretation: te ara represents a very formal approach to mathematics education. legitimation is very much about mastery of the techniques, language, and symbolic representations of the curriculum version of mathematics coupled with “proper” behavior and participation in public displays of mathematical performance. there is much tension between this approach and that of the school ethos expressed in a variety of ways, but all centered on te ara’s focus on the importance of mathematical competencies to life in a capitalist economic system and the urban context. in the perspective underpinning practices in this classroom, without mathematics, students are not only destined for poverty but also an inability to understand the world, which is conceived as inherently mathematical. this tension peaks and is resolved when students show distress, which allows the school ethos to reassert itself in te ara’s mathematics classroom. the dominant form of legitimation here is based strongly on a discursive epistemic relation (the engagement with mathematics mediated by the official discourse of curriculum mathematics education) and an interactive social relation (based on explicit formal rules of behavior/performance). in this class, being māori, being unique, only resurfaces in times of distress. tweed indigenous struggle with mathematics education journal of urban mathematics education vol. 14, no. 2 57 classroom 2, teacher: te mana te mana prioritizes real world skills that are needed to operate in the wider world. this usually means handling money and everyday activities like cooking and fishing. the usual practices and lessons are thought of in terms of pedagogical technique and mathematics knowledge acquisition. te mana mentions adjusting practices sometimes if the students are unsettled but otherwise there is a consistent sharp focus on mathematics learning and ensuring that students “stick to the point.” te mana indicates a strong influence from english-medium mathematics education. lessons focus strongly on what the teacher needs to do for students to “achieve the highest levels.” at the same time, there is an awareness of a potential injustice that lies within the universal status of mathematics. te mana had this to say on the matter: i haven’t thought about who wrote the curriculum and other resources in māori.... no doubt they are just translations of the english versions... it seems like assimilation... being pressured so that our ways of organizing things and thinking about things are just the same as pākehā [european new zealanders]... we don’t want that. the tension te mana feels here is strong; te mana would rather reject these imposed practices in favor of developing unique, iwi (tribal)-specific mathematics practices and language. this is desirable, but it may contradict a need to be able to measure progress of students’ learning and the need to “know whether you are at the national average.” the strength of this contradiction, which te mana calls a “weird situation,” is summarized so: “we try and think mathematics and we don’t relate it to the actual way we are.” te mana thinks that her own consciousness is changed when teaching mathematics in part because you “have to think like this (mathematically)” and “you can’t do what you want.” te mana is conflicted in several ways about mathematics and how it is learned. in terms of making progress through the levels of the curriculum, te mana professes impatience at not being able to “hurry up and finish tasks” but at the same time recognizes the need for patience and time to allow students to develop deeper understandings. for te mana, there is a strong tension between meeting targets and meaningful learning. te mana predominantly uses standard curriculum problems but sometimes attempts integrated work, which she believes will “increase the creativity.” however, te mana comments: when we do the integrated work, we sort of lose sight of the mathematics that we are supposed to be learning. it’s hard to tell what the children are learning sometimes, and they might go off and do all kinds of things... good things... but not mathematical. it doesn’t feel like mathematics anymore. tweed indigenous struggle with mathematics education journal of urban mathematics education vol. 14, no. 2 58 te mana comments that the overall purpose of the school is to produce a kind of person primarily defined in terms of personal human attributes. such a person is, ...a type of māori person: a gentle person, caring, open to all kinds of learning, with humility... so we should be producing a sort of person who in the first place will be following their gifts and enthusiasms and that makes them all different but at the same time they will all be the same in other personal qualities, like caring, hospitality, openness, respecting others. apart from a potential place in following their gifts and enthusiasms, te mana does not regard mathematics knowledge or any other kind of knowledge to be essential, but people should be open to all kinds of learning. te mana has a number of critical insights that appear to undermine a commitment to the way mathematics is being done. for example, the performance of mathematics merely to show that you have mastered what is expected (i.e., “doing mathematics to show you have done mathematics”) is rejected. te mana also contends that there is a destruction of the unique cultural understandings in traditional activities when they are treated as mathematics exercises. te mana explains how students examined number patterns in tukutuku panels (traditional geometric designs that express cultural understandings). here it was realized that the cultural purpose of doing tukutuku had been lost. te mana commented further: i’d like to get them to make a real tukutuku, an authentic one… i should ask nanny p (a māori elder)... not just do nice mathematics patterns as if they were tukutuku. in a similar way, te mana finds the notion of standardized assessment extremely problematic because of the idea that “a child should show predefined attainments at a certain age when the curriculum doesn’t have any idea about the child and the world they live in. this is a real affront to the child.” tweed indigenous struggle with mathematics education journal of urban mathematics education vol. 14, no. 2 59 classroom 3, teacher: te ao te ao is a young teacher who has been teaching at the school for four years. te ao is the only teacher who has been exclusively educated in kohanga reo (māori language preschool) and māori schools; all other teachers were educated in englishmedium schools and made the switch to the māori school after teacher training. te ao’s current view of mathematics focusses on informal knowledge useful in the world outside of the school. this includes providing tools that support, for example, dealing with finances and managing time. te ao prioritizes work that is relevant to things that will benefit students outside of the school. te ao expresses a belief in the universality of the basis of mathematics across languages and cultures. mathematics has a common basis for all people in the world. for te ao, this derives from universal needs that cause all people to develop practices such as navigating, cultivation and gathering of food, and creating shelter. te ao asserts that in modern times, mathematics has become similar all over the world because of national school systems. te ao suggests that before such systems, mathematical knowledge was more varied and dependent on context, language, and culture. te ao picks the parts of the curriculum that are most relevant for students. informal types of mathematics are associated with traditional cultural activities, such as rāranga (flax weaving), providing food for visitors, building canoes, and navigating by the stars. these activities are based largely on estimating measurements and the testing out of ideas through direct experimentation. this informality is interpretation: the mathematics classroom of te mana represents an “intermediate” position. the approach is similar to that of te ara, but there are many points at which tension is felt in a variety of ways. te mana has a growing sense of unease with the curriculum version of mathematics education, which is enacted and expresses a number of important insights. te mana recognizes how the curriculum prioritizes tasks, work completion, and progress through the levels of the curriculum and how this works against a natural acquisition of knowledge appropriate for each child. te mana conceptualizes curriculum mathematics education as a form of assimilation or colonization, which is eloquently expressed in epistemological and ontological terms as “being pressured so that our ways of organizing things and thinking about things” are changed. te mana also notices how seeing things in mathematical ways can erase the original, authentic māori meanings of cultural artefacts and practices. this connects with the discourse of power that surrounds mathematics—somehow it is allowed to explain everything in the world in mathematical terms and make everything inherently mathematical. although te mana’s mathematics classroom shows strong legitimation based on a discursive epistemic relation and interactive epistemic relation in a similar way to that of te ara, there is a simmering discontent that is a precursor to a shift in practice in which legitimation is more aligned with that of the school ethos. tweed indigenous struggle with mathematics education journal of urban mathematics education vol. 14, no. 2 60 contrasted with mathematics resources that are regarded as formal and in need of contextualizing. te ao locates being māori when learning mathematics in the nature of pedagogy. according to te ao, “the way you approach mathematics, that is the most important thing.” for te ao, there is no māori thinking in the curriculum resources themselves; rather, it is in the way the teacher works and relates to students: i quite like having the curriculum documents around… it’s good to know the assessment levels, but for me, it is always better to try to gauge the student’s achievements directly by what they are doing, materially, in front of me right then and there by listening to them and seeing what they are doing when we are working together. the construal of mathematics is strongly influenced by te ao’s own nature and the knowledge of students in the class, many of whom are related: i am a new teacher, but i have a lot of experience from other areas, and i know the students well because i am related to many of them. i use the curriculum resources sometimes, parts of them anyway, but i mostly rely on my own knowledge and my experience and what is working with the students. for te ao, the benefits for students from learning mathematics are an ability to be successful in the world based on a strong sense of māori identity derived from genealogy. in this view, it is essential that students, know who they are and have a strong sense of identity… it is important that they have all the necessary elements in place so that they are whole and know the structure of a person… their identity and origins. if this is all in place, students will be settled inside... if identity is good, the journey in life will be good as well. for te ao, a major benefit of a māori education is that students know their identity and their whakapapa (descent, genealogical origins). te ao interprets the social and spiritual well-being, the happiness of students, as a sign of the strong grounding of identity and extends this to be a sign that the foundational philosophical principles of the school are being enacted successfully. there is a location of success in the student-teacher relationship with reciprocal notions of give and take; respect must be given to be received. in this way, explicit links are made to the philosophical principles of the school in the mathematics learning. this justifies a particular view of learning as being primarily about relationship, identity, positive participation, and reciprocity. the activities designed by te ao usually involve physical movement or real material objects and are often unique, having been designed specifically for the particular lesson and students involved. there is no routine structure in te ao’s lessons; instead, students engage in a series of activities that may or may not relate to each tweed indigenous struggle with mathematics education journal of urban mathematics education vol. 14, no. 2 61 other in terms of mathematics learning. mathematics learning is another context in which the identity of students can be realized; the development of identity is the theme that relates different mathematics activities and gives coherence to them over time. a genuine, strongly felt tension between te ao’s mathematics practices and those of other teachers is felt. there is a vulnerability to criticism from other teachers and parents, and te ao expresses a need to be careful because being “too far out of the box is dangerous.” this may arise because such people cannot recognise that mathematics is happening in te ao’s lessons. in te ao’s view, …other teachers are trying to be the ideal mathematics teacher, use the curriculum resources and get the students to do what is expected… this makes it all predictable, boring, and routine. i just go on my own thinking and what i can see is the most useful for my students. people might look at my lessons and see the students yelling, excited and moving around, standing on chairs maybe in some kind of game, and they think there’s no mathematics going on… but it’s there, mathematics is happening, but you just have to look carefully… it might not be what you are used to seeing. because of this tension and perceived risk, te ao organizes a small part of each lesson in which bookwork or worksheets are done to show to others that “real mathematics is being done.” in all of the lessons in the data, students and te ao are highly collaborative, physically active, very vocal, and socially interactive. te ao does not maintain a separation from students and often participates in the activities as well. a sign of the success in te ao’s lessons is when students “come out of your class with a smile and are still keen on your lessons.” te ao does not attend strongly to how students are acquiring mathematics knowledge other than through participation in game-like activities. learning is assumed to happen spontaneously as a natural result of participation in activities. learning should be in context and have a real purpose. this involves integrating many learning areas in one activity. along with a focus on the practical and the integrated task, te ao elaborates the view that formal resources are not focused on the kind of math required in everyday life. te ao relies instead on personal thinking and resources. in all the lessons in the data, there is no emphasis given to the learning of the newly invented māori words for the mathematics curriculum. te ao assumes that the students’ language proficiencies are such that that they will learn any new terminology required. te ao asserts that “it’s up to each school to use their own words for mathematics; this is no big problem.” tweed indigenous struggle with mathematics education journal of urban mathematics education vol. 14, no. 2 62 student perspectives students tended to hold views that aligned with those of the teachers who taught them. for most students, mathematics was definitely about the mind and “sharpening the brain” so that mathematics activities, primarily “working out answers,” could be done quickly and efficiently. quickness is associated with sharpness of mind. students attribute the importance of mathematics to having essential skills in order to “succeed in a career or get a good career.” one student commented, if you want to have a good job, you need mathematics so that you will achieve in the world. mathematics will make you sharp... like on a scale of one to ten, you will get a ten. in all activities you have to use mathematics. like in rugby, you have to count the points, and in your career… you will need lots of subjects like english, māori... and brainy people will do science and mathematics. when asked about who real mathematicians were and what they were like, there was a very strong assertion that mathematicians would be male, european or asian. and probably “geeky loners” with poor social skills. it was apparent to students that māori people weren’t mathematicians. interpretation: te ao’s mathematics classroom represents a regime closely aligned with the school ethos and is based on strong ontic epistemic and subjective social relations. behavior and interaction between students and teacher are based on family-like relationships in which students and teachers interchange roles frequently and knowledge is generated largely through direct engagement and experience of unique activities. students behave in many different ways in activities that require direct involvement and engagement (often physical). mathematics learning is implicit to these activities, which has an overall purpose of supporting the identity development of each student as an individual and as māori. in this perspective, mathematics education is another context in which students may come to know themselves as unique and as māori. the legitimation is based on a strong ontic epistemic relation and subjective social relation in the ways understood in the school ethos—participatory epistemology and the person as a unique, māori being. this case also highlights how tensions with curriculum mathematics education manifest with this form of legitimation. te ao feels forced to include curriculum resources and demand certain behaviors from students (the reintroduction of a legitimation based on discursive epistemic and interactive social relations) for political reasons—to show other teachers and parents that this form of mathematics education is happening. tweed indigenous struggle with mathematics education journal of urban mathematics education vol. 14, no. 2 63 students tended to make somewhat circular arguments about the importance of mathematics, as these researcher-student interactions show: mahia (student): you will need mathematics in your career. researcher: to do what? mahia: to do the mathematics that is needed. ānaru (student): mathematics will make your brain sharp researcher: sharp in what sort of ways? ānaru: so that you can be good at games. researcher: what sort of games? ānaru: oh... mathematics games... like cool math games researcher: how will mathematics help you when you are older? tuki (student): it will be very useful. researcher: in what ways? tuki: i’ll be able to help my own children with the math they have to learn at school. commenting about mathematics work, one student offered the opinion that it wasn’t real math because there was “too much discussing and drawing pictures.” another student associated learning lots of mathematics strategies for calculating correct answers with an ability to “decide which pathway in your life is the good one.” students have a clear appreciation of mathematics as a challenge to the mind. without a challenge, there “could be no learning,” and through challenge, “correct mathematics learning” could be achieved. in contrast to this view of mathematics, a senior student, who had decided not to continue with mathematics learning in the senior school, explained that learning cultural knowledge from elders who, perhaps, would soon be lost was more important: i can pick mathematics up any time when i need it and know that it won’t be a struggle... that’s why i am knocking it out [not doing it anymore]... so i can concentrate on te reo māori [māori language] and doing kapa haka [māori performing arts] where i can express myself... and i am going to learn about my marae [tribal settlement and people] from my koro and kuia [elders]... they won’t be around for very much longer, and i want to learn from them. tweed indigenous struggle with mathematics education journal of urban mathematics education vol. 14, no. 2 64 conclusion the findings just presented and discussed paint a complex and nuanced picture of life in this māori school. at a high level of generality, and in the formal terms of the analytical concepts of legitimation, the school as a whole bases its practices on an ontic epistemic relation and a subjective social relation. in curriculum mathematics classes, on the other hand, legitimation involves a discursive epistemic relation and an interactive social relation. the former centralizes being māori and having a direct, participatory engagement with real events and the world in general. the latter centralizes being a mathematics student who has knowledge of the official mathematics education discourse. there are strong indications in the data about the desire to create a mathematics curriculum, and a full school curriculum in general, that is consistent with the legitimation prioritized in the school ethos. according to the analysis in this research, this is a mathematics curriculum based on an ontic epistemic relation and a subjective social relation (as understood in this indigenous context). it is here that a return is made to the ethical stance described earlier with respect to non-indigenous researchers working in indigenous contexts. at this point, it is very tempting to launch into a discussion of what is entailed in the creation of such a curriculum and to list recommendations. despite the generation of new knowledge that has taken place in this project, no recommendations can be made. in the spirit of ethical care for the other, in the sense of levinas, all that is possible is the unconditional offering of these findings with the understanding that they are possibilities or even a fantasy. interpretation: students generally have an instrumentalized view of mathematics education as something necessary for employment and necessary for participation in the education system itself. no students made strong links between their experiences in mathematics and being māori. societal discourses about the importance and power of mathematics and who does mathematics (brainy people, non-māori) were surprisingly strong given the fact that almost all students had been brought up from early childhood in māori schools and preschool centers. at the same time, students believed that they could learn mathematics but had an ambivalent attitude towards it; it was necessary and important for employment and careers but could be picked up as and when needed. in terms of specialization, students present a variegated picture. there are elements that can be interpreted as aligning with the discursive epistemic and interactive social relations of curriculum mathematics and somewhat contradictory elements that align with the school ethos. there may be some evidence that students are moving to a position that centralizes their own indigenous identity, language, and culture, which then allows them to operate strongly in social fields that do not legitimate these. curriculum mathematics certainly seemed to cause less angst for students than for their teachers. tweed indigenous struggle with mathematics education journal of urban mathematics education vol. 14, no. 2 65 recommendations, if any are to be made at all, are entirely an internal matter for māori schools and communities themselves, as are the following concluding comments. firstly, it seems likely that schools that may wish to embark on an exploration of mathematics education in a new paradigm will have to meet the challenge posed by the strong connection between mathematics attainment and success in urban, capitalist conditions. curriculum mathematics education plays a powerful role, if not a central role, in the creation of subjectivities aligned with neo-liberal social ontologies now deeply embedded in new zealand society. the identity formatting effect of mathematics education and complicity in the production of neo-liberal subjectivities is well documented in the critical mathematics education literature (see, for example, pais & valero, 2012; popkewitz, 2002; skovsmose, 1994; valero, 2018). encouragement can be garnered, however, from the phenomenon of māori resistance in urban conditions in general. as discussed earlier, māori have already successfully maintained cultural integrity whilst adapting to and adapting urban/capitalist conditions with respect to a number of domains of activity, which taken together might be thought of as a māori economy. if this wider view is taken, the creation of an ontic and subjective-based mathematics education might gradually be aligned with this māori economy rather than the existing mainstream economy. this accepts that education in general and mathematics education in particular is always prefigured by the nation’s economy; it seems plausible then that as the māori economy (itself based on its own ontological commitments) grows in strength, māori mathematics education may be pulled into its orbit. a second important and related challenge concerns the “gaze” of mathematics education, which dowling (1998) has labelled the myth of reference. in this gaze, mathematics education redescribes the contents of non-mathematical domains of all kinds in its own terms, recasting the whole world, in effect, as inherently mathematical in nature. as te mana recognizes clearly, there is a very significant problem when curriculum mathematics education casts its powerful gaze upon indigenous artefacts, practices, and the public and personal lives of indigenous people; in an uncritical engagement with curriculum mathematics education, and uncritical acceptance of its discursive epistemic relation, a cultural and historical revisionism, a form of symbolic violence, can result. this suggests that a form of perspective management might be employed in which the cultural origins and meanings (the ontologies and epistemologies) that underpin and create the artefacts and practices of māori life are protected from being redefined by this mathematical gaze. to do this, a critical awareness and careful distinction can be made between the mathematical perspective, which sees things in a mathematical way, and the authentic meanings inherent in indigenous artefacts and practices. in other words, we can understand that a mathematical gaze is possible of any entity but that the mathematical understanding generated is not the entity itself. the entity retains its own tweed indigenous struggle with mathematics education journal of urban mathematics education vol. 14, no. 2 66 ontological integrity, which may have nothing to do with mathematics. being explicit about both perspectives and when switches are made between them is necessary in order to avoid the very common assertion that an indigenous practice is “really” a mathematical one even when the participants themselves have no cognizance of any mathematics being enacted. there is a strong resonance here with the critique of culturally relevant and responsive pedagogical practices as asset-based pedagogies made by django paris and h. samy alim (paris & alim, 2014). these two authors build on the foundations laid by the earlier developments of culturally relevant and culturally responsive pedagogies (e.g., ladson-billings, 1995a, 1995b, 1997) to promote culturally sustaining pedagogy (csp) as a critical centering of the ways of being of indigenous communities and communities of color. csp should operate to automatically involve a disruption of dominant education systems that only legitimate “white, middle-class, monolingual, cisheteropatriarchal, able-bodied superiority” (alim & paris, 2017, p. 13). paris and alim critique asset-based approaches, such as funds of knowledge, culturally relevant/responsive pedagogies, and third space, as too easily defaulting in practice, if not in theory, to an instrumental usage of indigenous practices, language, and, indeed, indigenous bodies for the purposes of achieving success defined in terms of a panoptic white gaze. here, the connection with dowling’s myth of reference and the mathematical gaze so keenly felt by te mana becomes clear; we can understand the recontextualizing gaze of curriculum mathematics education as part of the panoptic white gaze—indigenous māori artefacts and practices are repurposed as assets to bring students into the legitimation regimes of curriculum mathematics education. paris and alim provide vital insights by recognizing the tendency of asset pedagogies to work on static, “archaic,” and reified forms of indigenous cultural practice, knowledge, and identities and neglect contemporary and emergent forms forged in urban, capitalist conditions such as those emerging in the māori school of this study. in the new zealand context, this manifests clearly as a persistent focus on traditional māori contexts and practices, such as flax weaving, ocean navigation, carving, and hāngi (traditional earth ovens). contemporary (urban) realities for māori communities are rarely present, and if they are, they tend to be indistinguishable from “whitestream” activities apart from being translated into the māori language, involving brown bodies, and with only their mathematical aspects emphasized. paris and alim (2014) also describe the necessity of developing an inward gaze, which critically addresses both the liberatory aspects of contemporary indigenous culture, language, and practices, and the restrictive. for the participants in this study, there was considerable critical reflection on their existing mathematics practices, the effects on identity, language, and māori knowledge, as well as the benefits of coming to know mathematics. for the researcher, the inward gaze is a constant critical reflection on his own position, the impact of his words, and the limits of what he can say and know, which has led to his position of ethical knowing and not-knowing. tweed indigenous struggle with mathematics education journal of urban mathematics education vol. 14, no. 2 67 in a further development of csp, mccarty and lee (2014) developed the concept of culturally sustaining/revitalizing pedagogy (csrp), specifically in indigenous contexts where indigenous sovereignty is a fundamental issue. mccarty and lee’s work resonates strongly with the situation in the māori school of this study. indeed, the desire to create a curriculum centering on māori ontology and epistemology is an expression of māori sovereignty, and the process that the māori school is going through may be thought of as an example of critical csrp. there is, it seems, a significant difference here in new zealand. the current strengthening position of māori in our small country is creating the possibility at least that rather than indigenous educational sovereignty being thought of as accountability to indigenous communities in the same way as to the national government, that education itself may eventually come directly under this indigenous sovereignty. despite the homogenizing cosmopolitanism and “melting pot discourses” prevalent in capitalist, urban conditions, māori in general, and this māori school in particular, are showing that a pluralistic society in which communities co-exist with deep level ontological differences is possible and necessary. references alim, h. s., & paris, d. 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(2012). neo-liberalism and the global restructuring of knowledge and education. routledge. yukich, r., & hoskins, t. k. (2011). responsibility and the other: cross-cultural engagement in the narratives of three new zealand school leaders. journal of systemic therapies, 30(3), 57–72. https://doi.org/10.1521/jsyt.2011.30.3.57 copyright: © 2021 tweed. this is an open access article distributed under the terms of a creative commons attribution-noncommercial-sharealike 4.0 international license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. journal of urban mathematics education december 2017, vol. 10, no. 2, pp. 52–65 ©jume. http://education.gsu.edu/jume hilary povey is a professor of mathematics education at sheffield hallam university, city campus, sheffield, s1 1wb, united kingdom; email h.povey@shu.ac.uk. her research centres on social justice issues in mathematics education with transformative learning in mathematics classrooms and the role of historical awareness in combating neoliberalism as current foci. gill adams is postgraduate research tutor at sheffield hallam university, city campus, sheffield, s1 1wb, united kingdom; email g.adams@shu.ac.uk. her research interests include mathematics teacher learning and identity, utilizing narrative approaches to explore social and political contexts. rosie everley is a mathematics education graduate of sheffield hallam university, city campus, sheffield, s1 1wb. her research interests include teacher professional development, social constructs of learning, and the continuing impact of performativity. public stories of mathematics educators “its influence taints all”: urban mathematics teachers resisting performativity through engagement with the past1 hilary povey gill adams rosie everley sheffield hallam university n england, globalisation and neoliberal2 political agendas have created an environment in which teachers are constantly measured and ranked and subjected to a discourse of marketisation, managerialism, and performativity. this measuring, ranking, and subjection is particularly strongly felt in urban schools, where a discourse that recognised the systematic disadvantages that many urban children experience has been replaced by a discourse of “failing” students, teachers, and schools. the effect is to erode teachers’ sense of independence and moral authority and to challenge their individual and collective professional and personal identities. the need to understand the current policy environment, to step aside and look on critically, becomes more important even as it becomes more difficult. many teachers are engaged in re-storying themselves against this audit culture. we argue that it is possible, through excavating the past, to offer current-day teachers’ stories to support this process of re-envisaging what they are, might be, and might become in their professional lives. here we offer a response from one currently serving teacher, the third author, to the experience of performativity and 1 an earlier unpublished version of this article was presented at the 13th international congress on mathematical education (icme13), hamburg, germany, 24–31 july 2016. 2 neoliberalism as a political philosophy stands for such ideas as privatization, austerity, deregulation, free trade and reductions in government spending in order to increase the role of the private sector in the economy and society, including within education. it brings the market into all spheres of human life and regards human beings fundamentally as market-driven consumers. i http://education.gsu.edu/jume mailto:h.povey@shu.ac.uk mailto:g.adams@shu.ac.uk povey et al. public stories journal of urban mathematics education vol. 10, no. 2 53 we illustrate some ways in which she is able to mobilise historical stories from a previous urban teachers’ curriculum project in her resistance to dominant, neoliberal discourses. social and political context in england, since the 1988 education reform act, education has been subject to constant reform. government interventions, particularly the intense monitoring of students, teachers, and schools (particularly those schools working with students of less advantaged socio-economic status) and the high stakes consequences of the judgements which are then made, have had consequences for teachers’ identities, subjecting them to increased surveillance and reducing their independence (day & smethem, 2009). the “audit ideology” (groundwater-smith & mockler, 2009, p. 5), evident in the school inspection system, and the accompanying league tables are key tools of the neoliberal political context, an environment within which teachers and schools are constantly measured and ranked and education itself becomes recast as a consumer good to be marketed and made available under market forces rather than conceived as a moral enterprise and a public service (macpherson, robertson & walford, 2014). this “epidemic of reform” (ball, 2003, p. 215) changes who teachers are as well as what they do. ball (2003) notes the three interrelated policy technologies of this epidemic: the market, managerialism, and performativity. the effect of government interventions in many countries, england included, is “to erode teachers’ autonomy and challenge their individual and collective professional and personal identities” (day & smethem, 2009, p. 142). we are indeed in the grip of the terrors of performativity and a struggle over the teacher’s soul (ball, 2003). in their study of one english urban comprehensive school, hall and noyes (2009) use the foucauldian notion of regimes of truth3 and note how these regimes, characterising what teachers do as “delivering” goods to their pupils, operate to normalise the use of data to determine the needs of staff and students, changing the nature of teachers’ work as they come under increasing pressure to document and justify performance. this delivery discourse4 has the ability to shape, order, position, and hierarchise those in the field through systems of comparison, evaluation, and documentation, making everything calculable: 3 foucault (1979) claimed that each society has a regime of truth which are the kinds of discourses it accepts and makes function as true. 4 discourse in the sense employed in this article draws on the work of foucault (1979). he defined discourses as thought systems composed of ideas, outlooks, beliefs, and practices that construct both subjects and the wider social processes that legitimate current taken-for-granted ways of seeing the world and the associated relations of power. povey et al. public stories journal of urban mathematics education vol. 10, no. 2 54 it is impossible to over-estimate the significance of this in the life of the school, as a complex of surveillance, monitoring, tracking, coordinating, reporting, targeting, motivating. (ball, maguire, braun, perryman, & hoskins, 2012, p. 525) currently, in england, pupil performance in mathematics examinations at age sixteen usually operates as the single most important item of data in judging secondary schools. as a result, mathematics teachers routinely experience greater pressure and come under more scrutiny than most, if not all, of their colleagues. the stakes are high. such pupil performance data are then used to terminate head-teachers’ employment and to convert their schools from publicly run local schools working under a democratically elected authority to privately run academies (reminiscent of the charter schools movement in the united states). urban schools working with disadvantaged communities are most vulnerable to such forced academisation and, even when vociferous local support from parents and the community is mobilised, protest is usually to no avail (see, for example, millar, 2012: the school was ultimately taken over by the harris multi-academy chain). we agree that “not all teachers are convinced by the rhetorics of performance, and many teachers are not convinced all of the time” (ball et al., 2012, p. 528). imagining an alternative to the role that falls to teachers in such an audit society, sachs (2001) calls for teachers to take on an activist identity, one that arises from democratic discourses and has social justice at its heart. the construction of reflexive self-narratives aids a critical examination of the policy environment; moreover, sachs proposes that such narratives, made public, may be a productive support for professional learning. as stronach, corbin, mcnamara, stark, and warne (2002) write: “professionals must re-story themselves in and against the audit culture” (p. 130). many teachers are engaged in this re-storying in a variety of ways; we argue that one way in which it is possible to support them is through excavating the past, creating a “public resource” (nixon, walker, & clough, 2003, p. 87) or, as this journal has it, a public story (bullock, 2014), available to current-day teachers to reenvisage what they are, might be, and might become in their professional lives. we are currently engaged in such an historical endeavour, centred on smile mathematics. the first author had been a founding member of the teacher-led smile project and the second author had also been a smile teacher. both knew they had experienced participation in a vibrant, democratic, autonomous community of secondary mathematics teachers working together to create similarly democratic educational spaces in their mathematics classrooms. relevant details about the smile project are presented in the section that follows. for those inhabiting the educational landscape, reforms following the education reform act have supplanted existing ways of understanding, destroying organisational memory (goodson, 2014). the current endeavour aims to counter this, povey et al. public stories journal of urban mathematics education vol. 10, no. 2 55 preserving such memory, by drawing on two interrelated narratives: a narrative of a mathematics curriculum initiative of the time (smile mathematics) and narratives of individual smile teachers’ professional life stories. the former “systemic narratives” are based on documentary analysis of historical documents (goodson, 2014, pp. 34–35), some already archived and some collected as part of creating the public story; the latter are being collected through reflective writing and conversational interviews. we will also be seeking to address the interconnected questions: if the purpose of the curriculum is to control, to limit teachers’ freedom (goodson, 2014), how was this subverted in the smile mathematics project, with its central tenet of authority of teacher and learner? and what conditions are necessary for such a project to flourish in the future? smile mathematics there is, to date, no socio-historical study that has explored the development of teacher-led curricular innovation in mathematics teaching in england during the period 1970 to 1990. the smile mathematics project, which was one of the most significant curriculum change projects of the era in england, has itself not been the subject of academic study although some contemporaneous accounts exist (not currently archived) and some retrospective descriptions are available (for example, povey, 2014). the inner london education authority (ilea), abolished in 1990 by the conservative central government, was a large urban authority serving some of the most deprived boroughs in the country. the smile mathematics project was financed and supported by the ilea and was innovative in embracing a commitment to all attainment teaching,5 teacher creativity, and an investigative, problem-solving pedagogy. smile saw itself as learner centred and gave considerable responsibility to students for organising and shaping their own learning and that of their learning community. it had its roots in the 1970s, a time of reconstruction in the english school system characterised by a commitment to social justice and building upon curriculum development projects of the preceding decade (goodson, 2014). although supported by the local education authority, smile was the result of teacher-initiated change. locally based, it nevertheless influenced thinking about mathematics teaching across the uk and internationally. teachers were released from school duties for one day a week over many years to form a working collective to create, refine, and publish imaginative and inspiring mathematics curriculum 5 we use the term “all attainment” to refer to the absence of any tracking, setting, streaming or other forms of segregation of pupils on the basis of their prior attainment in organising school mathematics classes. we prefer this to the term “mixed ability,” perhaps still more commonly used in the uk, because (a) we think any group of learners is going to be somewhat “mixed” and (b) “ability” suggests a pre-determined, fixed, and limited capacity. povey et al. public stories journal of urban mathematics education vol. 10, no. 2 56 materials for use in their own classrooms and beyond.6 equally distinctive and equally important, the structure of the project instilled a deep democracy, with decision-making resting with a consensus of those who participated. any, and all, were welcome and could contribute. fairly early on, the ilea chief inspector for mathematics argued with the assembly that a more conventional democratic structure consisting of elected hierarchies with committee members and so forth should be set up; but when this was rejected by the collective, he allowed its will to prevail despite his own misgivings. (that story alone speaks powerfully of a different worldview from that which is currently dominant.) with continuing support, both financial and philosophical, from the ilea from 1972 to the late 1980s, smile flourished with this open authority structure that placed the teacher at the heart of decision-making. in 1990, margaret thatcher’s administration abolished the ilea in “a grave error brought about through political spite” (mortimore, 2008, para. 8); this action and the beginnings of the neoliberal ascendancy led gradually and then increasingly rapidly to smile’s demise. our radical history of smile—not yet complete at the time of this writing— examines the contrast between, on the one hand, the opportunities that smile afforded for democratic professionalism, a concept that has collaboration at its core (whitty, 2006) and, on the other, the current dominant discourse that enshrines a managerial perspective enforced through compliance with teaching ‘standards’ (kennedy, 2007; sachs, 2001). through this history, we seek to illustrate and draw attention to the fact that this discourse draws on a historically contingent and fragile political rationality and to challenge its commonsensical appearance (ball et al., 2012, p. 514). the historical endeavour has initiated an archive using digital media.7 the archive includes contemporaneous and recent accounts including some group interviews analysed using narrative enquiry. some of those involved in smile, including those present during its inception, were invited to participate in unstructured group conversations at various inner london venues during the early part of 2016. participants were recruited through formal and informal mathematics education networks and by means of a snowball sampling process, with contacts proposing others who had a role in the project. they were offered several questions in advance of the meeting that asked them to reflect upon: how they became involved in smile; how they understood their role and responsibilities; the nature of authority and autonomy within smile; and the links to other events of the time. the group conversations involved between six and eight participants each, including the two main authors of this paper, and were supplemented by a single paired conversation. the group con 6 these have been digitised and are now available at the national stem (science, technology, engineering, and mathematics) centre at https://www.stem.org.uk/elibrary/collection/2765/smile-cards. 7 see https://smilemaths.wordpress.com/. https://www.stem.org.uk/elibrary/collection/2765/smile-cards https://smilemaths.wordpress.com/ povey et al. public stories journal of urban mathematics education vol. 10, no. 2 57 versations lasted approximately three hours and the paired conversation an hour and a half. all were audio recorded. these recordings have been transcribed, with initial narrative analysis shared with participants and with the wider mathematics education community. participants to the conversations and others who were unable to attend have also provided further personal commentaries, usually by email, and additional archive material. in addition to collecting this historical material, we are currently exploring how such tales from long ago may, or may not, speak to initial teacher education students and to practising teachers. we began in 2016 with a small number of recently qualified teachers. in this public story, we work with one of them, rosie, to offer phenomenological insights into her experience of performativity and then to illustrate how she has been able to use the past, in this case smile stories, to resist dominant, neoliberal discourses and to assert an alternative identity and set of practices in her classroom. we suggest that this account offers plausibility to our hopes for our historical endeavour. introducing rosie rosie entered teaching through a 2-year, post-graduate course in which the first year was spent studying undergraduate mathematics and the second on professional preparation for mathematics teaching. the first and second authors had known and taught rosie during her studies and she had kept in touch. during her course, and as she was aware, rosie was taught by several tutors, including but not limited to the first two authors, who had themselves been smile teachers and who saw themselves as working within a mathematics pedagogy that valued autonomy, independence, personal authority, and democracy. she later remarked: you could see how their teaching styles matched up with those used in the resources. (personal communication) in the first year, it was not uncommon for her tutors to take smile resources as a starting point for mathematical investigation: we … found the activities engaging, and they prompted us to think about different ways that we could present mathematics to students that would help them understand it more fully. (personal communication) this stating with mathematical investigation was built on in her second year when the smile resources were often used in professional sessions about the teaching and learning of mathematics in secondary schools. povey et al. public stories journal of urban mathematics education vol. 10, no. 2 58 performativity… in 2016, rosie worked in an urban school in the north of england, one that is perceived, and perceives itself, as a high-performing school with high “standards.” this reputation gave the school the right to offer leadership to lower performing neighbouring schools that work with more disadvantaged intakes. this school, therefore, functions as a link in the regulatory chain connecting neoliberal government policies with the practices of individual teachers, and through them to their objectified students, and the individualised performance and assessment of both. in her first year as a secondary mathematics teacher, as a prelude to some study at masters level, rosie was given stephen ball’s (2003) article “the teacher’s soul and the terrors of performativity” to read. performativity as it is used here is the condition that has been brought about by all the reforms driving for more teacheraccountability, external monitoring, de-regulation, standardised testing: that is, “performance” as interpreted by what can be measured and found desirable. rosie was asked to write about her own experiences of performativity in response to the article. the term “performativity” and the task immediately resonated with her: performativity is known to all teachers whether by name or not. its influence taints all the day-to-day activities of teachers and subconsciously affects the way they view their role. (pre-masters writing) rosie highlighted how demands of performativity absorb huge amounts of teacher time and energy, disciplining them through meticulous interaction with trifling and insignificant data (ball et al., 2012, p. 523), and leaving them less time and energy with which to engage creatively in the moral and interpersonal endeavour of education: the sheer amount of work involved causes a significant dilemma … i have to sacrifice a huge amount of my time in order to do my job, [but] much of this is dedicated to monitoring performance and meeting targets, not improving the learning experience of my students. (pre-masters writing) she experienced in very concrete terms that foucauldian technology of the self that permits individuals “to effect by their own means or with the help of others a certain number of operations on their own bodies and souls, thoughts, conduct, and way of being, so as to transform themselves” (foucault, 1988, p. 16, emphasis added). thus, the subject, rosie, acts on herself to conform to the demands of neoliberalism (lemke, 2001): for me, the pressure to be an outstanding teacher is ever present. you are constantly being compared to others and the standards, but you are also constantly comparing yourself and judging how you are progressing. there have been occasions when i per povey et al. public stories journal of urban mathematics education vol. 10, no. 2 59 sonally felt like i have been improving and doing a good job, but i still never feel like i am good enough. (pre-masters writing) as neoliberal subjects8 we are expected to be constantly remaking ourselves and to be doing this autonomously, but the experience of constant surveillance and judgment gives the lie to this. rosie wrote tellingly about this interplay of the judgment by self and the judgement of others: teachers now are responsible for making sure they are meeting the myriad of criteria to prove to others—and themselves—that they are a good teacher. having to constantly prove themselves drives teachers to invest huge amounts of time and energy into their job. the feeling of being constantly judged by uncertain criteria heightens the stress levels. all together it leads to a teacher who constantly questions their own ability to do their job and faces a daily personal battle over doing a good job and getting swallowed up by their work. …teachers may have responsibility for their own performance but they have very little control over it and, if they are anything like me, feeling that you are constantly chasing a moving target and coming up short. (pre-masters writing) rosie has a variety of strategies for resisting this neoliberal positioning. here we explore the extent to which and in what ways she has been able to use the excavated past and stories of smile as part of that resistance. … and resistance during her third year as a teacher, we asked rosie to write to us about her experiences of smile and the relationship of these experiences, if any, to the discourses of performativity and her resistance to neoliberal positioning. rosie wrote freely in emails to us about her encounters with smile resources and stories about smile she was told during the 2 years of her initial teacher education. she also wrote about her subsequent engagement with the digitised archived materials during her career as a teacher. we used this text to identify three ways9 in which rosie used smile stories in the complex process of shaping her sense of a professional self:  teaching in ways that were important to her and “against the grain” (cochran-smith, 1991); 8 neoliberal subjects are individual entrepreneurs who provide for their own needs and ambitions, rational, calculating and self-regulating. they are autonomous, competitive and exercise selfgovernance and self-responsibility—each becomes a company of one (verdouw, 2016). 9 we returned these themes to rosie who was happy that the analysis caught fundamentals of her experience. we then produced the original conference paper on which this article is based, and all three of us presented the paper at icme13. povey et al. public stories journal of urban mathematics education vol. 10, no. 2 60  fostering a more democratic way of knowing; and  acknowledging collaborative teacher professionalism. teaching in ways that are important and against the grain it is very clear from rosie’s writing that she was able to use the smile resources to resist the pressure of “our new performativity culture of teaching by level and showing linear progress” (pre-masters writing). for example, she wrote: the lessons we experienced at university really inspired me … they showed me the excitement of discovery and how that can be incorporated into teaching. … they also showed me a new approach to teaching mathematics, one that is more involved and engaging than i had experienced as a learner before. … it is something that i keep in mind now as i plan for my own classes. … the smile resources for me represent a huge ideas bank with examples of some great teaching practice. … i know that when i look through the activities i will find activities that will suit how i want to teach my students. (personal communication) rosie was able to draw on her previous experiences of smile to inform and develop her current practice, using questioning and ideas from the materials to inform her planning. she used the ideas embedded in the resources to make the mathematics accessible to a wide range of students and to enrich their learning. fostering a more democratic way of knowing rosie valued the resources because they gave students access to meaningmaking in mathematics. this access, in turn, promoted a more democratic epistemology (that is, our way of understanding what knowledge is, of how we come to know and of how knowledge is warranted) rather than subjugation to the “personal fatalism … servility …[and] negative self-esteem” (skovsmose, 1994, p. 189) so often encountered in secondary mathematics classrooms: a lot of the tasks are investigative and allow the students to discover relationships themselves, but all of them help foster deeper understanding of why things are happening. … i have a deep affection for [the smile resources] because their complete focus on teaching for understanding is something that is really important to me. … i can get [the students] to explore an area of mathematics themselves and discover something. (personal communication) in rosie’s school, there was “a very strange mix”: teaching for understanding was encouraged yet “testing and setting and the ‘best method’” were also relentlessly pursued. in the context of what rosie described as “an uneasy truce,” there were contradictory spaces within which she could teach more democratically (personal communication). povey et al. public stories journal of urban mathematics education vol. 10, no. 2 61 acknowledging collaborative teacher professionalism it is also the case rosie knew that the smile resources were created by teachers rather than for them and that these teachers worked together and with a sense of professional authority. she used this knowledge to see her current experiences from outside the currently taken-for-granted in schooling: i think of smile with a mixture of fondness and sadness; it reminds me that there are ways to include more engaging and investigative work in our maths lessons, but it also highlights just how limiting our current curriculum and testing system is. … it saddens me to know that all that time ago, teachers figured that this is a good way to teach children mathematics and yet there is still no room for it in most schools. it does give me hope though, because i know that my teaching is better as a result of my knowledge of smile. (personal communication) through the stories she had heard from her past smile teaching tutors, rosie was able to see the community of smile teachers as having had “a rationale for practice, [an] account of themselves in relationship to the meaningfulness of what they [did]” (ball, 2003, p. 222). therefore, they offered an alternative identity to that of performing the neoliberal self. discussion there is a long tradition that asserts that it is who the teacher is rather than simply what she does or what she knows that fundamentally shapes the educational experience (see, for example, dewey, 1957). the neoliberal project also takes seriously the need to shape the “soul” of the teacher. it redefines teachers as education technicians to bring about a change in how teachers experience their professional selves, to reform and regulate subjectivities so that teachers work intensively on the self and come to perform an individualised, enterprising identity (ball, 2003). the moral landscape of autonomy and professional judgement, of trust and co-operation (stronach et al., 2002, p. 130) is undermined by the new ethic of performance where everyone must strive to be above average and, indeed, all must become outstanding (see, for example, matthews, 2009). we have no doubt of the productive power (that is, the power to make things happen) of neoliberal discourse to colonise teacher identity; nor of the regime of truth which normalises that discourse. teachers are surrounded by ever increasing demands upon their time and by “meticulous, often minute, techniques” (foucault, 1979, p. 139) of surveillance. disciplinary coercion is exercised not directly but by requiring attention to petty minutiae, “a political anatomy of detail” (foucault, 1979, p. 139) with the ever-present threat of “underperformance”: “the microdisciplinary techniques of hierarchical observation, normalising judgements and examinations” (hall & noyes, 2009, p. 851) are used to classify, hierarchise, and povey et al. public stories journal of urban mathematics education vol. 10, no. 2 62 individualise. these minutiae dominate teachers’ lives as never before, leading them in turn to see their students as “data walking around, they no longer [seem] human” (lightfoot, 2016, para. 19). it is through the disciplinary techniques and the micro self-surveillance they are designed to provoke that the desired teacher identity under neoliberalism is “carefully fabricated” (foucault, 1979, p. 217). nevertheless, it is remarkable the extent to which teachers are able to resist the “combinatory and relentless effects” (ball et al., 2012, p. 528) of the technology of performativity and the fabrication of neoliberal selves, as the account in this public story illustrates. although based on the experiences of only one person, we offer evidence in this story that excavating the past for oppositional narratives can be a productive support for building teacher professional identities that challenge the current regimes of truth. we hope this public story, and the website of historic digitised archive material and current rememberings that we are creating, will prove to have catalytic validity (lather, 1986, p. 78); that is, to have value because of its capacity to re-orientate and energise in order to bring about change. rosie has kept in touch with her past smile university tutors and is able to use the currently circulating counter-hegemonic discourses (that is, those that seek to critique and dismantle the dominant discourses which legitimate and preserve the status quo) to re-story her current experiences and her understanding of how things might be: such is the power of performativity that, even after three years on the job, i still find it hard to acknowledge even to myself that i am a good teacher. i still have to justify every decision i make by demonstrating that it has led to progress. i still have to sneak out of the house at 6am to make sure i get all the tests marked and data sent in on time before i teach that day. ... but i am slowly learning to trust my own judgement, especially when deciding how to “present” mathematics to students. on those lessons where i do choose smile inspired activities and i see the sense of achievement in students’ faces as they connect key ideas together for themselves, i know how they feel. that’s how smile made me feel when i discovered how much fun learning maths can be. (personal communication) in understanding these writings from rosie, it is important to keep in mind that, not only did she sometimes teach mathematics using smile resources but also she met people from the original collective. her historical engagement with the project was thus personal and embodied. a major challenge for our current endeavour is to find ways to use digital media to make similar possibilities for resistance to dominant discourses available to a wider group of teachers through the excavated history of smile. at this early stage in our historical enterprise, these reflections from rosie give us courage to continue. different regimes of truth hold sway at different times and in different places (hall & noyes, 2009). rosie’s story shows that it is possible to live in the shadow of neoliberalism but nevertheless reject the impetus to think povey et al. public stories journal of urban mathematics education vol. 10, no. 2 63 economically rather than morally; and we know that “the subject who thinks morally, rather than economically, can powerfully undermine neoliberal subjectivities in generative ways” (verdouw, 2016, p. 526). concluding remarks as with most dominant ideologies, it is part of the neoliberal project to cut us adrift from our past and to de-historicise our lived experience of the present. berger (2016) uses the metaphor of no-fixed-abode to capture the absence of a sense of history: any sense of history, linking past and future, has been marginalised if not eliminated. people are suffering a sense of historical loneliness. (para. 9) it is the intention of our endeavour, of which telling rosie’s re-storying is a part, to challenge the subordination of the social (giroux, 2015) and to use a small part of the history of mathematics education in england as a meeting place (berger, 2016) from whence to understand, interrogate, and oppose the dominant discourses currently shaping society. the task before us as we continue with our historical endeavour is challenging. it is to find ways to make contemporary media—in this case a website— perform for others a role comparable to that of rosie’s personal experience. working within a radical history tradition (samuel, 1980), we assert that “history is about the present” (hodgkin & radstone, 2003, p. 1) and argue that to look backward is not backward-looking but forward-looking. our use of history in this endeavour is “present-minded” (samuel, 1980, p. 168). we have used documentary material, evocative stories, and aphoristic fragments (morson, 2003) to try to give an account of the recent past of smile and to do so in ways that can speak to the both the present and the future. we have argued that the currently dominant neoliberal discourses in education, the inspectorial judgments and assessments, and the auditing processes that the discourses endorse and the techniques of micro self-surveillance that they engender are indeed leading to a struggle over the teacher’s soul (ball, 2003). the pressure is most acutely felt in schools working in sites of systematic disadvantage, such sites being over represented in urban contexts. furthermore, we have noted that these discourses cut us off from our past. we come to suffer from “a profound sort of amnesia, where we can no longer quite remember how things became like this, or why, or whether anything was ever any different” (jardine, 2012, p. 97); and when we cannot remember what has been done to us, we also lose any sense of what the perpetrating causes might be. this not remembering has the effect of making the povey et al. public stories journal of urban mathematics education vol. 10, no. 2 64 present inevitable and timeless—we are at what has been called the “end of history.” it also, therefore, cuts us off from imagining a different future. this social amnesia matters for all young people but most especially for those who are poor and marginalised. it also matters for their teachers. we have described one way in which we might re-activate an engagement with the past and have offered the story of a single teacher who is struggling to resist the performativity agenda. amongst the strategies she has adopted is a mobilisation of stories from the past that allows her more clearly to see the current regime of truth as historically contingent and therefore subject to change. acknowledgements we are grateful to the british academy/leverhulme for financial support for the project (grant sg150824); to our colleague harry grainger for first drawing our attention, with their consent, to the recently qualified teachers’ writing on performativity and then putting us in touch with them; and to those new teachers who have given freely of their time to help us in this research. references ball, s. j. 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(1980). on the methods of history workshop: a reply. history workshop, 9, 162–176. skovsmose, o. (1994). towards a philosophy of critical mathematics education. dordrecht, the netherlands: kluwer. stronach, i., corbin, b., mcnamara, o., stark, s., & warne, t. (2002). towards an uncertain politics of professionalism: teacher and nurse identities in flux. journal of education policy, 17(1), 109–138. verdouw, j. j. (2016). the subject who thinks economically? comparative money subjectivities in neoliberal context. journal of sociology, 53(3), 523–540. whitty, g. (2006). teacher professionalism in a new era. belfast, united kingdom: general teaching council northern ireland. https://www.theguardian.com/education/2016/mar/22/teachers-plan-leave-five-years-survey-workload-england https://www.theguardian.com/education/2016/mar/22/teachers-plan-leave-five-years-survey-workload-england http://www.localschoolsnetwork.org.uk/2012/01/why-we-should-support-the-feisty-parents-at-downhills-primary-school http://www.localschoolsnetwork.org.uk/2012/01/why-we-should-support-the-feisty-parents-at-downhills-primary-school https://www.theguardian.com/education/2008/jun/03/schools.uk1 microsoft word 428-article text no abstract-2457-1-4-20210917 (proof 1).docx journal of urban mathematics education may 2022, vol. 15, no. 1, pp. 54–77 ©jume. https://journals.tdl.org/jume nickolaus a. ortiz, ph.d., is an assistant professor of mathematics education in the department of middle and secondary education at georgia state university, 30 pryor street sw, 6th floor, atlanta, ga 30303; email: nortiz1@gsu.edu; twitter: @professuhnao. nickolaus a. ortiz’s research focuses on how an ontological blackness is manifested and/or stifled during highquality mathematics instruction that emphasizes teaching for conceptual understanding and mathematical discourse. he studies how mathematics discourse for black children may be imbued with black linguistic patterns and is actively theorizing about what it means to create a black liberatory mathematics education that affirms these linguistic practices and black people writ large. terrell r. morton, ph.d., is an assistant professor of identity and justice in stem education in the learning, teaching, and curriculum department of the university of missouri–columbia, 303 townsend hall, columbia, mo 65211; email: mortontr@missouri.edu; twitter: @drtrmorton. terrell morton’s research focuses on identity as it informs students’ persistence and engagement in postsecondary stem education with a focus on race and an intersectional racegender identity for black students and black women in stem. he maintains a critical-phenomenological-ecological perspective with his work as he strives to transform the social positioning of race or race-gender regarding black students’ and black women’s stem engagement to promote stem learning spaces that are extensions of their identity rather than sites of hostility and alienation. empowering black mathematics students through a framework of communalism and collective black identity nickolaus a. ortiz georgia state university terrell r. morton university of missouri–columbia in this paper, we speak to the ways that black people have consistently strategized and advocated for rights and the education of their children through communal efforts related to a collective black identity. we use a framework of black x consciousness and ontological blackness to nuance the ways that blackness is internalized and taken up by members of this group and assert that although blackness is a unifying factor, diversity in ideologies and goals exist. the ways that black people have historically worked through these issues in hopes of achieving better education for their children is especially highlighted, along with the ways that these communal ideals may be utilized as forms of capital for black children learning mathematics. given this understanding, we also assert that what is generally culturally relevant for the collective may not be as prominent or relevant among various black identities. in addressing the question of how to cultivate a mathematics education that is culturally relevant for black children in particular, we discuss how our predecessors taught us that communal ontologies are instrumental in a) shaping the curriculum of a mathematics education worthy of black children and b) shaping the facilitation of teaching and learning to which black children are exposed.  keywords: black/african american, black political thought in education, communalism, culture, mathematics education, race ortiz & morton a framework of communalism journal of urban mathematics education vol. 15, no. 1 55 justice, is juxtapositionin’ us, justice for all just ain’t specific enough. (common & legend, 2014, 0:52) he words of grammy award-winning hip-hop artist common resonate with us as we reflect on our various experiences within education and deeply and critically consider this assertion that “justice for all just ain’t being specific enough” (common & legend, 2014, 0:55). in this paper, we call into question how black people have still found ways to achieve educational endeavors despite the obstacles placed on them and without the full promise of u.s. rights and privileges, specifically within the context of mathematics education. the current conversation is an important one within urban education, although it must be stated that “urban” is not synonymous with “black.” urban spaces, particularly those identified by milner (2012) as urban intensive or urban emergent, on average possess large populations of black people and students. for this reason, this conversation about black students in mathematics is especially important, as the field contends with the way that communalism and collectivism for black people, as explicated in this paper, seek to craft a more liberatory mathematics education for black children. we offer ideas toward this direction. thus, we speak directly to liberationseeking black teachers and those constituents who are critical players in the education of black children (i.e., black students, black parents). this is not to understate (or overstate) the role of white teachers but to suggest that there is a unique role that these black constituents play in educating black children. we start with, center, and solely focus on black people and the black experience in mathematics from a heterogeneous, intragroup perspective. taking this approach helps us identify the various strategies enacted by black people to gain their freedom, a perspective based on their individual or collective conceptualization of freedom and what it takes to be free. situated within a concept of black x consciousness, racialized metacognition (morton et al., 2019), we emphasize the political notion of blackness embraced by those who galvanized resources and worked together collectively as a community to advocate for better living and learning conditions for their children. in focusing on their political notions of blackness, noticed through collective strategies and action, we offer up communalism and a collective black identity as important principles for grounding and implementing culturally relevant processes in the policy development, teaching, and learning of mathematics for black children. communalism is operationalized here as collaboration and community rooted in reciprocal and interconnected relationships among african-descended peoples, and a collective black identity as an ontological position seeking to understand the intricacies and (dis)similarities among those who identify as black. as such, this paper uses these ideas to further conversations and research focused on black liberation in and through the context of mathematics education. t ortiz & morton a framework of communalism journal of urban mathematics education vol. 15, no. 1 56 a continued need to focus on blackness in mathematics education our rationale for engaging this work, though informed by many different scholars and experiences, primarily stems from examining the outcomes of white preservice teachers’ (psts) attempts at implementing culturally relevant pedagogy coupled with liberalistic ideologies of “for all.” in an earlier study examining psts’ approaches to culturally relevant mathematics lessons, the first author of this manuscript thought about some of the explanations that the psts provided as to why their mathematics lessons were culturally relevant (ortiz & davis, 2019). in short, the psts claimed that their mathematics lesson plans were culturally relevant “for all” of the students they taught and would teach. we find this perspective, though universally adopted by many, to be problematic as we consider how these psts homogenized people and their lived experiences. this perspective, that there is one homogenous understanding of people and life, is idealistic and often contested, as evident in a later excerpt from common’s (2014) verse when he claims that we “saw the face of jim crow, under a bald eagle” (2:27). in the same way that toni morrison (2017) has described whiteness as being tacitly understood in literature and accepted as the norm, particularly in the absence of identifying the race of literary characters (i.e., identifying them as black), u.s. culture, as hoelscher (2003) stated, more generally enables the unspokeness of whiteness to be the norm and all non-white identities to be “others.” we use the metaphor of the bald eagle and jim crow to underscore the ways that black people’s lives and histories are overlooked in the united states in general and specifically within education through a “justice for all” perspective; we extend this metaphor to the examination of black children in mathematics education. like common, we argue that justice for all just ain’t specific enough and use this theme as a guiding frame for how we approach an education meant to be beneficial for black children (e.g., ortiz et al., 2018; ortiz et al., 2019). broadly, curricula and educational policies assist the united states and its social institutions (like mathematics education) in overlooking the needs and ideologies, however diverse, of black students (asante, 1991), because they are not seen as ideal learners or even ideal citizens. whiteness is so far engrained in u.s. systems (bell, 1987; davis & jett, 2019; tate, 1997) that it is assumed that something culturally relevant for all students will take white norms as the default. a focus that does not begin and end with black children or center their needs and interests does not help them engage their greatest possible selves within (and outside of) mathematics education (gholson et al., 2012; martin, 2019; ortiz et al., 2018; warren, 2018). in setting the stage for a need to center black children in mathematics education, problematizing the notion of “for all,” we now discuss historical depictions of the goals shared among many black people manifesting through collective, ortiz & morton a framework of communalism journal of urban mathematics education vol. 15, no. 1 57 communal, and political notions of blackness, goals that respond to the issues outlined in the above metaphor. we discuss how black people have advocated for a better (mathematics) education, noting the specific resources requested. in highlighting collective, communal, and political notions of blackness enacted to make change and what this means for mathematics education, we further this work by nuancing the conversation, looking within the black racial identity group to discuss its heterogeneity. we end our conversation by detailing the pedagogical implications for how communalism among black people might impact both policy and pedagogy. black political thought among educators and parents to demonstrate variety in black political thought with communal blackness political strategy and action, we provide a brief overview of some well-known continuous debates taking place within the black community surrounding black education. we have reviewed literature that speaks directly to black education, allowing us in this section to build a case for principles related to black communalism and collectivity. in doing so, we attend not just to the differences discussed but the implications of said differences on the perspectives of black educators and parents and their perceived role in educating black children. additionally, we situate the ways these diverse goals for education have historically led to communal action. we feature the following ideas and scholars’ positions because of our shared histories and desires for improving black education, as well as their scholarly traditions and evidence of both persuading and achieving educational, civil, and economical advances for black people within their specific temporal and sociopolitical context. historical issues and concerns in black progress the prominence of some key scholars, educators, and activists are important for understanding black education and black intellectual thought, and we share these anecdotes to posit that both implicitly and explicitly these figures were foregrounding a communal approach to life and education. king (1994) suggested that the scholarly tradition of du bois and woodson reflect “the collective african american cultural ethos and social thought that evolved out of our common heritage and struggle” (p. 31). that is to say that this idea of a collective black identity is framed first and foremost around a common heritage and diasporic affiliation, but it does not end there. in his foundational text, carter g. woodson (1933/1990) articulated how negroes had successfully been miseducated. one claim he made regarding the postslavery era is that the debate over whether black people should have a classical or practical education was a very real one. many black families questioned whether a college education would be beneficial to what black families needed to survive, particularly those in the south. ortiz & morton a framework of communalism journal of urban mathematics education vol. 15, no. 1 58 tate (1995) gleans from woodson’s argument that an education that only catered to the needs, experiences, and desires of white individuals was a disservice to african american learners. for woodson (1933/1990), even science and mathematics teaching had to consider the complexities and context that accompanied black students’ learning in that contemporary moment, such as less exposure to family budgets and calculations than their white counterparts, as well as less access to qualified teachers. important here is that woodson makes the point that black people would be remiss to believe that individual success would be beneficial but that it was happening more often than it should have: “but the negro forgets the delinquents of his race and goes his way to feather his own nest, as he had done in leaving the masses in the popular churches” (woodson, 1933/1990), p. 37). this critique seems to advocate for an uplift in a communal and collective approach. also among those who raised questions about the structure of education for black americans was booker t. washington. it is common knowledge that w.e.b. du bois and booker t. washington clashed regarding their opinions about black social and economic progress. although somewhat of a reductionistic approach to their ideologies, some would sum up the major differences in these prominent black figures’ positions by saying that washington touted an industrial education for black people that stressed vocational skills, while du bois envisioned academic, philosophical inquiry as a way to understand and incite social change (bauerlein, 2004). du bois (1935) made the point that decisions about educating young black children is not an individual endeavor and is intricately connected to the important role of black parents: but in the case of the education of the young, you must consider not simply yourself but the children and the relation of children to life. it is difficult to think of anything more important for the development of a people than proper training for their children; and yet i have repeatedly seen wise and loving colored parents take infinite pains to force their little children into schools where the white children, white teachers, and white parents despised and resented the dark child, made mock of it, neglected or bullied it, and literally rendered its life a living hell. such parents want their child to “fight” this thing out, -but, dear god, at what a cost! sometimes, to be sure, the child triumphs and teaches the school community a lesson; but even in such cases the cost may be high, and the child’s whole life turned into an effort to win cheap applause at the expense of healthy individuality. (pp. 330–331) as crucial players in the decisions related to black children’s education, it is not uncommon for individual parents to have various ideas about how and where to educate their children. however, this too often translates into underacknowledging the beauty in our own culture in favor of a traditional or white framing of school curricula (asante, 1991; boykin, 1994; king & swartz, 2016; matthews et al., 2021), as black families alike rely on the curricula, its structures and norms, provided by k– 12 schooling (and postsecondary education) to educate (and even raise in some ortiz & morton a framework of communalism journal of urban mathematics education vol. 15, no. 1 59 circumstances) black youth. we see this in the work of delpit (2006), where parents did not necessarily agree on the progressive models of education but emphasized the need for their children to gain standard skills that would help level the playing field. we understand this tension to result from the desire to love and embrace one’s blackness while simultaneously engaging in learning that ensures survival (physically, socially, emotionally) and learning that promotes self-confidence and love. “we teach our babies to pursue equal status with those whose (access to) life is defined by our (vulnerability to) death” (woodson, 2020, p. 19). communal efforts must value the insight and concerns of black parents, believing and accepting that they have the best intentions for their own children but coming to a consensus that does not jeopardize one’s blackness or safety. notwithstanding, the importance of black educators cannot be trivialized in discussions about black education. a long-standing tradition of educating both oneself and the community is present among black people (mccluskey, 1994), notable in the efforts of educators like mary mcleod bethune who was committed to the academic needs of her people. johnson (2009) recounts the legacy of other black women educators, like nannie helen burroughs and anna julia cooper, who understood that their own progress was intricately connected to their students’ and the overall black community. as the leaders of classroom spaces, teachers play a crucial role in helping to create discourse around what is needed for the black collective and how it will be implemented. it is notable that what often made the difference in the lives of black children, pre and postsegregation, were dedicated black teachers who understood and acknowledged their humanity (ladson-billings, 1995; ware, 2006). frank (2018) and muhammad (2020) mentioned that these traditions of care and community are connected to a historical legacy of black teachers and the power black teachers wielded in classrooms (e.g., noblit, 1993). siddle walker (2000) recounted these black teachers as exemplars who were remembered for their high expectations and demanding teaching styles. these black teachers were a part of the collective that knew what value and joy lay in teaching these brilliant black children and helping to communicate a vision for their success. we share these points because what becomes true is that these actors seemed to realize that individualistic approaches were not going to fare well for the black masses who had recently become freed women and men, thus any theorizations and/or solutions had to account for large groups of black people and mobilize them in a way that was strategic and liberating. the ideologies surrounding integration-desegregation are worthy of a bit more attention, again with the undeniable presence of black educators and parents operating simultaneously. although many black people were in support of the victory in brown v. board of education and saw it as an iconic moment and win for civil rights, many believed that it was a disservice to black children and would prove more ortiz & morton a framework of communalism journal of urban mathematics education vol. 15, no. 1 60 harmful than good (ladson-billings, 2004; woodson, 2020). in asking the question of whether or not the negro needs separate schools, du bois (1935) so bluntly stated, “under such circumstances, there is no room for argument as to whether the negro needs separate schools or not. the plain fact faces us that either he will have separate schools or he will not be educated” (p. 329). he, like other black scholars, past and present, felt that although there might be access to more resources when attending schools with white children, the emotional, mental, and physical cost for black children might be too great and, equally concerning, there would be a teaching force that would not be as affirming or convicted to teach black children well. this is in stark contrast of former supreme court justice thurgood marshall, who fought voraciously for the desegregation of these schools, arguing that segregated schools contributed greatly to the unwholesomeness and despair of black children (love, 2004; woodson, 2020). still, siddle walker (2013) described a historical legacy of black educators during this time period who “supplemented their local and national advocacy for educational equality with a parallel pedagogical and curricular agenda designed to spur change by intentionally teaching generations of black children citizenship, democracy, and voting as a means to confront oppression” (p. 208). she painted the picture that these efforts were never accomplished alone and that even these educators’ membership within national protest efforts were aimed at doing what was best for the black children. further, she chronicled the actions of black educators who rarely received credit for what would eventually lead up to the brown v. board of education decision, all while acknowledging the collective efforts of black educators across the south advocating for better school houses and increased wages for teachers. contemporarily, black teachers like these still have much impact and influence (clark et al., 2013; mckinney de royston et al., 2021). an important lesson stemming from these debates around black education is that there is often overrepresentation among certain groups within the collective, such as the black elite and upper/middle class (smitherman, 2015; woodson, 1933/1990). for example, we must be cautious of how much power that the black elite and upper/middle class (smitherman, 2015) has in cultivating a meaningful mathematics education for black children. the particular moments in black education that we have shared also help us to realize that while black educators, parents, and students may aim to operate communally, sometimes the exact differences in the group’s desire to uptake any particular strategy might be made more salient when nuancing the ways that various black identities see the strategy as potentially useful. in other words, does the mathematics education we proffer benefit most black people or does it disproportionately marginalize some groups within the collective (e.g., low-income black families). ortiz & morton a framework of communalism journal of urban mathematics education vol. 15, no. 1 61 summary although solutions to these and other issues have never been clear cut for black people, the reality has always remained that within the united states, “black” was socially regulated to “other,” hence prompting collective and communal goals among black people as guiding principles in enacting change for black racial progress and, ultimately, steps towards a more liberating mathematics education. the debates described in this section helped clarify action and/or next steps for many black people, with the driving force of both debate and action being related to what the collective needed. woodson (2020) provided similar affirmation to this point by reminding us that achievement is “for the community but not on behalf of the community” (p. 20). ultimately, though there will be many decisions that must be made in regard to the education of black children, what we learn from each of these moments in history is that variance in perspectives and strategy are inevitable but must always respond to the needs of the collective and seek to foreground that which does not aim to commit harm. in building on this foundation, we extrapolate collectivism and communalism as guiding principles for advancing black liberation. to further nuance this conversation, specifically within the context of mathematics education, we leverage black x consciousness (morton et al., 2019) and ontological blackness (ortiz, 2020) to discuss anchoring black liberatory mathematics education in black communalism and collectivism in nonessentialist ways. this approach can build a mathematics education that empowers black children, deriving a consensus with the black collective that establishes a) a list of what we believe that black children should have learned before they graduate high school and b) ample opportunities to engage in mathematics learning in ways that prioritize the collective (i.e., other black students within their mathematics course). communalism and collectivism through black x consciousness and ontological blackness there is no monolithic black experience because blackness is not homogeneous; it is a social, cultural, historical, and political signifier for how individuals of a darker melanated hue who reside in or descend from continental africa perceive and engage life (johnson, 2003; woodson, 2017). blackness, as a construct, constitutes billions of people across the globe. how these various individuals coalesce, and in essence embody their blackness, occurs through either self-determined or socially regulated means (e.g., morton & parsons, 2018). from a socially regulated perspective, black is a racial identity ascribed to those who maintain shared ancestry and morphology with people of african descent (atwater & russell, 2015), where blackness, superficially defined, is subjected to the ideologies and outcomes of mainstream ortiz & morton a framework of communalism journal of urban mathematics education vol. 15, no. 1 62 colonialism (e.g., hegemonic whiteness; cabrera, 2018). the self-determined manifestation of one’s blackness—how one demonstrates to the world what black means to them, who one is, what one values, what one believes, who one associates with, and why—comes about after a series of psychological processing that involves perception, attention, information processing, and decision making all being regulated by a racialized metacognition, what was deemed black x consciousness (morton et al., 2019). black x consciousness thus details the ways in which black people operate as theorists of life, conceptualizing what it means to be black within and across various contexts given the sensory data they receive from interacting with their environment (morton et al., 2019). as a self-determined understanding of blackness, though situated within the mainstream socially regulated idea of black, how one understands and enacts their blackness (socially, culturally, and politically) falls on a spectrum that ranges from individualism to thin or thick conceptions of collectivism (shelby, 2002). individualism represents perspectives and actions that favor and privilege personal autonomy, meritocracy, and individual well-being (shelby, 2002). collectivism involves coordinated movement towards a black communal recognition. collectivist approaches to blackness vary from thin conceptions—basing one’s group identity off superficial notions of black limited to the hue of one’s skin and morphological structure—to thick conceptions that constitute cultural and political notions of black (i.e., building community around shared ideas, beliefs, values, norms, and strategies). in noting the expansive nature of black x consciousness to conceptualize how black people engage mathematics education, comprising metacognitive praxes such as stereotype management (mcgee, 2016), we draw on this concept to examine and emphasize the strategizing and action had by black people to revolutionize life and education. we specifically attend to the collective, communal strategies enacted by various politically black groups, noting how they theorized change given their interactions with their environment. in focusing on how and why they coalesced, detailing the justice they fought for, we provide a deeper conception of collective, communal black activism and the possibilities of embedding this perspective within the present and future teaching and cultivating of black children in and through mathematics. connectedly, ontological blackness can be understood through at least three forms of capital: communalism and a collective black identity, resistance, and linguistics (ortiz, 2020; ortiz & ruwe, 2021). these three forms of capital as described do not exhaust our understanding of blackness as an ontological reality but instead promote aspects that make blackness real to those who inhabit it. in delineating these various forms of capital, we attest to that fact that not all black people utilize these forms of capital in identical ways, as has been conceptualized in the thick and thin notions of blackness. despite this noted understanding, the experiences that black people have in education and in mathematics specifically are too often similar in disheartening ways (gholson & wilkes, 2017). these shared experiences have triggered ortiz & morton a framework of communalism journal of urban mathematics education vol. 15, no. 1 63 the many ways in which black people internalize ideological positions and strategize for solutions (morton et al, 2019, ortiz et al., 2019). given this understanding, situating ontological blackness within black x consciousness by attending to components that make blackness real as individuals engage their learning environment and how said blackness shapes their cognitive processing and behavioral applications, we operationalize communalism and a collective black identity as both political strategy and action to protect and advocate for black people in mathematics education. it is important to note that our focus here is on mathematics education, particularly because it is often regarded as the queen of the sciences, neutral in its implementation (kokka, 2020), and a gatekeeper responsible for rejecting or ascribing privilege and/or inequity (martin, 2009). notwithstanding, these conversations are relevant to other contexts and disciplines beyond the scope of this paper; we strategically make sense of nuancing the case in mathematics education. this framing of communalism and a collective black identity suggests that these key aspects of blackness, however understood and however enacted by black people, is a beautiful thing worth being celebrated, period! as part of that celebration, these concepts should be integral to educational curricula. as dumas and ross (2016) asserted, there is no one way to be black; the only recommendation they provided is just to love being black. we argue that with that love comes a desire to see all black people flourish and recognize that this desire, often connected to the communal efforts of black people, can be a unifying-communal experience in itself. in this way, we specifically tackle the vastness of communalism within blackness, suggesting that advancing black progress (even based on a thin description of black) is a uniting feature for black people and that historically it has been the foundation upon which their ontological realities are created and a priority in seeking group rights and privileges together. black collectivism and communalism— considerations for mathematics education in this section, we discuss our operationalization of black collectivism and communalism within the context of mathematics education. given the prior discussion on diversity in the strategies and needs for black people, we caution the reader to interpret our work as but one perspective that may serve to empower black students’ pursuit of mathematics and help educators see how to address their needs and self-proposed desires. we align our thinking with these scholars within black intellectual thought and more explicitly outline a commitment to education that permits black children to “choose collective liberation and survival as a goal and to see this as a part of a larger struggle for social change” (king, 1994, p. 30). ortiz & morton a framework of communalism journal of urban mathematics education vol. 15, no. 1 64 as we ideate the many possibilities in which we can transform mathematics education to ensure the thriving and success of black students, we attend to the notion of enhancing culturally relevant/responsive instruction for black children (ladsonbillings, 1997; leonard et al., 2009), advocating for conversations around the capital, ideologies, and needs of black children as central to political strategies and actions that advocate for black children as a collective. in other words, instead of a lesson being perceived as culturally relevant for all, we argue that one of the ways mathematics teaching and learning is made relevant for black children is by tapping into a capital that prioritizes communalism and afrocentric ontologies. by proposing that communalism and a collective black identity serve as guiding principles for the education, political advocacy, and action of black people (king & swartz, 2016), we again broach this concept with a perspective that blackness is heterogeneous in its embodiment, with the social regulation of blackness by u.s. society creating an ontological blackness that maintains a “barebones minimum” collective reality that is thinly situated. similar to hilliard (2001) and shujaa (1992), we acknowledge the temporal and geographical implications that accompany blackness and highlight how some iterations or understandings may be static (anderson, 1999; hooks, 1990). ross (2009) argued that these forms of conceptualizing blackness cannot erase black subjectivity because it allows for other forms of oppression to surface among black people. we call for a communal and collective approach that is sensitive to these needs and advocates for all black people. in iterating this idea, we extrapolate how the embodiment of blackness is situated within existing research. considerations of black heterogeneity in discussing a collective black identity, it is very important to acknowledge the “type” of black people featured, delineating differences in the embodiment of blackness in relationship to what martin (2009) has described as micro or macro levels of societal problems. in the context of this paper, we understand macro issues to be those that inherently impact the black x consciousness of black people, as macro issues dictate the ways in which black people navigate their various environments that are culturally, historically, socially, economically, and politically regulated. micro-level concerns, though also informing black x consciousness, thus reflect connections that might be specific to various groups within the collective black. for example, micro issues could include concerns, topics, or trends that appeal to specific subgroups of black people (e.g., black girls and women, low socioeconomic status, or individuals from atlanta). micro issues do not only reflect subgroups based on social identities; these can also reflect subgroups based on ideology. although the macro-level issues deal a lot with critiquing systems of inequity and privilege, we see more micro-level issues related to pedagogy and learning styles, home life, and interests. overall, the point we strive to make is that there are going to be issues that ortiz & morton a framework of communalism journal of urban mathematics education vol. 15, no. 1 65 pertain to black people just by virtue of them being black within the united states, and there will be more nuanced aspects of these phenomenological black realities (martin, 2012) as we survey the needs of subgroups of black people. communal approaches must take different meanings of blackness, that which we are suggesting under a collective black identity, very seriously. black perspectives—gendered, diasporic, and regional. scholars articulate the need for disaggregating black people in general and when analyzing different subgroups within the collective, such as black girls or young people (gholson & wilkes, 2017). we highlight this example as a way for us to consider how we frame capital for black students, acknowledging the need and possibility of providing a perspective that attends to the different subgroups among black people regarding age or gender and realizing that essentializing these identities is counterproductive (graven & heyd-metzuyanim, 2019; jurdak et al. 2016). when educators consider making instruction culturally relevant for black learners in particular (as opposed to a “for all” method as described in the introduction of this paper), they would do well to think about the nuanced understandings of blackness among subgroups, as well as the overarching theme of communalism. in proposing nine dimensions to a collective black cultural ethos, boykin and toms (1985) focused on “afro-americans” and linked the african american cultural experiences to a traditional african ethos. they suggested that black people express alignment with these dimensions—spirituality, harmony, movement, verve, affect, communalism, expressive individualism, orality, and time—at varying degrees. in leveraging black cultural ethos, we emphasize that how people approach and enact black cultural capital can differ. for example, black cis-gender women may utilize black capital in ways that differ from black men or non-binary folx given the multiplicative nature of identity and its influence on determining reality, a perspective that should be reflected in pedagogy and praxis (e.g., berry, 2010). a communal approach must account for how within-group differences could include leveraging frameworks that note collective cultural identities among black people but differences in the understandings of reality specifically for black girls and women given gendered experiences (e.g., hudson-weems, 2020). seemingly, there is little disaggregation had of black people within the diaspora within mathematics education as it relates to black cultural capital or culturally relevant interventions. some noteworthy exceptions do occur in bermuda (matthews, 2008), among black english language learners (leonard et al., 2009), and in research that examines domestic and international black students in engineering— a mathematically heavy discipline (burrell et al., 2015). aside from gender and diasporic (e.g., ethnicity and nationality) exemplars, blackness can be nuanced by geographical regions. caniglia (2003) discussed the role of detroit’s black history in the overall black history and how highlighting the specific contributions of detroit can make educational content relevant for black children within that locale because ortiz & morton a framework of communalism journal of urban mathematics education vol. 15, no. 1 66 of their connection to the city. regional pride and identification are important because what may be familiar to black children in one part of the country may not have the same meanings or intuitiveness in other areas; blackness may be performed much differently in detroit, michigan, versus atlanta, georgia, or even asheville, north carolina. this reflection regarding the approach to multiplicative identities and specificity within blackness among communalist approaches is also true when we think about religion or spirituality (e.g., black christians, black muslims, black agnostics, etc.) and black people across the spectrum of sexuality (e.g., heterosexual, homosexual, bisexual, pansexual, asexual). our positioning strives to think more about blackness as this multifaceted ontology and to prompt deeper conceptions about what unites black people and what makes us special. connecting this deeper, collectivebased introspection to aspects of culturally relevant pedagogy, we extend and deepen conversations about reflecting on the context of the classroom and how instruction can be culturally relevant to the heterogeneity of blackness present within the learning space. for example, though we see a lot of these discussions surrounding lower income black students (e.g., coleman et al., 2017; jackson & remillard, 2005; powell-mikle & patton, 2004), we raise questions around whether blackness differs in light of class or socioeconomic affiliation? more research that focuses on diversity within blackness alongside these notions of a collective black within the context of mathematics education is needed. additionally, mathematics education should focus on the implication of said research on both policy and practice within mathematics teaching and learning. in the same way that carter (2003) was explicit about how these poor african american youth living in yonkers, new york, used certain forms of capital, it may be worthwhile to see who is being left out of the conversation when we talk about black cultural capital. davis (2018) contended that a “liberatory paradigm is responsive to the distinct historical and contemporary needs of the collective black community in mathematics education and society at large” (p. 70). the major point worth taking from this part of the conversation is that the research community must include more explorations of the ways black children make sense of blackness and how this can be understood alongside cultural relevance, as situated within their experiences across multiplicative black ontologies. more succinctly, is our instruction culturally relevant for all black learners, or perhaps for a subgroup of these learners (e.g., black boys)? we suggest that this answer may vary according to context. principles towards mathematics instruction and assessment notwithstanding, it makes sense that communalism is prevalent in much of the research on black children’s mathematics learning and culture (e.g., coleman et al., 2021; leonard, 2018; thompson & davis, 2013). as defined by boykin (1994), communalism denotes “a commitment to the fundamental interdependence of people ortiz & morton a framework of communalism journal of urban mathematics education vol. 15, no. 1 67 and to social bonds and relationships” (p. 249). in short, the ontological position we foreground here celebrates collectivity, interdependence, and the well-being of the group over self-exaltation (boykin & toms, 1985; king & swartz, 2016). with this stated, the role of competition embedded in our mathematics classrooms (e.g., tracking efforts, standardized tests measures) is antithetical to how black children may be socialized outside of these classrooms (davis & martin, 2018; lim, 2008; moody, 1998; tate, 2005; yanisko, 2016). if, as we have argued throughout this paper, communalism and a collective black identity are noteworthy to many black people, then perhaps at least one way that communalism might be targeted in mathematics classrooms is through the use of group work. to be clear, we are not suggesting that all black students want to work in collaboration at all times, or that some do not prefer to work alone. what we are saying is that we have to be cognizant of the ways that we assume the existence of neutrality in the teaching of mathematics and the ways we assess our students (gutiérrez, 2013; jett, 2019; ortiz et al., 2018). weissglass (2002) described how evaluation and assessment have been used, particularly in mathematics, as barriers to social access and how these assessments may cater to cultural values and practices not shared by non-white student groups. we must be open to the fact that those who do not perform well on current assessments are still brilliant, and perhaps we are still utilizing far too many antiquated ways of evaluating students’ knowledge when alternative mathematics tasks can and do exist. there is much evidence in the mathematics education literature concerning the idea that advancement for black people groups has to be a collective effort (e.g., coleman et al., 2017; collins, 2018; davis, 2018; lewis et al., 2002). similar to the historical recap presented above, davis (2018) suggested that within historical movements of civil rights, the goal has been, and should remain, to evaluate ways in which to improve social conditions for black people and how to move the community forward collectively. he juxtaposed this with the ways in which a eurocentric ideal has established individualistic goals and a tendency towards competition within mathematics classrooms and in their mathematics trajectories, and in our analysis above, we showed how collaborative efforts were prioritized in these propositions on black education. eurocentric conceptions run counter to ideas of communalism, again creating a climate in which black students may face some dissonance. thus, even mathematical assessments should acknowledge the role that collective work and progress can serve black mathematics students, both in their learning or materials but also the application of these materials to real, relevant problems. davis (2018) is not alone in highlighting the role of collective effort, both in policy and in pedagogy (see battey & neal, 2018; bonner & adams, 2012; ladsonbillings, 1997). connectedly, clarkson and johnstone (2011) described the afrocentric connections that black students felt in one school. in the featured school, the leaders promoted ujima, a sense of collective work, because it aligned more with ortiz & morton a framework of communalism journal of urban mathematics education vol. 15, no. 1 68 afrocentric collaboration. these students explored mathematics learning in terms of group work, similar to ways described by jett (2013) and hubert (2014). when this happens, students are introduced to new ways of engaging in mathematics that bring new perspectives and arguments to the forefront. black educators are inherently connected to a legacy of black teachers who utilized these communal approaches and may thus be well positioned to utilize them and to advocate for their use by their nonblack counterparts. king and swartz (2016) provided some helpful principles (i.e., inclusion, representation, accurate scholarship, indigenous voice, critical thinking, and collective humanity) related to how we imagine some of these ideas being implemented in group work. although all equally important, these last two principles provide some tangible ways to think through mathematics pedagogy. they suggest that critical thinking should promote culturally authentic assessment such that it guides students to produce knowledge and proffer solutions through demonstration rather than selecting a predetermined right answer. this is powerful in that communalism would ensure that students are all able to think through scenarios together, an example being the best way to describe conceptually why 𝑠𝑖𝑛$𝜃 + 𝑐𝑜𝑠$𝜃 = 1 can be thought of as a pythagorean identity. multiple ways of approaching this explanation exist, but the communal responsibility that undergirds collective humanity would ensure that all students in a group make sense of this explanation before the class is over. we imagine that higher cognitive demand-type questions (mccormick, 2016; stein & smith, 1998) like this could help teachers to build an atmosphere where black (and other) students are responsible for helping one another to understand a geometric approach versus an algebraic one in explaining their rationales. in other words, a better approach to a conceptual question like this may result from helping students to recognize the power of not just providing a correct answer but collaborating in a group to make sure they can explain it to their peers and that everyone has mastered the concept. this is what we mean by fostering a communal approach in our classrooms. further, eglash (1997) described the brilliance of benjamin banneker as it related to a wider, african history of counting number systems, and tate (1995) described a scenario where black students’ reasoning behind purchasing a more expensive bus pass was because of the benefits it would bring to their family. their decisions were not based simply on them calculating the cost for themselves, but new perspectives were considered when the students explained that their family members could also use a bus pass to make it to the places they needed to go in a given month. in noting these exemplars, we assert that capital is invaluable in thinking about how mathematics can influence the lives and dispositions of black people (ortiz et al., 2019), or even the uptake of certain self-concepts that stem from collectivist views (woodland, 2008). some of the current assessments might be better completed together, and we borrow this approach from those black educators and parents who recognized that ortiz & morton a framework of communalism journal of urban mathematics education vol. 15, no. 1 69 solutions to the real-world problems they were working with (e.g., desegregation) were communal efforts. mathematics educators and policy makers might consider seriously the pros and cons to these types of assessments. would standardized tests completed within small student groups negatively impact the learning that is taking place among our students? boykin and toms (1985) and coleman and colleagues (2017) would seem to disagree, stating that communalism, which may be seen as dependency to the mainstream, exudes a range of social values. these values seem to align with the idea proposed by the scholars foregrounding afrocentric curricula that promote ujima and family (clarkson & johnstone, 2011; nyamekye, 2013). these scholars help to support the idea that mathematics teaching and learning can still be rigorous in these contexts and that mathematics can be taught alongside cultural values (grant et al., 2015). further, the benefits that might occur from allowing students different ways to demonstrate their knowledge, particularly in ways that may align more with the forms of capital that are well represented among black learners, speak to the different learning modalities that exist among diverse students. we proffer these ideas as necessary principles towards a more liberatory mathematics education for black learners that operates communally while still acknowledging the various needs within the collective. discussion with this said, culturally relevant and responsive pedagogies (brown et al., 2019; corp, 2017; ladson-billings, 1997; nolan & keazer, 2021) help us understand that communal approaches are at least one way that mathematics teaching and learning can be made relevant for black children. notwithstanding, group work is not something that is magically going to allow black children to exert their capital. teachers still have a responsibility to help facilitate what mathematical practices are occurring in these groups (sengupta-irving, 2014; webb et al., 2019). what may be different in the way that group work and collaborative assignments are implemented in classrooms with high populations of black children is that group work must still promote what is considered to be high-quality mathematics instruction (munter, 2014). these children can be pushed to higher levels and should not be exposed to group work only for trivial tasks; we see that this kind of low-level instruction occurs far too frequently in classrooms with black children when low expectations are abundant (stinson, 2008). communalism and this collective black identity have the power to inform our instructional activities without compromising or reducing the rigor of mathematics instruction. this conversation about communalism is an important one for thinking about the ways that whiteness holds power and how we might resist that power. if whiteness has dictated the eurocentric standards to which we still comply by promoting individual achievement of goals (davis & martin, 2018; tate, 1995), then how might ortiz & morton a framework of communalism journal of urban mathematics education vol. 15, no. 1 70 more afrocentric ideals of communalism challenge these dominant ways of mathematics instruction? we suggest that empowering the students in the way we have framed here is manifested through the uptake of a curriculum and pedagogy that values what many black students may regard as more consistent with the forms of capital they wish to utilize. it would stand to reason that if communalism is regarded highly by many of our black students, then it might offer some insight into how our assessments might better include opportunities for them to showcase their knowledge alongside other groups’ members. more pragmatically, as stated by ortiz (2020), one option might be to reconsider how we administer standardized testing. one thought might be to allow, for example, 1/3 of the assessment to be in communal activity. this might help to reduce the anxiety that is associated with the competitiveness of being compared to one’s peers and further validate the sense of collective goals for the group. this may also give a better indication of student knowledge and comprehension of the material. the reality is that standardized tests in the current form rarely tell us anything substantive about black children’s ability to do mathematics or whether they are actually “proficient” (larnell, 2019); they more explicitly reveal the exact disparities that exist in black children’s experiences and inherently provide part of the evidence for why black people must continue to advocate for a better mathematics education. we believe that the collective efforts of black people will help to see this type of change. next, we move to the voice that black parents and students can offer up in conversations about securing new teachers in their schools. being a citizen means also having a voice in important decisions, such as who will get the privilege of teaching any particular black child. again, the point we made earlier is that these goals and needs may differ as we think about who we want to teach black girls, black lgbtq+ children, and others, but the point is that these students should have some voice that advocates on behalf of the collective; protesting and opting out of a harmful system requires work in behalf of the collective. we see these communal efforts as a way to accomplish some of martin’s (2019) recommendations, such as refusing to engage in tracking policies that perpetuate competition and hierarchical systems, yet these efforts are contingent upon each of the constituents we have mentioned. lastly, black children deserve to learn mathematics from teachers who they also trust and feel comfortable with hiring, ones that help them fight against antiblackness and white supremacy in the u.s. empire (martin, 2019). they should feel that these teachers will present curricula in a way that does not marginalize their blackness but that seeks to celebrate their black identity(ies) and continue to move the collective forward. we see black students’ voices as left out of this conversation too often and recognize that empowerment can occur when these diverse black voices are included. thus, a communal approach in this context would allow black children to be a part of the hiring efforts for teachers in their schools. ortiz & morton a framework of communalism journal of urban mathematics education vol. 15, no. 1 71 conclusion toni morrison (2019) cautioned, “let us be reminded that before there is a final solution, there must be a first solution, a second one, even a third. the move toward a final solution is not a jump” (p. 14). in a similar way, martin (2009) cautioned us against a solution on demand for problems that have metastasized over centuries. this is important in understanding the goal of this paper. we have offered a critique of how black children are overlooked in mathematics education and how in looking for solutions, particularly by starting with the roots of black education, black people have strategized collectively and in ways that reveal their black x consciousness. we have argued that, in addressing the question of how to cultivate a mathematics education that is culturally relevant for black children in particular, our predecessors taught us that communal ontologies are instrumental in a) shaping the curriculum of a mathematics education worthy of black children and b) shaping the facilitation of teaching and learning to which black children are exposed. marginalization happens too often in education, especially when there are so many “others” underrepresented in conceptions of “all students.” if the curriculum was truly for all students, black children would not experience disparities and stereotypes within mathematics achievement at such alarming rates (ladson-billings, 1997; mcgee & martin, 2011; stinson, 2013). further, tension in this assumption of “for all” arises when one considers that the current state of mathematics education cannot be working and or relevant for all children (delpit, 2012; goffney, 2018; jett, 2013) if black children are not benefiting; furthermore, their neglect inherently means all children have yet to be considered. a huge point that we have made in this paper is that black people have historically advocated for the collective. we honor this tradition by suggesting that we must strategize with attention to the collective, constantly reevaluating whether some black identities are being overshadowed in what is to become a liberatory mathematics education, even within the propositions we have foregrounded in this discussion. empowerment for black people has always been connected to the communal efforts. thus, where justice (and mathematics) for all was not specific enough, we advocate for a more communal framing that is responsive to black children specifically. references anderson, v. 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(2016). negotiating perceptions of tracked students: novice teachers facilitating high-quality mathematics instruction. journal of urban mathematics education, 9(2), 153– 184. https://doi.org/10.21423/jume-v9i2a262 copyright: © 2022 ortiz & morton. this is an open access article distributed under the terms of a creative commons attribution-noncommercial-sharealike 4.0 international license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. math links: building learning communities to enhance prospective teachers’ attitudes and beliefs about inquiry-based mathemati journal of urban mathematics education december 2018, vol. 11, no. 1&2, pp. 118–141 ©jume. http://education.gsu.edu/jume jacqueline leonard is an associate professor of mathematics education in the college of education at temple university, 1301 cecil b. moore avenue, ritter hall 434, philadelphia, pa 19122; e-mail: jleo@temple.edu. her research interests are equity and access issues as they pertain to mathematics education and teaching for social justice and cultural relevance in the mathematics classroom. brian r. evans is an assistant professor of mathematics education in the school of education at pace university, 163 william street, 11th floor, new york, ny 10038;email: bevans@pace.edu. his research interests are social justice in urban mathematics education and international mathematics education. he is also interested in alternative certification and pre-service teacher preparation in mathematics. math links: building learning communities in urban settings1 jacqueline leonard temple university brian r. evans pace university learning mathematics in urban settings is often routine and decontextualized rather than inquiryand culturally based. changing prospective teachers’ attitudes about pedagogy in order to change this pattern is often tenuous. the purpose of this pilot study was to provide opportunities for teacher interns enrolled in a graduate certification program to interact with urban students in a community-based program called math links. twelve interns completed 30 hours of fieldwork at churchbased sites. prior to fieldwork, the interns participated in a 3-hour professional development and education session, in addition to their education courses. three interns’ work with urban children and youth reveal that community-based experiences changed their attitudes about practice and their capacity to teach urban children mathematics in culturally sensitive ways. one in-depth case study of an asian teacher reveals not only changes in her attitudes and beliefs about urban students but also changes in her pedagogy as she shifted from teaching by telling to guided inquiry. keywords: “at-risk” students, community-based programs, mathematics education, teacher interns or more than two decades, there has been an impetus of reform in mathematics education (martin, 2003, 2007). as mathematics teacher educators, we have focused on reform-based pedagogy in our elementary and secondary mathematics methods courses with the intent to inform preservice and beginning teachers in undergraduate and graduate teacher credential programs about the advantages and challenges of using reform-based practices. 1 originally published in the inaugural december 2008 issue of the journal of urban mathematics education (jume); see http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/5/5. f http://education.gsu.edu/jume mailto:jleo@temple.edu mailto:bevans@pace.edu http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/5/5 leonard & evans math links journal of urban mathematics education vol. 11, no. 1&2 119 reform-based practices in mathematics classrooms can be viewed in one of two ways: use of reform-based curriculum and/or use of reform-based pedagogy. studies on reform-based curriculum show that teacher educators can successfully guide preservice teachers in developing conceptual knowledge in mathematics (ebby, 2000; sherin, 2002; spielman & lloyd, 2004). in addition to the use of reformbased curriculum, reform-based pedagogy, such as teaching for understanding (ball, hill, & bass, 2005; ma, 1999; sherin, 2002), facilitating classroom discourse (cazden, 2001; o’connor & michaels, 1993), and engaging in culturally based practices (brenner, 1998; leonard, 2008; lipka et al., 2005) are common elements found in reform-based classrooms. yet, effective reform-based teaching of mathematics requires that preservice teachers learn by actively engaging students in the teaching-learning process (ambrose, 2004; ebby, 2000; lowery, 2002; sherin, 2002). thus, it is important for preservice and beginning teachers to have opportunities to apply the knowledge they gain from theory and research in education courses to real settings where they can implement reform-based practices with children. for the purpose of this article, reform-based teaching is characterized by inquiry and culturally sensitive approaches to teaching and learning. in an inquirybased approach, the roles of teachers and students are redefined. the teacher is no longer the sole authority for building mathematical knowledge in the classroom (national council of teachers of mathematics [nctm], 2000). instead students are encouraged to be proactive rather than passive, using their own knowledge and experience to justify solutions to mathematics problems. the principles and standards document (nctm, 2000) provides an impetus for reform-based teaching in mathematics. the document falls short, however, when it comes to cultural pedagogy (leonard, 2008; leonard, in press; martin, 2007). focusing only on content knowledge, without attending to pedagogy (ball, bass, & hill, 2005) or the students’ culture (ladson-billings, 1995; nieto, 2002), does not lead to the development of high-quality teachers (martin, 2007). because mathematics is not divorced from culture, teachers must also be culturally competent in order to be prepared to work with diverse student populations (ladson-billings, 1995). knowledge of diverse students’ learning styles and cultures helps teachers, especially those who are from different racial, ethnic, and/or social backgrounds, to develop strong teacher–student relationships with culturally diverse students (lipka et al., 2005; shade, kelly, & oberg, 1997; silverman, strawser, strohauer, & manzano, 2001). culturally sensitive approaches should also link mathematics to issues of social justice. teaching for social justice empowers historically marginalized students to use mathematics as a form of liberation (gutstein, 2006). diverse students are more likely to realize the importance of learning mathematics if it can be used to empower them to change their circumstances (gutstein, 2006; ladsonbillings, 1994; leonard, in press). leonard & evans math links journal of urban mathematics education vol. 11, no. 1&2 120 yet, teaching preservice and beginning teachers reform-based strategies is not a panacea. as methods instructors, we have found these teachers’ beliefs and prior experience cause them to be resistant, initially, to reform-based practices. among preservice teachers enrolled in our teacher credential program, a significant number did not experience any kind of reform-based teaching. essays written in our mathematics methods courses revealed that some preservice and beginning teachers continue to be taught mathematics in traditional ways as osisioma and moscovici (2008) had found in a similar study with science teachers. these preservice and beginning teachers were often taught to use rules and algorithms to solve problems and were not allowed to question the teacher or their peers in the learning context. excerpts of three preservice teachers’ reflections of their previous experiences in k–12 mathematics classrooms are presented for consideration: i remember my previous math teachers back in middle school, and they did not incorporate any hands-on activities into their lessons. it was simply learning off what was on the board. i think i would have been less intimidated by math if i had materials and engaging activities to help me to learn the concepts. (female student, fall 2007) for the most part, i don’t have a lot of clear memories of how exactly i learned math. this is probably because the teachers i had rarely did anything extraordinary to support their lessons. i do remember in second grade doing something called a mad minute, which were 30 addition and/or subtraction problems that we had a minute to try and complete. also, in second grade, we could earn fake money for doing certain things in class, and every other week or so we would have an auction where we could spend that money on small prizes. other than that, i honestly don’t remember anything specific from elementary school regarding learning math. (male student, fall 2007) what i can remember from math classes when i was younger involves a lot of scrap paper and many trips to the board. we would be given pages and pages of homework, sometimes without the concept even grasped. i just remember asking the question why a lot and never getting an answer. many math teachers are just concerned that you can actually solve the problem, rather than why it is solved like that. (female student, fall 2007) these excerpts reveal that despite almost 20 years since the publication of the curriculum and evaluation standards for school mathematics (nctm, 1989) the teaching-learning context in k–12 mathematics classrooms has not dramatically changed (martin, 2003). mathematics instruction continues to be disconnected from students’ culture and everyday experiences (silverman et al., 2001). mathematics, as a content domain, continues to be viewed as a complex series of algorithms with abstract entities that have nothing to do with the sociocultural context of students’ lives. these experiences are more pronounced among candidates in our graduate certification program, who tend to be older adults planning to teach as a second career or young adults with bachelor degrees in liberal arts who want to obtain a leonard & evans math links journal of urban mathematics education vol. 11, no. 1&2 121 teaching credential. these candidates are able to obtain jobs as teacher interns while completing the credential program. in this study, a teacher intern is defined as one who has a paid or unpaid field experience in an informal or formal school setting. in some cases, teacher interns have full-time jobs as teachers while they are taking courses at night. a recent study of science teacher interns revealed that it is possible to change teacher beliefs among this population (osisioma & moscovici, 2008). osisioma and moscovici (2008) examined nine science teacher interns’ beliefs about inquiry and reform-based methods of instruction before, during, and after taking two science methods courses. osisioma and moscovici collected data from written reflections, lesson and unit plans, interviews, observations, class discussions, and peer-teaching in the methods courses. they found most participants primarily used traditional methods of instruction and were more teacher-oriented. at the end of the two methods courses, the researchers found the number of interns who believed in the use of inquiry and student-centered instruction rose to seven from initially only one. the authors concluded that beliefs about science teaching and learning changed over the two semesters and recommended that this area of research receive greater attention. to address teaching mathematics from a cultural and social justice perspective, we studied the impact of reform-based practices learned in two graduate teacher education courses and the enactment of reform-based pedagogy in communitybased settings. it is in such settings that prospective teachers’ perceptions of urban students of color might change. too often, perceptions of african american and other underrepresented minority students are rooted in deficit theories that contend these students are “less than ideal learners and, therefore, in need of certain kinds of teachers” (martin, 2007, p. 8). more often that not, these teachers are strong in discipline but weak in mathematics content knowledge and cultural sensitivity. some african american scholars (gay, 2000; ladson-billings, 2006; leonard, 2008; martin, 2007; nasir, 2005) call for teachers to use students’ cultural experiences as a springboard for learning mathematics. in this article, we are particularly interested in how k–12 teacher interns interacted with urban students who were enrolled in an after-school program or a specialized program for “at-risk” high school students. the purpose of this article is to report on the enactment of reform-based practices in these non-traditional urban settings. because students in such settings are often marginalized, a framework that connects issues of social justice with education is needed. theoretical framework this pilot study is grounded in a framework that has social justice at its core. the ability to view the world through the eyes of marginalized persons is critical to developing culturally sensitive approaches to teaching (gay, 2000; nieto, 2002; leonard & evans math links journal of urban mathematics education vol. 11, no. 1&2 122 shade et al., 1997). paulo freire in pedagogy of the oppressed (1970/2000) described the importance of conscientização, which is the development of the skills necessary for critical consciousness and the teaching of social justice (apple, 2003; gutstein, 2003). to operationalize conscientização, however, a specific type of emancipatory pedagogy is needed. culturally responsive teaching/pedagogy is one such framework (gay, 2000).2 according to shade and colleagues (1997), culturally responsive pedagogy builds bridges between the culture of the school and the home. they contend that knowledge should be transmitted in three areas: general skills needed for survival (reading, writing, and mathematics); cultural information (art, science, and history); and cultural norms (behaviors and mores). yet, the debate continues over whose knowledge is valued and taught as official (apple, 1995). white-middle class values and examples dominate the american educational system while the contributions and values of persons of color are often neglected (blanchett, 2006; gutstein, 2006; leonard, 2008). freire’s construct of conscientização challenges this perspective from a class perspective while culturally responsive pedagogy grounds our work in racial and social justice. gay (2000) contends that culturally responsive pedagogy is crucial in motivating urban students of color to learn. culturally responsive pedagogy derived from multicultural education paradigm in the 1970s; it “simultaneously develops, along with academic achievement, social consciousness and critique, cultural affirmation, competence and exchange; community building and personal connections; individual self-worth and abilities; and an ethic of caring” (p. 43). one of the most consistent and powerful findings of research studies related to diverse students’ academic achievement is the ethic of caring (gay, 2000). the ethic of caring is demonstrated by teacher attitudes, expectations, and behaviors related to children’s intelligence and academic success (gay, 2000). caring teachers believe their students are competent and hold them in high esteem. students then live up to teachers’ expectations and exhibit appropriate classroom behaviors (gay, 2000; ladsonbillings, 1994). culturally responsive pedagogy, then, is an important aspect of the teaching-learning environment in urban school settings. thus, we use the constructs of conscientização and culturally responsive pedagogy to ground the math links study. research questions the research questions that emerged in the math links study were: (1) how does one teacher-researcher’s reform-based practices influence teacher interns’ be 2 gay (2000) uses the concept culturally responsive teaching; whereas, ladson-billings (1994) uses the concept culturally relevant pedagogy. for a brief discussion of the similarities and difference of concepts that might be positioned under the umbrella of culturally responsive/specific pedagogy see leonard (2008). leonard & evans math links journal of urban mathematics education vol. 11, no. 1&2 123 liefs about culturally responsive pedagogy? (2) how does interaction with urban youth in a community-based internship influence the development of culturally responsive pedagogy among teacher interns? to answer these questions, we conducted a year-long study with two cohorts of teacher interns to draw on data collected and analyzed on participants enrolled in two types of education courses in our graduate teacher certification program: elementary mathematics methods (fall 2006) and effective teaching (spring 2007). the math links study the math links pilot study grew out of a similar study in science that studied changes in preservice teachers’ science instruction when they engaged in community-based internships (leonard, boakes, & moore, in press). the math links pilot study was designed to obtain process data about the supports and resources needed to empower teacher interns to practice reform-based teaching in k–12 diverse school settings. the purpose of the math links pilot was two-fold: (1) to provide teacher interns with field-based experience to practice reform-based mathematics instruction; (2) to provide teacher interns with critical understanding of culturally responsive pedagogy. by exposing teacher interns to urban students enrolled in informal school settings, we aimed to reduce stereotypes about urban children and youth and to increase the capacity of prospective teachers to engage in reformbased mathematics instruction and culturally responsive pedagogy. study sample a total of 12 preservice teachers (4 undergraduate and 8 graduate) were recruited to participate in the study. six preservice teachers (1 undergraduate and 5 graduate) participated in fall 2006, and six preservice teachers (3 undergraduate and 3 graduate) in spring 2007. some of our participants were secondary majors and some were elementary majors. the variety of participants’ backgrounds adds important caveats to our data analysis. however, the population of interest was preservice and beginning teachers enrolled in a graduate certification program. as previously mentioned, the rationale for studying this population is studies on this particular population are scarce and these teacher candidates have few if any field experiences in education prior to student teaching or induction (osisioma & moscovici, 2008). the ages of the eight interns selected from the larger study ranged from 25 to 39 years of age. five were white women, two were white men, and one was a korean woman. leonard & evans math links journal of urban mathematics education vol. 11, no. 1&2 124 the teacher-researcher one of the teacher-researchers of this study was also the instructor of the courses in which the teacher interns were enrolled. one of the courses in which the interns were enrolled was an elementary mathematics methods course (fall 2006), and the other course was a general pedagogical course on effective teaching (spring 2007). the teacher-researcher will be referred to as bridget (pseudonym) for the sake of anonymity. although these were two different courses, bridget’s philosophy of education was consistent in both courses. her strong belief in equity and social justice influenced the texts and articles students read in the courses. students in the pedagogy course read texts that dealt with cultural relevance and social justice on a general level (i.e., gloria ladson-billings’ aera presidential address [ladson-billings, 2006], the dreamkeepers [ladson-billings, 1994] and diversity pedagogy [sheets, 2005]). students in the mathematics methods course read culturally relevant and social justice articles that were specific to mathematics education (i.e., gutstein, 2003; leonard, davis, & sidler, 2005; martin, 2003). the other teacherresearcher of this study, who did not have a teaching role in this study, shares a similar philosophy of education with bridget and also promotes equity and social justice in publications, presentations, and in the classroom. this researcher teaches at a university in which the college of education promotes teaching from a social justice perspective as its core mission. these courses provided a springboard to discuss issues of equity and social justice and to demonstrate pedagogical ways to infuse students’ culture into lesson plans, particularly in mathematics. in both courses, teachers had to demonstrate teaching effectiveness by presenting a micro-teaching lesson (short 20-minute lesson focusing mainly on one concept as opposed to a full lesson plan) to their peers. in the pedagogy course, students could present a lesson dealing with any of the core content areas specific to their major field of study (english, mathematics, science, social studies) or specialty areas (art, music, physical education). for example, a student in the general pedagogy course read a book by maya angelou to integrate art and literacy. in the mathematics methods course, students focused on teaching a mathematics topic to students in grades prek–8. an example of a lesson in the mathematics methods course consisted of using the faces of actors, such as will smith and sandra bullock, to teach about the golden ratio and symmetry. both of the above lessons made broad connections to american culture. micro-teaching to peers, however, does not provide preservice teachers with the field-experiences they need to teach. learning to teach involves practice with real students. an important part of any teacher credential program is providing settings for prospective teachers to work with actual children (ambrose, 2004; ebby, 2000). teacher interns, who were also students in bridget’s courses, had the privilege of not only delivering the content but also practicing culturally responsive pedagogy in urban settings in philadelphia. leonard & evans math links journal of urban mathematics education vol. 11, no. 1&2 125 the community-based sites the teacher interns worked at two african american churches that had longstanding relationships in the communities they served. both churches are located in urban neighborhoods in north philadelphia. both zion and haven churches have served the north philadelphia community for more than 100 years. in the last 5 years, zion collaborated with researchers at the university where the math links study took place by supporting preservice teachers’ work with children in saturday science programs (leonard, boakes, & moore, in press; leonard, moore, & spearman, 2007). furthermore, zion has served as a site for after-school and summer enrichments programs for early childhood and elementary students. in recent years, it became a site for an at-risk high school program supported by a grant from the city of philadelphia. haven, on the other hand, has not been as involved with educational endeavors. the recent addition of a computer lab and establishment of an after-school program has helped to thrust haven into the community spotlight, however. programs for children and adults have been developed. because of their educational initiatives and community efforts, the zion and haven sites were selected for the eight graduate student participants in the math links study to obtain field-based experiences. five of these interns (one man and four women) worked at the zion site during the fall of 2006. three of these interns (two women and one man) worked at the haven site during the spring of 2007. at-risk youth, 13 to 18 years of age, were enrolled at the zion site. children, 6 to 12 years of age, were enrolled in an afterschool program at haven. thus, we were able to collect data on teacher interns’ actions with elementary, middle, and high school students. it should be noted, however, that attendance at the two sites varied because both programs were relatively new and voluntary. methods we used qualitative research methods to collect and analyze data in the math links pilot study. because we report on two different cohorts of interns simultaneously, this study may be characterized as a study within a study. specifically, we use case study methodology to analyze and report our findings. case studies are often used for in-depth examination of processes that emerge from a small number of phenomena (bogdan & biklin, 1998). considerations were given to ethnicity and gender to obtain a diverse sample for the case studies. three of the teacher interns were selected for further study (one white man, one white woman, and one asian woman) because their backgrounds provide the research community with rich data about the cultural sensitivity of teachers from these specific backgrounds. it is important to understand how these teachers enact culturally responsive pedagogy in urban settings. whites, particularly white women, continue to choose teaching as a leonard & evans math links journal of urban mathematics education vol. 11, no. 1&2 126 career (martin, 2007; remillard, 2000). as a result, these teachers are more likely to work with urban students if african american and other teachers of color continue to decline. data sources consisted of the following for each of the cases: coursework, informal observations at project sites, logs, and interviews. we then used the constant-comparative method to compare and contrast the cases (glaser & strauss, 1967). each of the cases provided the researchers with rich data about the participants’ development of pedagogical content knowledge in mathematics and culturally responsive pedagogy. the results of this study will be presented in two parts. to answer the first research question about how the teacher-researchers’ reform-based practices influenced teacher interns’ beliefs about culturally responsive pedagogy, we analyze the results of structured interviews obtained from the teacher interns and classroom vignettes. to answer the second research question about how interactions in the community-based internship helped the interns to develop culturally responsive pedagogy, we analyze three case studies and examine one of these cases in depth. due to data source limitations, these participants qualify as a convenience sample. while not appropriate practice for a quasi-experimental study, a convenience sample served our purposes for this pilot self-study to inform future research. procedures prior to serving as an intern, participants completed a 3-hour professional development session taught by a mathematics education consultant while they were simultaneously enrolled in one of bridget’s courses (as previously described). the teacher interns were trained to use guided inquiry during professional development. windschitl (2003) characterizes guided inquiry by the level of student involvement. a hallmark of guided inquiry is that students investigate a prescribed problem using their own methods. while teaching the education courses, bridget modeled inquirybased instruction. teacher interns also watched episodes of kay toliver as she engaged students of color in inquiry-based mathematics instruction (foundations for the advancements in science and education productions [fase], 1998). thus, teacher interns were exposed to examples of inquiry-based instruction prior to working with students in the field. to provide participants in this pilot study with field experience prior to student teaching, we placed them in settings where they could obtain 30 hours of fieldwork in informal education settings. the local churches sites were located within a one-mile radius of the college. therefore, teacher interns could easily complete the required 30 hours over the course of the semester while simultaneously taking education courses. seven teacher interns, who were enrolled in the graduate teacher credential program in the college (one of the original eight graduate teacher interns dropped out of the study due to a schedule conflict with her job), were observed by the teacher-researcher and/or the graduate research assistant as leonard & evans math links journal of urban mathematics education vol. 11, no. 1&2 127 they worked with students in the community-based settings during the fall 2006 and spring 2007 semesters. the interns were required to keep a log of notes to document their activities with students each time they went to the site. these logs were analyzed by the researchers to determine not only how the teacher interns’ pedagogy was changing but also how their attitudes toward the students were changing. in other words, we examined the logs for evidence of caring (gay, 2000) and actions that exemplified behaviors that could be synonymous to having a culturally responsive or social justice stance (i.e., advocating for students when rules or regulations are unjust or unfair; teaching in a manner that informs students about the status quo and/or how to challenge such systems) (gutstein, 2003; tate, 1995). an interview protocol was also developed and administered to participants after they completed their community-based field experiences. in particular, we were interested in comments that reflected changes in practices or attitudes about the student population. teacher interns were given a four-digit id number for identification purposes. the structured interviews were read and coded to categorize the teacher interns’ responses. the teacher interns’ responses were then analyzed to find themes and patterns among their experiences. common elements informed the researchers about how to improve the field-based aspect of the project for future study. limitations one limitation in this pilot study is the sparse number of student participants in the community-based field settings and the variant amounts of data collected from the teacher interns’ logs. some teacher interns wrote a minimal amount in their logs, while one in particular (a korean woman) kept copious notes and detailed descriptions about the lessons and her interactions with students. thus, these data sources are uneven. the interview protocol, however, was used to fill in gaps in the data. thus, triangulation of data sources was used to increase the validity of our findings. a second limitation is the instructor of the general education and mathematics methods courses in which the participants were enrolled was also a participantobserver in the study. while teacher-researcher is common in qualitative studies in education, issues of power and researcher bias are threats to internal reliability. to minimize these threats, a graduate research assistant was also a participant-observer in each of the field-based settings, and a second mathematics educator (the other teacher-researcher in this study), who had previously taught mathematics methods courses at the same institution and currently teaches mathematics methods courses at a different institution, corroborated the interpretation of data and the results. thus, checks and balances were put into place to minimize bias and increase the integrity of our findings. leonard & evans math links journal of urban mathematics education vol. 11, no. 1&2 128 results structured interviews six of the seven teacher interns who participated in the community-based internship during the fall or spring semesters of 2006–2007 participated in the structured interview. two reported on their experiences at haven and four reported on their experiences at zion. as shown in appendix a, four categories emerged as a result of qualitative analysis: (1) lessons learned from the program, (2) teacher intern’s perceived strengths, (3) perspectives on the math links program, and (4) perspectives on urban students. analysis of intern responses in lessons learned from the program (category 1) reveal that three interns (2210, 0078, 1080) focused on classroom management issues (i.e., organized lessons, firm and consistent discipline, classroom management techniques) and three interns (3695, 9352, 0063) focused on care and/or relationships (i.e., diverse needs of children, building relationships, mutual respect, increased understanding of diverse students). teacher interns reported perceived strengths (category 2) by describing their commitment (0078, 9352), experience (3695), and lesson creativity (2210, 9352, 0063, 1080). perspectives on the math links program (category 3) reveal the interns at haven (2210, 0078) did not believe they had the supervision and oversight they desired. one intern (9352) at zion mentioned that organization and communication could have been better. however, teacher interns mentioned some of the benefits of the program, such as the resources (0078), exposure to work with students (2210, 3695, 0352, 0063, 1080), and learning from peers (0063). one intern (1080) who was student teaching at the same time that she participated in the study noted: “activities presented to youth in the math links program were used the next day with students during student teaching.” this comment highlights the importance of the teaching-learning process. one must actually engage in teaching in order to learn how to teach (ambrose, 2004; ebby, 2000). finally, we analyzed the comments that emerged in category 4: perspectives on urban students. three comments focused on students’ behavior (0078, 0063) or opportunities (3695), but three commented on how their own attitudes and perceptions of urban students changed (2210, 9352, 1080). one intern at haven remarked: “children were intelligent, focused, and dynamic.” one intern at zion stated the program “challenged myths about urban students: lazy, don’t want to learn; don’t care about education; don’t care about work. students were hard working, wanted to learn, and wanted to understand math.” because the college did not provide field-based experiences for graduate students prior to student teaching, the foregoing comments have important study implications. overall, comments about urban students were consistently positive. moreover, the community-based field experience allowed some of the teacher interns to become students of students and challenged negative perceptions and stere leonard & evans math links journal of urban mathematics education vol. 11, no. 1&2 129 otypes about students of color (nieto, 2002). in order to learn more about their interactions with students in the community-based settings, we present the case studies of three interns. pseudonyms are used for anonymity. these cases are presented as vignettes. the vignettes although a total of 12 teacher interns participated in community-based field experiences in the 2006–2007 academic year, only eight were graduate students, which is the focus of this article. the profiles of these eight interns reveal two were white men and six were women (5 white and 1 asian). of these eight participants only six were also enrolled in one of bridget’s courses: 1 man; 5 women. vignette 1: shawn. shawn was a 25-year-old european american man with a sociology degree from the university of michigan. shawn was the only male teacher intern who worked at zion. because of this, many of the predominantly male students perceived him as a mentor. he was observed teaching an inquiry-based lesson that integrated mathematics and space science in the fall of 2006. the lesson involved having students calculate the percentages of different elements found in a sample of playdoh used to simulate moon rock. students learned how to slice the rock samples like geologists and then estimated and extrapolated the data to determine what type of rock sample they had by counting beads of different colors. the vignette taken from one of shawn’s reflection papers is presented: it seems as if my undergraduate education in sociology has laid the groundwork for a deeper and more applicable understanding of social justice and equity, which i have been able to build upon both in theory and in practice. ultimately, my understanding of culture in mathematics education will be tested in the classroom, and that is why my experience at zion this semester has been so valuable. while the context that children are raised in may not be the sole determining factor of their success, it undoubtedly will impact the rest of their lives. students who have limited access to resources and effectual education will have limited opportunities to achieve success. this reality is clearer after one day at zion than it could ever be in a journal article or textbook. tutoring has become the ideal opportunity to apply what i am learning in the classroom to situations i will face as an educator. i am discovering that education, in particular my own, is a steady progression from abstract theory to more tangible concepts, concepts that have practical implications for the classroom. it is envisioning how to embrace the inquiry-based models of learning we are exposed to and relate them to every teaching opportunity we are presented with. shawn’s vignette reveals the math links experience was pivotal to his development of critical consciousness and his understanding of equity and social justice as it related to mathematics education. clearly, he exhibited culturally responsive pedagogy as he learned to mesh theory and practice. shawn, however, hints at the complex nature of inquiry-based teaching. how do teachers relate the pedagogies leonard & evans math links journal of urban mathematics education vol. 11, no. 1&2 130 they learn in teacher education programs to students’ lives? how do they include elements of student culture as a springboard for learning without watering down the curriculum or lowering expectations? these questions cannot be answered by participating in 30 contact hours of field experience. nevertheless, shawn had a better understanding of teaching diverse students after participating in the math links study. vignette 2: camille. camille was a 26-year-old european american woman who had received an undergraduate degree in english from cornell university. she had also lived and studied abroad in japan. she was an intern at the haven site. during her observation in the spring of 2007, bridget (recall this was the instructor and participant-observer) noticed that camille had an excellent rapport with the 6 to 12 years old african american students. they were learning about the story of sadako, origami, and how to make paper cranes, which they connected together to make a long strand. camille also brought pictures of her travels to japan so the students could see the lifestyle of the japanese people. impressed by camille’s ability to retell sadako’s story, a japanese girl’s fight with leukemia after being exposed to agent orange during world war ii, bridget loaned her the storybook by coerr (1993). to help the african american students at haven understand the gravity of sadako’s plight, camille described an event that her students could relate to. the vignette taken from one of her reflection papers on social justice is presented: i would like to address what to me was one of the main strengths of this article [leonard & hill, 2008]. the background material regarding analytical scaffolding and social scaffolding is extremely helpful and profuse as is its later exemplification within the context of the [lesson]. the following discourse occurred during one of my own sessions with six african american students. my boyfriend, c, and i gave a joint presentation about the blues (musically and historically) at the haven after-school program where i tutored once a week. c [stated], “a contradiction is when you say one thing and then do another. the united states contradicted itself when it took away african american rights. everybody, how would that make you feel?” “sad,” one said. “unfair,” said another. “has anyone ever heard of the blues?” i asked. another student said, “it’s when you are sad.” c and i found that when we quizzed the students later on, they remembered almost word for word what was said. i had a very valuable first-hand experience with scaffolding and intend now, further confirmed by this article, to use it as often as possible in future lessons. camille’s vignette reveals her ability to engage in culturally responsive pedagogy. she and her boyfriend engaged students by using their cultural capital to scaffold their learning. camille’s use of the blues as social scaffolding shows her ability to move from theory to practice. the mathematics in this lesson was culturally relevant as she tried to link learning geometry and spatial skills to sadako’s life with cultural beliefs about her illness. to make connections with sadako’s story, leonard & evans math links journal of urban mathematics education vol. 11, no. 1&2 131 camille used the blues, which is a part of african american culture. by using the context of the blues, the students were able to understand the affective nature of the story, and they were eager to make paper cranes and learn more about japanese culture. thus, camille’s lesson is an example of how to teach mathematics concepts within a social justice context. vignette 3: sun-lee. sun-lee was a 39-year-old korean woman attending the college on a visa. she was a graduate of ewha woman’s university in seoul, korea. she had an undergraduate degree in library science and was interested in teaching english to esl students. therefore, she chose to dually enroll in the tesol and graduate certification program at the college. sun-lee taught and integrated science and mathematics lessons to high school students in the fall of 2006. she was instrumental in helping the students learn geometry, measurement, and aerodynamics by constructing kites (leonard, 2002). students made tetrahedral kites out of tissue paper, straws, and string (center for engineering educational outreach [ceeo], 2001). they learned mathematics vocabulary (tetrahedron, faces, vertices, edges, etc.), used rulers to measure accurately, and learned how lift, gravity, drag, and thrust worked together to make a kite fly. sun-lee kept a meticulous journal of her experiences at zion. an excerpt from her journal describes her work with the students during the kite activity: there were three more girls, but i did not know their names yet. since we had to reattach two bridles, we had to measure two strings for the two small kites. i thought that the students needed to find information from the text for themselves. when they asked me how to do [it], i read the instruction with them while pointing out the part. after reading, i asked them what it meant. for example, part of the instruction was for [a] three-quarter inch of space between the loop knot and the straw. rickie showed me 3.4 inches by means of a ruler. therefore, i pointed to this part of the instruction, and we read together. rickie understood and made [a] three-quarter inch space. carolyn said her ruler was not big enough to measure the longer part of a bridle. i asked her how she could measure the longer string. she thought about it and said, “oh, moving like this.” she displayed the iteration on the ruler. sun-lee demonstrated structured inquiry in the foregoing excerpt (windschitl, 2003). rather than teaching by telling, she led one of the students to figure out the difference between 3.4 and ¾ inches and helped another student determine how to measure multiple pieces of string using a ruler. by helping them to construct their own knowledge, the students were more likely to remember the mathematics concepts they learned. this vignette hints at the complexity of the teaching-learning process. how much information should teachers tell students? how much should students be responsible for learning? knowing “when to provide an explanation, when to model, when to ask the rather pointed questions…is delicate and uncertain” (ball, 1993, p. 393). leonard & evans math links journal of urban mathematics education vol. 11, no. 1&2 132 the data presented above in vignette 3 as well as data presented in the interview protocol show the unique characteristics and behaviors that sun-lee exhibited during the study. moreover, the data she provided in her participant log provided rich accounts of her work with students at the zion site. while she was selected because of convenience, the case of sun-lee provided the researchers with a deep and informative account of her field experiences at zion. the case of sun-lee the case of sun-lee was quite unique and interesting. she was the only person of color who participated in the math links study. furthermore, she made tremendous strides in english fluency and literacy during the fall 2006 semester, and she became culturally competent as a result of her experience in the pilot study. her participant log consisted of 13 detailed accounts of teaching and learning mathematics within a cross-curricular context at the zion site from october 23, 2006 to december 13, 2006. appendix b summarizes the lessons, teacher actions, student actions, and teacher behaviors. analysis of sun-lee’s case study analyses of sun-lee’s journal entries, as shown in appendix b, reveal that she progressed rather quickly from using direct instruction to inquiry-based instruction over the course of the fall 2006 semester. on october 23rd, sun-lee used direct instruction to teach andrew a part-whole interpretation of fractions and subsequently the conversion of those fractions into percents: i asked [andrew] whether or not he knew how to get percentage. he said, “no.” he got the 41 white rocks, 13 red rocks, and 7 blue rocks. in order to explain [how] the numbers could be transformed into numbers less than one, i helped him to draw a pie chart. we sectored the pie into 61 pieces. in the comparison of the pie with pizza, i explained 61 pieces as a whole number 1. and i told him that the concept could be expressed 7/61, 13/61, and 41/61, which were less than the number one. on october 24th, sun-lee still wrote about “showing” students how to do things, but by october 25th, we have evidence that sun-lee began to use questioning techniques that allowed the students to develop their own understanding. rather than teaching by telling, she was beginning to help students take responsibility for their own learning: i gave short instructions and wanted them to read the procedure again. i thought that the students needed to find the information from the text for themselves. when they asked me how to do it, i read the instructions with them. … after reading, i asked them what it meant. leonard & evans math links journal of urban mathematics education vol. 11, no. 1&2 133 by november 1st, sun-lee attempted to engage students in discussion. yet, she was hesitant to do so because students’ work was at different stages and because of her perceived limited english proficiency. however, a breakthrough occurred on november 2nd when sun-lee had two students offer their own examples of newton’s third law of motion. by having the students experiment with a rocket launch from a lesson derived from mission mathematics (hynes & hicks, 2005) before introducing newton’s third law, students were able to make connections by collecting actual data and offering their own explanations of the law: “for every action there is an opposite and equal reaction.” i gave an example with a ruler and the edge of the desk. by hitting the ruler with weak force, the ruler dropped down. however, the ruler flipped over and dropped down when i used strong force. dante [used] a similar example. also, rickie [shared] his idea. toward the end of the semester sun-lee engaged students in investigations that led them to make discoveries about other theories as well. mathematical probability was connected to genetics when dante experimented with punnett squares using a coin to determine genetic outcomes. finally, it can be seen that sun-lee realized that inquiry, although time consuming, is paramount to good instruction. at-risk students at zion were engaged at high levels when they were given opportunities to investigate, discuss, and explain their reasoning. analyses of sun-lee’s journal show that she progressed over the course of the semester from using direct instruction to reform-based, context specific instruction. most importantly, she empowered at-risk students by helping them to be self-directed and to take charge of their own learning, which is one of the tenets of culturally relevant pedagogy (ladson-billings, 1995). discussion the results of this pilot study are promising. the major finding of the study is that during the graduate certification program five of the seven interns who participated in the pilot study, though older and more entrenched in their beliefs and values, engaged in reform-based practices during the internship. evidence for this claim is supported by the results of interview data and case studies. this finding has important implications for the field if these changes can also occur among beginning teachers. it is consistent with the finding by osisioma and moscovici (2008) in which they observed a shift in science teacher interns’ beliefs from traditional methods of teaching science to an inquiry and reform-based approach about teaching and learning over the course of two semesters. data, such as that published by the national association of educational progress (naep), continue to show dismal performance in mathematics, particularly in grade eight (national center for educa leonard & evans math links journal of urban mathematics education vol. 11, no. 1&2 134 tional statistics [nces], 2007). the ability to think and reason is critical if one is to achieve above basic and proficient levels in mathematics. a second finding is five of seven teacher interns who participated in the community-based internship changed their perceptions of the predominantly african american students participating in the math links program. they recognized that “typical” stereotypes about these children simply were not true. the students were eager and willing to learn mathematics and were receptive to one-on-one tutoring and whole group instruction. furthermore, these interns developed an ethic of care and exhibited teacher dispositions that martin (2007) characterized as racial competence and commitment to anti-oppressive and anti-racist teaching. nevertheless, additional studies are needed to learn whether the perception of hard work and the ethic of care might be transported into the traditional school setting. a third finding is the importance of providing teacher interns at the graduate level with field-based experiences prior to student teaching. because this population generally enrolled in evening courses at the college to obtain a teaching certificate, field experience was not a part of the credential program. five of the six teacher interns interviewed noted the value of the field experience. one specifically mentioned how she used materials and techniques learned in the math links program during student teaching. the community-based internship provided these interns with an opportunity to learn from their interactions with students. this finding concurs with the findings of our previous work with teacher interns (leonard, boakes, & moore, in press) and with ebby’s (2000) work with preservice mathematics teachers. prospective teachers’ pedagogical content knowledge in mathematics was dependent upon the teaching-learning process (ebby, 2000; sherin, 2002). additional research, however, is needed to learn whether inquiry-based practices learned during fieldwork can be sustained throughout induction. finally, this pilot study has implications for researchers attempting to link the teaching of mathematics to social justice. teaching preservice and beginning teachers about social justice in a vacuum was not meaningful, as two interns who participated in the case studies attested. actually working with students whose lives stood in stark contrast to their own privileged backgrounds was eye opening for these two interns. they realized firsthand the powerful impact poverty has on some urban students’ lives and the ramifications of the lack of educational opportunity. moreover, the structured interviews and case studies show how several teacher interns were moved by the potential (realized and unrealized) of students in the community-based programs. thus, learning to teach for social justice must include critical work with appropriate student populations. the results of the math links pilot study show that providing communitybased field experiences for teacher interns benefits both interns and students alike as relationships and rapport are forged during the mentoring process. not only did one of the interns continue for an additional semester but also site coordinators and leonard & evans math links journal of urban mathematics education vol. 11, no. 1&2 135 students requested additional interns the next semester. given limited research dollars, sustaining successful partnerships are challenging. however, the math links study shows the 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(2003). inquiry projects in science teacher education: what can investigative experiences reveal about teacher thinking and eventual classroom practice? science education, 87(1), 112–143. leonard & evans math links journal of urban mathematics education vol. 11, no. 1&2 138 appendix a comparison and contrast of teacher interns’ responses to interview protocol teacher id/setting lessons learned from program teacher intern’s perceived strengths perspectives of math links program perspectives of urban student 2210 haven learned that highly organized lessons that allow some flexibility are important; discipline needs to be firm/consistent. used background knowledge in art and history to blend a variety of media and context to lessons, which maintained student interest regardless of learning styles. prior teaching experience was limited so extra exposure was helpful. program allowed me to work one-on-one with children. little structure and guidance provided, which caused site director to have some uncertainty about the parameters of the program. children were intelligent, focused, and dynamic. struck by how poverty and family situation can undermine intelligent students, causing them to miss school and jeopardize their education. longed to exert more influence on parents when she becomes a full-time teacher. 0078 haven variety of resources available to teachers. classroom management techniques. commitment to teaching urban children. provided resources. more training and oversight needed. broader view of behavior issues. 3695 zion learned about the diverse needs of children. possessed patience and caring qualities; had prior experience working with pre-k inner city children. children have different needs; teacher must cater to the needs of all children; provided experiences beyond private and suburban settings. program provided opportunities for children. 9352 zion building relationships with children is important. mutual respect between teacher and students is key to academic success. consistency is vital with students. personal creativity came out during the internship. made connections with students despite differences in age and appearance. had prior experience teaching high school. commitment to working with children who have special needs. wonderful experience that exposed the intern to real-world setting and allowed her to build confidence. program provided opportunities to interact with students in an informal way. organization and communication could have been better. requested lesson templates or exact lesson plans to follow. program helped intern to become a better teacher by overcoming misconceptions and insecurities about teaching youth. learned how accepting students can be when given special attention. 0063 zion hands-on lessons can serve as motivator. increased understanding of the similarities/ differences between child in u.s. and korea. constructing hands-on lessons; ability to adapt what i learn to new situations. providing the opportunity to observe students’ learning and to learn from peers and mentors. hands-on activities provided learning opportunities for the students. worried about student motivation and attentiveness, but it changed. older students responded well to hands-on activities and were highly engaged in learning. 1080 zion learned how to control the pace of the lesson and make sure students are on task. learned how to use appropriate classroom management techniques. discovered ways to motivate students; oneon-one interaction led to direct involvement in one case. program allowed intern to tutor students in a small groups; experience allowed her to develop ideas for use in other settings. activities presented to youth in the math links program were used the next day with students during student teaching. program challenged myths about urban students: lazy, don’t want to learn; don’t care about education; can’t do the work. students were hardworking, wanted to learn, wanted to understand math. leonard & evans math links journal of urban mathematics education vol. 11, no. 1&2 139 appendix b journal analysis of one teacher intern’s pedagogy date and lesson type teacher actions student actions analysis of teacher behaviors october 23, 2006 rock doctors students cross cut playdoh representation of moon rocks to determine what type of rock they had by the percentages of minerals they found in the playdoh. i asked [andrew] whether or not he knew how to get percentage. he said, “no.” he got the 41 white rocks, 13 red rocks, and 7 blue rocks. in order to explain the numbers could be transformed into numbers less than one, i helped him to draw a pie chart. we sectored the pie into 61 pieces. in the comparison of the pie with pizza, i explained 61 pieces as a whole number 1. and i told him that the concept could be expressed 7/61, 13/61, and 41/61, which were less than the number one. by writing down the numbers in the format, i modeled how to compute the division 7/61. andrew computed the other fractions. anthony brought his multiplication knowledge to divide the fractions. sun-lee worked one-on-one with andrew. she uses direct instruction to help him understand parts of a whole and to teach how to calculate percentages. october 24, 2006 tetrahedral kite students used straws, string, and tissue paper to make tetrahedral kites. each student made one cell, and all of the cells were put together to make one large kite. since joshua said, “i don’t know how to put these straws,” i approached to help him. he was holding his sixth straw, which needed to support the tetrahedron. i showed how the straw could uphold the tetrahedron and said to him to run the thread through the straw and clip tightly. the runner and holder took their positions while releasing the string to fly it. the runner and the holder tried several times, but it didn’t work out. the kite kept falling down whenever the holder let go, although we changed the position of the bridle. sun-lee worked one-on-one with joshua. she supported his learning by helping him make one tetrahedral cell for the kite. leonard & evans math links journal of urban mathematics education vol. 11, no. 1&2 140 october 25, 2006 tetrahedral kite (cont.) students redesigned the tetrahedral kite. we started taking the kite apart. while making the knots, i looked for additional help. i called carolyn…. since we had to reattach two bridles, we had to measure two strings for the two small kites. i gave short instructions and wanted them to read the procedure again. i thought that the students needed to find the information from the text for themselves. when they asked me how to do it, i read the instructions with them…. after reading, i asked them what it meant. since we had two different kites, i suggested two groups of students pull each kite….the reason was their own experience of pulling the kite will make them think in depth. students suggested modifying the kite: place wax paper on the open cells, change position of bridle, move to larger space to fly the kite. rickie said he could make a new small kite. andrew and joshua flew the kites, and it stayed horizontal while the students were running. they were running all over the place, and it did not go up vertically. in this lesson, sun-lee employs some inquiry-based practices. she allowed the students to have some autonomy and encouraged them to find out information for themselves and to think in depth about how the kite flew. november 1, 2006 alka rockets students used fuji film canisters and index cards to make a rocket. after making predictions, students used different amounts of alkaseltzer and water into it to launch the rockets. they measured the height each time the rockets were launched. i told the students to check whether they had all the items [for] the procedure. also, i show two rocket pictures that i printed in color from the internet to talk about the force and direction of…the rockets. students were asked to read the first procedure. by referring to the procedure, i intended to enable students to practice applying necessary information to their own work. i asked what the function of the rocket fins and nose cones were. the discussion could not develop well partly because each student was working on a different stage. the other reason could be that i was not confident to lead the discussion because i was concerned about my english proficiency. although i understood preparation and practice in real classrooms could have reduced my anxiety, the anxiety in my mind still existed. if i use the lesson again, i will have the students read through the procedure first to grasp the whole process. also, they could discuss the functions of the fins and cones more than they did this time. the students completed their rockets and were [asked] to predict the height. a table was given to record each height. each student tried the initial variable (amount of water). the rockers went high up from the scale of 2 to 8. while trying the other variables (size of tablet). students figured out that the less amount of water and more amount of alkaseltzer went higher than the others. the lesson was a success. marie said it was the most interesting experiment. rickie said, “it was pretty cool.” analysis of lesson reveals sun-lee continues to utilize some aspects of inquiry. instead of teaching by telling, she wanted students to find the information and apply the knowledge learned to the task of making the alka rocket. although she was ambivalent about her english, she tried to lead the students in a discussion about how the fins and nose cones would impact the rocket’s flight. while she was not able to engage students in such a discussion at this juncture, sun-lee reflected about how she could do a better job the next time she taught the lesson. leonard & evans math links journal of urban mathematics education vol. 11, no. 1&2 141 november 2, 2006 alka rockets (cont.) followed up with data analysis of the results from previous day’s launch. i asked…how many inches are in a foot? some said 6 inches because they knew that the temporary ruler was…six inches. someone said 12 inches. i wrote 1 foot = 12 inches to help students transform inches to feet. after the measurements, i told them the experiment proves newton’s third law of motion: “for every action there is an opposite and equal reaction.” i gave an example with a ruler and the edge of the desk. by hitting the ruler with weak force, the ruler dropped down. however, the ruler flipped over and dropped down when i used strong force. dante [used] a similar example. also, rickie [shared] his idea. students recalled their highest results and discussed the influential factors for the highest rockets. everybody said that less water caused the rockets to go higher. they figured out the real lengths by measuring the heights and multiplying their results by six inches because the temporary ruler was segmented every six inches. dante thought one scale of the temporary ruler was a foot. so he wrote his rocket went, 6 ft., 8 ft., and 7 ft. but after the explanation, he changed 6 to 36 inches and 8 to 48 inches. sun-lee probed the students to determine their background knowledge on measurement. they knew 12 inches was one foot but the data reveal little experience with measurement tasks, scaling, and converting inches to feet. while sun-lee reinforced equations and laws to help students understand the activity, she also demonstrated newton’s law. this demonstration led two students to share examples and ideas about newton’s law, that is, the students offered alternative explanations, which is a hallmark of inquiry. november 15, 2006 genetics and punnet squares student used probability and punnett squares to make predicts about traits while leaving the classroom, i thought it would be better for us to go over the punnett square tomorrow because i felt that [dante] was not sure yet. in order to decide the components of genetic trait, students were asked to flip a coin. when dante flipped a nickel twice, he got a pair of n and n genes for each flip. that means his portrait has a round nose, which carried a recessive gene of a pointy nose. after deciding all other traits including the color of eyes and hair, he drew the portrait with color[ed] pens. students were asked to apply the rule to real life problems. dante created the punnett square but got 2 squares wrong. sun-lee worked one-on-one with dante as he determined genetic traits and used the punnett square. realizing his struggle with the punnett square, sunlee planned to review concepts with him the next day. december 1, 2006 measurement students learned customary and metric units of length. in a previous lesson, alka rockets activity, dante used the unit of length (3 ft., 8 in.) to record the measurement. dante uses the unit consistently [sic] in the right form, but i wanted to know whether or not he was aware of inches, square inches, and square feet. when i asked him to explain…the relationship between a meter and a yard, he did not explain, he draw [sic] a division calculation. how could i let him know the relationship between a meter and a yard? step by step! i asked dante to show and explain his thoughts a lot because i learned that teachers should ask students, “explain what you think.” however, this process takes time. in a real class with many students, it would not be easy to follow each student’s thought process. dante recognized an inch and said [to] himself the ruler was a foot when he saw the ruler. he measured a line of 3 inches on his own. when he measured 2 1/8, he asked [sun-lee] how to measure. (a book was provided for a reference.) he measured a yard and a meter of the table again by using the paper ruler. [he measured] a meter length with the paper inch ruler. he got 39 5/16 inches. also, he measured one yard with the paper centimeter rule. he got 91 cm. then, he referred to the conversion table and found that one meter is 100 centimeters. now, he could understand one yard is 90% of a one meter. during methods class, sun-lee watched the movie stand and deliver. in the film, jaime escalante helped his students learn calculus by teaching them step by step. sun-lee borrowed this phrase and helped dante develop mathematical knowledge by providing hands-on experiences with paper inch and centimeter rulers. she also learned the importance of allowing students to explain their thinking from the methods course. however, she was also aware of the challenges teachers face when they try to use the method in regular classrooms. journal of urban mathematics education december 2017, vol. 10, no. 2, pp. 39–51 ©jume. http://education.gsu.edu/jume megan nickels is an assistant professor of stem education at the university of central florida, 12494 university blvd., orlando, fl 32816; email: megan.nickels@ucf.edu. her research program focuses on the mathematical thinking and learning of critically and terminally ill children, applications of computer science and engineering for mathematics education, and analysis of inequities in mathematics education. craig j. cullen is an associate professor of mathematics education at illinois state university, campus box 4520, normal, il 61790-4520; email: cjculle@ilstu.edu. his research interests include children’s development of measurement understanding as well as the use of technology in the teaching and learning of mathematics. amanda l. cullen is an assistant professor of mathematics education in the mathematics department at illinois state university, campus box 4520, normal, il 61790-4520; email: almille@ilstu.edu. her research interests focus on children’s thinking and learning about geometric measurement concepts as well as elementary and middle school preservice teachers’ development of mathematical content knowledge. richelle joe is an assistant professor of counseling education at the university of central florida, 12494 university blvd., orlando, fl 32816; email: jacqeuline.joe@ucf.edu. her research interests include addressing hiv prevention and hiv/aids in counseling and counselor education, culturally responsive counseling services for diverse and underserved populations, and effective school-family-community partnerships. commentary separate and unequal: students with hiv/aids and mathematics education megan nickels university of central florida with craig j. cullen, amanda l. cullen, and richelle joe we must take sides. neutrality helps the oppressor, never the victim. silence encourages the tormentor, never the tormented. sometimes we must interfere. when human lives are endangered, when human dignity is in jeopardy, national borders and sensitivities become irrelevant. wherever men and women are persecuted because of their race, religion, or political views, that place must—at that moment—become the center of the universe. – elie wiesel1 uman immunodeficiency virus (hiv) and acquired immune deficiency syndrome (aids) constitute a true pandemic affecting 36.7 million people worldwide (unaids, 2016). this pandemic creates a complex set of problems regarding health and development, power and identity, gender and racial bias and inequalities, and justice and social ethics that aggregate as a singular phenomenon (coombe, 1 see the nobel peace prize acceptance speech delivered by elie wiesel in oslo, norway on december 10, 1986: https://www.nobelprize.org/nobel_prizes/peace/laureates/1986/wiesel-acceptance_en.html. h http://education.gsu.edu/jume mailto:megan.nickels@ucf.edu mailto:jacqeuline.joe@ucf.edu https://www.nobelprize.org/nobel_prizes/peace/laureates/1986/wiesel-acceptance_en.html nickels et al. commentary journal of urban mathematics education vol. 10, no. 2 40 2004). adolescents and youth represent a substantial number of individuals living with hiv/aids (hereinafter hiv2) in the united states. infected youth and young adults aged 13–29 (n = 100,724) made up 10% of all u.s. individuals living with hiv at the end of 2014 (center for disease control [cdc], 2016). this same age group comprises 42% of all new hiv diagnoses, with young black and hispanic/latin@/x people suffering at disproportionate rates (cdc, 2016). for young males aged 13–19 and 20–24, blacks and hispanic/latinos accounted for 84% and 76% of all new hiv infections in 2015, respectively, while cumulatively representing 35% of people these ages (cdc, 2016). similarly, young black and hispanic/latina females represented 83% of young teenagers aged 13–19 and 79% of young women aged 20–24 in the u.s. living with hiv in 2015, but they are 30% of the population of u.s. women in these age brackets (cdc, 2016). young black and hispanic/latino men who have sex with men are most at risk, with hiv diagnoses among both young black and hispanic/latino gay and bisexual boys and men aged 13–24 increasing 87% from 2005 to 2014 (cdc, 2015). despite the rising number of students with hiv attending u.s. schools, equitable educational policies and practices for these individuals are nearly nonexistent. students with hiv remain consigned to the ideologies and practices of segregation, if they are thought of at all. what is also significant, however, are the conversations “stirring” in the journal of urban mathematics education (jume; see stinson, 2016) and the tenor of equity these conversations are bringing to the collective black (cf. martin, 2015) and the possibility it presents for advancing the position of students with hiv. recently, the discussion of grand challenges (stephan et al., 2015) within the field have stressed the critical importance of achieving equity in mathematics education and of including all children—a call newly echoed by the national council of teachers of mathematics’ (nctm) endorsement of the joint equity statement put forth by the national council of supervisors of mathematics (ncsm) and todos: mathematics for all (2016), and tethered to prior position statements (e.g., nctm’s 2000 equity principle in principles and standards for school mathematics; nctm’s 2012 position statement on closing the opportunity gap in mathematics education; nctm’s 2014 access and equity principle in principles to actions: ensuring mathematical success for all). in this commentary, i take sides, furthering the issues of equity for marginalized populations articulated in martin’s (2015) and meyer’s (2016) prior jume commentaries. furthermore, i raise two issues. first, i call to question the authenticity of position statements and documents regarding equity that generalize their expectations to “all students.” such statements are ambivalent to each and every student and do not explicitly reference students with hiv and critical illness more broadly. to leave “all” undefined in this way creates a blindspot (e.g., winkle 2 hiv diagnosis refers to all individuals with an hiv diagnosis regardless of the stage of disease (stage 0, 1, 2, 3 [aids], or unknown; cdc, 2015). nickels et al. commentary journal of urban mathematics education vol. 10, no. 2 41 wagner, hinderliter ortloff, & hunter, 2009), a view obstructed from the center. i also build on martin’s and meyer’s commentaries to address how nctm’s (2014b) principles to actions (and other policy statements/documents aimed toward equity3) might be reimagined to bring the marginalized to the center and to ensure equitable practices for students with hiv (and other critical illnesses4), taking specific aim at the essential elements of access and equity, curriculum, and professionalism. separate and unequal “children with critical illness live under an educational apartheid” (nickels & cullen, 2017b, p. 23). despite the passage of 70 years since the landmark brown v. board of education (1954) decree that “separate educational facilities are inherently unequal,” students with hiv may be “physically, socially, and academically removed from their healthy peers” (nickels & cullen, 2017b, p. 23), even though significant strides in advancing technologies, clinical practices, and pharmacology for the diagnosis and treatment of hiv have made isolation unnecessary. quite simply, the practices of apartheidism affecting the education of students with hiv have far less to do with medical necessity than asserting prejudice and bigotry and perpetuating powered relationships (herek, 2014). students with hiv are distinguished in important ways from children with other disabilities and indeed from other populations of critically ill students. although all students with critical illness (e.g., cancer, sickle cell disease) suffer from inequitable mathematics education 3 while addressing nctm’s (2014b) principles to actions specifically, the challenges i raise regarding equity for students with hiv/aids and other critical illnesses are intended to problematize all recent policy documents and statements from equity organizations within mathematics education (e.g., association of mathematics teacher educators [amte], 2015; nctm, 2014a). 4 children, adolescents, and young adults (ayas) with critical illness (e.g., cancer, hiv/aids, sickle cell disease, traumatic brain injury) are an emerging, broad and complicated topic within mathematics education research. there are a variety of ways in which the literature of pediatric medicine describes critical illness including critical, chronic, and long-term health conditions; all can apply to children who have been diagnosed with life-threatening or life-limiting conditions that require lengthy treatments (hinton & kirk, 2015). as there is ambiguity within these terms, the number of children reported who are afflicted with critical illness can vary from source to source. the literature reports between 10% (7.4 million) and 43% (32 million) of children and ayas living in the united states suffer from some critical, chronic, and/or long-term malady (bethell et al., 2011; hinton & kirk, 2015; nabors, little, akin-little, & iobst, 2008). for my purposes, i define critical illness to be any disease or traumatic injury that is life threatening and requires treatment at a minimum of 3 months’ time. the choice to use the descriptor critical versus chronic is made to convey the insidiousness of such diseases and the virulent treatments that accompany them; alternatively, chronic is focused only on the length of time the child experiences the illness. while i choose to focus on students with hiv/aids in this research commentary, the vast majority of students with critical illness are affected by chronic and structural marginality in mathematics education (nickels & cullen, 2017a). nickels et al. commentary journal of urban mathematics education vol. 10, no. 2 42 practices (nickels & cullen, 2017b), the insidiousness of hiv stigma—both felt and enacted (herek, 2014)—renders students with hiv among the most disenfranchised. in contrast to the sacralized image of the innocent, white, bald child with cancer, the child with hiv is envisioned as depraved and blameworthy. moreover, the population of students with hiv reflects diverse racial and cultural backgrounds, cognitive profiles, physical and socioemotional needs, interests, and aims. hiv is comorbid with physical and cognitive disabilities (e.g., static or progressive encephalopathy), compounded by stigma (e.g., the cultural and gendered identities of lesbian, gay, bisexual, and transgender or minority youth), or geographically weighted to urban centers and low to middle income households. additionally, most children (aged 0–12) with hiv also have a parent with hiv (steele & grauer, 2003). these comorbidities further the enduring notion that individuals with hiv are inferior in status (herek, 2014) and may contribute to perceptions of a student’s mathematical capability or ableism mentality. these misconceptions or negative perceptions are communicated both through overt and subtle messages (nickels & cullen, 2017a), the effects of which may result in an insufficient or complete absence of meaningful and complex mathematical work and the debasement of the student’s mathematical power and identity. the trend in diagnosis rates predicts thousands of u.s. citizens who, without an equitable mathematics education, will be ill-equipped to enrich their lives and to live a life of their own choosing; likely they will be condemned to becoming the working poor and/or isolated from democratic and socially just opportunities for participation in society because their understanding of fundamental and complex mathematics will yield little else. although “calls for mathematics for all and the discourse of equity have become normative in the field of mathematics education” (lawler, 2005, p. 29), concerns of equality within mathematics education, however, are often only concerned with broadening students’ access to high-quality mathematics activities and opportunities (larson, 2016). although this aim is well intentioned, it stops short of extending equitable notions to the nature and use of mathematics and the student’s mathematical agency (lawler, 2005). strict equality for students with hiv would mean simply to provide them with a mathematics education identical to that of their peers. however, the mathematics education of students with hiv must necessarily be, in part, contextually bound to their disease, cognitive profile, and any number of comorbid issues. mathematics education is indeed an intricate process of a student’s developmental maturation and a social/cultural and institutional enterprise of transmitting our knowledge and, subsequently, our values (e.g., what counts as mathematics, who does mathematics, what mathematics matters). access alone does not guarantee equality. paradoxically, if we too severely delineate this population of students (i.e., grouping them only by disease and functional levels for educational purposes), we run the risk of furthering the segregation paradigm and marking them for nickels et al. commentary journal of urban mathematics education vol. 10, no. 2 43 stigmatization. consequently, i argue hiv, and critical illness more generally, must be regarded as a contextual variable that not only influences processes and student outcomes at all levels within mathematics education but also as a variable that must necessitate a shift in the goals, content, and role of mathematics. attention to what we deem as equitable mathematics for students with hiv further problematizes nctm’s (2014b) principles to actions—a document purposed with describing the five essential elements of effective mathematics programs to address “a range of troubling and unproductive realities that exist in too many classrooms, schools, and districts” (p. 3). here, i employ three of the essential elements—access and equity, curriculum, professionalism—to describe “what is” coming to know the current policies and practices for teaching mathematics to children with hiv, and thus revealing the limiting nature of principles to actions and other such policy documents and statements in addressing and attending to these concerns. access and equity viral load (i.e., the amount of hiv in a person’s blood) suppression—and consequently disease progression and mortality—for students with hiv can be well controlled by strict adherence to antiretroviral therapy (art; merzel, vandevanter, & irvine, 2008). art may include a dosing regimen of 15–20 different medications throughout a single day administered in relation to complicated dietary guidelines. the intensity and frequency of art constitute an appreciable challenge to students with hiv in schools. deviation from adherence to art is correlated with statistically significant progression of disease and central nervous system dysfunction; yet, here too, hiv stigma may limit students’ access to support (e.g., choosing not to disclose serostatus to teachers or school nurse; steele & grauer, 2003). hiv positive students with a suppressed viral load may experience little to no decline in cognitive function and present as presumably normal within the classroom. as viral load increases, young people with hiv develop cognitive and neuropsychological impairments that may include “insufficiencies in attention, concentration, expressive language, fine, and gross motor skills including oral-motor functioning and neuromuscular functioning” (collins, 2005, p. 49). public schools, however, are theoretically and functionally designed to educate healthy children (tseng & pluta, 2016). although federal law mandates a free and appropriate public education (fape) for all children (individuals with disabilities education improvement act [idea], 2004; u.s. department of education, 2007), children with hiv often occupy a gray space between federally established and recognized programs (gordon, 2015). for example, hiv does not often qualify as a disability until aids status is reached. although few youth and adolescents progress from hiv to aids within 12 months, many receiving early and continu nickels et al. commentary journal of urban mathematics education vol. 10, no. 2 44 ous stringent care will not advance to aids status until their early thirties (collins, 2005). historically, students with hiv whose viral loads are not suppressed or who have reached aids status have received mathematics education via hospital/homebound instruction (murphy, 1990). there are no federal guidelines for hospital/homebound instruction, and, as such, guidelines vary between states and often from district to district (tseng & pluta, 2016). the common model in the united states is to provide hospital/homebound students 1 hour of tutoring per week per core subject they are enrolled in, resulting in a modal value of 4 hours of tutoring per week as compared to the 25 hours of instruction per week healthy children receive (hull & newport, 2011). although the disparity in hours is certainly egregious, the quality of educational service is also called into question because tutoring bears little resemblance to school-based mathematics instruction. tutors hired to provide hospital/homebound services are often not required to be certified teachers, and instruction may be provided in a variety of formats including via telephone. thus, hospital/homebound practices regularly infringe upon a student’s rights to fape and entitlement to a highly qualified teacher (irwin & elam, 2011). furthermore, no general state aid or hospital/homebound reimbursement is made available to a school district for services provided to a student who is not enrolled in the district. thus, students with hiv can be made to un-enroll from the school district in which they live and enroll in the school district in which the hospital is located to receive services. in the case of intermittent absenteeism, the burden of un-enrollment and re-enrollment dissuades many parents from seeking services for their child. compulsory attendance and fiscal priority can also lead schools to delay educational services or enrollment or even withdraw children with hiv (often without parental consent) to remove them as impediments to annual yearly progress (irwin & elam, 2011). students with hiv who meet the requirements for special education services are not necessarily better served than those who do not meet the threshold requirements. despite having written accommodation plans, these students are presented with the same service model of hospital/homebound instruction. a final concern for access and equity involves the right for students with hiv to have their perspectives heard. silencing is one of the most potent and easily committed acts of inequity. there are many who will claim explicit recommendations for equitable practices for students with hiv are unwarranted because these individuals would be served by the extensive literature on special education, mathematics education reform, or students who identify as black, hispanic/latin@/x, or queer (nickels & cullen, 2017b). the faulty logic behind this suggestion is that adolescence is largely undifferentiated, at least concerning those dimensions that may affect mathematical thinking and learning; this reasoning furthers that disability and disease also present with analogous issues (e.g., research on educational attainment for children with autism would apply also to children with hiv). even if students with hiv were per nickels et al. commentary journal of urban mathematics education vol. 10, no. 2 45 fectly matched to another population represented in special education literature, this body of literature assumes that most individuals with disabilities will be served in a least restrictive environment integrated into the regular classroom. educational issues and outcomes for students with hiv including physiological, psychosocial, and neurocognitive effects are well documented within medical literature and delineate a clear path for educational research and practice specific to this population, whereby it becomes indefensible to produce equity documents that do little more than provide tokenistic platitudes for benefitting “all” students, when the reality is that the needs of diverse student groups cannot be addressed by blanket approaches. although very few studies and far fewer policy or practice documents have explicitly sought students’ perspectives, the voices of children with hiv are even less audible, and yet who better to provide the most valid interpretation of the beliefs and practices affecting their mathematics education than the individuals who have experienced the lived reality of marginalization and oppression. an argument, albeit deficient, for excluding the voice of students in these documents posits that researchers and practitioners as adults are knowledgeable of students’ perspectives, not only having a wealth of educational, sociological, and psychological literature at their disposal, but because they too were students (i.e., recalling their lived experiences). many counter arguments can attack this premise—chief among them, the vast majority of researchers and practitioners were not once students with hiv. that we as researchers, teachers, and policy makers have no basis for understanding what it is to live with hiv and to receive mathematics instruction concurrently, is sufficient reason alone for seeking the voices of students with hiv. an equity document that contains the perspectives of students with hiv with respect to mathematics education would serve to help develop a set of practices and polices that are appropriate to the students’ experiences and circumstances and, importantly, to create a record of the multiplicity and diversity of students with hiv. curriculum the aforementioned beliefs and practices related to access and equity are not discrete from concerns regarding high-quality mathematics curriculum. standards documents are often interpreted as a curricular checklist as opposed to a set of expectations guiding the teaching and learning of mathematics. this checklist mentality manifests as mere surface adherence to impersonal and decontextualized mathematical objectives rather than serving to promote deep conceptual understanding of key mathematics concepts. this manifestation is arguably a problem affecting most students, but students with hiv experience an even greater hardship when this teaching mentality is transferred into the hospital/homebound setting or reflects upon their experiences in regular mathematics classrooms. mathematics curriculum is highly fragmented for students with hiv in a hospi nickels et al. commentary journal of urban mathematics education vol. 10, no. 2 46 tal/homebound or bi-enrollment setting (i.e., receiving instruction in both the hospital/home setting and in the regular classroom) because often no specific guidelines for providing homebound instruction exist and tradition dictates the hospital/homebound instructor is given the curriculum from the student’s local school (daly-rooney & denny, 1991). hospital/homebound teachers may or may not have any familiarity with the curriculum, content, or the required materials needed to enact the curriculum; and vice versa, the regular classroom teacher may lack the required knowledge of the student’s disease, symptoms, and treatment to modify the curriculum to his or her academic, physiological, neurocognitive, and sociological needs (irwin & elam, 2011). often, all parties involved “remedy” this situation by swapping out the usual curriculum for worksheet packets that isolate problem types and provide repeated skill practice outside of meaningful contexts (nickels & cullen, 2017a). even if the student with hiv mostly remains in his or her classroom utilizing the regular curriculum, frequent absences or lack of facilities and resources make it difficult to keep up with the learning expectations of the teacher and school and district imposed pacing guides (fowler, johnson, & atkinson, 1985; thies, 1999). equally troubling is the value placed on the transience of childhood and adolescence—in which young people are nurtured and educated towards adulthood with little appreciation for the individual’s immediate being (nickels & cullen, 2017b). many students with hiv are thus left in a state of educational uncertainty, receiving little to no access to mathematics curriculum while their chances of entering adulthood wait to be seen (nickels & cullen, 2017a). this emphasis on adulthood supports the narrative of mathematics education as a market-driven practice; however, mathematics education has significance above and beyond socio-economic advancement. mathematics, many have argued (e.g., ernest, 2010; gutiérrez, 2012; gutstein, 2005; skovsmose, 1994), allows us to understand our natural and social worlds, our relationship to both worlds and to others within them. it advances both our human and social capacities. in relation to their disease, mathematics can allow children with hiv to regain control over their quality of life and the circumstances surrounding their treatment including leveraging powered positions that contribute to differential access to valued resources. this view of mathematics stands in direct opposition to the meritocratic culture of formal mathematics instruction. here the value lies in a student’s ability to learn and make sense of the real world and the opportunity he or she is then given to live authentically, creatively, and happily. professionalism in large part, policy and government regulations concerning the education of students with hiv have constrained professionalism in the practice of teaching these individuals (bogden, fraser, vega-matos, & ascroft, 1996; newhart, olson, warschauer, & eccles, 2017). little attention is paid to persons who volunteer or are nickels et al. commentary journal of urban mathematics education vol. 10, no. 2 47 hired to provide mathematics instruction to children with critical illness (nickels & bello, 2017; tseng & pluta, 2016). many individuals who are drawn to hospital/homebound programs out of compassion or religious affiliation believe that “feel good” activities should take precedence over core academic content or best pedagogical practices (richards, 1957/2013). this altruistic spirit is thus favored in teacher selection over actual merit (e.g., certification, experience; nickels & bello, 2017; tseng & pluta, 2016). however, hiv stigma often guarantees that this altruism is not extended to students with hiv (i.e., children with non-transmissible disease are favored or served first or only; synder, omoto, & crain, 1999). much too frequently schools are also complicit in providing a second-class education for students with hiv. hiring highly qualified teachers is expensive and thus deemed unaffordable for many school districts and hospitals that receive little to no state aid for hospital/homebound programs (tseng & pluta, 2016). the lack of qualified teachers also means that, either consciously or not, schools and hospital/homebound programs offer curricula based on the competency level of the teacher, not necessarily the curricula best fitting the needs of the student (ware, 1990). thus, the variability in teachers’ professional backgrounds and personal beliefs are not conducive to creating a community of mathematics educators for students with hiv that, “hold themselves and their colleagues accountable for the mathematical success of every student and for their personal and collective professional growth toward effective teaching and learning of mathematics” (nctm, 2014b, p. 5). teachers who are characterized as highly qualified within the regular classroom report feeling unprepared, unsupported, and fearful to teach children with hiv (collins, 2005; fishbein, 2003). these concerns are coupled with the conspicuous heterosexism of public schools. heterosexism and heterocentrism are communicated through teachers’ actions and their acceptance or lack thereof for different populations of students, most notably for students with hiv, a disease publicly focused on sex (collins, 2005). thus, even in the best-case scenario in which a student with hiv is positioned to receive mathematics instruction from a certified and competent teacher, he or she may presumably remain at a disadvantage and subject to further marginalization. periodic migration between hospital and home and frequent absenteeism also leave students with hiv poorly served by educational interventionists, programs, and facilities, placing an undue burden on the parents to step in as their child’s primary educator (newhart et al., 2017; nickels & bello, 2017). current policy and practices of mathematics education for students with hiv ignore the structural barriers that nuclear families face in the wake of a critically ill student’s diagnosis and treatment such as poverty, privilege, and lack of access to resources students need to achieve academically (bessell, 2001). the prospect of teaching mathematics to their child leaves many parents needing to find their bearings and reconstruct their own mathematical understanding in the context of a new, unanticipated scenario nickels et al. commentary journal of urban mathematics education vol. 10, no. 2 48 (nickels & bello, 2017; nickels & cullen, 2017a). parents who lack sufficient mathematical content knowledge or disposition for teaching may neglect engaging their child in mathematical work or employ dysfunctional pedagogy to teach standard algorithms and procedural fluency (nickels & bello, 2017). moving forward currently, little is being done to preserve the civil rights of students with hiv, including that of a meaningful and appropriate mathematics education (nickels & cullen, 2017a). similar to martin (2015) and meyer (2016), my purpose in writing this commentary is to further problematize nctm’s (2014b) principles to actions and other equity oriented policy documents and statements in service of students who arguably face some of the greatest educational disparities. if equity focused organizations mean to keep true to their assertion that every student has the right to equitable opportunities to learn mathematics, then expanding what is meant by “every” must be addressed. although many readers of jume may agree to this avowal, we must take seriously that the onus does not lie alone with organizations such as amte, nctm, ncsm, and todos. this reimagining must be a collective project. the question we face is—how and when? references association of mathematics teacher educators. 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(1999). identifying the educational implications of chronic illness in school children. journal of school health, 69(10), 392–397. http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/302/198 http://www.todos-math.org/socialjustice https://www.nctm.org/uploadedfiles/standards_and_positions/position_statements/opportunity%20gap.pdf https://www.nctm.org/uploadedfiles/standards_and_positions/position_statements/opportunity%20gap.pdf http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/308/193 nickels et al. commentary journal of urban mathematics education vol. 10, no. 2 51 tseng, m., & pluta, r. m. (2016). educating students with chronic illness: how the old service model fails. in m. gordon (ed.), challenges surrounding the education of children with chronic diseases (pp. 227–246). hershey, pa: information science reference. unaids. (2016). global aids update 2016. retrieved from http://www.unaids.org/sites/default/files/media_asset/global-aids-update-2016_en.pdf u.s. department of education. (2007). 25th annual report to congress on the implementation of the individuals with disabilities education act. washington, dc: u.s. department of education. ware, a. m. (1990). the influence of state reform in homebound/hospital instruction in the state of georgia (doctoral dissertation). clarke atlanta university, atlanta, ga. winkle-wagner, r., hinderliter ortloff, d., & hunter, c. (2009). the not-center? the margins and educational research. in r. winkle-wagner, c. hunter, & d. hinderliter ortloff (eds.), bridging the gap between theory and practice in educational research (pp. 1–14). new york, ny: palgrave macmillan. http://www.unaids.org/sites/default/files/media_asset/global-aids-update-2016_en.pdf journal of urban mathematics education december 2016, vol. 9, no. 2, pp. 1–4 ©jume. http://education.gsu.edu/jume david w. stinson is an associate professor of mathematics education in the department of middle and secondary education in the college of education and human development, at georgia state university, p.o. box 3978, atlanta, ga, 30303; e-mail: dstinson@gsu.edu. his research interests include exploring socio-cultural, -historical, and -political aspects of mathematics and mathematics teaching and learning from a critical postmodern theoretical (and methodological) perspective. he is a co-founder and current editor in chief of the journal of urban mathematics education. editorial absence of diversity in collegiate upper-level mathematics classrooms: perpetuating the “white male math myth” david w. stinson georgia state university atherine johnson, dorothy vaughan, and mary jackson are the names of three black women who are unfortunately relatively unknown—even though katherine was awarded the presidential medal of freedom by president barak obama in 2015, the highest civilian award in the united states. but familiarity with these three women’s names and, more importantly, with these three (and others) women’s stories will soon change when the movie hidden figures (gigliotti, chernin, topping, williams, & melfi & melfi, 2016) has its wide release on january 6, 2017. the movie, distributed by 20th century fox, features award-winning cast and crew members and producers, and shows promise in being a box office success and oscar contender (see buckley, 2016). the biographical drama is based on the nonfiction book, of the same name, written by first-time book author margot lee shetterly (2016). the dustcover of the william morrow harpercollins imprint reads: during world war ii, america’s fledgling aeronautics industry hired black female mathematicians to fill a labor shortage. these “human computers” stayed on to work for nasa and made sure america won the space race. they fought for their country’s future, and for their share of the american dream. this is their untold story. as a mathematics educator and researcher who works at deconstructing the “white male math myth” discourse (see, e.g., stinson, 2013), i am exhilarated when hidden—or more aptly, too often erased—histories such as these are brought to light. but sadly, such histories are a double-edged sword. cutting one way, these stories do assist in deconstructing the white male math myth discourse— mathematics is not just for white (and asian) boys and men. cutting the other way, however, these histories too often become the “exception story,” so to speak. that is to say, after learning of such histories as documented in hidden figures it is too easy for people to walk away and say: “yes, but they’re the exception, not just any girl (or not just any black or brown kid) can be a mathematical wiz. after all, k http://education.gsu.edu/jume mailto:dstinson@gsu.edu http://www.foxmovies.com/movies/hidden-figures stinson editorial journal of urban mathematics education vol. 9, no. 2 2 mathematics really is a discipline for, you know, white guys.” thinking such as this is too often undergirded by misogynist (see, e.g., mendick, 2006) and white supremacist (see, e.g., martin, 2013) ideologies. nonetheless, in the united states, and throughout most of the western world, people continue to imagine the mathematician as a white, middle-aged, balding or wild-haired man (see picker & berry, 2000). the einstein-ish silhouette readily comes to mind, which continues to position mathematics as the discipline primarily engaged by elite white men (leonard, davila, & stinson, 2012). the dominance of this discourse is so strong that when professor erica walker was casually speaking to another colleague in educational research about her then-forthcoming book beyond banneker: black mathematicians and the paths to excellence (walker, 2014), he puzzlingly asked: “are there any [black mathematicians]?” (p. x) unfortunately, the near absence of female and black and brown students in upper-level undergraduate and graduate mathematics courses continues to perpetuate the white male math myth discourse; keeping it alive, even for educators (and others) who should know better. the near absence of student diversity in terms of gender and race in upperlevel undergraduate and graduate mathematics courses is well documented in the field. even so, the number of female mathematics/statistics majors grew steadily, for the most part, throughout the 1980s and 90s reaching a high of 48.0% of bachelor’s degrees awarded in mathematics to women in 2001 (national science board [nsb], 2016). since then, however, there has been a steady, although not steep, decline. women earned only 43.1% of the mathematics bachelor’s degrees in 2013, which is the last year data are available (nsb, 2016). in comparison, women earned 57.3% of bachelor’s degrees in all fields, and 50.3% of science and engineering degrees in 2013 (nsb, 2016). women’s strongest showing in science and engineering degrees was in the biological sciences at 59.2% (nsb, 2016). at the graduate level, women were awarded 39.8% of the master’s degrees in mathematics/statistics and 29.1% of the doctoral degrees (nsb, 2016). women, in comparison, were awarded nearly half of the doctoral degrees in all fields in 2013 (nsb, 2016). taking these percentages to the classroom level means that a typical 4000/6000 level mathematics course of about 20 students will have 7 or 8 female students, with only 2 or 3 female students in a typical 8000/9000 level mathematics course of about 10 students. the near absence of student diversity becomes even more stark when accounting for black and brown students in upper-level undergraduate and graduate mathematics courses. in 2013, only 4.7% and 7.0% of the bachelor’s degrees in mathematics/statistics were awarded to black and brown students, respectively (nsb, 2016). but black and brown students represented 9.6% and 10.5%, respectively, of bachelor’s degrees awarded in all fields in 2013 (nsb, 2016). there were 2.8% black and 4.0% brown students who earned master’s degrees in mathematics stinson editorial journal of urban mathematics education vol. 9, no. 2 3 in 2013, and at the doctoral level the percentages were 1.5% and 1.9%, respectively (nsb, 2016). again, taking these percentages to the classroom level means that the typical 4000/6000 level (about 20 students) or 8000/9000 level (about 10 students) mathematics course is more times than not completely absent of black or brown students. given the ever-changing gender and racial demographics of u.s. colleges and universities (williams, 2014), administrators and faculty members in mathematics departments must begin to take the absence of student diversity in upper-level courses seriously and to develop plans of action. if these mathematicians are genuinely interested in “unleashing the possibilities,” so to speak, of the socially constructed discipline so named mathematics (ernest, 1998), they must create mathematics departments and classrooms that welcome the mathematical ideas, interests, and brilliance of every student—no matter their gender or race. shetterly’s (2016) historical account gives us all pause to think: if not for the women “human computers”1 at nasa, would it be stars and stripes flying on the moon or a hammer and sickle? in terms of racial diversity, administrators and faculty members in mathematics departments can no longer afford to practice a form of dysconscious racism: “an uncritical habit of mind (including perceptions, attitudes, assumptions, and beliefs) that justifies inequity and exploitation by accepting the existing order of things as given” (king, 1991, p. 135). refusing to accept the existing order of things as given will become increasingly important as more colleges and universities move introductory mathematics courses (e.g., college algebra) out of mathematics departments, allowing other departments to design and teach introductory “quantitative literacy” courses (joselow, 2016). to remain financially viable (in the lack of action of other more humane and just appeals to diversify), mathematics departments must begin to get their “fair share,” so to speak, of the students on u.s. colleges and universities campuses in their upper-level courses. given the decrease in the percentage of white men on campuses, perpetuating the white male math myth discourse will no longer “keep the lights on.” besides, just think of all the new mathematical possibilities when more (and different) human talent is given access to and invited to the discussions. —yes, just any kid can learn (and be taught) to be a mathematical whiz! 1 shetterly (2016) notes: and while the black women are the most hidden of the mathematicians who worked at the naca, the national advisory committee for aeronautics, and later at nasa, they were not sitting alone in the shadows: the white women who made up the majority of langley’s computing workforce over the years have hardly been recognized for their contributions to the agency’s long-term success. 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(2014, september 22). college of tomorrow: the changing demographics of the student body. us news & world report. retrieved from http://www.usnews.com/news/collegeof-tomorrow/articles/2014/09/22/college-of-tomorrow-the-changing-demographics-of-thestudent-body http://www.nytimes.com/2016/05/22/movies/taraji-p-henson-octavia-spencer-hidden-figures-rocket-science-and-race.html?_r=1 http://www.nytimes.com/2016/05/22/movies/taraji-p-henson-octavia-spencer-hidden-figures-rocket-science-and-race.html?_r=1 https://www.insidehighered.com/news/2016/07/06/michigan-state-drops-college-algebra-requirement https://www.insidehighered.com/news/2016/07/06/michigan-state-drops-college-algebra-requirement http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/168/105 https://www.nsf.gov/statistics/2016/nsb20161/#/report http://www.usnews.com/news/college-of-tomorrow/articles/2014/09/22/college-of-tomorrow-the-changing-demographics-of-the-student-body http://www.usnews.com/news/college-of-tomorrow/articles/2014/09/22/college-of-tomorrow-the-changing-demographics-of-the-student-body http://www.usnews.com/news/college-of-tomorrow/articles/2014/09/22/college-of-tomorrow-the-changing-demographics-of-the-student-body microsoft word 390-article text no abstract-2306-1-6-20210225-1 (proof 2).docx journal of urban mathematics education december 2021, vol. 14, no. 2, pp. 71–104 ©jume. https://journals.tdl.org/jume joseph dinapoli is an assistant professor in the department of mathematics at montclair state university, 1 normal avenue, montclair, nj 07043; email: dinapolij@montclair.edu. his research focuses on mathematics education, specifically perseverance in problem solving, educational technology, and professional development. hector morales, jr. is an assistant professor in the department of teacher education at northeastern illinois university, lwh 3010, 5500 north st. louis avenue, chicago, il 60625; email: h-morales3@neiu.edu. his research focuses on the teaching and learning of mathematics with multilingual learners, mathematical discourse, and equity issues in mathematics education. translanguaging to persevere is key for latinx bilinguals’ mathematical success joseph dinapoli montclair state university hector morales jr. northeastern illinois university recent reform efforts state the importance of providing students with opportunities to persevere with challenging mathematics to make meaning. we posit translanguaging practice as a vital option by which latinx bilingual students can sustain collective perseverance during problem solving. in this paper, we employ a constant-comparative overlay analysis to simultaneously study the discursive translanguaging and perseverance practices of latinx bilingual students and the corresponding classroom supports. we observed collaborative problem solving in two classrooms of 12th-grade latinx bilinguals working to make sense of an exponential function and the involvement of the same monolingual english-speaking teacher. working within similar, supportive classroom environments, we describe how one group of students spontaneously and dialogically leveraged communicative resources to help persevere with in-the-moment obstacles, while another group of students worked together across languages but did not engage in a translanguaging mathematical practice to persevere. we suggest that only establishing a classroom environment conducive for translanguaging and perseverance practice is insufficient and that teachers should not solely rely on students spontaneously engaging in these practices. to complement this environment, we recommend specific teacher moves and scaffolds that could help latinx bilingual students initiate collaborative translanguaging and support their ongoing perseverance to make meaning of mathematics. keywords: exponential functions, latinx multilingual learners, mathematics discourse, perseverance, translanguaging dinapoli & morales translanguaging to persevere is key journal of urban mathematics education vol. 14, no. 2 72 here is a pressing need to rehumanize mathematics education for latinx bilingual1 students in the united states. the act of rehumanizing fosters respect and dignity through privileging the viewpoint and experiences of the latinx student and the ways in which the student perseveres to develop personal understandings through their own disciplinary perspective on mathematics. in this context, a rehumanizing perspective positions a latinx student as central to the meaning-making process while engaging in the practice of doing mathematics (gutiérrez, 2018). a rehumanizing perspective rejects the notion that bilingual students must solely reproduce the teacher’s idea of productive mathematical activity (lawler, 2016; matthews, 2018). instead, a rehumanizing perspective adopts a social-cultural lens on learning, which positions bilinguals as agents in their language use capable of interacting and communicating while working collaboratively with a challenging mathematical task (cross et al., 2012; khisty & chval, 2002; vomvoridi-ivanović, 2012; waddell, 2010). in stark contrast to this rehumanizing perspective is a deficit perspective often applied to latinx bilingual learners. this deficit perspective has been widespread in education around the united states and suggests that bilingual students are passive recipients of mathematical knowledge and slowed by language barriers (razfar et al., 2011; rubel, 2017). educators who ascribe to this model tend to employ classroom practices that marginalize rather than privilege linguistic, social, and cultural capital, thus creating dehumanizing school norms (langer-osuna et al., 2016). these perspectives have persisted for years (cf. moll, 2001) and continue to create a distance between latinx students’ language, cultural knowledge, and opportunities to develop mathematical meaning. as mathematics educators, we are interested in researching ways to support latinx bilinguals in leveraging their bilingualism to persevere to make meaning of mathematical ideas. developing personal mathematical meanings requires student perseverance and teacher support. this notion of student perseverance exists in the moment at specific times during problem solving when productive struggle is required. productive struggle, or grappling with mathematical ideas that are familiar but not yet well formed (hiebert & grouws, 2007), is necessary to overcome obstacles along the path toward developing conceptual knowledge. however, not all struggle is guaranteed to be productive. students can struggle unproductively when working without a teacher support system in place to offer feedback and guidance. such unproductive struggle can 1 we use the term latinx bilinguals to refer to our research participants to encompass a more inclusive identifier that reflects the diversity of gender, sexuality, gender-nonconforming, and trans people who identify with a latin descent. we also use this term to bring attention to the complexity of the intersectionality of language, culture, and identity commonly associated with speaking named languages (e.g., spanish and english). we use the term bilinguals to encompass multilingual students who have access to or are in the process of acquiring access to multiple languages. we acknowledge that researchers should be purposeful about the labels they use and contribute to the conversation while paying attention to the political and social nature of these important conversations. t dinapoli & morales translanguaging to persevere is key journal of urban mathematics education vol. 14, no. 2 73 spur frustration, anxiety, and discouragement toward perseverance in problem solving (dinapoli, 2018; star, 2015). perseverance is so vital to the learning process that guiding texts like principles to actions: ensuring mathematical success for all (national council of teachers of mathematics [nctm], 2014) have asserted that all students must have access, opportunity, and support to persevere in their study of mathematics. further, the common core state standards initiative (ccssi, 2010) has explicitly noted the importance of cultivating perseverance for all students to encourage mathematical expertise. however, there is a concern that mathematics reforms have largely ignored the needs of latinx students (martin, 2015; moschkovich, 2000, 2015). supporting students to persevere to learn mathematics with understanding has often involved more social and verbal activities that require students and teachers to engage in more substantive mathematical discussions and collective practice (bass & ball, 2015). such communicative practice dangerously assumes a shared collaborative language between students and teachers (chval & khisty, 2009) and alternately requires careful consideration of all language repertoires in the mathematics classroom to truly support latinx bilingual students’ perseverance. while this is a generalized perspective of classrooms with latinx bilingual students, it nevertheless raises questions about marginalization and undervaluing latinx bilingual students’ learning resources in mathematics. much of the research concerning latinx bilingual students learning mathematics has focused on only what students cannot do and largely ignores what students can do. this kind of deficit perspective is often manifested by focusing on the relationship between spanish-speaking students’ struggles with english and their difficulties in learning mathematics (e.g., langer-osuna et al., 2016). this deficit perspective has also manifested by detailing the barriers faced by latinx bilinguals learning mathematics across the spanish and english languages (e.g., macswan & faltis, 2019). deficit perspectives emerge when the affordances of bilinguals’ linguistic resources are ignored in the classroom while only privileging the dominant school language (langer-osuna et al., 2016). garcía et al. (2017) argued that this view of language and academic discourse in schools acts as a hurdle to knowledge (e.g., mathematical knowledge) and only empowers those students whose linguistic repertoire mirrors the dominant school language. combatting this deficit perspective, research has shown that latinx students can use a wide variety of cultural resources to construct, negotiate, and communicate (verbally or in writing) about mathematics (chval & khisty, 2009; morales, 2012). cultural resources include linguistic scaffolds like a mathematics register and mathematical discourse, everyday experiences, life histories, and community funds of knowledge (celedón-pattichis, 2003; gutiérrez, 2002, 2017; moll, 2001; morales et al., 2011; moschkovich, 2015). more recent research has privileged the complex personal perspective of the latinx bilingual mathematics student to demonstrate that involving family and/or community members’ knowledge and experiences in school mathematics can play a critical role in a child’s dinapoli & morales translanguaging to persevere is key journal of urban mathematics education vol. 14, no. 2 74 learning (e.g., maldonado et al., 2018; mazzanti valencia & allexsaht-snider, 2018). in this paper, we extended prior research through our investigation of the nature of latinx bilingual students’ perseverance in problem solving and how it can be encouraged and supported. to do so, we employed the construct of translanguaging in our conceptual framework. translanguaging is a theory of language that shifts ideologies from language separation perspectives to one that values the complex and interrelated communicative practice that makes up bilingual students’ linguistic repertoires (cenoz, 2017). translanguaging illuminates the ways in which latinx bilinguals can be empowered by their linguistic and cultural resources and how leveraging such resources can inform persevering to make meaning of mathematics. as such, this study is grounded in the conceptual perspectives of both translanguaging and perseverance practice. frameworks for translanguaging and perseverance practice we have drawn on the concept of translanguaging to explore and reconceptualize bilingualism as an empowering and liberating practice. translanguaging is an ideological stance on bilingualism that goes beyond transitioning bilinguals to the dominant school language (garcía et al, 2017). garcía (2017) defined translanguaging as the authentic meaning-making practices of bilinguals; she posited, “…speakers use their languaging, bodies, multimodal resources, tools and artifacts in dynamically entangled, interconnected and coordinated ways to make meaning” (p. 258). thus, the act of translanguaging looks and sounds like bilinguals seamlessly leveraging two languages and linguistic features to make sense of concepts, contexts, relationships, and representations. adopting a translanguaging stance shifts us ideologically toward viewing bilingualism holistically and understanding that bilinguals have a single language repertoire on which they draw as a resource for learning. this is in contrast to the notion that bilinguals simply shift between two language repertoires as if their language use is mutually exclusive (cenoz, 2017). instead, “the act of translanguaging is in itself transformative, having the potential to infuse creative bilingual meaning into utterances” (garcía et al., 2017, p. 20). in other words, the symbiosis between or across languages becomes a mediating tool that bilinguals can use as an affordance to make meanings. some researchers have used a translanguaging framework to better understand language practices of multilingual persons in mathematics classrooms. for instance, latinx kindergarteners’ use of english and spanish occurred simultaneously as a way to make mathematical meanings (mazzanti valencia & allexsaht-snider, 2018). their meanings were not limited to just linguistic accomplishments but also used symbolic representations of numbers and visual models of the number problems to leverage these students’ communicative resources. other studies have focused on dinapoli & morales translanguaging to persevere is key journal of urban mathematics education vol. 14, no. 2 75 purposeful translanguaging of latinx bilingual mathematics students. morales and dinapoli (2018) found that when students were given the freedom to explore mathematics via their dynamic bilingualism, they were able to dialogically leverage communicative resources to help them overcome in-the-moment obstacles and make meaning. illuminating these translanguaging practices repositions these students as competent problem solvers and agents of their own learning while leveraging bilingual identities as learners of mathematics, which reflects a rehumanizing perspective for all students in the group. pedagogies that support translanguaging practice mathematics teachers can enact translanguaging pedagogies in their classrooms to help create powerful spaces in which latinx bilinguals can make meaning. for instance, consider ramirez and celedón-pattichis’ (2012) five guiding principles for teaching mathematics to latinx bilinguals. these principles articulated effective ways to engage latinx bilinguals to persevere with rigorous mathematics while engaging in translanguaging mathematical practice. ramirez and celedón-pattichis first recommend that teachers enact challenging mathematical tasks that provide students multiple entry points but non-obvious solution paths. within engagement with such tasks, teachers can offer mathematical tools and even model their use as a resource. while these pedagogical recommendations allow for and support student meaning-making with mathematics, ramirez and celedón-pattichis also urge for cultural and linguistic teaching moves to support latinx bilingual students. teachers can help students leverage their cultural and linguistic differences as intellectual resources, such as by approaching a mathematical task in their native language to initiate engagement. teachers can also offer support for learning english while learning mathematics by juxtaposing certain mathematical terms in both english and spanish, in the context of latinx bilingual students’ problem solving. in addition, ramirez and celedón-pattichis also urge for teachers to build a linguistically sensitive social environment by celebrating bilingualism and removing language separation. some ways teachers can do this is by encouraging students to present their work in several languages, using translators to help facilitate whole-class bilingual discussions, and by celebrating how bilingualism can be a mathematical resource used to better understand concepts and connections. teachers who enact translanguaging pedagogies challenge the deficit perspectives commonly associated with language separation (moschkovich, 2019). these teachers can leverage students’ linguistic repertoires to engage with complex mathematics content and texts, strengthen students’ linguistic repertoires in mathematical contexts, draw on students’ bilingualism for the purpose of expanding their ways of knowing, and support students’ bilingual identities that counter english-only ideologies (garcía et al., 2017). in conjunction with ramirez and celedón-pattichis’ (2012) call for translanguaging pedagogies, researchers (e.g., garcía-mateus & palmer, dinapoli & morales translanguaging to persevere is key journal of urban mathematics education vol. 14, no. 2 76 2017; maldonado et al., 2018) have urged teachers to develop a translanguaging stance to help enact these practices in their classrooms. an authentic embrace of a translanguaging stance involves the teacher’s belief that bilingual students have one holistic language repertoire on which they draw, not two mutually exclusive repertoires. when teachers create a climate where translanguaging is embraced, it can help reconceptualize mathematics learning and language practices of latinx bilinguals. still, more research is needed in bilingual mathematics classrooms to study how flexibility and fluidity with students’ translanguaging repertoires is centrally aligned to mathematical meaning-making practices. translanguaging mathematical practice the construct of translanguaging mathematical practice describes the ways in which flexibility and fluidity with students’ translanguaging repertoires is central to meaning-making practices in mathematics contexts. translanguaging mathematical practice captures how bilingual students holistically use their linguistic, multimodal, and mathematical artifacts repertoires to make meanings as they persevere with challenging mathematical ideas. translanguaging mathematical practice was informed by maldonado et al.’s (2018) work on a translanguaging stance. we extended garcía’s (2017) translanguaging practice framework to accommodate how bilingual students engage in mathematics discourse, including repertoires like everyday language, mathematics register, and mathematical artifacts, such as visual representations and mathematical notations (avalos et al., 2018; o’brien & long, 2012). bilingual students do not learn best through passive, one-dimensional mathematics instruction (moschkovich, 2015). instead, a multimodal approach can incorporate several dimensions of a powerful learning environment. such an approach incorporates mathematical symbols and visual images to enhance the view of understanding language relative to teaching and learning mathematics. the meanings of mathematical symbols are encoded with precise and condensed representations of mathematical ideas. similarly, visual images help us represent mathematical ideas and apply knowledge (o’halloran, 2015). o’halloran (2015) argued for a multimodal literacy, “a literacy which extends beyond language to include mathematical symbolic notation and mathematical images, and the relations between and integration of these three resources” (p. 73). translanguaging mathematical practice describes how bilinguals are engaged in this kind of learning environment. within the framework of translanguaging mathematical practice (see figure 1), students engaging within the mathematics discourse deploy fluid movement between mathematical and everyday speaking across spanish (s) and english (e) by using everyday linguistic features (ers and ere) and mathematics register resources (mrs and mre) in dialogically entangled ways. additionally, students draw on non-verbal mathematical representations, including visuals (mv), notations (mn), and gestures (mg), to complement their mathematics and everyday register. in translanguaging mathematical practice, dinapoli & morales translanguaging to persevere is key journal of urban mathematics education vol. 14, no. 2 77 these linguistic and semiotic resources work together with the intention to help bilingual students persevere with challenging ideas to make meaning. next, we introduce the conceptual perspective of perseverance in problem solving, which is an ideal outcome of bilingual students engaging in translanguaging mathematical practice. figure 1. a framework for translanguaging mathematical practice perseverance in problem solving in the context of working on a challenging mathematical task, perseverance is defined as “initiating and sustaining in-the-moment productive struggle in the face of one or more obstacles, setbacks, or discouragements” (dinapoli, 2018, p. 890). perseverance during problem solving is especially important for learning mathematics because students develop conceptual understanding as they wrestle with ideas that are not immediately apparent (hiebert & grouws, 2007). the self-regulatory actions amidst uncertainty as students navigate an obstacle during problem solving helps describe perseverance, and analysis of such engagement should consider the ways in which students first explore an uncertain mathematical situation and how (if necessary) they amend their initial plan to find a way to continue to make progress toward building understanding. perseverance in problem solving can be operationalized using the three-phase perseverance framework (3pp; dinapoli, 2018; see table 1 for a description of the framework and corresponding analytic codes for each component). the 3pp is an analytical perspective by which perseverance can be qualitatively described and measured. the framework reflects perspectives of concept (middleton et al., 2015; sengupta-irving & agarwal, 2017), problem-solving actions (pólya, 1971; schoenfeld & sloane, 2016; silver, 2013), self-regulation (carver & scheier, 2001; zimmerman & schunk, 2011), and making and recognizing mathematical progress (gresalfi & barnes, 2015). the 3pp has been used in several studies across different contexts to make explicit the perseverance process for an individual and/or group of dinapoli & morales translanguaging to persevere is key journal of urban mathematics education vol. 14, no. 2 78 students working towards a mathematical goal. findings have helped inform several outcomes relevant to mathematics education, including productive strategies for student disposition development (dinapoli, 2019), support systems for impasse learning (dinapoli, 2018), and making learning visible in non-formal science, technology, engineering, and mathematics learning environments (satyam et al., 2021). using the 3pp focuses the analysis on three chronological phases of perseverance: the entrance phase, the initial attempt phase, and the additional attempt phase. the entrance phase was designed to consider first if the task at hand warranted perseverance for students. essentially, the analysis of this phase establishes the appropriateness of a task for perseverance analysis with a particular group. for perseverance to be necessary, students must have understood the objectives of the task (c0) but did not immediately know how to achieve them (io-0). next, the initial attempt phase was designed to consider the ways in which students initiated and sustained productive struggle. because the students did not immediately know a solution pathway, evidence of perseverance includes deciding to engage with the task at all (ie-1). if the students decided to pursue solving the problem, evidence of perseverance includes the use of a problem-solving strategy to diligently explore the uncertain nature of the mathematical situation (se-1). as a result of these diligent efforts, evidence of perseverance includes mathematical progress that was made toward better understanding the mathematical relationships or the problem being solved outright (oe-1). last, the additional attempt phase was designed to consider the ways in which students reinitiated and resustained productive struggle if they encountered a perceived setback as a result of their initial attempt. to enter this phase (marking the end of a first attempt and the beginning of a second attempt), the group must first have encountered a perceived setback. evidence of a perceived setback occurs when a group collectively affirmed that they were substantially stuck and unsure how to continue with the task (vanlehn et al., 2003). at this point, evidence of perseverance includes deciding to reengage with the task using a different strategy—one that was not used during the first attempt (ie-2). assuming the students decided to pursue solving the problem with a new plan, additional evidence of perseverance includes diligent exploration of the mathematical situation with the new strategy (se-2) and new progress being made toward better understanding the mathematics involved in the task or the problem being solved outright (oe-2). students could continue on to start a new additional attempt phase(s) if their efforts produce another perceived setback(s) and require initiating a new effort(s) with a new strategy(ies). dinapoli & morales translanguaging to persevere is key journal of urban mathematics education vol. 14, no. 2 79 table 1 three-phase perseverance framework entrance phase clarity (c-0) objectives were understood initial obstacle (io-0) expressed or implied that a solution pathway was not immediately apparent initial attempt phase initiated effort (ie-1) expressed intent to engage with task sustained effort (se-1) used problem-solving heuristics to explore task outcome of effort (oe-1) made mathematical progress toward a solution additional attempt phase (after n perceived setback(s)) re-initiated effort (ie-n+1) expressed intent to reengage with task re-sustained effort (se-n+1) used problem-solving heuristics to explore task outcome of effort (oe-n+1) made additional mathematical progress toward a solution n = the number of perceived setbacks because student perseverance is vital for learning mathematics with understanding, it is important for educators to encourage perseverance via their classroom practice. as such, nctm (2014) called for teaching practices that “consistently provide students, individually and collectively, with opportunities and supports to engage in productive struggle as they grapple with mathematical ideas and relationships” (p. 48). one such practice specific to working with latinx bilingual students considers the structure of tasks. for latinx bilingual students, mathematical tasks necessitating perseverance should be complex enough such that students engage at all levels of language proficiency to help make their own connections (driscoll et al., 2012) yet invite engagement with multiple entry points and resources. however, it is still unclear how perseverance might manifest in students working across two languages. further, outside of task design, more research is needed to consider the ways in which teachers can provide opportunities for latinx bilingual students to leverage or capitalize their communicative and linguistic repertoires to build understandings. research questions in this paper, we addressed calls to action to examine translanguaging and perseverance practice in classrooms largely from the student point of view (e.g., maldonado et al., 2018; martinez et al., 2017; nctm, 2014). our research questions are grounded in the belief that translanguaging mathematical practice can encourage dinapoli & morales translanguaging to persevere is key journal of urban mathematics education vol. 14, no. 2 80 perseverance practice in latinx bilinguals when given the right pedagogical conditions. we carefully considered if and how latinx bilingual students used translanguaging practice while collaborating (as per maldonado et al., 2018; martinez et al., 2017) and if and how these students persevered with challenging mathematical ideas (as per ntcm, 2014) as they worked together. by studying these practices closely, we used inductive reasoning to discern important tenets of translanguaging pedagogy that can support these practices in mathematics classrooms. the research questions used to guide this study were as follows: 1. in what ways, if any, do secondary latinx bilingual students demonstrate translanguaging mathematical practice while collaboratively engaging with a mathematical task? 2. in what ways, if any, do secondary latinx bilingual students demonstrate perseverance practice while collaboratively engaging with a mathematical task? 3. by comparing the evidence of latinx bilingual students’ translanguaging mathematics and perseverance practice, what pedagogical insights can we glean the about creating productive (or unproductive) learning environments for latinx bilingual students? methods our observations and interviews come from a larger study of 12th-grade latinx students in mathematics classes in a school with an 80% latinx student population. each class was using year 4 of the interactive mathematics program (imp; fendel et al., 2015b) curriculum, which focused on topics such as statistical sampling, systems of linear equations and inequalities, geometric transformations, integral and derivative concepts, and deeper exploration of previously encountered mathematical ideas. we studied two 12th-grade mathematics classes taught by the same teacher. participants we focused on two purposeful groups of students—one group from each class. we chose to focus on these particular groups because they had collaborated together for the majority of the school year, were bilingual, had similar past experiences with mathematics, and demonstrated average achievement in mathematics as informed by their past grades and standardized test scores. the first group of students consisted of carina, jessica, elena, and ines (pseudonyms). these students were all children of mexican parents and grew up speaking both english and spanish at home. carina, elena, and jessica were born and have lived in the united states for their entire lives. these three students had been enrolled dinapoli & morales translanguaging to persevere is key journal of urban mathematics education vol. 14, no. 2 81 in bilingual programs during their elementary grades and had transitioned into mainstream classrooms by the time they were in middle school. they all stated that they felt comfortable speaking spanish but were not as comfortable reading or writing in spanish. ines arrived to the united states from mexico at age 12, returned to mexico at age 14, and ultimately came back to the united states to complete her high school education. ines had formal educational experiences around speaking, reading, and writing in spanish and reported being more comfortable speaking spanish than english. these students represented typical students in the school with a history of average achievement in mathematics. each student reported that spanish had always played a major role in their meaning-making process for mathematics. the second group of students consisted of yasmin, julia, bernardo, and lorenzo (pseudonyms). lorenzo, yasmin, and bernardo all arrived to the united states at ages 12, three, and seven, respectively, and all three of them stated that they speak only spanish at home. julia was born in the united states to an american mother and mexican father. julia reported some comfort in speaking spanish but was not as comfortable reading or writing in spanish. lorenzo was the only student who reported some discomfort reading words or sentences in english. similar to the first group, these students represented typical, average achieving mathematics students for whom spanish had always played a major role in their process for understanding mathematics. the teacher for both groups was ms. patrick (pseudonym). ms. patrick was a monolingual english speaker with 20 years of mathematics teaching experience. she had seven years of experience enacting the imp curriculum and was a proponent of student-centered mathematical activity and problem solving. via our preliminary observations, ms. patrick’s teaching philosophy incorporated many aspects of smith and stein’s (2011) 5 practices for orchestrating productive mathematical discussions, and she often explicitly encouraged her students to collaborate with challenging mathematics in their native language and use mathematical tools to explore multiple resources and representations while doing so. we selected her classroom because she had many bilingual students and often had them working in groups—a perfect setting to learn more about how latinx bilingual students use translanguaging to persevere. data sources and context we observed two groups of students with the same teacher every day for six weeks each for a total of 50 class sessions. we accumulated over 40 hours of videorecorded observation with accompanying field notes. we also collected copies of student-generated mathematical artifacts. our observations largely concentrated on the students’ discourse patterns. we paid particular attention to how students used their linguistic resources (linguistic, visual-graphic, or gestures) while negotiating dinapoli & morales translanguaging to persevere is key journal of urban mathematics education vol. 14, no. 2 82 meaning. we also focused on the ways in which students interacted with available resources as a method for aiding the development of meaning. each participant was interviewed after the observations were completed. the main purpose of these interviews was to member-check the field notes of our impressions of the observation data (guba & lincoln, 1994). such member-check interviews helped ensure that participants’ voices were prevalent and accurate in the findings (guba & lincoln, 1994). these interviews also helped us gather information about the students’ family background, language proficiency in both english and spanish, academic history, and impressions of their mathematical identity. to select the cases to report in this paper, we drew from a thematic analysis of the entire data set. a saturation of themes emerged from our analysis, and we chose vignettes that illustrated those themes. additionally, we purposely selected vignettes from each group that concerned the same mathematical task and that demonstrated students’ typical interactions with each other and with ms. patrick. the findings reported in this paper stem from each group working on an activity designed to necessitate perseverance. each group was assigned the function analysis task: a task of discerning ways in which mathematical functions were helpful (figure 2). this activity was part of the imp year 4 goal of deeper exploration of previously encountered mathematical ideas. each group collaboratively selected a unit and a function they had encountered from a past unit. groups could choose a past function with which they experienced success and possessed prior knowledge. this made it more accessible for group members to initially engage in the meaning-making suggested by the function analysis task. each group would eventually be asked to share about their work on the function analysis task with the class. figure 2. function analysis task (fendel et al., 2015b) both groups chose “all about alice” (figure 3) as their function, selected from a year 2 imp unit with which both groups reported that they experienced success and understanding. ms. patrick corroborated this report. this unit started with a modeling task based on lewis carroll's alice’s adventures in wonderland. from this situation came the basic principles for working with exponents and an introduction to function analysis task select a specific function from a past unit. 1. describe the problem context in which the function was used and explain what the input and the output for the function represent in terms of the problem context 2. describe how the function was helpful to you in solving the central unit problem or some other problem in the unit. 3. if possible, determine what family the function is from. dinapoli & morales translanguaging to persevere is key journal of urban mathematics education vol. 14, no. 2 83 exponential growth and decay problems. this task of analyzing a previously encountered function necessitated perseverance because the all about alice problem is familiar to each group, yet the analysis is open to student exploration. figure 3. “all about alice” (fendel et al., 2015a, p. 385) to solve this problem, one could represent this relationship in the form of a table of values and a graph and consequently discover that if alice’s height is doubling, then her height is multiplied by powers of two depending on how many ounces of cake she eats. for example, if alice’s height is originally four feet and she ate three ounces of cake, her height is now 32 feet: 4(2 ∙ 2 ∙ 2) = 4(2') = 4(8) = 32 this mathematical problem can also be represented by an equation of the form 𝑦 = 𝑎 ∙ 𝑏.. in this case, alice’s height is doubling, so the base b is 2 and alice’s original height is represented by the variable a given the equation 𝑦 = 𝑎 ∙ 2.. this problem can be difficult because students have to grapple with the distinction between doubling and multiplying by two, an often-confusing idea because they seem so similar (confrey, 1991). also, the key variable x is an exponent, which is uniquely different from all of the other functions they have studied thus far. data analysis we employed a three-tier analytic method designed to recognize the ways the two groups of latinx bilingual students used translanguaging practice to collaboratively persevere and construct meaning while engaging with a challenging mathematical task (morales & dinapoli, 2019). by studying the simultaneous evidence of translanguaging and perseverance practices, we used inductive reasoning to make inferences about creating productive mathematics learning environments for latinx bilingual students. video transcriptions of students’ dialogic interactions were the center of our analysis. this analysis helped reveal how the groups of students used translanguaging to persevere by identifying translanguaging mathematical practice, all about alice alice’s height changes when she eats the cake. assume as before that her height doubles for each ounce of cake she eats. 1. find out what alice’s height is multiplied by when she eats 1, 2, 3, 4, 5, or 6 ounces of cake. 2. make a graph of this information. dinapoli & morales translanguaging to persevere is key journal of urban mathematics education vol. 14, no. 2 84 perseverance practice, and the overlay of both practices simultaneously (see figure 4). the first tier of analysis captured evidence of students’ translanguaging via their verbal and non-verbal communications. the second tier of analysis captured evidence of students’ perseverance in problem solving. the third tier of analysis captured how evidence of students’ translanguaging occurred alongside their perseverance (see morales & dinapoli, 2019). figure 4. translanguaging to persevere: three-tier analytic method in tier 1, the main objective was to understand the meaning-making process and to identify the translanguaging resources to which students made connections a result of this process. we coded video transcripts using the translanguaging mathematical practice framework (see figure 1) to identify which linguistic resources students used across both english and spanish as well as non-verbal resources. these resources included the use of the everyday register (er) and mathematics register (mr) in both english (e) and spanish (s). given the nature of communicating mathematically to negotiate meanings, identifying moments when students transitioned to non-verbal mathematical representations that include visual-graphic (mv) and symbolic representations or notations (mn) are also key components to the meaningmaking process. in tier 2, the main objective was to capture the ways in which students collectively persevered, or did not, during their engagement with a challenging mathematical task. we used the 3pp to do so (dinapoli, 2018; see table 1). the entrance phase captured whether the group of students understood the entirety of what a task was asking (c-0) and if the group immediately knew how to solve the problem (io0). the initial attempt phase examined whether and how a group of students initiated (ie-1) and sustained (se-1) their effort and the outcome (oe-1) of such effort as they worked toward solving all parts of the problem. in the event the group did not solve dinapoli & morales translanguaging to persevere is key journal of urban mathematics education vol. 14, no. 2 85 the problem after making a first attempt, the additional attempt phase aimed to capture if (ie-2) and how (se-2) the students amended their original problem-solving plan and the outcome (oe-2) of such efforts as they worked to overcome any setbacks and worked toward solving all parts of the problem. if at this point the problem was not yet solved, a group of students could continue their effort in the additional attempt phase—making a second, third, or fourth additional attempt, and so on. in tier 3, we conducted a constant comparative overlay analysis to simultaneously analyze the latinx bilingual students’ translanguaging practice during key moments of their problem solving when perseverance was necessary. inductively, we analyzed the ways in which evidence of translanguaging practice occupied different phases and components of perseverance. such analysis results in a theoretical sampling of patterns and themes that help describe the meaning-making actions in those moments of collective problem solving. two primary themes concerning students’ translanguaging and perseverance emerged through this constant comparative overlay analysis, and such qualitative saturation directly informed the selection of two illustrative cases to report in the findings. findings in this section we will first describe aspects of the learning environment established in ms. patrick’s classroom as evidenced by our observations and interactions with her to help the reader interpret student data. then, we will present two cases of two different groups of students in two different classrooms exemplifying latinx students’ meaning-making processes and perseverance while engaging in advanced mathematics (with and without ms. patrick present). case 1 depicts carina, jessica, elena, and ines, and case 2 depicts yasmin, julia, bernardo, and lorenzo. the mathematical task at hand spanned three days. for both groups of students, ms. patrick introduced the task for about 15 minutes during day 1, provided about 30 minutes of work time during day 2, and provided about 30 minutes of work time during day 3. alongside the transcripts of the vignettes, we will depict our tier 1 and tier 2 impressions of the group’s translanguaging and perseverance, respectively (see tables 2, 3, 4, and 5 below). we then will unpack the translanguaging mathematical practice that took place across the groups’ phases of perseverance in our tier 3 analysis in the paragraphs that follow each table. we will share these vignettes from both cases to compare how students may or may not persevere and make meaning of a complex mathematical situation involving exponential growth. aspects of the learning environment in ms. patrick’s mathematics classroom through our observations and interactions with ms. patrick, we were able to make inferences about her opinions and expectations that helped describe aspects of the learning environment in her classroom. across the three days of both groups of dinapoli & morales translanguaging to persevere is key journal of urban mathematics education vol. 14, no. 2 86 students engaging with the function analysis task (fendel et al., 2015b) with “all about alice” (fendel et al., 2015a, p. 385), much of ms. patrick’s careful efforts were in preparation with intent to build a classroom culture that was conducive to exploratory problem solving with bilingual learners. the thoughtful selection of a challenging mathematical task designed with several supports for bilingual learners demonstrated ms. patrick’s commitment to nurturing her bilingual students toward mathematical understanding. assigning the function analysis task was an important decision because it allowed students to access mathematical ideas through their prior work and created an environment conducive for productive struggle. ms. patrick often encouraged her students to first read their textbook in english and then discuss the mathematical meaning in both english and spanish. she also encouraged her students to work in their native language when it felt necessary. this offered support for their language and literacy development by helping make the mathematical content more comprehensible. collaborative learning was a commonly used pedagogical strategy. ms. patrick frequently used collaborative learning strategies to allow her students to communicate with each other in their groups. such emphasis on student-to-student communication in any language evidenced ms. patrick’s belief that cultural and linguistic differences are an intellectual resource and valuable to the meaning-making process when learning mathematics. further, there was ample evidence that ms. patrick created a learning environment where using mathematical tools was the norm. students’ understandings were often mediated by the use of the graphing calculator, in/out tables, and other multimodal visual and symbolic representations. access to these mathematical tools were always available and encouraged in ms. patrick’s classroom. vignettes from case 1 – carina, jessica, elena, and ines consider the following vignettes from this group’s three-day collaboration around the all about alice problem. these exchanges exemplify latinx bilinguals’ translanguaging meaning-making processes and perseverance while doing mathematics. when student utterances are spoken in spanish, we set apart the english translation using italics within parentheses. just prior to this vignette during day 1, ms. patrick introduced the task at hand. the group demonstrated evidence of understanding the goal of the task (c-0), which was to describe the mathematical relationship at hand, but an immediate solution pathway was not known (io-0). thus, our analytic process began on day 2, when this group already passed through the entrance phase of the 3pp and were ready to enter and work within their initial attempt phase (see table 2). dinapoli & morales translanguaging to persevere is key journal of urban mathematics education vol. 14, no. 2 87 table 2 translanguaging practice in the initial attempt phase of perseverance for case 1 transcript (english translation) evidence of translanguaging evidence of perseverance jessica: ¿qué era la primera, se hace así? (what was the first one, do you do it like this?) if [alice] eats one ounce, that means that she grows twice, dos ¿qué? (two, what?) double, no double, two…see, so when two is four, and then three is six, and four is eight, y así, y así vamos hacer la graph (like this, and this is how we are going to make the graph). going like that (gesturing), para arriba (up). you get it? elena: um hmm. pero (but), how to times it? mre & mrs: jessica uses her mathematics register to describe and represent how height grows: “grows twice,” “dos,” “double, two,” “two is four, three is six, four is eight.” mg: jessica gestures graphically about the nature of doubling. mrs: jessica describes the shape verbally: “para arriba.” ie-1: making sense of context se-1: exploring doubling jessica: porque mira (look), two, times two. well no… double it by, nomas (just) double the number of ounces, so if she takes… elena: two times two, y luego (and then) four times two, y luego (and then) six times two, is that what you are saying? ers & mre: jessica combines everyday and mathematics registers: “porque mira”, “nomas double the number of ounces.” mre & ers: elena combines everyday and mathematics registers: “two times two, y luego, four times two…” ie-1: making sense of context se-1: exploring doubling jessica: más o menos como sumando el mismo número. (more or less like adding the same number.) carina: pero es lo mismo de sumando si lo multiplicas por dos. (but it is the same as adding if you multiply by two.) mrs: jessica represents doubling with operations: “como sumando el mismo número”. se-1: exploring doubling oe-1: better understanding doubling ines: lo que parece escomo hicimos un in/out table y ya lo sacamos. (it looks like we just did an in/out table and that’s it). [see figure 5] carina: yeah. in times two equal out… ¿ya no tenemos que hacer su altura? (we don’t have to use her height?) mv: the group draws on a tabular representation of the doubling relationship [see figure 5]. mn: carina expresses the equation symbolically. se-1: revisiting representations of function oe-1: recognizing interpretation mistake dinapoli & morales translanguaging to persevere is key journal of urban mathematics education vol. 14, no. 2 88 this group entered the initial attempt phase of the 3pp by initiating effort (ie1) toward differently expressing what doubling meant to them. jessica used her full linguistic and multimodal repertoire as she expressed what doubling meant to her. she moved fluidly between english and spanish, drawing on a variety of registers to express her understanding of doubling. she had multiple equivalent ways to describe the relationship: grows twice, dos, double, two, a sequence of number relationships (two is four, three is six, etc.), and finally made an upward, increasing hand gesture to represent the shape of the graph. the group did not, however, realize immediately that they were multiplying the number of ounces of cake by two instead of alice’s height. not sure how exactly to represent their doubling relationship, jessica and elena dialogically moved between everyday and mathematics registers (er & mr) to try to represent the relationship algebraically. carina and jessica finally agreed that it was similar to adding the same number or multiplying it by two, utilizing the mathematics register in spanish (mrs). they also quickly represented the relationship symbolically as an equation. exploration of these points of view is evidence of sustained effort (se-1) to make sense of the all about alice function. their equation correctly spanned the table of values (see figure 5), yet these representations did not model an exponential function. this provided the group a metacognitive opportunity to become aware of their error, empowering them to engage more deeply to recognize their error, overcome the setback, and make an additional attempt to make sense of the function. not completely agreeing with the other students’ mathematical representation, ines specifically studied the table of values and helped her peers realize that they were doubling the number of ounces of cake instead of alice’s height. speaking entirely in spanish, ines persuaded the group back to the problem context and reminded them that they must start with alice’s initial height. ines used her mathematics register in spanish (mrs) to describe how she doubled alice’s height as each ounce of cake was eaten. ines’ revelation showed how perseverance can emerge from a group dynamic. figure 5. student-made in/out table from case 1 near the end of day 2, ines’ individual metacognitive awareness of the mistake during their initial attempt phase of perseverance helped collectively move the discussion in a different direction that considered alice’s initial height. the rest of the dinapoli & morales translanguaging to persevere is key journal of urban mathematics education vol. 14, no. 2 89 group began to buy into ines’ point of view amidst moments of struggle, which was essential for perseverance and building equitable and effective learning communities. this change in strategy marked the exit of the initial attempt phase and entrance into the additional attempt phase (see table 3), which took place during day 3. table 3 translanguaging practice in the additional attempt phase of perseverance for case 1 transcript (english translation) evidence of translanguaging evidence of perseverance ines: empezamos de cuatro pies. si toma si come un pedacito son ocho, si come un pedacito son dieciseis, el tercer pedazo dieciseis y dieciseis. treintaidos ¿no? (we start at four feet. if she drinks, if she eats one piece it becomes eight, if she eats one piece it becomes sixteen, the third piece, sixteen and sixteen, thirtytwo, no?) jessica: pero, ¿cómo sacastes eso? (but how did you get that?) mrs: ines rethinks the problem in relation to the problem context representing a different doubling relationship. ie-2: changing strategies to consider initial height se-2: exploring how her height changes ines: porque si empezamos con cuatro pies, como yo les digo, si come un pedacito y sale, aumenta de altura de doble (gesturing up). (because, if we start at four feet, like i’m telling you, if she eats one piece and it comes out to, her height grows double (gesturing up)). jessica: ohh, her height doubles! mrs: ines begins with an initial height and doubles that instead of the ounces of cake—“aumenta de altura de doble.” se-2: exploring how her height changes oe-2: understanding how her height changes elena: you know it’s the same thing, mira (look). dos (two), you multiply one times two is two, two times four is eight, y si pones (and if you put) two times two is four, four times four is sixteen. carina: in squared, times two is equal to your out. [see figure 6] mv & mn: students express the new relationship with a table and new equation [see figure 6]. se-2: revising a representation of the function at the start of day 3, this group began to reinitiate their effort (ie-2) to rethink about the mathematics, crossed out their table, and began to think about starting a new table. ines helped resustain effort (se-2) in their additional attempt phase of the 3pp to understand that alice’s initial height is necessary to compute subsequent heights as she ate each ounce of cake. it is important to note that the problem was written in english, yet ines leveraged her home language of spanish to revoice the dinapoli & morales translanguaging to persevere is key journal of urban mathematics education vol. 14, no. 2 90 problem. she modeled mathematically (mrs) the concept of doubling and also used an upward, increasing hand gesture (mg) to demonstrate how alice’s height doubles for each ounce of cake she eats. immediately after ines spoke, jessica demonstrated a productive outcome of resustained effort (oe-2) when she realized they needed to double alice’s height, not the number of ounces of cake. following this discussion, the students created a new table of values that correctly modeled alice’s exponential growth (mv & mn). this new approach demonstrated the group amending their plan and making a second attempt to make sense of the all about alice function. the equation was not yet correct, however, and did not span all of their entries (see figure 6). figure 6. revision of student-made in/out table from case 1 still, this vignette showcased how this group was making progress understanding the ways in which the relationship in “all about alice” was not linear nor quadratic while engaging in translanguaging mathematical practice. this was an important opportunity for the group to recognize more mistakes and continue to persevere, which took place near the end of day 3 (see table 4). dinapoli & morales translanguaging to persevere is key journal of urban mathematics education vol. 14, no. 2 91 table 4 translanguaging practice in the additional attempt phase of perseverance for case 1 transcript (english translation) evidence of translanguaging evidence of perseverance carina: mira, la pongo en la calculadora y luego pido la “table” y me da otra answer de lo que nosotros tenemos aquí. (look, i put it in the calculator and then i push “table” and it gives me a different answer from the one we have here.) [see figure 7]. y equals x, quedamos al (we have) x squared times two, verdad (right)? elena: three es (is) nine… times two … is eighteen. carina: y es lo que sale aquí (that’s what comes out here). thirty-two. and out of three sale (gives) eighteen. ines: pero ¿por qué? (but, why?) ers, mv, & mn: students compare the student-made table with the table generated by the graphing calculator. se-2: creating another representation of the function se-2: comparing two representations of the function jessica: mhmm. entonces lo hicimos mal…pero esto, el out, tiene que ser así. (mhmm. well then we did it wrong… but this, the out, (referring to the output values) has to be like this.) ines: lo que está mal es esto. (this (the equation) is what is wrong.) ers, mre, & mv: jessica uses the word “out” to represent output values from the hand generated table. oe-2: recognizing equation mistake jessica: porque también el zero tiene que ser el cuatro, in tiene que ser cero y luego el out tiene que ser cuatro. así tiene que ser. y primero el zero cuatro uno ocho dos dieciseis. (well, the zero has to be four, in has to be zero and then the out has to be four. that’s how it has to be. and, first, the zero four, one eight, two sixteen.) elena: solo si tenemos que cambiar el formula, el equation, ¿no? (the only thing we have to change is the formula, the equation, no?) mrs: jessica explains about pairs of numbers that satisfy the relationship of doubling alice’s height. oe-2: understanding how her height changes oe-2: recognizing equation mistake ms. patrick: which one are you guys doing? alice growing? elena: yeah. we thought we had it but now we realize that we don’t. n/a n/a ms. patrick: what’s the base? the base is two. then what happens? she grows. you put the equation in the calculator. what do you do? carina: y equals. ms. patrick: y equals, so you have y equals two to the x power. y is equal to two to the x power. mre & mrs: ms. patrick tries to support their understanding of the problem using words like “base,” “equation,” and “x power,” which are all part of the mathematics register for exponential functions. n/a dinapoli & morales translanguaging to persevere is key journal of urban mathematics education vol. 14, no. 2 92 near the end of day 3, carina used a graphing calculator to make a table of values for 𝑦 = 2𝑥0 (se-2; mv & mn; see figure 7) and determined that it did not match their current table (se-2). carina, elena, and ines then questioned the source of this inconsistency (se-2) and ultimately realized and discussed, mostly in spanish interwoven with the mathematics register in english (ers & mre), the ways in which their equation was incorrect (oe-2). the students recognized the flaws in their attempt to model the situation with the function 𝑦 = 2𝑥0 (oe-2) and were ready to continue to explore ways to revise their equation and enter another additional attempt phase of perseverance. however, they had ran out of time and ms. patrick needed to move on to a discussion of solution processes. although this group did not yet find the proper equation to span their in/out table in this vignette, they were agents of their own translanguaging mathematical practice and leveraged this practice to help persevere and think more deeply about exponential functions. figure 7. calculator-generated in/out table from case 1 in summary, the findings from case 1 demonstrated a group of secondary latinx bilingual students leveraging their translanguaging mathematical practice to persevere in their efforts to better understand an exponential relationship. although carina, jessica, elena, and ines did not find the proper equation in the allotted class time, they persevered over several obstacles to facilitate more deep mathematical thinking about the all about alice function. in our view, it was within ms. patrick’s established, productive learning environment that these students spontaneously leveraged each other’s linguistic repertoires and viewed their linguistic resources as an asset to help persevere past obstacles and make progress toward a solution. ines’ strong academic discourse in spanish was particularly celebrated and played a crucial role in the meaning-making process for the group. vignettes from case 2 – yasmin, julia, bernardo, and lorenzo this case depicts a different group of students from a different class than described in case 1. ms. patrick was also the classroom teacher for case 2 students. ms. patrick introduced the activity to this group on day 1, and, similarly to case 1, dinapoli & morales translanguaging to persevere is key journal of urban mathematics education vol. 14, no. 2 93 this group of students demonstrated evidence of understanding the goal of the task (c-0), but an immediate solution pathway was not known (io-0). our analytic process began on day 2 when this group continued in the entrance phase of the 3pp (see table 5). interestingly, although this group of students were similarly poised as the group in case 1 to leverage their translanguaging mathematical practice to persevere, they never progressed to enter the initial attempt phase of perseverance. table 5 translanguaging practice in the entrance phase of perseverance for case 2 transcript (english translation) evidence of translanguaging evidence of perseverance lorenzo: ¿qué tenemos que decir? ¿qué tenemos que decir? (what do we have to say? what do we have to say?) what we have to say? ¿qué tenemos que decir? (what do we have to say?) bernardo: we have to explain this, man. lorenzo: pero. ¿qué hay que decir? (but what is there to say?) ers: lorenzo is confused about what he needs to explain. c-0: reviewing the task objectives io-0: aspects of solution not immediately apparent bernardo: alice in wonderland, which one do we have to do? julia & yasmin: “all about alice.” cuando crece come? toma o come? (when she grows, she eats? drinks or eats?) yasmin: page 385, we have to do this [reads task]. she was telling us yesterday. julia: yea, aqui lo tengo ya. (yes, i have it here already.) ere & ers: this group moves between languages to clarify the objectives. c-0: reviewing the task objectives ms. patrick: how are you guys doing? you got your equation? you got your function table? so how much is alice’s height going to grow by after one? julia: it’s going to double. ere & mre: ms. patrick attempts to support their understanding of the task. julia says that the function is doubling. c-0: reviewing the task objectives ms. patrick: after one ounce of cake? you need to write that out. y equals two to the…? julia & bernardo: four? … third? ms. patrick: follow your equation and write it out. bernardo & lorenzo: two to the… to the power… julia: but we do for every one? ms. patrick: yes. mre: ms. patrick provides some scaffolds to help the group to symbolize the equation for the function: “y equals two to the …?” n/a ms. patrick: are you doing it on your calculator too? yasmin: like that ms. patrick? ms. patrick: no, that’s not the general equation, what’s the general equation, julia? students: y equals two…? ms. patrick: two what? yasmin: to the second. ms. patrick: no, what is this equation? julia: x yasmin & julia: y equals two x. ms. patrick: no! it is not two x! y is equal to two to the x power. ere & mre: ms. patrick provides verbal sentence frames to help the students to symbolize the equation. n/a dinapoli & morales translanguaging to persevere is key journal of urban mathematics education vol. 14, no. 2 94 these students began work on day 2 by reminding each other that they would need to present their solution to the whole class and by rereading the all about alice problem to determine how they might proceed (c-0). in both spanish and english, they discussed details of the instructions and context and seemed ready to enter the first attempt phase of perseverance (ere & ers). at this point, ms. patrick intervened and asked the group about their progress obtaining an equation, a corresponding table of values, and a general understanding of the context (ere & mre). julia confirmed that they understood the problem’s context, that is, that alice’s height doubles for every ounce of cake she eats (c-0). ms. patrick then tried to use leading questions to help scaffold yasmin, julia, bernardo, and lorenzo toward writing an equation to model the scenario. this group seemed to understand that there was a connection between values like 22, 23, and 24 with alice’s changing height but were not sure how to generalize the function of height and ounces of cake. ms. patrick then shifted the conversation toward using a graphing calculator to help obtain the correct equation to solve the all about alice problem. still, the group did not seem to understand the meaning of their model and its values. lastly, the students began to guess at appropriate equations for the scenario, and eventually ms. patrick stated that the correct equation was 𝑦 = 2.. the group spent the rest of their time on day 2 and day 3 monitoring the routine steps to enter the equation that ms. patrick provided to them into the calculator to access the table of values. see table 6 for an excerpt of their discourse on day 3. table 6 translanguaging practice while monitoring routine calculator steps for case 2 transcript (english translation) evidence of translanguaging evidence of perseverance yasmin: tienes que escribir siempre el x la, la equation (you always have to write the x, the equation) the equation you have to write down, you get it? and then you just give, you… in case you want to an in/out table asi y no sabe los numeros que van el la (so and you don’t know the numbers that go) in the x in the y nomas ponla equation aqui (and just put the equation here) and then you just, ah um.. bernardo: table. yasmin: table. bernardo: second table. ers, ere, & mv: students describe across languages the technical steps to enter an equation and generate a table. n/a yasmin: second table, second table y te da la answer (and it gives you the answer), you get it? bernardo: yeah. lorenzo: a ver (show me). yasmin: good. julia: [copying table of values into her notes]. ers, ere, & mv: students double check the technical steps to generate a table with their calculator while julia copies the table. n/a dinapoli & morales translanguaging to persevere is key journal of urban mathematics education vol. 14, no. 2 95 during day 3, this group spent their time reviewing the procedures by which to enter the equation into their calculator, the equation ms. patrick provided them. in spanish and english, yasmin explained how the calculator can produce a table of values for them, as long as they can enter the equation into the calculator (ers, ere, & mv). bernardo reminded yasmin that they needed to push the buttons “second” and then “table” in order to access the table of values after they entered their equation. yasmin then showed the group the table of values on the calculator screen and confirmed that her partners understood how to execute these steps (ers & ere). while this was happening, julia copied the table of values into her notes (mv; see figure 8). figure 8. copied in/out table from calculator’s table of values from case 2 in summary, the findings from case 2 demonstrate little to no evidence of the students’ translanguaging mathematical practice to persevere to better understand an exponential relationship, compared to case 1. although yasmin, julia, bernardo, and lorenzo were working in a similar context as case 1—an environment with built-in supports for bilingual problem solving—the students in case 2 did not spontaneously leverage the available linguistic resources to help persevere past obstacles and make progress toward a solution. instead, much of the case 2 students’ engagement was in pursuit of rehearsing procedures modeled by ms. patrick without much meaning. these findings from case 2 suggest the over-scaffolding of a group activity in an unproductive way (stein et al., 1996). we find such over-scaffolding (i.e., modeling procedures without time for exploration) to be incompatible with a translanguaging stance that, ideally, should be facilitating opportunities for students to leverage their linguistic resources to make meaning and persevere. unfortunately, in this case, the consequences of teacher over-scaffolding resulted in very few perseverance opportunities for yasmin, julia, bernardo, and lorenzo and, thus, few opportunities to make meaning of the exponential relationship. dinapoli & morales translanguaging to persevere is key journal of urban mathematics education vol. 14, no. 2 96 discussion in this section, we interpret our findings relative to the research questions (i.e., evidence of translanguaging mathematical practice, perseverance, and resultant pedagogical insights). as we reflect on these cases and respective classroom vignettes, we suggest several pedagogical moves to substantially encourage the translanguaging and perseverance practices of latinx bilingual students. first, we draw on ramirez and celedón-pattichis’ (2012) five guiding principles for teaching mathematics to latinx bilinguals to help reflect on the presented vignettes from both cases. these principles articulate effective ways to engage latinx bilinguals to persevere with rigorous mathematics while engaging in translanguaging mathematical practice. recall ramirez and celedón-pattichis’ (2012) guiding principles: enacting challenging mathematical tasks, leveraging cultural and linguistic differences as intellectual resources, building a linguistically sensitive social environment, offering support for learning english while learning mathematics, and offering mathematical tools and modeling as a resource. these principles help illuminate two meaningful but conflicting insights from the shared vignettes. on one hand, our inferences of the classroom culture built by ms. patrick suggested an intentional focus on supporting latinx bilingual students’ mathematics learning. on the other hand, despite this classroom culture, a variance of demonstrated translanguaging mathematical practice and perseverance is evident when comparing the case 1 and case 2 vignettes. we discuss each of these insights in the following paragraphs. regarding the alignment of ms. patrick’s practice to ramirez and celedónpattichis’ (2012) five guiding principles, we first note the importance of selecting and enacting the function analysis task (fendel et al., 2015b). the thoughtful selection of this challenging mathematical task designed with several supports for bilingual learners demonstrates ms. patrick’s commitment to nurturing her bilingual students toward mathematical understanding. enacting the function analysis task was an important decision because it allowed students to access mathematical ideas through their prior work and created an environment conducive for perseverance. ms. patrick also used collaborative learning strategies to allow her students to communicate with each other in their groups. such emphasis on student-to-student communication in any language evidenced ms. patrick’s belief that cultural and linguistic differences are an intellectual resource and valuable to the meaning-making process when learning mathematics. it was evident in the way ms. patrick insisted her students use their native language to engage with the task that she genuinely intended to create a productive, inclusive, and linguistically sensitive learning environment. additionally, ms. patrick encouraged her students to first read their textbook in english and then discuss the mathematical meaning in both english and spanish. this supported their language and literacy development by helping make the mathematical content more comprehensible. finally, ms. patrick created a classroom environment where using dinapoli & morales translanguaging to persevere is key journal of urban mathematics education vol. 14, no. 2 97 mathematical tools was the norm. students’ understandings were mediated by the use of the graphing calculator, in/out tables, and other multimodal visual and symbolic representations. access to these mathematical tools proved to be instrumental in the ways in which the students in case 1 persevered past several mathematical obstacles. regarding the variance of demonstrated translanguaging mathematical practice and perseverance when comparing the vignettes from case 1 and case 2, we recognize that the alignment of ms. patrick’s practice to ramirez and celedón-pattichis’ (2012) five guiding principles is not enough to support latinx bilingual students’ inthe-moment meaning-making. as depicted in case 1, ms. patrick’s acceptance of students using their first language while collaborating can encourage students to engage in mathematical meaning making using their full repertoire of linguistic, symbolic, multimodal resources. however, what we saw in case 1 was the spontaneous actions of carina, jessica, elena, and ines using their translanguaging mathematical practice to persevere with the mathematical task. although the classroom environment was certainly supportive, we do not see much evidence of ms. patrick monitoring and facilitating this group’s progress to understand the function’s complex symbolic representations nor do we see an explicit attempt by ms. patrick to make space for students to express bilingually what they did or did not understand in those moments. as depicted in case 2, we saw latinx bilingual students collaborating using spanish but only to monitor instructional and routine steps—not to navigate meaning. although yasmin, julia, bernardo, and lorenzo seemed similarly poised (to those from case 1) to draw on translanguaging mathematical practice to help persevere with the exponential relationship, ms. patrick’s early interjection seemed to unintentionally derail any such progress. from a perseverance perspective, the students in case 2 never had the opportunity to invest effort in a first attempt at solving the problem and quickly became focused on rehearsing the procedures modeled by ms. patrick. on day 3, without ms. patrick interjecting, the group in case 2 had another opportunity to persevere to try to make sense of the function but instead reviewed in both spanish and english the procedures of how to enter an equation into the graphing calculator to make sure they obtained the correct answers. contrary to case 1, in which students spontaneously leveraged translanguaging mathematical practice to help overcome conceptual obstacles, the group in case 2 drew on their bilingualism to discuss procedures alone. when all group members seemed to understand how to conduct these procedures that their teacher seemed to value so highly, they stopped working and waited for further instructions from ms. patrick about what to do next. the comparison of these cases suggests, in part, that if latinx bilingual students are given a limited opportunity to productively struggle with important mathematical ideas, regardless of their collective prior knowledge, then they may be reluctant to use the ample resources at their disposal, including translanguaging mathematical practice. of course, the students in case 2 were different from the students in case dinapoli & morales translanguaging to persevere is key journal of urban mathematics education vol. 14, no. 2 98 1, so it is possible the groups’ collective content knowledge was different and might help explain why students in case 1 were able to make more progress than students in case 2. however, we do not think this is likely because each group chose to revisit the all about alice function because they had experienced success with it during year 2. this implies that the observed differences between groups is less about whether one group was mathematically stronger than the other and more about the different opportunities each group had during the function analysis task to make meaning. the support conditions set forth by ms. patrick across case 1 and case 2 were different, and it is apparent that case 2 students required more substantial support to help them make mathematical progress. teachers of latinx bilinguals must develop and enact a substantial translanguaging stance in their mathematics classrooms. in reflecting on the classroom environment examined in this study, we see several ways that teachers could more substantially develop a translanguaging stance and enact a translanguaging pedagogy in their mathematics classrooms (garcía et al., 2017), specifically by actively supporting translanguaging mathematical practice and perseverance practice in their students. to develop a translanguaging stance, teachers must believe that bilingual students have one holistic language repertoire on which they draw. although teachers can advocate for students’ native language use, we recognize it can be difficult for teachers, especially monolingual english-speaking teachers, to enact in-the-moment teacher moves that respond to student thinking and align with a belief in one holistic language repertoire. teachers would benefit from a mindset that goes beyond acceptance of dynamic bilingualism to being more celebratory of creating spaces that model and encourage productive struggle specific to the translanguaging practice of latinx bilinguals (palmer et al., 2014). teachers could also build on bilingual students’ linguistic resources and home language repertoire by revoicing, clarifying, and probing students’ understandings by asking questions about the content in their first language to inform instruction (ramirez & celedón-patichis, 2012). for the classroom environment examined in this study, we contend that there is room to grow to earnestly establish a translanguaging stance by finding ways to access and build on students’ cultural, linguistic, and community funds of knowledge (maldonado et al., 2018). to further substantiate the development of a translanguaging stance and pedagogy in mathematics classrooms, we iterate on garcía et al.’s (2017) translanguaging strategies for teachers to employ, which could help support translanguaging mathematical practice in classrooms. first, a teacher could have offered extended reading support for students as they engaged with the english imp textbook. students from cases 1 and 2 may have benefitted from rereading the problem with a teacher present, allowing students to make connections across languages and linguistic and multimodal repertoires. second, a teacher could have provided more in-the-moment opportunities for students to develop linguistic practices for academic contexts. dinapoli & morales translanguaging to persevere is key journal of urban mathematics education vol. 14, no. 2 99 students from cases 1 and 2 may have benefitted from a teacher prompting them to voice how drawing on both their english and spanish mathematics registers helps them make sense of a mathematical idea as they work. third, a teacher could do more to make space for students’ bilingualism in her mathematics class. despite encouragement for students to work in any language they prefer, the students’ bilingualism existed in a peripheral space in the classroom: students used their bilingualism in small groups but rarely shared their full linguistic repertoires with the entire class. in addition to creating the translanguaging stance above, we also suggest that teachers adjust the separative language nature of their classrooms. garcía (2017) posits that we cannot gauge bilinguals’ mathematical proficiency if we artificially separate the students’ language repertoires. much of the work involving bilingual students is framed around an ideology and policies of language separation that encourage teachers to create separate instructional spaces (physical and other) for bilingual students to communicate in either the dominant or minority language (palmer et al., 2014). teachers might consider creating a more substantial space in their classrooms that privileges bilingualism and supports the development of their bilingual mathematics identities. one step toward creating such a substantial space would be for teachers to encourage their students to present their problem-solving process to the entire class in spanish and have other students interact with their process in a bilingual way. in addition, students could help translate these presentations for monolingual english speakers (including the teacher), furthering the development of their bilingual mathematics identities. we see students from cases 1 and 2 benefitting from such intentional identity work. conclusion through this study we addressed calls to action by martinez et al. (2017) and nctm (2014) by examining the ways in which latinx bilingual students used translanguaging to persevere to make sense of challenging mathematical ideas. such work helps rehumanize mathematics education for latinx bilingual students in the united states by privileging their unique resources to learn mathematics with understanding and finding ways to better encourage the use of these resources (gutiérrez, 2018). we observed and conducted member-check interviews with two groups of 12th-grade latinx bilingual students in different classes working to make sense of an exponential function with the same monolingual, english-speaking teacher. we found that a group of students were capable of spontaneously translanguaging to persevere with a challenging problem, seemingly supported by peripheral classroom supports established by the monolingual teacher. however, we also found that a similarly poised group of students did not spontaneously collaborate in this way and required more substantial support from their classroom teacher to encourage translanguaging mathematical practice. unfortunately, such teacher support did not dinapoli & morales translanguaging to persevere is key journal of urban mathematics education vol. 14, no. 2 100 happen, and these students had few opportunities to persevere to make meaning. this suggests that teachers should not solely rely on students to spontaneously engage in these practices and instead must fully commit to having a translanguaging stance. having a translanguaging stance that manifests as native language encouragement around rich mathematical tasks is not enough; teachers must also plan for and enact in-the-moment moves to help latinx bilingual students leverage their linguistic resources and persevere to make meaning of mathematics. building from suggestions from ramirez and celedón-patichis (2012), garcía et al. 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(2011). handbook of self-regulation of learning and performance. routledge. copyright: © 2021 dinapoli & morales. this is an open access article distributed under the terms of a creative commons attribution-noncommercial-sharealike 4.0 international license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. journal of urban mathematics education july 2017, vol. 10, no. 1, pp. 32–51 ©jume. http://education.gsu.edu/jume terri l. kurz is an associate professor in the teachers college – arizona state university, 7271 e. sonoron arroyo mall, mesa, az 85212; email: terri.kurz@asu.edu. her research interests focus on the use of tools and technology to support teaching and learning in mathematics. conrado gómez (now retired) was previously a clinical assistant professor in the teachers college – arizona state university, 7271 e. sonoron arroyo mall, mesa, az 85212; email: conrado.gomez@asu.edu. his research focused on enhancing ell instruction with an emphasis in mathematics. margarita jimenez-silva is an associate professor in the teachers college – arizona state university, p.o. box 3700, phoenix, az 85069; email dr.mjs@asu.edu. her research focuses on providing access to content for english language learners. guiding preservice teachers to adapt mathematics word problems through interactions with ells terri l. kurz arizona state university conrado gómez arizona state university margarita jimenez-silva arizona state university in this article, the authors present a framework for guiding elementary preservice teachers in adapting mathematics word problems to better meet english language learners’ (ells) needs. they analyze preservice teachers’ ell adaptations implemented in a one-on-one setting. through qualitative methods, four themes regarding implemented adaptations are identified: language adaptations, mathematical adaptations, tool/visual adaptations, and structural adaptations. the authors conclude that the framework was successful in helping preservice teachers learn about adapting curriculum by interacting with ells. implications for teacher education are discussed. keywords: ells, preservice teachers, mathematics education, word problems or english language learners (ells), mathematics can be more challenging than other subjects, as there is an emphasis on both the language of words and the symbols of mathematics (freeman & crawford, 2008; harper & de jong, 2004; moschkovich, 2002; swanson, 2015). it has been argued that there is an interconnectedness of language, symbols, and visuals that are characteristic in learning mathematics and in learning the language of mathematics (o'halloran, 2008). nevertheless, meanings of words differ in common language versus mathematical language. for example, the word leg has two very different meanings: in mathematics it represents the sides of a right triangle, but commonly it is known as a limb used for walking (simpson & cole, 2015). because of the development of both mathematics skills and language skills, it is imperative that ells’ needs are considered when developing, implementing, and adapting lessons in mathematics (ernst-slavit & slavit, 2007; janzen, 2008; martinello, 2008; truxaw & rojas, 2014). ells should have access to high quality, effective mathematics instruction that supports f http://education.gsu.edu/jume mailto:terri.kurz@asu.edu mailto:conrado.gomez@asu.edu mailto:dr.mjs@asu.edu kurz et al. adapting word problems journal of urban mathematics education vol. 10, no. 1 33 their development and considers their needs (moschkovich, 2010, 2013). reforms have been encouraged that offer multiple approaches to mathematics for different types of learners; these multiple approaches are aimed at providing opportunities of success for all students (see, e.g., standards for mathematical practice in the common core state standards1). however, success for ells often requires specific accommodations. because of the limited language skills of the vast majority of public school teachers, making accommodations to lessons to assist ells is not easy and requires careful consideration of what to teach and how to teach (avalos, medina, & secada, 2015; celedón-pattichis & ramirez, 2012; goldenberg, 2013). furthermore, there is a movement to embrace mathematics as more student-centered with a focus on thinking, communicating, and reasoning by requiring more than just computational understanding but also conceptual understanding (bunch, 2013; santos, darling-hammond, & cheuk, 2012). adapting curriculum to meet students’ needs requires a skillset that must be developed and enriched over time through practice and experience (van ingen & ariew, 2015). and while researchers have encouraged the focus on meeting the needs of ells through teacher preparation courses and lessons (see, e.g., darling-hammond, 2010; keengwe, 2010; samson & collins, 2012), there are still challenges in that preservice teachers (most often) are not being prepared to meet the needs of ells through their university training and coursework (bunch, 2013). most mathematics education preparation programs do not emphasize the instructional skills mathematics teachers need to address and meet the needs of ells (de jong & harper, 2005; ernst-slavit & slavit, 2007; freeman & crawford, 2008). for instance, durgunoglu and hughes (2010) explored how prepared preservice teachers were to teach ells. they found that the participating preservice teachers of their study were neither well prepared to teach ells in their teacher education program nor were they provided with support in their placements to address their inexperience and lack of knowledge (also see siwatu, 2011; webster & valeo, 2011). in the university setting, preservice teachers must be provided with opportunities to grow as future teachers of ells by learning how to accommodate the needs of ells through lesson plan design (lucas, 2011). it has been recommended that preservice teachers be provided with opportunities to better connect theory learned at the university with practice out in the classrooms (grossman, hammerness, & mcdonald, 2009). furthermore, field experiences can be beneficial in guiding preservice teachers’ understanding of ells and their needs (coady, harper, & de jong, 2011). specifically, garcía, arias, murri, and serna (2010) suggest an emphasis on developing knowledge of ells through contacting and collaborating directly with community members. 1 see http://www.corestandards.org/math/practice/. http://www.corestandards.org/math/practice/ kurz et al. adapting word problems journal of urban mathematics education vol. 10, no. 1 34 with the goal of providing preservice teachers with practical experience working with ells, we designed a project that integrated the content of an elementary mathematics methods course with implementing adaptations for ells in mathematics. using a variety of guidelines described by researchers for adapting curriculum, preservice teachers in our study were asked to adapt text to better meet the needs of ells in mathematics. the specific emphasis was on mathematics word problems at the elementary school level. in this article, we aim to address the following question: when preservice teachers have the opportunity to adapt word problems to better meet the needs of an ell in a one-on-one setting, what adaptations are employed? word problems and ells word problems are often challenging for all learners because they encompass various cognitive processes. for example, learners need to access pre-stored information and to determine what algorithm to use and what information is pertinent and irrelevant (orosco, swanson, o’connor, & lussier, 2011). given the complexities of language, ells face unique linguistic challenges when approaching mathematics and word problems (abedi & lord, 2001; yeong & chang, 2014). with these challenges in mind, it is important that there are considerations with respect to the demands of word problems on ells’ mathematical and linguistic development. researchers have identified linguistic features that make a text difficult to read by slowing down the reader, making misinterpretation more likely, and adding to the reader’s cognitive load (see, e.g., abedi, hofstetter, baker, & lord, 2001; de jong & harper, 2005). these indices of language difficulty include word frequency, word length, and sentence length, in addition to the overall length of the mathematical item, which is unique to mathematics word problems. elsewhere (see, e.g., gómez, kurz, & jimenez-silva, 2011), we have provided a practice-based guide for adapting mathematics word problems for ells taking into account these described challenges. adapting by simplifying the language of the text does not distort nor dilute the content concepts (echevarria, vogt, & short, 2008). but rather, it reduces the readability demands by eliminating linguistic characteristics that get in the way of comprehension (dyck & pemberton, 2002). there are benefits to keeping language simple for ells. an ell who encounters familiar words will spend less time analyzing the task (gathercole & baddeley, 1993). ells perform better on mathematical items with shorter words and shorter sentence length because they are less morphologically and syntactically complex (abedi, lord, & plummer, 1995). lengthy items will take longer to complete given that ells on average read more slowly (lepik, 1990). adaptations of kurz et al. adapting word problems journal of urban mathematics education vol. 10, no. 1 35 word problems may help ells successfully engage in mathematics word problems by making the content more accessible (swanson, 2015). language acquisition experts as well as teachers use the term adapt to refer to the adjustments that need to be made to any type of text to make it comprehensible for ells. indeed, adaptation of content is one of the pillars of structured english immersion (sei; echevarria et al., 2008). there are generally two types of adaptations: accommodations and modifications. generally, accommodations are “changes to materials or procedures that provide students access to instruction and assessments,” while modifications are “defined as changes in…materials or procedures that do alter the content being measured” (thurlow & kopriva, 2015, pp. 333– 334). modifications change the content; accommodations do not. while modifications may be appropriate for students receiving special education services, most of the time educators should be providing accommodations for ells (hite & evans, 2006). in this article, we focus on making accommodations by adjusting word problems to best meet the needs of ells. accommodations can support ells’ access to curriculum; specifically, barriers can be removed so that opportunities to engage in the curriculum are provided to ells (lópez, scanlan, & gundrum, 2013). without language support provided by the teacher, ells could fall behind their peers (swanson, moran, bocian, lussier, & zheng, 2013; yeong & chang, 2014). to stay abreast of their peers, ells need access to a continuous languagefocused program across all subjects (gibbons, 2002), including mathematics. simple exposure to english does not guarantee that ells will learn the academic language and mathematics content. consequently, the teacher needs to understand that integrating content and language requires systematic planning (gibbons, 2002). adapting curriculum for the ell abedi and colleagues (2001) have identified indices to predict the difficulty of a text. besides word frequency, word length, and sentence length, they discuss additional linguistic features that may cause difficulty for readers, including: passive voice constructions, long noun phrases, long question phrases, comparative structures, prepositional phrases, sentence and discourse structure, clause types, conditional clauses, and concrete versus abstract or impersonal presentations. abedi, courtney, leon, kao, and azzam (2006) summarize research findings on adapting the language of word problems on tests. their findings were used to structure the processes followed to guide the preservice teachers in our study:  if the words are long, replace them with high frequency words that are easier to read;  if words are unfamiliar, replace them with familiar words, omitting or defining words with double meaning or colloquialisms; kurz et al. adapting word problems journal of urban mathematics education vol. 10, no. 1 36  if the sentences are long, retain subject-verb-object structure, begin questions with question words, and avoid clauses and phrases;  if the item is long, remove unnecessary expository material; and  if sentences are complex, keep to the present tense, use active voice, avoid the conditional mode, and avoid starting statements and questions with clauses. in our study, we were interested in guiding preservice teachers in implementing these techniques to make changes to their background knowledge relating to adaptation of curriculum in mathematics. part of the preparation involved some basic tenets for adapting material. rhine (1995) found that in-service teachers were unable to properly gauge their ells’ skills. in addition, the teachers had limited knowledge about how their ells think. we also wanted to address this disconnect by focusing on the ells’ thinking along with the preservice teachers’ understanding of the ells’ mathematical and linguistic needs based on their interactions. methods in a course deigned to prepare preservice teachers to meet the needs of ells, preservice teachers were asked to work one-on-one with any k–12 ell student in their student-teaching placements. because the preservice teachers were placed in such different school contexts, we worked with their specific needs based on their placement. the one-on-one ell interactions were structured to focus on adapting the content of mathematics curriculum to better meet the ell’s needs while emphasizing the learning and growth of the preservice teacher as a result of the interactions. participants the participants were elementary graduate preservice teachers working simultaneously on their elementary education degree and teacher certification in the state of arizona. the course in which this study was conducted was designed to prepare preservice teachers for linguistically diverse classrooms in which there were ells learning content supported by sei strategies. preservice teachers were prepared to address linguistic and cultural awarenesses by learning strategies designed to meet the individual needs of ells based on language acquisition research. because the course was open to all education majors (elementary, secondary, and special education), there was a diverse group of specializations. however, only those that focused on adapting curriculum in mathematics were included for analyses. six preservice teachers concentrated on mathematics and completed all kurz et al. adapting word problems journal of urban mathematics education vol. 10, no. 1 37 the components of the assignment with an ell. four were women; all were white; all were elementary education majors. setting states like california, arizona, and massachusetts that have adopted an english-only policy in k–12 education generally require that colleges of education build into their curriculum a place where mathematics education students acquire the knowledge and skills in language development to address the needs of both native and non-native speakers of english (guo & koretz, 2013; parra, evans, fletcher, & combs, 2015; rolstad, mahoney, & glass, 2005). for example, in arizona’s state-mandated sei courses, preservice teachers learn about the nature of language development and how language varies according to the context in which it is used. coursework for preservice teachers explains that it is easier to learn language that is embedded in the visual context provided by manipulatives, other visual cues, and hands-on demonstrations and activities (gibbons, 2002), which are commonly used in the mathematics curriculum. preservice teachers are also taught that ells are supposed to learn english as well as learn in english. there are two primary approaches to learning in regards to ells: englishonly or bilingual instruction. while bilingual instruction is more often supported by research studies (adetula, 1990; moschkovich, 2007; see also rolstad, mahoney, & glass, 2005 for a meta-analysis), states often discourage bilingual instruction preferring english-only (guo & koretz, 2013; menken, 2013; menken & solorza, 2014). with english-only instruction as a common occurrence in states with significant numbers of ells (menken, 2013), we approached our instructional framework with that in mind. because the children were placed in english-only classrooms, the framework that guided the preservice teachers’ data collection focused on meeting the needs of ells in an english-only setting. we recognized the importance of bilingual education but had to work within the framework required by the state. in arizona at the time of the study, there were two 3 hour-credit courses required of all teachers. the arizona department of education established the curricula for the two courses. the courses cover history, policy, research, theory, and practices. in addition, topics such as culture, family role, politics, and standards were embedded into the course. the bulk of the time was spent on teaching strategies, including the adaptation of content. as explained, the course content focused on a multitude of curricular ideas. the task at hand was designed to fuse the content in a meaningful way that provided an opportunity to learn about ells from ells; the primary objective was to contextualize the theory learned in a university setting with actual ells out in the classroom. preservice teachers were to learn theory in class and experience the theory in context with children. kurz et al. adapting word problems journal of urban mathematics education vol. 10, no. 1 38 data sources the preservice teachers were asked to identify and work with an ell in their student-teaching placements. the ell could be in any grade level. participating preservice teachers were asked to select an ell at the speech emergence level of language acquisition or below. ells at this stage of language acquisition have received english instruction for at least one year. their active vocabulary consists of around 3,000 words, and they generally have a good comprehension of contextualized information. they still make many pronunciation and grammar errors when producing simple sentences. also, they are capable of reading basic vocabulary and writing simple sentences. the preservice teachers in this study received guidance in analyzing the linguistic demands of written mathematics word problems. (the instruction they received in this area is more specifically described in the adapting curriculum for the ell section of this paper.) the ells the preservice teachers selected were from a variety of countries: china, croatia, korea, two from mexico and one not specified (first language was spanish). all of the ells the preservice teachers chose to work with were in elementary school. the oldest child was in fourth grade; the youngest was in first grade. the mathematical content area that the preservice teacher selected to implement with the ell considerably varied because the content was based on what the placement teacher was teaching. there was complete freedom in terms of the study design in relation to the selection of mathematics word problems; however, there were sometimes limitations provided by the placement teacher (based on curricular goals or the structure of content). none of the preservice teachers noted any interaction or changes of the word problem by the placement teacher; all indicated that they implemented the accommodations without the placement teacher’s feedback. data were gathered throughout the semester based on the preservice teachers’ reflective responses to adapting curriculum and working with the ells. data were based on the preservice teachers’ responses to the prompts described; both a pre-response and post-response were collected. the preservice teachers followed the process outlined in figure 1. when the preservice teachers wrote their reflections, they were asked to focus on the following prompts: 1. explain who this student is. what is his/her background? how long has he/she been in this country? what is his/her first language? what other schools has he/she attended, and where are the schools? what is his/her ell level? how old is he/she? what grade level is he/she in? any other pertinent information? provide details about your student. 2. analyze the student’s responses and/or actions to each of the four problems. 3. problem #1 4. problem #2 kurz et al. adapting word problems journal of urban mathematics education vol. 10, no. 1 39 5. problem #3 6. problem #4 7. did the student appear to understand the language of the problems? explain and supply supportive evidence to back up your statements. 8. did the student appear to understand the mathematics of the problems? explain and supply supportive evidence to back up your statements. 9. what do you think this student needs to better understand the word problems? 10. if you were this student’s teacher, how would you help him/her? what would you do? 11. staple the student’s work for each problem to the back of this reflection. figure 1. an outline of the steps followed by the preservice teachers during the semester. when the word problems were administered after the readjustment, the same prompts were asked; however, there was an emphasis instead on the rewritten word problems. for example, the new question #6 read: “revisit your answer to #6 in the first reflection. would you still answer this question the same? explain and support your stance.” the preservice teachers did not prepare an interview script but were instead asked to question and interview by asking the ell to solve the problem and kurz et al. adapting word problems journal of urban mathematics education vol. 10, no. 1 40 explain his/her reasoning by thinking aloud. they were also encouraged to ask follow-up questions of the ell as needed. when administering the word problems to the ell, the preservice teachers were encouraged to read the problem aloud when requested or when they felt it would be helpful. the preservice teachers were also encouraged to talk to the ell and to encourage think-alouds while they questioned the ell’s reasoning. they were discouraged from helping the ell solve the problem or guide the solution of the problem. data analysis using steps described by lecompte (2000), the data were qualitatively analyzed: (a) tidying up, (b) finding items, (c) creating stable sets of data, (d) creating patterns, and (e) assembling structures. the term data refers to the preservice teachers’ analyses of their interactions with the ells based on their preand postresponses to the prompts (see figure 1). first, the data were identified and organized. this identifying/organizing involved sorting through all of the data for preservice teachers who emphasized mathematics specifically and making sure that paperwork and data were in order. second, the process of finding items was initiated. we continually sifted through the preservice teachers’ responses to the prompts to look for items that were relevant to the research questions. next, we evaluated the data with an emphasis on both frequency and declaration with evidence (lecompte, 2000). for example, if a preservice teacher said that she or he decreased word count it was verified by analyzing the original question with the changed version provided by the preservice teacher. after the items were identified, they were organized into groups. we then compared and contrasted the statements of the preservice teachers looking for an organized structure to their adaptations and analysis of their interactions. patterns were then created in the fourth step. the items were reassembled into a coherent pattern to describe what adaptations were implemented and how they influenced knowledge of working with ells. these patterns were revisited and reevaluated throughout the data analysis until a cohesive taxonomy was identified. and finally, the structures were assembled to help build an overall description of the implemented adaptations (see lecompte, 2000, for the complete steps). findings reflections from six preservice teachers were collectively analyzed. after following lecompte’s (2000) procedures, four themes were employed or suggested after the rewrite by the preservice teachers based on the analysis of their preand post-interaction prompt responses. these themes were: language adaptations, math kurz et al. adapting word problems journal of urban mathematics education vol. 10, no. 1 41 ematical adaptations, tool/visual adaptations, and structural adaptations. every adaptation made by the preservice teachers fit into one of these four broad categories. the impact the adaptation had on preservice teachers’ learning in reference to their ells is discussed within each theme. to remain anonymous, the names of the ells were removed; they are simply referred to as the ell. commentary regarding success, failure, what worked or did not was from the participant (not us); it was not a discussion but simply results from the participants’ experiences. in terms of the mathematical content of the word problems, nearly all of the adjusted problems were number and operations (21/27). the other content came from algebra (3/27), geometry and measurement (2/27), and probability (1/27). (some students decided to implement five questions instead of four, so the total number of questions adjusted was 27, not 24.) language adaptations language adaptations were the most often implemented adaptation made by all six preservice teachers. when they implemented the word problems the first time they were able to observe difficulties the ells had with specific vocabulary words. one of the preservice teachers discussed the frustration he saw with his ell in terms of the word problem: “he greatly needed teacher assistance to help break up the problem to simplify which data to use in order to correctly solve the word problem.” this preservice teacher then made adaptations to the language (along with structural adaptations). he observed that “[the ell] showed improved understanding for simplified text of the word problem. the answer provided…was incorrect due to poor mathematics…he was able to decipher [the] information needed to…answer the problem.” another preservice teacher supported his observations. she stated, “the integrity of the math problem was not damaged, the math problem was simplified in that the reading was only simplified not the math.” another preservice teacher discussed issues with terms as well. she stated that the ell struggles “with…his…unfamiliarity with the words bought and brought. the words appear very similar but indicate a very different action.” she reduced the language demands for the ell. in addition, she stated that there was too much unnecessary information that distracted from the mathematics: “it made it hard for him to determine exactly what was happening in the story and what math operation represented it.” she found that her changes (simplifying vocabulary and removing unnecessary vocabulary) helped. the ell seemed “very relaxed and confident…he didn’t even ask for help or look at me.” an example of language adaptations can be seen in the following first-grade sample problem. the original problem was: “an emperor penguin ate 13 fish for breakfast. at lunch, she ate some more fish. she ate a total of 23 fish. how many fish did she eat for lunch?” (kyrene school district, 2009) the preservice teacher observed, “the wording of the problem prevented the ell from following the ac kurz et al. adapting word problems journal of urban mathematics education vol. 10, no. 1 42 tion of the problem. the complicated language made it more difficult for the ell to understand the math.” the revised problem read: “at sunrise, a bird eats 13 fish. at sundown, the bird eats some more fish. the bird eats 23 fish that day. how much did the bird eat at sundown?” the reflection of the preservice teacher stated, “i can see that the ell has very strong math skills and is perfectly capable of solving complicated problems. she just needs the chance to use them by being able to understand what the problem is asking.” mathematical adaptations there were various adaptations made in terms of mathematics as well—all six made adaptions relating to the terms. the preservice teachers adjusted the form of the numbers and the mathematical terms. specifically, a preservice teacher replaced the numeric words with numbers (6 instead of six). this adaptation seemed to help, “using numbers instead of number words is always easier for a first grader.” another preservice teacher questioned what to do when an ell struggles with a critical term in mathematics. the ell struggled with the term “quotient.” the preservice teacher debated whether or not the term should be removed. he felt “uncomfortable removing the word quotient because of its significance in math vocabulary.” in the end, he adapted the word problem removing the term quotient. he stated, “i felt that understanding the steps in the math calculation were more important than the labels used like quotient.” the ell was still unsuccessful. the preservice teacher believed that the lack of success was an issue with understanding what division means: “it is pretty clear…that his math skills are weak in understanding the components of a division problem.” he recommended support for the ell in the topic of division. some of the preservice teachers perceived difficulties in mathematical terms and concepts. for example, clarify mathematical terms is demonstrated in an adaptation of a fourth-grade sample problem. john has 10 pairs of white socks and 1 pair of blue socks in his drawer. there are no other socks in the drawer. without looking, he takes 1 pair out of the drawer. what are his chances of choosing a white pair of socks? (arizona department of education, 2009) a. certain b. impossible c. likely d. unlikely the preservice teacher adjusted the problem as follows: in john’s drawer, he has only 10 pairs of white socks and 1 pair of blue socks. without looking, he takes 1 pair of socks out of the drawer. what is the probability of choosing 1 pair of white socks? kurz et al. adapting word problems journal of urban mathematics education vol. 10, no. 1 43 a. 100% chance b. 0% chance c. 91% chance d. 50% chance in the preservice teacher’s reflection, she said, “[the ell] stated that he liked having numbers as a choice as opposed to vocabulary.” tool/visual adaptations some of the preservice teachers discussed the need for tools to support their ells while completing the word problems; this adaptation was made by three of the preservice teachers. one of the preservice teachers said, “he did not know the relationship between meters and centimeters. if i were his teacher, the use of [meter] sticks are the type of manipulatives this student could benefit by using to help learn the metric system.” the preservice teacher with the ell who struggled with division felt that “the use of manipulatives to better understand how to calculate a division problem” would be an important focus area for the ell’s teacher. another stated, “i would use play money as a tool and allow him to practice handling money and counting it back in practice scenarios.” money would provide a visualization to connect the numeric value with the visual representation using currency. moreover, some ells may not be as familiar with american currency and may need to gain experience. another preservice teacher used pictures to structure the simple addition problem. the first grade ell was supposed to total the two quantities in the word problem. a preservice teacher stated that the rewritten problems “included pictures to represent the numbers in the word problems.” the preservice teacher found that the ell was much more successful with this adaptation. another preservice teacher used pictures of the items described in the word problems. she stated, “adding pictures…could have been too much guidance. since he is an ell, i felt adding pictures would help him, but i am not sure if it helped too much.” a sample provided by a preservice teacher originally read: “solve. farmer dan had 37 rows of corn on his farm last year. this year, he has double that number of rows of corn. how many rows of corn does farmer dan have this year?” (charles, crown, & fennell, 2004) the preservice teacher kept the sentences the same but added 37 pictures of corn. the preservice teacher stated that supplying pictures allowed the student to “work through the problems with greater ease than [the ell] did with the first version…it took half the time.” structural adaptations the preservice teachers also changed the structure of the problems; this adaptation was made by three of the preservice teachers. for example, one preservice kurz et al. adapting word problems journal of urban mathematics education vol. 10, no. 1 44 teacher broke up a word problem into smaller sentences that were written with one sentence per line rather than in paragraph form. after making her adaptations, she stated, “this layout was much better [for the ell]; it was easier to read because the short sentences were broken up from each other. i just made it easier to pick out key phrases.” the ell was more successful when the preservice teacher altered the structure. another preservice teacher stated, “in the rewrite of the problem, the arrangement of numbers was reversed to help emphasize the question of how much more.” she stated that the adaptation helped the ell better understand the mathematics contained in the problem. for example, a problem read: “16 penguins were playing in the ocean. 10 more penguins jumped into the ocean to play. how many penguins are playing in the ocean?” (kyrene school district, 2009) the question was structurally adapted (along with language adaptations): 16 birds played in a tree 10 more birds came to play how many birds are playing in the tree? in the reflection, the preservice teacher stated that the child made progress when answering the question: the first time she worked the problem, [she] seemed to rush through the problem…and just picked out two numbers from the problem and put them in a number sentence…the second time, [she] spent a great deal of time thinking…i simplified the problem so that she could follow the action. discussion when the preservice teachers were provided with the chance to work one-onone with an ell, they were able to implement adaptations often discussed in theory. this implemented structure provided opportunities to move beyond theory written in a textbook to practice with an ell. preservice teachers were able to experience the complexities of making adaptations while noting the benefits (and sometimes obstacles as in the division example) the adaptations had on the ell’s understanding of mathematical word problems. mihai and pappamihiel (2012) have discussed the critical role of having preservice teachers engage with ells. the practices and insights learned in coursework are likely to be most effective once preservice teachers are working regularly with ells and have a clear understanding of the learning challenges ells face. in our case, the preservice teachers were clearly able to apply what they were learning in class in adapting the work for their ells. the preservice teachers of this study successfully implemented the language adaptations described by abedi and colleagues (2006). the preservice teachers were able to analyze a word problem and simplify the linguistic demands without kurz et al. adapting word problems journal of urban mathematics education vol. 10, no. 1 45 compromising the intent of either the word problem or the mathematics. while the intent of the word problems was not compromised, there were some issues with respect to compromising cognitive demand. for example, some of the preservice teachers seemed to guide the students too much (by producing pictures that really limited thinking and restricted multiple approaches or entry points) compromising students’ thought processes. there seemed to be a focus on making sure the ells were able to get the right answer rather than providing opportunities to challenge the students at an appropriate mathematical level (that may or may not lead to a correct answer; gillmor, poggio, & embretson, 2015; kapur, 2014). in this study, we found that it was a challenge for preservice teachers to learn how to finesse word problems (or other problems for that matter) while still maintaining an appropriate and meaningful level of cognitive demand (feldon, 2007; schnotz & kürschner, 2007). research has shown that mathematical items can be linguistically adjusted to reduce the language load without altering the construct being assessed (sato, 2008; swanson, 2015; swanson et al., 2013). however, doing so requires that preservice teachers understand the ell’s level of english proficiency; in particular, understanding which words may be unfamiliar or challenging (haag, heppt, stanat, kuhl, & pant, 2013). numerous researchers have emphasized the importance of understanding ells’ english proficiency levels in order to make such adaptations (carr et al., 2009; echevarria et al., 2008; mihai & pappamihiel, 2012; wright, 2010). preservice teachers were able to replace long and unfamiliar words with words that were easier to read or were more familiar. they also helped ells by breaking down sentences that were difficult in terms of their grammatical complexity and by using more familiar verb tenses (such as present tense). when preservice teachers changed the terms (e.g., the brought/bought example), they were able to experience the ells’ difficulties within the context of learning. the adaptation of mathematical terms implemented by the preservice teachers demonstrated their ability to make changes to try and meet the language needs of the ells. although it is important that ells develop mathematical academic vocabulary, it is also important that teachers learn how to distinguish between terms that comprise essential mathematical vocabulary (abedi, 2006). this form of adaptation posed some difficulty for the participants as evidenced in the quotient term analysis. the preservice teacher questioned whether the term was an essential mathematical term and whether or not it should be changed. adjusting mathematical terms requires that teachers understand issues of scope and sequence in mathematics given that the goal is mastery of the subject matter; it is complex (nutta, mokhtari, & strebel, 2012). for example, if a preservice teacher does not understand the trajectory of mathematical content, then it is difficult to identify what is important and relevant in the current context. if the focus is on understanding whether or not students can build or comprehend a number’s value, then exchang kurz et al. adapting word problems journal of urban mathematics education vol. 10, no. 1 46 ing 6 for six (as explained in the results) should not be problematic. of course, the eventual goal is for ells to be able to solve the problem regardless of how the numbers are presented. the use of tools was not a specific component of our framework (figure 1); however, the use of tools was discussed within the course. several of the participants discussed the potential benefits of using tools within the context of helping ells. specifically, the results showcased the perceived importance of tools as visual aids to support learning. while tools and visuals can benefit all, they are often advantageous to ells as tools help connect the language to an object (like connecting a meter stick with its actual length as one preservice teacher explained; furner, yahya, & duffy, 2005; garrison & mora, 1999). pictures and other graphical representations can also be used to demonstrate understanding (chamot, 2009), and several preservice teachers adjusted their word problems using visualizations (e.g., providing drawings of corn). however, it is also important to note that while ells can benefit from tools and visuals, language demands should also be reduced while simultaneously developing english skills (harper & de jong, 2004). while the preservice teachers recognized the usefulness of tools and manipulatives, they were able to voice the need for other supportive adaptations. the structural adaptations represented an understanding by preservice teachers of how the structural presentation of the problem impacted ells’ ability to understand what was being asked of them mathematically. by visually breaking apart the word problem, it allowed the ell to focus on the mathematical concepts as evidence by the preservice teacher who used a list of sentences rather than a paragraph. this type of adaptation also requires teachers to have a firm grasp of ells’ language proficiency in order to anticipate structural difficulties (echevarria & graves, 2010). rearranging how the numbers were presented in the word problem reflected one preservice teacher’s understanding of how beginning ells may be translating from english to their native language word for word and how that may impact how the ells process the information. it is important to emphasize that the eventual goal is for all ells to have enough mastery of english and mathematical concepts to solve problems regardless of how they are presented on various standardized tests (chamot, 2009). however, scaffolding word problems for ells in ways such as those discussed here can help ells on their way to that goal (carr et al., 2009; orosco et al., 2011). implications for teacher educators by having preservice teachers engage in fieldwork with ells, they are able to see the “real-world” application of what they are learning in their coursework (fitts & gross, 2012; mihai & pappamihiel, 2012). understanding ells is critical to better meeting their needs (rhine, 1995) and encouraging ells to talk can support their development (bielenberg & fillmore, 2004/2005). the adaptation of cur kurz et al. adapting word problems journal of urban mathematics education vol. 10, no. 1 47 riculum for ells is not an easy process; it takes time and practice (hite & evans, 2006; orosco et al., 2011). it also requires collaboration between content area specialists and, in this case, mathematics education faculty and faculty with expertise working with ells (nutta et al., 2012). together, faculty can effectively design a framework that meets the needs of the preservice teachers as well as the ells they will serve. furthermore, when preservice teachers see faculty collaborating, they can learn from that modeling for their own future collaboration with colleagues. the framework provided allowed preservice teachers to take a first step in learning how to adapt curriculum to better meet the needs of ells. preservice teachers should be provided with an opportunity to learn about ells from ells; doing so puts the learning into a context that will support theory aligning with authentic practice (mihai & pappamihiel, 2012). perhaps this structure will allow preservice teachers to gain experience that they will carry over into their careers as teachers. further research needs to be conducted to investigate this idea as well as how other content areas can benefit from such a framework. it is important to note that there were certain limitations to this study. first and foremost, the sample size is quite small and may not be representative of other preservice teachers (our study was mostly white women). also, the structure of the framework seemed to confine the preservice teachers to number and operations problems. while there was no restriction to the types of problems selected, for some reason, most were from this content area. because of the lack of mathematical diversity, this limitation can create issues in terms of truly understanding the adaptions. it seems that adaptions were more commonly implemented with number and operations, so measuring preservice teachers’ changes in thinking is limited to word problems that focused on this content area. and finally, the emphasis on helping the ell get the “right answer” rather than productive struggle with meaning was an issue. the preservice teachers seemed to measure success based on correctness of the problem and did not take the time to understand the student’s thinking. future implementation of this framework should include a discussion on the meaning of success in mathematics (see, e.g., gillmor, poggio, & embretson, 2015; kapur, 2014). concluding thoughts while preservice teachers are often in need of experiences working with ells, it is sometimes a neglected area of focus in education programs (ernst-slavit & slavit, 2007; freeman & crawford 2008). the framework provided here (figure 1) along with the analysis of preservice teachers implementation and reflection of adaptations demonstrate the complexities of teaching ells word problems. additionally, the potential of guiding preservice teachers in this area is also demonstrated. this framework provides an opportunity for preservice teachers to learn from kurz et al. adapting word problems journal of urban mathematics education vol. 10, no. 1 48 ells while providing opportunities to put theory learned in courses into practice that can ultimately impact ells’ opportunities to succeed in mathematics classes (furner et al., 2005). adapting the text is a sei technique that involves rewriting specific sections of a text containing critical concepts and information that remain intact in the process. adapting the text and adjusting readability is time consuming and requires effort and thought (walkington, clinton, ritter, & nathan, 2015). but if we are serious about meeting the educational needs of this student population, the added time and effort involved in the process of adapting the material will be beneficial because mathematics material will become more accessible to ells. references abedi, j. 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(2014). teaching mathematics for korean language learners based on ell education models. zdm mathematics education, 46(6), 939–951. http://ell.stanford.edu/sites/default/files/pdf/academic-papers/10-santos%20ldh%20teacher%20development%20final.pdf http://ell.stanford.edu/sites/default/files/pdf/academic-papers/10-santos%20ldh%20teacher%20development%20final.pdf http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/233/160 journal of urban mathematics education july 2017, vol. 10, no. 1, pp. 52–73 ©jume. http://education.gsu.edu/jume jessica morales-chicas is an assistant professor at california state university, los angeles in the department of child and family studies, 5151 state university drive, los angeles, ca 90032-8530; email: jmora163@calstatela.edu. her research interests include urban youth achievement motivation, equity and pathways to stem, and school ethnic diversity on school adjustment. charlotte agger is a lecturer at indiana university, bloomington in the kelley school of business and affiliated faculty in the school of education, 1319 e. 10th street, bloomington, in 47405; email: cagger@indiana.edu. her research interests include student motivation, underserved students, and pathways to postsecondary education. the effects of teacher collective responsibility on the mathematics achievement of students who repeat algebra jessica morales-chicas california state university, los angeles charlotte agger indiana university in this article, the authors use the national high school longitudinal study of 2009 (hsls:09) dataset to explore (a) if repeating algebra in the eighth grade was associated with overall mathematics grades and course-taking patterns by twelfth grade, (b) if repeating algebra in the eighth grade was associated with students’ final grade in algebra, (c) if the level of teacher collective responsibility of mathematics teachers in school predicted students’ who repeated algebra final grade in algebra, and (d) if this association differed by students’ gender. the authors’ analysis suggests that repeating algebra may bolster mathematics success for certain students; however, in schools with low perceptions of collective responsibility among teachers, final grades in algebra were lower for male students repeating algebra. implications for achievement and long-term course-taking patterns when students repeat algebra are discussed. keywords: algebra, course-taking patterns, mathematics achievement, mathematics policy, teacher collective responsibility lgebra represents an important gatekeeper to higher-level mathematics and science courses, high school graduation, and entrance into college (attewell & domina, 2008; liang, heckman, & abedi, 2012; schiller & muller, 2003). therefore, an increasing number of states are encouraging students to enroll in algebra; this initiative is known, by many, as the algebra for all educational reform initiative (domina & saldana, 2012). in addition to promoting algebra enrollment, this initiative also pushes students to take algebra earlier in middle school to give more students the opportunity to complete higher-level mathematics courses while in high school (allensworth & nomi, 2009). a http://education.gsu.edu/jume mailto:jmora163@calstatela.edu mailto:cagger@indiana.edu morales-chicas & agger students who repeat algebra journal of urban mathematics education vol. 10, no. 1 53 because of the algebra for all initiative, there have been steady increases in eighthand ninth-graders’ algebra course taking over the past several years across the united states (national center for education statistics [nces], 2005). in california, the percentage of eighth-grade student enrollment in algebra has almost doubled from 32% in 2003 to 59% in 2011 (liang et al., 2012). these drastic increases in algebra enrollment necessitate more investigation into the implications for students’ achievement and potential in mathematics, especially among subgroups of students. accordingly, the current study examined the characteristics and achievement of an often-overlooked subgroup of algebra students, those who repeat algebra, and the role that teachers play in these students’ achievement. while requiring students to take algebra is important, there is mixed evidence regarding whether there is an achievement benefit of taking algebra during middle school, as opposed to starting algebra for the first time in high school. some studies have found that exposing students to advanced mathematics curricula early is associated with larger achievement gains (attewell & domina, 2008; domina, 2014; gamoran & hannigan, 2000). other studies document greater enrollment in subsequent advanced mathematics courses for students who take algebra in the eighth grade (spielhagen, 2006; stein, kaufman, sherman, & hillen, 2011). on the other hand, competing evidence suggests that increases in algebra enrollment are associated with lowered achievement scores on tests such as the california high school exit exam (cahsee; clotfelter, ladd, & vigdor, 2012; domina, mceachin, penner, & penner, 2015) and early algebra enrollment does not predict more advanced mathematics course taking (domina et al., 2015; loveless, 2008). one explanation for these mixed results could be that this policy has focused on the timing of algebra enrollment and has spent relatively less effort on whether students who take algebra early on receive the same quality instruction, teacher support, or adequately learn the fundamental mathematics skills to move through a higher-level mathematics course trajectory. overview of the literature students who repeat algebra students who repeat algebra are often overlooked and merit more attention as the algebra for all initiative is implemented nationwide. there are many reasons why a student might repeat algebra, and these reasons largely depend on the student, school, and school district. despite the myriad reasons for repeating algebra, it is typically a result of low performance in earlier mathematics courses, teacher recommendations, and course grades (bitter & o’day, 2010; fong, jaquet, & finkelstein, 2014). it is also likely that recent increases in the number of students who repeat algebra may be a byproduct of the algebra for all initiative. structural shifts in morales-chicas & agger students who repeat algebra journal of urban mathematics education vol. 10, no. 1 54 schools, such as the reassignment of teachers to accommodate for increases in algebra enrollment, or students’ lack of preparedness for higher-level mathematics, could all contribute to increases in the number of students who repeat algebra. thus far, limited work has explored the experiences of students who repeat algebra; most existing studies only examine demographic characteristics. a study of 3,400 california students found that nearly 45% repeated algebra and that this percentage was higher for english language learners, latinas/os, and students in special education (fong et al., 2014). fong and colleagues (2014) also examined student achievement outcomes of students who repeated algebra in california. they found that overall student achievement improved for both lowand high-achieving students when they repeated algebra. however, other studies on the achievement of students who repeat algebra showed more discouraging results. finkelstein, fong, tiffany-morales, shields, and huang (2012) found only 21% of ninth-grade students who repeated algebra achieved proficiency on standardized mathematics assessments on their second attempt. waterman (2010) also found that among students who did well in eighth-grade algebra (i.e., received a grade of bor better) and had to repeat algebra again in the ninth grade, nearly half received the same or worse grades after repeating the course. the mixed results of prior studies, as well as the limited literature on students who repeat algebra, motivated our investigation of the achievement of students who repeat algebra and factors that could predict this achievement. a recent study by howard, romero, scott, and saddler (2015) aimed to better understand students who failed algebra by examining their test achievement after failure, motivation, and readiness for college. using the high school longitudinal study of 2009 (hsls:09), howard and colleagues (2015) found that students who failed algebra in the eighth grade were similar in terms of mathematics proficiency compared to students who passed lower-level mathematics courses but reported lower mathematics motivation. although this study provided more information on students who fail algebra and their subsequent motivation, there is still a lack of information about students who repeat algebra in particular. for example, little is known about whether retaking algebra from eighth to ninth grade (the transition to high school) helps mathematics achievement, especially over time. moreover, although howard and colleagues’ (2015) study documented differential motivational patterns for students who repeat algebra, scant studies have investigated whether these students differ in mathematics achievement or mathematics course enrollment by the end of high school. to help address these gaps in the literature, we used a diverse and national sample of united states high school students and investigated whether students who repeated algebra from eighth to ninth grade differed from students who took algebra for the first time in ninth grade on the following outcomes: (a) advanced placement (ap) mathematics enrollment by twelfth grade, (b) mathematics grade point average, and (c) grade point average in stem (science, morales-chicas & agger students who repeat algebra journal of urban mathematics education vol. 10, no. 1 55 technology, engineering, and mathematics) courses. the study reported here also extends prior research as we examine the predictive relations of an important teacher-based variable on final algebra grades between students who repeat algebra and first-time algebra takers. gender was used as a moderator in this association. the role of teachers in student achievement in exploring predictors of mathematics achievement among students who repeat algebra, we specifically focused on the role of teachers in shaping that achievement. over the last several decades, researchers have found that in addition to students’ individual characteristics, such as socioeconomic status (ses) and prior achievement, teacher influences can play a large role in shaping student achievement (caprara, barbaranelli, steca, & malone, 2006; goddard, hoy, & woolfolk hoy, 2004; goddard, logerfo, & hoy, 2004). the ways that teachers view their students’ abilities, and their own roles and responsibilities in educating children affect student outcomes. for example, teachers’ self-efficacy beliefs are shown to influence students’ school success (caprara et al., 2006; muijs & rejnolds, 2001). in addition, teachers’ expectations for their students influence student learning such that high teacher expectations are associated with gains in student learning (firestone & rosenblum, 1988). here, we are not interested in the specific influence one teacher could have but rather on the collective responsibility mathematics teachers have for students’ learning. teacher collective responsibility. teacher collective responsibility is defined as the degree to which teachers feel responsibility for student learning (lee & smith, 1996; logerfo & goddard, 2008). teacher collective responsibility also emphasizes the obligation and trust among teachers and school administrators (bryk & schneider, 2002), particularly regarding high accountability for student learning (bolam, mcmahon, stoll, thomas, & wallace, 2005). this construct is distinct from teacher efficacy or individual teacher responsibility because teacher collective responsibility captures school culture, and how teachers perceive their colleagues as accepting responsibility for student learning (logerfo & goddard, 2008). in schools where there is high teacher collective responsibility, teachers have a sense of shared responsibility for the success or failure of their students. on the other hand, in schools with low teacher collective responsibility, teachers might attribute student success and failure, not to themselves, but rather to student characteristics or schooling conditions (lee & loeb, 2000). although limited research has examined teacher collective responsibility, extant work has found that teachers’ collective responsibility is an important factor in shaping student achievement. lee and smith (1996) initially found that collective responsibility was positively associated with students’ achievement after controlling for ses. more recently, logerfo and goddard (2008) and lee and loeb (2000) similarly found that teacher collective responsibility was associated with morales-chicas & agger students who repeat algebra journal of urban mathematics education vol. 10, no. 1 56 increased student learning. research also suggests that in schools where teachers hold strong beliefs of collective responsibility, student-learning gains are more equitably distributed within the school across, for example, students from different socioeconomic groups (lee & loeb, 2000; lee & smith, 1996). in other words, schools with higher teacher collective responsibility are not only more likely to have higher teaching effectiveness but also tend to foster a more equitable learning environment for students. teacher collective responsibility may be an especially important part of school climate during times of change or reform (louis, marks, & kruse, 1996). because teacher collective efficacy encourages mutual support and responsibility for students’ learning (kruse, louis, & bryk, 1995), research suggests it could potentially act as a protective factor during times of organizational instability or change (whalan, 2010). as the algebra for all initiative has rolled out in schools, teachers’ tasks and responsibilities are being constantly adjusted, such as reassigning inexperienced teachers to teach algebra (clotfelter et al., 2012). these constant shifts, new assignments, and new regulatory guidelines likely influence teachers’ efficacy and responsibility beliefs. however, little is known about how mathematics teachers perceive their collective responsibility in the context of the algebra for all initiative and whether these beliefs affect student achievement. thus, our second aim of this study explored whether mathematics teachers’ perceptions of collective responsibility played a role in predicting algebra repeaters’ final grade in algebra. in conceptualizing the study reported here, we drew on ideas rooted in social cognitive theory (bandura, 1977). teacher collective responsibility is an extension of research on teacher beliefs and attitudes, stemming out of research on teacher self-efficacy and locus of control (logerfo & goddard, 2008). teacher selfefficacy involves teachers’ beliefs in their abilities to effectively engage in instruction and as a result, positively influence student learning (tschannen-moran & woolfolk hoy, 2001). in addition, teacher locus of control refers to teachers’ tendency to attribute student success or failure to their own performance (ross, 1995). the construct of teacher collective responsibility extends these ideas, yet is distinct from teacher self-efficacy and locus of control. teacher collective responsibility is a willingness to be proactive after efficacy beliefs are constructed and a locus of control is attributed to internal, rather than external, factors (logerfo & goddard, 2008). therefore, teacher collective responsibility is defined as exhibiting collective agency, or the shared beliefs of a group of people that they can collectively work to produce desired effects (goddard, hoy, & woolfolk hoy, 2000). because more teacher collective responsibility establishes a school culture where teachers assume the joint responsibility to help students succeed, we expect that more perceived teacher collective responsibility might motivate teachers to work harder to ensure that all students learn and excel in mathematics. more teacher collective responsi morales-chicas & agger students who repeat algebra journal of urban mathematics education vol. 10, no. 1 57 bility might be especially important for students attempting to pass a gate-keeper course like algebra because of its implications for future mathematics course taking and mathematics motivation (finkelstein et al., 2012; howard et al., 2015). gender and mathematics achievement we also investigated the role that gender played on the associations among algebra enrollment status (i.e., whether a student repeats algebra or is taking it for the first time), algebra achievement, and teacher collective responsibility. gender gaps in performance on both school-based and national mathematics assessments are now virtually nonexistent (agger & meece, 2015). despite these gains, stereotypes and perceptions about gender gaps in mathematics performance are still prevalent at all levels of schooling. for example, girls’ lower performance and ability in mathematics still permeate the beliefs of students and teachers (beilock, gunderson, ramirez, & levine, 2010; cvencek, meltzoff, & greenwald, 2011; hyde, lindberg, linn, ellis, & williams, 2008; nosek et al., 2009). also, boys continue to report more positive mathematics attitudes and affect when compared to girls (elsequest, hyde, & linn, 2010). given the more favorable ability beliefs, motivation, and stereotypes regarding the mathematics achievement of boys, we predict that having to repeat algebra may be more detrimental for boys than girls. the aforementioned gender-related stereotypes that surround mathematics performance and ability are often ingrained in teachers. teachers have been found to overrate mathematics ability, have higher expectations, and more positive attitudes about male students in mathematics (li, 1999). as a result of these nuances in student and teacher perceptions of mathematics ability and performance, we argue that gender is an important consideration when studying students’ mathematics trajectories. although boys seem to be stereotyped as more competent in mathematics, little is known about whether teacher collective responsibility differentially affects students based on gender, particularly among those who repeat algebra. research questions the study reported here used a large, united states nationwide sample that allowed for exploration of both student and teacher factors that predict shortand long-term mathematics outcomes among diverse students repeating algebra or taking it for the first time in the ninth grade. the research questions that guided the inquiry were: 1. does repeating algebra between eighth and ninth grade predict mathematics grade point average, grade point average in stem courses, and/or ap mathematics enrollment trends by the twelfth grade? morales-chicas & agger students who repeat algebra journal of urban mathematics education vol. 10, no. 1 58 2. is repeating algebra associated with receiving a higher final grade in algebra compared to not repeating algebra, when controlling for student demographic characteristics (e.g., race, gender, etc.)? 3. do mathematics teachers’ perceptions of collective responsibility play a role in the achievement of students who repeat algebra? 4. do these associations differ by student’s gender? based on prior research, we hypothesized that students who repeat algebra will look significantly different from students who do not repeat the course in terms of their mathematics and stem grade point average and in their ap mathematics enrollment by the time they are in twelfth grade. in addition, we hypothesized that repeating algebra would be associated with lower final grades in algebra and that teacher collective responsibility would moderate this relation. to be more specific, given that high collective responsibility for students is associated with increases in student learning and more equitable learning environments (firestone & rosenblum, 1988; goddard et al., 2000; lee & smith, 1996), we posited that for students repeating algebra, lower levels of teacher collective responsibility would be associated with poorer student achievement in algebra grades. however, predictive relations between gender and teacher collective responsibility remained exploratory. methods data source, participants, and sampling participants included students who took part in the high school longitudinal study (hsls:09). the hsls:09 dataset, housed by the united states department of education, national center for educational statistics (nces), includes an ethnically diverse sample of students from throughout the country. hsls:09 employed a two-stage, random sample design where schools were the primary sampling units (psu). after public and private schools were randomly sampled (1,889 schools were eligible and 944 schools participated), a random sample of ninth-grade students within the psus was selected to participate. of the 25,206 students who were eligible to participate, 21,444 were surveyed (about 27 per school). in the base year students were surveyed in the ninth-grade year (2009), then the spring of their eleventh-grade year (2012). a follow-up data collection occurred in 2013, during the students’ expected graduation year, which collected information on students’ postsecondary and occupational plans. the current study uses information from all three waves of data. the sampling pattern employed in hsls:09 resulted in an ethnically diverse sample consisting of american indian/alaska native 0.73% (n = 163), asian 3.46% (n = 1,673), black/african american 13.52% (n = 2,214), hispanic (no race morales-chicas & agger students who repeat algebra journal of urban mathematics education vol. 10, no. 1 59 specified) 1.69% (n = 204), hispanic, race specified 20.51% (n = 3,311), more than one race 7.74% (n = 1,912), native hawaiian/pacific islander 0.50% (n = 110), and white 51.85% (n = 11,837). this sample was evenly divided by gender, with 10,887 (50.8%) male students and 10,557 (49.2%) female students. procedures. in the base year of the survey, in the fall of their ninth-grade year (2009), students were administered a mathematics assessment and survey component. the first follow-up study was conducted in the spring of most students’ eleventh-grade year (2012). there is also a 2013 follow up that collected high school transcripts, a second follow-up which occurred in 2016, and a follow-up planned for 2021 to gather information about students’ postsecondary enrollment and later adulthood experiences. measures algebra course pattern. using students’ eighthand ninth-grade self-reported mathematics class, two groups were created: (1) first-time algebra students who took algebra for the first time in the ninth grade and took either math 8, advanced or honors math 8, or pre-algebra in the eighth-grade (n = 8,327); and (2) students repeating algebra who took algebra in the ninth grade and in the eighth grade (n = 1,805). mathematics teachers’ collective responsibility. this variable (x1tmresp) was a composite consisting of seven items that measured mathematics teachers’ perceptions of collective responsibility (e.g., teachers at this school feel responsible that all students learn) among fellow teachers, α = .65. responses ranged from strongly agree to strongly disagree, with higher values representing greater collective responsibility. the scale was created by nces using principal components factor analysis and was weighted and standardized to a mean of 0 and standard deviation of 1. final grade in algebra. students reported their final grade in algebra i (s2alg1grade) in the first follow up of the hsls:09 study, in the spring of participants’ eleventh-grade year. grades ranged from: 1 = a (between 90–100), 2 = b (between 80–89), 3 = c (between 70–79), 4 = d (between 60–69), and 5 = below d (less than 60). grade 12 gpa in stem courses. students’ grade 12 gpa in stem related courses (x3tgpastem) was calculated using high school transcript information. stem courses were defined as courses in mathematics, science, computer and information sciences, and engineering and technology. gpas ranged from 0.25 to 4.00. grade 12 gpa in mathematics courses. students’ grade 12 gpa in mathematics (x3tgpamat) was calculated using high school transcript information. gpas ranged from 0.25 to 4.00. morales-chicas & agger students who repeat algebra journal of urban mathematics education vol. 10, no. 1 60 grade 12 ap mathematics course enrollment. students reported whether they were taking or have taken an ap mathematics course by the twelfth grade (s3apmath). response options were 1 = yes and 0 = no. gender. participants’ gender (i.e., either male or female) (x1sex) was collected from one of the following: student questionnaire, parent questionnaire, or school-provided sampling roster. if any responses were inconsistent, a review of the student’s first name was conducted. control variables. race (x1race), socioeconomic status quintile (x1sesq5), locale (x1locale), and mathematics test achievement (x1txmscr) were used as control variables. race was taken from participants’ self-report and consisted of eight categories (described in the participants’ section). locale was measured by identifying whether students were from a city, suburb, town, or rural area. the ses quintile variable was a composite capturing parent/guardian’s education, occupation, and family income. mathematics test achievement was measured in the fall of 2009 by estimating the total number of items that a participant would have answered correctly if they responded to all 72 items in the hsls:09 mathematics assessment. an (item response theory) irt-based estimate score was then created for each participant using ability estimates and item parameters derived from the irt calibration. analysis plan to address our first research question, separate independent samples t-tests were run to investigate whether students repeating algebra (when compared to students who took algebra for the first time in the ninth grade) had significantly higher (a) grade point averages in mathematics, (b) enrollment in ap mathematics courses, and (c) grade point averages in stem courses by the twelfth grade. to address the remaining research questions, multiple regression analysis models were run. model 1 tested the main effect of teacher collective responsibility, algebra enrollment status (students repeating algebra as the reference group), gender (males as the reference group), and several control variables (i.e., mathematics test score in the ninth grade, ses, locale [city as the reference group], and race [white as the reference group] on final grade in algebra. model 2 further tested if the effect of algebra enrollment status was conditional on either gender or teachers’ ratings of collective responsibility, while adjusting for all control variables. our final model, model 3, tested the three-way interaction between gender, algebra course status, and teachers’ perceived ratings of collective responsibility. our analysis utilized both analytic and balanced repeated replication (brr) weights provided by nces (ingels et al., 2014). because our analysis incorporated both student and teacher data, we used the base year mathematics course enrollee weight (w1mathtch). the analytic weights were used because they accounted for the complex survey design of hsls:09 and accordingly, generated appropriate morales-chicas & agger students who repeat algebra journal of urban mathematics education vol. 10, no. 1 61 estimates for the target population, with properly adjusted standard errors. in addition, all 200 base year student-level brr weights (w1student001-200) were included in the analysis. the brr weights were used for variance estimation, through calculating standard errors that accounted for the random sampling of students clustered within schools. analytic sample the current study focused on students who self-reported taking algebra in the ninth grade and who also took one of the following courses in the eighth grade: algebra, pre-algebra, math 8, or advanced or honors math 8 not including algebra (n = 10,132). participants who did not fit these criteria were excluded from the analysis. we were also interested in students’ final grade in algebra, thus, of the 10,132 students eligible for the study, students who were in non-graded algebra courses (n = 11) or courses in which students did not receive a grade (n = 65), were omitted from analyses. the total sample eligible for this study consisted of 10,056 participants. from the sample eligible, missing values were found for only two measures: teacher collective responsibility and final algebra grade. these missing data resulted from participants either not responding to the overall questionnaire (i.e., unit non-response) or because they omitted one or more related questions (i.e., item nonresponse). to account for missing data due to unit non-response, two adjustments were applied to each set of base weights to reduce bias (ingels et al., 2014). additionally, listwise deletion was used in each regression model to drop cases that were missing due to item non-response (n = 1,131). after addressing missing data, the final analytic sample included a total of 8,140 students. results figure 1 shows ninth graders’ final grade in algebra by comparing students repeating algebra to first-time algebra students (i.e., students taking algebra for the first time in the ninth grade). an independent samples t-test was run to determine if there were any initial differences in algebra grade by algebra course status (i.e., whether the student was enrolled in algebra for the first time or was repeating the course). the descriptive results suggest that by the end of the ninth grade, students repeating algebra earned significantly higher final grades in algebra (m = 3.94; brr se = 0.05) when compared to first-time algebra students (m = 3.77; brr se = 0.03), t(8,908) = 3.19, p < .01. figure 1 illustrates this pattern, suggesting that the biggest difference between grades occurred when there was higher achievement, which was less the case for students with lower mathematics grades. morales-chicas & agger students who repeat algebra journal of urban mathematics education vol. 10, no. 1 62 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 a b c d below d p r o p o r ti o n o f s tu d e n t w it h t h is g r a d e final grade in algebra algebra achievement students repeating algebra first-time algebra students figure 1. the distribution of students repeating algebra when compared to firsttime algebra students across final grades in algebra. we also wanted to examine how students repeating algebra, relative to students who were taking algebra for the first time, fared in overall mathematics grade point average, grade point average in stem courses, and ap mathematics enrollment by the time they reached the twelfth grade. our descriptive analyses showed that students repeating algebra attained significantly higher grade point averages in stem courses (m = 2.34; brr se = 0.04) by twelfth grade when compared to students who took algebra for the first time in the ninth grade (m = 2.18; brr se = 0.02), t(9,677) = 3.82, p < .001. additionally, students repeating algebra showed higher grade point averages in mathematics by twelfth grade (m = 2.26; brr se = 0.04) than students who took algebra for the first time in the ninth grade (m = 2.08; brr se = 0.02), t(9,690) = 3.70, p < .001. of the respondents eligible for this study who took an ap course, only n = 1,406 reported taking an ap mathematics course. chi-squared analyses in table 1 demonstrated that the percentage of participants that repeated algebra when compared to students who took algebra for the first time differed significantly on whether or not they took an ap mathematics course by the twelfth grade, χ2 (1, n = 1,406) = 6.24, p < .01. in addition to the descriptive analyses, our main research question involved testing the interaction between algebra course status, teacher collective responsibil morales-chicas & agger students who repeat algebra journal of urban mathematics education vol. 10, no. 1 63 ity, and gender. because results in table 2 for models 1 and 2 were conditional on the results in model 3, only model 3 findings are presented and discussed. model 3 findings suggest that the predictors in the model explained 16% of the variance in mathematics achievement (r² = .16, f(19, 181) = 24.48, p < .001). beginning with the control variables in model 3, we found that when all variables were held constant, an increase in students’ ses (β = .04, p < .05) and standardized mathematics test score (β = .03, p < .001) significantly predicted a decrease in final mathematics grade in algebra. table 1 cross-tabulation of algebra course status and ap enrollment algebra course pattern ap enrollment repeating first time χ2 no 246 757 6.24* yes 125 278 note: *p < .05. furthermore, there were significant differences among ethnic groups in terms of final grades in algebra. table 2 indicates that asian students’ final grade in algebra was significantly higher than white students’ (β = .22, p < .05), whereas hispanic students’ final algebra grade (β = -.18, p < .01) was significantly lower than white students’ final grade in algebra. table 2 multiple regression results for final grade in algebra parameter final grade in algebra model 1 model 2 model 3 constant 2.34 (.14)*** 2.36 (.15)*** 2.36 (.15)*** suburb .05 (.06) .05 (.06) .05 (.06) town .09 (.08) .10 (.08) .10 (.08) rural .15 (.08) .15 (.08) .15 (.08) socioeconomic status .04 (.02)* .04 (.02)* .04 (.02)* mathematics test score .03 (.00)*** .03 (.00)*** .03 (.00)*** american indian and alaska native -.40 (.22) -.40 (.22) -.40 (.22) asian .22 (.09)* .22 (.09)* .22 (.09)* black/african american -.13 (.08) -.13 (.08) -.13 (.08) hispanic (no race specified) -.35 (.23) -.36 (.22) -.35 (.24) hispanic -.18 (.08)* -.18 (.09)* -.18 (.08)* more than one race -.17 (.09) -.17 (.09) -.17 (.09) native hawaiian and pacific islander -.21 (.29) -.20 (.30) -.20 (.30) taking algebra for the first time -.08 (.07) -.10 (.10) -.09 (.10) female .26 (.04)*** .21 (.10)* .23 (.10)* morales-chicas & agger students who repeat algebra journal of urban mathematics education vol. 10, no. 1 64 teacher collective responsibility .02 (.02) .04 (.06) .14 (.07)* taking algebra for the first time x teacher collective responsibility -.04 (.05) -.17 (.07)* taking algebra for the first time x female .06 (.11) .04 (.11) female x teacher collective responsibility .04 (.04)* -.19 (.09)* taking algebra for the first time x female x teacher collective responsibility .27 (.10)** *p < .05; **p < .01; ***p < .001 note: white, male, and city are the reference group. figure 2. the three-way interaction between algebra course type, teacher collective responsibility, and gender. (left panel = female students; right panel = male students) morales-chicas & agger students who repeat algebra journal of urban mathematics education vol. 10, no. 1 65 the main finding, however, was a significant three-way interaction between the algebra course type, participant’s gender, and ratings of teacher collective responsibility (β = .27, p < .01). this interaction is illustrated in figure 2. across all levels of teacher collective responsibility and algebra course type, female students (figure 2, left panel) held a higher expected algebra grade when compared to male students (figure 2, right panel). the biggest difference between male and female students’ algebra grades, however, was when teacher collective responsibility was low (i.e., 1 standard deviation [sd] below the mean); in this case, male students repeating algebra showed significantly lower grades in algebra than female students repeating algebra (b = -.42, z = -2.63, p < .01). furthermore, (figure 2, right panel) when teacher collective responsibility was high, male students repeating algebra had significantly higher grades in algebra than male students who were taking algebra for the first time (b = .27, z = 2.56, p < .05). lastly, male students repeating algebra who were low on teacher collective responsibility also had lower grades in algebra (b = -.29, z = -2.06, p < .05) when compared to male students repeating algebra with high ratings of teacher collective responsibility. discussion the recent rise in algebra course taking across the united states (liang et al., 2012; nces, 2005) has spurred increases in the incidence of students repeating the course. using a nationwide sample of contemporary high school students, the study reported here sought to compare students who repeated algebra to students who took it for the first time in ninth grade on various mathematics success indicators. our descriptive results suggest that repeating algebra was associated with a higher gpa in mathematics and stem courses, as well as an increase in the number of ap mathematics enrollment by the twelfth grade. while repeating algebra initially holds students back from taking the next mathematics level, our findings suggest that in the long term, taking algebra a second time does not hinder students’ future in mathematics and in stem course pathways. on the contrary, these findings are consistent with prior work suggesting that repeating algebra can bolster later mathematics achievement and future mathematics success (attewell & domina, 2008; fong et al., 2014; gamoran & hannigan, 2000). preliminary results also show differences in final algebra grades in the ninth grade. although the number of students who earned lower grades (e.g., grade c or below) was more similar between algebra status groups, students who repeated algebra tended to earn more “a” grades than first time algebra takers. it is important to caution that it remains unclear whether the achievement benefits associated with repeating algebra are truly a function of repeating the course or instead partly explained by the potential benefits of already starting on a more advanced curriculum morales-chicas & agger students who repeat algebra journal of urban mathematics education vol. 10, no. 1 66 (i.e., starting algebra in the eighth grade). prior research has suggested that being placed in an advanced curricular track tends to be associated with higher motivation and teacher engagement (oakes, 2008). therefore, it could be that students who repeat algebra (especially those who tend to earn higher grades) may already have adaptive tools (e.g., motivation) to succeed; consequently, they may be better inclined to reap the benefits of retaking the course. it may also be the case that increased exposure to algebra material, through taking the course a second time, could boost motivation and confidence in that subject. more research is needed to understand how repeating courses is linked to motivation, achievement, and curricular track. the results from our main analysis also show that students who repeated algebra significantly differed from first-time course takers in their final algebra grade; however, differences in final algebra grade depended on teacher-rated collective responsibility and students’ gender. as an example, for male students who were repeating algebra, as teacher collective responsibility decreased so did their final grade in algebra. additionally, when looking at differences between male and female students, male students who repeated algebra when compared to female students who repeated algebra with low collective responsibility had significantly lower grades in algebra. these findings are consistent with prior evidence showing that lower teacher collective responsibility is associated with poorer achievement (lee & smith, 1996). although our descriptive results suggest that repeating algebra may not be detrimental for mathematics success, the results from our main analysis highlight that in schools with low mathematics teacher collective responsibility, mathematics grades for students repeating algebra may suffer, especially for male students. why would being a male student repeating algebra (compared to a female student repeating algebra) in a school with lower teacher collective responsibility be more harmful for final algebra grades? we posit two different explanations. first, we know from the literature on grade retention that repeating a grade contributes to poor mental health, negative attitudes about school, and lower gains in achievement, which can create a stigma of failure (feldman, smith, & waxman, 2014). given that male students are historically stereotyped as better in mathematics, having to repeat algebra (regardless of the reason) makes them deviant from this stereotype, which could activate a stigma of failure and lead them to disengage in the course. we expect this to be the case in schools with low teacher collective responsibility, because this context tends to have teachers who are less willing to feel responsible for student learning. thus, a disengaged mathematics teacher and disengaged student (as a result of repeating a course) could be a double-edged sword. for female students, however, teacher collective responsibility level matters less for their final grade in algebra, regardless of whether girls are repeating algebra or taking it for the first time. collectively, these findings indicate that for male students, morales-chicas & agger students who repeat algebra journal of urban mathematics education vol. 10, no. 1 67 attending schools where there is more teacher collective responsibility, could serve as a buffer of achievement particularly for male students who are taking algebra a second time. another interpretation of these findings could be framed using an attribution theory of achievement motivation (weiner, 2000). research on attribution theory suggests that boys who experience failure in mathematics tend to attribute their failure to external forces (e.g., the teacher), whereas girls tend to attribute their failure to their lack of ability (stipek & gralinski, 1991). given these achievement attribution patterns, we might expect that boys will tend to view the teacher as less of a support system and more of a contributor to poor performance in algebra. this tendency may be particularly salient in the case of students who repeat algebra, given that they tend to already report lower interest and utility in mathematics (howard et al., 2015). thus, having teachers in the school who feel less responsible for student learning might exacerbate these negative feelings and in turn affect achievement. alternatively, when teacher collective responsibility is high in schools, male students repeating algebra may be more motivated to try because higher teacher support in the classroom may be encouraging for future success. prior research on the role of teachers in the classroom has focused predominantly on the positive implications (e.g., better academic performance and engagement) of more teacher support (goodenow, 1993; puklek levpušček & zupančič, 2008); however, our research draws particular attention to the collective influence that mathematics teachers play within a school. in our study, teacher collective responsibility is something that students have no control over, yet it was associated with students’ success in the classroom. given the present findings, it is noteworthy to recognize that in addition to the direct influence of teachers in the classroom, the overall practices and school culture teachers follow within a school could play an important role on students’ achievement. limitations and future research there are several limitations to the study reported here that are important to address. first, although we examined the influence of teacher collective responsibility, we only assessed this phenomenon from the teachers’ perspectives and not from the students’ points of view. future research should compare whether students’ perceptions of teacher collective responsibility in mathematics could differ from the effect that teachers’ perceptions of collective responsibility has on achievement. moreover, it would be interesting to examine whether a mismatch in these two perceptions of teacher collective responsibility could weaken the direct association on student achievement. second, although we could account for differences between schools by using weights in the analyses, we were not able to conduct multilevel analyses due to restrictions from the public-use data. public-use data suppresses information on the identity of responding schools and the individuals within them. as a result, we were morales-chicas & agger students who repeat algebra journal of urban mathematics education vol. 10, no. 1 68 not able to use hierarchical linear modeling, which would allow for tests of whether differences between schools exist in the effect of teacher collective responsibility on final algebra grades. this more advanced statistical approach could have also provided relevant information about students’ mathematics course-taking patterns, mathematics rigor in the school or classroom, and mathematics culture within departments that often differ between schools or even districts. missing data on both teacher collective responsibility and students’ final grade in algebra also posed a limitation in our research. another limitation was that despite administering important controls in our analysis, there could have been some variation in the type of students who repeated the course that we did not consider. for example, we did not explicitly investigate why the students repeating algebra retook mathematics. some could have repeated algebra due to a poor grade while others could have repeated algebra due to poor school policies that do not accurately place students in the next level of mathematics. although this concern is beyond the scope of this research, this is an important issue to consider in future research that explores algebra repeaters. finally, although our research focused on the importance of teacher collective responsibility as a moderator of mathematics achievement, we recognize that many other teaching-related factors could also shape students’ algebra learning. in the present study, teacher collective responsibility captured the collective mathematics teacher investment and obligation teachers feel for student learning (logerfo & goddard, 2008). however, more proximal measures of teacher influence (e.g., teaching style) also shape algebra achievement. for example, mathematics classrooms that featured more engaging instructional techniques, better class organization, and more emotional support are associated with higher student achievement, even after accounting for baseline achievement (allen et al., 2013). active engagement with mathematics has also been found to serve as a protective factor of interest in mathematics (martin, 2009). more active engagement in the classroom may be of upmost importance when students struggle in mathematics, especially when they are forced to retake a course. to illustrate, boaler and sengupta-irving (2016) administered a 5week teaching intervention focusing on active engagement, challenging algebraic problems, and student collaboration with students who previously failed in mathematics. these student-focused teaching techniques helped improve mathematics grades, engagement, and overall interest in mathematics. collectively, these studies highlight the importance of studying proximal teacher influences. future research should delve into how these other variables relate to performance in algebra. closing thoughts despite the caveats in our study, our findings provide novel information on the shortand long-term benefits of repeating algebra. furthermore, the results of morales-chicas & agger students who repeat algebra journal of urban mathematics education vol. 10, no. 1 69 this study shed light on the importance that teacher collective responsibility could play on algebra grades when students (particularly male students) repeat algebra. to help foster a culture of more collective responsibility among teachers, we recommend encouraging teacher collaboration and emphasizing responsibility for student learning, especially for male students who may be overlooked due to stereotypical beliefs that they are higher achievers in mathematics (beilock et al., 2010; nosek et al., 2009). drawing on the results of our study, we also recommend providing professional development opportunities where the school administration could help increase teacher morale and emphasize the critical role that teachers play on mathematics achievement. while a shift toward more teacher collective responsibility could benefit all students and teachers, this has important implications for male students who are re-taking an important gate-keeping course (i.e., algebra) related to stem pathways. shifting educational policies in mathematics may also shape the curricular landscape for teachers and students that could consequently alter when students take algebra and what teaching approach is used. for a few years now, the algebra for all initiative has been a prominent force in mathematics educational policy that has accelerated algebra placement; however, the change toward emerging policies like the common core state standards may create a deceleration of mathematics placement in algebra courses. one of the key shifts in common core standards for mathematical practice1 is to focus less on racing to cover all the material and instead focus on assuring an understanding of the depth of the material. with rising changes in mathematics policy and the uncertain future of common core, there may be direct implications on the conversations and collective teamwork mathematics teachers partake within their schools. as an example, the new common core standards in mathematics require a gradual progression and connection of content from grade to grade; for these connections to happen, mathematics teachers must be willing to work together, to communicate, and to embed a collective effort to help students build on previous grade material. as these standards change, it is important that researchers capture whether any changes in teacher collective responsibility result from these new policies and how these changes may affect student learning, especially for students struggling in mathematics (e.g., students who repeat algebra). lastly, although closing the gender mathematics achievement gap has been somewhat at the forefront of mathematics policy discussions since the 1970s (jacobs, 2005), it is essential that teachers and policymakers recognize that male students (especially those struggling in mathematics) also need a strong support system. these findings are particularly relevant given new data suggesting that the gender mathematics achievement gap is closing, and that female students are per 1 see http://www.corestandards.org/math/practice/. http://www.corestandards.org/math/practice/ morales-chicas & agger students who repeat algebra journal of urban mathematics education vol. 10, no. 1 70 forming at similar rates as male students (hyde et al., 2008). thus, for students (especially male students) who repeat algebra, teacher support in the classroom is especially vital in helping students persist and succeed in mathematics. references agger, c. a., & meece, j. l. 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(1991). gender differences in children’s achievement-related beliefs and emotional responses to success and failure in mathematics. journal of educational psychology, 83(3), 361–371. tschannen-moran, m., & woolfolk hoy, a. (2001). teacher efficacy: capturing an elusive construct. teaching and teacher education, 17(7), 783–805. waterman, s. (2010). pathways report: dead ends and wrong turns on the path through algebra. palo alto, ca: noyce foundation. morales-chicas & agger students who repeat algebra journal of urban mathematics education vol. 10, no. 1 73 weiner, b. (2000). intrapersonal and interpersonal theories of motivation from an attributional perspective. educational psychology review, 12(1), 1–14. whalan, f. (2010). an investigation of teachers’ collective responsibility for student learning (unpublished doctoral dissertation). university of newcastle, australia. retrieved from http://hdl.handle.net/1959.13/805553 http://hdl.handle.net/1959.13/805553 microsoft word 2 final swanson vol 10 no 1.doc journal of urban mathematics education july 2017, vol. 10, no. 1, pp. 7–15 ©jume. http://education.gsu.edu/jume dalene m. swanson is a senior academic at the university of stirling, scotland (and adjunct professor at the university of alberta, canada), fk9 4la scotland, united kingdom; email: dalene.swanson@stir.ac.uk. her research commitments are to social and ecological justice, and her work contributes to democratic and decolonizing perspectives in mathematics education and curriculum. commentary mathematics education and the problem of political forgetting: in search of research methodologies for global crisis dalene m. swanson university of stirling, united kingdom forgetting our intentions is the most frequent of all acts of stupidity. – friedrich nietzsche mathematics is created in the self-alienation of the human spirit. the spirit cannot discover itself in mathematics. the human spirit lives in human institutions. – giovanni battista vico ore and more, standardized, efficiencies-based, and surveillance-driven modus operandi are prescriptively defining the interests of the individual and collective in terms of market-driven imperatives in consonance with the demands of the nation state competing for resources, means, and power on a global stage (swanson, 2010a, 2010b, 2013). while trumpianism and the rise of popul(ar)ist nationalism has confused the straightforwardness of the “common sense” of neoliberalism, it is without undoing its expansionist effects in an increasingly unequal world (gamal & swanson, in press). acting in accordance with “(inter)national” relations of exchange, this dominant economic rationalism is reflected in the production of consumer-driven homo economicus for the new knowledge economy through the increasing trend towards techno-scientistic corporatist economic utilitarianism in education, of which mathematics education plays a leading role under a veil of political neutrality. this growth of techno-scientistic and managerialist instrumentality is, for hobart (1993), aligned with the growth of ignorance. it tends to facilitate what biesta (2005) has referred to as “learning” discourses, or the prevalence of “learnification.”1 this functionalism is concomitant 1 biesta (2005) differentiates learning from education, bemoaning the fact that the “new language of learning” heralds a trend toward education as a marketable commodity invested in economic relations of exchange rather than something whose purpose and value may be deeply and intellectually debated in terms of democratic principles, where trust, violence, and human relationships m swanson commentary journal of urban mathematics education vol. 10, no. 1 8 with increasing privatization, standardization, instrumentality, and commodification of mathematics education curricula and educational environments glocally. this trend belies the increasing global political and economic uncertainty, ecological fragility and human precarity that has become the hallmark of our anthropocenic age, masked by the dominant assumption of “common/global good” in the advancements of global capitalism and reliance on the “naturalness” of “market forces.” this trend is a normalizing condition pervading all aspects of our lives and increasingly threatens foreclosing the public sphere, in arendtian terms,2 leaching away imaginative and practical capacity with the intended effect of largely disaggregating political will for resistance. it instigates the question: in our incremental accommodation of this general depoliticized commonsense hegemony, our slow capitulation to a diminished public space, and our relinquishing of freedoms even with greater consumerist “choice” and networked transnational intercommunicative access, is this neoliberal spread a form of global “political evil”3 as hayden (2009) asseverates in drawing on the political thought of hannah arendt? or is it, following friedrich nietzsche (1878/1996), merely stupidity and ignorance on our parts4 in forgetting what our intentions were and what we were trying to do? (swanson, 2010a) are necessary features of such a debate and of education itself. this differentiation also applies well to arguments being developed in relation to mathematics education more specifically. 2 bowers (2006) refers similarly to this effect as enclosing the (cultural) commons. 3 hayden (2009), working in the field of international relations and drawing on the political theory of hannah arendt, notes: even as globalization shapes the horizon of current political thought and action, it does so at the risk of drawing that horizon ever tighter; it is less certain that the concept of ‘globalization’ continues to express transformative potentials rather than functioning as a token of the very effacement of the political. globalization has become not only the political foundation of the present, but also the suspect guardian of the future of the political itself. … i argue that neoliberal economic globalization is a form of political evil. (p. 92) 4 this stupidity is itself an effect and offset of the political evil of neoliberalism, a production of ignorance that contributes to a symptomatic erasure of history, a making unnecessary the historical in the constitution and vitalization of the human condition. the apolitical, ahistorical comportment of modernization permits the “forgetting,” and the stupidity of such forgetting is thus tolerable, hence an ignore-ing/ance of the necessity of our political/historical condition in understanding what it means to be human. this ignoring and forgetting is an attributable structuring of neoliberalism rather than just a side effect. the forgetting is precisely what is forgotten within modernist conceits. swanson commentary journal of urban mathematics education vol. 10, no. 1 9 these absences, blind spots, denials, and active acts of forgetting are conceited inheritances invested in mathematics education and with which mathematics education has played an important part. this forgetting has been representative in the way it has reified, and been reified in, particular paradigms of being and knowing. as adorno reminds us: “all reification is a forgetting: objects become purely thing-like the moment they are retained for us without the continued presence of their other aspects: when something of them has been forgotten” (as cited in bewes, 2002, p. 208). for adorno, when such forgetting shapes experience it becomes an “epic forgetting.” through this definition, we might argue that mathematics education, historically and in the present, has become in many cases a practice of epic forgetting. through the role it often has played in discourses of modernism, it has helped to hold in place rather than defeat existing global inequalities, injustices, prejudices, mentalities, fragilities, and imaginaries of being that support the current untenable global political condition. historical traces can be found in pervasive thinking and practices of mathematics education throughout the renaissance, enlightenment and (post)industrialization, and it has played its part in the colonial project or the paradigm of economic development as a modern extension of colonialism5 (swanson, 2010b, 2013). the areas of focus and their arguments and emphases in mathematics education research bear witness to the history of things as they have come to be, as well as how they have become “thingyfied” (verdinglichung). there is therefore some responsibility in what the practices of mathematics education research has enabled and prevented, how it has contributed to the current global political imaginary, as well as what it has produced as a legacy of political complicity. mathematics education has, in the past and present, embroiled itself in cognitive and constructivist obsessions. it has suffered (and gained) from intensely symbolic interactionist methodological approaches centered around, in large part, the individual self of the child/youth as an object of study and the self of the superior (mathematically) knowing subject, such as teacher/lecturer/researcher/mathematician. in the recent past, mathematics education research has tended to be somewhat inwardly focused and insular, convinced of the natural goodness of a relatively stable mathematics and mathematics education, and these conservativisms have tended to remain in fair part. this “natural goodness” has been enabled through the unconscious assumptions carried by researcher/mathematics educator, convinced of its wholeness and the wholesomeness of its effect on the world thereby (in)advertently or (un)intentionally contributing to the modernist global imaginary and neoliberal governance. furthermore, poststructuralism and critical theory came late to mathematics education research and its attendant theories, as did the sociology of mathe 5 development is a neo-colonial discourse. as kothari (1988) notes, “where colonialism left off development took over” (p. 143). swanson commentary journal of urban mathematics education vol. 10, no. 1 10 matics education. socio-cultural and cultural considerations, including ethnomathematics and localized/intergenerational and “indigenous knowledge systems,” have also been brought to the fray but still seem to suffer, to some degree, from a notion of “culture” as being constructed as a coherent otherness of being in ways that are often essentialist, exotic, and deterministic, as was sustained in the colonial project (said, 1979). social identity theory, social action theory, semiotics, and other sociological, reflexive feminist, socio-historical theories have contributed much to shifting the debates, but arguably there is still the euro-centered liberal conceptions of the rational, coherent, individual self deeply inherited in these theoretical positions even as they often attempt, from critical perspectives, to thwart them. complexity science, cultural historical activity theory, posthuman and bio-political theories, and similar, have been welcome contributions to decentering the human. these theories have also been appreciated for the way they recognize and further complexify the always-already complex situated shifting relationalities of networked systems of signs and interactions (both human and non-human) that produce and construct particular meanings and “regimes of truth” (foucault, 1975/1977) of the contemporary condition and the history of the present (appelbaum, 1995; swanson, 2016). but it might be argued that considerations of relations of power and their political and global effects have often been neglected, or rather not fully considered, in interactionist research within these frames, even as bourdieu, foucault, bernstein, butler, and similar theorists, and other power-oriented discourses have been brought to the conversation. while some attention has been given to anti-racist, anti-oppressive, diversity, and social justice mathematics education, and these are still somewhat under-researched, it could be argued that more could also be done on expanding our theories of ethics from non-euro-centered perspectives as well (appelbaum, 1995; maheux, swanson, & khan, 2012). i assert that our scales of research engagement need to shift. it is urgent that we expand our horizons further in the mathematics education research we do as well as increase the dimensionality of our research. we need to cast our research gaze beyond classroom and teacher education programmes and reflexive interactions within them, in ways that might shift theoretical, and hence methodological, perspectives and positionalities to more fully consider the global, glocal, and global-political-ethical-ontological dimensions. in learning to remember what it is we are trying to do with/in/for/about/through mathematics education, we need to actively and earnestly search for possible alternative political and ontoepistemological (swanson, 2015) ways of knowing that might open up different ways of being and radically hopeful futures. in this sense, while mathematics education has drawn more widely on the fields of social science and, to some degree, the humanities in recent times, including, to a lesser degree, arts-based methodologies and theories (swanson, 2010a, 2010b), perhaps it could productively turn to areas such as post/decolonial theories, critical development studies, international swanson commentary journal of urban mathematics education vol. 10, no. 1 11 relations, and political theory more substantively. at the same time, it would do well to actively consider the hybrid incorporation/inter-corporealization of indigenous thought and embodiments, pluralized/ing epistemologies, and alternative/alterglobalizing ontologies that may shed more light on and respond with what canadian indigenous scholar jonathan lear (2006) has referred to as “radical hope” to the current deleterious global condition in which mathematics and mathematics education is implicated. we need to remember the historical present. we need to remember ourselves out of forgetfulness and into the possibility of alternative futures by acknowledging, in the first instance, examples of mathematics education’s implicatedness in knowledge capitalism and the global surveillance-military-prisonindustrial complex. post/decolonial theories provide another important critical lens through which to unpack, gain perspective, and dialectically respond to some of the colonialisms inherent in much mathematics education research and the modernist development project they often facilitate. these colonialisms are enabled through absences as well as intentionalities that create and sustain symbolic and systemic violences within the broader social and ecological domains. we live within ongoing states of emergency and crisis—global, political, educational, ecological, economic, and paradigmatic. these crises have become permanent states of exception. few would render farfetched the assertion that we live in a global condition of a crisis of crises. i assert that it is both timely and critical to bring into play several postcolonial and decolonial theoretical concepts to bear on mathematics education in contexts of modernism and global development6 in providing a political, global orientation that more centrally considers the role of the nation-state, the geo-political imaginaries of empire, and the broader neocolonial/neoliberal global(izing) condition in respect of mathematics education in global context. certain post/decolonial ideas valuable to critiques and conversations in mathematics education can be understood as being inscribed within such non-exhaustive foci as: centre-periphery discourses, loss and exile, disavowal and dispossession, epistemic violence, epistemic suppression (quijano, 2000), epistemic racism (mignolo, 2011), abyssal thinking (de sousa santos, 2007), representation and voice in geo-political context, othering and exoticism (said, 1979), global social and ecological injustices, discourses on dominance and the subaltern (spivak, 1988), benevolence and salvationist discourses, global/local asymmetrical relations, cultural imperialism (said, 1993); and the problem of “dividing the world” (willinsky, 1998), whether these divisions are enacted east/west, south/north, developing/developed worlds, margins/centre, or majority/minority 6 as nederveen pieterse (2010) avers, “the crisis of developmentalism as a paradigm manifests itself as a crisis of modernism in the west and a crisis of development in the south” (p. 28). swanson commentary journal of urban mathematics education vol. 10, no. 1 12 worlds. these and other post/decolonial concepts offer opportunities to provide frames of reference with which to converse with mathematics education from wider geo-political and global justice-oriented perspectives. they also demand responses from us that help us remember what it is we were trying to do with/in/for/about/ through mathematics education. this remembering would necessitate our ability to think, act, and conceptualise otherwise than our current frame of reference permits, in ways that also demand the political will to do so. butler (2009), in reference to “the frame” in context of a “frame of war,” asks: how do we understand the frame as itself part of the materiality of war and the efficacy of its violence? … the frame does not simply exhibit reality, but actively participates in a strategy of containment, selectively producing and enforcing what will count as reality. it tries to do this, and its efforts are a powerful wager … this means the frame is always throwing something away, always keeping something out, de-realizing and delegitimating alternative versions of reality, discarded negatives of the official version. (p. xiii) the development theorist, jan nederveen pieterse (2010), in referencing the paradigm of economic development, draws attention to the reasons why it may be so difficult to think outside of its conceptual frame. for nederveen pieterse, this difficulty would resonate with a butlerian account of the frame and its strategy of containment, and he argues that much of the difficulty lies with the context of language. he notes: spatial metaphors are deeply embedded in everyday english and the language of social theory. ann salmond’s (1982), inquiry into the semantics of social theory, shows that knowledge is a landscape, that is, knowledge has a spatial existence, and that intellectual activity is a journey. related notions, that knowledge is territory and argument is war, are the basis of accusations of intellectual ‘imperialism’ in theoretical texts. understanding as ‘seeing’ and explanations as light sources (‘illuminating’) are related to the notion of intellectual activity as a journey. that theoretical systems are buildings are metaphors that are related to structuralist discourses. spatial distinctions of levels and of high and low further structure discourse. notions of intellectual advancement and the progress of science follow likewise. so the general conception of knowledge and social theory itself tends to be structured in terms of spatial or organic metaphors and of (linear) motion in space. knowledge itself ‘develops.’ developmentalism ‘grows’ out of the semantics of space/time. (p. 28) in the same way, much mathematics education research and praxis, and the metaphors considered salient to the field, have suffered from a frame that delegitimizes non-official other versions, derealizing alternatives as discarded negatives to official ones. an example is, as a dominant view, the blind disregard for the political nature of mathematics education in its complicity in systems of injustice and war; for the way in which mathematics education in its othering and often swanson commentary journal of urban mathematics education vol. 10, no. 1 13 dehumanising effects, contributes to epistemologies of ignorance and the stupidity of forgetting what its purposes are meant to be. instead, there is a general capitulation to the narrow economic rationalizing of modernism as a frame of war against alternative ways of being and knowing. a second example would be, as nederveen pieterse (2010) would agree, the indelibility of the linear model of ever-advancing “progress,” and this as a model or frame is often forcefully intentioned through mathematics education discourses, practices, and many of the research approaches applied to it. yet another example of a strategy of containment and the limiting context of metaphors of language in the production of a regime of truth is the way in which a futurist, modernist mathematics education is repeatedly spoken of as a natural “good,” a necessity, critically essential for “21st century skills.” bringing in jacques rancière’s (2009) notion of “radical equality” into consideration, rancière would recognize disobedience to the colonizing gaze of modernist mathematics education and its complicity with the globalizing development project as a democratic action, not deficit. refusal need not be automatically conceptualised as failure but as a critical position of radical equality in relation to mathematics education (swanson & appelbaum, 2012). the 17th century italian philosopher, giovanni battista vico, noted that mathematics was “created in the self-alienation of the human spirit” (as cited in davis & hersh, 1986, p. x). following from its master discourse, mathematics education too, to a large degree, has assumed this posture, as if neutrality were natural to it, denying its role in the global modernist project that has underscored mass inequality, prejudice, racism, and ecological disaster sustained through the ongoing hubris of empire. its conceits, kept in motion by economic development and modernist globalization, have led to the continuance of a dehumanizing and ecologically degrading project. mathematics education research approaches that, advertently or inadvertently, advance these positions are a dispiriting of mathematics education’s potential and responsibility in enabling a better world or multiple possibilities of one. … or perhaps it is an epic act of forgetting what we were trying to do, a form of gross stupidity and ignore-ance. perhaps the time has come for us to remember with mathematics education and research, how we might foster viable alternatives to burgeoning global injustices and ecological disaster. perhaps it is time for us to remember what the intentions of mathematics education should be, to live well with mathematics education in order to live well with others; to live and research well with mathematics education in order to make possible futures of radical hope. references appelbaum, p. 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(1998). learning to divide the world: education at empire’s end. minneapolis, mn: university of minnesota press. journal of urban mathematics education july 2017, vol. 10, no.1, pp. 16–31 ©jume. http://education.gsu.edu/jume megan h. wickstrom is an assistant professor in the department of mathematical sciences – montana state university, 2-235 wilson hall, bozeman, mt, 59717-2400; email: megan.wickstrom@montana.edu. her research interests include the teaching and learning of mathematical modeling at the elementary level, creating mathematical tasks that promote equitable learning opportunities for all, and investigating and supporting teachers’ applications of research into practice. susan a. gregson is an assistant professor in the curriculum and instruction and middle childhood education programs – university of cincinnati, 511d teachers-dyer complex, 2610 mcmicken circle, cincinnati, oh, 45221-0022; email: susan.gregson@uc.edu. her research interests include equitable classroom practice, political knowledge for teaching mathematics, and the preparation of mathematics teachers for effective teaching of marginalized students. public stories of mathematics educators responding to inequities in mathematics education: opening spaces for dialogue megan h. wickstrom montana state university susan a. gregson university of cincinnati ecently, mathematics educators have discussed the challenges of preparing teachers to effectively teach all students. when examining these challenges, researchers have acknowledged problems that often arise because “teachers–– largely white, female, monolingual, and middle class––are not effectively prepared to teach mathematics to an increasingly racially, ethnically, linguistically, and socioeconomically diverse student population with which they often have had limited previous interaction” (bartell, 2012, p. 113). this lack of preparation can be attributed, at least in part, to the fact that many teachers have limited personal experience with the types of inequities that exist across the educational system (stinson, 2004). when we consider teachers’ readiness to teach mathematics equitably, it is also important to step back and consider our roles as mathematics teacher educators. for many of us, our lived experiences in the educational system are not far from those of the teachers we instruct. therefore, we must ask how we can adequately advise and support preservice and inservice teachers to teach mathematics in equitable ways when many of us are still learning to navigate this terrain? there has been a recent call to offer cases and stories to support mathematics teacher educators in discussing inequities in the classroom (white, crespo, & civil, 2016). moreover, equity scholars have urged mathematics educators to “engage colleagues and friends in explicitly talking about race, class, gender, and other systems of privilege and oppression” (aguirre et al., 2017, p. 140). this engagement requires a willingness to enter a “brave space in which some of our assumptions are questioned” (p. 128). we hope to add to these conversations by narrating our experiences with confronting inequities in k–12 classrooms, including our personal efforts toward finding the courage to present these accounts. supporting teachers to r http://education.gsu.edu/jume mailto:megan.wickstrom@montana.edu mailto:megan.wickstrom@montana.edu mailto:susan.gregson@uc.edu wickstrom & gregson public stories journal of urban mathematics education vol. 10, no. 1 17 navigate inequities in mathematics classrooms involves continued uncertainty because no single solution works across all cases. it involves coming to understand the history and people in the communities in which we work, making a deliberate choice to address inequity, creating space for reflection and dialogue, and revising our strategies based on insights gained from working with others. moreover, it means expecting that we will make mistakes along the way and figuring out how to be simultaneously comfortable and uncomfortable with that fact. while a certain level of comfort is necessary to move forward, it can make us complacent in situations where our expertise is limited. a theme that connects our stories is our focus on creating spaces to “listen well” (powell, 2012, p. 26) and to learn as a critical first step toward confronting inequity. in this public story, we account our respective strategies for creating these spaces with teachers and with each other as colleagues. by listening to teachers and engaging with them in practice, we are working to become more adept at creating opportunities for productive changes toward equity. we also recognize that improving our practice requires creating opportunities to connect with other mathematics teacher educators to discuss challenges in our practice and how we might address them. we hope that our stories may provide other teacher educators with examples of what the process of opening spaces for addressing inequity with teachers and with other mathematics teacher educators might look like. we begin by describing our respective stances and a case involving inequitable mathematics teaching that we have encountered in our practice. we discuss our attempts to create spaces for productive change along with the teachers we mentor. each account includes acknowledgment of our remaining questions and tensions. we then connect our cases by describing how and why we came together to write this article. we discuss our collaboration process and provide examples of how this process opened spaces for our own learning. we conclude with remarks about both the nature of efforts to address inequity in our practices as mathematics teacher educators and the continued challenges we see for this effort. megan’s story i worked with mrs. cate,1 a fourth grade teacher with 6 years of experience, as part of a 2-year professional development (pd) project focused on integrating learning trajectories as a formative assessment tool in elementary classrooms. for the study, i interviewed her and observed her class throughout the project. mrs. cate taught at terrace elementary school located in the midwestern united states. terrace is an urban school where the majority of students identify as black or latina/o and 80% of students qualify for free or reduced-price school meals. at the 1 all proper names are pseudonyms. wickstrom & gregson public stories journal of urban mathematics education vol. 10, no. 1 18 time, i was a 28-year-old white female mathematics education doctoral student and pd provider. as a former middle school teacher, i respect the intensity of classroom teachers’ work. i was raised in a city that was similar to, and in close proximity to, the one in which my story took place. in addition, i student taught and volunteered in schools like terrace elementary, so i felt comfortable when working with students and teachers there. i believe that mathematics should be co-constructed by the teacher and the learners and that it becomes meaningful through shared experiences and interpretations. all students are capable of learning mathematics, but it is critical to find ways to relate mathematics to their lived experiences. i focus on mrs. cate here because i perceived components of her teaching as inequitable, and it was difficult for me, as a mathematics educator, to determine if and how i should respond. equity tensions initially, mrs. cate’s classroom seemed like a well-structured environment for student learning. during her first interview, she highlighted several features of her instruction that she felt promoted student discussion, reflection, and growth. mrs. cate explained that she organized students in groups to promote discussion. she also highlighted a bulletin board on the back wall lined with clipboards. she said that this was where she kept track of things each student was doing well and something for them to improve. in addition, mrs. cate implemented a mathematics journal to encourage students to express their ideas in writing. although mrs. cate articulated a solid rationale for the instructional strategies that she had put in place, i experienced them differently. even though students were arranged in groups, mrs. cate rarely allowed them to talk or work together. i did not perceive the clipboards in the back of the room as a tool for accolades, as mrs. cate had intended, but rather as a way to compare and to demean students. in front of the class, she often said things to students like “i wish i could find something good to put up here, but i haven’t seen any good work from you in 3 weeks.” lastly, the mathematics journal was often used as a punishment activity for when students’ behavior was not appropriate during class. my foremost concern was how she framed students’ intelligences. sternberg (2007) documented that although intelligence is often perceived as objective, it is very much subjective. students’ perceived performance in class is often tied to how the teacher perceives what it means to be “smart” and how well students’ behaviors align with the teacher’s expectations (hatt, 2012; wickstrom, 2015). mrs. cate evaluated and rewarded students based more on their behavior than their efforts toward mathematical learning goals. in this classroom, being good at mathematics was associated with listening, being quiet, not fidgeting or making faces, and speaking when called on. hence, mrs. cate favored students who worked on tasks quietly, answered questions quickly and correctly, and “behaved” during instruction. wickstrom & gregson public stories journal of urban mathematics education vol. 10, no. 1 19 when students behaved appropriately, she rewarded them with candy or prizes. students only discussed ideas or questions when directly asked, and students who acted out or misbehaved were sent outside or to the principal’s office. mrs. cate had two students––hadley, a white girl, and kirby, a white boy––on whom she called frequently and often used as exemplars for the rest of the class. she told me that these two students were her top students because they were able to answer mathematics questions quickly and often correctly. through observations, i noticed both hadley and kirby sat quietly in class. mrs. cate said, “look how nicely hadley is sitting and listening” to the rest of the classroom. in contrast, several students did not behave according to mrs. cate’s expectations. from the first day, i took note of her relationship with timothy. timothy was a black boy, and his desk was positioned next to mrs. cate’s desk and removed from the other students. timothy often sat with his arms crossed and head down. one day during instruction, timothy became bored and made faces at his friends across the room. when mrs. cate caught him, a confrontation erupted, and she sent timothy out of the room. mrs. cate had a special desk for timothy in the hallway where he would sit until she thought it was time for him to return. timothy missed most of the mathematics classes and was often forgotten out in the hallway for long periods of time, sometimes over an hour. when i asked mrs. cate about timothy she said, [timothy] is one of my highest testing math students, like the computer lab testing, which is kind of like standardized testing. but, those tests obviously don’t tell us everything because in class, he doesn’t get it [math]. mrs. cate’s statement surprised me, because, i had a different perspective on why timothy wasn’t learning; he was not allowed to participate in class. as an observer, i knew what was happening was not equitable. i often left observations feeling uncomfortable and concerned for students like timothy. whether knowingly or unknowingly, i felt mrs. cate’s approach was creating racial divides in her classroom. as reflected in the school statistics, most of the students in mrs. cate’s class were not white. in fact, there were only three white students in her classroom. mrs. cate positioned hadley and kirby as top students because they aligned with her expectations of what a well-behaved student should be. she allowed white students special opportunities such as explaining concepts and going to the chalkboard while often limiting opportunities or completely taking them away from black students like timothy. as mrs. cate continually equated “good” behavior with mathematical proficiency, both white and black students missed opportunities to grow mathematically. simultaneously, this approach unfairly marked black students as mathematically inferior to white students. as part of the pd, i encouraged mrs. cate to engage in mathematical discussions and activities with students. in the first week of observation, she would begin wickstrom & gregson public stories journal of urban mathematics education vol. 10, no. 1 20 with an inquiry-based task but quickly resorted to quiet work time after she felt students were becoming out of control. when debriefing these lessons, mrs. cate made statements like “that’s difficult with these kids” or “when working with these kids, i have to….” the language of “these kids” sparked my attention and i wondered if it referred to race, socioeconomic status, ability, or a combination of these. in getting to know mrs. cate, i found out that she commuted a half hour to terrace from a rural, primarily white community, which was not uncommon for teachers in the district. she discussed that her mother was a teacher and her mother’s passion for teaching motivated her to follow suit. she also revealed that even though her mother was an enthusiastic and hands-on teacher, she did not feel like she could use similar teaching strategies with “these kids.” when discussing the job at terrace, she said, “but coming into the city was a whole different ballgame for me as far as what i saw growing up as a student and i guess i have to be strict and more firm just because of the city.” she also asserted that she had never been around a “minority” child until college. in the first few weeks working with mrs. cate, i had the sense that she wanted to be an engaging teacher but felt she had to teach in a certain way because of her students’ backgrounds. mrs. cate’s story is not new in educational research. there is often a racial and cultural mismatch between teachers and their students (goldenberg, 2014), and instead of recognizing and engaging racial and cultural differences, many teachers take the stance that learning means working harder and behaving (haberman, 1991). in addition, i knew mrs. cate’s teaching style was not the only approach being used in her school; other teachers in her building had learned to navigate these tensions and to engage in inquiryand equity-based mathematics. creating space with mrs. cate it was difficult for me to know what to do in this situation, and i considered several possibilities. from the beginning, i knew i could not confront mrs. cate directly about her teaching practices for several reasons. first, mrs. cate perceived me as an outsider who did not understand the day-to-day realities of her teaching that made her teaching practices necessary and effective with her students. in addition, i was conducting research for my doctoral dissertation, so addressing my concerns with mrs. cate meant risking the study and my ability to work with her or other teachers in the district. i began to address what i observed by listening to mrs. cate in the interviews and asking her questions related to some of her comments and strategies. i hoped some reflection might allow her to consider her actions and to gain insight into her practices. for example, after she sent timothy out of the room, i asked her to talk about timothy informally after class and then more specifically during interviews. wickstrom & gregson public stories journal of urban mathematics education vol. 10, no. 1 21 in these conversations, she seemed more comfortable telling me about her students and her teaching. consequently, i realized mrs. cate’s perceptions were deeply engrained and reaffirmed daily. it seemed difficult to challenge these perceptions because as long as her students’ behaviors aligned with her preconceived expectations for them, my questions would not likely shift her beliefs. instead, i chose to demonstrate possibilities for more equitable instruction. i thought that if mrs. cate could see her students differently then she might begin to change how she viewed them. moreover, given that my frequent interviews with mrs. cate took up her time, i wanted to reciprocate by assisting her with her work in some way. i offered to teach for her, to free up time for her to reflect on her students’ thinking. glaser (1982) and lather (1986) describe this approach as reciprocity or “the exchange of favors and commitments, the building of a sense of mutual identification and feeling of community” (glaser, 1982, p. 50). my purpose was twofold. i wanted to give back to mrs. cate, but i also hoped that demonstrating other ways of interacting with her students might provide openings for talking about equity. when i proposed the arrangement to mrs. cate, she hesitantly agreed that i could teach one or two lessons a week. in the first few weeks, i stuck to mrs. cate’s lesson plans to gain her trust. although the lessons were not student-centered, i made a point to elicit multiple students’ perspectives, check with students to see how they were doing, provide scaffolding, and hold high expectations for everyone. as mrs. cate became more comfortable with me teaching, she asked if i would be interested in teaching mathematics intervention. intervention occurred several times per week and consisted of students practicing facts on computers or by playing games. she directed that students should practice math facts and concepts but gave me the freedom to choose what i wanted to teach and how. instead of having students independently practice facts, i designed mini-lessons for them to work cooperatively. for example, when students were studying area and perimeter, i asked them to help me design a backyard fence for a pet. this project led to discussions on how to use the space, the size of the pet, and whether the house could be used as a border. students were excited by the tasks and often discussed them with me days after the lesson. teaching with and for mrs. cate created an opportunity for dialogue. after the first few intervention classes, mrs. cate made comments like “i need to try that” or “i was really surprised how [a specific student] kept working on the task.” eventually, she tried some of the tasks from intervention in her own classes and asked me for help in designing similar tasks. i believe mrs. cate wanted to be an engaging teacher, but she did not believe that her students could engage in rich mathematics. providing students with challenging tasks gave us a glimpse of what they were capable of as well as concrete examples that highlighted particular students as creative and competent. it was difficult to discuss beliefs with mrs. cate directly, but i wit wickstrom & gregson public stories journal of urban mathematics education vol. 10, no. 1 22 nessed several incidents that challenged her beliefs about students’ mathematical abilities. remaining questions and tensions this case highlights the reality that opening much-needed spaces for conversations about teachers’ practices and beliefs about students is difficult. my main concern in working with mrs. cate was that students were not receiving equal opportunities for quality mathematics education. i attempted to address this by modeling approaches that allowed students in an intervention class to demonstrate mathematical competence beyond good behavior. i saw evidence that teaching for and working with mrs. cate opened space to discuss students’ mathematical thinking and provided concrete evidence that countered mrs. cate’s perceptions of struggling students. and yet, as our collaboration ended, i was left wondering, “did i do enough?” while i witnessed some positive change in mrs. cate’s instruction, my lingering tension is that by focusing my efforts on developing rich tasks accessible to all students, i skirted issues of race and equity. i continue to wonder if i could have done more to help mrs. cate challenge her assumptions about black students and to productively address the racial bias that i observed in her practice. more generally, i continue to grapple with how to broach topics like racism, classism, and ableism without fracturing relationships with my teacher partners. susan’s account as a mathematics teacher educator who has been a classroom teacher, schoolbased coach, and researcher in urban and rural schools, i respect the work of classroom teachers and believe that it is not possible to transform education to meet the needs of marginalized students without teacher knowledge, collaboration, and agency. i work in a teacher education program whose mission includes preparing teachers to work in urban schools. a significant tension of my practice is my desire to help early-career teachers challenge inequitable practices while avoiding the pitfall of portraying urban educators––especially those whose racial and economic backgrounds differ from their students––as the primary obstacle to equity. however, as a white middle class teacher, i routinely encounter and participate in situations where deficit notions of students of color––an intrinsic ideology of inequitable teaching––go unchallenged. i position myself as an equity researcher, yet i am troubled in situations like these in which i lack the tools or the courage to disrupt the status quo. moreover, i find that when i operate alone, outside of a community of educators regularly committed to equity issues, the tools i have acquired to resist deficit perspectives become dull. therefore, participating in communities where wickstrom & gregson public stories journal of urban mathematics education vol. 10, no. 1 23 educators with diverse experiences address tensions of equitable practice is essential for my own professional development. the context of my account is a voluntary professional development group for preservice and early-career educators, the mathematics equity group (meg). i both facilitate and participate in the meg. the meg’s goals include supporting teachers as they “identify and challenge discourses that further ingrain inequalities,” develop “political knowledge and experiences necessary to negotiate the system,” and develop “working networks of educators who share their emancipatory visions” (gutiérrez, 2013, p. 62). meg participants follow a modified version of gutiérrez’s (2012) in my shoes discussion protocol in which a teacher describes a problematic scenario from personal practice. after clarifying questions are addressed, the group discusses strategies for addressing the situation with follow-up questions such as: what would that strategy look like? is that something you can see yourself doing? teachers are encouraged to consider the scenario with respect to their own practice as educators working toward equity. this account focuses on mr. david’s in my shoes experience in the spring of 2014. mr. david, an african american, was a preservice teacher in a multi-level (grades 4–6) urban field placement. mr. david had significant pre-certification urban teaching experience as a full-time substitute in local schools with high poverty rates (over 98%) and large numbers of african american students (more than 95%). six other meg teachers, all white, and myself, participated in the discussion. equity tensions in working with the meg mr. david and i had previously discussed challenges in his field placement prior to this in my shoes experience, so i thought i knew what to expect. he was concerned with both the exclusively procedural nature of the enacted curriculum, and a potential personality conflict with his cooperating teacher, ms. marcus. i knew that he had worked through these issues to some extent, so i encouraged him to share his story in meg. however, as mr. david presented, it became clear that his concerns were more complicated than i thought. he described a learning environment where students largely worked in silence; where norms for behavior were rigid and enforced punitively; and where seating, participation, and discipline were highly racialized. mr. david told the group about jasmine, a black child who he saw as being frequently and unfairly disciplined. for example, when another child walked by jasmine’s desk and inadvertently knocked a piece of paper on the ground, ms. marcus noticed and pulled jasmine aside. mr. david began shouting to imitate ms. marcus’ tone: “i can’t believe you had a piece of paper under your desk. i told you last semester. i told you this semester. clean up under your desk.” according to mr. david’s account, the teacher “reams her for like two to three minutes. and then the wickstrom & gregson public stories journal of urban mathematics education vol. 10, no. 1 24 little girl has to take the [math] test.” in a voice that was both incredulous and outraged, mr. david went on to describe patterns he noticed in student seating: “all the little white girls sit in the front row in the middle. the black girls sit behind them. the black boys sit in the back right corner and the white boys sit in the front right corner.” another participant interjected, “it’s that clear cut and obvious?” mr. david acknowledged his own concern that perhaps he was simply imagining a bias. and so, he “started to keep track. [ms. marcus] only calls on the little white girls. i’ve seen the whole row [of black girls] raise their hand. and she calls on the one little white girl, ashley.” mr. david recounted another incident where he approached a black student to talk about a fraction worksheet. as soon as ms. marcus saw them talking, she reportedly said, “what did you say darnell? do this reflection!” reflections were a form of punishment in the class. mr. david described trying to respectfully defend darnell: mr. david: i talked to him. he wasn’t talking. i talked to him. ms. marcus: no! he knew what he was doing. he came in and he sat down next to you and he talked because he knew you would talk to him. mr. david: no ma’am. i asked him. ms. marcus: if you are a student in this class and you think you can come in and talk to another teacher about anything that is going on, you are going to get a reflection! as mr. david told his story, i wrestled with multiple emotions. first i was horrified. i remember saying, “this teacher does not belong in the classroom!” and it was hard for me to move beyond this immediate thought. the situation felt like an extreme case of a racialized, authoritarian, and repressive environment where “the achievement gap is a mirror image to the punishment gap” (yang, 2009, p. 51). but the example, though extreme, was not inconsistent with other situations i have encountered in schools that serve high numbers of marginalized students. i have witnessed both effective and ineffective colleagues and administrators forcefully reprimanding students. i have done some yelling myself over the years. without firsthand experience in the setting, and without knowing more about a teacher and her practice, it could be possible to mistake, for example, warm demander approaches that involve “mean-talk” (ware, 2006, p. 438) as oppressive. i trusted mr. david’s perspective of the situation, but i was uncertain how others in the group might interpret ms. marcus’ behavior. teachers with limited experience of the range of effective discipline practices may misread classroom situations. they may view even warm demander pedagogy as harsh and inappropriate. conversely, they may believe that discipline styles they would never choose for their own children are requirements for teaching marginalized students. i struggled with whether i should jump into the conversation to provide nuance. i also won wickstrom & gregson public stories journal of urban mathematics education vol. 10, no. 1 25 dered whether without my intervening, the participants might dismiss this example as so extreme that they would never likely face a similar situation in their own practice? my response that “this teacher does not belong in the classroom,” was genuine, but could also be used as an excuse to dismiss efforts to learn to confront similar behavior that teachers might encounter with peers in the future. i also worried that giving this example too much attention and treating it as typical might make participants hesitant to see themselves acting because they feel too inexperienced to tackle such a pervasive problem. despite these tensions, i fought the urge to share every concern that popped into my head. when you have experience, you want to share it––especially as a trained teacher educator. yet, i have learned from multiple in my shoes discussions, that when i limit my contributions, other participants’ questions and comments take the group in insightful directions. the meg protocol is designed to probe further, to connect participants with each other’s experiences, and to develop each person’s capacity for acting in related situations in ways that align with the kind of teachers we want to be. so, i decided to trust the protocol. i encouraged mr. david to phrase his concerns about this inequitable situation as a question. he responded with, “what is it that i can do in the limited time i have in this practicum to give those kids a different experience?” creating spaces through meg the participants began with clarifying questions such as, “who is ms. marcus?” and “what is her teaching background?” a generative moment came when ms. shelby, a first-year teacher, revealed that she was in the same teacher’s classroom for her first practicum, 2 years earlier. ms. shelby was eager to discuss her frustration in that placement where the way students were treated made her “feel terrible.” ms. shelby shared her earliest attempts at teaching in the placement describing how ms. marcus both discouraged her from trying student-centered approaches and appeared vindicated when ms. shelby tried them, and they did not go well. ms. shelby reported “feeling like a failure” in that experience. discussion turned to the suggestion phase as participants applied possible strategies developed in previous meg discussions to mr. david’s situation. for example, ms. cass, suggested “playing dumb.” mr. david might use his position as a novice to question ms. marcus’s methods, bringing attention to both her problematic behavior and potentially opening space for discussion through questions like, “what is the purpose of the reflections?” another approach was “claiming a requirement.” mr. david might claim that making sure all students’ voices are heard is an official component of his practicum, and therefore, he would have an excuse to use methods that insure students are called on randomly. a third suggestion was wickstrom & gregson public stories journal of urban mathematics education vol. 10, no. 1 26 that mr. david might “highlight the competence”2 of black students by amplifying their thinking with the cooperating teacher with comments like, “so and so has a really good idea.” mr. david asked the group to consider whether any of these ideas might backfire. following gutiérrez’s (2012) protocol and hoping to broaden participants thinking about this question, i posed an additional question to the group: “pretend you are in your first year of teaching and ms. marcus is on your team. what would you do?” the discussion opened to more carefully consider ms. marcus’s background, and to an extent, the sociocultural and institutional practices that may have shaped her views. ms. cass offered this speculation: maybe she really is racist, but maybe she has been trained that way? i am not saying it is good training, but what if having all the girls sit in front was her training? knowing this would put things into perspective to me about how you would handle things if this was your team teacher. this comment helped the group to step back and consider the institutional conditions under which teachers’ perspectives develop. discussion shifted slightly from how to “fix this colleague” to the circumstances under which teaching behaviors like those mr. david observed may have developed and could continue unchecked. as an example of institutional conditions that normalize inequity, i noted “culture of poverty” trainings that have been required in many districts and which promote racialized stereotypes of low-income children and their educational needs (gorski, 2008). other participants asked questions about the school climate and if other adults were aware of the atmosphere in ms. marcus’s classroom. we speculated about whether the lack of african american male teachers in both the building and as student teachers coming from our program might have affected this teacher’s perceptions of the appropriate mathematical goals and roles for black students and helped to make it seem acceptable to discipline them more harshly than white students. we talked about some of the reasons that a new teacher might be afraid to speak up against injustice. to close the session, mr. david explained that he had already implemented a “killing with kindness” approach with ms. marcus in which he strategically and deliberately praised her for everything she did in the classroom––even actions he disagreed with. he offered to help with everything from making copies to cleaning the board to tutoring challenging students. mr. david reported that this strategy allowed him to achieve key short-term goals despite the limits of this placement. ms. marcus became more approachable to him; she offered advice and mentorship, although that advice was sometimes questionable relative to equity. because she allowed him to take over the mathematics teaching when he was present, mr. marcus 2 meg participants revised this term from “assigned competence” cohen (1998). wickstrom & gregson public stories journal of urban mathematics education vol. 10, no. 1 27 gained experience teaching conceptually rich mathematics in a setting that was more racially and economically diverse than his pre-cohort experiences. while teaching, he highlighted the strengths of the range of students in the class and provided an example for all students of a black male mathematics teacher. he did not expect that his approach would change ms. marcus’s future behavior, but it allowed him space to provide some support for black and white students and also to hone his skills for future equity battles. remaining questions and tensions mr. david’s story provided an opportunity for preservice and early-career mathematics teachers to consider, question, and analyze one inequitable classroom environment from multiple perspectives. participants used the example to engage more nuanced questions about the nature of the institutional climate under which such inequitable conditions exist. the meg participants had the opportunity to think specifically about how they would handle similar situations in their own practice. for me, questions and tensions remain. ms. shelby’s revelation about her experience with ms. marcus surprised me and was a reminder that meg participants, like all learners, advance from their current understandings. at some point, i asked ms. shelby why she did not raise her concerns with this teacher in the group while they were happening. she reported that she had not thought about the problems as equity issues until she heard mr. david’s story. while i was pleased that the discussion opened space for ms. shelby to see her own field placement differently, i was disheartened to think that neither our preservice urban education program nor previous meg sessions had prepared ms. shelby to speak up about a classroom environment where she felt this uncomfortable. issues of power were likely at play. unlike mr. david, whose prior experiences in urban schools gave him perspective, without that experience, ms. shelby, may have been less willing to question her placement––trusting that our program would not have put her into a situation where she would experience inappropriate teaching. and, like it or not, i cannot separate my role as a professor in my program from my role as a participant in meg. i represent the institution that made ms. shelby’s placement. thus, it makes sense that she might not have felt comfortable questioning her placement in my presence. it took mr. david’s participation in meg to bring problems to light. likewise, many of my lingering questions involve discussing educational racism in an environment––mathematics teacher education––with low numbers of african american teachers. there is no doubt that mr. david’s story was powerful because his experience with racism in american life and schools provides him with authority about what is and is not normal. yet, he cannot speak for all black people. how can i, as a white facilitator of meg, do a better job of challenging participants to consider inequity beyond the superficial when they do not share mr. da wickstrom & gregson public stories journal of urban mathematics education vol. 10, no. 1 28 vid’s intimate knowledge of its effects? how can i provide direction for novice teachers without claiming to have all the answers? where should my models come from? for example, the fact that mr. david, a black male graduate student with advanced mathematics skills, had to go to such lengths simply to be accepted as a legitimate teacher in the eyes of his cooperating teacher was a lesson for all of us. the group had the possibility to directly consider how subject position, level of experience with racism, and amount of urban teaching experience, can influence how one views a similar situation. yet, i did little to explicitly help the group make these connections or to consider what such subjectivity means for promoting equity. moreover, i believe i had a responsibility as a facilitator to frame this reality with hope and possibility for change; but in this case, our discussions never got to that level. connecting our cases we (susan and megan) met at a conference for mathematics teacher educators. megan was inspired by a presentation about the meg because it was the first time she had heard open, detailed discussion of the challenges of learning to address inequitable teaching practices, especially racialized practices, in a professional setting. megan saw many similarities between mrs. cate and ms. marcus, but, until that point, had been nervous about discussing the situation with mrs. cate openly. megan was not sure what others could learn from her case and questioned the appropriateness of how she addressed the situation. susan admitted that she, too, was nervous, especially about revealing such details of teaching without having the example either seen as business as usual and therefore demonizing to urban teachers, or having the example be dismissed as an abnormal outlier and therefore not important to address. susan was also nervous about opening details of meg for scrutiny when she questioned her own expertise at addressing racism. but a central goal of meg is to learn to talk about and act against injustice even when it makes participants nervous. thus, we decided we should further discuss our stories together to see what we might be able to learn and to share with others. in discussing our work, we asked “is it appropriate to share these stories?” and “what is our purpose in sharing our stories?” we both felt inherent tensions within these cases. first, both of our accounts involved practicing teachers and the question of inequitable behaviors in their classrooms. it was challenging to retell these accounts without feeling like we were portraying the teachers as ineffective, uncaring, and racist while portraying ourselves as just and equitable researchers. we knew that both cases were complex and that publishing these accounts meant risking that both teachers’ practices and our own responses could be painted in binary “right or wrong” terms. in short, we did not want to paint classroom teachers from a deficit perspective, but working with teachers toward equity often creates wickstrom & gregson public stories journal of urban mathematics education vol. 10, no. 1 29 these awkward situations, and it is important to be honest about our struggles and to share our strategies, no matter how imperfect. we also had to face the fact that writing about inequitable teaching practices and our own ability to confront such practice would not be as clean cut or straightforward as reporting on traditional mathematics education research. this complexity made us both uncomfortable and forced us to confront our own uncertainty about what counts as important knowledge for the field and where in the literature stories like ours belong. lastly, as white, middle class teacher educators, we had concerns about whether our voices warranted being heard (megan) and whether we have enough perspective to do justice to the complexity of the issues (susan). we decided the best way to proceed would be to write down our cases and continually talk to work through our tensions as we wrote. as we worked, our discussion points repeatedly centered on the following questions:  why did we respond to these scenarios the way we did, and why did we feel, at the time, that our responses were appropriate?  what was and was not helpful in how we each responded to our situation?  how were our responses too safe, and how could we have pushed our boundaries further?  how would we respond now if faced with a similar situation? why? in writing our accounts and discussing these questions, we learned about each other and ourselves. for example, in discussing how and why we responded the way we did, megan wondered if she could have done more to explicitly discuss issues of equity with mrs. cate. susan helped megan to realize that the ways we address inequities are situated in our positionality and context. they can be improved over time. if faced with a similar situation now, megan would feel more comfortable approaching mrs. cate differently. for example, when megan witnessed mrs. cate favoring certain students, she might now ask, “how can we involve all students in the lesson and help them feel confident in their abilities?” she would consider saying directly, “in my view, it is not okay that all students are not working on the task.” she might also challenge mrs. cate to look for other students in the classroom who demonstrated competency. likewise, susan learned much from repeated discussions with megan. unlike megan, susan’s main research focus is equitable mathematics practice. thus, our experiences with the equity research base were quite different. there were times, especially at the beginning of our partnership, when susan’s experience with the literature got in the way of her own growth. for example, megan suggested early on that the pedagogy of poverty framework (haberman, 1991) might situate both of our experiences. at first, thinking that those ideas were dated and did not focus enough on racism to meet our needs, susan questioned megan’s idea. however, wickstrom & gregson public stories journal of urban mathematics education vol. 10, no. 1 30 megan’s thoughts about how the frame put her situation into perspective pushed susan to take a second look at what she thought she knew. after reflection, susan decided that haberman’s (1991) findings were still relevant and useful in many ways, especially for the novice teachers in her meg group. in future meg discussions, susan now plans to discuss the pedagogy of poverty frame by encouraging the participants to consider how the tenets of the original frame relate to racial injustice. as we participated in these conversations across time, our discussions opened spaces to process the equity challenges each of our stories raised. we found time to consider and talk through other approaches for acting the next time we are faced with such situations. and we developed a stronger understanding of each other’s perspective, which has allowed us to see each other as allies in in this work. a crucial element to having productive discussions was the fact that we both acknowledge that racial inequities exist and that addressing racism in the field is an essential component of our job as mathematics teacher educators. both of us have worked with colleagues who accept inequitable teaching practice as the status quo or believe it is a wasted effort to address inequities because teachers’ beliefs cannot be changed. moreover, we both have experienced instances where addressing inequity was treated as something to check off a list of expectations rather than an ongoing process. addressing inequities involves developing, refining, and rehearsing potential strategies. our collaboration allowed space to reflect and be better prepared when faced with a similar situation. it was through these conversations that we realized why our voices warranted being heard. the purpose of sharing our stories is to highlight that we all have the capacity to enact change, but we need the support and courage to start somewhere as well as the understanding that addressing inequity is not an all or nothing endeavor. advocating for equitable teaching practices takes continual reflection and dialogue. it involves reflecting on when and where to assert yourself and why and discussing the appropriateness of your actions with others. we realized that our conversations gave us space to feel like we were being heard and an outlet where someone else was acknowledging our tensions so that in the future we would have tools to assert ourselves in appropriate ways. we hope that our accounts motivate other mathematics educators to continue to discuss issues of inequity in their work as well as spark a larger conversation on the creation of mathematics education equity discussion groups at the post-secondary level. references aquirre, j., herbel-eisenmann, b., celedón-pattichis, s., civil, m., wilkerson, t., stephan, m., pape, s., & clements, d. h. (2017). equity within mathematics education research as a political act: moving from choice to intentional collective professional responsibility. journal for research in mathematics education, 48(2), 124–147. wickstrom & gregson public stories journal of urban mathematics education vol. 10, no. 1 31 bartell, t. (2012). is this teaching mathematics for social justice? teachers’ conceptions of mathematics classrooms for social justice. in a. a. wager & d. w. stinson (eds.), teaching mathematics for social justice conversations with educators (pp. 113–125). reston, va: national council of teachers of mathematics. cohen, e. g. (1998). making cooperative learning equitable. educational leadership, 56(3), 18–21. glaser, m. (1982). the threat of the stranger. vulnerability, reciprocity, and fieldwork. in j. e. sieber (ed.), ethics of social research: fieldwork, regulation, and publication (pp. 49–70). new york, ny: random house. goldenberg, b. m. (2014). white teachers in urban classrooms: embracing non-white students’ cultural capital for better teaching and learning. urban education, 49(1), 111–144. gorski, p. c. (2008). peddling poverty for profit: elements of oppression in ruby payne’s framework. equity & excellence in education, 41(1), 130–148. gutiérrez, r. (2012, october 8). developing political knowledge for teaching mathematics: one way of making classrooms more equitable for all students. a webinar presented to the association of mathematics teacher educators (amte). retrieved from https://vimeo.com/84610273 gutiérrez, r. (2013). the sociopolitical turn in mathematics education. journal for research in mathematics education, 44(1), 37–68. haberman, m. (1991). the pedagogy of poverty versus good teaching. phi delta kappan, 73(4), 290–294. hatt, b. (2012). smartness as a cultural practice in schools. american educational research journal, 49(3), 438–460. lather, p. (1986). issues of validity in openly ideological research: between a rock and a hard place. interchange, 17(4), 63–84. powell, a. (2012). the historical development of critical mathematics education. in a. a. wager & d. w. stinson (eds.), teaching mathematics for social justice conversations with educators (pp. 21–34). reston, va: national council of teachers of mathematics. sternberg, r. j. (2007). who are the bright children? the cultural context of being and acting intelligent. educational researcher, 36(3), 148–155. stinson, d. w. (2004). mathematics as “gate-keeper” (?): three theoretical perspectives that aim toward empowering all children with a key to the gate. the mathematics educator, 14(1), 8– 18. ware, f. (2006). warm demander pedagogy: culturally responsive teaching that supports a culture of achievement for african-american students. urban education, 41(4), 427–456. white, d. y., crespo, s. & civil, m. (2016). facilitating conversations about inequities in mathematics classrooms. in d. y. white, s. crespo, & m. civil (eds.), cases for mathematics teacher educators: facilitating conversations about inequities in mathematics classrooms (pp. 1–6). charlotte, nc: information age. wickstrom, m. h. (2015). challenging a teacher’s perceptions of mathematical smartness through reflections on student’s thinking. equity & excellence in education, 48(4), 589–605. yang, k. w. (2009). discipline or punish: some suggestions for school policy and teacher practice. language arts, 87(1), 49–51. https://vimeo.com/84610273 goals , aimed at forging links between the educational and commercial sectors journal of urban mathematics education december 2011, vol. 4, no. 2, pp. 1–6 ©jume. http://education.gsu.edu/jume david w. stinson is an associate professor of mathematics education in the department of middle-secondary education and instructional technology in the college of education, at georgia state university, p.o. box 3978, atlanta, ga, 30303; e-mail: dstinson@gsu.edu. his research interests include exploring socio-cultural, -historical, and -political aspects of mathematics and mathematics teaching and learning from a critical postmodern theoretical (and methodological) perspective. he is a co-founder and current editor-in-chief of the journal of urban mathematics education. editorial both the journal and handbook of research on urban mathematics teaching and learning david w. stinson georgia state university ver the past three decades or so there has been a proliferation of the edited handbook within the social sciences generally and education social science specifically (peruse your bookshelves). disregarding the anthropomorphism, we could say that (some of) these handbooks certainly have taken on a life all their own. rightly or not, the status and prestige awarded to (some of) these handbooks and their editor(s) and contributing authors are unmatched. these researchers and scholars are designated as “the” experts on a particular subject as “the handbook” often becomes the “go to” resource—or more aptly, the régime of truth (foucault, 1977/1980)—within its respective field. the growing influence of the handbook within the social sciences should not be underestimated; its “power” to produce and re-produce knowledge is substantial. within the science of mathematics education, a google scholar search of the 29 chapters of the handbook of research on mathematics teaching and learning (growns, 1992)—a national council of teachers of mathematics (nctm) project—returns nearly 12,000 scholarly references and the wider google web search returns about 114,000 hits. as a single resource, only the nctm 1989 curriculum and evaluation standards for school mathematics (about 156,000 hits) and 2000 principles and standards for school mathematics (about 101,000 hits) rival this web presence.1 but for many, the latter two lack “scientific” rigor, leaving the growns 1992 handbook as the most referenced (therefore the most influential?) single resource book of research on mathematics teaching and learning. from a foucauldian (cf. 1969/1972) perspective, the handbook as a discursive formation is readily apparent; it determines (too often?) what discourses and in turn what discursive practices are both possible and impossible within a particular field of science in a particular moment of time (to subvert the impossible is always possible, however). 1 google scholar and google web searches were conducted in december 2011; clearly, internet searches are not an exact science, only approximations of web presence. o stinson editorial journal of urban mathematics education vol. 4, no. 2 2 nevertheless, although dangerous,2 i like handbooks. handbooks have had and continue to have a significant influence on my development as a mathematics education researcher and teacher educator.3 handbooks in part determine how i conduct my science and what readings i assign to my students. that is to say, as a readily accessible, single resource my published articles and course syllabi nearly always contain references and readings pulled from handbooks. to speak more generally, as previously shown, the growns 1992 handbook has been undeniably influential in producing—and dangerously, re-producing—how the science of mathematics education and the teaching and learning of mathematics might be theorized and practiced. and the more recent second handbook of research on mathematics teaching and learning (lester, 2007), also a nctm project, appears to be destined to have a similar long-term impact on the field—taking on a life all its own (e.g., google web hits about 1,220). but who breathes life into the handbook? who “gives” it power? well, we do! i was taught several years ago, and have been reminded continuously ever since, that the multiple decisions we make concerning whose research and scholarship we reference and which readings we assign are not neutral, apolitical acts (e. a. st. pierre, personal communication, june 2001). but rather, acts of power (conscious or not) that hold uncertain possibilities for our own empowerment (or not) as well as the self-empowerment (or not) of our readers and students. yes, i know, it’s most tempting here to deny our own power, to say that power is actually held in the surveilling gazes of academia, the disciplinary processes of peer review, or the asymmetrical decisions of professional organizations. but to do so, although tempting and perhaps somewhat warranted, leaves us powerless—which, we, indisputably, are not! our scholarly and pedagogical decisions can be, if we so choose, powerful acts of scholarly activism (g. ladson-billings, personal communication, june 2010). so with the concept of scholarly activism in mind, i purpose a reenvisioning of the research and pedagogical possibilities of the journal of urban mathematics education (jume). might we envision jume not only as “a peerreviewed, open-access, academic journal published twice a year” (description found on the jume homepage) but also as a bi-annually updated and revised handbook of research on urban mathematics teaching and learning? this re 2 “my point is not that everything is bad, but that everything is dangerous, which is not exactly the same as bad. if everything is dangerous, then we always have something to do. so my position leads not to apathy but to hyperand pessimistic activism” (foucault, 1983/1997, p. 256). 3 for example, the first two editions of the handbook of qualitative research (denzin & lincoln, 1994, 2000), the handbook of research on mathematics teaching and learning (grouws, 1992), and the second international handbook of mathematics education (bishop, clements, keitel, kilpatrick, & leung, 2003). stinson editorial journal of urban mathematics education vol. 4, no. 2 3 envisioning certainly is not intended to suggest that the peer-reviewed journal somehow plays a lesser role to the handbook in producing and re-producing knowledge in a particular field. clearly, the journal is king. but rather, the intent here is to somehow seize the collective power of both the journal and the handbook by envisioning jume as a both-and rather than an either-or research and pedagogical resource: the journal of urban mathematics education is both a peer-reviewed journal and a handbook of research on urban mathematics teaching and learning. to use jume as a peer-reviewed journal, simply search its archives. and to assist you in using jume as a handbook, below is the table of contents of the most current “edition.” just remember, rather than waiting 5, 10, or even 15 years for an updated, revised edition, this handbook is updated and revised twice a year. enjoy and use (at no cost to you or your students) this new resource! note: all “chapters” are hyperlinked. handbook of research on urban mathematics teaching and learning table of contents part i: issues 1. putting the “urban” in mathematics education scholarship william f. tate – washington university in st. louis 2. the common core state standards initiative: a critical response eric (rico) gutstein – university of illinois at chicago 3. mathematics as gatekeeper: power and privilege in the production of knowledge danny bernard martin, maisie l. gholson – university of illinois at chicago jacqueline leonard – university of colorado denver 3.1 “both and”—equity and mathematics: a response to martin, gholson, and leonard jere confrey – north carolina state university 3.2 engaging students in meaningful mathematics learning: different perspectives, complementary goals michael t. battista – the ohio state university 4. changing students’ lives through the de-tracking of urban mathematics classrooms jo boaler – stanford university http://ed-osprey.gsu.edu/ojs/index.php/jume/issue/archive http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/19/2 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/88/43 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/95/57 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/95/57 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/108/53 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/108/53 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/115/58 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/115/58 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/138/85 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/138/85 stinson editorial journal of urban mathematics education vol. 4, no. 2 4 5. positive possibilities of rethinking (urban) mathematics education within a postmodern frame margaret walshaw – massey university part ii: theoretical perspectives 6. a metropolitan perspective on mathematics education: lessons learned from a “rural” school district celia rousseau anderson, angiline powell – university of memphis 7. mathematical counterstory and african american male students: urban mathematics education from a critical race theory perspective clarence l. terry, sr. – occidental college 8. caring, race, culture, and power: a research synthesis toward supporting mathematics teachers in caring with awareness tonya gau bartell – university of delaware part iii: teachers and teaching 9. comparing teachers’ conceptions of mathematics education and student diversity at highly effective and typical elementary schools richard s. kitchen – university of new mexico francine cabral roy – university of rhode island okhee lee, walter g. secada – university of miami 10. preservice teachers’ changing conceptions about teaching mathematics in urban elementary classrooms mindy kalchman – depaul university 11. evolution of (urban) mathematics teachers’ identity mary q. foote – queens college, cuny beverly s. smith, laura m. gillert – the city college of new york, cuny 12. when am i going to learn to be a mathematics teacher? a case study of a novice new york city teaching fellow michael meagher – brooklyn college, cuny andrew brantlinger – university of maryland, college park part iv: teacher education 13. teaching mathematics for social justice: reflections on a community of practice for urban high school mathematics teachers lidia gonzalez – york college, cuny 14. math links: building learning communities in urban settings jacqueline leonard – temple university brian r. evans – pace university http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/141/89 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/141/89 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/34/12 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/34/12 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/98/87 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/98/87 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/128/84 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/128/84 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/24/14 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/24/14 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/97/79 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/97/79 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/109/93 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/120/99 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/120/99 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/32/13 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/32/13 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/5/5 stinson editorial journal of urban mathematics education vol. 4, no. 2 5 15. learning to teach mathematics in urban high schools: untangling the threads of interwoven narratives haiwen chu – graduate center of city university of new york laurie h. rubel – brooklyn college, cuny 16. the mathematics learning discourse project: fostering higher order thinking and academic language in urban mathematics classrooms megan e. staples, mary p. truxaw – university of connecticut 17. collaborative evaluative inquiry: a model for improving mathematics instruction in urban elementary schools iman c. chahine – georgia state university lesa m. covington clarkson – university of minnesota 18. k–2 teachers’ attempts to connect out-of-school experiences to in-school mathematics learning allison w. mcculloch, patricia l. marshal – north carolina state university part v: student learning and identity 19. social identities and opportunities to learn: student perspectives on group work in an urban mathematics classroom indigo esmonde, kanjana brodie, lesley dookie, miwa takeuchi – university of toronto 20. exploring the nexus of african american students’ identity and mathematics achievement francis m. nzuki – the richard stockton college of new jersey 21. how do we learn? african american elementary students learning reform mathematics in urban classrooms lanette r. waddell – vanderbilt university 22. (in)equitable schooling and mathematics of marginalized students: through the voices of urban latinas/os maura varely gutierrez – elsie whitlow stokes community freedom public charter school craig willey – indiana university purdue university-indianapolis lena l. khisty – university of illinois at chicago part vi: policy 23. racism, assessment, and instructional practices: implications for mathematics teachers of african american students julius davis – morgan state university danny bernard martin – university of illinois at chicago 24. practices worthy of attention: a search for existence proofs of promising practitioner work in secondary mathematics pamela l. paek – university of texas at austin http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/50/60 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/50/60 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/74/49 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/74/49 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/44/38 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/44/38 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/94/92 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/94/92 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/46/35 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/46/35 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/45/68 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/45/68 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/62/72 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/62/72 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/112/91 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/112/91 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/14/8 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/14/8 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/6/1 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/6/1 stinson editorial journal of urban mathematics education vol. 4, no. 2 6 25. an examination of mathematics achievement and growth in a midwestern urban school district: implications for teachers and administrators robert m. capraro, jamaal rashad young, chance w. lewis, zeyner ebrar yetkiner, melanie n. woods – texas a&m university 26. compounding inequalities: english proficiency and tracking and their relation to mathematics performance among latina/o secondary school youth eduardo mosqueda – university of california, santa cruz part vii: international perspectives 27. learning mathematics in a borderland position: students’ foregrounds and intentionality in a brazilian favela ole skovsmose – aalborg university pedro paulo scandiuzzi – university são paulo states paola valero – aalborg university helle alrø – aalborg university bergen university college 28. transforming mathematical discourse: a daunting task for south africa’s townships roland g. pourdavood – cleveland state university nicole carignan – university of quebec at montreal lonnie c. king – nelson mandela metropolitan university 29. forging mathematical relationships in inquiry-based classrooms with pasifika students roberta hunter, glenda anthony – massey university references bishop, a. j., clements, j., keitel, c., kilpatrick, j., & leung, f. k. s. (eds.) (2003). second international handbook of mathematics education. dordrecht, the netherlands: kluwer. denzin, n. k., & lincoln, y. s. (1994). handbook of qualitative research. thousand oaks, ca: sage. denzin, n. k., & lincoln, y. s. (2000). handbook of qualitative research (2nd ed.). thousand oaks, ca: sage. foucault, m. (1972). the archaeology of knowledge (a. m. sheridan smith, trans.). new york: pantheon books. (original work published 1969) foucault, m. (1980). truth and power (c. gordon, l. marshall, j. mepham & k. soper, trans.). in c. gordon (ed.), power/knowledge: selected interviews and other writings, 1972–1977 by michel foucault (pp. 109–133). new york: pantheon books. (interview conducted 1977) foucault, m. (1997). on the genealogy of ethics: an overview of work in progress. in p. rabinow (ed.), the essential works of michel foucault, 1954–1984 (vol. i, ethics, pp. 253–280). new york: new press. (interview conducted 1983) grouws, d. a. (ed.). (1992). handbook of research on mathematics teaching and learning. new york: macmillan. lester, f. k. (ed.) (2007). second handbook of research on mathematics teaching and learning. charlotte, nc: information age. http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/33/20 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/33/20 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/33/20 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/47/48 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/47/48 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/47/48 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/4/4 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/4/4 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/18/15 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/18/15 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/100/81 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/100/81 journal of urban mathematics education december 2015, vol. 8, no. 2, pp. 11–22 ©jume. http://education.gsu.edu/jume marrielle myers is an assistant professor of mathematics education at kennesaw state university, md 0121, kennesaw hall, room 3111, 585 cobb avenue, kennesaw, ga 30144; email; mmyers22@kennesaw.edu. her research focuses on equity in elementary mathematics education and pre-service preparation of elementary teacher candidates. paola sztajn is a professor of mathematics education and department head for the department of teacher education and learning sciences at north carolina state university, 602 m poe hall, campus box 7801, raleigh, nc, usa, 27695-7801; email: psztajn@ncsu.edu. her research focuses on the mathematical knowledge and professional development of elementary school teachers. p. holt wilson is an associate professor of mathematics education at the university of north carolina at greensboro, 468 soeb, p.o. box 26170, greensboro, nc, usa, 27402; email phwilson@uncg.edu. his research focuses on the professional development of k–12 mathematics teachers. cyndi edgington is a teaching assistant professor of mathematics education at north carolina state university, 502j poe hall, raleigh, nc, usa, 27695-7801; email: cpedging@ncsu.edu. her research focuses on preservice and in-service teachers’ professional development. commentary from implicit to explicit: articulating equitable learning trajectories based instruction marrielle myers kennesaw state university paola sztajn north carolina state university p. holt wilson university of north carolina at greensboro cyndi edgington north carolina state university ver the last half century, mathematics education has seen numerous reform initiatives and standards. about every ten years, a new wave of documents offers recommendations on how to best teach mathematics. ellis and berry (2005) argued that although these “so-called reform” documents have increasingly attended to ideas such as mathematics for all (national council of teachers of mathematics [nctm], 1989) and equity (nctm, 2000), they have, in fact, only generated “revisions” of mathematics instruction because they “failed to change significantly the face of the mathematically successful student” (p. 8). the common core state standards for mathematics (ccssm; national governors association center for best practices & council of chief state school officers, 2010) offers the most recent set of recommendations for mathematics reform. the document builds on the concept of learning trajectories (lt)1 (daro, mosher, & corcoran, 2011) and outlines the mathematics content and practices to be addressed at particular grade levels. 1 while some scholars use the term learning progression, we use the term learning trajectory in this commentary to encompass both progressions and trajectories. o http://education.gsu.edu/jume mailto:mmyers22@kennesaw.edu mailto:psztajn@ncsu.edu mailto:phwilson@uncg.edu mailto:cpedging@ncsu.edu myers et al. commentary journal of urban mathematics education vol. 8, no. 2 12 with the widespread adoption of these standards, mathematics teacher educators have worked to share ideas about trajectories with teachers. because research on learning largely developed separately from research on teaching, our work used lts to link these two bodies of research. we theorized the concept of learning trajectories based instruction (ltbi) as a model of teaching where instructional decisions are grounded in research on student learning in the form of trajectories and we interpreted several highly developed domains of research on mathematics teaching in relation to these trajectories (sztajn, confrey, wilson, & edgington, 2012). since that time, we have worked to share this model with teachers in professional development settings, and our research has empirically examined and elaborated the affordances of ltbi. in our initial conceptualization, we considered ltbi to be situated within a broader landscape of equity in mathematics education because of its emphasis on the ways that instruction grounded in students’ mathematical thinking provides opportunities for learning and access to rigorous mathematics instruction based on individual students’ existing understandings of mathematics (civil, 2006; fennema & meyer, 1989). by articulating a relation between teaching and learning, ltbi foregrounded students’ thinking and suggested that students should be the primary consideration of instruction. for us, organizing instruction around these trajectories challenged the “correct/incorrect” dichotomy of ideas by acknowledging and honoring a variety of partial and alternative understandings that students have. it provided an organizational structure that aided in anticipating and explaining student learning. we proposed that ltbi pedagogical practices assisted teachers in engaging students in worthwhile tasks that engendered deep learning, eliciting and responding to kernels of important mathematical ideas, orienting students to one another’s ideas in classroom discussions. these affordances advanced the work of supporting teachers in improving mathematics learning for every student. as lts have proliferated across educational communities, some scholars expressed concerns with issues of equity and diversity inherent in their conceptualization, development, and implementation. some have argued that, though mathematics learning is multidimensional and occurs through connections across multiple domains, trajectories reduce learning to a hierarchical, linear path (empson, 2011; lesh & yoon, 2004). anderson and colleagues (2012) reported that researchers and other leaders in science and mathematics education have raised a number of concerns about trajectories. they pointed to theoretical framings that inadequately account for the ways culture, race, and context shape learning, challenging developers to expand the methodologies used for development and validation to ensure diverse student populations are represented in the trajectories. another concern was related to possible translation effects as trajectories move from research to policy and practice. myers et al. commentary journal of urban mathematics education vol. 8, no. 2 13 unresolved questions and concerns about the trajectories and their unintended uses led to a further examination of the ltbi model and its potential uses. in particular, we noted that our initial conceptualization explicitly attended to some aspects of equity (e.g., opportunity to learn) while leaving others tacit (e.g., race, culture, and language). in this commentary, we critically analyze the ltbi model using gutiérrez’s (2007) dimensions of equity as a comprehensive framework for equity in mathematics education. through this theoretical examination, we make explicit the assumptions inherent in the initial model and identify opportunities for ltbi to enhance equitable mathematics instruction. first, we briefly introduce current research on lts and highlight principles of lts that we contend are aligned with equitable instruction. next, we present gutiérrez’s framework and a rationale for its selection as a tool for our theoretical analysis, briefly describing each of its dimensions. we detail our analysis of ltbi and conjecture what equity-oriented uses of the model might look like in instruction. we conclude with an invitation to the mathematics teacher education community to discuss the potentials and challenges of using lts to support equitable mathematics instruction. learning trajectories mathematics education has increasingly attended to learning trajectories in recent years. as research-based representations of the ways students’ thinking in a particular domain develops over time with instructional opportunities and supports (clements & sarama, 2004; confrey, maloney, nguyen, mojica, & myers, 2009), learning trajectories are viewed by some as promising tools for aligning standards, assessments, curriculum, and instruction (confrey, 2012; daro et al., 2011). research in this area initially addressed two areas: development and validation (barrett, clements, klanderman, pennisi, & polaki, 2006; battista, 2007; confrey, 2012), and curriculum and assessment (battista, 2004; clements & sarama, 2009). scholars designed and empirically validated trajectories in different mathematics content areas, including whole number operations (clements & sarama, 2009; confrey, 2012; sherin & fuson, 2005); geometry and spatial thinking (battista, 2007; clements & sarama, 2009); length, area, and volume measurement (barrett et al., 2006; battista, 2006; clements & sarama, 2009); and functions (bernbaum wilmot, schoenfeld, wilson, champney, & zahner, 2011; lobato, hohensee, rhodehamel, & diamond, 2012). others have sought to design curricula and assessments based on lts (battista, 2004; clements & sarama, 2007; confrey, 2012). more recently, lt research expanded to include a focus on instruction through examining the ways lts might be useful in teacher education and mathematics classrooms (edgington, 2012; mojica, 2010; wickstrom, 2014), while others are beginning to examine student outcomes in lt-based classrooms (clements, sarama, wolfe, & spitler, 2013; sarama, lange, clements, wolfe, & spitler, 2012). myers et al. commentary journal of urban mathematics education vol. 8, no. 2 14 evidence is accruing that outlines positive effects of ltbi, including more learnercentered classrooms rich with mathematics conversations (clements & sarama, 2008; clements et al., 2013), instructional decisions based on student thinking (bardsley, 2006; mojica, 2010; wickstrom, 2014; wilson, sztajn, edgington, & myers, 2015), improved understandings of student thinking (mojica, 2010; wickstrom, 2014; wilson, 2009), the selection of developmentally appropriate activities (brown, sarama, & clements, 2007; edgington, 2012), and anticipations of the variety of students’ conceptions (edgington, 2012). clements and colleagues (2013) even argued that lt-based instruction could be especially beneficial for african american students based on results of standardized measures. several principles unify the various conceptualizations of lts in the field that, in our view, provide a foundation for equitable instruction. first, lts are grounded in empirical research with students and challenge more traditional approaches to curriculum development and instruction that focus on disciplinary knowledge. in contrast with a singular development of a concept based on the logic of the mathematics, trajectories acknowledge and build from variations in students’ conceptions as they engage in mathematics. second, specific learning goals for students are clear. though informal, partial, and alternative understandings are represented in them, lts outline general paths that expect these earlier understandings to become more sophisticated over time. third, trajectories are probabilistic in nature, suggesting only likely routes to learning while identifying key conceptual accomplishments along the way. this fluid nature allows for multiple points of entry for students and offers opportunities to engage ideas and coordinate them with other concepts, all the while building toward robust disciplinary understandings. finally, lts and student learning are necessarily dependent upon instructional opportunities. task quality and implementation are essential in supporting student learning. this last aspect is critical and lies at the heart of ltbi—students do not simply progress along a trajectory because of maturation (confrey et al., 2009). learning is a product of carefully designed learning environments, well-planned learning activities, and appropriate supports from teachers (daro et al., 2011). thus, by developing instruction that is guided by lts, ltbi can support more equitable instruction. teachers’ learning about students’ thinking and how it might evolve into formal mathematical concepts over time explains many of the improvements in instruction that ltbi supports. yet the strength of using lts in instruction—a focus on representing levels of thinking for all students—is also the cause for concerns. such a focus precludes consideration of how students’ social and cultural backgrounds shape learning and ignores the resources many students bring to instruction. though progress along a trajectory is critical, it should not come at the expense of students’ identities. our evolving awareness of the tensions between potential benefits and consequences of different uses of lts in teaching coupled with the critiques of lts raised by the field led us to seek a theoretical lens to explicate the strengths myers et al. commentary journal of urban mathematics education vol. 8, no. 2 15 of ltbi as well as illuminate opportunities for more equity-oriented uses of the model. gutiérrez’s dimensions of equity to critically examine our assumptions about equity in ltbi, we selected gutiérrez’s (2007) equity framework as a theoretical lens. four dimensions along two axes comprise the framework. access and achievement are associated with the dominant axis, which represents what students need to know to participate in mainstream mathematics. in contrast, identity and power are dimensions of the critical axis, which represents what students need to become critical members of society. four aspects of this conceptualization of equity supported its selection as a tool for our theoretical analysis of ltbi. first, its organization around the dominant and critical axes juxtaposed the progress toward formal mathematics content of lts with instructional commitments to the successful development and maintenance of students’ identities. second, its dominant axis specifically attends to access, positioning it as an independent pre-cursor of achievement, which allows for specific scrutiny of the ways lbti might be used to provide access while promoting student achievement. third, we view the framework as inherently including elements of culturally responsive pedagogy, such as high academic achievement through cultural competence; varied instructional strategies; and links between schools, homes, and communities (gay, 2000). lastly, for us, the dimensions of gutiérrez’s framework provide a comprehensive representation of research on equity in mathematics education as they address both the mainstream concerns about equity (e.g., opportunity to learn, standardized test scores) as well as issues of culture, language, and socioeconomic status. together, these aspects of the equity framework explain why we used it as a tool to elucidate strengths and identify underdeveloped areas of ltbi in relation to equity. access and achievement for a number of years, scholars in mathematics education conceived of access as opportunity-to-learn (elmore & furhman, 1995; fennema & meyer, 1989; tate, 1995). although opportunity-to-learn remains an important concept, it alone is insufficient in defining equity (flores, 2007; gutiérrez, 2007; silver & stein, 1996). thus, gutiérrez (2007) proposes that access, as one end of the dominant axis of equity, depends on resources that students physically have or do not have. it includes quality mathematics teachers, adequate technology and supplies in the classroom, a rigorous curriculum, reasonable class sizes, and supports for learning outside of class hours. access is a “precursor to achievement” (gutiérrez, 2007, p. 3). attending to access is insufficient if student outcomes are neglected. if students are provided with all of the resources mentioned above and traditional achievement patterns continue to myers et al. commentary journal of urban mathematics education vol. 8, no. 2 16 persist, then populations of students remain underserved. for gutiérrez, achievement is the opposite pole of the dominant axis, signaling the importance of supporting students with access in achieving. achievement includes participation in a given class, course-taking patterns, standardized test scores, and participation in the mathematics pipeline. identity and power gutiérrez (2007) stated: because there is a danger of students having to downplay some of their personal, cultural, or linguistic capacities in order to participate in the classroom or the math pipeline … issues of identity have started to play a larger role in equity research in mathematics education. (p. 3) equitable mathematics instruction, therefore, must provide opportunities for students to maintain and draw upon cultural and linguistic capacity, find a balance between self and others, see themselves in the curriculum, use the curriculum as a tool to view and analyze the world, find mathematics meaningful in their lives, and sense that they have become a better person (gutiérrez, 2007). it is not enough to provide students with access, support achievement at high levels, and maintain students’ personal identities if, “mathematics as a field and/or our relationships on this planet do not change” (gutiérrez, 2007, p. 3). the final dimension, power, is a call for using mathematics to bring about change and social transformation. gutiérrez suggests that this transformation may occur in a variety ways, including changes in who gets to talk in the classroom (voice), changes in who decides on curriculum, and creating opportunities for students to use mathematics to analyze and critique society. a critical examination of ltbi our analysis of ltbi using gutiérrez’s (2007) equity framework revealed a closer alignment of the model with the dominant axis than the critical. though the origins of lt research in assessment and curriculum development render this finding unsurprising, the analysis process highlighted assumptions implicit in our original conceptualization of ltbi as well as areas in need of greater specification. in particular, the lenses of identity and power from the critical axis highlighted areas unaddressed in the original model, identifying potential leverage points for more equityoriented uses of ltbi. in what follows, we conjecture the ways in which ltbi might be used for equitable mathematics instruction by grounding the model in relation to the dimensions of the equity framework. our conjectures represent both connections myers et al. commentary journal of urban mathematics education vol. 8, no. 2 17 to empirical findings and conjectures about how ltbi might be used to assist in achieving goals of equity. access and achievement through ltbi access requires quality mathematics instruction for each student, and we suggest that ltbi can improve this quality by: assisting teachers in attending to students’ logic (wilson, mojica, & confrey, 2013; wilson, sztajn, confrey, & edgington, 2014), supporting teachers in selecting or adapting rigorous and appropriate tasks within their curriculum or from other materials (edgington, 2012; myers, 2014; wickstrom, 2014), and making tasks accessible for each student based on the individual conceptions of the student (myers, 2014). ltbi supports instructional decision-making processes, including eliciting and building upon students’ ideas to facilitate productive mathematical discussions that are accessible to all students, encouraging full student participation (wilson et al., 2015). when using ltbi, teachers may set individual learning goals for students based on their current understandings and how these understandings relate to long-term mathematical objectives (myers, 2014). this shift from a purely disciplinary focus on mathematics to one that focuses on students’ conceptions opens possibilities for teachers to meet the needs of individual students in support of their achievement, an explicit ltbi practice. mosher (2011) stated, if children are to meet standards, “schools and teachers have to take responsibility for monitoring students’ progress and intervening on a timely basis when needed” (p. 1). further, ltbi promotes the use of cognitively demanding, open tasks to create spaces for eliciting evidence of student learning, allowing for the development of additional ways to assess what students know. identity and power through ltbi less apparent in our analysis were direct connections to the critical axis. this finding challenged us to envision how ltbi might be used as a platform to help students see themselves in mathematics and prepare students to use their mathematical knowledge to bring about social transformation. the long-term, developmental nature of lts assists teachers in understanding and valuing their students’ mathematical ideas, thus promoting the idea that all students are learners and doers of mathematics. such a view and understanding on learning promotes teachers’ acknowledgment of the mathematical contributions that all students may make in the classroom and bolsters students’ identities as doers of mathematics. by providing teachers with a framework for various mathematical strategies and how those strategies build toward refined mathematical concepts, ltbi sensitizes teachers to the variety of ways students solve problems and engenders respect for their approaches. this enhanced repertoire of students’ conceptions encourages teachers to be open to, and accepting of, myers et al. commentary journal of urban mathematics education vol. 8, no. 2 18 various forms of communication during mathematical discussions, connecting classroom mathematics to students’ experiences outside of school. similar to the identity dimension, we conjectured possibilities for ltbi to scaffold power. power in the classroom addresses the relations established among teachers, students, and society (gutiérrez, 2007, 2009). it addresses social transformation and the ways in which mathematics can be used as a tool to critique society (gutstein, 2003, 2006, 2007). because lts present a range of students’ mathematical understandings over time, teachers may utilize this knowledge to ensure that students positioned along the trajectory have a voice and are provided with the opportunity to share their thinking. in contrast with a view of students as “empty vessels,” ltbi assists teachers in viewing each student as knowledgeable in unique ways and in sharing that knowledge with their peers, positioning students as having expertise. in summary, our analysis indicated that ltbi was well aligned with the dominant axis of gutierrez’s (2007) framing of equity, and research evidence is accumulating in support of these facets of the model. our examination illuminated assumptions implicit in our original conceptualization and assisted us in strengthening the connections between ltbi and equity. furthermore, aspects of instruction that can support students’ identities as doers of mathematics and views of mathematics as a tool for social transformation unaddressed in the original model were foregrounded, leading us to develop initial conjectures about how ltbi might expand to encompass these goals. appendix a illustrates these refined conjectures as markers of equitable ltbi in classrooms. discussion our theoretical examination of ltbi with a lens for equity resulted from a tension between concerns from the field and positive findings from research. this tension led us to question the nature of an equity-oriented use of ltbi and illuminated implicit assumptions, strengthened connections, and identified new areas where ltbi might develop in relation to equity. this process confirmed many existing findings about ltbi and its potential in classroom instruction and allowed us to situate those findings in relation to the dominant axis. this process also allowed us to bring identity and power to the foreground and envision the ways in which ltbi could be used to support the critical axis. more important, we contend that this examination and the representation of equity-oriented implementation of ltbi proposed in appendix a can generate important discussion in mathematics education in relation to lts. we conclude by positing that it is not the ltbi model, but the use of the model that can be equitable or inequitable. for example, while trajectories support teachers to view student learning along a continuum, they also may allow for reifying of deficit views that justify pre-conceived ideas about “high” and “low” children, or ideas about students who do not follow the “typical” path as “deviants.” due to these po myers et al. commentary journal of urban mathematics education vol. 8, no. 2 19 tential challenges, more discussion and research are needed to understand teachers’ uses of ltbi in creating equitable classrooms and challenging potential inequitable assumptions about what students can or cannot do. we continue to argue that ltbi has the potential to foster equity, but suggest this “potential” requires further empirical validation. therefore, we invite our fellow researchers to engage in conversation to further explore this issue with us, to investigate the affordances and challenges of using lts as instructional tools, and to examine the potential uses of this tool in promoting equitable mathematics instruction. acknowledgements this research is supported by the learning trajectories based instruction project (ltbi) (nsf drl1008364) under the direction of paola sztajn and holt wilson. any opinions, findings, and conclusions or recommendations expressed in this 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(2009). teachers’ uses of a learning trajectory for equipartitioning (unpublished doctoral dissertation). north carolina state university, raleigh, nc. wilson, p. h., mojica, g. f., & confrey, j. (2013). learning trajectories in teacher education: supporting teachers’ understandings of students’ mathematical thinking. the journal of mathematical behavior, 32(2), 103–121. wilson, p. h., sztajn, p., confrey, j., & edgington, c. (2014). teachers’ use of their mathematical knowledge for teaching in learning a mathematics learning trajectory. journal of mathematics teacher education, 17(2), 149–175. wilson, p. h., sztajn, p., edgington, c., & myers, m. (2015). teachers’ uses of a learning trajectory in student-centered teaching practices. journal of teacher education, 66(3), 227–244. http://repository.upenn.edu/cpre_policybriefs/40/ myers et al. commentary journal of urban mathematics education vol. 8, no. 2 22 appendix a equitable ltbi in classrooms a c c e ss teachers engaged in ltbi ensure that students at various levels have entry points to the task. teachers engaged in ltbi identify relevant and research-based materials and technology that support the development of skills represented by lts (e.g., recognize what curricular materials are aligned with lts and thus supportive of their learning goal). teachers engaged in ltbi assess students’ current mathematical understandings and determine the level of support needed to ensure students are able to access and engage with the mathematics content of the instructional task. knowledge of lts informs teachers’ monitoring of their students’ work. teachers engaged in ltbi scaffold classroom discussions in ways that position all students to participate in the conversation and use knowledge of lts to build upon students’ current mathematical understanding and make connections among various mathematical ideas that arise. teachers engaged in ltbi diagnose students’ current understandings while focusing on future conceptions outlined by the lt to ensure that the work students are engaged in is rigorous and has the potential to help students progress toward more advanced mathematics. teachers engaged in ltbi allow students to work in ways that are comfortable to them and represent their work (written work and verbal descriptions) in ways that align with the students’ understanding of mathematics. a c h ie v e m e n t teachers engaged in ltbi set goals for students that are appropriate based on students’ current understandings. teachers engaged in ltbi distinguish what students have already learned from what they are learning and use that understanding to design instruction to advance the students’ learning. teachers engaged in ltbi think of a variety of ways to solicit evidence about students’ understanding. id e n ti ty teachers engaged in ltbi support students’ efforts and encourage movement along the trajectory. teachers use lts to acknowledge students’ current understandings as well as the knowledge that all students can progress. teachers engaged in ltbi create open tasks that are relevant to and affirm their students’ homes and communities. teachers engaged in ltbi recognize, encourage, and determine the validity of a variety of strategies, algorithms, and tools to solve problems. teachers engaged in ltbi assist students in making not only mathematical connections, but also real world connections (global, national, and local). p o w e r teachers engaged in ltbi include all students, allow all students to have voice, and ensure equitable ownership of the ideas and activities that are a part of the mathematics lesson. teachers engaged in ltbi position students as experts based on their usage of certain skills or strategies. teachers engaged in ltbi select or create tasks that impact the communities in which students live. teachers engaged in ltbi recognize various mathematical ideas present in the classroom and encourage all students to present, justify, and defend their ideas. teachers use lts to facilitate discussions, orient students to other, and make mathematical connections. teachers engaged in ltbi frame every student as a creator of mathematical knowledge, recognize what students already know, and self-empower students by helping them see themselves as doers of mathematics. journal of urban mathematics education december 2011, vol. 4, no. 2, pp. 131–139 ©jume. http://education.gsu.edu/jume cecilia henríquez fernández is a doctoral student in social research methodology at the graduate school of education and information studies at the university of california, los angeles, box 951521, moore hall 2027, los angeles, ca 90095; email: ceci08@ucla.edu. her research interests include the relationship between the learning and teaching of mathematics in a variety of social contexts. book review latinos/as’ mathematical experiences: a review of latinos/as and mathematics education1 cecilia henríquez fernández university of california, los angeles s a latina mathematics educator, i was eager to read latinos/as and mathematics education: research on learning and teaching in classrooms and communities. during my undergraduate training in mathematics, i developed a deep interest in sharing my passion for mathematics with children from nondominant communities, especially latino/a children. as one of three latinas in a group of nearly 300 mathematicians at the massachusetts institute of technology (mit), this commitment to my community was strengthened when i noticed how underrepresented latinos/as were within my chosen career path and the struggles we shared. my own difficulty in being successful at my undergraduate institution, coupled with the observation that many of my latino/a friends struggled to finish their careers in science, technology, engineering, and mathematics (stem) made me realize that something needed to be done to ensure future generations of underrepresented students continue to enter stem fields, and that the spaces in which non-dominant students learn mathematics needed to be attuned to meeting the needs of diverse groups of student learners. i became involved in mathematics education when, as an undergraduate student, i founded an outreach program for local middle school children who were from traditionally marginalized groups. this program exposed children from the local urban community to stem disciplines and matched them with undergraduate mentors that held similar backgrounds and interests. the program also held workshops for the parents of the participants that informed them of local resources and addressed concerns parents had about their children’s education. additionally, as an undergraduate, i went through a newly established teacher educa 1 téllez, k., moschkovich, j., & civil, m. (eds.). (2011). latinos/as and mathematics education: research on learning and teaching in classrooms and communities. charlotte, nc: information age. pp. 348, $39.09 (paper; web price), isbn 1-61735-420-5 http://www.infoagepub.com/products/latinos_as-and-mathematics-education a http://www.infoagepub.com/products/latinos_as-and-mathematics-education henríquez fernández book review journal of urban mathematics education vol. 4, no. 2 132 tion program and decided to follow a non-traditional path for a student at mit: a public school teacher. upon graduation, i headed back to my hometown to become a high school mathematics teacher. my experiences as a teacher led me to understand the necessity for better teacher education programs, the lack of resources for teachers (especially first year teachers) in urban communities, and the struggles some latino/a children face in their mathematical educational experiences. although i came from the same community as my students and grew up in a bilingual home, i had a different classroom mathematical experience because my mother was a mathematics teacher. i did not always experience the same struggles in school mathematics that my students did. hence, as a teacher, i often felt that i was not well prepared to deal with the challenges my students encountered in learning mathematics. i felt it necessary to return to graduate school to better prepare myself as an educator. as a graduate student, i have learned a lot from my mentors and from education research about learning and teaching. i have also discovered that we have so much more to understand about the school and everyday practices that inform the teaching and learning of mathematics. given the diversity that exists in our classrooms, we need to understand how teachers’ historical, social, and cultural experiences influence how they engage in the practice of teaching. likewise, we need to understand how students’ historical, social, and cultural experiences influence how they participate in the mathematics classroom. as such, i have continued my path as an educator in research that looks at the learning and teaching of mathematics. more specifically, i am interested in looking at home and community practices of mathematics students and thinking about how we can include these practices in classroom teaching. thus, i too adopt sociocultural theories of learning in my own research in order to make mathematics education accessible to students of diverse backgrounds. i have included this mini-autobiography to give you a sense of the multiple identities that shape my perspective. on one hand, i am a trained mathematician, on the other, a k–12 educator, and yet, on a third, if you will, a first generation mexican-american woman with a deep commitment to sharing her passion for mathematics. i exist as multiple identities at once. therefore, i read latinos/as and mathematics education and present my review with these different perspectives influencing my interpretation. what is this book about? latinos/as and mathematics education documents diverse mathematical experiences that some latino/a students have in and out of classroom settings. it is interesting to note that the diverse representation of mathematical experiences henríquez fernández book review journal of urban mathematics education vol. 4, no. 2 133 within latinos/as and mathematics education is analogous to the definition of what it means to be latino/a within the united states that the editors present in the first chapter. as what it means to be latino/a in the united states varies in multiple contexts, likewise, the contributing authors present multiple ways in which latino/a children experience mathematics within their latino/a and nonlatino/a communities. with this in mind, the use of a sociocultural perspective throughout the multiple analyses found in the volume is appropriate because it refuses the notion of painting all latino/a students in one brushstroke. it allows the authors to examine how the students’ unique home, school, local and nonlocal political communities influence their mathematics education. in the following sections, i give a brief summary of the three major themes that i see develop throughout the book: community experiences, classroom experiences, and the role of assessments. community experiences one of the important factors that latinos/as and mathematics education addresses is that latino/a children’s mathematical education is not limited to what they do in the mathematics classroom. the editors, téllez, moschkovich, and civil (2011), identify home and community mathematical practices and experiences that in various ways influence or can influence children’s classroom experiences. for example, domínguez (2011) illustrates how mothers with little or no prior schooling experience are able to identify mathematical practices in their everyday activity, and help their children make sense of classroom mathematics by situating school mathematics problems in contexts familiar to their children. previous research has shown that bridging home and community practices with classroom mathematics experiences helps students develop interest and motivation in engaging with a subject because it makes the subject matter culturally and socially relevant (lipka et al., 2005; nasir, hand, & taylor, 2008). additionally, drawing upon students’ community experiences helps researchers to continue to move away from culturally deficit views of students’ home experiences (gutierrez & rogoff, 2003). thus, identifying community mathematical practices is useful knowledge because it brings to light collaborative opportunities for schools and local communities to participate together in the students’ mathematical experiences (rogoff, 1994). classroom experiences latino/a children’s mathematical experiences differ from classroom to classroom, and the authors highlight the range of these experiences. in particular, the authors note the range of resources that students encounter in their daily classroom life. we see the importance of latino/a children’s native language in solv henríquez fernández book review journal of urban mathematics education vol. 4, no. 2 134 ing mathematical problems, the importance of creating social spaces where different kinds of mathematical problem solving can occur, the importance of tying subject-matter content to community agency, the importance of teacher preparation, and much more. for example, zahner and moschkovich (2011) attempt to understand the relationship between bilingualism and mathematical reasoning through analysis of the role of code-switching on social interaction during mathematical problem solving in a sixth-grade mathematics classroom in a dualimmersion program. thus, different cultural issues that concern not only latino/a students but also students from other non-dominant communities are presented to further document the diverse cultures that exist within mathematics classrooms. assessments and english language learners although mathematics assessments only account for a small portion of the book, the experiences that latino/a students have with mathematical assessments is an important theme. given the current climate of using testing for accountability, it is important to understand how assessments are written with bias against particular groups of students. solano-flores’ (2011) analysis of english language learner’s (ell) underperformance on standardized assessments in addition to his discussion on the development of standardized assessments causes the reader to question the purpose of creating assessments that are constructed to ignore ell’s linguistic needs and therefore do not accurately evaluate ell’s knowledge of mathematical concepts. in addition, mosqueda (2011) shows that exposure to rigorous mathematics courses mediates latino/a performance on assessments. however, he notes that perceived english proficiency can (does) limit latino/a students’ access to rigorous mathematics classes. thus, both of these authors illustrate that we have much to do in improving the functions of student assessments as well as in improving how well assessments capture student mathematical knowledge. why this book is important and what is missing? latinos/as and mathematics education gives us a glimpse of latino/a children’s experiences in their mathematical classrooms and draws attention to critical issues around mathematics learning that continue to impact latino/a students. for example, english as a second language continues to be an important issue for some children in mathematics classrooms. policies that enforce teachers to ignore linguistic and sociocultural capital have been linked to the continued underperformance of latino/a children on statewide assessments (gandara, hopkins, & martinez, 2011). in addition, latino/a children continue to struggle learning mathematical concepts in de-contextualized classrooms. throughout the volume, henríquez fernández book review journal of urban mathematics education vol. 4, no. 2 135 authors draw from mathematics education literature and latino/a education scholarship in a way that helps us further investigate the issues that latino/a students face contextualized within students’ mathematical experiences. the volume also demonstrates that educational researchers need to continue studying these issues as we think about how to make mathematics education in schools more accessible to latino/a children. latinos/as and mathematics education is important because it extends the discussions around latino/a community mathematical practices that empower latinos/as and mathematics educators to think critically about the communities in which they teach. it helps latinos/as and non-latinos/as to value the different mathematical knowledge that latino/a community members are constantly engaging. acknowledgement of our home mathematical experiences also empowers latinos/as because it gives agency to our children, as we consider the different but powerful knowledge that our children bring to their classrooms. this conversation is particularly important now, given the current anti-latino/a and anti-immigrant dialogue that is taking place nationwide which actively pushes latino/a students out of schools (robertson, 2011) and denies them of their basic human right to an education. an important discussion that could have been more explicit in latinos/as and mathematics education is what “mathematics education” means to the authors. perhaps this discussion was omitted given that mathematics education has many representations to different people; however, it would have been helpful if in each of the chapters, the authors clearly stated their own mathematical education values and beliefs. for example, quintos, civil, and torres (2011) are explicit that they view mathematics education as the teaching of mathematical practices that encourages students to be agents of change. this explicitly stated view allows us, as readers, to understand how and why quintos and colleagues choose the evidence that supports the arguments they make. clearly, articulating the researcher’s position in relation to mathematics education would help the reader understand why the researcher decided to conduct the study in a particular way. additionally, latinos/as and mathematics education covers a wide variety of topics within an equally wide variety of latino/a communities, and brings the reader to the realization that, when it comes to thinking about how latinos/as experience mathematics education, the social contexts of each educational setting need to be considered. given the centrality of social context within the volume, and within my own work, i think it is important to note that readers should not be alarmed if they feel that there is a lack of generality within latinos/as and mathematics education. the study of mathematics education is complex in itself, and there are a number of ways to examine many of the issues of mathematics education (such as content, problem types, classroom practices, classroom structures, language, classroom culture, etc.), which are always situated within a particular henríquez fernández book review journal of urban mathematics education vol. 4, no. 2 136 context. while it is important to gain a deep understanding of particular issues in mathematics education, it is equally important to first survey the landscape of such issues. through this survey, we see the need to continue research of mathematics education in context, so that we can get closer to an understanding of how we can work with teachers in creating mathematics classrooms that make mathematics accessible to all of its students. thus, latinos/as and mathematics education invites us to think about these mathematics education issues within context(s), particularly in the context of latinos/as mathematical learners. as editors, téllez, moschkovich, and civil (2011), provide the reader with a collection of powerful work that highlights the potential for classroom agency within the latino/a community. unfortunately, however, not all authors included in the volume discuss the issue of student mathematical agency, particularly within the sections that do not explore the intersection of classroom and community experiences. the chapters that document community experiences clearly articulate sources of student and family agency, even when students and families do not always realize the power of their personal knowledge and experiences. agency is noted in the sections on classroom experience and assessment; however, the connection between student agency in the classroom and mathematics learning is not always readily apparent. student agency within the classroom is an important topic that is related to teacher development and assessment. these sections would have been more powerful if the authors had made the discussion of agency a primary theme rather than a secondary theme. finally, latinos/as and mathematics education illuminates places where additional research is needed, and illustrates in what areas we need to create change so that latino/a students can have mathematical experiences that value the mathematical knowledge they bring to class. overall, this volume creates opportunities for educators and researchers to think about (different) ways in which we can continue to make mathematics education accessible to latinos/as in the united states and beyond. who should read this book? latinos/as and mathematics education is a great read for anyone who works with, is a part of, or just wants to learn about the latino/a community. more specifically, anyone in educational settings such as teachers and school administrators can read this book to see the kinds of opportunities that exist in relating mathematical learning to local community practices. i would hope that this book would encourage educators to reach out to their local community (especially parents) and to begin to dialogue about ways in which they might work together to help students make connections between what they learn in the classroom and what they do at home. educators can also find this book useful in terms of think henríquez fernández book review journal of urban mathematics education vol. 4, no. 2 137 ing about other social issues included in its pages, such as language and the organization of social participation structures within the schools and classrooms. further, educators can and should take into consideration their own professional development and think about assessments in relation to their students’ mathematical experiences and its direct impact on students’ mathematical social identities. additionally, i think this volume can be an excellent resource for latino/a parents. although most of latinos/as and mathematics education is written in academic language, the chapters that deal with the community and home experiences of latino/a students use language that is accessible to the general public. in doing so, téllez and colleagues (2011) give parents agency in many ways. first, latinos/as and mathematics education gives latino/a parents a voice by sharing their experiences and knowledge. second, the findings by domínguez (2011) recognize that regardless of the number of years that parents may have spent in school (or their success in school mathematics), parents are constantly engaged in complex mathematical activity in their everyday lives. third, acosta-iriqui, civil, díez, palomar, marshall, and quintos-alonso (2011) recognize latino/a parents’ ability to recognize gaps in their children’s education, and their concern for their children’s success. thus, téllez and colleagues not only give voice to latino/a parents, but they also recognize latino/a parents’ voices as legitimate, and in this way give latino/a parents agency in their children’s education. latinos/as and mathematics education is accessible by most readers; however, the intended audience is most likely educational researchers. this intent is evident through the use of academic language and the academic writing style by most of the authors. moreover, the authors present important issues on the mathematical experiences latinos/as have in their everyday lives in relation to research done on mathematics education. as a result, latinos/as and mathematics education brings together research on mathematics education and research on latinos/as educational experiences in an important way that not only informs the existing literature in mathematics and latino/a education but also allows the educational community to think about these issues in a practical way. final comments latinos/as and mathematics education is a riveting and foundational book that brings together the work on the latino/a educational experience with the work on mathematics education in a cohesive and unprecedented way. between domínguez’s (2011) work on mexican mothers’ desire and ability to help their children be successful in their mathematics classrooms, to morales, vomvoridiivanovic, and khisty’s (2011) study of parental participation in mathematical activities in informal settings, to solano-flores’ (2011) historical analysis of the assessment development process, téllez, moschkovich, and civil (2011) have skill henríquez fernández book review journal of urban mathematics education vol. 4, no. 2 138 fully created an opportunity for dialogue between the educational and local latino/a communities that will hopefully give agency to latinos/as in their own education. latinos/as and mathematics education brings to light issues that continue to be critical for mathematics education, and guides the reader to think about the work that we need to continue doing to improve the educational experiences of latino/a children. through this edited volume, téllez, moschkovich, and civil (2011) significantly contribute to both the work on mathematics education and latino/a education by bringing together a critical mass of researchers and scholars who force us to think differently about the ways that latino/a children experience mathematics education and how these experiences can shape what mathematical identities they take both in the classroom and in their social communities. given the current projections by the u.s. 2010 census on the growth of the latino/a population in the united states (u.s. census bureau, 2010), i would encourage you to read latinos/as and mathematics education to become more knowledgeable concerning meeting the needs of latino/a mathematics learners. the latino/a population is the fastest growing population in the united states, and we as a community need to be prepared to meet latino/a students’ needs in and out of mathematics classrooms. references acosta-iriqui, j., civil, m., díez-palomar, j., marshall, m., & quintos-alonso, b. (2011). conversations around mathematics education with latino parents in two borderland communities: the influence of two contrasting language policies. in k. téllez, j. moschkovich, & m. civil (eds.), latinos/as and mathematics education: research on learning and teaching in classrooms and communities (pp. 125–148). charlotte, nc: information age. domínguez, h. (2011). situating mexican mothers’ dialogues in the proximities of contexts of mathematical practice. in k. téllez, j. moschkovich, & m. civil (eds.), latinos/as and mathematics education: research on learning and teaching in classrooms and communities (89–124). charlotte, nc: information age. gandara, p., hopkins, m., & martinez, d. (2011). an assets view of language and culture for latino students. latino policy and issues brief, no. 25. gutierrez, k., & rogoff, b. (2003). cultural ways of learning: individual traits or repertoires of practice. educational researcher, 32(4), 19–25. lipka, j., hogan, m. p., webster, j. p., yanez, e., adams, b., clark, s., &, lacy, d. (2005). math in cultural context: two case studies of a successful culturally based math project. anthropology and education quarterly, 36(4), 367–385. morales, h. vomvoridi-ivanovic, e., & khisty, l. l. (2011). a case-study of multigenerational mathematics participation in an after-school setting: capitalizing on latinos/as funds of knowledge. in k. téllez, j. moschkovich, & m. civil (eds.), latinos/as and mathematics education: research on learning and teaching in classrooms and communities (pp. 175– 194). charlotte, nc: information age. henríquez fernández book review journal of urban mathematics education vol. 4, no. 2 139 mosqueda, e. (2011). teacher quality, academic tracking and the mathematics performance of latino english language learners. in k. téllez, j. moschkovich, & m. civil (eds.), latinos/as and mathematics education: research on learning and teaching in classrooms and communities (pp. 315–339). charlotte, nc: information age. nasir, n. s., hand, v., & taylor, e. v. (2008). culture and mathematics in school: boundaries between “cultural” and “domain” knowledge in the mathematics classrooms and beyond. review of research in education, 32, 187–240. quintos, b., civil, m., & torres, o. (2011) mathematics learning with a vision of social justice: using the lens of communities of practice. in k. téllez, j. moschkovich, & m. civil (eds.), latinos/as and mathematics education: research on learning and teaching in classrooms and communities (pp. 233–258). charlotte, nc: information age. robertson, c. (2011, october 3). after ruling, hispanics flee an alabama town. new york times. retrieved from http://www.nytimes.com/2011/10/04/us/after-ruling-hispanics-flee-analabama-town.html?scp=10&sq=alabama%20schools&st=cse. rogoff, b. (1994). developing understanding of the idea of communities of learners. mind, culture, and activity, 1, 209–229. solano-flores, g. (2011). language issues in mathematics and the assessment of english language learners. in k. téllez, j. moschkovich, & m. civil (eds.), latinos/as and mathematics education: research on learning and teaching in classrooms and communities (pp. 283– 214). charlotte, nc: information age. téllez, k., moschkovich, j., & civil, m. (eds.). (2011). latinos/as and mathematics education: research on learning and teaching in classrooms and communities. charlotte, nc: information age. u.s. census bureau. (2010). 2010 census shows america’s diversity [press release]. retrieved from http://2010.census.gov/news/releases/operations/cb11-cn125.html. zahner, w., & moschkovich, j. (2011). bilingual students using two languages during peer mathematics discussions: ¿qué significa? estudiantes bilingües usando dos idiomas en sus discusiones matemáticas: what does it mean? in k. téllez, j. moschkovich, & m. civil (eds.), latinos/as and mathematics education: research on learning and teaching in classrooms and communities (pp. 37–62). charlotte, nc: information age. http://www.nytimes.com/2011/10/04/us/after-ruling-hispanics-flee-an-alabama-town.html?scp=10&sq=alabama%20schools&st=cse http://www.nytimes.com/2011/10/04/us/after-ruling-hispanics-flee-an-alabama-town.html?scp=10&sq=alabama%20schools&st=cse http://2010.census.gov/news/releases/operations/cb11-cn125.html journal of urban mathematics education july 2009, vol. 2, no. 1, pp. 1–4 ©jume. http://education.gsu.edu/jume lou edward matthews is currently directing the institute for teaching excellence action and change located in bermuda; email lmatthews@myiteach.org. a former assistant professor of mathematics education in the college of education at georgia state university, his work as a public scholar, commentator, organizer, writer, speaker, and consultant has focused on excellence, innovation, and leadership in education at local, national, and international levels in the united states and bermuda for the past 15 years. dr. matthews is currently serving as the 2009 past president of the benjamin banneker association, a national nonprofit organization advocating excellence in mathematics for black students. dr. matthews is also co-founder and currently inaugural editor-in-chief of the journal of urban mathematics education. editorial identity crisis: the public stories of mathematics educators lou edward matthews institute for teaching excellence action and change (iteach) hamilton, bermuda each time that i am asked what it is that i do, i respond with the obligatory: “i am a professor of mathematics education at brand x university.” to which, and overwhelmingly so, the counter-response involves two juxtaposed statements. the first is usually something like: “wow, you must be a genius” (an assertion one should never deny in good company), and the second, usually more ominous, is: “i was never any good at mathematics.” reflecting on these two public statements one can sense that (a) our work perception is unwittingly attached to the privileged intellectual status that those who are successful in mathematics enjoy; and (b) we simultaneously inherit the legacy of deeply imbedded frustrations with the discipline mathematics, which most people in the public domain have also come to acknowledge. so even as we in the urban domain, and as editors of this journal, seek out a discourse agenda that addresses urban complexities, challenges, and excellence, we are simultaneously constrained by fearful, polarizing reactions to what people believe we do. in nearly 20 years as a mathematics and mathematics education teacher and professor at the middle school, high school, and college levels in bermuda, and several states in the midwest and southern united states, i have witnessed this scenario repeatedly being played out in my public conversations. notwithstanding, similar stories have been recounted by many of my students and colleagues who reside in the urban domain. many a student of mathematics education can summarize the very dynamic history of our field as a nexus of forces in psychology, mathematics, and society. but outside of our community, mathematics educators often live in obscurity relative to the public work that we really do—our public story. in light of the potential impact of the work of the recent national panel under president bush and the outcomes of stimulus-funded work in the urban do matthews editorial journal of urban mathematics education vol. 2, no. 1 2 main, i often worry that the present condition of our public story is akin to an identity crisis. this crisis was made none more clearly to me than when i returned home to bermuda to join more directly in education reform after 10 years in the united states. in my activities, collaborating with schools and districts, conducting and publishing research on bermudian education, speaking to parent and teacher groups, leading out locally and nationally through the benjamin banneker association and the national council of teachers of mathematics (the world’s largest mathematics education organization, emphasis added), in a country as affluently modern as mine, i found myself for the first time in a long time needing to explain (no justify) the depth and breadth of my experiences, talents, and ideas. my story had been effective for navigating the mathematics education world in which i resided, but was found wanting as it related to the realities of people i knew best. i saw this disconnect firsthand during the last summer course i taught at the university entitled: “public narrative and the empowered urban mathematics educator.” the course was the last of a series of three courses that i had taught over the last three years where my students and i had sought to better understand our mission in the public domain. as a course model in this last course, we drew on the inspiring leadership framework of the harvard professor marshall ganz, called public narrative. according to ganz (2008), public narrative is the leadership art that centers around central questions founded in the penultimate leadership story of moses: who am i; what am i called to do; who are these people; and what are we called to do, together. succinctly, it is our story of self, us, and now. for the transformative leader, leadership involves framing a new and different public narrative built around (a) the story of my values, my challenges, and my life, [self]; (b) the challenges that we have seen and face together, [us]; and (c) our mission in what we can accomplish together, [now]. ganz correctly and refreshingly argues that leadership involves moving people to achieve purpose in perplexing times and that this process uses narrative to appeal to the head (logic), the heart (emotions), and the hands (action). a most famous student of ganz, now president obama, recently captured the attention of the world with such an application of public narrative (recall stories of his mother and grandmother, and the mission of yes we can). yet, as skillful as obama accomplishes this narrative process, i must admit it appears to be an uneasy challenge for mathematics educators. in our first session o f the summer course, i asked each of my 15 graduate students of mathematics education who were taking the course to simply introduce themselves and tell a little bit about themselves—that is, tell their story. it wasn’t soon thereafter—as i had suspected—that the conversations progressed into venting about the kinds of frustrations and disconnect we were experiencing as we work as mathematics educators. in essence, i had asked them to share their story of self. now, why had our con matthews editorial journal of urban mathematics education vol. 2, no. 1 3 versations gone in this direction, becoming so uninspiring, in this meeting of progressive educators with impressive credentials? as we continued in the course, deliberately probing, refining, and retelling our stories to unveil more of our values and personal challenges in and around the discipline of mathematics education, the nature of our stories changed. i later learned that several students had overcome extreme personal hardships including health, expectation, and financial challenges. as the sessions progressed and we unveiled our “true” inspiring selves, we realized in reflection that our earlier stories—of frustration with gaps and administration—seemed to pale in comparison. the stark contrast rests with how we tell our public stories but points even more urgently to several challenges to our community. for starters, we must begin to redefine mathematics education as a movement of people’s stories—not merely content and curriculum. it is, however, far easier to describe the developments in our discipline’s journey as matters of change in curriculum and content (e.g., back-to-basics, new math, etc.) and political and technological challenges in society (e.g., sputnik, cold war, calculators, japan’s auto dominance, etc.). our history is less attached to social movements that stem from the stories of people, such as the civil rights movements for racial and gender equality, to name just two. as much as i have found myself in the company of “equity” researchers, even much of our work in mathematics education has focused on technical gaps in achievement through curriculum changes, increasing content knowledge, and/or adding to the strategy repertoire of teachers. these efforts have been important, but public narrative suggests, and as my original lead in indicates, that there is a public emotive element which we must draw upon in the framing of our public selves that is important for leadership. yet, no program in mathematics education i have come across does so. our discipline is famously framed around problematic premises: what some have said they know about human learning, the domain of framed mathematics, and certain developments in society. each of these premises point to an eclectic nature of human knowledge-building and community. each involves conversations with people to some degree or the other. hence, the opportunity to define a more people-centric mathematics education is our greatest challenge. it, however, will require us to tell our public story in the face of very perplexing times in urban education. so we must first open up spaces inside our undergraduate and graduate programs, our conferences, our meetings, symposia, and the like. that is, we must begin to connect the dots of our individual stories to, first, inspire ourselves. quite frankly, we suffer from storytelling confidence. our stories can connect, but as we discussed in the summer course, we will have to be deliberate about (a) our intentions to do so, (b) what we choose—or choice points—to share, (c) what the moral will be, and (d) our audience. we can share how we overcame personal chal matthews editorial journal of urban mathematics education vol. 2, no. 1 4 lenges in learning and experiencing mathematics to now take up its reform mission. secondly, i offer that we must focus very deliberately on our outward manifestations of our collective public self. we must allow stories of self to occupy prominent focus in structuring our research publications, professional development initiatives, reports, public meetings, and in general our public conversations. this focus connects us to people, but too often it has been reserved only for the “confessions” of qualitative researchers, banished to hastily framed sections called “role of the researcher,” “limitations,” or “appendix.” in this issue of the journal of urban mathematics education, you will see in several of the articles a very determined attempt on the part of scholars and editors to illuminate public stories to engage others in the urban domain by sharing their public story as a central focus of reporting their work. we hope it is a sign of things to come. references gantz, m. (2008). what is public narrative? retrieved july 22, 2009, from: http://www.cuac.org/documents/whatispublicnarrative08.pdf http://www.cuac.org/documents/whatispublicnarrative08.pdf journal of urban mathematics education july 2017, vol. 10, no. 1, pp. 74–94 ©jume. http://education.gsu.edu/jume annica andersson is an associate professor in mathematics education in the department for mathematics and science teaching, malmö university, 205 06, malmö, sweden; email: annica.andersson@mah.se. her research is located at the intersections of mathematics education, language, culture, and social justice with a particular focus on equity, power, discourses, and relationships in mathematics education school contexts. kate le roux is a senior lecturer in language development in the centre for higher education development  university of cape town, private bag x3, rondebosch, 7701, south africa; email: kate.leroux@uct.ac.za. her research is located at the intersection of language, mathematics, and the learning of disciplinary knowledge in science and engineering, with a particular focus on equity, power, and identity in the higher education context. toward an ethical attitude in mathematics education research writing annica andersson malmö university sweden kate le roux university of cape town south africa in this article, the authors propose a set of multi-level questions as a guide for developing an ethical attitude in researcher–participant and researcher–researcher relations during the research writing process. drawing on the sociopolitical turn in mathematics education, the authors view these relations in terms of power and positionings, in the dialectic between the micro-level of research writing and the wider, macro-level context of mathematics education. the authors illustrate the use of the proposed questions through a back-and-forth dialogue. the dialogue draws on experiences from a writing collaboration in which the authors—“the researchers”—wrote up for publication research conducted in their respective contexts of the political north and political south. both research projects focused on how mathematics students—“the participants”—narrate and hence position themselves and are narrated and positioned by mathematics education and sociopolitical discourses in research publications. keywords: ethical attitude, research writing, research relations, sociopolitial turn ara is a 17-year-old boy who repeatedly referred to his background in interviews that focused on his mathematical identities of failing in mathematics. he grew up with eight siblings in a kurdish immigrant family in sweden. at home he speaks (one of the) kurdish languages; none of his parents speak swedish. but ara learns mathematics in swedish. ara says that he, after failing year nine mathematics, had to attend a compulsory summer school “som min farbror undervisade” [that my uncle taught]. with an improved grade, he qualified for upper secondary school. however, now in upper secondary school, he says that he also needs to work late nights at his brother’s pizza restaurant. so, due to less time for homework and sleep, he says that he is failing again. annica – researcher, teacher, female, multilingual swedish-speaking, middleclass, white, swede luthando, a university student who identifies as black african, talked in his interview about his home in an urban, south african “township,” which was not “very advanced.” he said his “coloured” high school was “disadvantaged” as it lacked computers and maths teachers and had large, http://education.gsu.edu/jume mailto:annica.andersson@mah.se mailto:annica.andersson@mah.se mailto:kate.leroux@uct.ac.za andersson & le roux ethical research writing journal of urban mathematics education vol. 10, no. 1 75 noisy classes. he speaks isizulu at home but learnt school mathematics in english. using the community library, studying alone, and “cutting out” classroom noise, luthando was the “top student” in his school. furthermore, he claimed other students “knew me by my marks.” kate – researcher, teacher, female, monolingual english-speaking, middle class, white, south african ecent mathematics education research publications point to a concern with power and positionings in relations between mathematics education participants in and across contexts characterised by social, cultural, racial, and other differences. this concern may focus on teacher–student relations (e.g., amidon, 2013; bartell, 2011), relations between teacher educators and future mathematics teachers (e.g., herbel-eisenmann et al., 2013), and researcher–participant and researcher– researcher relations (e.g., adler & lerman, 2003; bartell & johnson, 2013; d’ambrosio et al., 2013; foote & bartell, 2011). in this article, we focus on researcher–participant and researcher–researcher relations within mathematics education research, with a specific focus on the research writing process. our discussion is presented at a period in history characterised by related ethnic, racial, class, and linguistic tensions between people within particular contexts. these contexts are foregrounded, for example, by the black lives matter movement in the united states, by the rhodes must fall movement and decolonization debates in south africa, in anti-immigration sentiments in europe and the united states, in the shift toward nationalist and populist parties such as the sweden democrats [sverigedemokraterna] in sweden, and in recurrent xenophobic attacks across the globe. in a time of internationalization and international conferences, the ease of communication in many countries provides enabling conditions for collaborative research relations across countries and continents. indeed, a researcher’s international collaborations in english, the lingua franca in the mathematics education research community (meaney, 2013), convey a level of status. however, internationalization in mathematics education research brings with it conflicting discourses concerning equity, plurality, complexity, and values (atweh & clarkson, 2002). ernest (2016) has problematized the effects of the global knowledge economy on education in terms of ideology, recruitment, appropriation, and dominance. other researchers express reservations about what they have to offer participants in (e.g., hand & masters goffney, 2013) and across (e.g., ernest, 2016; valero, 2014; wagner, 2012) contexts. for some, collaboration requires publishing in english as a second or third language (meaney, 2013). here, we pursue an argument that researchers can, in a powerful way, use theory to write about research participants and their experiences (cf. gutiérrez, 2013; valero, 2014), and to work with one another as writers. following walshaw (2013), we use ideas from poststructuralism as a language to talk about “ethical practical action” (p. 101) in mathematics education research relationships. in particular, we use concepts from what has been identified as the sociopolitical turn in r andersson & le roux ethical research writing journal of urban mathematics education vol. 10, no. 1 76 mathematics education (gutiérrez, 2013; valero, 2004); we explain our particular choice of using upper and lower case p/p in socio-p/political in the next section. these concepts have been used to foreground researcher–participant relations (chronaki, 2004; meaney; 2004); we extend their use here to include researcher– researcher relations. we view these relations in terms of power (fairclough, 2001; foucault, 1977) and positionings in intertwined discourses as suggested by andersson and wagner (2016). we use the term positionings as it points us to the distribution of power within discourses (harré & van langenhove, 1998) in the dialectic between the micro-level of the research writing process and the macro-level of the wider research context. these concepts enable us as collaborating researchers to talk about and account for our political writing choices of what and how we write about participants in our research in and across contexts of social, cultural, language, and political difference. our contribution here is a theoretically informed set of multi-level questions that can act as an ethical guide for mathematics education researchers as they reflexively work with the challenges of power and positionings in researcher– participant and researcher–researcher relations during the research writing process. we illustrate this framework using experiences from a writing collaboration reported in andersson and le roux (2015) as we wrote up for publication our research conducted in our respective contexts: annica in the political north, and kate in the political south.1 both research studies (described later) focus on how students such as ara in sweden and luthando in south africa narrate and hence position themselves and are narrated and positioned by mathematics education and sociopolitical discourses as included and/or excluded. the student interview and other data in our research projects offer a remarkably rich opportunity to listen to student voices on being mathematics students in the two contexts of sweden and south africa. however, as researchers we both experienced writing about the student data in our projects as deeply challenging. this writing challenge relates to power and positionings of the researcher and participants in mathematics education in our particular research contexts. our different cultural, linguistic, and social experiences (suggested in the introductory quotes) have the potential for new or further stigmatizations or harm of certain labelled student groups. as researchers, our common research interests and writing challenges pointed to the potential for research writing collaboration. however, the references for our introductory quotes point to differences in power and positionings within our writing collaboration itself. our concern about difference in and across contexts is not just technical but personal and contextual. in our writing collaboration—in which we communicat 1 our naming of these different geopolitical contexts draws on janks (2010). andersson & le roux ethical research writing journal of urban mathematics education vol. 10, no. 1 77 ed via written feedback on our writing and discussions in video conference calls—we aimed for an attitude of carefulness both in our writing about the students’ experiences as represented in our data and in our writing with one another as researchers. the set of multi-level guiding questions for developing an ethical attitude in researcher–participant and researcher–researcher relations in contexts of difference that we present in this article emerged out of these efforts. we begin by presenting the theoretical concepts underpinning our guiding questions, with a specific focus on research practice. we next present the questions that developed during our writing collaboration. we describe more about our individual research projects, and then illustrate an ethical attitude in use in a dialogue between us (annica and kate) about writing for these projects. concepts from the socio-p/political turn the term sociopolitical turn in mathematics education has been used by valero (2004) and gutiérrez (2013) to describe the move beyond sociocultural theories to the use of explicit theories of power and positionings (also referred to as identities and subject positions). this move has recontextualized concepts from wider social theory, mainly from poststructuralism, critical theory, and critical race theory. indeed, walshaw (2011) has proposed these concepts as productive for understanding urban mathematics education. what or who is included in this move in mathematics education is not conclusive. here, we present those concepts that we find productive for the challenges of research writing in and across contexts of difference, drawing on the work of mathematics education researchers and other social theorists as appropriate.2 these concepts provide a framework for thinking about power, social relations, positionings, and ethical action in the micro-level activity of collaborative research writing. they also allow us to locate this writing activity in wider social systems and the power relations that sustain them, and to consider how our writing activity is both shaped by and shapes this macro-level context. power and positionings in the socio-p/political turn we view the macro-level context of mathematics education as a network of social practices as suggested by valero (2007). school mathematics, university mathematics, assessment, policy, mathematics teacher professional development, urban schooling, mathematics education research, and students’ homes and communities are examples of practices in this network (valero, 2007; walshaw, 2011), and we take the notion of research practice forward as an example in this section. according to fairclough (2003), a practice is characterised by a relatively stable, 2 we refer the reader to the referenced work for more detailed exposition of the concepts and their antecedents. andersson & le roux ethical research writing journal of urban mathematics education vol. 10, no. 1 78 recognizable combination of objects, activities, participants (e.g., researchers and research participants), social relations, values, time, space, and language use (e.g., written research papers and turn-taking in interviews). these elements of a practice are re-created in social, cultural, material, and discursive conditions (valero, 2007). mathematics education practices are political as they (re)produce relations of power (fairclough, 2001; valero, 2007). in this use of political, power is viewed as situational, relational, and in constant transformation, and not as an intrinsic, permanent possession of a person or practice (gutiérrez, 2013; valero, 2004, 2007). according to fairclough (2001), power is (re)produced at two levels of the social world and this distinction underpins our use of the terms political and political. we discuss the macro-level power relations here and introduce micro-level power relations later in this section.3 according to fairclough (2001), power relations between practices in a network maintain boundaries between these practices and control the movement of meaning across practices. these relations have implications for who has access to the conventions of the dominant practices. for example in mathematics education, research publications written in english may hold more value than those written in other languages (meaney, 2013). here we choose to capitalize the word political to signify power working in this way at the macro-level. the relation between the wider network of socio-political practices and what happens at the micro-level of the research interview or the research writing process is dialectical (de freitas & zolkower, 2009). on the one hand, the practice of research and the wider network of which it forms part is not just background to the research. rather these practices actually shape and give meaning to what the researcher and participant say in an interview, what the researcher writes in a research article, and how researchers collaborate on their writing. these practices offer subject positionings for both the researcher and the participant within the available discourses (davies, 1991; fairclough, 2003). or in the words of walshaw (2013): “an individual’s performance as a member of a social group occurs differentially in relation to his or her positioning within each social context” (p. 101). on the other hand, what a participant says in an interview and the researcher’s choices—the questions, the theoretical concepts, what and how to write and in what language, the knowledge produced—are not neutral but give meaning to the practice (d’ambrosio et al., 2013). participants act agentically (biesta, 2009; stinson, 3 the use of upper and lower case letters to distinguish between macroand micro-levels of the social such as presented here has been used by, for example, janks (2010) to distinguish between political and political and by gee (2005) to distinguish between discourse and discourse. we find two levels or scales—the microand macro-levels—appropriate for our purposes, but acknowledge the description by others in the context of mathematics education using multiple scales (e.g., herbel-eisenmann, wagner, johnson, suh, & figuera, 2015). andersson & le roux ethical research writing journal of urban mathematics education vol. 10, no. 1 79 2008), positioning themselves in ways that are reflexive, relational, and contextual (wagner & herbel-eisenmann, 2009; walshaw, 2013). thus, at the micro-level, discourse itself is “a place where relations of power are actually exercised and enacted” (fairclough, 2001, p. 36). a discursive event—such as a research interview or research writing event—is a site of both reproduction and resistance (gutiérrez, 2013), as participants seek to control the content of what is said/done, on the language form used, on the social relations that the participants enter into, and on the positionings available for participants (fairclough, 2001). we use political to refer to power working in researcher–participant and researcher–researcher relations in the writing process at the micro-level of discursive events. this use recognizes that power to act agentically may not be equally distributed between participants in a practice (fairclough, 2001). the concepts of power and positionings presented here point to the potential affordances and constraints of research writing. on the one hand, given that how power works to position participants in a network of socio-political practices is opaque to those participants (fairclough, 2001; walshaw, 2013), this writing has the potential to bring the workings of power in mathematics education into view. this writing, however, is unavoidably structured by dominant discourses (apple, 1995) and is subjective and contested (walshaw, 2011). this does not mean that all research writing is “equal, but it does imply an ethically responsible engagement” (walshaw, 2011, p. 9). in the case of this article, the research writing is in and across contexts of difference. thus, we turn next to the concept of ethics within the socio-p/political turn. ethical action in the socio-p/political turn consistent with others in mathematics education (e.g., atweh & brady, 2009; boylan, 2013; radford, 2008), we do not use the word ethical for a set of normative codes. rather we use the term as an adverb to describe caring and responsible attitudes and action in relations between researcher–participant and researcher– researcher in contexts of difference. others here are “totally other” than the self (atweh, 2013, p. 8, emphasis in the original) and not the same as “i.” acknowledging atweh’s (2013) concerns about the relations between poststructuralist concepts such as power and ethical decisions, we suggest, consistent with walshaw (2013), that the notions of power and positionings provide a lens to view ethical action, with the focus here on our ethical writing practice. first, we note that because a socio-p/political practice by definition includes particular ways of being for participants, relations between these participants, and values on what is “right,” a practice necessarily includes ethical attitudes, action and relations as described above. similar to the classroom described by radford (2008), we argue that research practice and the writing that gives it meaning is an “ethico-political space of the continuous renewing of being and knowing” (p. 229). andersson & le roux ethical research writing journal of urban mathematics education vol. 10, no. 1 80 we believe that mathematics education research, and here in particular the writing process, is a socio-p/political space where ethical and political considerations need to be carefully acknowledged. second, the notion of power reminds us that our research writing is not neutral, it always advocates a moral-political argument (adler & lerman, 2003). this perspective is not a reductionist view that any writing will do. maxwell (1992), working from a critical realist perspective, suggests that the validity of an account (such as our research writing) should be based on “the implications and consequences of adopting and acting on a particular account,” with not all accounts, “equally useful, credible, or legitimate” (pp. 282283). adler and lerman (2003) argue that it is the duty of the mathematics education researcher to engage continually in the struggle to get descriptions “right” and make them “count.” for them, this ongoing ethical action is about producing research that not just meets the quality and ethical requirements of the mathematics education research community (e.g., validity, rigour, confidentiality, anonymity) but that also represents the researched in a comprehensive, respectful manner that matters in the empirical context. guiding questions for ethical action in research writing in this section, we list multi-level questions that we propose as a guide toward an ethical attitude in the research writing process. these questions arose in the interaction between the concepts of the socio-p/political turn and our research writing collaboration. this collaboration was characterized by relations of cultural, language, social, and p/political difference in researcher–participant and researcher– researcher relations in and across contexts. guiding questions about researcher–participant relations how does the macro-level socio-political context shape the researcher’s political choices when writing about the narrated experiences of the participant? more specifically:  what positionings does the socio-political context offer for the participant in terms of what counts for mathematical participation?  how does the socio-political context position the researcher and participant relative to one another? how do the researcher’s micro-level political choices position the participant? more specifically: andersson & le roux ethical research writing journal of urban mathematics education vol. 10, no. 1 81  which positionings are (in)visible in the writing, and with what implications for the participant? might these positionings reproduce existing stereotypes and do harm?  how does the researcher work across different data sources and different timescales to give meaning to the complex ways in which the participant positions herself or himself?  how does the researcher attend to the circulation of power between the researcher and participant during the research process? guiding questions about researcher–researcher relations how does the socio-political context shape the collaborating researchers’ political choices when writing for publication? more specifically:  how does the socio-political context position the researchers relative to one another?  what positionings does the socio-political context offer for researchers in terms of what matters for participation in the wider mathematics education research community?  how do collaborating researchers communicate the socio-political context of their research to readers in different contexts to avoid essentializing and stigmatizing individuals in marginalized contexts and practices? how do micro-level political choices of collaborating researchers position each other and the collaboration? more specifically:  how do collaborating researchers’ political choices position the researchers within the research community, and with what implications?  how do collaborating researchers best respect their differences and similarities in a caring way, while asking uncomfortable questions that (re)position one another out of her or his comfort zones? before illustrating the use of these questions in the form of a dialogue, we contextualise the dialogue by presenting more about our individual research projects. the extracts presented in these descriptions represent particular moments in our attempts to address the challenges of writing about students in our individual research projects. andersson & le roux ethical research writing journal of urban mathematics education vol. 10, no. 1 82 different studies, different contexts annica’s research in the political north the narrative about ara in this article originates in annica’s research on students’ identities and relationships with mathematics education in a context of being exposed to critical mathematics in their first year in upper secondary school (andersson, 2011a, 2011b). the ethnographic data comes from interviews, classroom blogs, students’ and teachers’ logbooks, everyday conversations, and field notes. the study aimed to explore how students narrated identities—specifically those students’ who talked about themselves as “disliking,” “not feeling well,” or even “hating” mathematics—changed in relation with different contexts that played out in the classrooms. it was particularly problematic, however, to write about ara’s narratives and relationships with mathematics due to the political climate and risk for further stigmatization of a specific group of students. his narratives therefore became foregrounded during the collaboration with kate. ara is a student who talks about himself as being an immigrant, male, muslim, poor, pizzabaker, and war-experienced. hence the research writings may induce stigmatization because certain aspects of him as a whole individual become focused while others are not visible. ara could, from the researcher perspective, be labelled as struggling to succeed, fighting with his siblings to be able to prioritize schooling, but also trying to position himself in discourses he does not really grasp or have access to. when re-listening to the recorded interviews, it is evident how he tries to cope and to be polite, and he strives, at least initially, to talk about his experiences and himself in a way he might believe he wants the researcher to hear. he talks about wanting to be good in maths, wanting his parents to be proud of him and to be a good muslim. his actions, however, are also problematic to address, as he later in our relationship shares stories about stealing video-films, prior school experiences of cheating on maths tests to pass, and so forth. every wednesday when the local market outside the school is on and students buy their fruits, ara comes and sits beside annica and asks if she can share because he is hungry as “i didn’t have time to eat breakfast because i worked late again, to midnight” [jag har inte hunnit äta frukost idag eftersom jag arbetade till midnatt igen]. all these different present identities and actions are his, as told at those particular points in time. annica – researcher, teacher, female, multilingual swedish-speaking, middleclass, white, swede kate’s research in the political south4 the narrative about luthando used in this article was produced in a longitudinal study of students’ transition to and through undergraduate studies at an elite, former “white,” urban university in south africa (kapp et al., 2014; le roux, 4 this project is financially supported by the andrew w. mellon foundation. andersson & le roux ethical research writing journal of urban mathematics education vol. 10, no. 1 83 2017).5 the study aimed to understand the ways in which the transition of “black” students—from mainly first generation, working-class, single parent families and for whom english is generally not a first language—may be enabled or constrained by the socio-political context in south africa, as well as students’ agentic action with the positionings in this context. the analysis of interview texts conducted annually during a student’s undergraduate degree was informed theoretically by the socio-p/political turn and performed using tools from critical discourse analysis. luthando’s application to study engineering at an elite, urban, english medium, former white university in south africa was not successful, but he was accepted to study his second choice of science. on account of his home and schooling background he was placed in an extended academic programme designed to provide him with the foundations for studying university science. initially, luthando resented his not being given a choice with regard to his positioning in this programme and he talked about his “struggles” in university mathematics. however, within a few months he said he was “doing good” and said the additional support was an “advantage.” finding money to travel home for the vacation was difficult, but on campus he could use his financial aid to buy food and books. after completing 2 years in the extended programme, luthando enrolled for advanced mathematics which he described as “just definitions, it’s proofs.” he talked about being “totally lost” while “really smart students, students who really, really love maths” interacted with the lecturer. he gradually lost his “love” of mathematics, stopped attending classes, and failed one of his final courses. he felt the extended programme had been a “disadvantage” for his progress in mathematics. luthando said that he “did not come from a very privileged background and my mom had to make do every day,” but he described other university students as “really, really disadvantaged.” nonetheless, the finances of his family and for his studies recurred in his interviews. in his fifth year at the university, luthando was permitted to enroll in an engineering degree on condition he passed certain mathematics courses. he secured a bursary to finance his studies, he felt motivated to pass mathematics, and he spoke about providing his family and himself with a “good life” when qualified as an engineer. kate – researcher, teacher, female, monolingual english-speaking, middle class, white, south african talking about our writing, participants, and collaboration in this dialogue between the two of us (annica and kate), we illustrate how we use the questions informed by the concepts from the socio-p/political turn to guide what we refer to as a caring, ethical attitude toward the researcher–participant and researcher–researcher relations during the research writing process. due to the nature of a dialogue, this exchange does not follow the order of the guiding questions as listed, so we explicitly use the language of the questions to foreground their use. we also bring other mathematics education researchers into our conversation 5 racial classifications such as “white” and “black” (for “black african,” “coloured” and “indian”) are still used in discourses about educational performance in south africa, despite a growing recognition of how the construct of race works with class, language, and geography in constituting performance. andersson & le roux ethical research writing journal of urban mathematics education vol. 10, no. 1 84 as necessary. we acknowledge that the research writing process is not stand-alone but is informed by all the other choices made in the research process. we point to this as necessary. annica: kate, i am interested in your political choice in your writing about participation in university mathematics in south africa, to write about luthando’s talk about money and food. what are you expecting the audience—for example a reader such as myself, positioned in the political north—to know about the macrolevel socio-political context that shapes your research? kate: annica, your question challenges me to reflect on what positionings in south africa shape my writing choices. it also alerts me to my positioning relative to others in the mathematics education research community and how i make my choices explicit to the international community. my description of luthando recognizes the material, discursive, and psychological load on students’ lived experiences of positionings such as “working class,” “black african,” and “english second language.” in south africa, positionings such as these have been shown to matter with respect to accessing mathematical practices and the related symbolic and material rewards (soudien, 2012; spaull, 2013). so, my choice to write about these positionings is not idiosyncratic but identifies the participants in ways that matter in mathematics education in south africa. annica: in the swedish context, ara’s positioning as a kurdish first language speaker may matter, and his related positioning as “blatte” 6 [immigrant] in turn positions him as marginalized relative to mathematical practices (marks, 2005; svensson, meaney, & norén, 2014). these power relations between language discourses are also present in swedish mathematics teacher education (skog & andersson, 2014). my political choice to position myself as a swedish-speaking citizen of sweden in my writing about ara positions me as having different experiences of mathematics education to the research participants in my socio-political context. while my self-identification is necessarily selective (bartell & johnson, 2013), this positioning signals that i have not experienced ara’s load in ways that matter when learning mathematics in sweden. saying who we are in our writing (as we do in naming ourselves as researchers in the introduction to this article) is not just about being transparent but part of accounting for our own positionings in the research process as suggested by chronaki (2004) and valero (2004). our selfidentification serves as a constant reminder of the asymmetries in the political choices of researcher and participant in our respective socio-political contexts. 6 can be interpreted as derogatory, “a dark foreigner” (svenska akademin, 1998). andersson & le roux ethical research writing journal of urban mathematics education vol. 10, no. 1 85 kate: yet, the political choices about our writing (and indeed, other aspects of our research) that we discuss here position us as researchers in particular ways in the mathematics education research community, and we should be alert to the implications. first, by “zooming out” (lerman, 1998, p. 67) to the wider sociopolitical contexts in which our research is located, we run the risk that our research will not be considered mathematics education research (adler & lerman, 2003; bartell & johnson, 2013; martin, gholson, & leonard, 2010). this risk may result in us being positioned as writers who have to justify—within space constraints— why context matters. it may mean that we encounter significant challenges in the review process (parks & schmeichel, 2013). it may position us as researchers who contribute only to journal special issues and particular conference strands (bartell & johnson, 2013; bullock, 2014; stinson, 2010). secondly, our political choice to self-identify by occupation, gender, race, class, language and geographical region in our writing resists the dominant referencing style of author surname and date. our agency in this respect has implications for where we publish and, again, our positionings in the mathematics education research community. annica: we also need to be alert to how our different socio-political contexts position us—in asymmetrical ways—in this research community, and what these positionings mean for our collaboration and our participation in the community. for example, your positioning as a south african researcher positions you within the asymmetrical political north/south power relations within the publishing space of the mathematics education research community (adler & lerman, 2003; ernest, 2016). yet, my positioning as a swedish-speaking researcher and our political choice to write for publication in english, signals asymmetries in the power relations in our research collaboration in the english research writing space. i am positioned as less knowledgeable than you with respect to writing in english in this space (meaney, 2013). these asymmetrical power relations have required careful political choices in our writing collaboration. for example, in writing articles such as this, i have required a greater proportion of the word length to convey my meanings and to include both ara’s swedish transcripts and the translations thereof. doing so has meant that we have had to negotiate the precious and tight writing space between us. on the other hand, our conversations have encouraged me to write more boldly in the english writing space. kate: for me, as someone who is positioned differently relative to you in terms of linguistic resources for publishing in english, an ethical attitude in our collaboration is not just about what questions we pose to one another. crucially, it is about making careful discursive choices about how i ask these questions, particularly in my written feedback to you. how do i word in-text comments and ques andersson & le roux ethical research writing journal of urban mathematics education vol. 10, no. 1 86 tions, when there is a possibility for misunderstanding that may undermine our ethic of care? how do i contribute to an article such as this in a manner that allows both our voices to come to the fore? certainly, the visual cues afforded when communicating via video-conferencing has enabled us to address these challenges of our collaboration, yet even this affordance has meant us being alert to inequities in access to online communication tools between the political north and south. annica: kate, i want to return now to your writing about luthando, and explore what positionings the socio-political context of south africa offers for this mathematics student. the word “disadvantage” is visible in your writing about luthando’s interview text. whose word is this? maybe luthando is “advantaged” in ways that are not valued by the dominant socio-political practices? gutiérrez (2013) challenges us to consider whether the positionings we use in our research are consistent with those the students themselves would choose. kate: my choice to use quotes for the word “disadvantage” in my writing signals that this is the word used by luthando in the interview to describe his positioning in the socio-political context of south africa. the term “disadvantage” is commonly used in south africa to acknowledge the material effects on educational performance of certain students’ past and current positionings. however, its use is critiqued as stigmatizing certain students as in deficit. my choice to reproduce luthando’s talk of his relative “disadvantage” and to quote this rather than paraphrase signals how his own political choices in the research interview are shaped by this wider socio-political context. thus, responding to gutiérrez’s (2013) challenge is not just about asking for luthando’s perspective on my writing to confirm “interpretive validity” (maxwell, 1992). indeed, the construct of “disadvantage” has traditionally been used by the university to select and place students in what are called “regular,” “mainstream,” or “support” academic development programmes. yet institutional statistics and qualitative research (e.g., kessi, 2013; le roux & adler, 2016) show that positioning as an extended programme student may in turn define an individual’s opportunities to be a university mathematics student, and also how she or he sees herself or himself more generally. my use of scare quotes in this dialogue to describe the positionings of south african university students shows how my writing is “heavily laden with the meanings of others” (walshaw, 2013, p. 116). while these positionings might matter in my context, i need to be alert to how my political writing choices may become reified and reproduce existing stereotypes. annica: yes, my choice of quotes from ara’s talk indicates that positionings as an immigrant, kurdish speaking, young male pizza-baker are part of ara’s talk andersson & le roux ethical research writing journal of urban mathematics education vol. 10, no. 1 87 as a student who “kämpar för att få ihop det” [struggles to get it all together]. these positionings as such do not qualify him for what is called “special education” in sweden, but he indicates that he might fail just because of these positionings. at the micro-level, luthando and ara’s choices are political as they act agentically within the set of power relations to position themselves in various practices. these actions may involve reproducing, redefining, or rejecting the positionings they identify in their respective contexts. ara comes to school hungry and tired, as he does not get enough hours in between work end and school start. he knows he has not done his homework. he says it is hard for him to concentrate. however, he also shares the school culture knowledge capital (bourdieu & passeron, 1990) that “för att inte misslyckas igen så måste jag göra alltihop och sitta där framme med dom som kan istället för där bak” [to not fail again i have to do it all and also sit in front with the clever students instead of in the back]. yet, similar to you, i find the ethical responsibility of not writing about certain positionings analytically in ways that become reified and have the potential to contribute to further stigmatization and harm so hard to practice in my writing. it is so difficult for me to write about ara’s home background (e.g., “eight siblings” and in particular the impact on him from older brothers). he is and takes the actions he does because he is who he is and hence marginalized in both open and subtle ways in our society and in school by teachers and peers. ara cannot oppose his brother’s work demands, as he is the younger one. he talks about “tänk att bara få rymma och liksom bestämma allt själv” [wanting to escape and take all decisions myself], but he cannot do that for several reasons. he needs to help out with the support of his family and younger siblings. this helping suggests a recognition of the load that is both visible and invisible. for me, he is one of those students so easy to label and hence stigmatize because certain aspects of a whole individual are focused while others are not visible in our research texts—as valero (2004) reminds us. yet, the notion of positionings within the socio-p/political turn helps me to move forward in these writing challenges. as suggested by walshaw (2013), doing so means that we are not attempting to write on behalf of the research participants but aim to understand the complex ways in which they work with the available positionings in their contexts. in addition, our use of critical theoretical concepts positions us as researchers who bring into view how power and positionings work to include/exclude students from participation in mathematics. this view tends to be opaque to participants in mathematics education (fairclough, 2001). kate: yes, my positioning as an english-speaking, middle class, white, south african suggests that, as a researcher, i cannot draw on my “lived experiences” (gutiérrez, 2013, p. 57) as marginalized in my context. yet, valero (2014) reminds us that as researchers we can bring theory to this task of understanding the complex positioning work of the researched. indeed, my “bearing witness” and “orienting” andersson & le roux ethical research writing journal of urban mathematics education vol. 10, no. 1 88 experiences (foote & bartell, 2011, p. 52) in south africa suggest that making this work visible is my only choice. annica, you have mentioned the challenge of different positionings being (in)visible in your writing about ara, and the need to consider the implications thereof for the participant. because my interview study was longitudinal, i face the challenge of writing about how luthando, over a period of 5 years, variously represents the socio-political context and positions himself therein. for example, his description of the extended programme varies as an “advantage”/“disadvantage” over time, as does his positioning of himself as “disadvantaged” in the sociopolitical context. it is a challenge to write—in the linear manner that counts for publication by the community—respectfully and comprehensively about luthando’s complex positioning work across timescales. annica: how to attend to the circulation of power between ara and myself during the research process is a consideration that i have been dealing with. the ways we position each other through the research is not static, and here i give a couple of examples to illustrate different positionings in our relationship. as suggested by wagner and herbel-eisenmann (2009), the positionings were reflexive and developed through our conversations. i give three examples. first, ara positions me as somebody he can trust and, for example, can share his rougher family and other stories with. however, he also positions me as an authority that knows what to do, especially in mathematics education but also where to ask for societal health and youth support. a last example may be the way he positions me as “different” to him as the you in the quote; “we don’t talk about it with you.” here, i become positioned as one of the swedes as opposed to the immigrants. ara accepted positionings as a student who possesses knowledge about mathematics education that is important to share with others. he also positioned himself as a student about to fail in mathematics, as an immigrant and as a student from a low socio-economic background (just to give some examples). i accepted and also mirrored these different positionings proposed by ara, as he mirrored my positionings. as van langenhove and harré (1999) point out: positionings are the ways in which people use action and speech to arrange social structures. the examples i have given of the different positionings in my conversations with ara illustrate well the reflexive, relational, and contextual nature of the power relations between researcher and participant. yet, i emphasize that the nature of these power relations shifts again when the researcher comes to write about the interviews. through recognizing my power as a researcher in this context, i strive to make further careful choices that aim at building a rich and caring description of the participant. first, in my wider study i asked participants to select their own pseudonyms. however, ara chose not to. thus, i turned to a kurdish language teacher at ara’s school who suggested and explained different kurdish male andersson & le roux ethical research writing journal of urban mathematics education vol. 10, no. 1 89 names. i chose “ara,” meaning wind, which represented the elusiveness of my research relationship with ara at that particular moment in time. second, following meaney (2013), i carefully represent ara’s talk both in swedish and in english. this move positions ara and me in the context; ara speaks kurdish as his first language, i interviewed ara in swedish and transcribed in swedish (ara’s second language), i then translated the swedish transcripts into english for publication. as we have already discussed, the fact that english is not my first language means that this final translation itself positions me in another set of power relations with you as research collaborator and within the research community. kate, i want to end with one question about your writing about luthando. i wondered about the “happy ending” to your story about luthando. i think we, as researchers tend to tell the good news stories. why? is there something in your context that shapes the story you tell? kate: annica, your short summary, as a reader, of my writing about luthando is an eye-opener to me as a writer. it alerts me to what positionings are (in)visible in my writing. it also talks to my struggle of communicating the complexity of my socio-political context to readers in other contexts. how do i signal, for example, what words such as “dis/advantage” and “success/failure” mean materially, socially, and psychologically over time for students in this context? certainly, luthando could be positioned as “successful” by the institutional statistics, in that he was one of the few students with his background who enrolled for a third-year undergraduate mathematics course. indeed, some researchers have argued that it is important to write about the experiences of academically “successful” students like luthando (e.g., berry, 2008). in addition, methodologically, it is these students who participated in the longitudinal study for the longest time, and our attempts to keep contact with the students who left the study for various reasons had limited success. however, the concepts of the socio-p/political turn mean that knowing in mathematics cannot be separated from who one is and how one relates to others (radford, 2008; valero, 2014) and that the experiences of students positioned as “disadvantaged” and “successful” cannot be placed in “narrow boxes” (erwin, 2012, p. 97). the longitudinal study provides the opportunity for me to write about how luthando positions himself relative to what his performance in mathematics assessments and his relations with others over time say about him. for example, he positions himself variously as a school mathematics student who is known “by my marks,” as a student who “struggles” in first-year university mathematics, as a second-year student who is “doing good” relative to others who share his background, as a student who is “totally lost” and excluded from the “nice conversations” in his mathematics classroom in his third year, and as a student who has to “separate my personal life from my studies” so that he can fulfil his family’s expectations of him. andersson & le roux ethical research writing journal of urban mathematics education vol. 10, no. 1 90 descriptions such as these show the complex agentic positioning work enacted by students like luthando as he bumps up against institutional structures. not getting the description “right” (adler & lerman, 2003) in the sense that luthando’s experience of learning mathematics at the university seems like a “good news” story as you suggest, means that these structural constraints which we as researchers seek to make visible, remain opaque. concluding thoughts the dialogue presented in this article represents a particular form of research collaboration between researchers interested in inclusion/exclusion in mathematics education. we also share a concern about how to write about research participants whose cultural, linguistic, social, geographical, political, and so on experiences may differ from our own. yet our locations in the political north and political south respectively, mean that we ourselves bring different experiences to this writing collaboration. we suggest that our conversations about our individual research writing as well as the collaborative writing to which this led opened the space for our “looking closely at [our] own work” (bartell & johnson, 2013, p. 42) and our asking “uncomfortable” (p. 41) or troubling questions about the microand macrocontexts of our research production. walshaw (2013) argues that adopting a poststructuralist perspective allows the researcher to “begin to ask questions we have not previously thought to ask” (p. 116) and provides a language to talk about “ethical practical action” (p. 101). we argue that our writing collaboration—informed by theoretical concepts from the socio-p/political turn—allows us to adopt an ethical attitude in researcher– participant and researcher–researcher relations in the research writing process. this attitude involves asking multi-level guiding questions of oneself and the research collaborator:  how does the macro-level socio-political context shape the researcher’s political choices when writing about the narrated experiences of the participant?  how do the researcher’s micro-level political choices position the participant?  how does the socio-political context shape the collaborating researchers’ political choices when writing for publication?  how do micro-level political choices of collaborating researchers position each other and the collaboration? these four questions (and their sub-questions) prompt us to think about power, social relations, positionings, and ethical action in the micro-level activity of andersson & le roux ethical research writing journal of urban mathematics education vol. 10, no. 1 91 collaborative research writing. they also allow us to locate this writing activity in wider networks of social practices and the power relations that sustain them, and to consider how our writing activity is both shaped by and shapes this macro-level context. although these questions derived out of discussions about our research writing challenges, we note that they also apply to other aspects of the research process. we suggest that researchers should reflect on these questions in a reflexive state of mind. we suggest that it is questions such as these which allow us in an ethical and caring way to bring into view the need to do the “risky work” (parks & schmeichel, 2013, p. 248) of zooming out beyond the mathematics and writing about issues of race, class, religion, and immigration in mathematics education but also to support one another in this work. in addition, the use of these guiding questions in our writing collaboration has meant “unpacking what seems ‘natural’” in our writing, and “reflecting on what we are today, how we have come to be this way, and the consequences of our actions” (walshaw, 2013, p. 116) and hence to think differently about our political choices when writing with and about others in and across contexts of difference. references adler, j., & lerman, s. 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(1992). understanding and validity in qualitative research. harvard educational review, 62(3), 279–300. meaney, t. (2004). the fly on the edge of the porridge bowl: outsider research in mathematics education. in p. valero & r. zevenbergen (eds.), researching the socio-political dimensions of mathematics education: issues of power in theory and methodology (pp. 167–183). boston, ma: kluwer. http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/206/133 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/212/131 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/95/57 andersson & le roux ethical research writing journal of urban mathematics education vol. 10, no. 1 94 meaney, t. (2013). the privileging of english in mathematics education research, just a necessary evil? in m. berger, k. brodie, v. frith, & k. le roux (eds.), proceedings of the seventh international mathematics education and society conference (vol. 1, pp. 65–84). cape town, south africa: mes7. parks, a. n., & schmeichel, m. (2013). obstacles to addressing race and ethnicity in the mathematics education literature. journal for research in mathematics education, 43(3), 238–252. radford, l. (2008). the ethics of being and knowing: towards a cultural theory of learning. in l. radford, g. schubring, & f. seeger (eds.), semiotics in mathematics education: epistemology, history, classroom, and culture (pp. 215–234). rotterdam, the netherlands: sense. skog, k., & andersson, a. (2014). exploring discursive positioning as an analytical tool for understanding becoming mathematics teachers’ identities. mathematics education research journal, 27(1), 65–82. soudien, c. 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(1998). svenska akademiens ordlista över svenska språket. (saol) (12. uppl.). stockholm: norstedts ordbok svensson, p., meaney, t., & norén, e. (2014). immigrant students’ perceptions of their possibilities to learn mathematics: the case of homework. for the learning of mathematics, 34(3), 32–37. valero, p. (2004). socio-political perspectives on mathematics education. in p. valero & r. zevenbergen (eds.), researching the socio-political dimensions of mathematics education: issues of power in theory and methodology (pp. 5–23). boston, ma: kluwer. valero, p. (2007). a socio-political look at equity in the school organization of mathematics education. zentralblatt für didaktik der mathematik (zdm), 39(3), 225–233. valero, p. (2014). cutting the calculations of social change with school mathematics. in p. liljedahl, c. nicol, s. oesterle, & d. allan (eds.), proceedings of the joint meeting of pme 38 and pme-na 36 (vol. 1, pp. 73–77). vancouver, canada: pme. van langenhove, l. & harré, r. (1999). introducing positioning theory. in r. harré & l. van langenhove (eds.), positioning theory: moral contexts of intentional action (pp. 14–31). oxford, united kingdom: blackwell. wagner, d. (2012). opening mathematics texts: resisting the seduction. educational studies in mathematics, 80(1&2), 153–169. wagner, d., & herbel-eisenmann, b. (2009). re-mythologizing mathematics through attention to classroom positioning. educational studies in mathematics, 72(1), 1–15. walshaw, m. (2011). positive possibilities of rethinking (urban) mathematics education within a postmodern frame. journal of urban mathematics education, 4(2), 7–14. retrieved from http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/141/89 walshaw, m. (2013). post-structuralism and ethical practical action: issues of identity and power. journal for research in mathematics education, 44(1), 100–118. http://www.section27.org.za/wp-content/uploads/2013/10/spaull-2013-cde-report-south-africas-education-crisis.pdf http://www.section27.org.za/wp-content/uploads/2013/10/spaull-2013-cde-report-south-africas-education-crisis.pdf http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/91/47 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/141/89 microsoft word 7 final myers vol 9 no 1.doc journal of urban mathematics education july 2016, vol. 9, no. 1, pp. 117–123 ©jume. http://education.gsu.edu/jume kayla d. myers is a phd student in the department of early childhood and elementary education in the college of education and human development, at georgia state university, p.o. box 3978, atlanta, ga 30302; email: kmyers@gsu.edu. her research interests include mathematics teacher education and development; teachers’ transitions from the university setting to the mathematics classroom; and issues of power, identity, and subjectivity through discursive practices in mathematics learning communities. book review keeping the “welcome sign” lit: a review of building mathematics learning communities: improving outcomes in urban high schools1 kayla d. myers georgia state university he unfortunate and dangerous assumption that “urban” high school mathematics students have a “deficit” continues to be held by too many educators, school leaders, researchers, policy makers, and others. such a deficit mindset ignores urban youth’s potential for mathematics engagement and achievement. within mathematics education research, this deficit-based agenda holds a heavy, harmful focus on “achievement gap” research between middle-class, white students and black, latino/a, first nation, english language learner, urban students (gutiérrez, 2008; parks, 2009). there is a dire need for a shift in attitude about students, teaching, and learning mathematics in urban settings—from a deficit model to one of capability. erica walker’s (2012) book building mathematics learning communities: improving outcomes in urban high schools is an example of a text that honors students’ mathematical capabilities. in particular, the book is a study of lowell high school (pseudonym), an urban high school in new york city consisting of mostly black (41%) and latino/a (56%) students. the purpose of her research was to explore students’ attitudes, participation rates, and performance outcomes in mathematics; another purpose was to understand the impact of mathematics learning communities as well as the networks of peers, teachers, families, and others on mathematics achievement in this urban high school setting. building mathematics learning communities expands on walker’s findings from her 4-year research project, which used mixed methods including statistics on the school’s demographics (ethnic, racial, and financial) and various test scores; teacher and student surveys on beliefs about student potential in mathematics, as well as their perceptions on peer influence; field notes and observations at lowell high school throughout the project; and interviews of students who were chosen using teacher nomination and 1 walker, e. n. (2012). building mathematics learning communities: improving outcomes in urban high schools. new york, ny: teachers college press. 168 pp., $34.95 (paper), isbn 0-80775328-9 http://store.tcpress.com/0807753289.shtml t myers book review journal of urban mathematics education vol. 9, no. 1 118 snowball sampling to find high-achieving mathematics students to discuss their successes and key influences of their mathematics achievement. using these data, walker designed and implemented a peer tutoring collaborative at lowell high school. she analyzed the program to assess the impact it had on teachers’ pedagogical beliefs as well as students’ growth in mathematical knowledge and learning. this peer tutoring program provided a space for students to collaborate with each other and with teachers, to learn from each other, and to grow in their mathematical thinking—a program that lights walker’s mathematical “welcome sign” for urban high school students. my review of walker’s (2012) book is situated within three professional roles that inform my subjectivity. i am a former urban mathematics elementary school teacher, a doctoral student at an urban research university, and an emerging elementary mathematics teacher educator and researcher. furthermore, i am interested in aspects of community in the mathematics classroom for young learners and the discursive practices that shape mathematics identities for students and teachers. my ideas about mathematics learning communities have been broadened in reading walker’s book, which maintains a focus on the high school setting but is easily accessible across primary and secondary education. the elementary teacher’s support and facilitation of young mathematical minds through rich, engaging tasks is just the beginning, just one piece, of a mathematics learning community. the community extends to classmates, friends, family members, and any other peers or adults that a child may learn with or from. walker writes with conviction and dedication to ensure that children in urban high schools have the means for mathematics success, are met with mathematical challenge, and engage in strong mathematical learning communities. building mathematics learning communities – an overview building mathematics learning communities is organized into six chapters, each addressing various elements of urban mathematics education in the high school setting, paying particularly close attention to aspects of mathematics learning communities as identified by the teachers and students at lowell high school. just as milner (2012) discusses ideas and misconceptions of the term “urban education,” walker challenges and seemingly embraces the term urban, discussing the rhetorical baggage that comes with it while also seeing the positivity and possibilities for urban students. similarly, lubienski (2006) addresses ideas and misconceptions of the term “equity” in reform mathematics education, and gutiérrez (2006) emphasizes the need to (re)define “equity” in mathematics teaching and learning, as both seem to focus too heavily on equal as being equitable and the achievement gap as being worthy of our attention. in a similar manner, walker (2012) redirects attention away from achievement gaps and toward equitable mathematics education op myers book review journal of urban mathematics education vol. 9, no. 1 119 portunities for all students. ultimately, walker highlights mathematical successes by bringing these ideas (i.e., urban and equity) together—challenging the urban in mathematics education with equitable and community-based teaching and learning of critical mathematics. chapter 1 “urban high school students and mathematics: myths and realities” provides a detailed outline of the popular beliefs about students in urban high schools, including beliefs that teachers often hold about teaching their students mathematics. fictions and actualities about students’ attitudes toward mathematics and their participation are addressed. in addition, the damaging teacher dispositions toward urban high school students associated with race and poverty leave too many in the dark, unguided due to a presumed disinterest and deficit. in walker’s (2012) documented reality, however, urban high school students want to be challenged, want to learn mathematics, and treasure those teachers that provide such a welcomed endeavor. the next chapter examines the community constructs in place for students in urban high school settings to engage in mathematics. “understanding students’ communities and how they support mathematics engagement and learning,” chapter 2, highlights family involvement, peer influence, and peer tutoring groups at lowell high school. walker (2012) argues that both low-achieving and highachieving students report benefits of positive academic behaviors, both in and out of school, which implies the potential impact of family members, peers, and others in the non-academic community. this chapter aims to “demonstrate that it is very important for students of color to experience mathematics success in the context of working together as a group” (p. 50), emphasizing the important role of peer influence on students’ mathematics success. chapter 3 “facilitating and thwarting mathematics success for urban students” explores the policies and practices of schools, their resources, and classroom dynamics. this exploration happens through analysis of student learning, achievement, student and teacher interactions, and mathematics teaching. as done in other chapters, walker (2012) uses the student interviews to provide an intriguing lens. students discuss their perceptions of teachers at lowell and their teachers’ practices and treatment of students as learners of mathematics. this unique and important perspective, from the mouths of students, so to speak, not only gives credibility to the words but also brings genuineness to the study. walker acknowledges the unfortunate thwarting of mathematics success for marginalized students, but she chooses to focus on the facilitation of mathematics success, highlighting the practices and perceptions at lowell high school as an example. “engaging urban students’ mathematical interests to promote learning and achievement,” the fourth chapter, examines non-engaging and engaging mathematics experiences of the students of lowell high school through their own voices. these actual accounts shine a light on the realities of school mathematics class myers book review journal of urban mathematics education vol. 9, no. 1 120 rooms, as well as out-of-school mathematics learning experiences. walker (2012) presents a framework to bridge these in-school and out-of-school contexts for mathematics student engagement. this framework includes four components she advocates should be included in any program aimed to promote student engagement in mathematics: “attention to rigor, attention to and validation of students’ everyday experiences and interests, focus on community, and out-of-school/in-school mathematics/experience connections—content and socialization” (pp. 86–87). chapter 5 “developing a peer tutoring collaborative” details the development of the peer tutoring program created and instituted at lowell high school. included are the components of the program, general impressions from teachers and students, features of the interactions between tutors and tutees, and how this collaborative program challenged traditional models of mathematics. the effectiveness of such a program, both in its growth and continued success, serves as an exemplar for urban high schools. walker (2012) considers the effectiveness of the peer tutoring collaborative program by analyzing students’ and teachers’ testimonials that illuminate the benefits and successes of the program, as well as comparing initial reactions to the results of such a peer tutoring collaborative. again, walker lit her welcome sign by initiating the peer tutoring program, but it was the students who kept it glowing through dedication to each other and to the mathematics. chapter 6, “conclusions,” leaves us with important questions to consider as urban educators: what opportunities are present in urban schools to build students’ interest and excitement about math? what opportunities are present in urban schools to facilitate communities of learning around math? and how often are these opportunities ignored? what implications do these opportunities, and the ignoring of them, have for students’ success? (walker, 2012, p. 112) walker posits implications for urban teachers, teacher educators, administrators, and policy makers. after considering both phases of her case study, the early data collection and the implementation of the peer tutoring collaborative and its effects, walker concludes with a question to push toward a reconceptualization of mathematics teaching and learning: what is student achievement in mathematics? connections there are far too many damaging myths in existence about the ways mathematics is taught and learned, ranging from misconceptions about the abilities of young mathematics learners (carpenter, fennema, franke, levi, & empson, 1999), to illusions of simple and seamless identity construction for prospective elementary mathematics teachers moving from the university to the classroom (walshaw, 2004), to fictions about who can learn mathematics (stinson, 2013). the goal is not myers book review journal of urban mathematics education vol. 9, no. 1 121 to prove these ideas false—after all, they are fictions—but rather, we must challenge these assumptions and (re)consider “how and where mathematics teaching and learning occur, what denotes talent and interest in mathematics, and who can be excellent in mathematics” (walker, 2012, p. 119). by creating mathematics learning communities in urban environments, we can keep the welcome sign of mathematics well lit for all students. my own teaching experiences in elementary mathematics learning communities feel relevant and connected to walker’s (2012) high school mathematics learning communities, specifically the peer tutoring collaborative program. for example, creating a safe space for elementary students to ask questions, much like walker’s learning communities, cultivates an open and empowering environment for maturing mathematical thinkers. providing that space in elementary school to engage with the mathematics in collaborative and supportive ways can set a strong foundation for young learners as they develop a mathematics identity. i believe that all students can learn mathematics, and a way to promote that engagement is with positive peer, family, and teacher interactions. walker advocates the same message throughout her book. in an effective mathematics learning community, students engage in the foundational knowledge and skills for thinking mathematically, practice problem solving and mathematical reasoning, and develop conceptual understandings of the mathematics. this community is made manifest in an environment that supports all learners as having potential for success, with peer interactions that inspire mathematical excellence as they work and learn together. as an emerging teacher educator and novice researcher at an urban research university, i understand the need to prepare mathematics teachers to dismiss negative dispositions toward urban students and adopt the empowering attitudes illustrated in building mathematics learning communities. i reiterate that my work in preparing elementary teachers to teach mathematics is undeniably connected to secondary mathematics preparation. while mathematics content knowledge is important, pedagogical content knowledge and teaching mathematics effectively require so much more (hill, 2010). all teachers need to be prepared to challenge all students, to set high expectations for student achievement and success, and to focus on positive attitudes and beliefs about mathematics learning, especially teachers in urban settings. therefore, i would recommend this book to those looking for ways to trouble the achievement gap in urban schools—whether elementary or secondary—and cultivate a mathematics learning community for all students. conclusions all students want to engage in challenging mathematics. throughout building mathematics learning communities, there are numerous examples of innovative and effective mathematics teaching practices, and the model of the peer tutoring myers book review journal of urban mathematics education vol. 9, no. 1 122 program is relevant for urban (and non-urban) mathematics educators and administrators. walker’s (2012) insightful ideas are an attempt to free teachers and students from the negativity surrounding achievement gap rhetoric, instead striving for building mathematics learning environments that support the potential for all learners. these communities, similar to the one built at lowell high school, promote collaboration, eradicate hierarchies, and encourage budding mathematicians of all sorts. as an advocate for such a collaborative, i was left feeling a joyous respect after reading such raw and truthful endorsements for a peer tutoring program that relies on student volunteers. walker (2012) calls for a reconceptualization of mathematics teaching and learning, specifically for urban high schools. this involves a reconsideration of urban mathematics students, which walker initiates by emphasizing the strong student interest in arduous mathematics at lowell high school. furthermore, contrary to extant damaging ideas of urban school communities, walker’s urban students “come from communities and networks that are committed to their education and mathematics development… [which] have gone unnoticed and unacknowledged” (p. 112). equitable mathematics education means focusing on students’ strengths and their possibilities and potential for success rather than on perceived weaknesses and deficits to which the achievement gap pays so much attention. what happens when we leave the gap-gazing behind us (gutiérrez, 2008; parks, 2009)? in building mathematics learning communities, walker (2012) challenges us to abandon the gap-gazing fetish and summons teacher practitioners to infiltrate urban schools and positively influence mathematics teaching practices. as a novice researcher, i wonder about the potential impact of walker’s research on practice and scholarship. yes, teachers and researchers should abandon the achievement gap negativities and adopt philosophies of all students’ potential for mathematics success, but how do we silence those negativities and bring voice to those who abandon it? overall, walker highlights the effect of this abandonment within one school, and i agree that we should employ such an ideological shift in all teachers. with regard to urban settings, where are these gap-gazing fetishes originating, and how do we move beyond them? how can we, both teachers and researchers, illuminate the mathematics welcome sign for all students when too many urban students often find that light burned out? walker’s book holds the potential to be a beacon in this search for such a light by refusing to gaze at the achievement gap and instead working to build up urban mathematics teaching and learning. references carpenter, t. p., fennema, e., franke, m. l., levi, l., & empson, s. b. (1999). children’s mathematics: cognitively guided instruction. portsmouth, nh: heinemann. myers book review journal of urban mathematics education vol. 9, no. 1 123 gutiérrez, r. (2006). (re)defining equity: the importance of a critical perspective. in n. s. nasir & p. cobb (eds.), improving access to mathematics: diversity and equity in the classroom (pp. 37–50). new york, ny: teachers college press. gutiérrez, r. (2008). a “gap-gazing” fetish in mathematics education? problematizing research on the achievement gap. journal for research in mathematics education, 39(4), 357–364. hill, h. c. (2010). the nature and predictors of elementary teachers’ mathematical knowledge for teaching. journal for research in mathematics education, 41(5), 513–545. lubienski, s. t. (2006). research, reform, and equity in u.s. mathematics education. in n. s. nasir & p. cobb (eds.), improving access to mathematics: diversity and equity in the classroom (pp. 10–23). new york, ny: teachers college press. milner, h. r. (2012). but what is urban education? urban education, 47(3), 556–561. parks, a. n. (2009). doomsday device: rethinking the deployment of the ‘achievement gap’ in equity arguments. for the learning of mathematics, 29(1), 14–19. stinson, d. (2013). negotiating the “white male math myth”: african american male students and success in school mathematics. journal for research in mathematics education, 44(1), 69–99. walker, e. n. (2012). building mathematics learning communities: improving outcomes in urban high schools. new york, ny: teachers college press. walshaw, m. (2004). pre-service mathematics teaching in the context of schools: an exploration into the constitution of identity. journal of mathematics teacher education, 7(1), 63–86. microsoft word reviewers final vol 4 no 2.doc journal of urban mathematics education december 2011, vol. 4, no. 2, pp. 140–141 ©jume. http://education.gsu.edu/jume 
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 reviewer acknowledgment january 2010–december 2011 noor aishikin adam joshua oluwatoyin adeleke, institute of education shuhua an, california state university lorraine baron tonya bartell, university of delaware dan battey, rutgers university clare bell, university of missouri robert berry, university of virginia denise brewley, georgia gwinnett college angela brown, piedmont college david brown, texas a&m university, commerce lecretia buckley, jackson state university stephanie byrd, clayton county schools patricia campbell, university of maryland theodore peck-li chao, university of texas, austin carrie lynn chiappetta, stamford public schools haiwen chu, graduate center, cuny karen cicmanec, morgan state university marta civil, university of arizona nicholas cluster, university of georgia lesa covington clarkson, university of minnesota cynthia cromer jaime curts, university of texas, pan american ubiratan d'ambrosio julius davis, morgan state university sandy dawson irene duranczyk, university of minnesota indigo esmonde, university of toronto gheorghita faitar mary foote, queens college, cuny cassie freeman, university of chicago joseph furner, florida atlantic university imani goffney, university of michigan lidia gonzalez, york college, cuny susan gregson, university of illinois, urbana-champaign rochelle gutierrez, university of illinois, urbana-champaign eric gutstein, university of illinois at chicago victoria hand, university of colorado, boulder deborah harmon, eastern michigan university cigdem haser crystal hill roberta hunter, massey university mine isiksal, middle east technical university andy isom, center for literacy laura jacobsen, radford university martin johnson, university of maryland shelly jones, central connecticut state university joyce king, georgia state university richard kitchen, university of new mexico courtney koestler, university of arizona della leavitt, rutgers university shonda lemons-smith, georgia state university jacqueline leonard, temple university julie livingood danny martin, university of illinois, chicago journal of urban mathematics education vol. 4, no. 2 141 donna mccaw, western illinois university jennifer mccray, erikson institute eduardo mosqueda, university of california, santa cruz nirmala naresh ellen pechman, emp consulting gerard petty, henry county public schools arthur powell, rutgers university laurie rubel, brooklyn college, cuny walter secada, university of miami megan staples, university of connecticut william tate, washington university dante abdul-lateef tawfeeq, adelphi university la mont terry, occidental college lanette waddell, vanderbilt university anita wager, university of wisconsin, madison dorothy white, university of georgia kimberly white-fredette, griffin regional educational service agency candace williams, dekalb county public schools desha williams, kennesaw state university curt wolfe, mt. carmel christian school jamaal rashad young, texas a&m university journal of urban mathematics education december 2016, vol. 9, no. 2, pp. 33–48 ©jume. http://education.gsu.edu/jume zandra de araujo is an assistant professor in the department of learning, teaching, and curriculum at the university of missouri, 121 townsend hall, columbia, mo 65211; email: dearaujoz@missouri.edu. her research examines teachers’ use of curriculum, particularly with emergent bilingual students. erin smith is a doctoral candidate in the department of learning, teaching, and curriculum at the university of missouri, 119 townsend hall, columbia, mo 65211; email: emsxh3@mail.missouri.edu. her research interests include equity and identity in mathematics education, the academic success of emergent bi/multilingual learners, and the use of positioning theory to examine discursive practices. matthew sakow is a former graduate student at the university of missouri and currently an algebra and geometry teacher at columbia river high school in vancouver, wa; email: matt.sakow@gmail.com. his interests include equity in mathematics education, funds of knowledge, and personalized tasks. public stories of mathematics educators reflecting on the dialogue regarding the mathematics education of english learners zandra de araujo university of missouri erin smith university of missouri matthew sakow university of missouri eachers’ beliefs are intertwined with their instructional practices and influence the ways they act and interact with students in the classroom (philipp, 2007; thompson, 1992). researchers have found that teachers may hold a number of unproductive beliefs about english learners (els), such as els cannot meet learning goals and their families do not highly regard education (mcleman & fernandes, 2012; pettit, 2011). teachers who hold these types of unproductive beliefs may perceive teaching mathematics with els as a problem to overcome, thus affecting student learning. such views, however, may ultimately encourage a perspective that devalues or fails to recognize the knowledge and skills els possess (see, e.g., civil, 2007; moll, amanti, neff, & gonzalez, 1992). to counter these types of beliefs, we argue that it is important to understand the context in which these beliefs are developed. unproductive beliefs about els reflect wider societal views. educational research comparing els to other “higher performing” student populations, who are frequently monolingual and white, continues to pervade the literature (gutiérrez, 2008). nonetheless, a focus on the achievement gap between students without regard to the social and political structures that contribute to these differences in achievement (gutiérrez, 2013) upholds a subconscious narrative that the standard to which all should be held is that of monolingual, white students and that other students (at the lower end of the gap, such as els) need to be “fixed” in order to “catch up” with their peers. parks (2009) cautioned: t http://education.gsu.edu/jume mailto:dearaujoz@missouri.edu mailto:emsxh3@mail.missouri.edu mailto:matt.sakow@gmail.com de araujo et al. mathematics education of english learners journal of urban mathematics education vol. 9, no. 2 34 the phrase achievement gap highlights some aspects of equality, but not others. focusing on achievement highlights the performance of individual students, and, because of the way it has been used in the united states, particular kinds of students—typically those who are poor and have dark skin…. closing the achievement gap implies that our work ought to be on students (or possibly schools and teachers) who do not measure up. it does not implicate definitions of mathematics, economic inequity, cultural or language hegemony, or other practices that many of us on the desired side of the “gap” benefit from daily. (p. 19) this implicit, invisible bias perpetuates beliefs that els are less capable and may interfere with a teacher’s ability to effectively work with els in the classroom because it encourages a need for emphasizing basic mathematical facts or procedures rather than equitable mathematical conceptual understanding. as a result, academically successful students from a marginalized population may even be considered as non-representative of her or his group. for example, a student perceived as an “exceptional” el may be granted “non-el” status because her or his abilities do not align with the expected characteristics of els. along with achievement gap research, the dominant narrative regarding available resources for els suggests they require support rather than challenge. currently, much of the dialogue surrounding the mathematics education of els focuses on linguistic support. this focus is visible throughout the myriad resources and publications widely available online, which answer the calls made by teachers for help teaching els. these resources number in the tens of thousands and can easily be found using a search string such as “supporting els” or “challenging els.” the number of resources uncovered by each search is indicative of society’s views of these students (fairclough, mulderrig, & wodak, 2011). assumedly, if the teacher seeks to challenge students, then she or he implicitly believes that students can meet this challenge. however, the common language of “supporting els” may have consequences for students; it may perpetuate a perspective of a population who needs to be fixed. consider the following three pairs of google searches1 that solicit resources for how to support or challenge “english learners,” “gifted students,” and “students” in general (see figure 1). the number of links related to supporting or challenging students in general shows approximately 2.8 results for the former for every 1 result of the latter. the search results for gifted students—approximately 0.4 results for support for every 1 result for challenge—reflect the identification of these students by ability and position as more academically autonomous. in contrast, the links for els exhibit a stark contrast between those of the other two groups. there were 11,000 returns related 1 we recognize the limitations of google searches and acknowledge that they are not an “exact science,” so to speak. they do, however, provide a sketch of how els are being discursively framed within the resources and articles available on the world wide web. de araujo et al. mathematics education of english learners journal of urban mathematics education vol. 9, no. 2 35 to supporting els in our search and no results for “how to challenge english learners.” to ensure that we were not missing a way to refer to els, we also searched with the terms english language learners, ells, and els in lieu of english learners. it was only in this last case (els) that we found exactly one result, in all other instances there were none. we do not argue that els do not need support; however, we recognize that a need for language support does not signal some sort of deficiency in mathematics, therefore there remains a need to challenge and extend students’ thinking. the extreme disparities in search results reveal aspects of the current narrative that underlie our approach to teaching els. the focus of online resources for teaching els overwhelmingly assumes they are more likely to need support rather than challenge. because of the focus on support, far less attention is given to challenging els mathematically or extending their thinking. it is perhaps not surprising that much of teachers’ focus in working with els surrounds remediation. figure 1. google search of web resources containing “how to challenge” or “how to support” students, gifted students, and english learners (august 15, 2016). in this public story, we discuss possible repercussions that extend from this narrative and the ways in which we, as mathematics teacher educators, are trying to be more critical of ourselves as we seek to change the discourse in our courses. we illustrate this effort through the presentation of a preservice elementary teacher’s (pst) work with a fifth grade el. we share how our analysis of interactions be de araujo et al. mathematics education of english learners journal of urban mathematics education vol. 9, no. 2 36 tween the pst and the student led us to deeply reconsider our practice as mathematics teacher educators, particularly regarding the mathematics education of els. sparking reflection our story begins with our analysis of data from a study that examined an elfocused mathematics field experience. the first author, zandra de araujo, was teaching an elementary mathematics methods course and had piloted the field experience as a means of providing psts with opportunities to work with els. in the field experience, four psts were paired one-on-one with an elementary el. prior studies (fernandes, 2012; kasmer, 2013) have found that experiences for psts that allow them to work with els can enable them to gain a stronger understanding of the connections among language and mathematics and may also challenge deficit views of els. nevertheless, these studies have also cautioned that such experiences need to be carefully designed because they may also lead to the reinforcement of unproductive beliefs about els. de araujo designed the field experience with this cautioning in mind. the field experience provided psts opportunities to enact cognitively demanding mathematics tasks (e.g., stein, grover, & henningsen, 1996) with an el. the psts planned for these sessions and met regularly with de araujo to reflect and detail plans for subsequent sessions. the explicit focus on cognitively demanding task was important because prior work has shown that teachers tend to avoid enacting such tasks with els (e.g., de araujo, 2012). for each week of the field meetings, the psts worked with the els on mathematics tasks for about 45 minutes. prior to each of the first three weekly meetings, the psts planned a lesson around a provided cognitively demanding task. in the final week, the psts selected or created their own tasks. in each of their weekly meetings, the psts would enact their plans, speaking with the research team before and afterwards to discuss their plans and reactions to the meeting. when analyzing the data, we noted that most of the psts took steps to remove obstacles for their students. for example, one pst often color-coded and bolded relevant information to help her student identify and keep track of key information (for further discussion of psts’ strategies see de araujo, i, smith, & sakow, 2015). one pst, kimberly, however, consistently did the opposite by raising a number of linguistic obstacles for her student, kyeong-tae.2 in one task, for instance, she replaced numerals with written out numbers (i.e., 24 was replaced with “twenty-four”). initially, we were perplexed by this action and other adaptations, which often left us asking ourselves, why was she doing this? 2 kimberly and kyeong-tae are pseudonyms. de araujo et al. mathematics education of english learners journal of urban mathematics education vol. 9, no. 2 37 kimberly regularly expressed her feelings of fear and anxiety that arose as she worked with kyeong-tae. her openness and comfort in sharing such feelings pushed us to critically reflect on what we were doing in our methods courses that might contribute to such feelings. moreover, kimberly’s experience forced us to reflect on our own teaching experiences, the feelings we ourselves had, ways we had worked to overcome such feelings, and repercussions of teachers’ fear and anxiety on el learning. we begin by sharing an abbreviated version of kimberly’s experiences throughout her meetings with kyeong-tae. to create this abbreviated narrative, we drew on kimberly’s preand post-survey data, video recordings of her meetings with kyeong-tae, lesson plans, and written reflections. we also drew from video recorded interviews with kimberly before and after each meeting and a final debrief. to analyze the data we first coded the lesson plans and interview transcripts using open coding to identify initial themes using analytic memos. throughout the coding process we focused our attention on excerpts in which kimberly described her thoughts about working with kyeong-tae and the specific strategies and accommodations she made for him. we used these excerpts to construct a brief portrait of kimberly’s experiences working with kyeong-tae throughout all 4 weeks. we focus this retelling on the ways in which kimberly tried to accommodate kyeong-tae and her feelings and beliefs regarding the experience of working with an el for the first time. we think it important to highlight her beliefs and the ways in which these beliefs manifested in the encounters. thus, we are grateful that kimberly was open in sharing her fears with us and emphasize that she was doing what she thought best for kyeong-tae by drawing on the resources to which she had access. through kimberly’s experiences we can learn how to improve our own practice as mathematics teacher educators. throughout the following four sections, we also bring up questions that arose for us as we analyzed kimberly’s experiences. thus, we use kimberly’s example to not only illustrate the consequences of the dominant deficit narrative of els on mathematics teaching and learning, but also to bring to light the ways in which these episodes led us to question our approach to preparing teachers to work with els. the initial encounter i’m, i’ve got nothing but worries. i know, that’s not fair, but i’m concerned. i’m worried that he won’t be able to understand me. that any and all inflection that i have in my voice or the way that i talk will be confusing, because i know that i mumble and i know that my tone isn’t as clear as it could be, so that’s a concern. i really, i have no idea what level kyeong-tae’s going to be on his english because i don’t know how we’re going to be able to communicate. (kimberly) de araujo et al. mathematics education of english learners journal of urban mathematics education vol. 9, no. 2 38 at the time of the study, kimberly was a junior elementary education major enrolled in the second of two mathematics methods courses. kimberly was a monolingual, white woman in her twenties. on her pre-survey, kimberly noted that she had no prior teaching experiences with els and no explicit instruction in teaching els. she agreed to participate because she thought the experience would be beneficial as she entered student teaching the following year. kimberly was paired with kyeong-tae, a fifth grade student from south korea. at the time of the study, kyeong-tae had been in the united states for about six months. prior to her first meeting with kyeong-tae, kimberly was candid in expressing her concerns, many of which, as she stated, related to potential communication barriers she anticipated. we see evidence in her statement of two assumptions related to els: (a) they are homogenous in their language proficiencies, and (b) they are most likely unable to communicate with teachers given their lack of language proficiency. we were not surprised by these beliefs, but questioned in what ways we were challenging psts to consider their own views of els. despite her concerns about being able to communicate, kimberly had created a plan for the first meeting: okay, so i really want to see what he is capable of without a whole lot of support first, so i am going to see where he is at. so i am going to present the problem, i intend to read it to him very clearly and slowly as many repeats as he requires…. i am trying to get to know a little bit because i want him to be comfortable. i think this is incredibly uncomfortable process for kids sometimes. kimberly had thought about ways to support kyeong-tae’s language by attending to her own language, as evidenced in her two quotes. she also brought along manipulatives to support him mathematically. within the first few minutes of the meeting, however, kimberly realized that kyeong-tae’s mathematical knowledge far exceeded her expectations. although she had anticipated about 30 minutes to implement the given task with kyeong-tae, he finished the task within 10 minutes. we talked with kimberly immediately after this initial session: when he put his strategies down, it was things i never would’ve thought of…like that last thing he did with the—i mean, i remember learning about it, but i also remember failing every test on it, so like that was impressive to me at least. just how he would get there so quickly and in his head…. that was so fast, the way he did that in before i even started thinking about the first number in a line, he was done, so kind of concerned about being able to keep up with him in future sessions…. i mean i don’t want to say that math is not my strong suit because i’m not terrible at it, but it’s—i am no kyeong-tae. we knew that the concerns kimberly raised regarding her lack of confidence in mathematics are not uncommon among elementary psts. however, her fear raised de araujo et al. mathematics education of english learners journal of urban mathematics education vol. 9, no. 2 39 a number of questions for us as mathematics teacher educators: what could we have done to better prepare kimberly to accept and effectively work through her fear? how could we challenge kimberly’s view of els as students who need mathematical support? how would kimberly respond to a gifted, non-el student? after this initial meeting with kyeong-tae, kimberly altered how she approached subsequent meetings. her plans changed from a focus on supporting kyeong-tae’s language and mathematics to challenging him. however, the ways in which she attempted to challenge him revealed much about the need to support mathematics teachers in this work. the second encounter so the main thing that i’m afraid of is that he is going to blow through everything again. and that we [teacher and researcher] are just going to be sitting here staring at him while he’s, he’s looking at me like i’m an idiot. so that was a big fear of mine. (kimberly) we found that the second week was characterized by kimberly’s sense of conflict over how to work with kyeong-tae. the quote above highlights the worry and apprehension she felt in anticipation of the second meeting. instead of being afraid he would not understand her, this week she was afraid that she was not prepared mathematically for kyeong-tae. for the second meeting, we had provided kimberly with two tasks and she had selected an additional task of her own. kimberly made two modifications to the tasks we gave her prior to enacting them with kyeong-tae. the first change she made was to remove a picture of mangoes from a task. initially, we were puzzled by her decision to remove the picture. a common strategy to help els is to add in pictures, so why would she take the picture out? kimberly explained her reasoning— i took out the picture. i just got a bad vibe about it. if the picture is on there, he might try to count the picture…it seems like he’s really advanced and i don’t want to pander to him a whole lot. the picture she removed was the only picture on the task and did not represent a correct solution to the problem. it is worth noting that kyeong-tae had not relied on pictures to solve the previous week’s task. however, while assuming that kyeong-tae would be mathematically proficient on this week’s task, kimberly may have assumed that he would avoid engaging in the prompt if he had a choice. thus, kimberly thought that engaging with the task’s english prompt would increase the task’s level of difficulty for kyeong-tae. it seems that kimberly thought de araujo et al. mathematics education of english learners journal of urban mathematics education vol. 9, no. 2 40 a linguistic challenge would even slow down his mathematical processes so that he had to think more deeply through the task. as she continued to describe her accommodations for kyeong-tae, kimberly again described a modification that ran counter to common supports for els: she replaced all numerals in one task with words. like the removal of the picture, kimberly did so to ensure that kyeong-tae read the problem for understanding. both of these modifications concerned linguistic aspects of the tasks, but were implemented to challenge kyeong-tae mathematically. the goal of many instructional strategies for els is to reduce or eliminate potential obstacles (e.g., chval & chávez, 2012; harper & de jong, 2004), particularly in terms of language. however, we noted that kimberly seemed to raise language barriers for kyeong-tae and this observation was confirmed as we examined the third task kimberly used. kimberly selected the third task on her own. this task was comprised of a set of arithmetic problems about the titanic (figure 2). the task featured many words and a context—social studies—which kyeong-tae described as his most difficult subject. kimberly explained that she had selected the task because she wanted more challenging tasks for kyeong-tae. although the worksheet stated it was for sixth grade, the problems involved basic operations and were low in cognitive demand. it is evident that she either failed to consider the level of mathematics of the task or selected it primarily for the amount of written text. figure 2. excerpt from the third week’s task kimberly selected for kyeong-tae. in the post-interview, kimberly acknowledged the problems did not challenge kyeong-tae mathematically, but she did believe they challenged him linguistically: interviewer: so, [the task] challenged him linguistically but not mathematically you said. de araujo et al. mathematics education of english learners journal of urban mathematics education vol. 9, no. 2 41 kimberly: not as much, so i might actually look for some word problems at a high school level, like a lower high school level, to see, in the same vein, of not being too complex but, i don’t know, you know, i don’t know because i don’t want him to feel frustrated on two fronts is all…. yeah, and i am not sure if that is right. i’m not sure if i should be feeling that way, because i feel like, he can probably handle it. i don’t know, what do you think? as evidenced in the excerpt above, kimberly was grappling with how to provide appropriate mathematics tasks for kyeong-tae. although she wanted to challenge kyeong-tae, kimberly hesitated to challenge him simultaneously in both linguistic and mathematical aspects. with this in mind, she opted to challenge kyeong-tae linguistically, perhaps based on her expertise in english and low-self efficacy in mathematics. said another way, we believe kimberly felt competent in her ability to linguistically challenge kyeong-tae, but unprepared to do so mathematically. when considering this belief we reflected on how we approached els in our methods courses. have we talked about how to extend els’ mathematical thinking or have we only discussed how to accommodate their language? how can we change our discourse of els to ensure psts have strategies to meet the diverse needs of all students? after the interview, it was clear kimberly was wrestling with her conceptions of els in light of her experiences with kyeong-tae: i almost don’t want to lump kyeong-tae in with what i was thinking about for my own classroom. because he is so exceptional i feel like, there is a certain extent to which kyeong-tae needs less support than your average el. maybe not in english, but certainly in math…. i don’t know, maybe that’s what the average el situation looks like; maybe it’s all about encouraging them to communicate when they’re confused. for the remainder of the field experience, kimberly continued to try to reconcile her beliefs about “average” els with her experiences with kyeong-tae. the third encounter if you would have asked me that question [whether the task should be implemented differently for els than non-els] on the very first problem that we did with kyeongtae, the very first time i met with him, i would have said, “yes, the task needs to be [modified], they need to have manipulatives, they need to have this and that.” kyeongtae has really sort of been a different experience. i think that he is probably not typical. (kimberly) in the third week, we have evidence that kimberly perceived her experiences with kyeong-tae in opposition with her general beliefs about els. this perception was evident when she referred to kyeong-tae as an el, but continued to describe de araujo et al. mathematics education of english learners journal of urban mathematics education vol. 9, no. 2 42 him as “not typical” or “exceptional” given that he did not exhibit characteristics she had associated with els. by compartmentalizing kyeong-tae as an atypical el, kimberly was able to retain her beliefs of els more broadly. one of the third week’s tasks focused on rounding and required kyeong-tae to determine if certain students’ claims of rounded values were true. the truth of these statements depended on the place value to which a student rounded. the task required kyeong-tae to explain his thinking and to critique others’ arguments. thus, the task was potentially more linguistically and cognitively demanding than prior tasks. kimberly anticipated some challenges with the transition to this type of task and stated in the pre-interview, “i think he’s just a little nervous about having to [explain his thinking] and think about it in english because he probably thinks in korean, doesn’t he?” this revelation was profound for kimberly who had not considered how an el might reason with the mathematics in a different way or even a different language. this revelation led us again to consider in what ways are we discussing els in our courses? much of our discussion with psts has focused on providing access to mathematical content by minimizing language obstacles. this focus on supporting els linguistically is a common theme present in a larger el narrative in mathematics education. we began to further consider the ways that we differentiate among various levels of english proficiency in our courses. were our general references to els contributing to the notion that they all had similar linguistic needs? if so, how can we talk generally about els without making broad generalizations? furthermore, were we discussing how to leverage els’ understanding of mathematics in their first languages? as kimberly continued to reflect on her beliefs about els and her experience with kyeong-tae, she expressed similar doubts and fears as in the previous weeks: i talked to the other [psts] who are doing these exercises and they talk about how their kiddos are struggling and i’m like, am i doing this wrong? is he supposed to be struggling? kyeong-tae’s success continued to surprise kimberly and did little to combat the belief that he should struggle. although she began to ask meaningful questions as she reflected on the experience, she remained hesitant to let go of previously held assumptions of els. this “holding on” caused us to reflect on the ways we can challenge such assumptions when discussing els in our courses. the final encounter i’m a little super nervous; i’m not going to lie, because i’m pretty sure that kyeong-tae is better at math than i am…. i’m sort of afraid that i’m going to mess him up by being wrong about something. (kimberly) de araujo et al. mathematics education of english learners journal of urban mathematics education vol. 9, no. 2 43 prior to her final encounter, kimberly continued to express fear and apprehension regarding her work with kyeong-tae. the quote above suggests the root of these fears lies in kimberly’s perceptions of her mathematical abilities in relation to kyeong-tae’s. as she grappled with these perceptions, she considered the implications for her teaching, namely that she might “mess him up.” instead of giving kimberly tasks as in the prior weeks, in this final week we asked her to select or create a task. she selected a task focused on the relationship among circumference, diameter, pi, and area of circles. the task utilized objects in the classroom, was conceptually based, and was unlike the other tasks she had enacted prior. in addition, kimberly chose to provide instructions orally. she explained these instructional decisions in the pre-interview: i wanted it to be engaging and fun for kyeong-tae since he’s already so good at math and i just figured, the other thing i thought about was that because it would be verbally—i would be verbally giving him directions, that this might be more challenging for him to pick up on the cues, but it’s probably a lot more similar to what he sees in school. her task selection was also motivated by a familiarity with it—a similar activity was done in her mathematics methods course—and an upcoming exam would cover related content. as they met, kimberly found that kyeong-tae had prior knowledge of the mathematical content of the task. he was familiar with the formulas for computing area and circumference of a circle. kimberly had not expected this prior knowledge as she reported in the post-interview, “he came in knowing all the formulas, which i didn’t expect.” she also expressed surprise at kyeong-tae’s mathematical thinking: “i never even thought about that with the 4r2 (kyeong-tae’s method of deriving the area of a circle), but i can totally see what he was thinking there and i wonder where he got that.” these statements illustrated how her conceptions of kyeong-tae, in conjunction with her lack of prior teaching experience and work with els in her teacher preparation program, influenced her ability to anticipate possible student solutions. in her final interview, kimberly reflected on this last encounter and the overall experience as it related to her future teaching: i was a little spoiled in this with kyeong-tae, because he’s so talkative and i think we did establish a rapport fairly quickly…. i don’t want to say that i maybe didn’t get as much out of this as i would have with a student that wasn’t as advanced, because i think i got a lot out of working with kyeong-tae, but i think that if i continue working with ell students at [school], i almost certainly will, i’m going to see more challenges and probably have to adapt in other ways. de araujo et al. mathematics education of english learners journal of urban mathematics education vol. 9, no. 2 44 her quote evidences her conception of els as not as advanced mathematically as kyeong-tae. she described kyeong-tae as an exception to the norm and she did not expect future els to be like him. a later statement reiterates this conception: i really expected him to not be on grade level, and he was beyond grade level, he was a gifted student…all of my preconceptions and all of my adaptations i had made were for taking it down and i would have never in a million years would have expected to have to take it up. these two quotes provided us with some ideas for why kimberly struggled to challenge kyeong-tae mathematically. she did not expect him to need more challenging problems. instead, she thought that would have to lower the difficulty of tasks for him because he was an el. this thinking made us wonder, “did we not talk about els when discussing challenging students? did we always associate els with low ability? in what ways did our methods courses frame els as mathematical thinkers?” truly, if all of kimberly’s preconceptions of els were deficit-focused, then her fears and apprehensions when working with a high-achieving el are not surprising. kimberly’s beliefs about els impacted her ability to appropriately challenge kyeong-tae mathematically. this case was particularly eye opening because although we realize the need to address els as individuals and know it is important to challenge them, we realized we might not be providing our students with the right opportunities to grow their understanding in these areas. perspectives on the experience how did we interpret kimberly’s experiences with kyeong-tae? the evidence suggests that she found kyeong-tae to be a non-representative el. although kyeong-tae was audibly an el, he did not perform mathematically in accordance with kimberly’s preconceptions. kimberly expected kyeong-tae to struggle and even asked after the third meeting whether he was supposed to be struggling. instead, kyeong-tae succeeded in every task he was given and surpassed kimberly’s own perceived abilities in mathematics. these realities fueled kimberly’s anxiety and, over the course of the 4 weeks, she continued to have “nothing but worries” concerning how best to support kyeong-tae. in large part, this anxiety may have stemmed from kimberly feeling unequipped in her role because she thought her mathematical knowledge was less developed than kyeong-tae’s; as she noted, “i am no kyeong-tae.” moreover, kimberly’s fourth week comment that expressed concern for “messing him up by being wrong about something” is evidence of this anxiety. she seemed to have an ideal notion of a teacher who would be able to avoid such a situation. de araujo et al. mathematics education of english learners journal of urban mathematics education vol. 9, no. 2 45 when her sessions with kyeong-tae did not go as she had imagined, kimberly responded in two ways. first, she sought to position kyeong-tae as a struggling learner by raising linguistic barriers. she introduced linguistic challenges to tasks intended to slow kyeong-tae down and to challenge him. doing so may have helped her reconcile her field experience with her preconceptions of els as struggling students. when reflecting on her introduction of linguistic challenges, we found that her actions were sensible within the greater narrative surrounding els and lack of available resources. namely, if lowering linguistic obstacles is how we support els, then to challenge els one might simply raise those obstacles. however, these modifications did little to further his mathematical growth and served to position both her as an expert or authority and kyeong-tae as a student in need during their interactions. second, kimberly figuratively relieved kyeong-tae of his el status by positioning him as “not typical.” she failed to recognize that teaching kyeong-tae was an authentic experience and may be characteristic of future experiences teaching els mathematics. thus, her positioning of kyeong-tae as an el-anomaly was detrimental to her success as an equitable teacher. specifically, this view led her to ignore kyeong-tae’s mathematical development in favor of his language. this view runs counter to researchers’ suggestions that it is more important for teachers to attend to what students do and say in mathematics, rather than how they do or say it (moschkovich, 1999). it also came as a surprise to kimberly that kyeong-tae thought in korean, suggesting that she had not considered how his own experiences and language might provide a supportive linguistic context (barwell, 2005) with which kyeong-tae could use to solve the tasks. that is, an el’s native language should be seen as a tool, rather than an obstacle. although we see an increased focus on els in teacher preparation programs (including our own), we question whether this preparation contributes to existing deficit narratives of els. in our own experience, we see evidence through our own feelings of disbelief. although we were surprised by kimberly’s responses and reactions to kyeong-tae, we see them as evidence of our failure to counter or challenge deficit narratives. such behaviors should be expected if we perpetuate current deficit narratives. moreover, the implications of deficit views or the impulse to position els in deficient ways are great and, in kimberly’s case, manifested in her selection and modification of tasks and a failure to hold kyeong-tae accountable for demanding mathematical practices. we anticipate similar patterns of behavior can be found in other teachers who hold comparable views of els. while examining kimberly’s case, we found ourselves asking, “what might we have done differently in our preparation courses to more effectively counteract deficit beliefs and actions?” first, we believed that we needed to talk about els every time we discussed working with students. marginalizing this group to a special chapter or specific lesson may only serve to promote a sense of peculiarity that de araujo et al. mathematics education of english learners journal of urban mathematics education vol. 9, no. 2 46 may lead to deficit views. when talking about gifted students, for instance, we should talk about students like kyeong-tae. neglecting to include this diverse population in each of our discussions surrounding mathematics education prevents our psts from normalizing their presence in the classroom. we expect that familiarity may lessen the surprise and discomfort surrounding working with els. secondly, els, like all students, are mathematical thinkers. we should consider this identity first and foremost in the mathematics classroom. perhaps in our methods courses, we need to change our approach in how we talk about els. we can talk first about the goal of developing students’ mathematical knowledge. then and only then, we can consider how to enact math tasks that draw on their cultural and linguistic resources. in other words, we might try to encourage psts to use the student’s language to increase access to mathematics. ultimately, if we are to change the discourse and feelings surrounding els, we must change how we talk with our psts about their status as mathematical thinkers. our conclusions what we learned from kimberly is that mathematics educators must consciously and continuously challenge psts’ unproductive beliefs about els. if psts enter mathematics methods courses with the idea that els—or any student population—are homogenous, they may be limited in how appropriately they can respond to students. therefore, mathematics educators must continue to consider how psts learn to engage els in mathematical discourse and practices through supports designed to improve access, rather than to lower difficulty. graphs, pictures, and diagrams, for example, can be powerful tools in communicating mathematically and can be used by students of any language (chval & chávez, 2012). in reflecting on kimberly’s experiences and our own practice, we have found that the common narrative of supporting els has unintended consequences. the resources available to students and teachers online overwhelmingly fail to position els as in need of challenging tasks and activities that extend their thinking. we need to make clear to psts that els do not inherently struggle with the ideas we teach in mathematics. we must not forget that els are also mathematical learners and thinkers. perhaps this is further evidence for rethinking the label el as others have proposed (e.g., kibler & valdés, 2016). one step we will commit to as we continue to grapple with these issues is to be conscious of our own language as mathematics educators and researchers as we challenge the common narrative and shift our perspectives from supporting els to engaging them in appropriate mathematics. de araujo et al. mathematics education of english learners journal of urban mathematics education vol. 9, no. 2 47 acknowledgments we would like to acknowledge ji yeong i for her work collecting and analyzing data for the larger project of which this study is a part. we would also like to thank kimberly for her participation and candor. references barwell, r. (2005). working on arithmetic word problems when english is an additional language. british educational research journal, 31(3), 329–348. chval, k. b., & chávez, ó. (2012). designing math lessons for english language learners. mathematics teaching in the middle school, 17(5), 261–265. civil, m. (2007). building on community knowledge: an avenue to equity in mathematics education. in n. nassir & p. cobb (eds.), improving access to mathematics: diversity and equity in the classroom (pp. 105–117). new york, ny: teachers college press. de araujo, z. (2012). transferring demand: secondary teachers’ selection and enactment of mathematics tasks for english language learners (unpublished doctoral dissertation). university of georgia, athens, ga. de araujo, z., i, j. y., smith, e., & sakow, m. (2015, november). preservice teachers’ strategies to support english learners. in t. g. bartell, k. n. bieda, r. t. putnam, k. bradfield, & h. dominguez (eds.), proceedings of the 37th annual meeting of the north american chapter of the international group for the psychology of mathematics education (pp. 648–655). east lansing, mi: michigan state university. fairclough, n., mulderrig, j., & wodak, r. (2011). critical discourse analysis. in t. a. van dijk (ed.), discourse studies: a multidisciplinary introduction (pp. 357–378). thousand oaks, ca: sage. fernandes, a. (2012). mathematics preservice teachers learning about english language learners through task-based interviews and noticing. mathematics teacher educator, 1(1), 10–22. gutiérrez, r. (2008). a “gap-gazing” fetish in mathematics education? problematizing research on the achievement gap. journal for research in mathematics education, 39(4), 357–364. gutiérrez, r. (2013). the sociopolitical turn in mathematics education. journal for research in mathematics education, 44(1), 37–68. harper, c., & de jong, e. (2004). misconceptions about teaching english-language learners. journal of adolescent & adult literacy, 48(2), 152–162. kasmer, l. (2013). pre-service teachers’ experiences teaching secondary mathematics in englishmedium schools in tanzania. mathematics education research journal, 25(3), 399–413. kibler, a. k., & valdés, g. (2016). conceptualizing language learners: socioinstitutional mechanisms and their consequences. modern language journal, 100(s1), 96–116. mcleman, l., & fernandes, a. (2012). unpacking preservice teachers’ beliefs: a look at language and culture in the context of the mathematics education of english learners. journal of mathematics education, 5(1), 121–135. moll, l. c., amanti, c., neff, d., & gonzalez, n. (1992). funds of knowledge for teaching: using a qualitative approach to connect homes and classrooms. theory into practice, 31(2), 132–141. moschkovich, j. (1999). supporting the participation of english language learners in mathematical discussions. for the learning of mathematics, 19(1), 11–19. parks, a. (2009). doomsday device: rethinking the deployment of the ‘achievement gap’ in equity arguments. for the learning of mathematics, 29(1), 14–19. pettit, s. k. (2011). teachers’ beliefs about english language learners in the mainstream classroom: a review of the literature. international multilingual research journal, 5(2), 123–147. de araujo et al. mathematics education of english learners journal of urban mathematics education vol. 9, no. 2 48 philipp, r. a. (2007). mathematics teachers’ beliefs and affect. in f. lester (ed.), second handbook of research on mathematics teaching and learning (pp. 257–315). reston, va: national council of teachers of mathematics. stein, m. k., grover, b. w., & henningsen, m. (1996). building student capacity for mathematical thinking and reasoning: an analysis of mathematical tasks used in reform classrooms. american educational research journal, 33(2), 455–488. thompson, a. g. (1992). teachers’ beliefs and conceptions: a synthesis of the research. in d. a. grouws (ed.), handbook of research on mathematics teaching and learning (pp. 127–146). new york, ny: macmillan. journal of urban mathematics education december 2015, vol. 8, no. 2, pp. 23–26 ©jume. http://education.gsu.edu/jume diane j. briars is president of the national council of teachers of mathematics (nctm), 1906 association drive, reston, va, 20191; email: dbriars@nctm.org. she is a mathematics education consultant, supporting schools and districts in systemic improvement of their mathematics programs; previously, she was mathematics director for the pittsburgh public schools. matt larson is the k–12 curriculum specialist for mathematics for the lincoln public schools, lincoln, ne and president-elect of national council of teachers of mathematics; email: mattlarson94@gmail.com. he is the author of several books focused on the effective implementation of professional learning communities. marilyn e. strutchens is an emily r. and gerald s. leischuck endowed professor and a mildred cheshire fraley distinguished professor of mathematics education in the department of curriculum and teaching at auburn university, 5010 haley center, auburn, al 36830; strutme@auburn.edu. her research interests include equity in mathematics education, secondary mathematics teacher education candidates’ field experiences, teacher leadership, and reform mathematics professional development for grades k–12 teachers. david barnes is the associate executive director for research, learning and development at the national council of teachers of mathematics, 1906 association drive, reston, va 20191; dbarnes@nctm.org. his interests include leadership development, the development and support of early career teachers, and linking research and practice coupled with association management. response commentary a call for mathematics education colleagues and stakeholders to collaboratively engage with nctm: in response to martin’s commentary diane j. briars nctm president matt larson nctm president-elect marilyn e. strutchens nctm board of directors david barnes nctm associate executive director research, learning and development n his commentary “the collective black and principles to actions,” martin (2015)1 offers a thought-provoking critique of principles to actions: ensuring mathematical success for all (national council of teachers of mathematics [nctm], 2014). first, we want to thank dr. martin for his continuing contributions. the questions, perspectives, and voices of those such as dr. martin and others who are critical of nctm’s positions and initiatives are some of the ones that the council needs to hear, both to help address persistent challenges and to inform our framing of the contexts and opportunities to make significant, real, and lasting change. we rec 1 the commentary referenced is a published version of danny martin’s remarks made at the national council of teachers of mathematics research conference plenary session “turning the common core into reality in every math classroom,” delivered on april 15, 2015 in boston, massachusetts. (other invited plenary panelists included deborah loewenberg ball, dan meyer, and steven leinwand.) i http://education.gsu.edu/jume mailto:dbriars@nctm.org mailto:mattlarson94@gmail.com mailto:strutme@auburn.edu mailto:dbarnes@nctm.org http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/270/169 briars et al. response commentary journal of urban mathematics education vol. 8, no. 2 24 ognize the need to listen more broadly and hear voices and perspectives that are beyond our immediate framework. we invite dr. martin and others who are interested in undertaking the work to improve mathematics education for students, particularly marginalized students, to engage with the council. initially, we need to come together to raise issues, identify opportunities, and increase understanding. more importantly, we need to work together to understand, build, disseminate, and implement specific strategies and actions to address shared challenges and concerns. martin (2015) states that the mathematics education community, in general, and the council, in particular, has not made significant progress in addressing the opportunity gap (flores, 2007); that is, it has not made significant progress in changing “the conditions of african american, latin@, indigenous, and poor students in mathematics education” (martin, 2015, p. 22). we agree. while mathematics achievement of all groups of students has increased substantially over the past 25 years, in terms of national assessment of education progress scores, advanced placement participation, and so forth, significant achievement gaps between subgroups remain (nctm, 2014; u.s. chamber of commerce foundation, 2015). while we have made progress, we have not made enough progress. we all need to do more. nctm needs to do more. in particular, we need to expand our understandings of the complexities inherent in equity and access with respect to issues of experience, identity, and agency. this need is one of the major reasons why we created principles to actions (nctm, 2014). we commit to addressing these concerns through our advocacy and policy work, and challenge others to join us in taking action toward changing the status quo. courageous and collaborative actions are necessary by university and classroom teachers; university, district, and school leaders; mathematics educators, researchers, and policy makers, working together to impact meaningful and sustained change. we recognize the significant challenges in producing systemic change to the status quo, such as those that martin (2015) articulated. we invite all (mathematics) educators and stakeholders to work with us and other mathematics education organizations, such as the benjamin banneker association, todos, national council of supervisors of mathematics, association of mathematics teacher educators, association of state supervisors of mathematics, and the algebra project, as well as other organizations, such as the council of great city schools, the national alliance of black school educators, the white house initiative on educational excellence for african americans, the u.s. chamber of commerce foundation, the national association for the advancement of color people, and the education trust, to address these challenges and impact the educational system so that it provides quality education and parity in opportunity and outcome. as the public voice of mathematics education, nctm has worked to raise the consciousness of all of those who are involved in mathematics education through its standards documents (e.g., nctm, 1989, 1991, 1995, 2000, 2009, 2014), other publications, and initiatives; yet, there is much work to be done. nctm must move be briars et al. response commentary journal of urban mathematics education vol. 8, no. 2 25 yond isolated projects, publications, and other initiatives aimed at promoting equity and access. as the public voice, we have significant work to do to make the complexities of equity and access part of the conscience, the discussions, and the daily actions of our membership. we cannot do it alone. we need and invite mathematics educators and other stakeholders, including those who have been critical of our work, to not only engage with the council but also to challenge the council with advice on specific frameworks, methodologies, practices, and policies, and so forth, and join us in actions that we can undertake to systemically change the experience and opportunity for marginalized learners. martin (2015) was correct in stating that change cannot come fast enough and has been too long in coming. his statement just reinforces the need for all parties to work together more closely and more diligently. as an immediate first step, nctm will be convening leaders of our national affiliates to begin a dialogue around issues of equity and access. furthermore, we intend to build on the framework of equity and access put forth in principles to actions (nctm, 2014) by working to foreground the complexities of identity, agency, and opportunity in our messaging and activities, including ongoing work associated with principles to actions. these efforts include online professional development toolkits and a research companion; both are in development. in addition, the 2016 nctm annual meeting and exposition program will explicitly address equity and access through the “instruction and policies that promote equity and access” and “promoting productive dispositions about mathematics” strands and the iris m. carl equity address. nctm is also considering how our commitment to equity and access is institutionalized in our strategic planning, initiative development, and the voice of nctm. initial steps include greater diversity among our leadership appointments and the planning for one of our newly formed “innov8” conferences to specifically address equity and access for 2017. principles to actions (nctm, 2014) and the dialogue dr. martin opened surrounding nctm’s historical treatment of equity and access issues can serve as catalysts for us to constructively collaborate. nctm commits itself to partnering with mathematics education colleagues and other stakeholders so that together we can improve the educational system. collaboratively, we can change our educational system to provide access and opportunity to students who are marginalized and build a positive mathematics identity and sense of agency. we invite dr. martin and others to engage with us, to identify needed courageous actions, and to take the actions necessary to help ensure meaningful and lasting change. those interested in collaborating in this work should contact nctm at change@nctm.org. references flores, a. (2007). examining disparities in mathematics education: achievement gap or opportunity gap? high school journal, 91(1), 29–42. mailto:change@nctm.org briars et al. response commentary journal of urban mathematics education vol. 8, no. 2 26 martin, d. b. (2015). the collective black and principles to actions. journal of urban mathematics education, 8(1), 17–23. retrieved from http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/270/169 national council of teachers of mathematics. (1989). curriculum and evaluation standards for school mathematics. reston, va: national council of teachers of mathematics. national council of teachers of mathematics. (1991). professional standards for teaching mathematics. reston, va: national council of teachers of mathematics. national council of teachers of mathematics. (1995). assessment standards for school mathematics. reston, va: national council of teachers of mathematics. national council of teachers of mathematics. (2000). principles and standards for school mathematics. reston. va: national council of teachers of mathematics. national council of teacher of mathematics. (2009). focus in high school mathematics: reasoning and sense making. reston, va: national council of teachers of mathematics. national council of teachers of mathematics. (2014). principles to actions: ensuring mathematics success for all. reston, va: national council of teachers of mathematics. u.s. chamber of commerce foundation. (2015) the path forward: improving opportunities for african-american students. retrieved from https://www.uschamberfoundation.org/reports/path-forward-improving-opportunitiesafrican-american-students http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/270/169 https://www.uschamberfoundation.org/reports/path-forward-improving-opportunities-african-american-students https://www.uschamberfoundation.org/reports/path-forward-improving-opportunities-african-american-students journal of urban mathematics education december 2015, vol. 8, no. 2, pp. 87–118 ©jume. http://education.gsu.edu/jume melva r. grant is an assistant professor in the stem education & professional studies department at old dominion university, 257-b education building, norfolk, va 23529; email: mgrant@odu.edu. her research interests include exploring student mathematics identity development and developing teacher leaders and coaches capable of supporting mathematics teachers working with underserved and underrepresented mathematics learners. helen crompton is an assistant professor in the department of teaching & learning at old dominion university, 145 education building, norfolk, va 23529; email: crompton@odu.edu. her research interests include k–12 teacher preparation and technology integration for mathematics learning and teaching. deana j. ford is a doctoral student in the department of teaching & learning at old dominion university, 145 education building, norfolk, va 23529; email: dford005@odu.edu. her research interests include secondary students’ mathematics learning and the intersection with english language literacy. black male students and the algebra project: mathematics identity as participation melva r. grant old dominion university helen crompton old dominion university deana j. ford old dominion university in this article, the authors examine the mathematics identity development of six black male students over the course of a 4-year the algebra project cohort model (apcm) initiative. mathematics identity here is defined as participation through interactions and positioning of self and others. data collection included nearly 450 minutes of video recordings of small-group, mathematics problem solving in which student actions, coded as acts of participation, were tallied. these tallied actions were conceptualized descriptively in terms of mathematics identity using the lenses of agency, accountability, and work practices. the analyses suggest that the apcm students’ confidence in self and peers increased over the 4 years, they consistently chose to engage in mathematics, and their reliance on knowledgeable others lessened. opportunities for future research and implications for policy makers and other stakeholders are discussed. keywords: black male students, mathematics identity, mathematics teaching and learning, the algebra project any urban high school mathematics classrooms have disproportionate numbers of students who are often described in policy reports and media as “at risk” (durbin, 2012). this imbalance is especially true for black male students who are often labeled as learning deficient, targeted for disciplinary action, and positioned for future incarceration (booker & mitchell, 2011; gregory, skiba, & noguera, 2010). many black male mathematics learners have been historically, and continue to be, underserved by schools and society at large, especially those attending urban schools and qualifying for reduced-price meals (anyon, 2006; haberman, 1991/2010). nevertheless, research has shown that when black male students become aware of and have opportunities to learn mathematics in culturally receptive climates they take on productive mathematics identities (berry, ellis, m http://education.gsu.edu/jume mailto:mgrant@odu.edu mailto:crompton@odu.edu mailto:dford005@odu.edu grant et al. identity as participation journal of urban mathematics education vol. 8, no. 2 88 & hughes, 2014). conversely, black male students positioned in restrictive school climates with limited learning opportunities often experience negative outcomes (gibson, wilson, haight, kayama, & marshall, 2014). furthermore, when black male students take on productive mathematics identities they are better equipped to self-advocate for more positive learning opportunities and improved outcomes for themselves, their communities, and society at large (hope, skoog, & jagers, 2015). to understand how black male students might take on productive mathematics identities, we explored the mathematics identity development of six black male students who chose to participate in the algebra project cohort model (apcm) initiative during their 4 years of high school. the overarching research question and accompanying sub-questions that guided the exploration were: how did the mathematics identity of six black male students participating in the algebra project cohort model initiative develop over their 4 years of high school? i. what types of agency were students observed exercising and how did their agency evolve? ii. how did students’ observed work practices (i.e., small-group problem solving) influence their mathematics identity development? iii. to whom were students observed being accountable to and how did their accountability evolve? review of literature there has been substantial scholarship over the past 20 years that explores identity from many perspectives. cultural and social psychologists, anthropologists, sociologists, and social scientists in general have reframed how we think about identity. in the mathematics education literature, this reframing has been driven by concepts derived out of a variety of theories such as critical theory, critical race theory, feminist theory, sociocultural theory, poststructural theory, and so forth (see, e.g., berry, 2008, gutstein, 2007; mcgee & martin, 2011b; stinson, 2013). nonetheless, for the study reported here, we take a narrower view of identity. we define mathematics identity simply as participation. specifically, we explore how six black male students’ mathematics identities developed over 4 years of high school using nearly 450 minutes of video recordings of small-group, mathematics problem solving. to contextualize our study, we discuss two connected areas of research: (a) “reform” in mathematics education, and (b) black male students and mathematics identity. grant et al. identity as participation journal of urban mathematics education vol. 8, no. 2 89 reform in mathematics education to position black children in reform efforts, we drew on berry, pinter, and mcclain’s (2013) critical review of k–12 mathematics education reform efforts from the mid 1950s to the early 2000s. their review of reform mathematics focused on what was taught, how it was taught, who taught it, and, most importantly, who got access to it. they concluded that the needs of black children in mathematics education reform efforts have not been attended to over the decades. segregation has been re-enacted through testing and tracking in many schools, and the brilliance of black children has been largely ignored by the majority of mathematics educators and researchers. recently, martin (2015) argued that mathematics education reform for several decades has yielded few benefits for the collective black1 as he critiqued the national council of teachers of mathematics’ (nctm) latest policy document principles to actions: ensuring mathematics success for all (nctm, 2014) at the 2015 nctm research conference held in boston, massachusetts. his critique included categorizing the long-standing rhetoric about equity and “mathematics for all” as political, questioning the audience for whom the document was written, and calling for mathematics educators to consider revolutionary reform designed for the collective black.2 for the most part, extant reform efforts have neither targeted nor yielded substantive improvements for the collective black, in general, and black male students, in particular. in this article, we discuss aspects of mathematics education reform in spite of this oversight because that is what exists (for now) and these efforts are pertinent for situating our project. over the last several decades, national organizations such as the nctm (e.g., 1991, 2000, 2014) and the national research council (2001) have called for significant cultural changes in mathematics classrooms. the nctm, for example, called for classrooms that are co-created by teachers and students, “where students of varied backgrounds and abilities work with expert teachers, learning important mathematical ideas with understanding, in environments that are equitable, challenging, supportive, and technologically equipped for the twenty-first century” (nctm 2000, p. 4). the latest national call for change is embedded in the common core state standards for mathematical practice (ccss, 2010). specific recommendations for mathematics education reform efforts have also emerged from mathematicians and mathematics educators. mathematicians have suggested that black students, in particular, need opportunities to engage in doing mathematics in ways that 1 this term was used by martin (2015), defined as african american, latin@, indigenous, and poor; he attributes the term and definition to eduardo bonilla-silva. 2 it appeared that the predominantly white audience received his remarks with loud silence. grant et al. identity as participation journal of urban mathematics education vol. 8, no. 2 90 are both social and cultural (fullilove & treisman, 1990; maton, hrabowski, & greif, 1998). similarly, mathematics educators have advocated for pedagogical approaches that are student-centered and collaborative versus the traditional didactic approaches that have persisted for decades (franke, kazemi, & battey, 2007; hiebert et al., 1997; lampert, 1990/2004). the consensus among many: students participating with peers in ways that fosters sense making while using their cultural experiences and ways of knowing within and beyond school settings supports mathematics learning (e.g., moses & cobb, 2001; walker, 2006). critical mathematics educators have also stipulated that these student-centered, collaborative approaches are more effective for black students when they are carried out by teachers who are culturally aware versus those who believe that learning and teaching are race neutral (martin, 2012; matthews, jones, & parker, 2013; mcgee & martin, 2011a; stinson, jett, & williams, 2013). in addition to student-centered, collaborative pedagogical approaches, mathematics educators have advocated for using high-level, cognitively demanding tasks (e.g., stein, smith, henningsen, & silver, 2000). these educators claim that the cognitive level of the task affords different types of teaching and learning opportunities. high-level tasks that require students to engage mathematically, to seek connections to other mathematical ideas, and to prove their approaches, require teachers to facilitate learning differently than low-demand tasks that only require students to recall memorized facts that teachers, in turn, validate. the types of pedagogies needed for facilitating high-level tasks are typically more student-centered, such as examining student work and listening to their explanations to inform instructional decisions, and requiring students to use mathematical processes and practices in learning (ccss, 2010; doerr, 2006; henningsen & stein, 1997; nctm, 1991, 2000). however, classrooms where students engage collaboratively in cognitively demanding tasks are not available to all students, in particular black students (ladson-billings, 2006). in fact, too many black students attend poor performing schools. according to balfanz and legters (2004), in 2002 almost half (46%) of black students attended high schools with weak promoting power3 where graduation was not the norm; most of these schools were in urban areas with high poverty. few reform efforts have been meaningfully enacted in schools, in general, and urban high-poverty schools, in particular, for many reasons that are beyond the scope of this article (for a complete discussion see marrus, 2015). mathematics education in urban, highpoverty schools typically manifests as perpetual remediation, discipline, and other authoritative actions (bracey, 2013; ladson-billings, 2006; love & kruger, 2005; patterson, 2014). bracey (2013) captured the essence of mathematics education re 3 promoting power is an indicator of high school dropout rates, calculated as a percentage comparison of seniors to freshmen 4 years earlier; 60% fewer seniors than freshmen represent weak promoting power (balfanz & legters, 2004). grant et al. identity as participation journal of urban mathematics education vol. 8, no. 2 91 form within the current climate of accountability: “the net result is a lack of opportunity to engage black children beyond prescriptive remediation to pass annual yearly performance … mandates” (p. 173). in summary, effective mathematics education reform efforts for black students must provide opportunities for and access to: (a) pedagogies that are student centered and collaborative; (b) teachers who are culturally aware and well prepared; and (c) high-level mathematics courses with cognitively demanding tasks. black male students and mathematics identity supporting positive and productive mathematics identity development for black male students requires they have access to teachers who: (a) explicitly and publicly hold high expectations for them to learn rigorous mathematics; (b) create receptive, engaging, and supportive learning environments; and (c) are culturally aware and responsive while exercising decentralized teaching authority (ladsonbillings, 1994, 1997; stinson, jett, & williams, 2013). from this perspective, we review literature about black male students’ mathematics identity development. martin (2009, 2013) argued that discussions about black students’ mathematics identities cannot be independent of discussions about race and racism in the united states. the historical rhetoric in the united states around mathematics teaching and learning often positions black students implicitly and explicitly as mathematically deficient compared to white students who are positioned as the norm. this positioning, unfortunately, is often supported by mathematics education research and educational polices (martin, 2013). therefore, for martin (2009), mathematics identity refers to the dispositions and deeply held beliefs that individuals develop about their ability to participate and perform effectively in mathematical contexts and to use mathematics to change the conditions of their lives. a mathematics identity encompasses a person’s self-understanding and how others see him or her in the context of doing mathematics. (pp. 136–137) central elements of mathematics identity that emerge from this definition include perceptions by others and beliefs about self in relation to mathematics learning and doing. perceptions by others influence the ways we think about ourselves and the actions we take. one perception about black male students held by others is the stereotypical image of the non-academic, street “thug.” this stereotypical image not only influences but also can threaten black (male) students’ mathematics identity development (steele, 1997, 2006; steele, spencer, & aronson, 2002). steele (2006) referred to this phenomenon as stereotype threat and defined it as “the threat of being viewed through the lens of a negative stereotype, or the fear of doing something grant et al. identity as participation journal of urban mathematics education vol. 8, no. 2 92 that would inadvertently confirm that stereotype” (p. 253). researchers have reported ways that black male students have navigated the peril of stereotype threat in mathematics learning contexts (e.g., berry, 2008; mcgee, 2013a; stinson, 2008). for instance, mcgee (2013a) analyzed interviews with 11 successful black male high school juniors and seniors. mcgee described the students as having defiant reactions to the stereotype; some ignored the threat and persevered and others worked harder to attain academic achievement. in either case, students developed productive mathematics identities using internal coping mechanisms when faced with negative perceptions by others. watson (2012) uncovered another type of perception that is more covert by nature. she studied mathematics teachers whom she described as norming suburban when asked to describe their students. the act of norming suburban uses middleclass, white cultural perceptions as the standard by which all other groups “are compared, judged, and subordinated” (p. 987). neither innocent nor objective comparisons emerge when norming suburban because it requires one “to posit, either implicitly or explicitly, that teaching in suburban schools is better, and base this belief on the perceived inferiority of urban students,” all the while not using “race language” (p. 988). watson outlined a three step suburban norming process: (a) assume groups are monolithic with respect to behaviors, values, and beliefs; (b) decide if these cultures are negative or positive; and then (c) establish hierarchies among groups. norming suburban appears to be a form of stereotype threat that does not attend directly to characteristics such as race and class. students, particularly those in the lowest hierarchical group, however, are likely to notice teachers who adopt norming suburban practices and discourses (berry, 2008). stinson (2006) reviewed historical and theoretical perspectives surrounding black male students schooling experiences and presented three discourse clusters often used by others when discussing black male students: the discourse of deficiency, the discourse of rejection, and the discourse of achievement. the discourse of deficiency is the perception that black children are products of genetics, families, communities, and sociocultural spaces that are historically lesser than and not sufficient. this discourse leads to perceptions by others that black male students, in particular, are incapable, lacking, and otherwise deficient with respect to mathematics learning and achievement. school officials and policy makers who adopted deficiency perceptions for black students often select intervention options that are typically segregating and anti-intellectual, such as labeling, tracking, isolating remediation, and authoritative pedagogies. the discourse of rejection is the perception that black male students reject either a productive intellectual identity or the collective black identity; the intervention here is often nurturing support programs, such as african-centric rites of passage programs. the discourse of achievement is the perception that black students are able to achieve intellectually and mathematically. leonard and martin (2013) took up the discourse of achievement to compile their grant et al. identity as participation journal of urban mathematics education vol. 8, no. 2 93 edited volume the brilliance of black children in mathematics: beyond the numbers and toward new discourse, which approaches mathematics learning and identity development of black children from the perception of brilliance, omitting the discourses of deficiency and rejection altogether. beliefs about self as articulated by successful black male students starkly contrast the mathematical identity descriptions presented about them received via educational research, policy reports, and media outlets (ladson-billings, 2006; martin, 2012); and there are too few of these stories told within extant literature (martin, 2013; mcgee, 2013a). berry (2008) reports one such student’s self-account who described mathematics as “an easy subject for him to learn because he likes it and he loves the challenge of problem solving” (p. 464). this mathematically talented and engaged black male student’s account was shared during middle school; he had been identified as academically gifted in the fourth grade. the account reported by berry described the student’s relationship with his father that included mathematical challenges with games and puzzles done at home. in sixth grade, however, he encountered a teacher with whom he did not connect. this teacher appeared set on removing him from her class and presumably the gifted program. the student with parental advocacy persevered and passed the teacher’s class earning a b. accounts of black students’ mathematics learning experiences that include social and cultural influences using students’ “voices” (e.g., berry, 2008; jett, 2010; mcgee, 2013b; stinson, 2013) or strongly influenced by students’ voices (e.g., grant, 2014; mcgee & martin, 2011a, 2011b) are adding new positive perceptions and characterizations for how black students see themselves in relation to doing and learning mathematics. conceptual framework: mathematics identity as participation mainstream education scholars have explored the notion of identity development to better understand how people think about themselves or how others perceive them in relation to learning (e.g., cobb & hodge, 2002; gee, 2000; gilpin, 2006; greeno, 1997). in these cases, mathematics identity is conceptualized as mathematics participation: the ways that students interact with others and position themselves and others in relation to mathematics engagement. these mainstream conceptualizations, however, most often do not consider the socio-cultural and -political contexts of learners and of learning. with this limitation in mind, varelas, martin, and kane (2013) used a socio-cultural and –political critical lens to develop the content learning and identity construction framework for researching learning in mathematics and science classrooms. this framework considers content learning and identity construction as requisite. they described identities as “lenses through which we position ourselves and our actions and through which others position us” (p. 324). positioning influences learning opportunities in which students may en grant et al. identity as participation journal of urban mathematics education vol. 8, no. 2 94 gage, and changes in positioning result in different learning opportunities. opportunities are essential for an exploration of identity as participation. students’ self-perceptions are central to the actions (or inactions) they pursue within social systems, such as mathematics classrooms (gresalfi, 2009; nasir & hand, 2006; nasir, mclaughlin, & jones, 2009; varelas et al., 2013). within the lens of mathematics identity, self-perceptions, the perceptions of others, and the situated contexts converge and influence identity related processes (esmonde, 2009; esmonde, brodie, dookie, & takeuchi, 2009; nzuki, 2010). esmonde and colleagues defined three identity related processes, referred to as work practices for cooperative groups: collaborative, individual, and helping. in this study, we explore student mathematics identity development within the context of small-group, mathematics problem solving, and through observation we sought to interpret their mathematics identity development in terms of mathematics agency, accountability, and work4 practices (with a focus on collaborative and individual practices only). mathematics identity and participation this study characterizes participation as observable mathematics engagement and uses participation as the overarching construct for students’ mathematics identity. this two-tiered construction of students’ mathematics identity has foundations in educational psychology and mathematics education literature: (a) exercised agency (bandura, 2006; gresalfi, taylor, hand, & greeno, 2009; gutstein, 2007; hand, 2010) and (b) student accountability (ares, 2006; cobb, gresalfi, & hodge, 2009; cobb & hodge, 2002; yackel & cobb, 1996). these constructs, agency and accountability, manifest as observable student actions (i.e., agency) or inactions in mathematics learning contexts, and students chose participation or non-participation based on afforded opportunities that are influenced by feelings of accountability. mathematics identity and agency. our perspective of agency is grounded in bandura’s (2005) agentic perspective of social cognitive theory: “to be an agent is to influence intentionally one’s functioning and life circumstance. in this view, people are self-organizing, proactive, self-regulating, and self-reflective” (p. 9). in other words, people make intentional choices in their self-interests, which, from our perspective, are manifestations of identity as (observable) participation, or nonparticipation, which is also agency exercised. gresalfi and colleagues (2009) explain the possession and exercise of agency: it is important here to dispel the notion that people “have” or “lack” agency. in virtually any situation, even the most constrained, people are able to exercise agency; at the basic level, by complying or resisting. the ways that agency can be exercised, and the 4 the work in this study is mathematics problem solving within a small group of three to four students. grant et al. identity as participation journal of urban mathematics education vol. 8, no. 2 95 consequences for doing so, are what change in a particular context. said differently, an individual can always exercise agency, it is the nature of that exercise that differs from context to context. (p. 53; emphasis in original) here, gresalfi and colleagues are suggesting that close attention be paid to issues of power and authority within the mathematics classroom when considering distribution of agency. many critical researchers have acknowledged that mathematics classrooms and mathematical tasks are not neutral or without power dynamics, equitable access, and opportunity for engaging (e.g., esmonde & langer-osuna, 2013; mcgee & martin, 2011a; tyner-mullings, 2012; varelas et al., 2013). while power dynamics and equity are not prominent in this study, we recognize that these dynamics influence agency and accountability and the importance of being mindful of such within both research and practice. otherwise, there is no commitment to social justice and the status quo continues. mathematics identity and accountability. accountability is a prominent element of construction of competency as participation (cobb et al., 2009). cobb and colleagues articulated competency in terms of curricula with respect to agency distribution (i.e., accountable for what) and in terms of the culture for discourse in terms of accountability (i.e., accountable to whom). similar to the other participation components described thus far, this component is observable and interpretations can be made to categorize what was observed. the second portion of this participation component, accountable to whom, includes five levels: (a) teacher only or class only; (b) teacher and peer; (c) small group only; (d) teacher and small group; and (e) teacher, small group, and class. for this study, as students were situated in small groups for problem solving and the proctor followed a non-helping protocol (discussed later), our focus for whom students were accountable included: (a) expert: directs discourse to knowledgeable other, in this case, proctor or a peer positioned by the student as expert or more knowledgeable; (b) peers: expressed concern for peer in relation to mathematics at hand; or (c) self: positioning self as expert/knowledgeable or expressed disinterest in peer or others’ perspectives. in summary, mathematics identity as participation was framed using agency and accountability. observable incidents of participation were used as the overarching construct that situated actions of agency and accountability related to mathematics problem solving. we connected our study to recommended reforms for improving mathematics teaching and learning and to extant understandings about black male students and their mathematics identity development. methods interpretive qualitative analyses were employed for the purpose of understanding student mathematics identity development as related to mathematics participation (schwandt, 1994). descriptive statistics were also used to aid in pattern grant et al. identity as participation journal of urban mathematics education vol. 8, no. 2 96 discovery and recognition. the social and cultural contexts selected for exploration were the small-group, problem-solving assessments observed throughout the 4 years of the apcm initiative. video recordings of these assessments were the primary data source. the algebra project the underlying genesis of the algebra project was influenced by concerns of mathematical equity and access (moses & cobb, 2001). therefore, the primary goal for the apcm initiative was to transform urban and rural students’ perceptions of themselves from adopting mathematical identities given to them by others (e.g., at risk students) to mathematics learners and leaders who possess mathematics literacy. in other words, “young people finding their voice instead of being spoken for is a crucial part of the process” (moses & cobb, 2001, p. 19). the apcm initiative was designed for accelerating mathematics understanding for mathematics students who are likely to be underserved by schools and society at large. it was comprised of three parts: a cohort structure, curriculum and pedagogy, and community outreach. worth noting explicitly, the algebra project consistently seeks to work with students from the lower quartile,5 but “interventions” neither advocate for nor include remedial approaches, and students are not positioned as deficient. instead, the algebra project curriculum begins with students sharing an experience from which mathematical understandings are developed and abstracted, an experiential learning approach (kolb, 1984). the apcm initiative is built on 15 years of experience in middle and high school pilot programs that included instructional materials development funded by the national science foundation (moses, dubinsky, henderson, & west, 2008). a robust discussion of the algebra project curriculum6 would likely be interesting, but is beyond the scope of this article. the apcm initiative endeavors to create opportunities for students to actively engage in mathematics that develops mathematical identities while building mathematical literacy. participants and context the first author, in years 1 and 2 of the project, visited participants’ classroom several days per month to work with the apcm teacher and the local university mathematician, the principal investigator for the local project. in years 3 and 4, 5 how one measures and determines hierarchies that order students and relegates some to the lower quartile is of no consequence because the algebra project seeks to work with all students perceived as underserved or otherwise labeled through deficiency discourses. 6 the algebra project curriculum is available for inspection and comment through a public curriculum portal accessible at http://www.algebra.org/curriculum/. http://www.algebra.org/curriculum/ grant et al. identity as participation journal of urban mathematics education vol. 8, no. 2 97 she visited the classroom once or twice per year, and attended two of the four summer institutes.7 during site visits, she supported the teacher and sometimes took an active role during instruction with the students and she collected data to support the research project. through these activities, she got to know the students and they got to know her. in year 1, she conceptualized the need for and developed the smallgroup assessment protocol (discussed later). the study reported here was part of a larger the algebra project research project that spanned five urban and rural sites across the united states. the goals for the larger apcm research project included: (a) students graduating from high school in 4 years; (b) students, upon graduation, enrolling in credit-earning mathematics courses for those choosing post-secondary options; and (c) students developing and participating in productive peer cultures for learning mathematics (moses et al., 2008). the research reported here focuses on one of the sites from the large project; a small, urban community located in the midwestern united states. the apcm student cohort was comprised of 19 students in their first year of high school, the only high school in the community. the students, with parental or guardian consent, agreed to take two, 50-minute classes of mathematics each day with the same teacher for all 4 years of high school. most of the students and their parents (or guardians) knew the apcm teacher as a member of their community prior to entering high school. the apcm teacher is white, but she raised her bi-racial (black) son, who was academically successful and a star on the football team, in the community. her son was about two years older than the apcm cohort students. however, all of the children in the community who engaged in sports did so within the community leagues, and the majority of the male cohort students were also on the high school football team. the first author, on many occasions, observed students gravitating to the apcm teacher in times of need. several of the black male students referred to her using familial terms, such as “school mom” or “second mother.” the apcm teacher was observed reciprocating the students’ affections. for instance, she maintained a snack cabinet to feed hungry students; admonished poor decision making, in or out of school, while encouraging and expecting better in the future; and returned many unsolicited hugs. after her son graduated, the apcm teacher continued to participate in the community with students and to attend extracurricular events. the state department of education designated 16 of the 19 cohort students as “not proficient” as freshmen based on a score received on the eighth-grade, statemandated mathematics achievement test. the school and society at large, from our 7 summer institutes were held each summer to provide students with opportunities to engage in mathematics and to develop leadership and other positive dispositions. the institute locations alternated between a large urban university and a moderately sized rural university, where the students lived on the campus of the hosting university for 2 weeks each year. grant et al. identity as participation journal of urban mathematics education vol. 8, no. 2 98 perspective, underserved these students; our project endeavored to serve them differently with respect to developing mathematics literacy. six black male students from the cohort were purposefully selected for this study because they represented a sample cohort. the first consideration for selection, and most obvious, was to represent the cohort demographically; the cohort was predominantly comprised of students who identified as black (90%) and male (71%).8 a conscious decision to select only black male students was made for several reasons: (a) the vast majority of cohort students identified as black and male during the 4 years of high school; (b) in year 4, there was only one student who identified as not black and that student was new to the cohort; and (c) a personal interest of the first author to study black students and their mathematics identity development. the other important consideration for selecting the sample was to restrict selection to 4-year participants and to those participants with sufficient data. thus, the small-group assessment videos (discussed later) were viewed to identify an initial list of eligible students, and two factors were used to cull that list: (a) the student was present for at least one small-group video segment for each of the 4 years;9 and (b) an equal number of students positioned by peers as leaders. using these as criteria, six black male students were selected. it is worth noting that no students who identified as female appeared in more than two years of the smallgroup assessment videos. reasons for absences were twofold: either the student was absent on a particular group assessment day or there were technical challenges while videotaping. it was not uncommon for more students to be absent on assessment days, especially during the earlier years. we always announced research related data collection activities and allowed students to opt out without penalty, per the institutional review board agreement. moreover, the research team was responsible for videotaping, and especially in the earlier years, unintended errors occurred such as failure to turn the camera on or uncharged batteries. the six black male students selected included: (a) three students who were regularly positioned by peers as class leaders; (b) two students who were more outspoken during class, one was positioned as a leader by peers and the other was not; and (c) two students who tended to be less vocal during class, one was positioned as a leader by peers and the other was not. of the six students, only one was not an athlete, but all students engaged in extracurricular activities at school. a descriptive summary of the six students was compiled from the first author’s experiential knowledge and relationships with the students, class observations, and informal conversations with the apcm teacher (see table 1). 8 the demographic percentage calculations represent averages calculated using cohort enrollment data over the 4 years of the apcm initiative. 9 there was one exception, ray was not in a year 1 video, but he appeared in two year 2 videos, and one was used for his year 1 assessment. grant et al. identity as participation journal of urban mathematics education vol. 8, no. 2 99 the local university mathematician, mentioned earlier, was task developer and proctor for all of the small-group assessments. as the principal investigator and participant researcher, he participated in curriculum development with other mathematicians from the algebra project, and supported the teacher as a local consultant for the algebra project curriculum during the 4 years she taught the apcm students. in that capacity, he became well known by the students because during their freshman year he regularly visited (2 to 4 days per week) and participated in mathematics instruction in collaboration with the teacher. table 1 student descriptions assessment protocol the small-group assessment protocol had three components: (a) pose the problem, answering only questions related to task clarification; (b) provide no hints or validation during problem solving; and (c) encourage students to rely on peers for support. the mathematician, in most every instance, faithfully executed this namea leaderb achievementc summary of in-class persona hal no moderate a gregarious personality, a collaborative, confident, and enthusiastic mathematics engager; he identified passing the state test for mathematics as impactful neo yes high a hard worker, soft spoken student leader, logical defender of mathematical ideas, ready mathematics participant and collaborator; teacher calls him dependable ray no low a hard and persistent worker who puts forth great effort, encourages peer participation and focus; vocal in class, may have undiagnosed learning disability reg yes high confident, loyal, and success oriented, supportive of peers, focused and driven, and non-judgmental; recognized student leader rex no low a hard worker, willing to work with others, ready participant, may have undiagnosed learning disability ted yes high class leader who led by example, willing to work with anyone, ready participant; passing the state test for mathematics was impactful a all names are pseudonyms. b leader as positioned by peers. c estimate of achievement as measured by state test scores and school grades. grant et al. identity as participation journal of urban mathematics education vol. 8, no. 2 100 protocol over the 4 years, as evidenced by the video recordings. the purpose of the group assessments was to gain insights about: (a) the algebra project curriculum effectiveness in relation to students’ mathematics understanding, 10 and (b) the students’ sociocultural development for mathematics learning. we assert that the small-group, problem-solving assessment protocol and the established relationship between the mathematician and the students afforded a narrow and fertile context for addressing the research questions. over the years, the proctor and students built an amicable and trusting relationship, based on observations by the first author. gillen (2014), a long-time veteran teacher of the algebra project and social justice advocate, found from his extensive experience that an environment where students have sufficient opportunity to engage mathematically within a receptive climate affords freedom for them to engage through a myriad of roles. all of the assessments were proctored using the same protocol and students were free to choose to work individually or collaboratively with no negative consequences. because of these factors and the nature of an algebra project classroom as described by gillen, we posit that the small-group assessment context minimized inherent power dynamics that exist in typical learning environments. the environment afforded students opportunities to engage with limited or no barriers, and therefore afforded an unobstructed view of these students’ mathematics identity as it emerged and evolved. data collection data collected for this study included six different tasks, captured using 15 video recordings of small-group assessments; the average length of the recordings was about 30 minutes (in total, approximately 450 minutes). the assessment proctor also created analytic field notes and collected document artifacts for each of the problems. the video recordings were captured using a stationary camera or nonprofessional videographer. the camera was placed or held near the small groups in order to record the participants’ discourses, interactions, and body language. the local university mathematician proctored all of the assessments and generated analytic field notes about the group members and their work; these notes were used to add clarity to the video recordings. for example, if a student had a misconception about the mathematics, the proctor’s analytic notes may have described the misconception; or if a student created a drawing and they were pointing out a particular 10 all of the problem-solving tasks, after the first problem, where designed using a context different from that used for instruction. this design component is significant because the algebra project curriculum situates mathematics learning within student shared experiences. the types of tasks developed for the small-group assessments were not unique and could be characterized as worthwhile tasks (stein et al., 2000). grant et al. identity as participation journal of urban mathematics education vol. 8, no. 2 101 aspect of it while talking, the proctor may have kept a copy of the drawing with his notes. the apcm curriculum addresses mathematics topics from the high school content standards outlined by the common core state standards for mathematical practice (ccss, 2010). the group assessment tasks used for this investigation targeted mathematics content from several ccss high school content standards, including algebra, functions, and geometry. for example, one of the year 1 tasks focused on algebraic reasoning, a subset of the ccss algebra standard. reg, rex, and ted worked on that task; reg and ted were in the same group for their year 1 assessment, they have the same group #, while rex worked on the same problem, but in a different group (see table 2). table 2 mathematics topics and student groupings by year reg rex ted hal neo ray year 1 math topics algebraic reasoning algebraic reasoning algebraic reasoning linear functions (continuous) linear functions (continuous) linear functions (piecewise) group # 1 2 1 3 3 4a year 2 math topics linear functions (piecewise) linear functions (piecewise) linear functions (piecewise) geometric construction (with paper) geometric construction (with paper) geometric construction (with paper) group # 5 6 4 7 8 7 year 3 math topics area of composite shapes area of composite shapes area of composite shapes area of composite shapes area of composite shapes area of composite shapes group # 9 10 10 11 11 9 year 4 math topics direct variation (velocity comparison) direct variation (velocity comparison) direct variation (velocity comparison) direct variation (velocity comparison) direct variation (velocity comparison) direct variation (velocity comparison) group # 12 13 14 15 15 15 a ray’s year 1 task actually occurred during year 2. task selection for each year was done in a way to align the mathematics topics of his peers. video recording data analysis the analysis of video data was done using qualitative software (nvivo version 10) that allowed multiple researchers to analyze the same data, which simplified comparative analysis of coding, and diminished the amount of transcription required. the protocol used for video analysis was as follows: (a) segment each video recording into time segments of about 1 to 2 minutes; (b) watch each segment, and assign descriptive themes (i.e., nodes) that captured observed phenomena, paying specific attention to each target student; and (c) for recordings with two grant et al. identity as participation journal of urban mathematics education vol. 8, no. 2 102 or more target students in the same small group, recordings were watched multiple times, at least once for each target student (counting re-watched video recordings, nearly 700 minutes of recordings were analyzed). the group assessment video recordings were coded and analyzed for recurrent themes, allowing inferences to be made and supported (or disputed) via the data—warranted claims were made (wolcott, 2001). the analysis process was multilevel. the first-level analyses assigned thematic categories (i.e., nodes) to recording segments. the second-level used descriptive statistics and graphical representations of coded node frequencies to search for patterns. heavily coded nodes (i.e., those with relatively high frequency counts) and patterns were used to draw inferences related to the research questions. then the video data corpus was searched to find representative examples, evidence that confirmed or disputed inferences—the thirdlevel analyses. the evidentiary data served to warrant the inferences from which claims were made (erickson, 1986). a second researcher, a doctoral student whose worldview and biases differ from the first author’s, was recruited to work collaboratively with the first author to improve the validity of findings while increasing the efficiency of the video analyses—a somewhat forth-level of analyses. using multiple researchers to analyze the video data increased the trustworthiness of the analyses, which strengthens the validity of the findings (lather, 1986). the first author used the video analysis protocol to code the first 2 years of video recordings after creating an initial codebook that defined thematic nodes based on the grounding literature. the nodes in the codebook were organized hierarchically beneath the primary constructs: agency, accountable for what, and accountable to whom. analyzing the video segments led to defining emergent nodes during the analysis process to capture unanticipated phenomena observed that had not been included in the initial codebook; an approach described by schwandt (1994) as an emic perspective. additionally, mathematical work practices as an organizing category was added late in the analysis process and after revisiting the literature. the emergence of this category is described later in the results section, as it was not part of the original analysis plan. late during the analyses, we reorganized the a priori categories because of patterns in the data, which lather (1986) described as face validity, which also strengthens qualitative research findings. the videos were watched several times by one or two researchers to establish and maintain an acceptable inter-rater reliability standard (landis & koch, 1977) with the kappa coefficient > .70 and percent coding agreement > 85% throughout the coding process. to that end, the two researchers’ coded one video recording from year 4; comparison analyses were run that showed coding did not meet the pre-established standard. the researchers met to review the coding and collaboratively re-coded and refined the codebook definitions; refining the codebook during the analysis process ensured shared understandings for node definitions and consistency of coding between researchers. collaborative coding continued until the grant et al. identity as participation journal of urban mathematics education vol. 8, no. 2 103 two researchers reached coding reliability agreement of > 85%. the two researchers then independently coded videos from years 2 and 3; comparison analyses were run and the inter-rater reliability standard was met. overall, the first author analyzed video recordings for years 1 through 3, and the second researcher analyzed video recordings for year 4. results the purpose of this study was to determine how mathematics identity developed for six black male students who choose to participate in the apcm initiative; students agreed to take two periods of mathematics taught by the same teacher for all 4 years of high school.11 the six black male students participated in the apcm for all 4 years, passed the state’s graduation achievement test, and graduated high school in 4 years. according to exit interview data, these six students’ paths after graduation included college (2-year or 4-year), work, or military service. their exit interviews also revealed that each student claimed mathematical readiness to pursue their planned path for the future. we turn our attention to the results from analyzing the video data. mathematics agency three themes were evident from analyzing the 4 years of small-group, problem-solving assessments: confidence, collaboration, and personal effort. the most heavily coded thematic nodes aggregated over the 4 years from the categories agency and accountable for what, the primary constructs for the analysis, are shown in table 3. interestingly, we noticed that several of the heavily coded thematic nodes might have been categorized as discourse practices. manouchehri and st. john (2006) characterized discourse for mathematics learning as being comprised of both reflection and action for the purposes of gaining understanding of their peers’ perspectives and influencing them. the heavily coded nodes align with these characterizations and purposes: the discourses were reflective and action oriented for the purpose of gaining understanding of peer’s conceptions or garnering peer support. 11 scheduling conflicts for credit attainment for graduating precluded students from taking the double periods of mathematics during their final year of high school. grant et al. identity as participation journal of urban mathematics education vol. 8, no. 2 104 table 3 the predominant codes aggregated over year 1 to year 4 one example of this type of discourse practice occurred as hal worked to understand how to fold a circular piece of paper in a way to construct parallel lines. as hal grappled with the problem, he engaged in a mostly nonverbal way with a peer (student 1, not a study participant) in his group: hal: [explains to the proctor why the lines he has constructed are not parallel. as the two other group members continue to work on folding their papers. he glances at student 1’s folding a couple of times as he continues to contemplate his work.] student 1: [after making several folds and examining his paper closely, he rotates the paper twice to examine the lines] “i don’t know if it’s right” [student giggles.] hal: [reaches over and picks up the paper his peer had been folding for closer examination.] it’s not bad, dog. student 1: [nods in acknowledgement of hal’s praise.] (video recording, year 2) in this interaction, there is little dialogue, but after taking student 1’s paper, hal goes on to explain why he believes student 1’s constructed lines are parallel. this interaction clearly depicts a discourse practice in which hal was reflective— by comparing his approach to student 1’s—and led to action—explaining why student 1’s lines were parallel. the purpose of the discourse was an example of hal attempting to understand student 1’s mathematics. a second example depicts hal’s effort to garner peer support for an idea, using the reflection/action discourse practice. this example occurred in year 4 in a small group comprised of hal, neo, and ray. in this episode, hal and ray are engaged in making sense of the problem involving two moving cars, a and b, travelthematic nodes hal neo ray reg rex ted totals engaged in peer collaboration a 22 23 5 8 10 25 93 engaged in individual process a 5 9 26 19 12 15 86 collaborative sense making 17 16 2 10 3 14 62 explaining ideas 13 12 7 10 7 12 61 listening for understanding 9 13 6 3 7 22 60 sharing ideas with peers 14 13 2 5 6 16 56 consulting expert source 13 7 7 12 6 10 55 listening to peer 11 13 8 3 7 10 52 asking clarifying questions 7 7 6 4 7 13 44 acknowledging contributions by others 7 6 2 5 1 14 35 a work practices for mathematical engagement grant et al. identity as participation journal of urban mathematics education vol. 8, no. 2 105 ing at different rates, and their locations are described in terms of the other car and a referent red line at specific time intervals: hal: there is no doubt, that it [car a] was one meter past the red line and car a was two meteres past the red line, and if they’re both exactly at four meters at three [sec], that means that it [car b] caught up, and passed, not passed it, but it just equaled up with it. so that it’s [car b] clearly faster. they’re going the same speed. neo: car b is at first, like already ahead of car a; and then car a tied it up. and car a is going faster. hal: no, it says [referring to the problem], that it’s [car a] two meters past it [car b] already, so it’s [car a] in front of car b, it’s one meter past the red line. neo: oh, ok. i thought… hal: so, it’s kind of lined up like this [begins drawing and talking softly about his sketch.] (video recording, year 4) in this episode hal again reflects on input received from a peer as he listens to neo’s explanation, but then disputes neo’s position using evidence from the text of the problem, which convinces neo of hal’s position. this reflective process leads to hal taking action: depicting his thinking via a sketch. hal’s purpose in the discourse appears to be garnering peer support: hal is seeking neo’s support before investing in creating a pictorial representation. this action is representative of agency as articulated by bandura (2005); hal’s exercised agency was self-regulated and negotiated within the group’s social system. the two most heavily coded thematic nodes for participation represent two distinct student work practices for mathematics engagement, individual and collaborative (esmonde, 2009). the total coding for these nodes engaged in peer collaboration (93) and engaged in individual process (86) are much greater than the total for the next most heavily coded node, collaborative sense making (62; see table 3). these two most heavily coded nodes were interpreted as students choosing to engage mathematically, as they did not opt to engage in off task behaviors or otherwise not participate. close examination of this analysis led to two things: (a) reorganizing the nodes hierarchically around these two heavily coded nodes; and (b) examining how the students’ participation acts split across the two nodes. the qualitative analyses were done using software (nvivo version 10), which allowed for easily restructuring of nodes at any point within the analyses without disturbing prior analyses. the node restructuring led to additional analyses using this emergent perspective—looking at the students’ participation through the lens of their observed work practices. immediately obvious, we found that some students mostly worked collaboratively (i.e., hal, neo, and ted), while others opted to work independently (i.e., ray and reg), and one student (i.e., rex) worked almost equally across the two work practices (see table 3). grant et al. identity as participation journal of urban mathematics education vol. 8, no. 2 106 when we reorganized the heavily coded thematic nodes by student work practices for engagement, individual and collaborative, some nodes were combined beneath an existing node or a new node was defined. in the end, the most heavily coded nodes were found primarily under the collaborative work practice. node descriptions are provided to make clear the meanings used for the coding process: categorizing what was observed (individual or collaborative) and students’ verbal utterances, actions, and gestures (see table 4). table 4 definitions for heavily coded thematic nodes organized by categories mathematical identity development over 4 years the reorganized codebook provided the foundation for our response to the primary research question: how did the mathematics identity of six black male students participating in the apcm initiative develop over their 4 years of high school? we found the aggregate coding frequencies (i.e., totals for the six students by years) for the heavily coded nodes; doing so afforded an overall perspective of the students’ mathematics identity development across the 4 years. summaries depicting the evolution of the students’ identity are shown using summary line charts, using the following lenses: (a) students exercising individual problem-solving practices, (b) students exercising collaborative problem-solving practices; and (c) for whom students were observed being accountable. individual problem-solving practices. with respect to individual work practices, there were only two heavily coded nodes asking clarifying questions and consulting expert source that emerged from the analyses (see figure 1). these two participation behaviors were observed most during years 2 and 3. interestingly, stuheavily coded thematic nodes definitions of nodes category: individual work practice asking clarifying questions independently initiates questioning, clarifying and probing questions consulting expert source seeks help and/or support from someone perceived as expert category: collaborative work practice acknowledging contributions by others making public recognition of a peer’s mathematical contribution collaborative sense making expressing efforts to understand while engaging with others listening to peers making public the effect of a peer’s verbal expression passive peer interactions nonverbal response to another’s action sharing ideas with peers making public, verbally or by action, ideas, explanations, and artifacts grant et al. identity as participation journal of urban mathematics education vol. 8, no. 2 107 dents consulting a perceived expert source was highest in year 2, and then declined for each year after. however, the students’ questioning increased from year 1 and peaked in year 3. we posit that this summary suggests that these students transitioned from reliance on a knowledgeable other to a greater reliance on self and collaborations among peers. two examples from the asking clarifying questions node from years 2 and 3 are presented to illustrate this finding. in year 2, rex was observed asking more questions than he did in all of the other years combined. in the first example, students are given prices for purchasing jelly bracelets from an online vendor. they are given a variety of information, such as the price for a specific number of bracelets and volume shipping costs. the information, however, is not presented simplistically in a way that suggests a linear relationship. rex asks several questions, such as: “isn’t it [jelly bracelet cost] going up?”; “what did you get?”; and “so, how much is it for one bracelet?” (video recording, year 2) these questions were asked of the group and are often focused on getting to the answer or seeking confirmation. rex’s group members offered little in response to his questions, which led him to pose questions to the proctor about the given information. the proctor acknowledges that there is sufficient information to solve the problem, to which rex replied, “well, i didn’t find it.” (video recording, year 2); a response that suggests that rex is done, unable to solve the problem and has no other options. rex’s questioning was focused narrowly on getting support for finding the right answer to the problem. the second example illustrates a different perspective that shows the evolution of questioning among the students. in year 3, reg asked the most questions. the problem presented was about finding the linear measure, width, on each side of figure 1: summary of heavily coded nodes categorized by individual practice for all students by years. grant et al. identity as participation journal of urban mathematics education vol. 8, no. 2 108 a rug centered in a room, given the room dimensions and the area of the rug. in addition to asking fewer questions seeking support or confirmation, reg poses questions to the proctor: “has any group solved this problem, yet?”; “have we learned what we need to know in order to solve this question?”; and “are we over thinking this?” (video recording, year 3) reg’s questions appear to be seeking understanding about his preparedness to solve the problem. these questions are not searching for hints or support, but rather validation that he possesses all that is needed for success. what underlies these questions is an inherent trust that exists between the student and proctor given the student’s willingness to ask such questions and then to accept the response without question, even when he was faced with uncertainty about his solution. reg persevered in problem solving after this exchange. collaborative problem-solving practices. there were four heavily coded nodes for student collaborative work practices that emerged from the analyses (see figure 2), three of which are aligned with discourse practices (e.g., manouchehri & st. john, 2006): collaborative sense making, listening to peers, and sharing ideas. the shape of the lines that show a summary of coding for these three nodes are similar in shape, the lines are relatively flat between years 1 and 2, peak in year 3, and then fall in year 4. these trends suggest that student identity development related to discourse practices followed an increasing trajectory and peaked in year 3; however, we hesitate to consider year 4 because of previously stated reasons and the lack of observed participation in year 4. ted was observed as the most collaborative participant as measured by observed behaviors in this study. therefore, we selected examples from video segments featuring ted to show a progression over the years as an illustrative case for figure 2: summary of heavily coded nodes categorized by collaborative practice for all students by years. grant et al. identity as participation journal of urban mathematics education vol. 8, no. 2 109 all students (see table 5). the year 1 discourse practice is, for the most part, not collaborative, even though students are talking to one another. in year 2, ted gives up on the problem until he is lured back by a question posed by the proctor to ray. in year 3, ted listens to peers and without invitation he alerts them of an error in their mathematics strategy for solving the problem. while he does not know the correct solution at the time, he was sufficiently engaged to recognize the potential pitfall and share his perspective with peers to redirect their trajectory. each year, ted’s level of mathematical engagement and discourse with peers seemed to gain in complexity in the sense that year 1 was more collective than collaborative—kids voicing ideas but not using them to improve mathematical understanding—and by year 3 discourses were unsolicited peer supportive for mathematics learning. while year 2 was clearly between the two extremes, greater perseverance emerged and participation continued after claiming, “done.” thus, over the years, the discourse frequency and complexity increased. table 5 ted’s progression of discourse practice: an illustrative example of one aspect of mathematics identity development year #: problem/context description of discourse year 1: a trip is represented using a linear model ted initiates conversation by sharing his idea; reg responds, makes no comment about ted’s idea, and shares his approach. ted listens to reg explain his idea more than once with little insight to further his solution; reg disengages, “can i work on my own?” year 2: find the total cost to purchase and ship jelly bracelets, given complex pricing information ted declares himself done; the proctor poses a question to ray, “why are the price differences the same and then different?” ted did not appear to be listening, but in response asks, “which one?” ted reengages with the problem. year 3: find the width around a rug centered in a room, given room dimensions and area of the rug ted watches as rex and the other group member discuss an idea; ted recognizes an error in their logic. ted takes rex’s paper and by drawing on his paper, shows the group members the width they seek. the growing discourse practice may also explain the continual rise in students acknowledging contributions by others, the fourth heavily coded node, which should not be overlooked (see figure 2). by far, ted was the most observed engaging in this behavior, with the greatest number of instances observed in year 4, making this node not as representative of the group, but ted, the individual. accountability. the aggregated coding of nodes from the thematic category accountable to whom is shown in table 6. these nodes were coded less heavily in comparison to those coded for the participation nodes because interpreting whom one is accountable is not always transparent to an observer. nonetheless, there were grant et al. identity as participation journal of urban mathematics education vol. 8, no. 2 110 patterns; notice that the students were observed as most accountable to peers when looking at aggregate totals across the years. however, the aggregate totals for observed accountability to expert and self are almost evenly split. interestingly, hal, neo, rex, and ted were most accountable to peers, which is supported through analysis results that hal, neo, and ted were observed being the most collaborative among the students. according to the apcm teacher, reg mentored ray and ray modeled himself after reg (informal communication, year 4), and interestingly, they were the only two among the sample with the least amount of coding for accountable to peers and with the most coding for accountable to expert and self. the descriptions used for this study for accountability to whom from the researchers’ perspective follow. accountability to expert most often suggests that the learner is not sufficiently empowered and lacks autonomy for mathematics or mathematical understanding. there were many instances of this across the years where students requested validation from the proctor and their peers by asking questions such as: “so it took her 10 seconds to get to the end of the field?” (hal, video recording, year 1); “what’s up with this, [proctor]?” (reg, video recording, year 2); “with your math knowledge is that correct?” (hal, video recording, year 4) table 6 aggregated coding of nodes from one category showing analyses of smallgroup, problem-solving assessment videos over 4 years accountable to whom hal neo ray reg rex ted total expert: seeks other to model or guide knowledge construction and validate 8 6 7 15 4 4 44 peers: constructs understanding and validity collaboratively 17 22 2 5 7 18 71 self: confident and autonomous constructs and validates independently 5 8 8 11 4 7 43 accountability to self, means a level of learner confidence and belief with sufficient mathematical autonomy that he readily shares ideas with others so they may value, critique, or dispute them. from this stance, the proctor may be considered a peer at times; the proctor was positioned to share facts and not give hints or validation. the proctor was observed not exercising his authority within the groups’ power hierarchy. accountability to peers is between accountability to expert or self if one viewed the three along a continuum. from a student perspective within a social system of a small group, being accountable to peers suggests sufficient confidence for overcoming the risk of being grant et al. identity as participation journal of urban mathematics education vol. 8, no. 2 111 wrong. the difference between peers and self is that for those who are accountable to peers, they are sufficiently free to rely upon others for support and adjust their ideas based on collaborations. on the other hand, those accountable to self may be more autonomous yet less open to “hear” critiques from others, which may be cognitively limiting, based on the situation, of course. from the summary line chart shown in figure 3 we can see that observed instances of reliance on peers peaked in year 3 and was maintained in year 4. observed instances of reliance on self grew steadily from year 2 through year 4 and over the same period observed instances of reliance on expert decreased steadily. these observations taken with the accountable to whom perspectives, we conclude that students made a shift from relying on knowledgeable others to relying on themselves and/or peers and they were sufficiently confident to risk being wrong, yet free enough to be influenced by collaborations. summary of findings: mathematics identity development the first point to make is that the apcm initiative created a receptive climate and we posit the freedom and nurturing was fertile ground for the six black male students’ mathematics identity development during high school. the students expressed, their teacher described, and the researchers observed students’ mathematics confidence. one student described his mathematics confidence as, “i can do anything that i put my mind to” (hal, interview, year 4). this simple statement is emblematic of the way these students saw themselves mathematically (i.e., their mathematics identity), and it suggests that their confidence was connected to personal effort. what is not captured by this particular statement is the value the stufigure 3: summary of heavily coded nodes categorized by accountable to whom for all students by years. grant et al. identity as participation journal of urban mathematics education vol. 8, no. 2 112 dents placed on their peers and collaborations for learning mathematics, an aspect that was evident across the observations. another key aspect of these students mathematical identity was their intentional choice to engage mathematically. the students opted to engage in individual and/or collaborative practices; however, even for those who favored individual practices they were observed in discourse through asking questions, which often led to verbal discourses with others. overall, we observed students engaged in rich discourse practices that were reflective and action oriented for the purpose of gaining understanding or garnering peer support. the evolution of these six black male students included transitioning from reliance on knowledgeable others to reliance on self and peer collaborations or mathematical participation, which reified the observed increase in the number of discourses and their complexity within their smallgroup, problem-solving assessments. therefore, it was not surprising to realize that confidence appears to be the foundation for our students’ mathematics identity development when viewed through the lens of mathematical agency, a relationship established by bandura (2002). discussion and conclusion research on reform-based mathematics calls for classrooms environments where mathematical autonomy and freedom abound and are available for all learners (carpenter, fennema, franke, levi, & empson, 2000; carpenter & romberg, 2004; hiebert et al., 1997; west & staub, 2003). such autonomy and freedom, however, cannot be taken for granted, especially from those students (urban and rural) who are underserved (gillen, 2014). we agree that pedagogical content knowledge and mathematical knowledge for teaching (e.g., ball & bass, 2003; ball, hill, & bass, 2005; hiebert et al., 1997; hufferd-ackles, fuson, & sherin, 2004) are necessary conditions in creating effective learning environments; however, these knowledges are not sufficient conditions for creating equitable learning environments for all children (e.g., martin & herrera, 2007; nctm 2000, 2014). although nctm (2000) established the equity principle long ago, equity continues to elude many mathematics classrooms, especially those with large numbers of racial and ethnic minority students (berry, 2008; martin, 2008). further research is needed to understand the full impact of practices and policies on student mathematics identity development, and to articulate specific reforms needed to free our children and our classrooms from those practices and policies that inhibit mathematics literacy, leadership, and freedom (hope et al., 2015). we must serve underserved students differently so that they are afforded opportunities to choose mathematics literacy. as the research reported here demonstrates, historically underserved students develop different mathematics identities when provided access to classroom environments that are not reliant on traditional remediation ap grant et al. identity as participation journal of urban mathematics education vol. 8, no. 2 113 proaches. we know that mathematics efficacy and confidence are essential dispositions for exercising mathematics agency (bandura, 2002), which supports our finding that the students’ confidence played a role in the ways they engaged in the small-group problem solving. we also posit that the receptive climate and freedom in the groups contributed to the ways their participation manifested in relation to their mathematics identities. these findings were not unanticipated. in another study about apcm students, grant (2014) examined students’ verbal and written reflections about the ways they interact with peers while learning mathematics. she found that apcm students from two cohorts, one urban and the other rural, described productive classroom culture for mathematics learning as students getting along with peers, working hard, and supporting one another. in the end, the study reported illustrates the development of mathematics identity as participation of six black male students; the findings add to the literature extolling the virtues of black learners. the documented participatory freedom exercised by the students may be useful for those looking for new approaches in transforming the culture of participation and agency in mathematics classrooms. the study, however, is limited in the sense that it looked closely at only six black male students in one mathematics teacher’s classroom, over a 4-year time period. moreover, the structure of the apcm initiative—students studying with one teacher for all 4 years—worked for the students and teacher reported here. it is important to note, however, that there were instances with other apcm cohorts where that was not the case. some teacher–student relationships were not synergistic, and did not promote effective mathematics learning. nevertheless, one implication of this study is that the apcm initiative provides guidance for those interested in creating equitable and receptive environments for underserved students generally, and for black male students specifically. policy makers and other stakeholders have claimed interests aligned with equity as evidenced by names such as “no child left behind” and “race to the top.” these and other initiatives funded through public and private organizations are all innocently labeled as accountability measures. these labels, however, are misleading and have been far reaching with many unintended negative consequences for u.s. schools and mathematics classrooms. these mandates manifest in education systems as hierarchies where teachers and students are at the bottom, with little or no choice or autonomy (gillen, 2014). gillen and others (e.g., leonard & martin, 2013) argue, and we concur, that students need different learning environments and opportunities if they are to develop the types of positive mathematics identities described here. acknowledgments a special thank you is extended to the apcm cohort students, mrs. amanda tridico clawson, the apcm teacher, and dr. lee mcewan, the mathematician and proctor for the group assessments, grant et al. identity as participation journal of urban mathematics education vol. 8, no. 2 114 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(2012). norming suburban: how teachers talk about race without using race words. urban education, 47(5), 983–1004. west, l., & staub, f. c. (2003). content-focused coaching: transforming mathematics lessons. portsmouth, nh: heinemann. wolcott, h. f. (2001). writing up qualitative research. thousand oaks, ca: sage. yackel, e., & cobb, p. (1996). sociomathematical norms, argumentation, and autonomy in mathematics. journal for research in mathematics education, 27(4), 458–477. http://ed-osprey.gsu.edu/ojs/index.php/jume/article/download/45/68 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/download/45/68 journal of urban mathematics education december 2017, vol. 10, no. 2, pp. 8–24 ©jume. http://education.gsu.edu/jume rochelle gutiérrez is professor of mathematics education in the department of curriculum and instruction and latina/latino studies at the university of illinois at urbana-champaign, 1310 south sixth street, champaign, il 61820; email: rg1@illinois.edu. her research interrogates the roles that race, class, language, and gender play in teaching and learning mathematics, creative insubordination in teaching, and new possible relationships between living beings, mathematics, and the planet. commentary why mathematics (education) was late to the backlash party: the need for a revolution rochelle gutiérrez university of illinois at urbana-champaign our lives begin to end the day we become silent about things that matter. – martin luther king, jr. hen our pedagogy or scholarship involves challenging the status quo, especially on behalf of students who are indigenous,1 latinx,2 and black,3 some people will go to extreme measures to silence us. those extreme measures became apparent this year when i came under attack from the alt-right as a result of a “news” story that campus reform and fox news produced.4 in a chapter of a book on scholarly practices in mathematics methods courses, i wrote about the relationship between whiteness and mathematics (gutiérrez, 2017b). my argument had two key points: (a) mathematics operates as whiteness when we do not acknowledge the contributions of all cultures, and (b) mathematics operates as whiteness when it is used as a standard by which we judge others. the message that made headlines, 1 i use indigenous and aboriginal interchangeably. u.s. authors tend to use the term indigenous or native, whereas authors from canada, australia, and new zealand tend to use the term aboriginal. in canada, aboriginal includes first nations, métis, and inuit peoples. 2 i use the term latinx (as opposed to latino, latina/o, or latin@) as a sign of solidarity with people who identify as lesbian, gay, bisexual, transgender, queer, questioning, intersexual, asexual, and twospirit (lgbtqia2s). latinx represents both a decentering of the patriarchal nature of the spanish language whereby groups of men and women are normally referred to with the “o” (male) ending as well as a rejection of the gender binary and an acceptance of gender fluidity. 3 i use the term black, as opposed to african american, to highlight the fact that many black students living in the united states have ancestry in the caribbean, south america, and asia, among other places. nonetheless, black students who attend schools and live in the united states are often racialized in similar ways, regardless of country of origin. 4 rather than giving traffic and further legitimacy to conservative websites, you can read a mirrored page of these stories here https://equitymathed.wordpress.com/. w http://education.gsu.edu/jume mailto:rg1@illinois.edu https://equitymathed.wordpress.com/ gutiérrez commentary journal of urban mathematics education vol. 10, no. 2 9 however, was simply “mathematics operates as whiteness,” and it got morphed into “white privilege is bolstered by teaching mathematics,” which people took to mean that i was claiming either that only whites5 could do mathematics or that we should stop teaching mathematics altogether because it was racist. within the first few days of the campus reform and fox news stories, i was inundated with hundreds of hate-filled email and voicemail messages; trolls invaded my twitter space; a facebook page about mathematics education was closed for comments after being flooded with vitriol; individuals wrote to my university arguing that i needed to be fired; alt-right groups produced podcasts and additional stories that slandered me and my work; and turning point usa6 (a conservative group known for their bigotry and witch hunting of left-leaning professors on college campuses) placed me on the front page of their professor watch list. my college and campus also received a flood of harassing emails and calls. some of the email messages sent to me were carbon copied to random students and faculty, thereby inflicting violence upon them, for no apparent reason other than to instill fear or have them question my scholarship. these attempts to censor faculty through swarm attacks is becoming more common in this era of “coercive efforts to control political debate” (wu, 2017); where there is increased incivility in public spaces (kamenetz, 2017); and where professors are unfairly blamed for indoctrinating students into more leftist views (jaschik, 2014). yet, a national attack on mathematics education scholars is new. in some ways, this kind of national attack was surprising. over the past 17 years, i had produced a number of other writings on mathematics that touched on issues of power in general and white supremacist capitalist patriarchy, in particular (see, e.g., gutiérrez, 2000, 2002, 2009, 2010/2013,7 2012a, 2012b, 2013, 2015, 2016, 2017a, 2017b). drawing on the work of bell hooks (2004), i use the term white supremacist capitalist patriarchy to highlight the interlocking systems of oppressions at work in society. in fact, i am not alone in making these connections to mathematics. there is a robust domain of scholarship dedicated to chronicling the relationship between mathematics and power/domination in society stemming back more than 50 years (see, e.g., bishop, 1990; burton, 1994; frankenstein, 1989, 1990; knijnik, 2007, 2011; mellin-olsen, 1987; o’neil, 2016; popkewitz, 2002; 5 similar to the term black, i capitalize the term white throughout this commentary to highlight the racial category that has been constructed in society. 6 for a better understanding of turning point usa, see bascunan simone (2017). 7 i cite this article as 2010/2013 because it was published online through journal for research in mathematics education (jrme) in 2010 and some researchers began citing it as such then. it was not released in print until 2013, and some researchers have cited it as such since. because the focus of the article is on a particular point in history, the work should reflect the earlier date. gutiérrez commentary journal of urban mathematics education vol. 10, no. 2 10 skovsmose & valero, 2002). moreover, a growing number of scholars have written eloquently about the connections between whiteness or white supremacy and mathematics education (see, e.g., battey, 2013; battey & leyva, 2016; bullock, 2017; joseph, haynes, & cobb, 2015; leyva, 2017; martin, 2013, 2015; stinson, 2013; warburton, 2015, 2017). the kind of attack that is launched through social media today, partly because it can be done anonymously (with stolen ip addresses or anonymous accounts), is dehumanizing and is launched against the person as opposed to the ideas or argument that a scholar has conveyed. few of the negative responses i received through email, voicemail, or twitter actually attended to my argument or mathematics education. instead, the messages tended to convey white pride, pro-trump, or antilatinx sentiments, such as “you are the perfect argument for why we need to build the wall!” or “leave our country now!” or “i’m white privilege. i’ll be here anytime you need to blame someone for your poor life choices.” elsewhere, i analyzed how the attack related to mathematics teacher educators (gutiérrez, in press-b). here, i focus more exclusively on possible reasons for a delay in attacking mathematics education and what that may mean for our work as mathematicians and mathematics education scholars. first, i need to put this attack into a broader context. only a few months earlier, alt-right groups also targeted dr. luis leyva (a latinx scholar) after campus reform and breitbart published stories about his research. his work highlighted the intersecting nature of whiteness and gender as contributing to racialized and gendered spaces that produce inequities in mathematics education (battey & leyva, 2016; leyva, 2017). following these stories, he received online harassment via email, blog post comments, and tweets. sadly, i (and many others) had not heard of that attack until i was in the thick of mine. around the same time, campus reform had produced another story, singling out two national professional organizations, the national council of supervisors of mathematics (ncsm) and todos mathematics for all (todos) for their 2016 joint position statement on social justice in mathematics education. moving beyond typical calls for equity, nscm and todos had come out strong, suggesting that mathematics teachers needed to take on an advocacy stance that “interrogates and challenges the roles power, privilege, and oppression play in the current unjust system of mathematics education—and in society as a whole” (para. 1). shortly thereafter, campus reform produced another story, targeting texas state university for posting a job announcement for a mathematics education professor with preference given to a candidate who addresses social justice issues.8 although it is standard practice for job announcements to indicate they welcome candidates from un 8 see https://equitymathed.wordpress.com/2017/10/29/history-of-istandwithrochelle/ for a mirrored webpage of this story. https://equitymathed.wordpress.com/2017/10/29/history-of-istandwithrochelle/ gutiérrez commentary journal of urban mathematics education vol. 10, no. 2 11 derrepresented groups, the story made sure to highlight this phrase, implying “those” people will bring their social justice agendas with them. most recently, campus reform produced a story on brian lawler, andrea mccloskey, and theodore chao who jointly wrote a chapter for the same mathematics methods book where my chapter appeared. although these groups suffered no apparent backlash as a result of the stories, the trend is clear that mathematics education is now on the radar of alt-right groups. earlier this year, dr. piper harron, a mathematician who is black and studies number theory, suggested in her blog9 that white cis10 men consider leaving the profession to make room for more women and people of color in mathematics (harron, 2017). campus reform produced an article11 that launched a national attack against her, including additional stories by breitbart and other right wing media and web outlets. she was heavily trolled and made to feel unsafe on and off campus. these are some of the recent cases of public scholars under attack within mathematics and mathematics education. they raise issues for why our field is coming under attack now, and why not before? from tinkering to overhauling much of what currently counts as scholarship in mathematics education assumes we will work within the given system or expand what we currently count as the status quo. within mathematics education, we have convinced ourselves that “equity” is a strong enough agenda when maybe revolution should be the goal. elsewhere (gutiérrez, in press-a), i have argued that equity harms our cause in mathematics education for a number of reasons, including: (a) it assumes we all mean the same thing and does not promote dialogue or creativity; (b) it only guarantees we are addressing equity when we are far from reaching the goal; (c) it tends to privilege images of teaching and learning that are universal, or that only attend to issues of access, achievement, or a bit of identity but not power in any serious way; and (d) because the term is bogged down in history. dr. harron points out the irony in trying to tinker with a system in mathematics that needs overhauling. she states: 9 see https://blogs.ams.org/inclusionexclusion/2017/05/11/get-out-the-way/. 10 cis gender means when an individual’s gender identity matches the one they were assigned at birth. 11 see https://equitymathed.wordpress.com/2017/10/29/history-of-istandwithrochelle/ for a mirrored webpage of this story. https://blogs.ams.org/inclusionexclusion/2017/05/11/get-out-the-way/ https://equitymathed.wordpress.com/2017/10/29/history-of-istandwithrochelle/ gutiérrez commentary journal of urban mathematics education vol. 10, no. 2 12 most of us do not have good role models for what a feminist math department would look like. i have this talk that i give and afterwards, i will often get concerned white men asking me what they can do to fight sexism. but they’re not really thinking about ending sexism. they’re thinking about progress. they want to know which benefits the cis male hoarders-of-power can offer to women so that we don’t feel so bad and complain so much and contribute to such dismal numbers. this is natural, reasonable even, but sexist all the same. (blog post may 11, 2017) similarly, we do not have good models for what a feminist, pro-black/indigenous/latinx, socialist mathematics education would look like, or if even such a thing could exist. what would a mathematics education program that took decolonization seriously look like? when i say decolonization, i do not mean something tacked on to anti-racist approaches or an approximation of other oppressions (tuck & yang, 2012). i mean a program that takes seriously land, sovereignty, and the history of erasure of people through culture and language. i acknowledge the ways in which mathematics teaching and learning contributes to the denial of language and history for indigenous students primarily. how do we create a system that does not just give some of the breadcrumbs to others but emancipates people from colonialism? first, we must begin by acknowledging settler colonialism and ask whose history and whose language is part of mathematics? we cannot claim as our goal to decolonize mathematics for students who are black, latinx, and aboriginal while also seeking to measure their “achievement” with the very tools that colonized them in the first place. when we consider the relationship of power to mathematics, we cannot be content with notions of power that are limited to solving difficult problems in mathematics classrooms. we must be open to deconstructing power dynamics, challenging authority, restoring peace and dignity, repairing settler colonialism, and positing new questions that need to be asked. dismantling white supremacist capitalist patriarchy is one step in the right direction and, perhaps, one reason we have caught the attention of the alt-right who seek to protect the status quo. mathematics education as a broad field has tended to be complacent with tinkering toward utopia. documents such as the curriculum and evaluation standards for school mathematics (national council of teacher of mathematics [nctm], 1989), professional standards for teaching mathematics (nctm, 1991), assessment standards for teaching mathematics (nctm, 1995), principles and standards for school mathematics (nctm, 2000), and principles to actions: ensuring mathematical success for all (nctm, 2014) have tended to produce the same kinds of recommendations—grade level scope and sequences, processes important for learning, and attention to equity. but there is little in these documents that acknowledges the role of mathematics education as a field that creates the very inequities it seeks to address (martin, 2015). the standards for preparing teachers of mathematics, produced by the association of mathematics teacher educators (amte, 2017), however, take a more gutiérrez commentary journal of urban mathematics education vol. 10, no. 2 13 directed approach in addressing issues of identity and power. similar to the aforementioned ncsm/todos position statement on social justice, amte states: well-prepared beginning teachers of mathematics understand the roles of power, privilege, and oppression in the history of mathematics education and are equipped to question existing educational systems that produce inequitable learning experiences and outcomes for students. … they are prepared to ask questions as needed to understand current policies and practices and to raise awareness of potentially inequitable practices. these practices are particularly important related to students who are black, latinx, american indian, emergent multilingual, or students living in poverty. (p. 23) even so, the mathematical content and much of what counts as pedagogy within the amte standards continues to align with the status quo. there is acknowledgement that the system has failed latinx, indigenous, and black students in their mathematical learning and, therefore, requires attention to such things as the history of mathematics. however, the standards are not written in a way that puts those students first, something that would require overhauling the system. in some ways, we were surprised by the attack on me, and others, in mathematics education because we had not embraced the fact that our work could have that upsetting effect. perhaps we believed, like others, that our scholarship was neutral or absent from politics. many of us have certainly been conditioned to think that way. scientists and engineers for social and political action (sespa) have highlighted how one problem with adequately understanding and addressing social and political issues through science is the way scientists are trained: consider first the training of the scientist. in the classroom and laboratory the myth of an apolitical, benevolent science prevails. graduate school, and often undergraduate education, involves a near total submersion of the student in technical material with little if any historical or philosophical perspective. research productivity is the measure of worth as the student acquires skill in a specialized field. technical questions are isolated from their social and economic context (e.g., the use of science) except for perhaps consideration of the prestige and financial status of the researcher. thus the end product of this training is a narrow specialist—one taught to perform scientific miracles without considering their political implications—a reliable tool of the power structure. another aspect of this training is an ingrained sense of elitism. courses are designed to select and separate out potential scientists from their fellow students. those who succeed are led to view themselves as members of an elite intellectual class. (boston sespa teaching group, 1974, p. 7) the group is describing the training of a scientist over 50 years ago, but the description easily fits the mathematician and many mathematics education professors today. few are ever required to think deeply about the ethical implications of their work when funded by the department of defense, silicon valley (garcía martínez, 2016), other industries (o’neil, 2016), or even universities (hersh, 2014). gutiérrez commentary journal of urban mathematics education vol. 10, no. 2 14 this idea of mathematics separate from politics is perpetuated even in these attacks. mathematics is the last place one would expect someone to claim that it relates to white supremacy. the white genocide project suggests the ridiculousness of the claim simply because it involves mathematics, a discipline that could not reasonably have anything to do with whiteness: “now, even mathematics is being thrown to the floor and buggered by the anti-white louts who’ve taken over the universities.” in a similar vein, the american thinker blog adds, “it’s finally happened: professor claims math perpetuates ‘white privilege!’” by claiming, “even mathematics is being thrown to the floor” and “it’s finally happened,” these alt-right groups are relying upon a view shared by many that mathematics is objective, pure, and culture-free (burton, 1994). by focusing on mathematics, they are using our work to prove that anti-white sentiments must abound in universities if comments like these can arise in disciplines that have nothing to do with power. as the title of this commentary infers, i argue that mathematics was late to the backlash party because we have not been making institutional or structural changes that would impact society; our agendas have been aligned too closely with the status quo, with a system of education that is not meant to support students who are black, indigenous, or latinx (especially womyn and people who identify as queer). as such, we are not seen as a threat. what if we began with the premise that our work would call for such radical and systemic changes that it would threaten those who currently benefit from the system? rather than being surprised, we would anticipate the backlash. but history shows us that we cannot look within mathematics or mathematics education to imagine such radical changes; we need to look elsewhere for those models. looking outside of ourselves when we look elsewhere, we can reimagine a new way of doing things, based on different values/principles, with different goals, and different systems for considering how well things are working (e.g., how well our field and its related practices reflect the principles upon which we say they rest). rather than beginning with the principles of logic, proof, and universality as indicative of mathematics, we might look to see ourselves in others and others in us, providing windows and mirrors, reflecting reciprocity, uncertainty, and pattern as problem solving and joy, something i have referred to as mathematx (gutiérrez, 2017a). i have suggested that we no longer hold humans as the center, that we challenge where learning should take place and with what guiding principles, and that we rethink whom our new teachers might be. this rethinking is not simply a call for a broadened view of ethnomathematics, but a radical shift in how we do mathematics, for what purpose, and how that influences us and our relationships with others in this universe. this gutiérrez commentary journal of urban mathematics education vol. 10, no. 2 15 rethinking is not simply a challenging of dehumanizing practices in a system that already exists but a re-envisioning of our discipline. when we look to other disciplines, we see that their goals are not simply to have historically oppressed people viewed as legitimate participants in the discipline as defined. they are seeking to radically change the discipline itself, partly through putting the needs, views, and contributions of historically oppressed people first. that is, in contrast to “equity” or “social justice” language within mathematics education, other disciplines have been more forthright in addressing identity and power head on. for decades, ethnic studies programs, departments of english, and even literacy professionals in k–12 schools have been: questioning what counts as the “canon;” expecting to see different kinds of authors/perspectives on the discipline; choosing to teach “controversial” texts; recognizing there is no one “truth” but only interpretations; and rethinking literacies and knowledge. perhaps as a result, there is greater infrastructure for such disciplines to deal with backlash. see, for example, the national council of teachers of english’s intellectual freedom center that supports english teachers under attack for the texts they choose to use.12 in doing this work, ethnic studies programs have experienced a long history of backlash from those who seek to protect the status quo, including from administrators at the very universities that house them (chang, 1999) or from conservative politicians who want to control the k–12 curriculum (calacal, 2017). opponents have presented la raza/mexican american studies (mas) programs as “courses that promote the overthrow of the united states government” or that “promote resentment toward a particular race or class of people” and as being taught by “radical instructors teaching students to be disruptive” (calacal, 2017). what arises from these debates is the political clarity of the teachers. proponents of mas programs have rejected current versions of culturally relevant curriculum because it lacks the critical focus on race theory and pedagogy that were at the heart of the 1960s protests to create such programs. we can benefit from understanding how other disciplines (e.g., ethnic studies) have used their backlash to their benefit. after house bill 2281 was passed in 2010 to ban the mexican american studies course in tucson, arizona, teachers pushed back and led the movement to strengthen the program, including librotraficantes— people who smuggled banned books into mobile underground libraries where students would have access—and to expand the courses nationally (phippen, 2015). students supportive of mas chained themselves to each other and desks in the district boardroom in a similar manner to the east la 13 high school students who organized thousands of chicanos to walkout of their schools in 1968 (smith, 2013). in addition to filing a lawsuit against the state board, some teachers began holding 12 see http://www2.ncte.org/resources/ncte-intellectual-freedom-center/. http://www2.ncte.org/resources/ncte-intellectual-freedom-center/ gutiérrez commentary journal of urban mathematics education vol. 10, no. 2 16 sunday gatherings to teach the courses through a liberal arts college where students would receive college credit. meanwhile, students used social media to broadcast to the world the fact that officials were collecting textbooks in order to enforce the ban on mas. all this attention created solidarity among teachers, students, and scholars across the nation, further strengthening their resolve to keep mas curriculum and expand it to other places. although the threat to dismantle these programs remains, as a result of the backlash and community organizing, now every school in the city of san francisco offers ethnic studies. more importantly, there is a network of scholars and community members ready to act if another ban arises. we need to create alliances with such groups who can teach us about how they have negotiated backlash on their work over the years. the field of science, technology, and society (sts) has paved the way for many advances in our field by taking seriously intersectionality and colonialism. that is, sts has documented how science and society co-construct. the importance of the history of mathematics is not just to show that certain racial or cultural groups contributed to the knowledge we have today but to also highlight the ways in which settler colonialism or white supremacy are linked to scientific projects (e.g., astronomy being developed to help europeans identify the location of slaves and to make efficient the export of their labor [prescod-weinstein, 2017]). in looking to other disciplines, we need to ask ourselves how we can take more and higher risks in our everyday work? that is, we need to be arguing for a deep sense of education, not compliance. mathematics education researchers need to be embracing activism, as opposed to seeing it as activity that only bad, trouble-making, unprofessional scholars, or ones who do not really “know their mathematics,” engage (gutiérrez, 2017b; picower, 2011). a revolution in mathematics (education) having been raised in a chicanx activist family, i already understood the power of organizing. one thing that has been underscored from this attack is that we cannot create a revolution by ourselves; we need accomplices (not allies) in this work (indigenous action, 2014). that is, we need people who are willing to stand with us, around us, so that those who attack us will need to go through them (first). having accomplices is different than having allies who support with solidarity, cheer loudly from the sidelines, or who safely stand on the sidewalk with their signs. accomplices do what delores huerta called for when organizing for the rights of chicano farmworkers: “walk the street with us into history. get off the sidewalk.” mathematicians are one group who are showing some promise in the arena of being our accomplices. gutiérrez commentary journal of urban mathematics education vol. 10, no. 2 17 an accomplice as academic would seek ways to leverage resources and material support and/or betray their institution to further liberation struggles. an intellectual accomplice would strategize with, not for, and not be afraid to pick up a hammer. (indigenous action, 2014, p. 5) rather than absolve themselves from the politics or try to speak for us as mathematics education researchers, when the attack on me occurred, many mathematicians deferred to us as the experts of our domains. they also communicated to other mathematicians the value of our work and the impossibility of separating politics or domination from mathematics and thereby used their privilege to lend credence to our position. they blogged and tweeted about our issues and pointed their readers to our work (karp, 2017; katz, 2017; lamb, 2017). they wrote letters to our administration and op-ed pieces to newspapers. they invited us to give talks at mathematical conferences, in their mathematics departments, and in their colloquia series. they read and discussed our work in their lab spaces and gave us feedback. they invited several of us in mathematics education to the joint mathematics meetings and organized receptions and small gatherings to help continue the momentum and to educate their members on these issues. many of the mathematicians i know did not shy away from the “politics” or ask me not to use words like white supremacy when naming the relationship between mathematics and power. i saw a different relationship between mathematicians and mathematics education researchers than what happened over a decade ago to jo boaler (boaler, 2012), whose main opposition came from mathematicians. the attack on our field was meant to discredit all of us who work to document and abolish the relationships between current forms of mathematics education and white supremacist capitalist patriarchy. instead, it seemed to spark greater allegiance and create or strengthen networks that had not existed or were previously weak. in some ways, the attack gave us greater exposure and legitimacy, as more thoughtful citizens chose to read our work. one question that looms is how will this play out going forward? how will mathematics education (and mathematicians) act and interact in the future when further attacks arise? for those who were paying attention, there was great concern for the health of our field and the scholarship that attends to dismantling white supremacist capitalist patriarchy in connection to mathematics. but, more should be feeling moral outrage at the current state of affairs (spencer, 2015). throughout history, we have seen how some mathematicians have done this political work themselves. scientists and engineers for social and political action (sespa), a group that later became science for the people during the vietnam war, is one example. they produced a magazine that chronicled their efforts and the successes they achieved. another group, founded in 1973, alongside of the chicano movement and the native american civil rights, the society for advancement of chicanos and native americans in science (sacnas) sought to highlight gutiérrez commentary journal of urban mathematics education vol. 10, no. 2 18 how indigenous mathematicians/scientists have carried out their work while negotiating a racist society. sacnas did not assume that members could attend their national conferences and only talk about “pure” science. they embedded native worldviews and rituals that celebrated what traditionally was erased from society. how might we draw lessons from these professional groups to understand how mathematicians and mathematics education researchers might join forces to radically reimagine a more humane practice? neither perfect articulations of the problems with the current system nor literature to back it up will be enough to sway policy makers or school officials to alter what they are currently (or have been) doing. instead, it will require an insurgency by the people; a rejection of a bill of goods that suggests being a winner in a colonizing system is any prize at all. we need a rejection of the hyper evaluation processes that occur in mathematics classrooms, and perhaps an elimination of classrooms altogether. we need a movement like the civil rights or science for the people to catalyze social transformation. yet, unlike the civil rights movement that argued for obtaining a share of the settler wealth that was stolen, our goals should be to return the wealth to their proper owners. critique or critical consciousness cannot be the goal. we must be willing to reject domination and exploitation in various forms. such exploitation includes forms of packaging and selling social justice programs to districts that require people of color to pay for their own internalized colonialism (gutiérrez, 2017b) or for the right to count in a stem (science, technology, engineering, and mathematics) based society (e.g., bullock, 2017). some of the things we might consider in future work include:  reframing of scholar in the 1960s and 1970s, the black panther party inspired health activism (e.g., desegregating hospitals, sickle cell anemia screening, free breakfast for children). at the time, health was seen as a basic human right. which groups might help support activism in mathematics? the algebra project (e.g., moses, kamii, swap, & howard, 1989) is well on its way in reimagining mathematics education by paying black and brown students in “eduprises” to teach each other mathematics and develop into leaders (gillen, 2014; j. gillen, personal communication, november 29, 2017). what other re-visions could we imagine? in framing this work, what might be the role of scholar activists? in the 1830s, black scientists such as martin delany and sarah mapps douglass used their discipline to support their arguments for emancipation (rusert, 2017). among other things, they delivered scientific lectures that included astronomy and maps to help others chart a course for freedom and had black girls use their own bodies as anatomical models. activists in the civil rights movement won battles through legal means, negotiations, petitions, and protest demonstrations. what might these look like in mathematics education? gutiérrez commentary journal of urban mathematics education vol. 10, no. 2 19 what would a mathematics education scholar look like that serves not the university but the most vulnerable/dehumanized?  decolonizing readings lists and resources chonda prescod-weinstein has created a readings list for decolonizing science education.13 we need to create one in mathematics. and then, we should consider how to include those pieces in readings lists for ph.d. qualifying exams, national science foundation program officers, and mathematics department chairpersons but also local school councils, leadership institutes, book clubs for teachers/students/community members, journalists, policy makers, and others. what other resources would be critical to overhauling the system, rather than tinkering?  rethinking programming and professional development what forms of graduate education would better prepare mathematicians and mathematics education scholars to understand the relationship between the humanities and the sciences? to see the relationship between white supremacist capitalist patriarchy and mathematics? to question the current state of affairs and know how to channel their outrage into new research projects that are designed to honor reciprocity with the communities they most wish to engage? which mathematics programs serve as models for effectively preparing their graduates to understand issues of colonization and oppression as it relates to mathematics? what are the mechanisms through which they accomplish this work? which graduate programs in mathematics education do the same? and what are their mechanisms? which programs require their graduates to consider the ethical implications of the mathematics they practice?  networking among mathematicians and mathematics education researchers rather than superficial naming of committee members from outside of mathematics education onto mathematics education dissertation committees, what might it look like to have meaningful engagement with other fields (e.g., more in line with minoring in another field)? should mathematics education graduate students be required to read outside of one’s field for qualifying examinations? might there be a role for mathematics education researchers to guest blog on mathematician-led blogs, and vice versa? what kinds of coalitions can develop between mathematicians and mathematics education researchers to address social and political action in society? between these groups and teachers? and community members? what would it look like to have local 13 see https://medium.com/@chanda/decolonising-science-reading-list-339fb773d51f. https://medium.com/@chanda/decolonising-science-reading-list-339fb773d51f gutiérrez commentary journal of urban mathematics education vol. 10, no. 2 20 chapters of an organization like sespa that was geared to mathematicians and mathematics education researchers? to whom would these groups address themselves? to colleagues? students? community groups? political groups? policy makers? journalists?  developing a set of principles to guide the field other movements have begun with a set of unwavering principles. see, for example, those of the black lives matter movement.14 if womyn scholars of color in mathematics education were to write a manifesto or develop a statement of principles relating to a decolonized or rehumanized mathematics, what might it include? how might we use such a set of principles to hold ourselves accountable to a standard? how might that set of principles guide the field on the aforementioned areas (e.g., reframing of scholar, decolonizing reading list, rethinking programming and professional development, networking among mathematicians and mathematics education researchers)? given those principles, what is the next set of critical conversations we must have in and about our field? the attack on me, and others, has shown that when we present arguments that challenge white supremacist capitalist patriarchy head on, we pose a threat. the alt-right will use their power to launch personal attacks on individuals, co-opting our language and twisting our words to make it seem as though we are the ones who are discriminating or doing evil. they will tell their readers explicitly how to use social media and what to say to flood our inboxes. however, if we stand side by side as accomplices, not allies, we will be stronger. it is much more difficult to launch an attack on a field than on an individual. understanding how these attacks work and having a game plan for responding is key. i am reminded of a mexican proverb with which i was raised: quisieron enterrarnos sin saber que éramos semilla (they tried to bury us; they didn’t know we were seeds). when the attack on me was launched, i did not suffer alone. instead, leaders of professional organizations wrote public letters in supporting scholars whose research addresses issues of power, privilege, and oppression and who challenge the status quo; many individuals wrote to my campus administration with support for me and my work; educators blogged or responded on my behalf on twitter. and, someone created the website that housed the timeline and resources for other scholars under attack.15 14 see http://blmdetroit.org/principles. 15 see https://equitymathed.wordpress.com/2017/10/29/history-of-istandwithrochelle/. http://blmdetroit.org/principles https://equitymathed.wordpress.com/2017/10/29/history-of-istandwithrochelle/ gutiérrez commentary journal of urban mathematics education vol. 10, no. 2 21 it was a proud moment for mathematics education researchers and mathematicians who stood together and communicated, “not on our watch.” however, we were, nevertheless, caught off guard because for so long we had slid under the radar of alt-right groups who have been busy targeting critical scholars in other disciplines. instead of seeing the attack as an isolated incident, we need to be planning for the next one—organizing ourselves, strengthening our networks, building resources and models, setting precedents, and creating infrastructure and policies in universities and professional organizations to better support future scholars under attack. and, in those efforts, we must begin with a focus on womyn of color. there will certainly be more attacks and the next time we need to be savvy about how to respond. if there is one lesson i believe mathematics education should take from this attack, it is that we were late to the backlash party because we were not doing enough in our field to get noticed. now, if we take that idea seriously, what do we plan to do about it? 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(2017, october 27). how twitter killed the first amendment. the new york times. retrieved from https://www.nytimes.com/2017/10/27/opinion/twitter-first-amendment.html https://www.theatlantic.com/education/archive/2015/07/how-one-law-banning-ethnic-studies-led-to-rise/398885/ https://www.theatlantic.com/education/archive/2015/07/how-one-law-banning-ethnic-studies-led-to-rise/398885/ https://griid.org/2013/03/03/this-day-in-resistance-history-1968-chicano-students-walk-out-in-protest-of-racist-policies/ https://griid.org/2013/03/03/this-day-in-resistance-history-1968-chicano-students-walk-out-in-protest-of-racist-policies/ https://www.nytimes.com/2017/10/27/opinion/twitter-first-amendment.html journal of urban mathematics education july 2012, vol. 5, no. 1, pp. 21–30 ©jume. http://education.gsu.edu/jume christopher c. jett is an assistant professor in the department of mathematics in the college of science and mathematics at the university of west georgia, 1601 maple st., carrollton, ga 30118; email: cjett@westga.edu. his research interests are centered on employing a critical race theoretical perspective to mathematics education research, particularly, at the undergraduate mathematics level. his research has been published in the journal of negro education, journal of black studies, and journal of african american studies. critical race theory interwoven with mathematics education research christopher c. jett university of west georgia eflecting on the theme for the 2011 benjamin banneker association conference at georgia state university the brilliance of black children in mathematics, i posit that it is important to emphasize the coupling of “black” and brilliance. those concerned with the mathematics education of all children, i believe, must bring issues of race to the fore and investigate the racialized experiences of black (and all) students in mathematics in such a way that celebrates the brilliance that black children possess rather than their perceived “at-risk” status. as it stands, critical race theory (crt) is a theoretical framework that has been used to bring forward issues of race in education research in general and mathematics education research in particular. in this paper, i focus my discussion exclusively on crt as a theoretical framework. in so doing, i describe the foundational tenets of crt while simultaneously addressing the theoretical and methodological appropriateness of crt. next, i discuss how crt has informed mathematics education research with examples from my work and that of another mathematics education researcher, and suggest that other researchers and practitioners employ crt in their efforts. finally, i conclude by challenging us all to (continue to) spread critical race messages of hope and brilliance to black children concerning their mathematics achievements. critical race theory historically, race has been used as an unjust construct in the united states and throughout the world. 1 as west (2001) states, “it goes without saying that a profound hatred of african people (as seen in slavery, lynching, segregation, and second-class citizenship) sits at the center of american civilization” (p. 106). since the enslavement of the peoples of africa, black americans have experi 1 the executive board of the american anthropological association endorsed a statement on race that seeks to deconstruct race as a social construct that inequitably categorizes different racial groups (american anthropological association, 1998). the executive board argued that racial inequities exist not because of biological (scientific) reasoning but due to historical and modernized institutionalized “racial” practices. r jett crtical race theory and mathematics bullock, e. c., alexander, n. n, & gholson, m. l. (eds.). (2012). proceedings of the 2010 philadelphia and 2011 atlanta benjamin banneker association conferences – beyond the numbers [special issue]. journal of urban mathematics education, 5(2). 22 enced this hierarchical race system that places europeans at the top and people of color at the bottom (dubois, 1903/2003). african americans are still grappling with issues of race and racism (cleveland, 2004). nonetheless, it has been suggested by some people that race has been minimized with the election of president obama as the united states has entered a post-racial era. racism, however, still exists in various forms. there are some who suffer from dysconscious racism where they accept white normative ideologies or possess a distorted ideological paradigm concerning race or racism (king, 1991). hilliard (2001) argued that race is a political construct and that race in education is designed “to teach african inferiority and european superiority” (p. 25). taken together, these arguments demonstrate that racism is an institutionalized force that has been used both historically and currently to dismiss and oppress people of african descent as well as other people of color. while race has been used overtly and covertly to marginalize people of color, crt has emerged to critically analyze the ideological power structure embedded within racial hierarchies. crt has its genesis in critical legal studies (cls) (delgado & stefancic, 2001). crt developed in the mid-1970s from the work of legal scholars such as alan freeman, richard delgado, and derrick bell (delgado & stefancic, 2001) as a response to the lack of diversity among the faculty at harvard university and the marginalization of students of color from the law school’s curriculum (carbado, 2002). derrick bell, an african american law professor, is considered the father of crt (delgado & stefancic, 2001). this movement also consisted of other activists and scholars interested in investigating and transforming the injustices that were brought about because of issues of race, racism, and power in our society (delgado & stefancic, 2001). although crt began in legal studies, it has expanded to other disciplines, including (mathematics) education. solórzano and yosso (2002) argue that crt in education advances a strategy to foreground and account for the role of race and racism in education and works toward the elimination of racism as part of a larger goal of opposing or eliminating other forms of subordination based on gender, class, sexual orientation, language, and natural origin. (p. 25) additionally, crt includes african american epistemology, which gordon (1990) defines as “the study or theory of the knowledge generated out of the african-american existential condition, that is, of the knowledge and cultural artifacts produced by african-americans based on african-american cultural, social, economic, historical, and political experience” (p. 90). historically, crt builds off of four foundational principles. some scholars have added different tenets to these four principles while others have modified them to reflect their own cultural inclinations (see brayboy, 2005; decuir & jett crtical race theory and mathematics bullock, e. c., alexander, n. n, & gholson, m. l. (eds.). (2012). proceedings of the 2010 philadelphia and 2011 atlanta benjamin banneker association conferences – beyond the numbers [special issue]. journal of urban mathematics education, 5(2). 23 dixson, 2004; solórzano & yosso, 2002). nonetheless, the four principles are consistently noted within most versions of crt in education research and provide the philosophical underpinnings. first, critical race theorists argue, “racism is normal, not aberrant, in american society” (delgado & stefancic, 2000, p. xvi). because the united states has historical structures, institutional structures, and schooling practices that perpetuate racism, critical race theorists assert that racism seems “normal” to people in the united states. second, crt does not follow traditions of positivist scholarship because it allows researchers to employ storytelling as an epistemological resource to “analyze the myths, presuppositions, and received wisdoms that make up the common culture about race and that invariably render blacks and other minorities one-down” (delgado & stefancic, 2000, p. xvii). through writing and storytelling, critical race theorists seek to speak against rules and processes that continue to give power to european americans and allow racism to thrive in american society with the hope of contributing to social justice by breaking down (some of) these racist barriers. third, crt asserts a critique on liberalism (ladson-billings, 1999). i embrace delgado and stefancic’s (2001) definition of liberalism: a “political philosophy that holds that the purpose of government is to maximize liberty…the view that law should enforce formal equality in treatment” (p. 150). critical race theorists critique this philosophy by arguing that the dominant culture does not fully understand how liberty and equality function. under the notion of liberalism, critical race theorists have also critiqued “colorblindness, the neutrality of the law, and incremental change” (decuir & dixson, 2004, p. 29) as phenomena that simultaneously minimize and/or disregard race and do not espouse liberalism in its truest sense. fourth, crt argues that whites, particularly white women, have been the major beneficiaries of affirmative action and civil rights legislation (ladsonbillings, 1999). this alignment reifies that whites will accept and/or encourage policies that advance people of color only when these policies can also help whites advance themselves (delgado & stefancic, 2001; ladson-billings, 1999). critical race theorists critique the motives behind white support of legislation and question whether the policies were designed to benefit people of color in the first place (delgado & stefancic, 2001). collectively, these four tenets provide the groundwork for how crt can be applied theoretically or methodologically to education research in general and mathematics education research in particular. theoretical and methodological appropriateness of crt crt has been underutilized as a theoretical and methodological framework to investigate the mathematical experiences of african american students. further, issues of race and racism have been underexplored in education research. ladson jett crtical race theory and mathematics bullock, e. c., alexander, n. n, & gholson, m. l. (eds.). (2012). proceedings of the 2010 philadelphia and 2011 atlanta benjamin banneker association conferences – beyond the numbers [special issue]. journal of urban mathematics education, 5(2). 24 billings and tate (1995) argue that while race issues seem to be prevalent in society, race has been untheorized, especially in education research. solórzano and yosso (2002) further argue, “substantive discussions of racism are missing from critical discourse in education” (p. 37). as hilliard (2001) questions, what can be done about the way that we deal with race in education, particularly in education research? these scholars do not suggest that scholars have not explored race or even considered race when examining social inequality; instead, they argue that this theorizing has not been systematically employed when analyzing educational inequality. in an attempt to address the shortage of research in mathematics education approached from a critical race perspective, they propose that researchers employ crt to investigate race and racism in education. all of these issues emphasize the importance of including the construct of race in educational research and integrating crt as a framework to examine educational settings. theoretically and methodologically, crt can be viewed “as a way to link theory and understanding about race from critical perspectives to actual practice and actions going on in education for activist social justice and change” (parker & lynn, 2002, p. 18). crt allows researchers to critically analyze historical and current racial concerns through a critical race lens (parker, 1998). solórzano (1998) claims, crt “in education challenges the dominant discourse on race and racism as they relate to education by examining how educational theory, policy, and practice are used to subordinate certain racial and ethnic groups” (p. 122). by not fully taking into account race and culture when investigating the issues students of color face in schools, educational research has not adequately examined the educational complexities students of color confront theoretically (parker & lynn, 2002; solórzano, 1998; valenzuela, 1999). ladson-billings (2000) posed the following question: “where is ‘race’ in the discourse of critical qualitative researchers?” (p. 272). parker (1998) answers with his own methodological question for critical thought and reflection: rather than ask what can this theory do for qualitative studies in education, an alternative inquiry i would propose is what can qualitative research in education do to illuminate and address the salient features of crt with respect to race and racism in educational institutions and the larger society? (p. 46) drawing upon parker’s argument, we need to employ crt in qualitative educational research to ascertain these racialized discourses. one of the main ways to accomplish this goal, along with one of the methodological tenets of crt, is the use of “voice” (dixson & rousseau, 2005). with voice, there is “the assertion and acknowledgement of the importance of the personal and community experiences of people of colour as sources of knowledge” (p. 10). borrowing from this tradition, critical race theorists believe in and use personal narratives and personal sto jett crtical race theory and mathematics bullock, e. c., alexander, n. n, & gholson, m. l. (eds.). (2012). proceedings of the 2010 philadelphia and 2011 atlanta benjamin banneker association conferences – beyond the numbers [special issue]. journal of urban mathematics education, 5(2). 25 ries as forms of knowledge to document inequity, injustice, and/or discrimination (dixson & rousseau, 2005). when using the term voice, it is important not to essentialize 2 or assume that one person’s voice speaks for the entire group or culture (dixson & rousseau, 2005). delgado (1989) asserts that many who tell stories are those “whose voice and perspective—whose consciousness—has been suppressed, devalued, and abnormalized” (p. 2412). allowing their voices to be heard through the usage of devices such as personal stories, narratives, counterstories, and autoethnographies, critical race theorists seek to offer a counter-perspective to the disparaging narratives about marginalized groups that circulate throughout the discourse. additionally, delgado (1989) contends, “members of the majority race should also listen to stories, of all sorts, in order to enrich their own reality” (p. 2439). my argument is that listening to the success stories of african american male students in mathematics can continue the legacy of mathematical brilliance among african american students. implications of crt in mathematics education research bell (1992) charged scholars to employ a critical race theoretical perspective in their scholarship. he wrote: “with what some of us are calling critical race theory, we are attempting to sing a new scholarly song—even if to some listeners our style is strange, our lyrics unseemly” (p. 144). employing crt as a theoretical framework also allows education researchers to explore the culture of k–16 institutions, explore the nature of racist acts, behaviors, and/or utterances to students of color, and examine the disciplinary acts (if any) of students who engage in these racial undertakings (decuir & dixson, 2004). i, along with other scholars (see, e.g., berry, 2008; carter, 2008; duncan 2002; harper, 2012; martin, 2009a; stinson, 2008; terry, 2011), am seeking to sing a new scholarly song in mathematics education research regarding african american male students. martin (2009b) brought race to the forefront of his analysis. seeking to change the way that race is used as a means to categorize students in mathematics education research, he proposed that race be understood as a “sociopolitical, historically contingent construct” (p. 298). further, his analysis included (re)conceptualizing the mathematics education goals of marginalized groups such as african americans, latinas/os, and native americans to adequately reflect race. by doing so, martin called for a deconstruction of the racial hierarchy of mathematical ability that places asian and white students at the top and african 2 leistyna, woodrum, & sherblom (1996) defined essentialism as a fundamental nature or biological determinism to human rights through attitudes about identity, experience, knowledge, and cognitive development. within this view, categories such as race and gender become gross generalizations, and single-course explanations about individual character. jett crtical race theory and mathematics bullock, e. c., alexander, n. n, & gholson, m. l. (eds.). (2012). proceedings of the 2010 philadelphia and 2011 atlanta benjamin banneker association conferences – beyond the numbers [special issue]. journal of urban mathematics education, 5(2). 26 american, latina/o, and native american students at the bottom. martin’s analysis is an example of employing a critical race theoretical lens in mathematics education research, and his work is seminal because it provides the backdrop from which recent crt mathematics education scholarship is grounded. i utilized crt to investigate the mathematical experiences of four successful african american male graduate students in mathematics or mathematics education who obtained undergraduate degrees in mathematics (jett, 2009). i conducted three semi-structured interviews and honed in on their racial experiences in mathematics as african american male students. i shared their stories in narrative form. participants were given a copy of the article beyond love: a critical race ethnography of the schooling of adolescent black males (duncan, 2002) to read and reflect on before the final interview. it is interesting to note that the article was not used to steer the participants down critical race lane, but rather to highlight the “voices” of black male students themselves embedded within the manuscript. in other words, i shared the article because i wanted the participants to feel comfortable sharing their stories with me as a researcher and to simultaneously expose them to crt. i sought to examine their experiences with the discipline of mathematics as african american male students. more specifically, i explored the culture of the participants’ undergraduate institutions as well as how they negotiated race and racism as african american men in a society entrenched with racism. conclusion crt is a theoretically and methodologically sound framework that allows us, as researchers, to bring race to the forefront of our differing analyses in various interdisciplinary, educational research studies. if the goal of educational institutions is to recruit and retain students of color in multiple disciplines, then these students must be recognized as “holders and creators of knowledge” (bernal, 2002, p. 106) even though their knowledge might not reflect the beliefs held by the dominant population. crt maintains that the histories, experiences, cultures, and languages of students of color are acknowledged and respected (bernal). in the context of mathematics, this charge would constitute capitalizing and building on the rich history of africana mathematics. while crt should be employed to investigate the educational experiences of african american students, new and different theoretical frameworks should be used in tandem with crt to better address the needs of students of color. (for information concerning other theoretical constructs, see, e.g., stinson & bullock, 2012.) these frameworks must allow researchers to investigate educational phenomena through a critical race lens, examine and act upon racial change in order to eradicate (as much as possible) racial inequalities and injustices, and provide jett crtical race theory and mathematics bullock, e. c., alexander, n. n, & gholson, m. l. (eds.). (2012). proceedings of the 2010 philadelphia and 2011 atlanta benjamin banneker association conferences – beyond the numbers [special issue]. journal of urban mathematics education, 5(2). 27 access to equitable schooling practices, especially in (undergraduate) mathematics (parker & stovall, 2004). such scholarship that draws upon frameworks such as crt in education is long overdue (gordon, 1990). in addition, such work can be done by scholars from all racial and ethnic backgrounds (see, e.g., brayboy, 2004, 2005; fernández, 2002; villalpando, 2003). scholars are invited to the “critical race table” to problematize, challenge, deconstruct, and work to eradicate issues of race and racism. furthermore, theoretical frameworks like crt are also needed to contribute to knowledge production. ladson-billings (2000) asks, “but how can the full range of scholarship be explored if whole groups of people are systematically excluded from participating in the process of knowledge production?” (p. 271). similarly, how can the full spectrum of mathematics be studied if african and african american contributions are excluded from mathematics curricula, standards, and the like? how can we sustain mathematical communities if african american male students are still viewed as mathematically incompetent from the onset (i.e., pre–k/k) of their schooling experiences to the outset (16–graduate/professional school)? additionally, how can we remain silent and not act while racist mathematical practices continue to hinder african american (male) students from entering the mathematics pipeline? something has to be done to critically analyze such spaces and work to foster positive racial mathematical environments for african american male students; crt is a viable mechanism to critically analyze environments where black male students are not promoted as the mathematically talented beings that they are. as forestated, i make use of crt in my work because it does not contribute to the marginalization of african american male students but rather provides a space for them to contribute to knowledge production, even in mathematics enclaves (parker & stovall, 2004). educational researchers should be more willing to listen to scholars and students of color and seek to better understand the meaning(s) behind their racialized stories. this ideological paradigm reinforces what i did with the four participants in my study. i listened to their stories of how being african american and male played out in their educational experiences, especially with regards to college mathematics. in sum, brilliance goes hand and hand with the longstanding mathematical africana/african american tradition. brilliance! brilliance! brilliance! these words should be the main chorus of the scholarly song that we sing to and about black children concerning their academic success in general and their mathematical success in particular (bell, 1992). this message is one that i “shout out” to my two nephews, cousins, church members, students in school settings, community children, and all with whom i connect. my hope is that black children both near and far will hear these messages, internalize them, and turn them into selffulfilling prophecies. such positive affirmations with critical race messages are jett crtical race theory and mathematics bullock, e. c., alexander, n. n, & gholson, m. l. (eds.). (2012). proceedings of the 2010 philadelphia and 2011 atlanta benjamin banneker association conferences – beyond the numbers [special issue]. journal of urban mathematics education, 5(2). 28 needed for the sustainment of our mathematical communities as well as the production of more critical race educational research in the african tradition. what critical race messages are you sending to black children concerning their mathematics experiences? at the symposium during the symposium, i shared with conference attendees this overview of crt and how i have used it in my own work. in the breakout session, conference participants narratives and stories were shared that aligned with the ideas expressed during my presentation. the testimonies were rich and powerful, and they could have been used as research data to substantiate why race should be brought to the forefront in mathematics education scholarship. additionally, critical race theoretical ideas were further extrapolated and expounded upon to challenge, critique, and continue the work in this domain. references american anthropological association. 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(2008). negotiating sociocultural discourses: the counter-storytelling of academically (and mathematically) successful african american male students. american educational research journal, 45, 975–1010. stinson, d. w., & bullock, e. c. (2012). critical postmodern theory in mathematics education research: a praxis of uncertainty. educational studies in mathematics, 80(1&2), 41–55. terry, c. l. (2011). mathematical counterstory and african american male students: urban mathematics education from a critical race theory perspective. journal of urban mathematics education, 4(1), 23–49. retrieved from http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/98/87. valenzuela, a. (1999). subtractive schooling: u.s.-mexican youth and the politics of caring. albany, ny: state university of new york. villalpando, o. 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(2001). race matters. new york: vintage books. http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/98/87 journal of urban mathematics education december 2016, vol. 9, no. 2, pp. 185–194 ©jume. http://education.gsu.edu/jume susan o. cannon is a phd student in the department of middle and secondary education, mathematics education unit, in the college of education and human development at georgia state university, p.o. box 3978, atlanta, ga 30302; email: scannon5@gsu.edu. her research interests include uncertainty in mathematics classrooms, counting practices and measurement, and teacher education. kayla d. myers is a phd student in the department of early childhood and elementary education in the college of education and human development at georgia state university, p.o. box 3978, atlanta, ga 30302; email: kmyers@gsu.edu. her research interests include mathematics teacher education and issues of subjectivity in teachers’ beliefs about teaching and learning mathematics. book review radical reconfiguring(s) for equity in urban mathematics classrooms: lines of flight in mathematics and the body: material entanglements in the classroom1 susan o. cannon kayla d. myers georgia state university georgia state university n mathematics and the body: material entanglements in the classroom by elizabeth de freitas and nathalie sinclair (2014), the authors call for a “radical reconfiguring” (p. 225) of mathematics education. similarly, barad (2012) argues that “theories are living and breathing reconfigurings of the world” (p. 207) and that putting new materialist and posthumanist theories to work in mathematics education could open up a space for this radical reconfiguring. this book review is based on our developing thinking about the use of theory and its possibilities within urban mathematics education. we situate ourselves in what de freitas and sinclair call a “stretchy space of continuous transformation” (pp. 90–91)—continuing to think and re-think issues like equity with this text, letting the words wash over us (st. pierre, 2003), opening up and questioning urban mathematics education research. as two white women mathematics educators and emerging scholars in the field, we do not assert that theory alone or theory removed from practice can address the complex, prevalent, and long-lasting inequities present in mathematics education; however, we view theory in concert with practice as having potential to advance the field. de freitas and sinclair’s theories, which they use to question school mathematics in general, could be built upon and deployed to expose the problematic presence of white rationality (martin, 2015) in urban classrooms. we encourage the reader to join us in this stretchy space and embrace the potentialities of the book as an as 1 de freitas, e., & sinclair, n. (2014). mathematics and the body: material entanglements in the classroom. new york, ny: cambridge university press. 284 pp. isbn 978-1-107-03948-3 (hb) $99.00, isbn 978-1-139-95040-4 (e-book) $79.00. http://www.cambridge.org/us/academic/subjects/psychology/educational-psychology/mathematics-and-body-materialentanglements-classroom?format=ar i http://education.gsu.edu/jume mailto:scannon5@gsu.edu mailto:kmyers@gsu.edu http://www.cambridge.org/us/academic/subjects/psychology/educational-psychology/mathematics-and-body-material-entanglements-classroom?format=ar http://www.cambridge.org/us/academic/subjects/psychology/educational-psychology/mathematics-and-body-material-entanglements-classroom?format=ar http://www.cambridge.org/us/academic/subjects/psychology/educational-psychology/mathematics-and-body-material-entanglements-classroom?format=ar cannon & myers book review journal of urban mathematics education vol. 9, no. 2 186 semblage, a process of making and unmaking, arranging and fitting together of theories (jackson & mazzei, 2012), as it allows us (i.e., researchers and educators) to disrupt and perturb taken-for-granted assumptions about urban mathematics education. putting theory to work: inclusive materialism and equity to think potentialities of equity with mathematics and the body, like de freitas and sinclair (2014), we pay attention to particular parts of the text, recognizing that they are in assemblage with other parts of this text, other texts, and us. we highlight what we consider to be the main concepts within each chapter, and we consider these concepts as points of departure for potential flights that mathematics educators could embark on with and through their area(s) of interest. although equity is not addressed explicitly by de freitas and sinclair in all of the chapters, one of the flights that we take involves thinking this text and equity together as “knots and meshworks…less an image of different bits and pieces glued to each other than an image of a tangled set of paths, each with its own mobility and degree of freedom” (p. 225) that might “push the field into new uncharted terrain and allow for new conjectures about teaching and learning” (p. 5). de freitas and sinclair (2014) pull the threads of various (and differing) theories, drawing on barad, deleuze, rotmann, ranciere, and, most heavily, on châtelet to put forward a new form of materialism that they term inclusive materialism, which troubles traditional humanist and rationalist notions and takes up the aesthetic, affective, and material as mattering. de freitas and sinclair put this theory, and others, to work to rethink school mathematics, proposing that inclusive materialism might “alter the way we think about embodiment of mathematical concepts, offering alternate ways of studying how students learn concepts and how we might choose and order concepts as part of a curriculum sequence” (p. 12). in addition, de freitas and sinclair (2014) address inequity in education through the entanglement of the political and the material and by proposing the reconfiguring of school mathematics, inviting readers to conceive of a minor mathematics that is “not the state-sanctioned discourse of school mathematics but that might be full of surprises, non-sense and paradox” (p. 226). they recognize that this mathematics might be “at odds with current institutional demands. however, a minor mathematics is likely to engage students and teachers in more expansive ways, and our hope is that it would engage more students in mathematics” (p. 226). de freitas and sinclair are not interested in theory separate from the material and political reality of schools. their minor mathematics would entail shifts in the way we theorize and practice mathematics education. cannon & myers book review journal of urban mathematics education vol. 9, no. 2 187 theoretical flights mathematics and the body is divided into eight chapters, each of which uses theory to push the bounds of what has become sensible in mathematics education. all the chapters are densely packed and take some time to “chew and digest” (jett, 2015, p. 14). thinking with even one chapter at a time could (and should) take the reader on endless “lines of flight” (deleuze & guattari, 1980/1987). a line of flight is a place of possibility and change, an opportunity to shift. it implies the type of movement and flow across paradigms that would be necessary to move toward a minor mathematics. flight(s) 1: when does a body become a body? bell hooks (1994) notes that the legacy of the cartesian cogito in educational practice has meant the erasure of the body that we may “give ourselves over more fully to the mind” such that the normalized governing assumption is “that passion has no place in the classroom (p. 192)”. (as cited in taylor, 2016, p. 202) de freitas and sinclair (2014) begin by asking—when does a body become a body? we ask—when do students’ bodies become mathematical bodies? which bodies are assumed to be mathematical and which are resisted in their mathematical becoming? de freitas and sinclair acknowledge materiality of bodies as an intraactivity, an inexhaustible dynamism that (re)configures relations of space and time and matter (barad, 2007), across and between bodies and materials. in (re)visioning the body and its boundaries, they also destabilize traditional views of knowing and consider knowing that “extends beyond the boundary of the skin” (p. 16). they question the individual learner and knower as a separate and self-contained entity that has (or does not have) mathematical knowledge. if the body is not the singular fixed container of knowledge with known abilities, and instead the body is in process of becoming in engagement with materials and others, how might our classroom and pedagogical processes change? de freitas and sinclair (2014) consider the assemblage of child-materialsconcept as a body that emerges through the intra-action, a view that might allow us to consider learning in new ways. in this view, the materials matter, and the concept is invented through the intra-action rather than being viewed as preexistent and waiting to be “discovered” by the child. the body is assembled in relation to nonhuman components and is “always in a process of becoming that belies any centralizing control” (p. 24). if students “bodies” are in the process of becoming, our views of them as particularly abled (tracked, sorted, low, high, gifted, (dis)abled…) do not work anymore. what happens to mathematics education when we begin to think classroom with assemblage? how might research fall apart, so that we can create new ways to understand urban mathematics education? what opportunities cannon & myers book review journal of urban mathematics education vol. 9, no. 2 188 might there be in this view to think mathematics education differently in ways that acknowledge students and teachers as in process as becoming? barad (2007) takes up the concept of cuts, boundary-making practices that categorize and classify: “cuts are enacted not by willful individuals but by the larger material arrangements of which ‘we’ are a ‘part’” (p. 178). once a cut gets made what are all the mechanisms that get produced that reify and solidify that cut so it becomes the only way that we can think? in mathematics education, cuts are being made around black and brown bodies with the production of the achievement gap (gutiérrez, 2008). how might a different conception of bodies undermine these cuts? if there is no discrete knower, then it becomes unthinkable that a single number derived from a test score could (or should) represent knowledge of mathematics. flight(s) 2: the “ontological turn” of inclusive materialism as de freitas and sinclair (2014) draw the reader through the works of niels bohr and karen barad, they build a case for the consideration of mathematical concepts as material and inventive, not as larson (2016) states as something to be given. with barad, the authors question the apparent immobility of matter and the construction and fixed bodies of knowledge. de freitas and sinclair animate matter and concept in material intra-action; the traditional view that— learning is assumed to have a teleological trajectory towards fixed and immovable mathematical concepts. concepts are said to emerge through activity, but there is no troubling of the specificity of the concepts—in other words, the mathematical concepts (multiplication, cube, zero) are taken for granted, while students collaboratively move towards them. (p. 40) this traditional view is upturned in favor of sensational (not just sensible) learning that is inventive and intra-active. in this chapter, de freitas and sinclair (2014) outline their framework, inclusive materialism, noting four crucial aspects: 1. it is not reductive, seeing all matter as the same; instead it privileges “difference and multiplicity” (p. 42). 2. the socio-political and the material are seen as “inextricably entangled” (p. 42) and in this viewing inequity issues in education can be addressed within a broader framework. 3. affect and aesthetics and nonsense are central and rationality is not privileged. 4. humanist notions and human agency are decentered (not as anti-human) but to distribute agency across the assemblage. cannon & myers book review journal of urban mathematics education vol. 9, no. 2 189 in laying out this theory, de freitas and sinclair (2014) move through theories of language, considerations of discourse as material with barad, and views of boundary making practices that led them to question the taken-for-granted curriculum in school mathematics. when we begin to understand the curriculum that we have taken up as constructed and having effects on mathematical learners, we might be willing to “rethink learning as an indeterminate act of assembling various kinds of agencies rather than a trajectory that ends in the acquiring of fixed objects of knowledge” (p. 52). if we can begin to think mathematical concepts as quivering spaces of potential rather than “tool[s] that we give students” (larson, 2016, ¶7), more students might be invited into mathematical thinking. flight(s) 3/4: diagrams, gesture, movement, and inventiveness in the mathematics classroom de freitas and sinclair (2014) use châtelet’s (2000) ideas to connect materiality, gestures, diagrams, and inventiveness in mathematics. châtelet views gestures and diagrams as interrelated, inseparable, and inventive. in figuring space, he traces mathematical invention (through gestures and diagrams) and argues that mathematical concepts are material. this argument is counter to the prevailing belief that mathematical concepts are abstract and static. for châtelet, mathematical concepts are mobile and have multiple latent virtualities that can be actualized. in one classroom example, de freitas and sinclair read multiple student diagrams not as to whether the diagrams approach a definitive answer, but as to how the diagrams and gestures were put to use to think and how they mobilized potential readings of a shared experience in the classroom. here, students became in material intra-action with mathematics. de freitas and sinclair view this type of student gesturing and diagramming as a “disruptive and innovative practice” (p. 84), a minor mathematics. assemblage, gesture/diagram, the virtual, and inventiveness are intertwined by de freitas and sinclair (2014) in chapter 4 as they consider several additional classroom episodes that reassemble and reconfigure the world. the authors begin a conversation about the interplay between the actual and virtual. they build on this conversation to consider how the “new,” mathematical inventiveness or creative acts, come into being in the mathematics classroom. they argue that a creative act: 1. introduces or catalyzes the new, 2. is unusual, 3. is unexpected or unscripted (not directly caused by the software the teacher or any individual student), and 4. is without given (or predetermined) content. cannon & myers book review journal of urban mathematics education vol. 9, no. 2 190 these characteristics are not held within individuals as creative (or not creative) people. creativity in this framework “flows across the learning assemblage in a somewhat impersonal way” (p. 86). these impersonal creative acts are seen as ontological reassembling, radical reconfigurings of the world. de freitas and sinclair are careful to draw a line between gesture/diagram and representation. the diagram and gesture are powerful in that they do not represent but interfere, link, pleat, and crease matter, keeping matter and concepts mobile. they do not try to fix or make mathematics static. rather, they mobilize mathematics. flight(s) 5/6/7: materialist approaches to mathematics classroom discourse, the sensory politics of the body mathematical, mapping the cultural formation of particular interest to us, in this section, is de freitas and sinclair’s (2014) use and conception of data. given their new materialist theoretical frame, the question of how to do research takes on new meaning, forcing the researcher into a position of negotiation with the limits of methodology. if we consider the classroom as assemblage, then we cannot isolate variables or pin down particular aspects to be studied independently. de freitas and sinclair conceive of a momentary stabilization of the data in order to think with it, yet they acknowledge that doing so is impossible and imperfect. in chapter 5, de freitas and sinclair (2014) attend to and decenter language in new materialist terms, “exploring how classroom discourse, in particular speech, is coupled to other materialities and affective forces” (p. 111). moving away from the focus on classroom discourse as being what is written or spoken, this chapter highlights the integration of the body, diagrams, and gestures into the assemblage of materialist approaches. they note, “our aim here is not to fetishize speech as the immediate expression of embodied presence, but in fact to recombine speech and thinking in new, material ways and to show how this recombinatorial logic operates in classrooms” (p. 138). for de freitas and sinclair, speech is not the disconnected representation of thought. thought is embodied and intra-acting with speech production through the body. the field’s privileging of student production of linear and sequential explanations of mathematical thinking is problematic if seen in relation to this theory as it excludes expressions of thinking that are not sensible within the normative frame of school mathematics. building on this thinking, in chapter 6, de freitas and sinclair (2014) decenter privileged forms of knowing/sensing in their thinking about (dis)ability: “learning disabilities in mathematics…are constructed using narrow definitions of what counts as acceptable mathematics and what counts as evidence of proficiency in mathematics” (p. 160). by taking a posthumanist approach, de freitas and sinclair make a move to reconceptualize the body as collective. and in chapter 7, de freitas and sinclair discuss acts of dissensus: “the subject comes into being through both consensus (alignment with common sense) and through dissensus (divergent cannon & myers book review journal of urban mathematics education vol. 9, no. 2 191 individuation)” (p. 175). they actively question what we conceive of as making sense and put forward dissensus as a way to radically reconfigure the world. de freitas and sinclair claim: “dissensus eventually produces new consensus. the question thus becomes: how might dissensus-producing ideas be kept lithe and fleeting, so that they escape becoming part of the common sense while remaining meaningful for a community of practice?” (p. 199). this question is of particular importance in rethinking school mathematics: “the aim is to perturb, if even only temporarily, what is taken to be common sense and who is assumed to possess it” (p. 199). how do we continue to perturb school mathematics and “equitable” practices that have become normalized but are not working? flight(s) 8: the virtuality of mathematical concepts in the final chapter of mathematics and the body, de freitas and sinclair (2014) dive deeply into châtelet and his use of the virtual and the actual in thinking mathematical invention particularly with respect to mathematics curriculum. they move through examples of linking mathematical concepts with the virtual deploying châtelet’s conception of the “virtual as the necessary link that binds the mathematical and the physical together in mutual entailment” (p. 201). this mutual entailment allows concepts to “sustain a certain vibrancy and vitality. in other words, a concept of this kind must be a multi-purpose device that resists reification while carving out new mathematical entities” (p. 217). if the concept remains vibrant and multi-purpose, then it cannot become static and fixed. it remains mobile, which is crucial for châtelet. de freitas and sinclair point out that keeping mathematical concepts vibrant and mobile is difficult, as mathematicians tend to focus on the real and the possible rather than the virtual. the tendency is to nail down and know, so they ask, “can we find the latent virtualizes in the mathematics curriculum and also reanimate the bodies of ossified mathematical concepts?” (p. 213). what are the effects of a bodiless mathematics on students and teachers that engage with it? critique if we are going to pursue the goal of equity in mathematics classrooms and consider how minor mathematics might function in the field in ways that matter, how might the thought experiments that de freitas and sinclair (2014) share be “technologies we create not only to remember but to think” (p. 91)? similarly, how might theories like poststructuralism, posthumanism, and new materialism as diagrams, or material actions work “as prosthetic devices that become vehicles of intuition and thought” (knoespel, 2000, p. xiii)? the centering of movement, potentialities, and creativity that de freitas and sinclair put forward in mathematics and the body could be taken up to work toward more equitable thinking and practices in cannon & myers book review journal of urban mathematics education vol. 9, no. 2 192 mathematics education, and we propose that thinking with the decentering of language, the body, and the linear curriculum sequence has significant implications for urban mathematics education. changing what makes sense to us about mathematics education by building a minor mathematics could make for more equitable conceptions of learning and research for students, practitioners, and researchers. while we see this potential in the text, we note that de freitas and sinclair (2014) spend little time and space on issues of race and gender, as they acknowledge in their own conclusion and implications. given the attention devoted to race and gender in urban mathematics education scholarship, we wonder what it might have looked like for these researchers to take up race and gender in this text. could examples have foreclosed or opened up lines of flight? what gets ignored or silenced by not attending to such issues? can we tend to race and gender as well as other oppressive structures sufficiently? paying so little attention to race and gender feels irresponsible, but to what degree is the researcher the authority in terms of responsibility? do they feel authority to raise these issues in more specific and straightforward ways? race and gender certainly deserve attention in mathematics education, but who has the authority to take these up? perhaps, by not including those explicit labels as boundaries, de freitas and sinclair (2014) opened a space for their readers to bring their agendas, views, and experiences in assemblage with this text and to think those (and other) topics alongside the provocative theories they present. de freitas and sinclair’s resistance to boundary-making practices (barad, 2007) allows readers to take up critical issues, like equity, in their own lines of flight. however, by not addressing them explicitly, what (and who) gets silenced? what cuts are de freitas and sinclair making and what effects might they have on their readers’ thinking? how might readers of this text assemble its concepts and enact new potentialities? echoing stinson and bullock (2012), “we challenge mathematics education researchers to claim (and articulate) their own theoretical space—pure or hybrid—that might activate a praxis of uncertainty within their research passions as well as inspire those passions not yet known” (p. 52). mathematics and the body provides fertile ground for the development of theoretical space and concepts with potential to decolonize (urban) mathematics education. conclusion we came into assemblage with this text, entangled with our computers, our other readings, and bodies whose boundaries had recently become strange to us. susan had just broken her leg, and it was rebuilt with a plate and screws. therefore, she was bedridden, wondering what had become of her body. kayla was nine months pregnant with her first child and wondered about the now blurry borders of her body. reading the first chapter in this state made us think about theories and cannon & myers book review journal of urban mathematics education vol. 9, no. 2 193 ourselves differently. our bodies mattered to us in new ways. we were both reading elsewhere, together and separately, in poststructuralism and posthumanism and new materialism. we waded into mathematics and the body, spent hours and hours marveling at the beauty of the language, wrestling with what de freitas and sinclair (2014) “meant,” wondering how it would be possible to take up all these ideas at once and knowing that we could not, and asking countless questions of each other and the text. one thing became clear very quickly, though this text specifically questions school mathematics, the theories cross boundaries and make boundaries between fields porous. additionally, the gestures that de freitas and sinclair were making were productive to our thinking. there were so many questions to be rethought, so much work to be done out of this text. how does mathematics look when it is thought alongside the body, mind, materials, teacher, students, concept, gesture, and/or diagram? how do labels like standardized test scores fail students and teachers when mathematics is (re)considered as distributed and in assemblage, especially within urban contexts? how might our conceptions of sense and ability enact violent cuts between students and mathematics? and how might our research practices reify those cuts? mathematics and the body opened us up to think taken-for-granted concepts differently and iteratively. in our discussions about the later chapters and in the margins of our books, equity came up again and again. concepts in this text had the potential to challenge assumptions, beliefs, biases, and practices that could open up a space for us to consider mathematics in a way that could allow access for more students. yet, de freitas and sinclair (2014) resisted telling and closure; instead, they questioned and invited readers on lines of flight. perhaps they harken back to archimedes and “intentionally chose an obscure and ‘jumpy’ presentation so as to ‘inspire a reader with the shocking delight of discovery, in proposition 24, how things fit together, so as to have them stumble, with a gasp, into the final, very rich results of proposition 27” (p. 189). in this text, readers are inventors and creators of new thinking. with that said, we bring this review to jume in an attempt to invite urban mathematics educators and researchers to join together in this “stretchy space” with permeable boundaries and to find places for creation, innovation, and practicality within (various) theoretical texts. although de freitas and sinclair (2014) point to this book as a move to challenge inequities in mathematics education, they do not speak directly to the kinds of inequities that it might function to undermine. this ambiguity has been criticized, yet it leaves a wide-open space for the “radical reconfiguring” (p. 225) necessary to improve mathematics education for underserved students. we wonder what possibilities exist if readers couple radical reconfiguring together with martin’s (2015) critique of principles to actions: ensuring mathematical success for all (nctm, 2014), which calls for a revolution of values, a new way of thinking, and a decolonizing education for the cannon & myers book review journal of urban mathematics education vol. 9, no. 2 194 collective black. whatever the case, mathematics and the body presents possibilities, and the ideas expressed in it should be taken up in a critical manner by urban mathematics educators to produce a more equitable mathematics body. references barad, k. (2007). meeting the universe halfway: quantum physics and the entanglement of matter and meaning. durham, nc: duke university press. barad, k. (2012). on touching: the inhuman that therefore i am. differences, 23(3), 206–223. châtelet, g. (2000). figuring space: philosophy, mathematics, and physics (r. shore & m. zagha, trans.). dordrecht, the netherlands, kluwer. de freitas, e., & sinclair, n. (2014). mathematics and the body: material entanglements in the classroom. new york, ny: cambridge university press. deleuze, g., & guattari, f. (1987). a thousand plateaus: capitalism and schizophrenia (b. massumi, trans.). minneapolis, mn: university of minnesota press. (original work published 1980) gutiérrez, r. (2008). a “gap-gazing” fetish in mathematics education? problematizing research on the achievement gap. journal for research in mathematics education, 39(4), 357–364. jackson, a. y., & mazzei, l. a. (2012). thinking with theory in qualitative research: viewing data across multiple perspectives. new york, ny: routledge. jett, c. c. (2015). an urban mathematics education book review?: considerations for jume book review authors. journal of urban mathematics education, 8(1), 14–16. retrieved from http://ed-osprey.gsu.edu/ojs/index.php/jume/article/viewfile/271/168 knoespel, k. j. (2000). diagrammatic writing and the configuration of space. in g. châtelet (ed.), figuring space: philosophy, mathematics and physics (pp. ix-xxiii). dordrecht, the netherlands: kluwer. larson, m. (2016, september 15). a renewed focus on access, equity, and empowerment. retrieved from http://www.nctm.org/news-and-calendar/messages-from-the-president/archive/mattlarson/a-renewed-focus-on-access,-equity,-and-empowerment/ martin, d. b. (2015). the collective black and principles to actions. journal of urban mathematics education, 8(1), 17–23. retrieved from http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/270/169 national council of teachers of mathematics. (2014). principles to actions: ensuring mathematical success for all. reston, va: national council of teachers of mathematics. st. pierre, e. a. (2003). elan 8045 the postmodern turn: theories and methods. course syllabus, the university of georgia, athens. stinson, d. w., & bullock, e. c. (2012). critical postmodern theory in mathematics education research: a praxis of uncertainty. educational studies in mathematics, 80(1&2), 41–55. taylor, c. (2016). close encounters of a critical kind: a diffractive musing in/between new material feminism and object-oriented ontology. cultural studies ↔ critical methodologies, 16(2), 202–212. http://ed-osprey.gsu.edu/ojs/index.php/jume/article/viewfile/271/168 http://www.nctm.org/news-and-calendar/messages-from-the-president/archive/matt-larson/a-renewed-focus-on-access,-equity,-and-empowerment/ http://www.nctm.org/news-and-calendar/messages-from-the-president/archive/matt-larson/a-renewed-focus-on-access,-equity,-and-empowerment/ http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/270/169 journal of urban mathematics education  december 2008, vol. 1, no. 1, pp. 35–59  ©jume. http://education.gsu.edu/jume  ole skovsmose  is a professor in mathematics education at the department of education, learning  and  philosophy,  aalborg  university,  fibigerstraede  10,  9220  aalborg  east,  denmark;  e­mail:  osk@learning.aau.dk. he has a special interest  in critical mathematics education and has investigated the  notions of landscape of investigation, mathematics in action, students’ foreground, and ghettoising.  pedro paulo scandiuzzi is a professor at the department of education, mathematics pratical edu­  cation, university são paulo states, brazil. he has a special interest in ethnomathematics education, indigen­  ous education, mathematics in different social­cultural groups, and teacher formation.  paola valero is an associate professor in mathematics education at the department of education,  learning and philosophy, aalborg university, fibigerstraede 10, 9220 aalborg east, denmark; e­mail: pao­  la@learning.aau.dk. her research interests are the political dimensions of mathematics education at all levels.  helle alrø is a professor in interpersonal communication at the department of communication and  psychology, aalborg university, kroghstraede 1, 9220 aalborg east, denmark; and professor ii at bergen  university college, norway; e­mail: helle@hum.aau.dk. she has a research interest in interpersonal commu­  nication and learning in helping relationships.  learning mathematics in a borderland  position: students’ foregrounds and  intentionality in a brazilian favela  ole skovsmose  aalborg university  paola valero  aalborg university  pedro paulo scandiuzzi  university são paulo states  helle alrø  aalborg university  bergen university college  in this article the authors introduce a theoretical framework for discussing the  relation between favela students’ life conditions in relation to their educational  experiences and opportunities. a group of five students from a favela in a large  city in the interior of the state of são paulo in brazil was inter­viewed. the stu­  dents were invited to look into their future and explore whether or not there could  be  learning motives  relating mathematics  in  school and possible out­of­school  practices, either in terms of possible future jobs or further studies. four themes  were identified: discrimination, escape, obscurity of mathematics, and uncertainty  with respect to the future. students in a favela could experience what the authors  call a borderland position, a relational space where individuals meet their social  environment and come to terms with the multiple choices that cultural and eco­  nomic diversity make available to them.  keywords: students’ foregrounds, borderland position, brazilian favela  the permanent  growth  of  shanty  towns  (favelas  in  brazil,  invasiones  in  colombia and ecuador, townships in south africa, or grecekondu in turkey) is  characteristic of the unequal growth of modern society in many countries in the http://education.gsu.edu/jume mailto:osk@learning.aau.dk mailto:paola@learning.aau.dk mailto:helle@hum.aau.dk skovsmose et al.  brazilian favela  journal of urban mathematics education vol.1, no.1  36  world.  the  brazilian  word  favela  refers  to  an  urban  area  formed  when  large  groups of people moved from the rural areas to the big cities in search of work,  took possession de facto of large empty extensions of land, and started construct­  ing dwellings by putting together plastic, cardboard, wood, concrete, or whatever  material could offer shelter from the inclemency of sun and rain. when the roof is  completed, the house is finished. a favela is always transient and in permanent  construction, even if time seems to have regularised it. the older red brick houses  are rough, built side by side and on top of one other in layers that remind one of a  fragile domino rack. the red bricks remain exposed and uncovered as they are;  they  never  get  dressed  with  cement,  and  the  walls  never  get  painted.  the  en­  trenched network of small, almost impenetrable streets is a labyrinth where vul­  nerable electricity installations meet flying water pipes and open sewers. for an  outsider, a favela signals resignation.  the film cidade de deus (city of god) provides an impression of life, and  of criminal life in particular, in one of the most famous favelas in rio de janeiro. 1  that is the picture that many people have when thinking of a favela. however,  favelas in other cities  in brazil, with different conditions,  look more like slums  where disadvantaged people struggle to make a living.  the metropolis of today includes a patchwork of neighbourhoods and eco­  nomic extremes. one finds squatter settlements beneath highway junctions where  the  passing  of  speedy,  fashionable  new  cars  almost  blow  poverty  away.  rich  neighbourhoods and favelas are separated by only a few streets. the patchwork of  diversity is kept together by invisible threads that also maintain radical forms of  separation. rich condominios (gated communities) are surrounded by high walls  topped with electric wires. a guarded gate separates the outer reality from the ap­  parently  protected,  wealthy  life  inside  a  condominio,  which  looks  more  like  a  small city surrounded by a wall than a neighbourhood. here, unlike most houses  in  brazilian  cities,  no  walls  separate  the  houses  and  windows  are  not  barred.  green lawns and gardens, crystal blue swimming pools, and well­dressed families  certainly contrast with the air of messiness that emanates from a favela only a few  streets away from the outer walls of the condominios.  that  students  coming  from  different  neighbourhoods  experience  different  educational  opportunities  is  no  new  eye­opener  in  educational  research.  many  studies focusing on students’ backgrounds and their influence on education have  provided evidence of the fact that there is a strong relation between students’ ma­  terial and cultural life conditions and their experience in an educational system. it  1 cidade de deus is an oscar­nominated brazilian film, released in its home country in 2002 and  worldwide in 2003. it was adapted by bráulio mantovani from paulo lins’s novel city of god  (1997/2006), which is based on the true story of the parallel lives of two young men from a favela  in rio de janeiro. http://en.wikipedia.org/wiki/academy_award http://en.wikipedia.org/wiki/brazil http://en.wikipedia.org/wiki/2002 http://en.wikipedia.org/wiki/2003 http://en.wikipedia.org/wiki/paulo_lins http://en.wikipedia.org/wiki/city_of_god_(novel) skovsmose et al.  brazilian favela  journal of urban mathematics education vol.1, no.1  37  is beyond the scope of this article to provide a thorough account of research do­  cumenting this relationship because there have been many in different countries in  the world. researchers such as cooper and dunne (1999) in england, zevenber­  gen (2001) in australia, vithal (2003) in south africa, and oakes and collabora­  tors (2004) in the usa, have provided an analysis of this issue in operation in ma­  thematics and science education.  our intention in this article is to bring into the discussion a set of different  theoretical tools to cast light on the relation between students’ life conditions and  their educational experiences and opportunities. students coming from different  neighbourhoods can experience and foresee very different life opportunities. stu­  dents belonging to disadvantaged and marginalised social groups are faced with  the stark question of who they are and who they can become. students’ percep­  tions of their future life possibilities are full of conflicting experiences, realities,  dreams, and hopes for the future. all of these can impact students’ motives for  engaging in schooling and learning in general, and in learning mathematics in par­  ticular. in what follows, we start by introducing the notions of foreground, inten­  tions for learning, and borderland position. we explore the potentiality of these  notions by relating them to a conversation with a group of brazilian students in a  favela. we highlight some of the issues that we see emerging from the interview  in relation to the notions, and we conclude by discussing the potentialities of the  concepts in relation to mathematics education.  foregrounds, intentions for learning, and borderland position  we have been developing the notions of students’  foregrounds and  inten­  tions for learning over a longer period of time, while only recently have we try to  explore the notion of borderland position. we define a person’s foreground as his  or her interpretations of life opportunities in relation to what appears to be accept­  able  and  available  within  the  given  socio­political  context  (see,  e.g.,  alrø  &  skovsmose, 2002; skovsmose, 1994, 2005a, 2005b). this notion emphasises that  students’ engagement in learning is deeply rooted in the meaning they attribute to  learning with respect to their future life. in this sense, the intentions for learning  might be connected not only to the “past” or the background of a student, but also  to his or her “future” or foreground. seeing meaning in learning as related to the  future, rather than to the past, emphasises that students’ making sense of school­  ing in general, and of mathematics education in particular, is not only cognitive in  nature  but  also  socio­political.  meaning  given  to  learning  is  bounded  by  the skovsmose et al.  brazilian favela  journal of urban mathematics education vol.1, no.1  38  learner’s social, political, cultural, and economic conditions and how the learner  interprets them. 2  the notions of students’ foregrounds and intentions for learning have been  used  in  interpreting  a  variety  of  educational  phenomena.  some educational  re­  search has located certain groups of students as having problems with mathemat­  ics. a grotesque example of such a stigmatization is found in the so­called white  research in black education, conducted during the apartheid period in south af­  rica (see khuzwayo, 2000, for a critical discussion of this research). this research  identified black children as low achievers in mathematics and suggested an expla­  nation of this observation in terms of a deficit discourse. this discourse could take  different racist formats: the weak performances of black children are due to their  biological origin; or:  the weak performances are due to the structures of  black  families. however, do we consider the black children’s foregrounds in apartheid  south africa; it simply appeared ruined due to the very apartheid regime. a socio­  political and economic destruction of opportunities for a certain group of people is  a tremendous obstacle for learning. considering the students’ foregrounds might  reveal the limitations of deficit  interpretations of school performances, and turn  the attention to the socio­political and economic formatting of life opportunities,  and, as a consequence, of conditions for learning.  in  previous  studies,  we  have  illustrated how the way  students  experience  learning  may  relate  to  their  foregrounds.  in  alrø,  skovsmose,  and  valero  (in  press) we interviewed 8th grade students in a multicultural school in denmark. in  one inter­view, razia, an iraqi refugee, clearly points out how, in her perception  of her school mathematics experience and her hopes for the future, discrimination  is present. her reaction to this discrimination  is  incarnated  in her head­scarf, a  symbol of muslim womanhood that she herself has decided to keep and defend  fiercely as a way of showing who she is, where she comes from, and what she  wants to become. valero (2004) illustrates how the mathematical school experi­  ence of colombian students in poor public schools is deeply rooted in the socio­  political context where the students act as human beings. escaping a harsh life  might be a reason to learn, however, not powerful enough to give full meaning to  school mathematics. in skovsmose, alrø, and valero (2007), we have explored  how a group of indigenous students in brazil see their foregrounds, and the mean­  ing they attribute to the experience of learning mathematics. the apparent lack of  significance of mathematics is replaced mainly with an instrumental significance.  2 such a definition of foreground allows thinking about the similarities and differences with other  powerful notions such as “identity,” which has been increasingly used in educational research. the  discussion of identity and foreground deserves an article on its own. suffice it to say, here, we see  similarities and differences with the notion of identity as presented by, for example, sfard and  prusak (2006). skovsmose et al.  brazilian favela  journal of urban mathematics education vol.1, no.1  39  baber (2007) has studied how pakistani families in denmark see mathematics as  playing a central role in their participation as citizens of the country, and he points  to the uncertainty about the future that characterises their current situation.  we see the students’ foregrounds and their intentions for learning as closely  related. furthermore, we find that both foregrounds and intentions are structured  differently for different groups of students. here, we will pay particular attention  to the notion of borderland position, which refers to a position from where the in­  dividual can see his or her current life conditions in relation to other life possibili­  ties. the “borderland” metaphor has been used in research dealing with cultural  diversity to signal the vicinity and overlapping, as well as the conflict between  people’s participation  in different cultural worlds. 3  we see the  borderland as a  space of individual and social exchange where the meaning of difference is nego­  tiated. a borderland position is a relational situation where individuals meet their  social environment and come to terms with choices that diversity makes available  for them, as well as with the many choices that are beyond reach.  borderland positions exist for all people. for a person placed in a marginal  position in relation to the dominant culture or establishment, however, the border­  land position  shows the sharp and  clear contrast between  his or her world and  other worlds, particularly those belonging to the participants in the dominant cul­  ture. being in a borderline position allows that person to experience social, cul­  tural, and political differentiation and the stigmatization that operates through the  stories that the dominant culture constructs about his or her life. focusing on peo­  ple  in  borderland  positions  allows  us  to  have  an  insight  into  how  exclu­  sion/inclusion mechanisms operate and, more important, are experienced by those  deeply affected by them.  we now turn to the streets, houses, and people in a brazilian favela with the  intention of illustrating the significance of the notions in relation to how a group  of youngsters experience their mathematical learning.  inter­viewing students in a brazilian favela  in what follows, we will meet five students from a favela located in a large  city in the interior of the state of são paulo in brazil. pedro paulo scandiuzzi has  known them for some time and invited them to look into their future: how would  they like to see themselves in the future? could there be any “learning motives”  relating  mathematics  in  school  and  possible  out­of­school  practices,  either  in  terms of possible future jobs or further studies? ole skovsmose has also met the  five students and spoken with them. paola valero and helle alrø have never met  3 for further discussion of related notions see chang, 1999, and macdonald and bernardo, 2005. skovsmose et al.  brazilian favela  journal of urban mathematics education vol.1, no.1  40  the  students  personally  but  have  read  pedro  paulo’s  inter­view  transcript  and  ole’s accounts of his meetings with the students and with pedro paulo.  the  five  students  that  pedro  paulo  inter­viewed 4  were:  júlia,  mariana,  natália, argel, and tonino. 5 mariana was 14 years old at the time, while júlia and  natália were16 years old. the two boys, argel and tonino, were both 16 years  old. argel was eager to present what he wants in life, while tonino remained qui­  eter. mariana and natália talked rather freely, while júlia was normally relatively  quiet. but given that the inter­view took place in júlia’s house, she might have  taken upon herself the responsibility of being a hostess, and in this respect, she  participated eagerly.  júlia, mariana, natália, and argel attended a public school called floriano  paixoto. this school is comprehensive, containing primary, secondary, and upper  secondary levels. the school is surrounded by high walls. the gate of the school  is locked and watched by a guard who ensures that only those who are supposed  to enter, in fact do enter. in this city, even a poor school  is  in danger of being  robbed. the walls might also help to protect the students when they are in school,  as well as preventing them from escaping before they are allowed to leave. the  school is located in a densely populated and rather poor area of the city. part of  the area includes the favela cidade de são pedro, where four of the students come  from; tonino is from a nearby favela. tonino does not attend floriano paixoto but  an agricultural school called esperança verde, which is located on the outskirts of  meiadia, a neighbouring town. this school is surrounded by fields and has a vari­  ety of animals. the students have the opportunity to learn  farming through the  praxis of farming. the agricultural school applies an alternative educational pro­  gramme, where students have to be at the school for 2 weeks, and then work at  home for another 2 weeks. this alternating attendance ensures better possibilities  for students from poor families to go to school, given that their financial support  could be needed at home. in esperança verde, 5 hours per day are dedicated to  regular school subjects, while 4 hours are reserved for practical subjects. pedro  paulo and ole visited the schools, floriano paixoto and esperança verde. the  4  following  previous  studies  where  our  task  was  the  empirical  exploration  of  students’  fore­  grounds (see skovsmose, alrø, & valero, 2007; alrø, skovsmose, & valero, in press), pedro pau­  lo orchestrated a conversation with the students where questions about their life, their imagina­  tions for the future, their like for mathematics, and their perception of mathematics in their current  and future life were discussed. we use the term inter­view, inspired by kvale’s (1996) concept of  a semi­structured inter­view that develops as a conversation about selected topics. thus, a semi­  structured inter­view is “an interview whose purpose is to obtain descriptions of the life world of  the interviewee with respect to interpreting the meaning of the described phenomena” (p. 5). this  description also  implies an active asking of questions and exploring of answers between  inter­  viewer and inter­viewee that emerge through the conversation.  5 all names of students, schools, and locations are pseudonyms. skovsmose et al.  brazilian favela  journal of urban mathematics education vol.1, no.1  41  head of floriano paixoto showed them around and told about the stressful life of  directing  a school.  at  esperança  verde  two  students  showed them  around  and  talked about the organisation of the school.  pedro paulo has had contact with people from cidade de são pedro for a  long period of time. he knows many people there, and he is known by many. be­  cause the neighbourhood is nearby the university where he works, and the univer­  sity library is available for schools and students from the neighbourhood, pedro  paulo has had the chance to help these students when they have needed a hand  with homework or an activity in the university. in this way, pedro paulo has be­  come a friend, a person who is allowed in the favela even though he does not live  there. he has often visited júlia’s family, and júlia was happy to invite her friends  to her house for the inter­view. the inter­view was scheduled for the evening to  make it possible for the students to participate.  in what follows, we turn to the inter­view and listen to how the students de­  scribe their situations and their expectations and hopes for the future, and their  wishes for further education. 6  what do you not want to do with your life?  the small room in júlia’s house accommodates enough chairs to seat every­  one. some of the chairs have seats made of braided plastic strings, originally of  different  bright  colours.  time  and  use,  however,  have  made  them  appear  the  same. pedro paulo breaks the ice and tells a bit about himself:  pedro  paulo  (pp):  when  i  was  your  age,  14  and  16,  i  studied  in  a  public  school in a town close to here [...] i went to school, played ball, went fishing, took  small jobs, and dreamed of travelling, and that’s why i studied a lot. i dreamed of at­  tending good schools. and that was my life. i studied a lot. and afterwards, i left and  went to work in ubatuba 7  as a mathematics teacher. now i have returned, and i’m  working here at the university […] they say that i’m at the end of my life, being  over 50. so i’m getting to the end.  6 after pedro paulo conducted the inter­view, a transcript of the session was produced and trans­  lated into english. readings of the transcript were discussed between ole and pedro paulo, who  provided additional information and contextualization about the students’ ideas, based on pedro  paulo’s knowledge of them and their situation. all members of the research team discussed differ­  ent interpretations of the students’ words and of what seemed to be behind them. we do not use  the inter­view as an empirical documentation of the students’ actual thinking, motives, and inten­  tions. we use what they express as a window into a reality that triggers our reflections on the con­  cepts that we want to explore. the quotations from the inter­view are presented in the original  order. parts of the transcription, however, have been omitted.  7 ubatuba is a town along the coast between são paulo and rio de janeiro. skovsmose et al.  brazilian favela  journal of urban mathematics education vol.1, no.1  42  the  first  to  be addressed  is  argel,  who  is  in  his  2nd  year  of  upper  secondary  school. in addition to his regular classes, he takes a course in electronics and a  course to prepare for a military career. 8 such a career includes much competition,  but argel is ready to face the challenge. he says that he likes geography, history,  and biology, and also art education, although less so. he also likes mathematics  and portuguese a little. he prefers physics and chemistry, however. let us listen  to his remarks about mathematics:  pedro paulo (pp): what are you learning in mathematics?  argel (a): uh, i’m studying, now at this moment, matrixes; i’m studying ma­  trixes and i’m studying all the definitions—reverse functions, inverse of the matrix  […]  pp: and what have you thought about doing with these matrixes?  a: well, last year, other calculations appeared; this year for me, what am i going to  do with this […] another course that i’m taking is electronics. the matrixes i’m go­  ing to use—they have a binary sequence.  argel  is working with  matrixes:  their definition  and  formal properties. he also  refers  to  possible  connections  between  matrix  calculations  and  the  electronics  course he is taking. the calculus of matrixes might well be included on the exam  argel needs to pass in order to get started on his military career. he tries to clarify  connections between matrixes and binary numbers. it  is obvious, however, that  the possible applications of matrix calculus are not clear to argel.  crucial to argel is his choice of career. he is interested in the military, and  this priority provides meaning to many other activities in school:  a: i’m taking a preparatory course for the military.  pp: you want to be in the military?  a: i like it, the army or the naval air force.  pp: the army or the naval air force?  a: i’m not sure yet, where i’ll go.  pp: is that what you want to do with your life?  a: yes.  pp: what do you not want to do with your life?  a: hanging out here without doing anything, making a living doing what? i’m not  going to keep depending on my parents for the rest of my life.  argel has not made up his mind if he prefers the army or the naval air force. but  his overall decision is made: he wants to pursue a military career. argel certainly  does not want to hang out in the neighbourhood doing nothing. and he does not  want to be financially dependent on his parents.  8 the course prepares students for the entrance exams of the military schools. skovsmose et al.  brazilian favela  journal of urban mathematics education vol.1, no.1  43  argel’s comments touch upon the notion of meaning. learning about ma­  trixes  might  not  be  experienced  as  meaningful  because  of  applications  that  he  knows about; but rather, it might have instrumental significance, if it is significant  for passing tests crucial for his future career. in fact, students might be ready to  accept an instrumental significance as a preliminary resource of meaning as they  assume that what they learned could later turn out to be relevant.  to escape from the city a little  tonino opted to study at esperança verde, near meiadia. but why did he  choose to do so?  tonino (t): ah, to escape from the city a little.  pp: escape from the city a little? are your parents agricultural workers?  t: my mother is a seamstress, and my father works in a factory.  pp: yes, but were they agricultural workers before?  t: my mother lived in the country. i don’t know about my father.  pp: what led you to want agricultural school?  t: employment, you know—leave there with employment guaranteed.  tonino wanted to escape from the city. however, he does not seem to have con­  nections to rural life, except that he knows that his mother once lived in the coun­  try. it might not be the content of agricultural work that provides the main attrac­  tion for tonino. it seems important to him to change location, and maybe, first of  all, to be able to secure a job. this choice could provide stability in life, different  from being a seamstress or a factory worker. tonino seems to believe that an ag­  ricultural education would lead to “guaranteed employment.”  people from certain neighbourhoods in the city are not considered to be “re­  liable,” and they have difficulty getting a job. so in order to get a job, it is not  only important to get an education that could lead to a permanent  job;  it might  also be important to change location, in order to get rid of the stigmatization that  people from certain neighbourhoods, like cidade de são pedro, suffer. 9  which school subjects does tonino like the best? could his preferences in  school have something to do with the choices he has made?  pp: what are the courses you like the least?  t: history and portuguese.  pp: do you like mathematics?  t: more or less.  pp: what have you studied in mathematics?  t: i don’t remember.  9 students comment on this issue later in the inter­view. skovsmose et al.  brazilian favela  journal of urban mathematics education vol.1, no.1  44  pp: you don’t remember? what are you going to do with this subject matter that you  don’t remember?  t: i don’t remember anything.  tonino  might  be  referring  to  the  mathematics  from  the  secondary  school.  he  might also be referring to mathematics at esperança verde. as mentioned before,  the schedule in the school is organised around 2 weeks of work in school and 2  weeks of work at home. but mathematics, as well as any other school subjects, is  out of tonino’s memory. he does not remember anything. then the conversation  includes argel again:  pp: argel plans to do what in the military, be a soldier?  a: yeh, i suppose, you know […] there you start out as a soldier and then you pass  the exams, tests, to rise in rank to captain, sergeant […] something to grow there  within—but it’s a course there. you go through a public exam, and if you pass, you  have technical courses, and after you leave there, you can even work in a  large  fac­  tory. because the highest salary is for colonel, retired earning double, in these indus­  tries.  pp: you want to work in a factory?  a: no, i want to take the course, because in addition to the military high school, i  have technical courses in the morning, and in the afternoon, i practice and earn a  salary like the ita [instituto técnico da aeronautica – technical institute in aero­  nautics],  the  espcex  [escola  preparatória  de  cadetes  do  exercito  –  preparatory  school for the army], or aman [academia militar das agulhas negras – special  force academy].  pp: you want to do one of those?  a: i want to do the ita.  pp: do you study a lot?  a: at least 2 hours a day; if not, i don’t pass the tests.  pp: two hours a day. you work, too, or just study?  a: not me, i don’t have time. i study during the week and on saturday. i only have  sundays free.  argel knows about career possibilities and about how to obtain them: studying,  despite the fact that studying “at least 2 hours a day” seems to be considered at  lot. argel expresses  his  interest clearly. but what about tonino? is his  interest  limited to getting out of the city and getting a job? are there more reasons for  argel to “remember” mathematics in light of his desire to enter the military, than  for tonino who wants only to get a stable job? it might well be that stronger de­  sires for the future bring better reasons to want to remember school mathematics.  what do you remember?  at this moment, the girls enter the conversation; first júlia, who is kind of a  hostess. the subjects she likes include art education and physical education, while  she does not like portuguese, which she finds to be very difficult. skovsmose et al.  brazilian favela  journal of urban mathematics education vol.1, no.1  45  pp: do you like mathematics?  júlia (j): more or less.  pp: what are you studying in school right now in mathematics?  j: i’m reviewing the subject matter from the 3rd quarter for the test.  pp: what subject matter, do you remember?  j: delta, sets, images, things like that.  pp: in the future, what do you plan to do with this mathematics that you are learn­  ing?  j: i don’t know what i’m going to choose as a profession. i think it [mathematics]  will help.  júlia’s  first answer to what she  is  studying does not concern the  mathematical  content. she studies for the test. asked directly about the subject matter, júlia re­  fers to topics like delta, sets, and images (of functions). delta is the expression ∆  = b 2 – 4ac used when solving the 2nd­degree equation ax 2 + bx + c = 0. and what  to make of this when one thinks of future education? júlia, certainly a polite host­  ess, confirms that although she does not know what she will choose as profession,  she thinks that mathematics will turn out to be helpful. thus, júlia also seems to  believe in the instrumental significance of mathematics.  later in the conservation, júlia emphasises that she does not want to become  a housewife and do housework. she does not want to stay at home preparing food  for her husband. she says that she might want to study healthcare or medicine.  these are ambitious wishes, and it might well be that júlia knows that mathemat­  ics  composes  part  of  such  studies,  although  she  does  not  know  in  what  way  mathematics will be useful.  a housewife, in my opinion, is a slave  natália is 16 years old. she is in the 2nd year of the upper secondary school.  júlia and natália are in the same grade, although they are not in the same class.  pp: but you’re not in the same class? what are you studying in mathematics?  natália (n): we’re doing […] seeing some things about 2nd­degree functions,  the delta. these 2nd­degree things.  pp: what do you like least about school?  n: the teachers.  natália remembers the “2nd­degree things.” she seems to remember more than  tonino, but a bit less than júlia. natália expresses clearly her dislike for teachers.  then she is asked what she would not like to be:  n: a housewife.  pp: you don’t want to be a housewife? skovsmose et al.  brazilian favela  journal of urban mathematics education vol.1, no.1  46  n: a housewife, in my opinion, is a slave.  pp: even if she owns her house?  n: even if she owns her own house.  pp: why do you think that?  n: ah! because everything you tell her to do, she does. she doesn’t avoid doing it.  even if she doesn’t want to, she does it. it’s like being a slave; you’re giving the or­  der, and she’s following it.  in natália’s view, a housewife is given orders and follows orders. this compli­  ance is  like the life of a slave, even if she is the owner of the place. natália’s  words resonate with júlia’s. in the favela, girls have seen many women, starting  with their own  mothers, and they  express their  positive rejection of  a  life as  a  housewife. studying and choosing a profession seems to be a way of escaping that  frightening  scenario. natália therefore dreams of becoming a psychologist or a  veterinarian. she likes animals very much, and she likes psychology because she  likes to listen to people talk about their lives and to give them advice. when asked  if  mathematics  has anything to do with  veterinary  medicine or psychology  she  answers:  n: nothing.  pp: it has nothing to do with it? júlia, tonino, argel, do you know what psychology  and veterinary medicine would have to do with mathematics?  t: i don’t have the faintest idea.  pp: no idea. so that means that what she’s learning in mathematics will not be very  useful to her?  n: i think it will, because when you go to a university, you have to study all the sub­  jects.  natália seems not to see the instrumental significance of mathematics with respect  to psychology and veterinary medicine. she, however, sees clearly that when one  gets to the university, one must “study all the subjects,” including mathematics.  that perspective might be reason enough to engage in school mathematics, above  all, to avoid being a housewife.  delta is just a formula  mariana wanted to be the last to talk. she lives in a neighbourhood near by.  she goes to the same school as argel, júlia, and natália. mariana is 14 years old  and she is in the 8th grade, the last year of secondary school. she likes the school  and the teachers, and she likes to study. but she does not like the school when  there is much quarrelling and disorder.  mariana intends to study law and become a lawyer, or maybe she wants to  study medicine. and what about mathematics? skovsmose et al.  brazilian favela  journal of urban mathematics education vol.1, no.1  47  mariana (m): ah! i’m in 2nd­degree delta, these 2nd­degree things.  pp: and what will you do with these 2nd­degree things in medicine, or as a lawyer  or a judge?  m: ah! i think, for sure i’ll need it to go to the university. i’ll need it.  pp: to go to the university. in your profession, you don’t think you’ll use it?  m: ah! i don’t understand it a lot. but i don’t think so. i don’t know.  like everyone else, except for argel, mariana does not know what to do with the  delta. well,  it might be necessary knowledge for entering the university or the  faculty of medicine. mathematics per se does not seem to be considered impor­  tant.  later in the inter­view, mariana mentions that she does not like portuguese,  and grammar in particular: “what sense to make of issues like subordinate clause  and punctuation?” mariana does not think of portuguese as being important for  studying law. however, if it turns out that it is, she will be ready to study it. then  pedro paulo returns to mathematics, and the students comment again on the delta  formula.  a: […] delta is just a formula. but you use it for the rest of your life.  m: you keep deepening it, complicating it, more and more.  pp: the delta gets complicated, just like life?  m: i think so.  j: more or less.  pp: more or less?  j: all is the same.  pp: as times goes it gets more complicated.  j: yes.  m: in first grade you learn 2 + 2 and then it gets more complicated, you learn to di­  vide.  delta is just a formula, but it seems to stick with you, as argel emphasises: “you  use it for the rest of your life.” it will appear in more and more complex situa­  tions, as all mathematics do. you start with simple things like addition, but it al­  ways gets more and more difficult. but as things get complicated, it seems as if  the meanings of mathematical expressions and techniques do not emerge in the  context  of  learning.  their  meaning  might  (or  might  not)  be  revealed  later  in  school  or  in  life.  students  seem  to  be  struggling  with  what  we  could  call  the  “delta syndrome,” a weird  kind  of  disease  in  which  the patients  are  presented  with some mathematical formula or technique, which they are supposed to master  in order to get on with their education, but whose significance will not be revealed  until later.  the inter­view then turns to a discussion of what the students’ parents are  doing. it is clear that tonino, argel, júlia, natália, and mariana are hoping they  will not become like their parents. mariana does not want to become a maid or skovsmose et al.  brazilian favela  journal of urban mathematics education vol.1, no.1  48  cleaning woman, a type of job that many women in the favela have, doing house­  work in other neighbourhoods. mariana would like to become a housewife, how­  ever. she cannot follow júlia and natália who think that a housewife is a slave,  even in her own house:  m: but a housewife, yes, because i like to do the housework at home. i’m the only  one who does it because my mom and dad work.  mariana’s mother works in the butcher shop owned by mariana’s father. natália’s  father works as a truck driver, and her mother is a seamstress. júlia’s father works  as a driver for the local government. part of his work is to assist in repairing the  roads. júlia’s mother works as a kitchen assistant. tonino’s father is working in a  furniture factory, while his mother is a seamstress. argel’ father has retired, and  his mother is a housewife.  pp: and she likes being a housewife?  a: she likes it, because she didn’t know her mother and father. she was raised by  her aunts, so she was their slave. at our house, we tell her not to do stuff, but she  ends up doing it. she likes to do things. i want to help her, but she doesn’t let me.  students’ expressions of their future profession are far from being inspired  by their parents’ current occupations. even when júlia and natália express their  dislike of the housewife life, they seem to do so in relation to the situation of their  own families and relatives. they hope for something different, probably better.  the exams are very complicated  the students  come to talk  about  the possibility  of  realising  their  dreams.  they believe it is possible to achieve what they hope for, but that there are many  difficulties. one is the tuition at private universities; another is the cost of the pre­  paratory courses for the college entrance exams. it seems particularly difficult for  those who dream of enrolling in some of the most expensive programmes (such as  medicine). for example, a driver like natália’s father earns about 800 reais per  month and one could expect that the study costs for natália would be around 400  reais per month; having children engaging in higher education puts a huge eco­  nomic demand on a family. a student could do some work in addition to their  studies, but a student’s salary would cover only a minor part of the study costs.  only if one chooses to study at night and work during the day is  it possible to  make a reasonable amount of money. another option is to enrol in shorter techni­  cal or vocational programmes; however, those programmes are less prestigious. it  is also possible to get some kind of scholarship; but then one must be an excep­ skovsmose et al.  brazilian favela  journal of urban mathematics education vol.1, no.1  49  tional student and have very good grades. anyway, the cost of engaging in further  studies is certainly a huge obstacle for making the students’ dreams come true.  the public universities are free but very difficult to enter. in brazil, each  university applies its own entrance exam, which applicants are charged for. they  can register (and pay) for as many exams at different universities as they want,  which  are  typically  administered  during  the  months  of  december  and  january.  the results, often published in early february, take the form of a ranking list of all  students that participated in the test. on the internet, one can see one’s position  and also where the cut­off for entry was made. naturally, the most attractive pub­  lic universities are the most difficult to enter. many hopeful applicants take the  exams, and the most attractive universities need only to select the top 10% of ap­  plicants. if one does not succeed one year, one can pay for a one­year study pro­  gramme to prepare for the next year’s exams. and so on, until one enters, or until  one gives up on the idea of doing further studies. again, the need for good exam  results seems to go against the realisation of their hopes and future expectations.  a: … the exams are very complicated.  n: there aren’t many people admitted, either.  students from a public school  like floriano paixoto  are unlikely to be as  well­prepared for the college entry exams as students from private schools. brazil  has  a  large  number  of  private  elementary  and  high  schools,  always  better  equipped than the public schools, and usually  more focussed on ensuring their  students  good  possibilities  for  pursuing  further  studies.  so  the  private  schools  provide the very best preparation for students entering the attractive public uni­  versities. the situation could be very different with respect to the public school,  as argel explains:  a: the classes they give, they are the same in the private high schools, and in the  public schools it’s the same. but the teachers are slow. they’re not too concerned.  some are concerned; others don’t even care about you. you, who are from the public  or municipal school.  n: in the public school, the teacher doesn’t care about what he does.  a: in public universities, it’s very difficult to find people like us who studied in the  public schools. in the public universities, they only have daddy’s little kids going  there. they are in no need of going to public universities.  pp: so what are you going to do? you are in the public schools. you depend on a  salary, and the salary isn’t high. you have the wish to get into a good program. what  are you going to do? are you going to say, like, we’re just going to stop here?  a: we have to study, to fight.  n: we have to make an effort.  the problem  is  clearly  formulated by  argel:  in  public universities,  there  is  no  room for many students from public schools. it is mostly well­off students who skovsmose et al.  brazilian favela  journal of urban mathematics education vol.1, no.1  50  manage to get in. the students really find their opportunities restricted by their  economic situation. some try to compensate by doing some extra courses. thus,  júlia does extra studies in english, and argel takes a course in electronics, includ­  ing computation. the lack of access to computers at home is a problem, so it is  important for students to take courses where they are able to get experience with  computers. the situation at home does not facilitate any form of study. most of  the time there are not adequate resources to study; normally there are many peo­  ple around, and it is difficult to find a quiet place to concentrate on studying. be­  sides, many other characteristics of life in a favela—such as violence, struggles  related to drug trafficking, and even sexual assaults—are not the most nurturing  for youngsters who want peace of mind and who are probably in need of getting  rid of the “delta syndrome.”  we’re discriminated against  it is not difficult to list obstacles that these students have to face in their life.  but are they able to find reasons for optimism as well?  pp: do you guys see this desire of yours with optimism/excitement or not?  j: ah! i get pretty excited when i think about what i want to be.  pp: and you, tonino?  t: you have to go after it.  pp: argel?  a: you have to fight. and if you get discouraged, feel down—you can’t get discouraged.  pp: why do you get discouraged, argel?  a: well, it’s kind of different when it’s time to study there. i feel discriminated against.  j: sometimes people fail; give up, too. you have to persist.  m: because public schools, the teaching is weak. not that it’s weak, it’s that the teachers  don’t care, and the students even less.  […]  n: we keep getting left behind.  m: i have a friend, he studies in the seta. he’s in the 8th grade. he knows five times  more than i do.  seta (sociedade educacional tristão de andrade – educational society tristão  de andrade) is an expensive private school, located in the city centre. according  to mariana and júlia, who know people attending this school, the students there  are far ahead of those who study in public schools,  including those who attend  floriano  paixoto.  for  them,  it  is  really  necessary  to  fight.  as  emphasised  by  argel,  even  during  education,  one  gets  discriminated  against.  they  perceive  schooling as a form of establishing and maintaining inequalities, rather than pro­  moting equity. skovsmose et al.  brazilian favela  journal of urban mathematics education vol.1, no.1  51  pp: argel, you said that you feel discriminated against sometimes. why do you feel  discriminated against?  a: ah! because they feel—they’re better than us, you know?  pp: who?  a: these people who are daddy’s little kids and are protected by their parents. then  they want to give us the cold shoulder. they think they’re better than we are.  the students  experience discrimination,  not only  in  terms  of  attitudes,  like  the  “daddy’s  little children”  who  think  they  are  better;  they  are  also  discriminated  against in real measures. in the private school, there are better teachers with more  commitment, and the students have better conditions for  learning. pedro paulo,  however, points to a fact that might serve as a counter­balance to their experience  of being left behind:  pp: did you know that in the universities a lot of people are entering that studied in  public schools? and those students are getting into a good study habit,  facing all  those people who had college preparation courses.  a:  they’re  the  bigger  schools  there  downtown,  aren’t  they?  […]  those  schools  downtown where the teachers are stricter.  j: they get after you more, demand more.  pedro  paulo  points out  that one finds  many  students  from  public schools  studying in the universities. argel, however, stresses that they are from the bigger  public schools  in the city centre, where teachers are  more strict, demand  more  and, therefore, prepare students more adequately for further studies than schools  in a poor favela neighbourhood. then mariana and argel add:  m: they [the students from downtown schools] don’t have the needs that we have.  they [the teachers] discourage us, too.  a: because of two or three in the class, she discriminates against everyone […] eve­  ryone pays for it; everyone is a trouble­maker. this is not true. just because of two  or three that are like that, everyone gets into trouble.  mariana emphasises that there are differences among students in public schools.  different students could have different needs. she indicates that teachers discour­  age students from poorer neighbourhoods to try to pursue further studies. argel  follows up by pointing to teachers who exercise discrimination and stereotyping.  there might be some students from their neighbourhood who might cause trouble  for the teacher, but “everyone pays for it” and all are discriminated against.  then  pedro  paulo  turns  to tonino  who  attends  the agricultural  school  in  meiadia. how are things experienced in this place?  pp: is it like that in meiadia, too, tonino?  t: we’re discriminated against in meiadia.  pp: you’re discriminated against in meiadia. skovsmose et al.  brazilian favela  journal of urban mathematics education vol.1, no.1  52  t: it’s the agricultural school they talk about. leaving to go to another city is diffi­  cult.  pp: and why did you choose a school that is discriminated against?  t: i didn’t know, either, right—i arrived there believing it was a wonderful place.  pp: ah! did they take you to visit?  t: it was my mom who visited the school.  pp: what school there has a good reputation?  t: ah! i don’t know.  now tonino realises that esperança verde might be a school that is also regarded  as having a very low status. it is a rural school, and according to tonino, they are  discriminated against in meiadia. the same is the case for floriano paixoto, lo­  cated in cidade de são pedro.  j: they think, like, that it’s poor suburbs. even we who live in the poor suburbs,  we’re discriminated against, if you look.  a: there when i arrive at home—cidade de são pedro is the worst neighbourhood  in our city, a favela. to get a job, it depends on courses. you get there to enrol, and  they’re even afraid to meet you.  j: because of two or three, we get it because of that. i already tried to get a job, and i  didn’t get one.  the students address not only the problem of being stigmatised by coming  from  the  favela of  cidade  de são  pedro,  but  also  teachers  might  exercise dis­  crimination. it might be difficult to get a job in other parts of the city. people in  general might feel afraid of someone coming from cidade de são pedro, as argel  says. the stereotyping of favela life as portrayed in the media falls on all its in­  habitants. júlia expresses it clearly: a few people get in trouble, but not all; still,  that affects her own possibilities for employment.  issues of life, learning, and mathematics in a favela  the inter­view between pedro paulo and argel, júlia, mariana, natália, and  tonino reflects different aspects of the life conditions of students in a favela as  they  perceive and  experience  them.  let  us  highlight  some themes  that  we  see  emerging  from  the  inter­view.  these  themes  are  related  to  the  students’  fore­  grounds being in a borderland position and they seem to influence their motiva­  tion for learning mathematics.  the first theme is discrimination. the students feel they are being discrimi­  nated against due to the fact that they come from a favela, a poor neighbourhood.  there is no doubt that the socio­economic conditions strongly limit the possibili­  ties for people from cidade de são pedro. favela life is a life in poverty, and pov­  erty stigmatises people. it affects many aspects of life: the clothes one is wearing  and one’s habits (young people from a favela do not go to the cinema, but they skovsmose et al.  brazilian favela  journal of urban mathematics education vol.1, no.1  53  might hang out at a gas station convenience store). it affects possibilities of doing  homework, of accessing books and other resources for doing homework, and of  studying. however, poverty not only sets a range of life conditions; it also frames  the way others look at one. based on their experience, the students feel it is better  not to reveal that they come from cidade de são pedro. they could be discrimi­  nated against, not only economically speaking, but also in terms of attitude: peo­  ple could look down on them, look at them as potential criminals. somehow pov­  erty also frames the way one looks at oneself.  the students fear being trapped in some stereotype, and there could be good  reasons for this fear. a dominant theme of the news in brazil is violence, often  associated with the favelas, particularly the famous ones  in rio de janeiro and  são paulo. as already mentioned, cidade de deus (the city of god) is both a  name of a favela in rio de janeiro and the title of a film about the meeting of life,  crime, and violence in this favela. this violence includes the wars between gangs,  the war to expand or keep the drug markets, and the war against the police. but it  also includes the everyday  life of  thieves who systematically assault  the trucks  that deliver tanks of propane gas to the households in other areas of the city. it  certainly  includes  the  struggles  of  many  workers  like  the  students’  parents  to  make a living, and the struggle of the students themselves to have a chance in the  future. all such common “knowledge” about life in a favela is the basis for the  construction  of  stereotypes  that  stigmatise  favela  inhabitants.  so  when  the stu­  dents react to the possibility of being discriminated against, they might well have  good reasons for doing so.  the second theme is escape. there is a strong motivation to begin a new life  away from the favela. however, it is not clear to what extent this “new life” is ex­  perienced by the students as something they, realistically speaking, could work  for, or as just something they dream about. there is a strong motive for escaping  the neighbourhood. it could be taken in a strict sense as expressed by tonino. but  “escape from the city” could also be taken as a metaphor for getting out of the life  conditions the students know all too well, such as júlia’s and natália’s reactions  towards being a favela housewife. they all acknowledge that the best way to es­  cape is through further education. therefore, the discussion of tuition fees for en­  tering the university becomes at the same time important and fatal.  a third theme concerns the obscurity of mathematics. is seems clear to eve­  ryone that education is relevant for ensuring a change in life. the role of mathe­  matics in changing life, however, is less visible. mathematics lessons do not pro­  vide any clue of how mathematics might function in this respect. one could see  an instrumental significance of mathematics, while the content of mathematics in  itself appears  meaningless. the mathematics curriculum  in brazil  is a manifest  representation of the school mathematics tradition. this tradition defines the cur­  riculum  with  strong  references  to  mathematical  ideas,  notions,  and  structures. skovsmose et al.  brazilian favela  journal of urban mathematics education vol.1, no.1  54  everyday examples might be included, but mainly to illustrate mathematical con­  ceptions, and not as situations to be explored in greater detail. the school mathe­  matics tradition places a particular emphasis on the teacher’s presentation of the  mathematical  content  (and  not  on,  say,  communication  among  students  about  mathematical problems). naturally, the teacher’s presentation takes on a particu­  lar significance when the students have no textbooks, and have to rely only on the  notes they take themselves. and in order to make reliable notes, what could be  better than carefully copying down what the teacher writes on the blackboard? a  nice pedagogical contract could be established between teacher and students. as  long as the teacher makes a careful presentation and students copy down the pres­  entation, then everybody has done their job properly, and good order can prevail  in the classroom. still the obscurity of mathematics prevails.  the inter­view indicated clearly that it was very difficult for the students to  point to any relationship between mathematics and their future studies and work.  argel gave it a try, but was not very successful. the only relationship they could  openly express was instrumental: mathematics is a necessary ingredient for pass­  ing required university entry examinations. at the same time, they did not deny  that mathematics  might  turn out to be significant;  they were  just unable to see  what this could be like. the “delta syndrome” was part of their experience.  this brings us to a fourth theme, namely the uncertainty with respect to the  future. the students are remarkably aware of what they do not want from the fu­  ture: argel does not want to hang out and be financially dependent on his parents,  tonino does not want to stay in the favela, and natalia does not want to become a  housewife.  and they agree that education  could  be an entry point  into another  kind of future life. the students find that they might have difficulties in compet­  ing  with  privileged  children.  they  find  that  the differences  are  established  be­  cause of  differences  in  schooling,  their  teachers  and  the resources  available  to  them. if one considers the ranking of the different schools in brazil, there is no  doubt that wealthy private schools top the list with respect to ensuring their stu­  dents’ access to private and public universities and colleges. schools  located in  favelas are very seldom found on such lists. the students also felt that teachers  might treat them as inferior; as someone who is not capable of completing further  studies.  the students could easily formulate very optimistic but almost unattainable  aspirations, while reality might set some heavy limitations. how, then, to get out  of such uncertainty? one way of getting out is simply to stop dreaming and hop­  ing,  and  instead  become  “realistic”  and  renounce  one’s  ambitions.  one  could  simply face that one is doomed to a poor modest life. so it may be better to get  out of school and get a job, a permanent job, if possible. leaving school, how­  ever, is not what the students want to do and actually seem to do: “you have to skovsmose et al.  brazilian favela  journal of urban mathematics education vol.1, no.1  55  fight,” “you have to go after it,” “make an effort,” “persist” are all expressions of  their feelings that they can influence their future life.  what is the significance of these issues of discrimination, escape, obscurity  of mathematics, and uncertainty with respect to the future for understanding the  way in which students decide to engage in learning mathematics? in what follows,  we explore a bit further the notion of students’ foregrounds, intentions for learn­  ing, and borderland position in order to address this question.  generating intentions for learning while constructing  foregrounds in a borderland position  we consider learning as an act, and as such it requires intentional engage­  ment on the part of the learner. this claim does not apply to all forms of learning;  thus  many  habits  may  be  adopted  without  much  intentional  engagement,  and  some forms of learning may be forced on people. when we see learning as action  we have in mind forms of learning as they might take place in school, for exam­  ple,  learning mathematics. students might get involved in solving mathematical  problems  or  be engaged  with  mathematical  investigations;  but  they  might  also  find the classroom activities to be without meaning and occupy themselves with  other things. a decision about being involved in the mathematical tasks, or not, is  not simply the result of a conscious individual choice, but rather a decision that is  strongly associated with the intricate relationship between the student, the teacher,  and  the  context  for  learning  in  the  social­political­cultural  environment.  the  meeting  between  the  individual  and  the  social  is  a  space  where  intentions  for  learning emerge and grow, or might be destroyed. in that space, the  individual  constantly  constructs  and  re­interprets  both  previous  personal  experiences  and  actual life conditions in dynamic relation to his or her wishes for life and dreams  for the future. in other words, the individual’s consideration of his or her back­  ground in relation to his or her foreground is a powerful source of reasons and  intentions to decide to engage in learning as well as a cause for giving­up to be  engaged in learning. 10  while the notion of background has been central in much research trying to  establish a connection between students’ learning experiences and students’ social  environment, the notion of foreground is relatively new. we find that the notion  of foreground has a close relationship with intentions for learning, which in turn  represents the broader meaningfulness that students might associate to processes  of  learning.  the  students’  foregrounds  are  constructed  through  different  social  10 for an in­depth discussion of the notions intentionality in learning, background, and foreground  see skovsmose (1994, 2005); see also alrø, skovsmose, and valero (in press). skovsmose et al.  brazilian favela  journal of urban mathematics education vol.1, no.1  56  processes. in a profound way they are constructed through economic conditions;  thus poverty is a highly influential factor. the construction of foreground, how­  ever, includes many other elements. in this article, our interest has been focused  on students who are constructed by others, and even by themselves, as marginal­  ized and excluded from dominant cultural practices and forms of life. when stu­  dents experience discrimination, they perceive that it will be difficult, if not im­  possible, for them to cross the line and become part of the dominant culture. this  experience strengthens their awareness of their own stigmatized position. we find  discrimination to be a powerful social factor, which might ruin the foregrounds of  certain groups of people.  for  the  five  students  from  cidade  de  são  pedro,  argel,  júlia,  mariana,  natália, and tonino, their borderland position allows them to constantly weigh a  set of favela­life opportunities against, for example, a set of “city­centre life” op­  portunities, or “condominio­life” opportunities. they can see what it would take  for them and for their education to cross the line to enter other ways of life. one  reaction to the experienced discrimination turns into a dream of escape. education  is clearly one possible way of doing so and, therefore, learning (mathematics)—  even if the reasons are purely instrumental—makes sense and represents a more  or less meaningful investment in the future. at the same time, however, they can  also see and experience the enormous barriers to a successful jump over the bor­  der.  their  borderland  position  makes  evident  the  harshness  of  social  division,  stratification, and stigmatization.  we could imagine a borderland school as the site of learning that provides  an opening for radically different life opportunities. (it might also be that such a  school would jail students in their current positions.) borderland schools should  be able to establish opportunities for a transition from one way of life to a differ­  ent one. at least students in a borderland school might consider such transitions to  be possible. what transitions, realistically speaking, a borderland school might be  able to prepare for is another question. the obscurity of mathematics has a strong  implication for the students’ experience of the opportunities a school might pro­  vide. there seems to be some agreement among the five students from cidade de  são pedro that mathematics might play a role in further education, but it is not  clear to them what role mathematics in fact could play. this lack of clarity means  that it is simply impossible for students to relate their activities in the mathematics  classroom to any more specific features of their foregrounds. as students’ fore­  grounds  are  associated  to  their  construction  of  meaning,  the  activities  in  the  mathematics classroom remain meaningless, or, as best, instrumental. this con­  struction is a huge  learning obstacle  for students  in a borderland position, who  experience an uncertainty with respect to their future. skovsmose et al.  brazilian favela  journal of urban mathematics education vol.1, no.1  57  in previous studies, we analysed brazilian, indigenous students’ perceptions  of  their  educational possibilities and priorities. 11  one student had made a clear  choice: he wanted to study medicine. completing such study would certainly es­  tablish a new life situation for him. however, his priority did not include a break  with his indigenous background and life in the indigenous village. he wanted to  study medicine with the particular aim of being able to return to the village and  contribute to the effort to improve the health situation of the indian community.  therefore, one has to be aware that possible transitions can be thought of in very  different ways. when one talks about transitions, one should not assume any sim­  plistic scale of  preferences.  for  example,  it  should  not  be assumed  that  white,  middle­class priorities and life opportunities are, by definition, “better” than some  other forms of priorities. one should not assume that the scale of priorities reflects  a scale of economic wealth. nor should one try to romanticise poverty. we try to  avoid  assuming  any  simplistic  scaling,  and  instead  to  listen  to  how  priorities  might be expressed,  how students  might  think of possible transitions, and  how  they can be related to their learning motives.  postscript: the fragility of dreams  almost  3  years  after  the  inter­view,  pedro  paulo  and  ole  again  visited  júlia’s  family.  júlia,  natalia,  argel,  and  tonino  were  all  there;  mariana  had  moved to another city. it was a nice evening. the four students told about what  had happened to them during the past 3 years and about their current situations.  they told about what had become of their dreams and aspirations.  júlia’s family had moved to a house in the countryside. three dogs barked  and wagged their tails welcoming the visitors, together with chickens and ducks,  while cows grazed in the field next door. the garden had vegetables, and pedro  paulo picked a small bag of lemons from a tree to bring home. there were also  some other friends around in the house. júlia’s mother had cooked the food, and  her father showed the guests around.  júlia was not talking very much, and when her boyfriend arrived—he was  even more taciturn than júlia—they spent the rest of the evening holding hands.  júlia had stopped her studies, and she was now working as an assistant in a law­  yer’s office. she was considering  starting studying again  in order  to become a  nurse specialized in radiology, which is a programme that can be completed in  only 2 years.  tonino had left the agricultural school. he had become much more talka­  tive, and now he wondered why he had started at the agricultural school at all.  farming was not really something he found interesting, which he expressed while  11 see skovsmose, alrø, and valero in collaboration with silvério and scandiuzzi (2007). skovsmose et al.  brazilian favela  journal of urban mathematics education vol.1, no.1  58  pressing his long thin fingers firmly together. he liked the city, and he had found  a job. he was working in a goldsmith’s shop, and one of his jobs was to put to­  gether different components of the jewellery. did he like the work? he was not  sure. he said that he would like to become a policeman. he believed this profes­  sion would bring him better opportunities in life.  natália had begun studying to become a nurse. she helped her mother with  the housework. she also helped her mother in her work as a seamstress. natália  had entered a private institution, and she had to pay for her programme of study.  she was receiving a small scholarship, but the largest portion of the money she  needed came from her parents.  during the evening, argel was the one who spoke the most. he had stopped  his studies and was no longer considering a military career. he had arrived that  evening with his wife and their small baby. it was a smiling baby who, in a good  mood, said hello to everyone who wanted to touch and tickle him to make him  smile, which he did. it was a happy family, and argel took perfect care of his son.  he was considering moving to a city in the neighbouring state of minas gerais  where he saw some better opportunities for getting a job. he hoped to work with  computers.  when one considers students’ learning of mathematics in a borderland posi­  tion, one sees many factors in operation. we have pointed to discrimination, es­  cape,  obscurity  of  mathematics,  and  uncertainty  with  respect  to  mathematics.  meaningfulness (or lack of meaningfulness) of learning cannot be analysed if one  concentrates on particular elements of the situation. intentions in learning have to  be related to students’ backgrounds as well as to their present situation and fore­  grounds. argel, júlia, mariana, natália, and tonino are still on their way, seeking  a better future. the complexity of  the situation,  however, renders  their dreams  fragile.  acknowledgments  this paper is part of the research project “learning from diversity,” funded by the danish re­  search council for humanities and aalborg university. we want to thank the students for partici­  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beyond the numbers: the brilliance of black children in mathematics d e p a r t m e n t o f m i d d l e s e c o n d a r y e d u c a t i o n a n d i n s t r u c t i o n a l t e c h n o l o g y p.o. box 3878 atlanta, ga 30303-3978 phone: 404/413-8060 fax: 404/413-8063 november 11, 2011 dear conference participants: on behalf of georgia state university and the benjamin banneker association, we welcome you to the 2011 benjamin banneker association (bba) conference beyond the numbers: the brilliance of black children in mathematics, sponsored and generously funded by the national science foundation (nsf). the conference theme—the brilliance of black children—aims to motivate a different discourse about black children and mathematics teaching and learning. in general, the conference represents the steadfast commitment of bba and nsf to positively influence mathematics access and excellence for black (all) children. the conference participants include mathematicians and mathematics teachers, administrators, educators, and researchers who strongly support this commitment toward mathematics access and excellence for all children. many agree, we must continue to demand equitable opportunities to learn so that, even among the injustices of racism, black children, like benjamin banneker himself, might be purposeful and innovative in their learning of and engagement with mathematics. the conference speakers represent researchers and scholars throughout the united states who are visible advocates for the teaching and learning of rigorous, meaningful, and culturally relevant mathematics for black children. during the symposia and breakout sessions, you will learn about some of the most up-todate research on teacher preparation and professional development for mathematics teachers of black children, mathematics persistence and achievement among black children, mathematics identities and intellectual communities of black children, and how black children might acquire critical mathematics literacy through social justice and cultural relevant pedagogy. we organized the conference to provide you with several opportunities to meet others who hold similar beliefs about the brilliance of black children so that you might share ideas and some of the work in which you are engaged. overall, we hope the conference provides ongoing opportunities for networking, collaborating, and developing professional learning communities and relationships. again, we extend a warm welcome to you and all our colleagues in the education community who share our passions in assisting, as dr. asa hilliard wrote, all children in reaching ―levels of [mathematics] excellence.‖ we are grateful for your presence and active participation in what promises to be a most stimulating and enjoyable event! warmest regards, david w. stinson, ph.d. conference co-host pier a. junor clarke, ph.d. conference co-host erika c. bullock conference administrator c o l l e g e o f e d u c a t i o n o f f i c e o f t h e d e a n p.o. box 3980 atlanta, ga 30302-3980 phone: 404/413-8100 fax: 404/413-8103 november 11, 2011 dear colleagues: welcome to our college, georgia state university, and this national conference on the brilliance of black children in mathematics. i am grateful to all for your attendance, the sponsorship of the national science foundation and benjamin banneker association, and the conference co-hosts drs. david stinson and pier junor clarke. each day of the conference features two symposia where education scholars, researchers, and teacher educators present current trends and up-to-date research on the mathematics teaching and learning experiences of black children. and on both days, these symposia are followed by small breakout sessions where conference participants can discuss in-depth how all children might be provided learning opportunities to reach levels of mathematics excellence. given our college’s explicit focus on urban education, we are pleased to provide the critical mathematics educators and classroom teachers assembled the space for two intense days of engagement and reflection about how we might all contribute to achieving excellence in the mathematics education for black children. please enjoy your stay and take time to learn more about our college’s commitment to urban education. we are at your service. wishing each of you a successful conference experience, r.w. kamphaus, ph.d. dean and distinguished research professor presumption should never make us neglect that which does not appear easy to us, nor despair make us lose courage at the sight of difficulties. –benjamin banneker president’s message cheryl adeyemi, ph.d. virginia state university november 11, 2011 greetings conference participants: on behalf of the board and membership of the benjamin banneker association, i welcome you to the 2011 benjamin banneker association conference beyond the numbers: the brilliance of black children in mathematics – sponsored by the national science foundation and georgia state university, atlanta, ga, november 11–12, 2011. the benjamin banneker association (bba) wishes to thank dr. jacqueline leonard, bba past president, and drs. peter appelbaum, erika davila, and david stinson for their vision and leadership in acquiring the generous national science foundation funding for the two mini-conferences (philadelphia 2010 and atlanta 2011) and summit (denver 2010) series. bba also extends its gratitude to dr. pier junor clarke and ms. erika bullock, co-host and conference administrator, respectively, for the atlanta conference, as well as the faculty and student supporters and volunteers from georgia state university (gsu). we are very encouraged by and appreciative of the intimate involvement of gsu graduate students—our next line of defense in our advocacy for black children in mathematics. this conference provides bba members and supporters with yet another opportunity to come together, share our research and expertise, and learn more about the most effective and promising ways of mathematics teaching and learning for black children. the symposia and breakout sessions during the next two days provides our village of mathematics educators with a greater individual and collective understanding and appreciation of the multiplying successes that we might celebrate and the continuing challenges that we will face—thus far, and those to come. the research presented during the conference has the specific aim to highlight and honor the great potential and brilliance of black children in mathematics and to increase the growing numbers of advocates for the teaching and learning of mathematics for black children. bba wishes to thank the conference presenters whose scholarship and research provides a foundation for the important equity work to which we and others have committed ourselves. without your dedication and concern for the teaching and learning of mathematics for black children, this conference and the advocacy work of bba would not survive nor thrive. the benjamin banneker association heralds this conference as another important event in our yearlong celebration of our 25th anniversary. as we celebrate 25 years of advocacy, we are reminded of the work yet to be done. the inadequate number of high-quality environments of mathematics teaching and learning in u.s. schools for black children continues to be a key factor contributing to their underperformance. as a result of these conditions, black children and other children of color are denied access to challenging mathematics, advanced mathematics courses, and stem academic and professional opportunities. as those who advocate mathematics education reform continue to explore and create ways to improve the teaching and learning of mathematics, we will find ourselves critically examining new initiatives and programs to determine their possible impact on the teaching and learning of mathematics for black (and all) children. we want to ensure that our advocacy will be focused and truly make a difference in the lives of black children. for instance, currently, bba is promoting an initiative to analyze the potential impact of the common core standards (ccs) on black children. we will be an important voice in the discussion that determines how ccs can be effective with black students so that we don’t leave our children ―even further behind.‖ bba asks those of you assembled here this weekend to join our efforts to critique this new curricular initiative through your research agendas and expertise. at the october 16th unveiling of the statue of dr. martin luther king jr. in washington, dc, reverend bernice king reminded those in attendance that regardless of the countless challenges and struggles we face today and tomorrow, we must not give up…we must not become weary…there is much work to be done. in the spirit of dr. king, 25 years ago, seven courageous visionary mathematics educators founded bba to provide a forum for discussing the successes and challenges and needed advocacy work for the learning and teaching of mathematics with respect to african american children. this work, began by the founders of bba—similar to dr. king’s work—is not yet complete. there is much work to be done. the benjamin banneker association is confident that this conference will stimulate further scholarly research that will advance the vision and work of bba. as we raise the bar in terms of advocacy, membership, services, programs, publications, and collaborations with other organizations, we encourage all conference participants to become active bba members. we ask each of you to learn more about the history of bba and its advocacy work over the past 25 years. on this, our 25th anniversary, we challenge you to join and get (and stay) involved with your bba! in closing, on behalf of the bba board and members, i wish to once again thank everyone who will make this conference a success. we say, ―job well done!‖ (in advance). think deeply and reflectively on your experiences during these two days and ask yourself what more will you do and what more can we do together. remember that benjamin banneker members are committed to removing the obstacles that keep black children from achieving parity of opportunity to study and excel in mathematics. become a part of this quest. if you are not currently a member—join bba by the end of this conference! –if not now, then when… and if not us, then who? sincerely and respectfully, cheryl adeyemi cheryl adeyemi, ph.d. president – benjamin banneker association founding members benjamin dudley edgar edwards, jr. william greer harriett haynes marie jernigan genevieve knight dorothy strong 6 the brilliance of black children in mathematics beyond the numbers: the brilliance of black children in mathematics friday november 11, 2011 11:00–12:15 registration student center lunch urban life 12:15–1:00 welcome:  dean randy kamphaus  dr. david stinson  dr. pier junor clarke georgia state university logistics: ms. erika bullock speakers auditorium symposium i black children and mathematics teacher education 1:00–2:00  dr. shonda lemons-smith georgia state university  dr. dorothy white the university of georgia  dr. christine thomas georgia state university speakers auditorium 2:00–2:15 break symposium ii black children and mathematics success 2:15–3:15  dr. robert berry university of virginia  dr. brian williams georgia state university  dr. christopher jett georgia state university speakers auditorium 3:15–3:30 break 7 the brilliance of black children in mathematics breakout sessions* 3:30–5:00 presenters from symposia i and ii will facilitate five (5) breakout sessions (see table 1) student center rooms 5:00–5:15 break program celebrating 25 years of the benjamin banneker association 5:15–6:15  ms. linda gojak nctm president-elect john carroll university  dr. cheryl adeyemi bba president virginia state university speakers auditorium 6:15–7:45 networking reception urban life saturday november 12, 2011 8:00–9:00 breakfast keynote address:  dr. joyce king georgia state university urban life symposium iii black children’s identities and communities 9:00–10:00  dr. james earl davis temple university  dr. danny martin university of illinois at chicago  dr. erica walker teachers college columbia university speakers auditorium 10:00–10:15 break 8 the brilliance of black children in mathematics symposium iv black children and critical mathematics literacy 10:15–11:15  dr. jacqueline leonard university of colorado denver  dr. natasha brewley georgia gwinnett college speakers auditorium 11:15–11:30 break breakout sessions* 11:30–1:00 presenters from symposia iii and iv will facilitate six (5) breakout sessions (see table 1) student center rooms 1:00–2:00 lunch wrap up:  dr. david stinson  dr. pier junor clarke georgia state university closing remarks:  dr. jacqueline leonard bba past president university of colorado denver urban life *note: questions to explore during breakout sessions 1. how might the ideas presented in the symposia be readily implemented in schools and classrooms? 2. how might the ideas presented in the symposia be readily accessible to family and community members, leaders, groups, and organizations? 3. how might the ideas presented in the symposia be readily integrated into education policy at the local, state, and national levels? 4. how might the ideas presented in the symposia be readily used for local, state, and national advocacy and activism? 5. what is missing at conferences? what is missing in research? what are some suggestions for future conference themes and research agendas? 9 the brilliance of black children in mathematics table 1: breakout sessions day s presenter title of presentation (partial) room floor fri i s. lemons-smith tapping into the intellectual capital of black children (with b. williams) capital 2nd fri i d. white preparing preservice teachers to educate black students lanier 2nd fri i c. thomas influence of an online learning community on teacher retention lucerne 2nd fri ii r. berry identities of black boys who are successful with school mathematics sinclair 2nd fri ii b. williams the value of early access to mathematics (with s. lemons-smith) capital 2nd fri ii c. jett critical race theory perspective on ―race‖ in mathematics education 460 4th sat iii j. davis understanding black student identity at the intersections capital 2nd sat iii d. martin the making of black children in mathematics education lanier 2nd sat iii e. walker mathematics engagement and socialization within and across generations lucerne 2nd sat iv j. leonard enacting social justice and culturally relevant pedagogy sinclair 2nd sat iv n. brewley mathematics literacy for liberation and liberation in mathematics literacy 460 4th 10 the brilliance of black children in mathematics speakers & abstracts 11 the brilliance of black children in mathematics keynote speaker joyce e. king, ph.d. benjamin e. mays endowed chair for urban teaching, learning, and leadership georgia state university jking@gsu.edu in her keynote presentation, professor king addresses the importance of both academic and cultural excellence in education and the contributions of african and african american people’s heritage of mathematics excellence can make to transformative education for human freedom. dr. joyce e. king is professor of social foundations of education and holds the benjamin e. mays endowed chair for urban teaching, learning, and leadership at georgia state university. the former provost and professor of education at spelman college and associate provost at medgar evers college, dr. king is recognized here and abroad for her contributions to the field of education, including the concepts of ―dysconscious racism,‖ ―diaspora literacy‖ and ―heritage knowledge.‖ her publications include four books: preparing teachers for diversity; teaching diverse populations; black mothers to sons: juxtaposing african american literature with social practice; and black education: a transformative research and action agenda for the new century. dr. king is a graduate of stanford university where she received a doctor of philosophy degree in social foundations of education and a bachelor of arts degree (with honors) in sociology. she also holds a certificate from the harvard graduate school institute in educational management. mailto:jking@gsu.edu 12 the brilliance of black children in mathematics symposium speakers & abstracts robert q. berry iii, ph.d. associate professor university of virginia charlottesville va robertberry@virginia.edu identities of black boys who are successful with school mathematics: the follow-up study dr. berry presents findings of a follow-up study to one conducted 4 years ago. the initial study investigated the constructions of mathematics and racial identities among 32 black fifththrough seventh-grade boys who were successful in school mathematics. the boys attend school in a southern rural school district; data collection included focus group interviews, mathematics autobiographies, review of academic records, and observations. in the initial study, four factors were identified that positively contributed to the boys’ mathematics identity: (a) the development of computational fluency by third grade, (b) extrinsic recognitions, (c) relational connections, and (d) engagement with the unique qualities of mathematics. in the initial study, racial identity in school was connected to perceptions of others’ school engagement; this sense of ―otherness‖ led to a redefinition of the boys’ mathematics and racial identities. in the follow-up study, interviews were conducted to investigate shifts in the boys’ identities and perceptions of self. twenty-four of the initial 32 boys reviewed transcripts and viewed short video snippets from interviews and focus groups collected during the initial study; 18 of the 24 boys showed significant shifts in their mathematical and school identities. d. natasha brewley, ph.d. assistant professor georgia gwinnett college lawrenceville ga dbrewley@ggc.edu mathematics literacy for liberation and liberation in mathematics literacy dr. brewley discusses the persistent efforts of two african american young people’s project mathematics literacy workers. the young people’s project (ypp) chicago is a youth empowerment and afterschool mathematics initiative created by young people for urban youth with the goal of expanding how mathematics is experienced in urban communities. the mathematics literacy workers are known as college mathematics literacy workers (cmlws). the discussion aims to provide an understanding of how membership in a community of practice, the ypp chicago, influenced how cmlws worked toward youth achieving mathematics literacy. the community of practice and modes of belonging are used to explain ways in which cmlws participated in ypp chicago and subsequently, how this participation influenced their identity and their efforts of achieving mathematics literacy for and with urban youths. mailto:robertberry@virginia.edu mailto:dbrewley@ggc.edu 13 the brilliance of black children in mathematics james earl davis, ph.d. professor and interim dean temple university philadelphia pa jdavis21@temple.edu understanding black student identity at the intersections: lessons to learn and un-learn dr. davis describes how the analytic potential of intersectionality has been underdeveloped and, in turn, has limited methodological and theoretical approaches to studying black students’ experiences in school. the notion of separate identity categories (e.g., ―race,‖ gender, class) is being challenged by more nuanced treatment of black students that incorporates meaning informed by the interdependence, confluence, and community context of various identity categories. for a variety of reasons, traditional strategies for examining black student identity continue to limit the use of intersectionality. in general, these limitations are related to two fundamental tendencies in the education literature: (a) the reluctance of researchers to fully explore how the analytic strength of intersectionality can inform identity and what is known about black students, particularly boys and young men’s engagement in mathematics and other subjects; and (b) an overreliance on simplistic uses of race to capture the cultural complexities of black students’ educational experiences. based on quantitative and qualitative data from education-based research studies and national databases, the critical importance of intersectionality is highlighted. examples are provided of how the analytic tool of intersectionality can offer insight about black students and their identity development and context. christopher c. jett, ph.d. clinical assistant professor georgia state university atlanta ga cjett2@gsu.edu critical race theory perspective on “race” in mathematics education dr. jett explains how ―race,‖ currently and historically, is used as a construct to place various ethnic groups in a hierarchical system in the united states. since the enslavement of africans, african americans in the united states have experienced this hierarchical race system that places europeans at the top and people of color at the bottom. additionally, race has been used to dismiss and marginalize the intellectual activity of people of african descent as well as other people of color, especially in the context of mathematics. here, critical race theory (crt) is used to examine the experiences of four young african american male students who successfully completed undergraduate degree programs in mathematics. employing crt as a theoretical lens, the ―voices‖ of these young men are presented to better understand their mathematical experiences as racialized beings and to combat the dominant discourse surrounding african american male students as mathematically inferior. issues of race and/or racism are brought to the forefront when investigating the experiences of the young men as african americans in a society and institutional spaces entrenched with racism. mailto:jdavis21@temple.edu mailto:cjett2@gsu.edu 14 the brilliance of black children in mathematics shonda lemons-smith, ph.d. assistant professor georgia state university atlanta ga slemonssmith@gsu.edu tapping into the intellectual capital of black children in mathematics: a critical look at mathematics teacher preparation dr. lemons-smith discusses how each year teacher preparation programs certify teachers of mathematics and decree they are capable of providing high-quality mathematics instruction to all students, regardless of race, class, gender, language, culture, or other characteristics. these programs frequently rely on a standalone multicultural course as a mechanism for addressing issues of diversity, equity, culturally responsive pedagogy, and social justice. too often, prospective teachers do not grasp the significance and application of these ideas in core content areas like mathematics. mathematics teacher educators must ensure that teachers not only espouse positive perspectives about black children but also possess the pedagogical skills to tap into the valuable capital they bring to the mathematics classroom. in other words, prospective teachers must be encouraged to engage in contextual anchoring—using students’ backgrounds, families, communities, lived, and out-of-school experiences to ―anchor the mathematics.‖ anchoring draws on children’s informal knowledge and experiences, makes connections, and facilitates understanding of mathematical concepts throughout instruction. jacqueline leonard, ph.d. professor past president – benjamin banneker association university of colorado denver denver co jacqueline.leonard@ucdenver.edu enacting social justice and culturally relevant pedagogy in mathematics classrooms dr. leonard claims that linking social justice issues and cultural relevance to mathematics instruction is far from the norm in u.s. classrooms. moreover, when teachers attempt to make linkages, too often they do so superficially with prescribed and scripted mathematics lessons. providing teachers with practical and meaningful examples of how to use social justice and culturally relevant pedagogy is important if they are to latch onto these pedagogical strategies and enact them in everyday mathematics classrooms. while social justice and culturally relevant pedagogy are different and serve different purposes, both are important when it comes to teaching mathematics for self-determination and empowerment. presented here are some theoretical underpinnings and practical examples that might assist in moving teacher educators toward pedagogies of teaching for social justice and cultural relevance. these examples emerge from the findings of a two-part teacher-research study that explored explicitly the effects on preservice and inservice teachers when teacher educators make social justice and cultural relevant pedagogical strategies a primary focus in mathematics methods courses and professional development workshops. mailto:slemonssmith@gsu.edu mailto:jacqueline.leonard@ucdenver.edu 15 the brilliance of black children in mathematics danny bernard martin, ph.d. professor university of illinois at chicago chicago il dbmartin@uic.edu proofs and refutations: the making of black children in mathematics education dr. martin discusses how the identities of black children as black children and as mathematics learners are socially constructed through a process involving conjecture, evidence, verification, and proof. based on the ways that it has typically been used in mainstream research and policy contexts, this process has led to a commonsense and largely uncontested understanding that black children are maladaptive in their everyday behaviors and intellectually inferior to white and asian children in mathematics. this process is also accompanied by a logic that demands one must prove that black children are brilliant. as a result, the brilliance of black children in mathematics is rarely the starting point in research and policy discussions but is often framed as a counterexample. this model of framing black children’s identities and competencies makes it very difficult for researchers, teachers, policymakers, and the public at large to accept and invest in the idea that black children truly are brilliant. christine d. thomas, ph.d. associate professor georgia state university atlanta ga cthomas11@gsu.edu influence of an online learning community on teacher retention in urban schools dr. thomas provides details of the robert noyce: urban mathematics educator program (umep) at georgia state university, a program currently in its seventh year. the goal of umep is to increase the number of high-quality secondary mathematics teachers who seek jobs in urban school districts and are committed to remaining in urban school environments. providing high-quality mathematics education for all students, however, goes beyond the recruitment of knowledgeable teachers. retention focuses on new teachers, especially those in urban areas and sustaining high-quality mathematics teachers in hard-to-hire settings. efforts to deepen our understanding of the complex and multifaceted picture of why teachers leave and why they stay, and how efforts to retain teachers impact their work in the classroom and their decisions to stay or leave are developed through the sharing of research designs, data collection, and ongoing results. over the duration of the project, various components of the project have been studied with respect to the influence on retention. the mentoring and support for the project’s teachers are executed in a variety of formats including an online professional learning community (plc); aspects of the online plc with respect to the influence on teacher retention are shared. mailto:dbmartin@uic.edu mailto:cthomas11@gsu.edu 16 the brilliance of black children in mathematics erica n. walker, ed. d. associate professor teachers college columbia university new york ny ewalker@tc.columbia.edu african americans’ mathematics engagement and socialization within and across generations dr. walker, drawing from a longitudinal, multi-sited study of black high achievers in mathematics, describes how academic communities facilitate mathematics engagement and socialization for both high school students and mathematicians. the study offers an intergenerational look at academic communities and how they contribute to mathematics success in multiple sites—within schools, outside of schools, and within ―in-between‖ spaces. these communities and the engagement and socialization experiences that mathematicians and high school students describe as integral to their mathematics success operate within important racial/ethnic, social, and cultural contexts, and for mathematicians, historical contexts as well. for both sets of participants, findings are presented that demonstrate the power of academic communities across multiple settings to facilitate identity development and mathematical excellence. given the preponderance of research that reifies the status of african americans in mathematics education as ―low achievers,‖ this work offers new perspectives on how communities broadly defined can and do support mathematical excellence. it sheds light on the often unrecognized and undervalued communities that students bring with them to school which support their academic and mathematical work. finally, to foster better teaching and learning for all children, this work encourages a rethinking of how and where mathematics education occurs. dorothy y. white, ph.d. associate professor the university of georgia athens ga dywhite@uga.edu preparing preservice teachers to educate black students: the role of multicultural mathematics dispositions dr. white argues that to educate all students in general and black students in particular, preservice teachers need to develop culturally receptive and critical dispositions in mathematics. these dispositions are termed multicultural mathematics dispositions (mcmd). mcmd are based on three dispositional factors: (a) openness to the role of culture in the teaching and learning of mathematics; (b) self-awareness/selfreflectiveness of one’s own culture, its relation to other cultures, and the mathematics classroom cultures experienced; and (c) commitment to using culturally responsive pedagogy to teach mathematics. here, the construct of mcmd is introduced and described with particular attention to the role of mcmd in the preparation of teachers of black students. findings from a study to discover preservice teachers mcmd during a mathematics methods course are then presented to illustrate how these dispositions are evidenced in preservice teachers. to conclude, a discussion of the importance of mcmd in the preparation of teachers and implications for teacher education programs and research is provided. mailto:ewalker@tc.columbia.edu mailto:dywhite@uga.edu 17 the brilliance of black children in mathematics brian a. williams, ph.d. assistant professor georgia state university atlanta ga bawilli@gsu.edu cultivating success: the value of early access to mathematics in the lives of african american children dr. williams discusses how many scholars identify the ―gap‖ that exists on standardized assessments between african american students and their white and asian peers as the ―achievement gap,‖ while others have chosen to point to gaps in the services provided by the educational systems that limit the opportunities for african american children to perform at their highest potential. the historical and persistent lack of opportunities has prompted some researchers to critically examine those factors that contribute to the success of african american students in mathematics in spite of the lack of opportunities. here, a study designed to explore the early childhood experiences of successful african american mathematicians is outlined. specifically, the study answers the following questions as they pertained to mathematical successful african american students: what factors were perceived to have contributed to the students’ early interest in mathematics? what factors, related to early childhood schooling experience, were perceived to have contributed to the students’ success in mathematics? findings of the study have direct implications for the educational community’s understanding and utilization of culturally relevant pedagogy, early access to mathematics, positive personal interventions, and successful experiences in mathematics. other platform speakers presidents cheryl adeyemi, ph.d. associate professor president – benjamin banneker association virginia state university petersburg va cadeyemi@vsu.edu linda gojak president elect – national council of teachers of mathematics james carroll university cleveland oh lgojak@sbcglobal.net mailto:bawilli@gsu.edu mailto:cadeyemi@vsu.edu mailto:lgojak@sbcglobal.net 18 the brilliance of black children in mathematics conference organizers erika c. bullock doctoral student conference administrator georgia state university atlanta ga ebullock1@student.gsu.edu pier junor clarke, ph.d. clinical associate professor conference co-host georgia state university atlanta ga pjunor@gsu.edu david w. stinson, ph.d. associate professor conference co-host georgia state university atlanta ga dstinson@gsu.edu student speakers nathan n. alexander teachers college columbia university nna2106@tc.columbia.edu maisie l. gholson university of illinois at chicago mghols2@uic.edu curtis v. goings emory university cgoings@emory.edu patrice l. parker georgia state university pparker12@student.gsu.edu mailto:ebullock1@student.gsu.edu mailto:pjunor@gsu.edu mailto:dstinson@gsu.edu https://pod51010.outlook.com/owa/redir.aspx?c=57p8ox2xc0qdfqhnetiamra9hh3fzs4ioabho54furrrckktscjur9rbl4n-tftz5hi832sxca8.&url=mailto%3anna2106%40tc.columbia.edu https://pod51010.outlook.com/owa/redir.aspx?c=57p8ox2xc0qdfqhnetiamra9hh3fzs4ioabho54furrrckktscjur9rbl4n-tftz5hi832sxca8.&url=mailto%3amghols2%40uic.edu https://pod51010.outlook.com/owa/redir.aspx?c=57p8ox2xc0qdfqhnetiamra9hh3fzs4ioabho54furrrckktscjur9rbl4n-tftz5hi832sxca8.&url=mailto%3acgoings%40emory.edu https://pod51010.outlook.com/owa/redir.aspx?c=57p8ox2xc0qdfqhnetiamra9hh3fzs4ioabho54furrrckktscjur9rbl4n-tftz5hi832sxca8.&url=mailto%3apparker12%40student.gsu.edu 19 the brilliance of black children in mathematics conference volunteers gsu mathematics education: faculty  iman chahine, ph.d. doctoral students  stephanie byrd  jacqueline hennings  alanna johnson  kori maxwell  patrice parker master’s students  lauren frazier 20 the brilliance of black children in mathematics campus map special issue: journal of urban mathematics education proceedings of the 2010 philadelphia and 2011 atlanta benjamin banneker association conferences beyond the numbers guest editors:  erika c. bullock  nathan n. alexander  maisie l. gholson due: summer 2012 edited book: beyond the numbers and toward new discourse: the brilliance of black children in mathematics editors:  jacqueline leonard  danny bernard martin due: winter 2012 tconcerns-anhalt-rodriguez-april08 journal of urban mathematics education december 2013, vol. 6, no. 2, pp. 42–61 ©jume. http://education.gsu.edu/jume cynthia oropesa anhalt is an assistant specialist in mathematics education and is director of the secondary mathematics teacher preparation program in the department of mathematics in the college of science at the university of arizona, 617 n. santa rita avenue, tucson, az 85721; email: canhalt@math.arizona.edu. her research interests include teacher professional development in pedagogical strategies for problem solving and mathematical modeling, in particular, with a focused interest in english learning students. maría elena rodríguez pérez is a teacher-researcher in the behavioral research center at the university of guadalajara, 180 francisco de quevedo st., guadalajara, mexico 44130; email: rpm08428@cucba.udg.mx. her research interests include learning, in particular, the role of verbal mediation in learning, teacher formation for scientific education and the acquisition of scientific abilities. k–8 teachers’ concerns about teaching latino/a students1 cynthia oropesa anhalt the university of arizona maría elena rodríguez pérez universidad de guadalajara in this article, the authors examine elementary and middle school mathematics teachers’ concerns about teaching latino/a student populations across three regions in the united states: southern arizona, northern new mexico, and central california. surveys were administered to 68 teachers who participated in professional development activities on language and culture diversity. survey questions consisted of items from three domains: (a) concerns about social issues central to teaching latino/a students, such as discrimination, multiculturalism, and stereotypes; (b) concerns about the task of teaching latino/a students focusing on methods, strategies, materials, and new ideas for teaching; and (c) concerns about latino/a students’ learning, which dealt with factors that impact student performance in school, such as home environment, family culture, and expectations. in general, the authors found that the surveyed teachers were highly concerned with issues about teaching latino/a students and their learning and were less concerned about social issues in teaching latino/a students. keywords: english learning students, latinos/as, mathematics education, urban education he ethnic and linguistic diversity of u.s. schools has grown significantly in the past 30 years (u.s. census bureau, 2010). the increase in diversity exists due to many factors including students’ place of birth; length of residence in the 1 this article was supported through the center for the mathematics education of latinos/as (cemela). cemela is a center for learning and teaching supported by the national science foundation (nsf), grant number esi-042983. any opinions, findings, and conclusions or recommendations expressed herein are those of the authors and do not necessarily reflect the views of the nsf. t http://education.gsu.edu/jume mailto:canhalt@math.arizona.edu https://horde3.math.arizona.edu/horde3/imp/message.php?index=34645 anhalt & rodríguez pérez teaching latino/a students journal of urban mathematics education vol. 6, no. 2 43 united states; linguistic backgrounds (varying levels of proficiencies in english and non-english languages); prior school experience; socioeconomic status; child nurturing practices; family configurations; and communication patterns, including code switching and varying levels of bilingualism (garcía & gonzález, 1995). in 2004-05, the latino/a student enrollment in the u.s. k–12 education system was approximately 19% and approximately 21% in 2009 (national center for education statistics, nces, 2013a). in some states, the latino/a 2 student enrollment was above the national average; for example, in arizona, california, and new mexico, it was 38%, 47%, and 53%, respectively. current reports indicate that white students score higher than their latino/a peers on standardized tests at a national level; the “achievement gap” between hispanic and white students in 2009 at grades 4 and 8 in mathematics was between 21 and 26 points on the naep scale (nces, 2013b). this so-called achievement gap—the difference in performance between “racial” groups of students—has long been linked to a difference in family socioeconomic status (ortiz-franco, 1999). recent findings (see nces, 2013b) show that the difference in academic achievement between ethnic groups is more than an issue of poverty versus wealth. gándara (2005) reported that high achieving latino/a students are not likely to come from economically and educationally advantaged backgrounds. these recent findings call for a reexamination of the nature of the educational vulnerability of linguistically and culturally diverse students. effective teaching for linguistically and culturally diverse students supporting latino/a students in the past decade or so there has been a growing body of research that has explicitly explored how to best support latino/a students’ mathematical experiences in a variety of in-school and out-of-school contexts (see, e.g., the edited volume latinos/as and mathematics education: research on learning and teaching in classrooms and communities, edited by téllez, moschkovich, and civil, 2011). much of this research documents how linguistic and cultural diversity can be a valuable resource for mathematics teaching and learning—for students and teachers alike. for instance, zahner and moschkovich (2011) found that multilingual students who use two (or more) languages while doing mathematics possess a set of linguistic resources for managing the social and cognitive demand of group mathematics discussions. they concluded that these students’ participation 2 we use the term latinos/as to refer to the student population in the united states whose origins are of cuban, mexican, puerto rican, south or central american, or other spanish cultures regardless of “race.” anhalt & rodríguez pérez teaching latino/a students journal of urban mathematics education vol. 6, no. 2 44 in mathematics discourse in classrooms is critical to their understanding of mathematical ideas, and that mathematics learning is mediated by participation in a community where discussions of mathematics take place. this growing body of research in general supports the fact that when schools view linguistic and cultural diversity as a resource rather than a deficiency and hold high expectations for latino/a students, they more times than not experience consistent academic growth in achievement (see, e.g., jesse, davis & pokorny, 2004). turner, varley gutiérrez, and díez palomar (2011) explored out-of-school mathematics learning experiences. turner and colleagues successfully worked with latino/a elementary students in problems grounded in community settings that gave the students new perspectives on seeing mathematics in their everyday world outside of school mathematics. they framed their work in community mathematization, where students collaboratively use mathematics to make sense of their environment of familiar contexts in an afterschool setting. the contexts for the mathematics problems included single and multi-step computation, geometry, area, and volume measurements in rich modeling problems. turner and colleagues found that students were able to capitalize on their background knowledge to solve problems and explain solutions through their understanding and ownership of the mathematics. teacher preparation and professional development eliciting and making sense of students’ cultural, home, and communitybased knowledge, and its relevance to mathematics instruction, is a complex practice that takes special attention by teachers. this process should begin in teacher preparation and continue to develop as teachers enter the field (civil, 2007). recently, turner, drake, mcduffie, aguirre, bartell, & foote (2012) proposed a vision of effective mathematics teaching for diverse learners where pre-service teachers developed lessons that reflected meaningful connections to diverse students’ cultural, home, and community-based knowledge that supported mathematics learning. the pre-service teachers created lessons for elementary students inspired by what they learned about the mathematical practices and skills used in a familiar hub for the local latino/a community. within the lessons, they created challenging problem-solving tasks situated in a familial context, which had cultural relevance for the students. in addition to the socio-cultural perspectives for teacher preparation, mathematics education involves helping teachers consider strategies that incorporate multiple modalities and representations of mathematical ideas for the classroom setting. anhalt and ondrus (2011) worked with middle school mathematics teachers in a professional development course in addressing algebraic concepts using multiple representations: algebra blocks for the concrete representations, relevant contextual representations, pictorial representations, linguistic representations, anhalt & rodríguez pérez teaching latino/a students journal of urban mathematics education vol. 6, no. 2 45 and abstract mathematical symbolism. one goal of the course was for the teachers to see mathematical concepts through concrete representations to expand their understanding of abstract decontextualized mathematics symbolism. they found that the teachers were able to make connections between the concrete, abstract symbolism, linguistic, and contextual representations of mathematical situations. in building this fluency between representations of mathematical ideas, the teachers saw the value of incorporating multiple representations in their teaching for all students, and especially crucial for their latino/a and english learning (el) students. these findings echo those of another similar study where the use of multiple representations designed for understanding a target language was found to be an effective instructional practice (téllez & waxman, 2006). because the use of language plays a crucial role in understanding mathematics, especially for latino/a el students, explicit and deliberate linguistic and intellectual support during cognitively demanding tasks is vital. effective teachers and schools recognize that any attempt to address the needs of latino/a students in a deficit or “subtractive” mode is counter-productive (garcía & gonzález, 1995; valenzuela, 1999). therefore, we argue that the relevance of teachers’ everyday positive personal interactions with latino/a students is critical in helping students succeed academically. valenzuela (1999) suggests that teachers’ use of an “additive” approach when teaching linguistically and culturally diverse students influences students’ academic success. an additive approach would include, among others: a school climate free of prejudices, school methods and materials that appeal to all students regardless of their cultural background, and high expectations for students from teachers and parents. these topics should be of high interest to schools with a significant number of latino/a students enrolled. in the study presented here, teachers’ concerns while teaching latino/a students were assessed in order to learn the importance given to issues regarding school climate, methods, materials, and expectations for students. the target participants of this study were k–8 teachers enrolled in professional development programs across three u.s. geographical regions. researching teachers’ concerns concerns are defined as an emotional undertone that signals insecurity and resistance to new situations and changes (van den berg & vandenberghe, 1995). concerns also can be interpreted as feelings, thoughts, or reactions to certain things (mok, 2005). research on teachers’ concerns draws heavily on the work of fuller (1969). fuller and bown (1975) suggest that pre-service teachers start their careers with idealized ideas about students and teaching. this idealization changes with the first teaching experience and a central question becomes important: will i be able to manage the class? fuller and bown name this kind of concern as anhalt & rodríguez pérez teaching latino/a students journal of urban mathematics education vol. 6, no. 2 46 a “survival concern” or “self-concern.” as teachers become more experienced, they become concerned about methods and materials and start looking for new ideas for their teaching. still, these are concerns about their own performance as a teacher and not concerns about students and their learning. they name these as “concerns about the task” or “teaching concerns.” finally, fuller and bown referred to “concern about the pupils,” “impact concerns,” or “learning concerns” when teachers have an eye for students’ social and emotional needs and they become more focused on their relationships with individual students. initially, fuller argued that concerns would change according to teachers’ development. that is, self-concerns would appear mainly at the beginning stage of teacher development, in which teachers have anxiety about their ability to survive in the classroom. at a second stage of teacher development, the task of teaching is the largest concern. teachers are concerned about the performance of their teaching tasks, which include resources, strategies, and time management. at the third stage, the impact concerns relate to the teachers’ apprehensions about social and learning needs of pupils. studies have found that concerns do not necessarily develop in a sequential manner in the stages of teacher development (see, e.g., adams, 1982; ghaith & shaaba, 1999). any kind of concern may increase or decrease suddenly (swennen, jörg & korthagen, 2004), overlap (pigge & marso, 1987) or play a central role from the very beginning of the professional development without changes (smith & sanche, 1993). mok (2002) explained that the differences in findings across studies suggest that the concerns in fuller’s (1969) model are framed in very broad terms and hence it is not surprising that task concerns and impact concerns occur in similar stages. these findings may imply that task and impact concerns, which are highly associated with the job of teaching, naturally are concerns in most stages of teachers’ careers. therefore, charalambous, philippou and kyriakides (2004) argued that fuller’s types of concerns could be considered in terms of levels, not stages. hence, those concerns related to self-survival (i.e., awareness, information-seeking, and personal relationships) are categorized as firstlevel concerns, those concerns related to teaching (e.g., management, methods, curriculum, and resources) are categorized as second-level concerns, and finally, those concerns related to student impact (e.g., consequences of effective teaching, collaboration with other teachers, making suggestions for improving student learning) are categorized as third-level concerns. an interesting finding from studies on teacher concerns is that self-concerns are normally found to decrease with increase in years of experience (adams, 1982; pigge & marso, 1997; veenman, 1984). additionally, ghaith and shaaban (1999) found that teaching concerns, which include performance, curriculum, resources, and strategies, are very low in teachers with more than fifteen years of experience. this evidence reveals the complex patterns of personal development, anhalt & rodríguez pérez teaching latino/a students journal of urban mathematics education vol. 6, no. 2 47 professional identity, and the emotional dimensions of the teaching profession (poulou, 2007). overall, the issue of what concerns teachers is an important one. although studies have been done on general teaching concerns, few studies have attempted to document teachers’ concerns while teaching linguistically and culturally diverse students. the study reported here addresses this gap. methodology participants and contexts sixty-eight k–8 mathematics teachers of latino/a students from three geographical contexts participated in this study. approximately two-thirds of the participants taught at the elementary level and one-third taught at the middle school level. the regions represented an urban area in arizona, a rural and urban area of california, and rural and urban areas of new mexico near large local universities. the teachers from the three regions participated in a variety of professional development activities during their partnership with their local universities (the university of arizona, the university of new mexico, and university of california, santa cruz). the teachers from new mexico participated in summer institutes with a focus on teaching strategies for teaching mathematics to el students. teachers from california participated in professional development activities that incorporated mathematics content and pedagogy specific to the context of latino/a students. and teachers from arizona engaged in additional coursework and also participated in a variety of professional development activities including a teacher study group (9 teachers), professional development courses on various mathematics topics with an emphasis on teaching el students (22 teachers), and lesson study (4 teachers). while the professional development activities at each site differed, the premise under which the cemela professional development activities functioned was the same across the three sites: all activities centered on ways to turn language and cultural diversity into educational assets for the mathematics education of latino/a students. the three regions in which this survey was administered have different political contexts and differing policies and state laws that govern the language of instruction in their schools. california and arizona both have legislation requiring the use of english during instruction, while new mexico allows bilingual education programs for students identified as english language learners (ells). the various school districts from which the teachers come all have a high percentage of latino/a student populations. specifically, each of the schools in which the participating teachers work has approximately an 85% latino/a student population and approximately one-third of the students are identified as ells. the teachers volunteered to participate in the cemela-associated professional development activities at their local university because they were seeking to learn about ways anhalt & rodríguez pérez teaching latino/a students journal of urban mathematics education vol. 6, no. 2 48 to have a positive impact on their students, who were predominantly latino/a, and were not paid either to participate in the professional development activities or to take the survey. instruments the instrument used for the study was a 20-item survey designed by mj young & associates. items, in a likert scale format, addressed teacher concerns regarding teaching latino/a students. table 1 describes the 20 items (table 1: appendix a). survey items were categorized in three broad types of concerns: (a) concerns about one’s own promotion of a school climate free of discrimination and prejudices, or “self-concerns”; (b) concerns about the use of methods, materials, and strategies in class of specifically designed to cover the needs of linguistically and culturally diverse students, or “task concerns”; and (c) “appropriate” adult role models at students’ homes and parents’ high expectations for their children, or “impact concerns.” in general, the categorization of items borrows from the work of fuller (1969) and swennen, jörg and korthagen (2004). in order to evaluate the reliability of the survey, a chronbach alpha coefficient was calculated. it yielded 0.9102, which indicates that the instrument is reliable. each survey item was placed in one of the three categories using a factor analysis after survey administration. as an exploratory tool, factor analysis can be used to extract “factors,” that is, statistical entities that serve as classification axes. this technique is useful when reducing a dataset to a more manageable size while retaining as much of the original information as possible (field, 2005). the major assumption in factor analysis is that factors represent real-world dimensions. thus, researchers have to interpret statistical analyses and define the clusters of variables aided by theoretical assumptions. in the study reported here, a factor analysis was carried out using spss software. five factors were extracted from our data. questions 6, 7, 10, 11, and 16 comprised one cluster, which we identified as “selfconcerns” because these questionnaire items refer to discrimination and prejudices. questions 12, 17, and 18 defined another cluster, which were associated with “impact concerns.” the other three factors were clustered in what we labeled “task concerns,” taking into consideration that these items refer, globally, to class methodology. the factor analysis suggested that our variable “task concerns” might be susceptible to a finer categorization; however, we decided to keep the three concern classifications as identified in the research literature (see table 1: appendix a for final classification of survey items). chronbach coefficients for the three subsets of survey items were calculated and they suggest good reliability: 0.894, 0.891, and 0.920, respectively. in order to learn about the sample size effect, a kaiser-meyer-olkin (kmo) measure of sample adequacy was included in the factor analysis. the kmo statistic varies between 0 and 1 indicating the degree of anhalt & rodríguez pérez teaching latino/a students journal of urban mathematics education vol. 6, no. 2 49 common variation (1 being a perfect communality among variables) and serves as an indicator of generalizability of the research data collected. in the present analysis, kmo measure yielded a factor of 0.768 and suggested a good sample size for generalization of research findings. procedure surveys were administered at the three regions during the last session of the professional development activities in either december 2006 or january 2007. the directions for the survey were: below are questions some teachers have posed about working with culturally diverse students (e.g., latinos). these questions may or may not be of concern to you at this point in your professional career. read each question and then circle the number that represents the degree of concern the question holds for you (1 being extremely unimportant and 5 being extremely important). teachers rated items individually using numbers from 1 to 5. at the end of the survey, some background information questions were included regarding years of teaching experience and personal ethnicity. this background information about the teachers was collected for the purpose of determining the correlation between years of experience, ethnicity, and concerns. data analyses two different analyses were carried out. the purpose of the first analysis was to characterize the teacher concerns as a group of 68 teachers of latino/a students. to do so, we calculated sums of teacher responses from 1 to 5 for each survey item. these sums were divided by total responses to compute percentages of teacher responses to each option. we used a procedure similar to mau and kings’ (1996) to calculate a weighted average to indicate a level of concern for each survey item. therefore, the level of concern can vary from an average rating of 1 to 5 and would indicate how teacher responses distributed along the unimportant– important scale of the instrument used (table 1: appendix a). the purpose of the second analysis was to explain the differences within the data according to three variables that may have an impact on teacher concerns. in this study, the variables examined were years of teaching, teacher ethnicity, and geographical region of the teachers. to do so, we conducted several analyses of variance (anova) tests to determine how well these three variables accounted for data variance. because anova requires a normally distributed interval dependent variable, we carried out a shapiro-wilk w test, and the test resulted in a anhalt & rodríguez pérez teaching latino/a students journal of urban mathematics education vol. 6, no. 2 50 value of 0.97 with a p-value greater than 0.1; therefore, normality was fairly assumed. limitations it is important to discuss some limitations of the research reported here. first, we recognize that there was only one item related to mathematics specifically. however, research literature has pointed out that content teachers are well-aware of the issues related to learning the content itself (fletcher, mountjoy & bailey, 2011). therefore, we assume that the inclusion of more content items specific to mathematics would probably have responses with a high degree of concern. a second limitation is the lack of follow-up surveys to the participating teachers after they completed the professional activities. however, research has found that teachers’ concerns are stable along large periods of times (melnick & meister, 2008), and most often do not change even within a reform context (charalambous & philippou, 2010). findings teacher concerns characterization table 1 (appendix a) represents how the 68 teacher responses distributed along the unimportant–important scale. for each survey item, a percentage of responses of 1, 2, 3, 4, and 5 ratings of the unimportant-important scale were calculated. the highest percentage for each survey item appears shadowed in table 1. the majority of teachers rated self-concern items as “extremely unimportant.” on the contrary, almost all of the task and impact concerns were rated as “extremely important.” the last column of table 1 (appendix a) has the weighted average level of concern for each survey item. the level of self-concern ranged from 2.1 to 2.7, the average ratings of the task concerns ranged from 3.9 to 4.6, and the average ratings of the impact concerns ranged from 3.5 to 3.6. the most important concern referred to the methods and techniques that appeal to all students regardless of their cultural background (item 15 of task concerns). the least important concern referred to being accused of discrimination by latino/a students (item 10 of self-concerns). in general, these teachers were highly concerned with issues about the appropriate methods and materials for linguistically and culturally diverse students and appropriate parent models, but they seemed to be less concerned about their promotion of a prejudice-free school climate. anhalt & rodríguez pérez teaching latino/a students journal of urban mathematics education vol. 6, no. 2 51 teacher ethnicities and concerns in reporting ethnicity, 24 teachers reported being “white,” 35 reported “hispanic/latino,” 3 reported “asian or pacific islander,” 1 reported “african american,” 3 reported “other,” and 2 did not answer this question. because teacher concerns referred to teaching latino/a students, data were broken down into two broad categories: hispanic and non-hispanic. anova analysis for selfconcerns suggests that non-hispanic teachers are more concerned about their promotion of a prejudice-fee school climate than hispanic teachers and this difference is statistically significant (f = 4.23, p = 0.04). there were no significant differences between hispanic and non-hispanic teachers for task and impact concerns (see table 2: appendix b for anova summaries). to estimate the importance of the effect in the sample and, therefore, the likely importance of the effect in the population given that sample, a measure of effect size was calculated for the differences found for self-concerns between hispanic and non-hispanic teachers. an effect size is an objective and standardized measure of the magnitude of the observed effect. a common measure of effect size is pearson’s correlation coefficient (r). it is widely accepted that a correlation coefficient greater than 0.30 represents a medium effect and greater than 0.50 constitutes a large effect (field, 2005). in this case, r was calculated using the between-group effect (ssm) and the total amount of variance in the data (sst) from the spss output for anova (field, 2005). thus, r 2 = ssm / sst = 5.557 / 89.650, r = 0.25 represents a small effect size. years of teaching and concerns as previously discussed, research on teacher concerns has linked types of concerns with teacher developmental approaches (mok, 2005). previous research findings have not been conclusive regarding how teacher concerns vary along teaching experience, so it seemed important in this study to relate level of concern with years of teaching experience. using a factor analysis (component analysis defining 3 components), three groups of teachers were identified: new teachers with 0–7 years of teaching, more experienced teachers with 8–20 years of teaching, and most experienced teachers with more than 20 years of teaching. our anova analysis shows no significant differences among teachers in the types of concerns about social issues and teaching issues. there is, however, a difference among teachers when comparing concerns about student learning at a level of confidence below 0.10 (see table 2: appendix b). post hoc test of bonferroni was used in order to determine how groups compare among each other. while this test shows no differences between teachers with 0–7 years of experience and those with 8–20 years of experience, the most experienced teacher group (20+ years of experience) differed from the other groups of teachers. thus, data sug anhalt & rodríguez pérez teaching latino/a students journal of urban mathematics education vol. 6, no. 2 52 gests that more experienced teachers are less concerned about student learning issues. this finding may reflect a more confident attitude that is developed through years of teaching experience. however, an effect size measure indicates a small effect (r = 0.29). the different regions and concerns table 3 (appendix c) summarizes percentage of teachers and level of concern when taking into consideration the regions where the survey was administered. teacher responses were categorized as being unconcerned (1 or 2 on the likert scale), being neutral (3 on the likert scale), and being concerned (4 or 5 on the likert scale). table 3 indicates percentages of teachers by region (arizona, new mexico, and california) as unconcerned (un-c), neutral (n), or concerned (c) for each of the survey items. the highest percentage for each item is shaded in table 3 in order to more easily view similarities and differences at the three regions. all the items regarding task and impact concerns were rated similarly at all regions; all items were items of concern (see table 2: appendix b for anova summaries). however, teachers at the various regions rated self-concerns differently. arizona and new mexico teachers indicated no concern on all items in the category of self-concerns. in contrast to california teachers, 92% of new mexico teachers indicated no concern for being accused of discrimination, while 45% of california teachers indicated no concern for being accused of discrimination (item 10). the majority (61%) of california teachers were concerned with latino/a students perceiving them as biased because the teachers’ backgrounds may be different than the students’ (item 6). additionally, 56% of california teachers were concerned with parents of latino/a students being prejudiced against them (item 7). forty-eight percent of california teachers were neutral with the issue of engaging in reverse discrimination (item 11) and 41% were concerned about stereotyping students on the basis of race (item 16). anova analysis for self-concerns suggests that these differences in rating among teachers of the three regions are statistically significant (f = 10.77, p = 0.000091). this effect is large (r = 0.5). the last section of table 3 (appendix c) has the level of concern (average rating) of each survey item at arizona, new mexico, and california. the most important concern in arizona and california referred to the most effective methods for teaching mathematics to latino/a students (item 1). the most important concern in new mexico referred to the methods and techniques that appeal to all students regardless of their cultural background (item 15). the least important concern at the three regions referred to being accused of discrimination by latino/a students (item 10). anhalt & rodríguez pérez teaching latino/a students journal of urban mathematics education vol. 6, no. 2 53 discussion in the research reported here, a teacher-concern survey was administered to 68 k–8 teachers of high latino/a population schools in arizona, new mexico, and california to characterize items as concerns about social issues in teaching latino/a students, concerns about teaching latino/a students, and concerns about latino/a student learning. all teachers were participating in a variety of professional development experiences associated with cemela at the three regions. we analyzed teacher concerns taking into consideration years of teaching and teachers’ ethnicities. similarities and differences among regions were of special interest because cemela is a multi-university consortium, for which working with teachers of latino/a students is a focus. overall, teachers seemed to be highly concerned with teaching and learning issues independent of region, ethnicity, or years of teaching. issues of teaching and learning as illustrated in the survey are about effective strategies and techniques that can be used for teaching, strategies to motivate culturally diverse students, relevant content, and meeting academic needs in addition to questions about expectations for students. these findings are consistent with past research reports on teacher concerns. for example, melnick and meister (2008) have reported that there are eight global issues that have worried teachers at all levels and in all disciplines in the last 30 years. in order of importance, these issues are: classroom discipline, motivating pupils, dealing with individual differences, assessing pupil’s work, relations with parents, organization of class work, insufficient materials and supplies, and dealing with problems of individual pupils. all eight of these major issues can be classified under teaching and learning concerns (or task and impact concerns, as labeled in this study and in research literature on teacher concerns). teacher concerns on social issues centered on students’ perceptions of teachers from a different cultural background, issues of prejudices, and discrimination in general were examined. concerns on social issues seemed to be impacted by teachers’ ethnicity and region. because a concern has been defined in the field as a kind of emotional undertone that signals insecurity (van den berg & vandenberghe, 1995), it seems reasonable to suppose that concerns on social issues involve a higher degree of emotional charge and, therefore, can vary more easily with contextual variables and social perceptions than teaching and learning concerns might. according to our results, teachers from arizona and new mexico were similar in that they were unconcerned with social issues. however, teachers from california were highly concerned on item 6 (“will latino/a students perceive me as biased simply because my background is different than theirs?”), item 7 (“will parents of latino/a students be prejudiced against me?”), and item 16 (“will i ste anhalt & rodríguez pérez teaching latino/a students journal of urban mathematics education vol. 6, no. 2 54 reotype students on the basis of their race?”). these differences could possibly be explained when we take into consideration ethnic proportions of teachers at each region. in arizona, 67% of the teachers were hispanic/latino and 27% were white. in new mexico, 70% of the teachers were hispanic/latino and 15% were white. in california, 26% of the teachers were hispanic/latino and 61% were white. we speculate that teachers’ ethnicity is an important variable in considering teacher concerns on social issues in teaching mathematics to latino/a students. these differences in teachers’ ethnicities may be responsible for the differences that we found from teacher responses from the regions. it is necessary to replicate the survey with sample populations using a random layered sampling in order to test this hypothesis. as previously noted, in the results, all new teachers (0–7 years of teaching) reported low concerns on social issues such as cultural background, discrimination, multiculturalism, and stereotype issues. we speculate that this may be the case due to new teachers’ general low awareness of the implications of these social issues. new teachers may not consider these issues to play a role in their everyday lives of teaching or they may consider social issues far removed from their mathematics classroom environments, as if the classroom exists insulated from society at large. these findings concur with evidence found in studies of teacher candidates that show that those who become teachers tend to be young people who are typically not politically active in social issues or are distant from social issues, and, therefore, have a limited firsthand awareness of or engagement in many of the nation’s major social issues (howey & zimpher, 1996). gutiérrez and dixon-román (2011) note that students of color continue to be framed in comparison to their white counterparts, and this comparison then becomes normalized, as if it is a “natural” way of thinking about achievement, rather than focusing on the excellence of students of color or the many other ways subordinated students may make sense of their experiences with mathematics. because this is a highly political and unfortunate “common picture,” we wonder if it lends itself to unintentional, unexamined, or unwitting prejudice by educators. this topic merits a discussion at a more in-depth level than can be provided within this context. concluding thoughts worries and concerns have been reported to play a role in teachers’ work (boz, 2008; boz & boz, 2010) and is, therefore, an important area for research. theoretical foundations on teacher concerns has been grounded on fuller’s (1969) work which distinguishes three types, levels or phases of concerns: self-concerns, task-concerns, and impact-concerns. empirical evidence, especially with teachers in urban schools, suggests that teachers tend to concern themselves with issues of task and impact (fletcher, mountjoy & bailey, 2011; melnick & meister, 2008). anhalt & rodríguez pérez teaching latino/a students journal of urban mathematics education vol. 6, no. 2 55 the research reported here represents an effort to learn about urban teachers’ concerns in contexts of latino/a student populations. moreover, mathematics teachers of latino/a students are an important population to investigate given that research has found that meeting the needs of culturally diverse students requires high expectations and an “additive approach” to their education (valenzuela, 1999). in summary, urban mathematics teachers of latino/a students considered self-concerns about social issues as globally unimportant. as a reminder, selfconcerns about social issues pertain to teachers’ anxiety about their ability to successfully undertake demands stemming from social issues on cultural diversity such as racism, discrimination, or prejudices. this finding could mean that prejudiced labels do not bias teachers’ perceptions, and that these teachers therefore do not consider latino/a students as lacking skills to perform successfully in school. additionally, teachers may feel that because they have personal relationships with their students that social issues, such as prejudice or discrimination, could not possibly enter in their everyday teaching lives. although teachers’ concerns about social issues were of low importance overall, there was a significant difference between the hispanic/latino and non-hispanic/latino teachers. it could be that non-latino teachers were more concerned about social issues in teaching latino/a students than latino/a teachers because they feel like cultural strangers to their latino/a students; these teachers, therefore, may have more concern about social issues, such as being accused of discrimination or being accused of having biases against latino/a students. further research needs to be done with emphasis on mathematics teacher concerns and their interaction with other kinds of variables such as teacher performance, impact on student learning, and beliefs about teaching. teacher ethnicity seems to be an important variable to take into account in future analyses of teacher concerns with respect to self-concerns on social issues. we can speculate that non-hispanic/latino teachers may fear being rejected by hispanic/latino students. non-hispanic/latino teachers may also feel unprepared to cope with a high proportion of hispanic/latino students. it would be worthwhile to examine the reverse—non-white teachers’ concerns in schools of a high percentage of white students—to further 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(2011). bilingual students using two languages during peer mathematics discussions: ¿qué significa? estudiantes bilingues usando dos idiomas en sus discusiones matemáticas: what does it mean? in k. téllez, j. moschkovich, & m. civil (eds.), latinos/as and mathematics education: research on learning and teaching in classrooms and communities (pp. 37–62). charlotte, nc: information age. anhalt & rodríguez pérez teaching latino/a students journal of urban mathematics education vol. 6, no. 2 59 appendix a table 1 percentage of teachers that rated survey items from 1 (unimportant) to 5 (important) survey items percentage of responses on unimportant – important scale level of concern (average rating) 1 2 3 4 5 self-concerns about social issues in teaching latino students 6. will latino students perceive me as biased simply because my background is different than theirs? 31 18 21 18 12 2.6 7. will parents of latino students be prejudiced against me? 31 18 22 15 14 2.6 10. will latino students accuse me of discrimination? 42 28 17 8 5 2.1* 11. in attending to multicultural issues, will i be engaging in reverse discrimination? 29 17 27 13 14 2.7 16. will i stereotype students on the basis of their race? 39 17 19 8 17 2.5 task-concerns about teaching latino students 1. what are the most effective methods for teaching mathematics to latino students? 3 1 9 34 53 4.3 2. what strategies should i use when working with latino students? 3 4 6 37 50 4.3 3. what specific techniques and materials motivate latino students? 4 1 10 30 55 4.3 4. how does the home environment of latino students impact their receptivity to school? 3 6 13 28 50 4.2 5. in what specific ways does family culture affect latino students’ performance in school? 0 6 12 38 44 4.2 8. how do i make lessons and content relevant to latino students? 4 0 10 32 54 4.3 9. what kinds of things can i do to meet both the academic and emotional needs of latino students in my class? 3 1 6 34 56 4.4 13. how should i vary my teaching methods when dealing with culturally diverse students? 2 2 8 32 56 4.4 14. how do i effectively teach a class of students whose ability and experiential levels are widely diverse? 0 3 6 23 68 4.5 15. what are the methods and techniques that appeal to all students regardless of their cultural background? 0 5 0 26 69 4.6** 19. what criteria do i use in selecting materials related to latino culture? 5 6 15 41 33 3.9 20. how can i help all students relate to those who have different backgrounds in my classroom? 2 2 17 25 54 4.3 continued on next page anhalt & rodríguez pérez teaching latino/a students journal of urban mathematics education vol. 6, no. 2 60 continued from previous page appendix a table 1 percentage of teachers that rated survey items from 1 (unimportant) to 5 (important) impact-concerns about latino student learning 12. do latino students have appropriate adult role models? 3 16 31 23 27 3.5 17. do parents of latino students possess high expectations for their children? 10 11 23 26 30 3.6 18. are latino students’ home environments an adequate model for academic study? 9 13 24 27 27 3.5 j. m. young & associates (2005) * least important concern ** most important concern anhalt & rodríguez pérez teaching latino/a students journal of urban mathematics education vol. 6, no. 2 61 appendix b table 2 anova results for teachers’ concerns taking into consideration teacher ethnicity, years of teaching, and region self-concerns about social issues in teaching latino students teacher ethnicity mean standard deviation f-ratio p non-hispanic teachers 2.8 1.0 4.23 0.04** hispanic teachers 2.2 1.2 years of teaching mean standard deviation f-ratio p 0 – 7 2.5 1.2 0.247 0.78 8 – 20 2.5 1.3 20 + 2.2 1.0 region mean standard deviation f-ratio p arizona 2.2 1.0 10.77 0.000091*** new mexico 1.8 0.8 california 3.3 1.1 task-concerns about teaching latino students teacher ethnicity mean standard deviation f-ratio p non-hispanic teachers 4.3 0.4 0.04 0.85 hispanic teachers 4.3 0.8 years of teaching mean standard deviation f-ratio p 0 – 7 4.4 0.4 1.025 0.365 8 – 20 4.2 0.8 20 + 4.2 0.7 region mean standard deviation f-ratio p arizona 4.3 0.7 1.9 0.15 new mexico 4.1 0.7 california 4.5 0.4 impact-concerns about latino students learning teacher ethnicity mean standard deviation f-ratio p non-hispanic teachers 3.4 1.1 0.8 0.37 hispanic teachers 3.6 1.2 years of teaching mean standard deviation f-ratio p 0 – 7 3.5 1.3 2.636 0.08* 8 – 20 3.8 1.0 20 + 2.9 1.1 region mean standard deviation f-ratio p arizona 3.2 1.2 2.78 0.07* new mexico 3.9 1.0 california 3.8 1.0 * p<0.01 ** p < 0.05 *** p<0.1 anhalt & rodríguez pérez teaching latino/a students journal of urban mathematics education vol. 6, no. 2 62 appendix c table 3 percentage of teachers from arizona (az), new mexico (nm) and california (ca) either unconcerned (un-c), neutral (n) or concerned (c) for each survey item item # percentage of teachers & level of concern unconcerned (un-c), neutral (n) or concerned (c) levels of concern at different regions (average rating) arizona new mexico california un-c n c un-c n c un-c n c az nm ca self-concerns about social issues in teaching latino students 6 55 29 16 72 14 14 26 13 61 2.3 2.1 3.5 7 55 29 16 76 15 8 26 17 56 2.3 1.9 3.4 10 76 17 7 92 0 8 45 32 23 1.9* 1.4* 2.8* 11 51 17 31 69 23 8 19 48 34 2.6 1.9 3.3 16 67 17 16 75 8 17 32 27 41 2.1 2.0 3.3 task-concerns about teaching latino students 1 0 10 90 21 21 57 0 0 100 4.5** 3.4 4.7** 2 6 3 90 21 14 65 0 4 96 4.3 3.6 4.6 3 6 10 84 14 7 79 0 13 87 4.3 4.0 4.5 4 13 16 71 14 13 74 0 9 91 4.0 4.0 4.4 5 6 19 74 14 14 72 0 0 100 4.1 4.0 4.4 8 3 13 84 14 14 72 0 9 91 4.4 3.7 4.6 9 3 10 87 14 0 85 0 4 96 4.3 4.2 4.6 13 3 7 89 7 14 79 0 5 96 4.4 4.1 4.6 14 7 10 83 0 7 93 0 4 96 4.4 4.7 4.7 15 10 0 90 0 0 100 0 0 100 4.5 4.9** 4.6 19 13 13 73 7 14 79 9 18 72 3.8 4.1 4.0 20 3 13 83 7 21 71 0 18 82 4.3 4.1 4.4 impact-concerns about latino students learning 12 27 33 40 8 23 69 14 36 50 3.2 3.9 3.8 17 34 17 50 10 20 70 9 32 59 3.3 3.8 3.9 18 31 31 38 16 8 75 14 23 63 3.1 3.8 3.9 * least important concern ** most important concern journal of urban mathematics education july 2012, vol. 5, no. 1, pp. 55–65 ©jume. http://education.gsu.edu/jume brian a. williams is the director of the alonzo a. crim center for urban educational excellence and an associate professor in the department of early childhood education in the college of education at georgia state university, 30 pryor street, atlanta, ga 30302, e-mail: bawilli@gsu.edu. his work is situated at the intersection of science education, urban education, and education for social justice. more specifically, he is interested in the ways in which equity issues related to race, ethnicity, culture, and class influence science teaching and learning and access to science literacy. early mathematical experiences of successful african american scientists, engineers, and mathematicians brian anthony williams georgia state university n 2000, the national council of teachers of mathematics (nctm) published the principles and standards for school mathematics (nctm, 2000). these standards, created by a team of researchers, educators, and mathematicians, provided the united states with a detailed framework for teaching mathematics in schools. the first principle in the document, the equity principle, emphasizes the value of diversity and equity in mathematics. recognizing that success in mathematics is not reserved for one group of students, the nctm attempted to address the specific needs of students of color who, traditionally, have had limited access to educational opportunities in mathematics. the authors contend: all students, regardless of their personal characteristics, backgrounds, or physical challenges, must have opportunities to study—and support to learn—mathematics. equity does not mean that every student should receive identical instruction; instead, it demands that reasonable and appropriate accommodations be made as needed to promote access and attainment for all students. (p. 27) inherent in the nctm standards is the idea that, when given the opportunity, all children can learn mathematics content as well as the problem-solving and critical-thinking skills involved in the study of mathematics. by investing in and utilizing these standards, schools (supposedly) ensure that all students, regardless of race, ethnicity, gender, or class, have equitable access to and opportunity to participate in a high-quality educational system that will provide them with a sound foundation in all subjects, including mathematics and science. unfortunately, the united states continues to struggle in its effort to meet the goals outlined in the standards despite the efforts of the educational and research communities. findings from the third international mathematics and science study (timss) indicate that students from other industrialized nations continue to outperform u. s. students in mathematics and science as indicated by proficiency during the twelfth grade year of high school (national science board, 2012). furthermore, a significant gap exists between african american, latina/o, i williams early mathematical experiences bullock, e. c., alexander, n. n, & gholson, m. l. (eds.). (2012). proceedings of the 2010 philadelphia and 2011 atlanta benjamin banneker association conferences – beyond the numbers [special issue]. journal of urban mathematics education, 5(2). 56 and native american students and their white and asian peers on standardized measurements of mathematics proficiency. although some may argue that improving the performance of people of color in science, technology, engineering, and mathematics (i.e., stem) as a means of ensuring the nation’s economic viability is sound, it creates another question: if the us were not experiencing a shortage of trained individuals in these fields and the scientific and technological future and related economic system were not in jeopardy, would there still be a need to improve the performance and representation of people of color in stem? some researchers (e.g., perry, moses, wynne, cortés, & delpit, 2010) are encouraging the research community to look at the problem as one related to ethics, morality, and basic human rights. they argue that despite the economic, scientific, or technological needs in the united states, those responsible for shaping the educational system must make every attempt to ensure that all students, regardless of race, culture, class, or gender, have equitable access to stem. the aforementioned researchers view equitable access and opportunity as an inherent civil right. moses and cobb (2001), for example, argue that “math literacy in urban and rural communities throughout the country is an issue as urgent as the lack of registered voters in mississippi was in 1961” (p. 5). in my opinion, positioning stem access as a civil rights issue offers a more valid argument for the need to improve the performance and representation of people of color in stem than the argument related to the needs of the nation’s scientific and technological workforce. however, steps toward improvement can only be taken by providing adequate and equitable access to educational opportunities in these fields. unfortunately, many existing educational programs are failing to attract and retain those who currently account for a large share of the workforce: people of color and women. consequently, many educators, researchers, and policymakers are seeking to understand the factors related to the under-representation of people of color in stem fields. in spite of current educational research regarding this issue, there remains a significant omission. a review of the literature reveals that much of the current research focuses on simply describing the problem of poor representation and performance among african american students. what is lacking is research that focuses on the characteristics of african american students who are successful in stem. instead of focusing on the factors that keep these students out of stem, we must focus on the elements of the lives of african american students that have allowed many to persist. this study reported here represents a part of a larger study designed to identify and investigate the critical life elements related to the development of successful african americans in stem. in this paper, i present my investigation of the students’ early experiences that influenced their successful participation in williams early mathematical experiences bullock, e. c., alexander, n. n, & gholson, m. l. (eds.). (2012). proceedings of the 2010 philadelphia and 2011 atlanta benjamin banneker association conferences – beyond the numbers [special issue]. journal of urban mathematics education, 5(2). 57 stem fields throughout their academic careers. specifically, i explored the following question as it related to african american graduate students: what factors were perceived to have contributed to the students’ initial interest in science, engineering, or mathematics? the findings provide valuable information to schools, educators, policymakers, and researchers on how to effectively prepare some children, including those of color, for induction into tomorrow’s scientific community and all children for life in our science and technology driven society. methods study participants the participant group included 32 african american students, equally divided by gender (16 women, 16 men), who were engaged in graduate work in science, engineering, or mathematics. eight of these students were pursuing terminal degrees in mathematics. the balance of the participants was pursuing graduate degrees in fields that relied heavily on mathematics (e.g. physics, computer science, mechanical engineering). in addition to race and gender, the participants were selected based on socioeconomic status and geographic region. in terms of socioeconomic status, 13 students (roughly 41%) self-identified as lower class; 15 students (roughly 47%) self-identified as middle class; the remaining four students (roughly 13%) self-identified as upper middle class or upper class. in terms of the regional diversity of the participant group, 23 (71.9%), 4 (12.5%), 2 (6.3%), and 2 (6.3%) were from the southern, midwestern, northeastern, and western regions of the united states, respectively. one student did not identify a particular state or region of origin based on a military background. because i examined life histories in the study, these classifications are based on the participants’ perceptions of their socioeconomic status and geographic region during their adolescence rather than categories that i imposed upon them. data collection and analysis the study relied heavily on each participant’s personal narrative, as interviews were the sole source of data. each student participated in one 60 to 90 minute interview. the interview had to be sufficiently open-ended to capture each student’s unique life history. however, i had to maintain some degree of consistency across the interviews. in an effort to meet both of these criteria, the study used an open-ended, semi-structured interview protocol. initial coding revealed a set of unorganized first-level codes. these first-level codes were then grouped and organized to develop second-level codes, also known as pattern codes, which allow for a more in-depth understanding of the themes and concepts involved in the study and the ways in which they interacted with and related to each other (miles williams early mathematical experiences bullock, e. c., alexander, n. n, & gholson, m. l. (eds.). (2012). proceedings of the 2010 philadelphia and 2011 atlanta benjamin banneker association conferences – beyond the numbers [special issue]. journal of urban mathematics education, 5(2). 58 & huberman, 1994). finally, these second-level codes were used to develop a set of themes related to the success of african american students in stem. findings the majority of the students involved in the study indicated that their early lives (ages 4–12) were characterized by an emerging interest in stem and concepts related to these areas (e.g. problem solving). furthermore, students perceived fewer obstacles to their success during this period of their development. it is most interesting that the majority of the students’ experiences with stem took place within their communities and was mediated by family members and other community members as opposed to the formal school environment. four major themes emerged from the analysis of the data. the first was that all students were involved in experiences that allowed a significant level of participation in stem. this theme refers to any experiences that created access to stem as well as an opportunity to develop interest in stem. second, all of the students experienced some form of positive personal intervention by another person. these interventions were aimed at developing, encouraging, and refining the students’ interest and performance in science, engineering, or mathematics. third, all students possessed perceptions of these fields that involved some sort of positive outcome. these outcomes ranged from actual (e.g., “good” grades on a mathematics test) and immediate (e.g., feels of accomplishment after solving a mathematics problem) to perceived (e.g., admiration from family members) and anticipated (e.g., access to college or employment in the future). finally, all of the students believed they possessed intrinsic qualities that qualified and prepared them for their involvement with stem. the participants often characterized this theme as a genetic connection to their parents, a god given gift or blessing, or a calling to stem. the remainder of this paper focuses on two of these themes: level of participation and positive personal intervention. level of participation level of participation refers to the availability and quality of experiences aimed at developing the students’ interest and competence in stem. early in their lives, students took advantage of activities that exposed them to knowledge and skills related to stem and gave them access to projects that allowed them to apply the knowledge they possessed. as children, 31 of the 32 students interviewed were introduced to the concepts that they would later study in graduate school during these early experiences. some of these experiences were fairly formal and similar to those one would have in a science or mathematics classroom. williams early mathematical experiences bullock, e. c., alexander, n. n, & gholson, m. l. (eds.). (2012). proceedings of the 2010 philadelphia and 2011 atlanta benjamin banneker association conferences – beyond the numbers [special issue]. journal of urban mathematics education, 5(2). 59 for example, jackie explained how she was first introduced to biology when, at the age of six, her mother gave her a microscope for christmas: what i do remember clearly that day is raiding the refrigerator and putting everything under the scope. making slides of onions. making slides of lettuce. pulling out hairs and making slides and spending literally all day making slides and looking at stuff under the scope. and the microscope is still, to this day, what i really like. i like being able to see things that you can’t see otherwise. this example demonstrates a level of access that would not normally be found in a child’s home and illustrates the importance of out-of-school experiences in shaping a child’s interests. other students described early experiences with stem that were also not as formal, but did involve equipment and skills related to stem. some students were introduced to and became interested in stem through play. in most cases, these experiences were not related directly to the content involved in any of these fields, nor did the students explore any of the skills employed by scientists, engineers, and mathematicians. instead, the instruments and experiences related to their play inspired the students to begin to explore stem. angela, a doctoral student in mathematics, explained how a simple contest with her grandfather evolved into an interest in mathematics. during these contests, she would race against her grandfather in an attempt to solve a mathematics problem: i think i got to use a calculator and he just got to use pen and paper. he would give me problems and we would race to see who could get it the fastest. and that’s when my first interest in math came because i always wanted to beat him. this informal experience allowed angela to explore mathematics and to develop a degree of familiarity with the discipline. it should be noted, however, that these less formal experiences were not always as explicitly linked to stem as angela’s was. some students reported experiences in their play that introduced them to stem in a more implicit way. for example, derrick described the ways in which his mother helped him develop the skills he would use as an engineer later in life by insisting that he design and build his own toys. the examples presented are similar in that they all occur in the students’ homes or communities, which was common to most of the participants. however, there were two students who recalled early experiences with mathematics, science, or engineering that occurred within the elementary school classroom. reginald, currently conducting research in human computer interactions, recalled the influence of the apple 2e computer stations in his third grade classroom. he stated that the computer’s capacity to perform functions ranging from teaching phonics to word processing sparked his interest in computer science. it should be williams early mathematical experiences bullock, e. c., alexander, n. n, & gholson, m. l. (eds.). (2012). proceedings of the 2010 philadelphia and 2011 atlanta benjamin banneker association conferences – beyond the numbers [special issue]. journal of urban mathematics education, 5(2). 60 noted, although reginald’s teacher was in the classroom, reginald did not describe her or him as a mediator of those experiences. instead, he was responsible for exploring his interest in computers on his own. derrick, pursuing a master’s degree in electrical engineering, also recalled his early experiences with science and mathematics taking place within his school. interestingly, he attended a preschool that focused specifically on mathematics and science. his experiences at this school provided him with advanced preparation in mathematics and science. consequently, when he entered elementary school, he was already familiar with much of the mathematics and science content. many of the students, particularly those from low socioeconomic backgrounds, explored science and mathematics through books or television. once they learned to read, many of the study participants became avid readers. in his interview, rodney described how his love of reading provided him with access to information about computers: at an early age, i was reading about computers. i saw that reading satisfies your desires to learn about something. so i started reading about computers and i had subscriptions to pc magazine….i got in trouble because i didn’t know how to pay for the stuff and after the free trial was over, i let my mom deal with all of that….so i put the connection together that if there was something out there that i wanted to learn, it was available in these forums that i was reading. the love of reading and the influence of books were particularly significant in the lives of students from low-income homes. the presence of books in the home or public libraries in the neighborhood provided these students with inexpensive access to information about stem. other students were introduced to stem through television. jared explained how he was influenced by a television series related to science: i use to watch [the television show,] mr. wizard’s world. that was one thing that i remember. i don’t know if that was my inspiration as for why science became fascinating to me, but that’s one of my earliest memories of being fascinated with how the world works. [mr. wizard would do] things that would incorporate physics and chemistry….i just thought it was cool. it was one of my favorite shows. although television programs did not offer students direct, hands-on interactions with stem, the medium did allow students to engage in limited exploration of these fields. consequently, television programming similar to mr. wizard’s world was particularly valuable for students who may not have had the financial resources or parental guidance needed to incorporate stem activities into the everyday pastimes of the home. each of these examples demonstrates the importance of both in-school and extracurricular participation in activities both explicitly and implicitly linked to stem. williams early mathematical experiences bullock, e. c., alexander, n. n, & gholson, m. l. (eds.). (2012). proceedings of the 2010 philadelphia and 2011 atlanta benjamin banneker association conferences – beyond the numbers [special issue]. journal of urban mathematics education, 5(2). 61 positive personal intervention although all students, regardless of ethnicity, gender, or class have stemrelated experiences in the home, many fail to see these experiences as such and to connect what occurs in the home with their school lives. this statement was not true for the students involved in this study. in each of their lives, there were people who were able to not only forge these connections but also encourage the development of the child as a scientist, engineer, or mathematician. positive personal intervention refers to the role of specific contributors to the development of the student as a scientist, engineer, or mathematician. according to the interviews, these contributors possessed two distinct characteristics. first, all interventions involved some level of interaction between the student and the intervener. high levels of interaction are defined as those that involve direct and often extended contact between the student and the intervener. low levels of interaction, on the other hand, were not as intimate and often were distant and brief. the second characteristic was the intervener’s level of knowledge of stem and the associated careers. similar to the level of interaction, level of knowledge also ranged from high to low. a contributor with a high level of knowledge possessed intimate knowledge of stem and the associated careers as well as the ways in which one prepared for and achieved success in these careers. a low level of knowledge was characterized by a rudimentary understanding of stem and the associated careers. the students described two types of interveners in their early experiences: role models and supporters. they characterized each type of positive personal intervention by level of knowledge (high to low) and level of interaction (high to low). role models were those who students saw as similar to themselves in terms of a variety of characteristics such as socioeconomic status, race, or gender. however, the direct interaction between a role model and the student was minimal. role models possessed a high degree of knowledge concerning stem and the professions associated with these areas. however, the student usually only observed a role model’s participation in these areas. supporters were defined as those who were not involved with science, engineering, or mathematics, but did participate in direct interactions with the student. during their early lives, parents and community members fulfilled the roles of supporters and role models in the students’ lives. the students most often reported personal intervention that consisted of direct, nonverbal encouragement from supporters in their lives. the term direct, in this case, refers to interaction solely between the child and the supporter. the supporter encouraged the student by mediating activities related to stem. for example, michael described the ways in which his family supported his interest in mathematics: williams early mathematical experiences bullock, e. c., alexander, n. n, & gholson, m. l. (eds.). (2012). proceedings of the 2010 philadelphia and 2011 atlanta benjamin banneker association conferences – beyond the numbers [special issue]. journal of urban mathematics education, 5(2). 62 [my family members] weren’t the ones saying, “you need to go to college.” they weren’t saying that. but in a way they were saying that by making sure i got to be in a position to do everything i needed to do. they didn’t say it, but they did it. it was by their actions. in this example, the family was able to offer direct encouragement through their involvement with michael. however, it should be noted that michael did not recall his family making any explicit verbal statements aimed at persuading him to study mathematics. direct nonverbal encouragement also occurred when supporters gave students the opportunity to demonstrate their abilities. by doing this, the supporters expressed their confidence in the students’ proficiencies in a given area. joseph, later to become an electrical engineer, explained how people in his community encouraged his early interest in mathematics: people…whenever they dealt with the money or anything that dealt with numbers, i was just real quick. i don’t know who the person was that picked up on it but at some point, people, when they had math stuff to do, they just said, “well, get joseph to do it.” this example was similar to michael’s in that joseph did not receive any explicit verbal encouragement aimed at motivating him to develop an interest in mathematics. instead, community members encouraged his interest in mathematics by providing him with opportunities to demonstrate his abilities. furthermore, this type of encouragement also reflected the value that the larger community placed on joseph’s abilities, and gave him the perception that, because of his abilities, he had a valuable role in the community. a number of students spoke about the influence of role models their early development as scientists, engineers, and mathematicians. most often, students who interacted with role models described their families as middle or upper class. these students often spoke of family members who demonstrated success in stem or related fields. adrienne explained that her uncle, a medical doctor, influenced her decision to pursue science. in this case, the student did not have a close relationship with her uncle. however, her perceptions about his career and her family’s appreciation for his profession had a profound influence on her later decision to pursue a career in stem. another student, travis, mentioned the influence his older brother had on his development as a mathematician: i really looked up to my brother…and i just remember him one day saying how much he liked math. i…remember him saying something about learning trigonometry and i wanted to know what that was. i remember very shortly after that i grew to pick up math and discover that i had an aptitude for it. there were other students in the participant group who had parents, siblings, or other family members who were involved in science, engineering, or mathemat williams early mathematical experiences bullock, e. c., alexander, n. n, & gholson, m. l. (eds.). (2012). proceedings of the 2010 philadelphia and 2011 atlanta benjamin banneker association conferences – beyond the numbers [special issue]. journal of urban mathematics education, 5(2). 63 ics. however, most of them described the relationships they had with these individuals as more direct and therefore, more indicative of a supporter relationship. in each of these examples, the student was involved in a positive interaction with a person acting as a supporter or role model. these people were responsible for mediating many of the students’ early experiences with stem. through their actions, they were able to encourage the interests the students were developing in these areas. discussion three factors characterized most of the students’ early experiences with stem. first, experiences, although related to stem, were also relevant to the students’ lived experiences. for example, at six years old, jackie did not use her microscope to examine the phenotypic expression (i.e., red eyes versus black eyes, wings versus no wings) of drosophilae (the fruit fly)—a common curricular objective in biology. instead, she was creating slides of familiar objects found within her home. as opposed to a tool of science, the microscope was seen as a tool for investigating the world around her. the same can be said of the other students’ experiences: the majority of their experiences built on their prior knowledge and were relevant to their world. some researchers are urging teachers to utilize this finding in their instruction. ladson-billings (1994) argues that effective teachers scaffold or build bridges between the knowledge students’ carry with them from their communities and what is being taught in the classroom in an effort to facilitate the learning process. by using cultural referents to impart knowledge, skills, and attitudes, teachers not only empower their students academically but also socially, emotionally, and politically. as opposed to being placed at a disadvantage because of the discontinuities that exist between their culture and the culture of the school, students who work within the context of a culturally relevant curriculum find that their cultural knowledge actually serves as a foundation for the knowledge they learn in the classroom. second, in addition to exposure to stem, the science and mathematics experiences in the home also provided hands-on and direct access to these subjects. for example, the contests between angela and her grandfather did not simply introduce her to mathematics. in addition to presenting mathematical concepts, the game also allowed her to apply knowledge she possessed, hone the skills she was developing, and cultivate both new knowledge and skills. this particular characteristic of the experiences, however, was most apparent in the lives of students from middle and upper class home and communities. students from low-income homes had limited access to these types of experiences. instead, they relied on the more intangible stem experiences presented in books and television programs. williams early mathematical experiences bullock, e. c., alexander, n. n, & gholson, m. l. (eds.). (2012). proceedings of the 2010 philadelphia and 2011 atlanta benjamin banneker association conferences – beyond the numbers [special issue]. journal of urban mathematics education, 5(2). 64 finally, in each of these examples, the exploration and subsequent learning were directed, at least in part, by the student. consequently, students not only experienced significant levels of participation in areas related to stem but also they were involved in the direction and development of the activities that facilitated their learning. it should be noted, however, that the adults involved in each of these situations played a very critical role. in all of these cases, family and community members were responsible for encouraging the student’s interests and strengthening the connections between the child’s experiences and the areas of stem. it is important to note that only two of the 32 students (both from schools outside of the southeastern region of the united states) involved in the study recalled any influence from their schools during their early lives and only one of these two students remembered the positive influence of teachers at the school. this finding in addition to the ones summarized above points to two important ideas concerning the early preparation of african americans in stem. first, the findings suggest the positive influence that early involvement in science and mathematics at home and in the community can have on children. informal experiences mediated by family members in the students’ homes likely prepared the students in the study for success in these fields later in life. second, these findings support other studies, which have revealed a severe lack of effective mathematics and science preparation at the elementary school level. for example, research on science and mathematics education at the elementary school level revealed that that less than half (49%) of elementary school classes use science and mathematics objects each day (national education association, 2002). according to malcolm and anderson (2000), elementary school teachers deem instruction related to science and mathematics as less critical to child development when compared to teaching that focuses on language development, play, creative arts, and motor activities. most students, regardless of race, receive little mathematics instruction and even less science instruction during elementary school. research has shown, however, that structured mathematics and science lessons in the elementary classroom improve students’ performance in high school (novak & musonda, 1991). a renewed commitment to mathematics and science preparation at the elementary level may result in a larger pool of potential african american scientists, engineers, and mathematicians to the nation’s middle and high schools. these findings have implications in educational curricula, teacher training, classroom practice, educational policy, and informal education programs (e.g., afterschool and museum programs). if the united states is to continue to move forward into its scientifically and technologically based future, it must begin to cultivate a scientifically and mathematically literate populace. however, in order to do so, the educational community must improve its capability to provide effective stem education to all students; including those that are often marginalized williams early mathematical experiences bullock, e. c., alexander, n. n, & gholson, m. l. (eds.). (2012). proceedings of the 2010 philadelphia and 2011 atlanta benjamin banneker association conferences – beyond the numbers [special issue]. journal of urban mathematics education, 5(2). 65 by the current educational system (i.e. student of color, poor students, and immigrant students). furthermore, these changes must be systemic and incorporate early childhood and elementary education as critical areas of improvement. only then will we begin to provide our children with the skills and knowledge necessary to authentically participate in the world in which they will one day live. at the symposium the conversation during the symposium focused on two key points. first, participants were interested in the idea of stem education and its connections to social justice and democratic participation. this interest represented an important framework for the rest of the symposium. second, the participants spent a great deal of time discussing the systemic and structural changes that are needed in order to create stem education that prepared all children for participation in tomorrow’s world. specifically, the participants explored classroom practice, educational policy, and teacher development. the conversations resulted in several key recommendations for improving stem education, specifically, as it applies to marginalized communities. references ladson-billings, g. (1994). the dreamkeepers: successful teachers of african american children. san francisco: jossey-bass. malcolm, s., & anderson, b. (2000). entering the education pipeline. in j. g. campbell, r. denes, & c. morrison (eds.), access denied: race, ethnicity, and the scientific enterprise (pp. 49–60). oxford, united kingdom: oxford university press. miles, m. & huberman, a.m. (1994). qualitative data analysis. thousand oaks, ca: sage. moses, r. p., & cobb, c. e. (2001). radical equations: math literacy and civil rights. boston, ma: beacon press. national council of teachers of mathematics. (2000). principles and standards for school mathematics. reston, va: national council of teachers of mathematics. national education association. (2002). 2002 instructional materials survey: report of findings. retrieved from http://www.publishers.org/press/pdf/2002%20instructional%20materials%20report.pdf. national science board. (2012). science and engineering indicators 2012. arlington, va: national science foundation. novak, j. d., & musonda, d. (1991). a twelve-year longitudinal study of science concept learning. american educational research journal, 28, 117–153. perry, t., moses, r. p., wynne, j. t., cortés, e., jr., & delpit, l. (eds.). (2010). quality education as a constitutional right: creating a grassroots movement to transform public schools. boston, ma: beacon press. http://www.publishers.org/press/pdf/2002%20instructional%20materials%20report.pdf journal of urban mathematics education december 2012, vol. 5, no. 2, pp. 44–52 ©jume. http://education.gsu.edu/jume maggie lee mchugh is an associate lecturer and learning center director at the university of wisconsin-la crosse, 1007 cowley hall, 1725 state street, la crosse, wi, 54601; email: mmchugh@uwlax.edu. her research interests include critical pedagogy, social justice mathematics, and equitable uses of technology in education. jennifer kosiak is an associate professor of mathematics education at the university of wisconsin-la crosse, 1004 cowley hall, 1725 state street, la crosse, wi 54601; e-mail: jkosiak@uwlax.edu. her research interests include the mathematics content knowledge for teaching, the integration of technologies to support the teaching and learning of mathematics, and social justice in mathematics. public stories of mathematics educators critical transformation of mathematics educators maggie lee mchugh university of wisconsin la crosse jennifer kosiak university of wisconsin la crosse hat can be done to enhance pre-service educators’ knowledge of social justice and its role in the elementary mathematics classroom? i (maggie) pondered this question as a graduate student and mathematics educator. following the advice of cochran-smith (1991) who advocated for connecting reform-minded teachers and pre-service teachers, i approached my colleague dr. jennifer (jenn) kosiak, an associate professor of mathematics education, about assisting in my research. although jenn had little background knowledge in social justice, her strong passion to continually learn and grow made her an ideal traveling companion as we set off on a journey to enhance per-service mathematics teachers’ awareness of social justice issues and their relevance in the mathematics classroom. this public story focuses not only on the pre-service educators we worked with but also, and more so, on our own professional and personal growth through the journey. we found that we could not speak of the transformation that we witnessed in these future teachers without addressing our personal transformations. throughout this story, i will share our journey from my perspective. as part of the research involved written reflections from jenn, i will incorporate her voice throughout the article in arial font to address her growth in her own words. before embarking on this journey, i read literature about how to best implement a social justice curriculum. i found that, in education, social justice themes are enacted to enhance students’ learning and their life chances by challenging the inequities of school and society (michelli & keiser, 2005). yet, social justice education cannot be implemented without the dedication of critically aware teacher educators who have examined their own identity and privilege so w mchugh & kosiak public stories journal of urban mathematics education vol. 5, no. 2 45 that they are able to guide pre-service teachers in the process of developing critical consciousness. quin (2009) explicitly states the goal of social justice educators is “to empower [both] educators and learners to act in anti-oppressive ways for social justice” (p. 110, emphasis in original). to implement such pedagogy, social justice educators must create what freire (1970) calls horizontal relationships, relationships where teacher and learner work together toward a common goal. this is the type of relationship that developed between jenn and me. no longer were jenn and i limited to our respective roles of professor and graduate student; rather, we were both mathematics educators with the desire to create a dynamic classroom where pre-service educators could learn about social justice through critical dialogue. analyzing identity through my initial readings in critical pedagogy, i learned that our first step as educators was to reflect upon our own identities and privileges. as two white females from small towns with little diversity of race, ethnicity, or language, we had a lot of examining to do. reflecting upon our own personal and educational upbringing, we asked the following questions:  as a teacher, what is my role in the classroom?  as a white person, what innate biases do i hold regarding student learning?  as a female, what privileges may be affecting my teaching practices?  as a white female, how do the tasks that i select for my students either include or exclude students in my classroom?  overall, how do these lenses affect my view of social justice in a mathematics classroom? many of my biases come from my upbringing and education. most of my peers were from two-parent, middle-class homes. having experienced little diversity, i believed that all students had the ambition to go to college, to become educated, and to find a fulfilling career. it was not until i started working in public schools while in college that i came to understand that many students, whether urban, rural, or suburban, come from backgrounds much different from mine. desiring to engage students through relevancy in learning, i began to search for resources, conferences, and like-minded colleagues who could help me to examine my privilege in order to become a critical educator who can embrace all students. although i summarize my desire to change in a few sentences, this transformative process occurred over multiple years (and continues). in the midst of learning more about critical pedagogy, i approached jenn about my research idea. mchugh & kosiak public stories journal of urban mathematics education vol. 5, no. 2 46 when maggie first approached me about incorporating social justice into my elementary mathematics methods course, i was apprehensive because i thought that social justice merely meant making my classroom equitable to diverse student populations. with the national council of teachers of mathematics (nctm, 2000) equity principle in mind, my beliefs about social justice suggested that i should create a classroom environment that was both challenging and supportive of student learning regardless of their background. “i already do this, right?” i thought to myself. however, listening to maggie’s discussion about social justice and critical pedagogy caused me to reformulate my understanding of social justice as a mechanism to investigate injustice in the world and create change. immediately, i thought of issues such as distribution of world wealth and poverty rates and how i could use statistics and number facts to model math concepts to my pre-service students. i believed that purposefully modeling pedagogical practices would help these future teachers understand the role that social justice could play in the classroom, especially the mathematics classroom. defining social justice mathematics much of the planning for this project occurred over the summer preceding the mathematics methods course. when the first class assembled, i had some jitters. were the students willing and ready to embark on this journey with jenn and me? would they believe in the value of transforming the mathematics classroom into a place where all students can feel empowered to enact social change? we were about to find out. we administered a pre-survey to all of the pre-service educators in three sections of a mathematics methods course in order to ascertain their beliefs and abilities regarding social justice and mathematics. like jenn, the students began the semester with little knowledge of what social justice meant, specifically what it meant in the mathematics classroom. similarly, these students were also influenced by the nctm equity principle, which was introduced the week before, as much of their pre-survey definitions focused on supporting all students in the classroom. this finding made me realize that these students needed a firmer grounding in general social justice literature before i could move them toward integrating social justice and mathematics. however, given the limitations of a onesemester course, jenn and i decided to focus on the concept of social justice within the mathematics classroom, all the while maintaining that the principles of social justice can be utilized in every classroom regardless of content. when introducing social justice mathematics to the pre-service teachers, i was apprehensive about how they would perceive this different course expecta mchugh & kosiak public stories journal of urban mathematics education vol. 5, no. 2 47 tion. i tried to find concrete examples of what social justice pedagogy would look like in the elementary mathematics classroom. i believe these examples were critical based upon the students’ initial difficulty in defining social justice education both inside and outside of mathematics. overall, i was relieved that no one questioned why i was including social justice in the mathematics classroom. in fact, many of the students expressed their satisfaction with the practicality of including social justice in a content area given that no other content methods course addressed this mode of teaching. to engage the pre-service educators in creating a more formal definition of social justice mathematics, jenn and i asked them to critically reflect upon the article “a social justice data fair: questioning the world through math” by alexander and munk (2010). this article showcases how a canadian elementary school introduced social justice through an activity called the welfare diet. as part of their reflections, we asked these students to develop a working definition of social justice as well as an idea of how they might use this kind of activity with their future students. i was happy that these pre-service teachers took the definition of social justice beyond equity to include connecting mathematics to real-world problems as well as to the students’ lives and communities. many of my students found that the article helped them to see a new approach to teaching mathematics. additionally, this article allowed the pre-service teachers to see that mathematics is not a singular course; it can be effectively integrated across the curriculum into areas such as social studies and language arts. their course reflections continually shaped my own vision of social justice in the mathematics classroom. perhaps, not having a preset notion of what social justice is and is not allowed all of us to explore, investigate, and conjecture about social justice through the lens of mathematics. with our building concept of social justice mathematics, i introduced critical mathematics pedagogues such as gutstein (2006) and gutiérrez (2007). using my research to spur a more formalized definition, we decided that teaching mathematics for social justice hinges on a teaching and learning environment where  students are introduced to the various issues of equity, diversity, and social injustices;  students increase and strengthen their mathematical content knowledge; and  students learn to use mathematics to identify and examine social issues with the intent to enact change. mchugh & kosiak public stories journal of urban mathematics education vol. 5, no. 2 48 modeling social justice mathematics the association of teacher educators (2003) states: “in order for teacher educators to impact the profession, they must successfully model appropriate behaviors in order for those behaviors to be observed, adjusted, replicated, internalized, and applied appropriately to learners of all levels and styles” (p. 1). with this charge in mind, our first task was to slowly but intentionally model social justice pedagogy in the math classroom. embracing our love of children’s literature, jenn and i chose to read the book if the world were a village (smith & armstrong, 2006) that scales the world population to a village of 100. this book provides the reader with demographics and statistics such as “20 villagers earn less than one dollar a day” or “60 are always hungry.” as i listened to jenn read this book aloud to the pre-service teachers and heard all of the descriptions of the world’s population in terms of percents and fractions out of 100, i wondered if they, too, were reacting to the reality behind those numbers. after reading if the world were a village, we asked students to pick a page from the book related to language, ethnicity, age, or wealth and to make mathematical models to represent the data presented. i wondered if these pre-service teachers would concentrate solely on the numbers and ignore the real-world inequities that were so vividly portrayed. jenn and i were delighted when they discussed not only the mathematics but also the inequities across the world’s population. for example, they were surprised that roughly one-third of the population is illiterate and almost 40% of the population does not have access to running water. this activity led to a richer discussion of how percents and fractions can be used in the mathematics classroom to illustrate larger social issues. we discussed ways in which they could extend such an activity into their future classrooms, including creating a real-life version of the book entitled if wisconsin schools were a village. this project would entail investigating the make-up of the classroom and comparing that to statistics in wisconsin including the number of minority students, ell students, and students with special needs. experiencing social justice mathematics in addition to modeling social justice mathematics, we wanted to develop a purposeful learning experience where these pre-service teachers could apply their definitions of social justice to a mathematical concept. this desire led to the implementation of the accessible playground project, adapted from a social justice lesson plan developed by the centre for urban schooling. i walked into the classroom one day and declared, “all children have the right to relax and play!” after some initial wonderment about my strong declaration, jenn and i outlined the accessible playground project. groups of students mchugh & kosiak public stories journal of urban mathematics education vol. 5, no. 2 49 were given the task to research both playground equipment and accessibility requirements. they then completed a scale 2d blueprint and a 3d model using mathematical concepts such as geometric shapes and solids, area and perimeter, scale and proportional reasoning. additionally, they calculated an estimated cost for the playground. these pre-service teachers summarized their findings and outlined the criteria they established in a formal report. at the end of this 3-week activity, the groups presented their models in the classroom linking their presentation to not only why their model was safe and accessible but also to the unicef (1990) convention on the rights of a child, article 31, which prompted the activity. through this activity, groups of pre-service teachers determined what accessibility meant not only in a playground situation but also their own classroom. our conversations began with the appropriate angle of an accessible slide and the cost function to model this task, but quickly shifted. we found ourselves discussing accessibility is a classroom-based issue focused on things like location of white boards, heights of desks and lab stations as well as instructional materials that would meet the needs of all students regardless of ability. jenn and i facilitated discussion amongst the pre-service teachers about including bilingual signs in schools, conservation of the environment, and recycling both inside and outside of the classroom. the idea of going out to play at recess was linked to childhood obesity and school lunch programs. all of these discussions spurned from the initial concept of playground equity. additionally, part of our established criteria for a social justice mathematics lesson is to use the mathematics to critically examine an inequity, therefore, we asked the pre-service educators to analyze cost differentials between an accessible and traditional playground. almost every group found that the cost of accessible playgrounds was fairly equal to the cost of traditional school playgrounds. they also found themselves examining the playgrounds at their clinical field sites to give them an accessibility ranking. creating social justice mathematics jenn and i used discussion, article reflections, purposeful modeling, and learning experiences to teach our students about social justice pedagogy and its relevance to the mathematics classroom. finally, we implemented a pedagogical project that allowed the pre-service educators to explore their understandings of social justice. the culminating activity was the creation of a mathematical concept plan that embedded a social justice theme. initially, students were apprehensive about the type of activity that they would use in order to satisfy this course assignment and jenn and i did not know what to expect. they continually asked, mchugh & kosiak public stories journal of urban mathematics education vol. 5, no. 2 50 “is this a social justice idea?” to which we replied, “you need to decide that for yourself.” as the semester progressed, students would reflect on different ways they could include social justice in their concept plans and the idea of linking mathematics through socially relevant activities emerged. some students discussed how they could link measurement to the inequities of fresh water supply in the united states versus india. other discussions evolved into how they could use percents and fractions to investigate how school lunch programs are related to childhood obesity. they used numbers and operations to model poverty in the united sates and its impact on student achievement. even at the earliest grade levels, these preservice teachers were able to not only have students use the mathematics but also engage in the discussion of the social justice themes. jenn and i were pleasantly surprised at the depth of mathematical connections to social justice these pre-service teachers created. mathematical concepts of fractions and percents were tied to child labor; coordinate planes and graphing helped locate the native american tribes of wisconsin; data regarding cyber bullying was graphed to find the increasing rate of change. tasks began with critical questions such as: how many steps do you take to get a glass of fresh water? do you think all people on earth can measure the distance to fresh water in steps? these pre-service teachers found ways to critically embed social justice into mathematics lessons. their growth throughout the semester was reflected in their post-survey where all students could define social justice in the mathematics classroom, but more so in their final reflections about the value they found in not just “doing” mathematics but “using” mathematics to explore local, regional, national, and global concerns. yet, just as important to the research was the unintended outcome of our transformation as critical educators. overall, the experience of incorporating social justice into the mathematics class was rewarding as i saw not only growth in my pre-service teachers’’ understandings of social justice but also in my own. the main challenge was that i did not fully grasp the concept of social justice in education at the onset of the course. indeed, i often felt that i learned more about social justice as i processed article reflections with the students and read their concept plans. i also wonder if my own identity might have influenced how i modeled social justice to my students. as a white female educator teaching mainly white female students, i think it is important for all of us to examine how our beliefs shape our teaching and the types of activities we give to our students. if you asked me a year ago if i would incorporate themes of societal inequalities into my mathematics methods classroom, i would have probably said, “don’t they get that in social studies?” i have always been a believer of purposeful modeling about math concepts including integrat mchugh & kosiak public stories journal of urban mathematics education vol. 5, no. 2 51 ing reading and writing into the mathematics classroom. this project has helped me understand how all curricula can be integrated in a manner that will enhance students’ academic and social learning. though i had studied social justice practices and had begun my transformation as a critical educator, i had worked with others in their exploration of these difficult concepts. while moving with these pre-service educators towards a deeper mathematics pedagogy guided by social justice principles, i realized just how difficult it is to maintain an atmosphere appropriate for critical pedagogy. i would constantly critique and reflect upon everything that occurred in class from the tasks we chose to the language we used. sometimes, i could tell the preservice educators were frustrated with jenn and me because we never “gave them the answer.” the perception of mathematics is that all problems have a direct solution. however, one of the most difficult tasks was attempting to open the preservice teachers’ perception from a purely quantitative view of mathematics to a qualitative, narrative understanding that highlights problem solving and process more than the solution. indeed, this view of mathematics as a dynamic set of knowledge still plagues me. i fall into the trap of teaching to skill building without providing a critical context for the learning. nevertheless, in small and large ways, social justice pedagogy has transformed my teaching. my students engage in an activity that examines the wage gap between white men and white women, black men, black women, latinos, and latinas. i ask my students not only to engage in the mathematics of finding models of equations and when white men and afore mentioned groups would earn equal pay, but i also encourage them to reflect upon their future career. i recognize that the experience of leading others to an awareness of social justice truly transformed me into an emerging critical educator. i realize i will continue to explore my identity and reflect upon my teaching practices; however, with the lens of social justice, i know that my awareness of social justice will lead to praxis in my teaching and daily living. one year after engaging in the research project, jenn and i have come to realize that we are critically transforming as mathematics educators. it is hard to believe that prior to this research project, we viewed mathematics and social justice as separate entities. the idea of enacting change through examining societal inequalities in mathematics continues to transform both our classes and the preservice teachers with whom we engage. today, we share a formulaic definition of social justice in the mathematics classroom: critical mathematics + societal inequity investigations = transformative change for all. references alexander, b., & munk, m. (2010). a social justice data fair: questioning the world through math. rethinking schools, 25(1), 51–54. mchugh & kosiak public stories journal of urban mathematics education vol. 5, no. 2 52 association of teacher educators. (2003). standards for teacher educators. retrieved from http://www.ate1.org/pubs/uploads/tchredstds0308.pdf. cochran-smith, m. (1991). learning to teach against the grain. harvard educational review, 61, 279–309. freire, p. (1970). pedagogy of the oppressed. new york, ny: continuum. gutiérrez, r. (2007). (re)defining equity: the importance of a critical perspective. in n. s. nasir & p. cobb. (eds.), improving access to mathematics: diversity and equity in the classroom. (pp. 37–50). new york, ny: teachers college press. gutstein, e. (2006). reading and writing the world with mathematics: toward a pedagogy for social justice. new york, ny: routledge. michelli, n. m., & keiser, d. l. (eds.) (2005). teacher education for democracy and social justice. new york, ny: routledge. national council of teachers of mathematics. (2000). principles and standards for school mathematics. reston, va: national council of teachers of mathematics. quin, j. (2009). growing social justice educators: a pedagogical framework for social justice education. intercultural education, 20(2), 109–125. smith, d. j., & armstrong, s. (2006). if the world were a village: a book about the world’s people. toronto, canada: kids can press. unicef. (1990). the conventions on the rights of the child. retrieved from http://www.unicef.org/crc/files/survival_development.pdf. http://www.ate1.org/pubs/uploads/tchredstds0308.pdf http://www.unicef.org/crc/files/survival_development.pdf journal of urban mathematics education december 2017, vol. 10, no. 2, pp. 106–139 ©jume. http://education.gsu.edu/jume cathery yeh is an assistant professor in the college of educational studies at chapman university, 1 university drive, orange, ca 92866; email: yeh@chapman.edu. her research examines issues of equity in mathematics. in particular, she is interested in capturing teachers’ efforts to disrupt language, gender, and dis/ability hierarchies in mathematics classrooms. math is more than numbers: beginning bilingual teachers’ mathematics teaching practices and their opportunities to learn cathery yeh chapman university in this article, the author provides results from a 3-year, longitudinal study that examined two novice bilingual teachers’ mathematics teaching practices and their professional opportunities to learn to teach. primary data sources included videotaped mathematics lessons, teacher interviews, and field notes of their teacher preparation methods courses. findings revealed that the teachers were oriented toward differing views of learning that shaped how they organized students’ learning of language and mathematics during classroom instruction. while both teachers used similar teaching strategies to support students’ development of mathematics specific literacies, there were variances in how the learners were positioned within the classroom community and how and which repertoires of language practices were available and used during mathematics instruction. the teachers’ differing orientations toward learning are traced to their own professional opportunities to learn to teach. the significance of recognizing both the acquisition and participation metaphors of learning and the development of linguistically and culturally relevant teacher education are discussed. keywords: bilingual education, mathematics education, novice teachers mergent bilinguals comprise the fastest growing demographic group and represent a fourth of all urban public school students in the united states (garcía, kleifgen, & falchi, 2008; menken, kleyn, & chae, 2012). as the population of emergent bilinguals grows across the country, the prevailing view about bilingual education has become increasingly more negative and debates around language policy more contentious (see, e.g., flores & bale, 2016; menken et al., 2012; ovando, 2000; sleeter, neal, & kumashiro, 2015). despite 40 years of research on the benefits of bilingual education, the mathematics learning experiences of emergent bilinguals have mirrored u.s. language politics of xenophobia and assimilation (aquino-sterling, rodríguez-valls, & zahner, 2016; khisty & willey, 2008; valdés, 2015; valenzuela, 2002). emergent bilinguals educated in western schooling practices have learned to perform mathematics using algorithms that are not their own, in a language different than their native tongue, and solving mathematics problems e http://education.gsu.edu/jume mailto:yeh@chapman.edu yeh math is more than numbers journal of urban mathematics education vol. 10, no. 2 107 irrelevant to their interests and experiences (aquino-sterling et al., 2016; garcía et al., 2008; moschkovich, 2012). in the current educational backlash against public education, and bilingual education in particular, political and financial groups such as the english-only movement are growing and their primary goal is to maintain euro-american dominance (see, e.g., aquino-sterling et al., 2016; garcía et al., 2008; khisty & willey, 2008; valdés, 2015; valenzuela, 2002). nonetheless, there exists a growing counterinsurgence of bilingual teachers (achinstein & aguirre, 2008; achinstein & ogawa, 2011; athanases, banes, & wong, 2015; sleeter et al., 2015; villegas & irvine, 2010). the very nature of bilingual education represents a resistance toward cultural and linguistic hegemony. in the research project reported here, i examine the teaching and learning experiences of two socially conscious1 novice bilingual teachers serving in the same urban2 elementary school in a state where english-only policies have been codified into law. specifically, the purpose of the 3-year longitudinal study, which followed two bilingual teachers from their initial teacher preparation program through their first 2 years of teaching, was to gain an understanding of their mathematics teaching practices and their own opportunities to learn to teach. the emphasis of the work presented is not only to understand their mathematics teaching in bilingual settings but also to understand how their teaching practices are negotiated and grounded by lived experiences and the perspectives developed over time and through their everyday teaching. studies that examine beginning teachers’ experiences, especially with bilingual mathematics curricula, are scarce (ingersoll & strong, 2011; sleeter et al., 2015) and important in advancing understanding of the processes in which teacher preparation programs can increase their potential to prepare critical bilingual educators. literature review conceptually, i build the project beginning with the idea that different ways of understanding learning contribute to our knowledge of the relationship between language and mathematics in bilingual classrooms. therefore, i consider both cog 1 the term socially conscious refers to the two teachers’ expressed commitment to social justice and issues of equity. these teachers entered the profession with the goal to challenge english hegemony in schools and to leverage the cultural and linguistic resources of children and their communities. 2 the term urban takes on multiple meanings in public discourse and in educational research. here, urban refers to a place with high population density, and urban schools take on two characteristics. first, urban schools most often serve a diversity of students across racial, linguistic, and socioeconomic backgrounds (among other characteristics). second, urban schools are most often part of a large school district characterized with bureaucratic leadership structures, emphasis on standardized testing, and high teacher turnovers. yeh math is more than numbers journal of urban mathematics education vol. 10, no. 2 108 nitive and sociocultural theories and pedagogical understandings on bilingualism in mathematics education (e.g., cummins, 1984; gutiérrez, 2002; gutiérrez, sengupta-irving, & dieckmann, 2010; halliday, 1978; moschkovich, 2007, 2012; o’halloran, 2005; téllez, moschkovich, & civil, 2011) and in the critiques and theories on the politics of language in the united states (cummins, 2000; khisty & willey, 2008; setati, 2008; valenzuela, 2002). in the following section, i provide a historical account of bilingualism in mathematics education as well as research on beginning bilingual teachers of mathematics to provide context for the tensions, contradictions, and different motives that shape language and mathematics learning in bilingual classrooms. language and mathematics learning in bilingual classrooms the relationship between language and mathematics in research began in the field of psychology and examined the cognitive functioning of bilingual learners during arithmetic computations and the solving of word problems (see moschkovich, 2007 for a review of early research). these studies brought to the fore the importance of language in mathematics education. simultaneously learning a second language and mathematics is cognitively demanding and can slow down the process of mathematics learning. language demands, and even the absence of a common language, can generate tensions and impact who is (and is not) participating in classroom interactions (cummins, 1984; gutiérrez et al., 2010; halliday, 1978; pimm, 1987; setati, 2008; spanos, rhodes, dale, & crandall, 1988). contrary to popular beliefs, math is more than just numbers; school mathematics is linguistically complex, with multiple language modalities—reading, writing, listening, speaking, and representing—in play (aguirre & bunch, 2012; cummins, 1984; dutro & moran, 2003; halliday, 1978; kress, 2010; o’halloran, 2005). over the last three decades, there has been growing attention in the mathematics education research community on bilingual and multilingual learners. the majority of past and current studies have examined the effects of students’ language proficiencies on academic performances and how teachers’ facilitation and scaffolding can support students’ acquisition of language and mathematics content (e.g., cuevas, 1984; cummins, 1984; dutro & moran, 2003; halliday, 1978; o’halloran, 2005; pimm, 1987; quinn, lee, & valdés, 2012; spanos et al., 1988). however, language is only one of many semiotic resources (e.g., physical control of space, gesture, gaze) at play in bilingual/multilingual classrooms (aguino-sterling et al., 2016; dasilva iddings, 2005; domínguez, 2005; gutiérrez et al., 2010; turner, dominquez, empson, & maldonaldo, 2013; zahner & moschkovich, 2011). a growing number of studies demonstrate that language use goes beyond language proficiency. in a study of student interactions within a reform-based calculus class, gutiérrez (2002) found bilinguals fluent in english spoke in spanish not as a necessity but as a way to bond with others. moschkovich’s (2007) study of code yeh math is more than numbers journal of urban mathematics education vol. 10, no. 2 109 switching in mathematical conversations similarly builds on language hybridity. her analysis of student discussions found that code switching between english and spanish, using everyday colloquialisms and school-based discourse, was most prevalent when students were negotiating ideas and engaging in mathematical arguments. these studies suggest that language and choice of language, more than reflecting proficiency, are integral to the identity work students engage as they learn. bilingual speakers use language not only on the basis of proficiency but also as an expression of identity (cummins, 2000; khisty & willey, 2008; setati, 2008; valenzuela, 2002; white, crespo, & civil, 2016). validation and maintenance of students’ linguistic identities are intricately linked to academic performance. bilinguals who can read, write, and communicate in their home language are more likely to enroll in advanced mathematics courses and continue to higher education (garcía et al., 2008; khisty & willey, 2008). instruction in a student’s home language and its validation play significant roles in achievement for bilingual learners (aquinosterling et al., 2016; celedón-pattichis & ramirez, 2012). the literature reviewed offers important contributions to this study on developing principled instruction for bilingual mathematics learners. first, teachers must facilitate students’ acquisition of both the mathematical knowledge and language skills needed to engage in meaningful disciplinary discourse practices (aguirre & bunch, 2012; aquino-sterling et al., 2016; celedón-pattichis & ramirez, 2012; cummins, 1984; moschkovich, 2012; o’halloran, 2005; pimm, 1987; quinn et al., 2012). second, mathematics education research shows that patterns of classroom interaction as it relates to student agency, status, and positioning affect students’ access and achievement in the classroom (gutiérrez, 2002; khisty & willey, 2008; turner et al., 2013; zahner & moschkovich, 2011). third, bilingual teachers need to leverage the varied linguistic repertoires bilingual students bring into classrooms and can employ to learn and do mathematics (celedón-pattichis & ramirez, 2012; domínguez, 2005; gutiérrez et al., 2010; téllez et al., 2011; zahner & moschkovich, 2011). this body of work, encompassing over three decades of research, demonstrates the advantages of bilingual education; bilingual education still serves as the most divisive approach to teaching the 12 million emergent bilingual students in the united states. the politics of bilingual education in the united states the question arises, why is there so much controversy over bilingual education? tension in bilingual education involves complex issues of power, identity, and social status tied to an american history of xenophobia (arce, 2004; garcía et al., 2008; ovando, 2000). as arce (2004) argues, anti-bilingual movements are grounded on racialized discourse that, “to be ‘american’ means using only one language english and accepting the dominant culture’s norms and values” (p. 230). it is no coincidence that federal government and state efforts toward banning bilingual yeh math is more than numbers journal of urban mathematics education vol. 10, no. 2 110 education in california (1998), arizona (2000), and massachusetts (2002) occurred during the largest wave of non-english speaking immigration in american history. the very nature of bilingual education, with the embracing of diverse cultures and recognition of other languages as significant to american mainstream institutions, serves as counter-resistance to euro-american and english hegemony. schools are transmitters for capitalist interests as well as locations for opposition to the economic and political interests of the dominant (giroux, 2001; mclaren & jaramillo, 2007; monzó & mclaren, 2014). bilingual educators have the potential to break the power structures in schooling by challenging hierarchical mandates and leveraging the cultural and linguistic resources of children and their communities. bilingual teachers of mathematics the need for well-prepared bilingual teachers is critical, which helps to explain why a growing body of research has focused on teacher recruitment and the practices of experienced bilingual teachers. studies spanning over two decades indicate that bilingual teachers often enter and remain in teaching driven by a humanistic commitment to give back to their own communities and to redress inequities in schooling for students who share their cultural backgrounds (achinstein & aguirre, 2008; achinstein & ogawa, 2011; cavazos, 2009; gomez, rodriquez, & agosto, 2008; sleeter et al., 2015; villegas & irvine, 2010). a growing body of scholarship suggests that bilingual teachers may be more capable of supporting language learning, tapping into the cultural resources in themselves and their students, and positioning bilingual learners as capable contributors to the classroom learning community (e.g., cavazos, 2009; celedón-pattichis, musanti, & marshall, 2010; gutiérrez, 2002; musanti, celedón-pattichis, & marshall, 2009; remillard & cahnmann, 2005; sleeter et al., 2015; turner et al., 2013; zahner & moschkovich, 2011). while these studies justify teacher diversity initiatives, the larger research community currently knows little about the experiences of novice bilingual teachers, particularly in mathematics, and what actually happens when graduates of teacher preparation programs begin professional teaching (ingersoll & strong, 2011; sleeter et al., 2015). what is known, however, is that the current orientation toward language and the structured sets of expectations and accepted teaching practices that dominate u.s. schools and teacher education programs have largely perpetuated language policies of assimilation (gomez et al., 2008; khisty & willey, 2008; lópez, scanlan, & gundrum, 2013; ovando, 2000; sleeter et al., 2015; téllez et al., 2011; valenzuela, 2002; white et al., 2016). the majority of u.s. teachers, including bilingual teachers, received the majority, if not all, of their own k–12 schooling and teacher preparation in english with a focus on supporting students to “acquire english” rather than maintaining bilingualism. mathematics is still often seen as consisting of just numbers; rarely do bilingual teachers have professional development yeh math is more than numbers journal of urban mathematics education vol. 10, no. 2 111 opportunities centered on discussions of language and mathematics (aquinosterling et al., 2016; celedón-pattichis et al., 2010; musanti et al., 2009). sleeter and colleagues (2015) contend that the teacher-education research community has concentrated on examining teacher preparation when more attention should be placed on what actually happens when graduates of teacher preparation programs begin to teach. a close examination of novice bilingual teaching in relation to teacher preparation is critical for school leaders and teacher educators to consider in our work to better support our teacher workforce and our students. methods purpose and research questions in the 3-year, longitudinal study reported here, i explore the teaching and learning experiences of laura and elise (pseudonyms, as are all proper names throughout), two socially conscious novice bilingual teachers in a state where english-only policies were codified into law and with strict accountability measures. both teachers attended a post-baccalaureate teacher education program at a west coast public university and were purposefully selected (patton, 2014) because of their expressed commitment to bilingual education and equity-oriented teaching. after completion of the credential program, both teachers were hired to work at the same dual language program (spanish and english) in valadez elementary, an urban elementary school in which they also student taught during teacher preparation. their shared commitments and teaching context offer a unique opportunity to examine their teaching practices and how such practices are negotiated and grounded by their experiences and orientations developed over time. studies that examine beginning teachers’ experiences, especially with bilingual mathematics curricula, are scarce (ingersoll & strong, 2011; sleeter et al., 2015) and important in advancing understanding of the processes in which teacher preparation programs can increase their potential to prepare critical bilingual educators. specifically, i explore two research questions: 1. how do two novice bilingual elementary teachers organize mathematics learning for their emergent bilingual students? 2. what opportunities arise for the teachers to learn to teach mathematics in bilingual settings during teacher preparation and in the first 2 years of professional teaching? data sources in the study, i take a situated lens and view mathematics teaching, neither as static nor fixed, but rather as evolving as teachers learn to participate in and across communities (wenger, 1998). as such, this study was longitudinal in nature, and eth yeh math is more than numbers journal of urban mathematics education vol. 10, no. 2 112 nographic methods (patton, 2014) were used to collect data over a 3-year period encompassing 1 year in the teacher preparation program and the teachers’ first 2 years of professional teaching after graduation. to examine their teacher preparation, i attended and took field notes at the two-quarter (10 weeks per quarter) elementary mathematics methods course. in addition to the field notes, all teaching artifacts from the mathematics methods course (e.g., course syllabi, powerpoint slides, handouts, assignments) were gathered. additional publically available programmatic data (e.g., program design, course sequence, syllabi of courses) were collected that may have emphasized pedagogies for working with emergent bilinguals: child development, educational equity, and theories and methods in english language development. as i sought to investigate their mathematics teaching after graduation, the main source of data gathered during the teachers’ first 2 years of professional teaching were classroom observations. during the classroom visits, i observed their 50 to 75-minute mathematics lessons and followed each lesson observation with a 30 to 60-minute post-lesson interview, during which i interviewed the teachers about their mathematics lesson, their teaching, and the ways they saw themselves developing as teachers. all classroom lessons were videotaped and later transcribed and interviews were audiorecorded. classroom artifacts (e.g., lesson plans, students’ assignments, class generated work), e-mail exchanges, and informal participant interviews with teachers and other school personnel were gathered and served as secondary data sources. data analysis data analyses occurred in two phases. the first phase examined how the teachers organized mathematics learning for their emergent bilingual students, drawing on prior literature and the three dimensions of principled instruction that framed the study: types of language supports, authority of knowledge, and repertoires of language practices. the software program dedoose3 was used to review video data repeatedly for close attention to the details in classroom interactions. dedoose allows for microanalysis of video interactions and provides a time mark and key code deduction so that the data can be organized into themes. data were coded according to the aforementioned three dimensions of focus. each videotaped lesson was analyzed systematically, identifying aspects of the lesson and specific teaching practices related to types of language supports, authority of knowledge, and repertoires of language practices (strauss & corbin, 1998). data were also coded for the type of semiotic resources deployed by the teacher and student during whole classroom interactions. multimodal analysis (jewitt, 2009; kress, 2010) provides insight to the manner in which verbal language is enacted as well as the nonverbal in social inter 3 for details about dedoose see http://www.dedoose.com. yeh math is more than numbers journal of urban mathematics education vol. 10, no. 2 113 actions. this type of analysis is particularly helpful in research examining bilingual/multilingual learners as students often rely upon semiotic resources other than spoken language to participate (cummins, 2000; dasilva iddings, 2005; domínguez, 2005; turner et al., 2013). appendix a provides the practices found in the data at this stage. analysis was conducted on all videotaped classroom data. a research assistant and i reviewed the data independently and met regularly to discuss reoccurring themes, coding categories, and interpretations. when divergent opinions existed, data were reviewed multiple times until an agreement was reached (miles, huberman, & saldaña, 2014). themes and interpretations were analyzed and compared across data sources: videotaped lessons, interview transcripts, and field notes (fielding & fielding, 1986). while the videotaped mathematics lessons allowed us to examine the classroom interactional architecture, the interviews and field notes provided a window into the intentions and reasons behind the teachers’ actions. teachers’ socialization into the teaching profession begins as students in k– 12 classrooms, throughout teacher preparation, and continues as they encounter the reality of schools as they begin their professional lives (feiman-nemser, 1983; lortie, 1975; zeichner & gore, 2010). the goal of the second phase of analysis was to examine the professional opportunities that arose for the teachers during teacher preparation and in professional developing settings; however, their earlier k–12 schooling experiences are included as these experiences also shape their perceptions of teaching. the second phase builds from earlier analysis of the interview data. the interview data, field notes of observations at the school site and the mathematics methods courses, and programmatic data (e.g., teacher education program design, learning outcomes, course syllabi) were reviewed to examine learning opportunities that arose for the teachers during their k–12 setting, teacher preparation, and professional development settings related to bilingual mathematics teaching and learning. analysis consisted of identifying the teaching repertoire privileged in the data sources. the notion of repertoire is taken from bernstein (1996) and refers to the set of symbolic and material resources selected and configured to shape classroom practice. in this case, i examined the specific instructional materials, pedagogic resources and strategies, and arrangement of task sequence privileged in the professional learning settings as identified in the interviews or in the programmatic data (strauss & corbin, 1998). teacher portraits were written for each teacher highlighting their learning experiences in k–12 schools, within teacher preparation, and after graduation (miles et al., 2014). to verify and confirm interpretations of the data, i triangulated the various data sources (i.e., field notes, interview transcripts, field notes, and teaching artifacts) and teacher portraits were shared with each teacher to ensure accurate reflection of their experiences (lincoln & guba, 1986). yeh math is more than numbers journal of urban mathematics education vol. 10, no. 2 114 findings in this section, i present the findings in two parts. the first part presents findings from the study organized in three episodes to demonstrate the array of interactional encounters the students navigated and the role of the teacher in shaping the classroom interactions at three timepoints during their second year of professional teaching. while their teaching repertoires were relatively consistent, both elise and laura felt more confident in their mathematics teaching by their second year. in the second part, teacher portraits that connect their teaching practices to their opportunities to learn are shared. organization of mathematics learning both elise and laura utilized similar pedagogical strategies that leveraged students’ cultural and linguistic identities, but how these strategies were used revealed differing ideologies about learning. these differing ideologies are discussed in relation to the three dimensions of principled instruction that guided the study: types of instructional supports, authority of knowledge, and repertoires of language practices. types of instructional supports. laura and elise created a classroom environment where talking about language was as important as the mathematics lesson itself. in their lessons, laura and elise structured instructional time for language to serve as the topic of classroom discussion. in their practices, they used various strategies from research. they encouraged students’ use of their native or hybrid language practices, modeled the target language in the lesson, and connected language with mathematical representations (e.g., pictures, words, numbers, gestures; de jong & harper, 2008; dutro & moran, 2003; garcía et al., 2008; moschkovich, 2002, 2007, 2012; o’halloran, 2005). while both teachers used similar strategies, there were variations in the way these strategies were enacted. an analysis of typical classroom interactions is illustrative. in laura’s class, mathematics instruction followed a consistent routine. the class began with a math talk activity in which students explored mathematical concepts or number patterns and relationships. the class then engaged in a problemsolving activity with students working independently or in pairs. a public sharing of student solutions followed. figure 1 and the discussion to follow are excerpted from the public sharing of a lesson taught in laura’s second year. here, her first-grade students were introduced to a join change-unknown word problem (see figure 1). yeh math is more than numbers journal of urban mathematics education vol. 10, no. 2 115 nostros tenemos __ fotos en el altar. (los estudiantes ponen algunas fotos más.) ahora hay ___ fotos in el altar. ¿cuántas fotos ponen los estudiantes? [we have ____ photos on the altar. (the students put more photos on the altar.) now, there are ___ photos on the altar. how many photos did the students put on?] 3/10 13/20 83/100 figure 1. join change-unknown word problem. excerpt 14 laura [november, year 2]: (sammy walks up and stands next to laura as she writes his name below the problem.) t: sammy, explicanos esta idea. ¿como contaste? / sammy, explain the idea to us. how did you count? s: yo conté en mi mente (points to forehead) que 3 y 1, 2, 3, 4, 5, 6, 7 más y 3, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. (counts with fingers) / i counted in my mind 3 and 1, 2, 3, 4, 5, 6, 7 plus 3, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. t: ¿oh sammy, entonces empezaste con el número tres? / sammy, so you started with the number three? s: si (laura writes 3 below sammy’s name.) / yes t: ¿por qué empezaste con el número tres? / why did you start with the number three? s: porque tenemos tres (points to the 3 written in the word problem displayed) y los estudiantes pone algunos mas (points to the problem again). / because we have three and the students put some more. t: ¿oh tenemos tres fotos (finger circling the 3 in the word problem) en el altar y los estudiantes ponen algunas mas? / oh we have three photos on the altar and the students put some more? 4 the following transcription symbols are used in all transcript excerpts / english translation follows ( ) description of nonverbal communication such as gestures, gaze, movement, and so forth capital emphatic stress  indicates rising tone t = teacher; s = student; ss = students yeh math is more than numbers journal of urban mathematics education vol. 10, no. 2 116 t: dígale a su amigo lo que recién dijo sammy. / tell your friend what sammy just said. (walks around listening to student partner talks.) t: ¿por que sammy empezó con el número tres? ¿quién nos puede decir? / why do you think sammy started with the number three? who can tell us? s1: porque los otros tienen tres fotos y los estudiantes ponen algunos mas, don’t know how much (lea points to “ponen algunas fotos más” in the word problem). pero, al final es 10 fotos / because we have three photos and the students put some more, don’t know how much more. but at the end, it’s 10 photos. t: ¿oh al final es diez? okay, entonces voy a escribir diez (writes 10 on the same line as the 3). y sabemos que ahora hay diez fotos. ¿cómo llegaste de tres a diez? ¿qué hiciste? / so at the end there is ten? okay. i’m going to write ten. we know that now there are ten photos. how did you get from three to ten? what did you do? the vignette above documents the starting conversation around sammy’s strategy. figure 2 provides an image of laura’s documentation of the student’s strategy on chart paper as the class examines his problem solving together. the whole class discussion around sammy’s strategy lasted for over five minutes. multiple students explained sammy’s strategy and connected it to the problem’s context. by doing so, students were required to explain sammy’s sense-making and the underlying characteristics of a change-unknown problem type. exemplified here is laura’s integrated approach to learning language and the math content. students arrived at meanings and definitions as they engaged in analysis of each other’s reasoning and problem solving. figure 2. documentation of sammy’s strategy. yeh math is more than numbers journal of urban mathematics education vol. 10, no. 2 117 elise’s classroom interactions followed a different pattern. elise set a specific time before the math content instruction to teach language. dutro and moran (2003) term the teaching of vocabulary prior to content instruction “frontloading vocabulary.” the excerpt that follows captures the beginning of math instruction from a lesson observed during a similar time to laura’s; this lesson occurred during elise’s second year of teaching. here, elise’s second-grade students are learning to tell time: (students are standing and holding one arm up. elise is standing in front of the class with her right arm extended out.) t: vamos a aprender acerca de las horas del reloj. / we are going to learn about the hours on the clock. ss: vamos a aprender acerca de las horas del reloj. / we are going to learn about the hours of the clock. t: punto / point ss: punto / point t: quince / fifteen (left hand is extended straight out to represent the minute hand at the 15-minute mark on a clock) ss: quince / fifteen (models the teacher with left hand extended straight out to represent the minute hand at the 15-minute mark on a clock) t: treinta / thirty (left hand moves clockwise to the position of the 30minute mark on a clock) ss: treinta / thirty (follows after the teacher with left hand moves clockwise to represent the 30-minute mark on a clock) t: cuarenta-cinco / forty-five (left hand moves clockwise to the position of the 45-minute mark on a clock) ss: cuarenta-cinco / forty-five (left hand moves clockwise to the 45-minute mark on a clock) elise organized instruction into two parts. instruction began with students engaging in a vocabulary activity using total physical response—in this case, using their hands to represent the minute hand and identifying the parts of an analog clock. the clock vocabulary activity lasted for 10 minutes in which students engaged with mathematical terms across representations (e.g., gestural, visual, graphical, oral). after the song, elise and the students read from the vocabulary word bank posted on the wall with students echoing back in choral response as elise yeh math is more than numbers journal of urban mathematics education vol. 10, no. 2 118 pointed to and read the written math vocabulary word or phrase. in other activities, students created a human number line (choral response of double digit numbers), searched for geometric shapes in the classroom (identification of geometric structures), and recited the place value chart (naming the value based on its place, or position, on the chart). elise’s goal for her vocabulary activity was to familiarize students with the vocabulary for the lesson. this format of front-loading the vocabulary followed by engagement with the mathematical content occurred in the majority of the math lessons observed in her class. authority of knowledge. students in all classrooms traverse an array of interactional practices where power and status impacts students’ access to learn and to demonstrate their learning (cazden, 2001; dasilva iddings, 2005). analysis of classroom interactional patterns provides insight to issues of agency and positioning in the classroom (hand, 2012; jewitt, 2009; kress, 2010). another difference identified during analysis was the positioning of competence in the two classrooms. laura attended closely to the nature of the classroom culture and tried to center instruction on inviting students to become a part of the classroom community. in the episode shown earlier, laura positioned herself as the facilitator as well as a learner within the classroom community. as shown in the vignette above, laura often adopted a posture of uncertainty (“¿por que sammy empezó con el número tres? ¿quién nos puede decir? / why do you think sammy started with the number three? who can tell us?”) to open up space for students to take on the expert role (“dígale a su amigo lo que dijo sammy ahora. / tell your friend what sammy just said.”). in a later component of the public solution share, laura called on a third student, jesus, to explain his strategy. jesus was a quiet student, who had never spoken out during class discussions on previous visits. what follows is the whole class discussion after jesus shares his strategy with the class: t: explicale a su amigo lo que hizo jesus. / explain to your friend what jesus did. (the students talk amongst themselves, then laura claps them to attention.) t: okay. alguien puede explicar porque jesus empezo con veinte y terminó con trece? por que hizo esto? jesus, escoge a otra persona que pueda explicar lo que hiciste. / who can explain why jesus started with 20 and ended with 13? why did he do this? jesus, pick another person who can explain what you did. (jesus points to a student sitting near the back, who comes and stands next to jesus.) yeh math is more than numbers journal of urban mathematics education vol. 10, no. 2 119 s3: habían trece al principio (uses finger to circle the thirteen crossed out dots that represent a visual of jesus’s strategy shown on chart paper) y luego habían veinte y estos son los que ponen. / there were thirteen in the beginning and then there were twenty and those are the ones they put there. t: hmm, okay. / (puts finger on cheek pensively) alguien. / hmm, okay. anyone… s4: porque jesus did it backwards a sammy… so it’s twenty take away thirteen is seven. / but jesus did it backwards to sammy… so it’s twenty take away thirteen is seven. t: veinte quita trece es igual a siete y tú dijiste él lo hizo al revés a sammy. qué quieres decir? (writes 20–13=7 under the dot representation) / twenty minus thirteen is equal to seven. you said that he did the reverse to sammy. what did you mean? s5: sammy counted up. sustracción es opposite. / sammy counted up. subtraction is opposite. in the scenario above, laura demonstrated some of the instructional moves used to distribute the knowledge authority between herself and among the students. one way is her consistent questioning, displaying a sense of wondering (“alguien puede explicar porque jesus empezó con veinte y terminó con trece? por que hizo esto?” / who can explain why jesus started with 20 and ended with 13? why did he do this?”). laura regularly asked the class to explain the reasoning behind a particular strategy. this practice was a common move used by laura to open the floor to peer mediation, putting the onus of explanation and analysis on the students. students’ share-outs were not always fully articulated ideas, partial explanations were common, and students built upon each other’s explanations. during lesson reflections, laura regularly discussed the intentional selection and sequencing of specific solution strategies as well as strategies to reconfigure the classroom social order for more meaningful and respectful dialogue. in another example, i highlight a lesson that took place during a similar period to illustrate differences between the two classrooms. during march of elise’s second year, the class was on day two of working on regrouping in subtraction. the day prior, the class used concrete models to represent the process of regrouping: working with place-value blocks and place value charts to connect the procedural skill of regrouping (the decomposing of a 10 unit for 10 ones) to its conceptual basis (each place value position is related to the next by a constant multiplier of 10). each student had on their desks a set of base-ten blocks and the place-value chart. the following dialogue is from the start of the lesson during which elise yeh math is more than numbers journal of urban mathematics education vol. 10, no. 2 120 demonstrated the process of regrouping from concrete to representational (see figure 3)—connecting the written recording of each step of the procedure to the modeling with the base-ten blocks: figure 3. teacher modeling of the problem on the overhead. t: con los bloques de diez, les voy a dar un cuento sobre algo, así que tienen que escuchar calladitos. ¿por ejemplo, si digo diego tiene treintacuatro pájaros (elise places the three base ten rods in the tens place and four units in the ones place of the place value chart, figure 3), cuantas decenas voy a poner? / with the blocks of tens, i’m going to give you a story. so, you all must listen quietly. for example, if i say diego has thirtyfour birds, how many tens am i going to have? ss: tres (choral response) / three. t: ¿cuantas me faltan para ser treinta-cuatro? (elise points to the four ones units on the ones side of the place value chart.) / how many more do i need to make thirty-four? ss: cuatro (choral response) / four t: (a few students have the base ten blocks in hand.) no es necesario tocar nada en este momento. (elise looks around the room and waits for students to put down the base ten blocks.) pero su hermanito, allen, le abrió la jaula, y se le fueron veinticinco. se fueron veinticinco pájaros ¿que voy hacer? se fueron veinticinco pájaros. ¿qué vamos hacer? / you don’t need to touch anything right now. but his little brother, allen, opened the cage, and twenty-five left. twenty-five birds left. what are we going to do? yeh math is more than numbers journal of urban mathematics education vol. 10, no. 2 121 ss: vamos a quitar una decena. (choral response) / we are going to take away a ten. t: no es necesario tocar nada en este momento. (elise waits for students to put down the base ten blocks in their hands.) / you don’t need to touch anything right now. t: ¿si, vamos a quitar una decena y que se va hacer? / yes, we are going to take away a ten and what is that going to make? ss: pone unas unidades. / put ones. (elise takes away a ten rod from the ten place-value column.) t: ¿las pongo en unidades? (elise places the ten rod into the ones place value column.) / i put them with the ones? ss: no (choral response) t: ¿las voy a poner aquí? ¿qué va a pasar? (elise lifts the placed ten rod from the ones place-value column and holds it up for students to see.) / am i going to put them here? what is going to happen? s1: vas a quitar dos decenas. / you are going to remove two tens. t: ¿dos decenas? (elise holds up the ten rod in hand and shakes it.) / two tens? s1: no ss: una decena. / one ten. t: una decena. (elise holds up the ten rod in hand and shakes it.) ¿dónde se va a ir esta decena? / one ten. where is this ten going to go? ss: en las unidades. / in the ones. (elise places ten individual units in the ones column.) t: ¿y vas a cambiarlas porque? / and why are you going to change them? ss: por unidades, se fueron / for ones, they left t: ¿veinticinco se fueron entonces ya puedo quitar veinticinco? / twenty-five left and now can i take twenty-five? ss: si (choral response) / yes this vignette illustrates elise’s developmental approach to linking a procedural skill to a conceptual base. elise was building on the concrete experience of yeh math is more than numbers journal of urban mathematics education vol. 10, no. 2 122 the day prior addressing the representational stage—linking each step of regrouping to the written recording. she represented the minuend with the place value blocks and made trades (i.e., exchanging 1 for 10 in the position to the right) before subtracting. this process of “trading all” before subtracting has been shown to help prevent errors in subtraction (van de walle, karp, & bay-williams, 2010) and demonstrated elise’s attention to students’ mathematical development. elise regularly used physical and visual models to connect procedures and concepts; however, she also determined which models were used as well as the process for problem solving. at one point, a student appeared to suggest a different way to subtract (“vas a quitar dos decenas. / you remove two tens.”). it’s possible that the student may have wanted to subtract by place value—taking the two tens of 25 from 34 first. elise, however, redirected the students’ attention back to the tens rod in hand as she wanted students to regroup before subtracting. elise, unfortunately, often started lessons modeling a solution path for students. by providing a solution path, elise unintentionally reinforced a hierarchy and dichotomy between the teacher and student and knower and learner. repertoire of language practices. the last theme in the examples is related to the previous and documents the repertoire of language practices utilized. both teachers created lessons that encouraged multi-modal and hybrid language practices; however, there were variances in how the students’ language practices were leveraged during instruction. the following scenario illustrates how laura built upon students’ language practices over time. i discuss a visit made 2 months after the photo-problem lesson described in the earlier vignettes. the class regularly engaged in student-strategy share. in prior lessons, solution strategies were labeled with the students’ names (e.g., rosa’s strategy). in this visit, the lesson focus was “estrategias para juntar / strategies for adding” to link each student’s addition strategy to its mathematical term. however, the definition was not given by laura but unpacked by the students as they analyzed the strategy’s features: t: cada vez que hacemos un problema hay personas que comparten su idea, verdad? / every time we do a problem, there are other people who share the same idea as you, right? ss: si / yes t: si, cada vez que lo hacemos. pero estaba pensando que sus ideas no tienen nombre. entonces si yo le digo, as la idea de caleb, no se de qué estoy hablando. entonces hoy quería hacer una lista de las ideas que tenemos. entonces llamaré esta hoja estrategias para juntar. ¿que es una estrategia? ¿quien ha escuchado esta palabra? (laura points to the words “estrategias para juntar/strategies for adding” on the chart paper. another yeh math is more than numbers journal of urban mathematics education vol. 10, no. 2 123 sheet of chart paper covers this one. laura lowers the top paper to reveal the first strategy “modelar de uno en uno (modeling one by one).” the strategy title and the visual example are shown.) / yes, whenever we do it. but what i was thinking that your ideas do not have a name. therefore, if i tell you all to do caleb’s idea, i don’t know what i am talking about. today, i would like to do a list of the ideas that we have. therefore, i call this page strategies for adding. what is a strategy? who has heard this word? t: la primera estrategia es de ella. su estraegia el nombre de su estrategia es modelar de uno en uno. esta es una manera que muchos de ustedes están usando modelar de uno en uno. y que están observando de la estrategia de ella? que ha hecho. hmmm. (laura puts her hand to her face pensively. many students in the class follow the same pose.) / the first strategy is ella’s. her strategy’s name is modeling one by one. this is one strategy that many of you use. what do you observe about ella’s strategy? what has she done? hmmm. t: todos piensa su estrategia. ¿que están observando? ¿que están observando de la idea de ella, alex? / everyone think about her strategy. what do you observe? what do you observe about ella’s idea, alex? s1: (alex walks up to the poster and points at ella’s strategy) um, tiene quince y catorce-trece the same one as this (alex walks over to the class math wall) because quince y trece and then right there (points to strategy on a poster on the math wall where the strategy shown is also a modeling all strategy) he put quince y trece mas, but then she counts each one – one, two, three, four, five, six, seven, eight, nine, ten, eleven, twelve, thirteen, fourteen, fifteen, (touches each drawn unit and counts out loud) and then one, two, three, four, five, six, seven, eight, nine, ten, eleven, twelve, thirteen, fourteen, fifteen, and then you also could do this. / um, there’s 15, and 14, 13, the same one as this because 15 y 13 and then right there (points to strategy on another poster where the student is also modeling all) he put 15 and 13 more, but then she counts each one, two, three, four, five, six, seven, eight, nine, ten, eleven, twelve, thirteen, fourteen, fifteen, and then one, two, three, four, five, six, seven, eight, nine, ten, eleven, twelve, thirteen, fourteen, fifteen. t: oh, entonces estas diciendo, el está diciendo que en la idea de ella cuenta todas las cosas. entonces alex dijo que cuenta cada palo, ¿verdad? cuenta de uno en uno, voy a escribir eso. (she writes “cuenta todas las cosas / counts one by one” under the modeling one by one representation.) / okay, then what you are saying, he is saying that ella’s idea is to yeh math is more than numbers journal of urban mathematics education vol. 10, no. 2 124 count every thing. then, alex said to count each of the lines, right? count one by one, i am going to write this. t: algo más que observan de su idea de ella? ana? / is there anything else you observe about ella’s idea? ana? s2: una manera fácil también is to take quince first and add uno, dos, tres, cuatro, cinco, seis, siete, ocho, nueve los demás más los demás le va hacer fácil solo hacer números. / an easy way to also do this is to first take the first fifteen plus the other one, two, three, four, five, six, seven, eight, nine and then the rest of them plus the rest of them. it will be easier to just use those numbers. t: estas diciendo que sería más fácil usar números? / you are saying that it would easier to use numbers? s2: si porque te vas a hacer cansada de escribir muchas muchas líneas. / yes because you will get tired of drawing so many, many lines. t: en esa tienes que escribir muchas cosas. voy a escribir eso también, (laura writes “escribes muchas cosas / writes down a lot” on the chart paper.) addy algo más? / in this one, you must write many things. i am going to write this one too. addy, anything else? the interaction above highlights some of the ways laura drew on students’ language practices. first, she used students’ ways of thinking and reasoning about the strategies as the focus of discussion. laura’s classroom walls were filled with posters of student strategies and their representations (e.g., drawings, words, equations) gathered from lessons past. as in the vignette above, these posters were often referenced by laura and the students during instruction. the students also moved in and out of a range of language practices (spanish, english, and code switching between spanish and english). this flow of language practices was most observed when students were trying to communicate their reasoning and sense-making. it appeared laura’s students often articulated the academic language of mathematical concepts and numbers using the school-based discourse heard in class and would use their everyday, native language (spanish or english) to provide illustrations (“una manera fácil también is to take quince first and add uno, dos, tres, cuatro, cinco, seis, siete, ocho, nueve los demás más los demás le va hacer fácil solo hacer números”). notice in the vignette above, laura accepted alex’s response but rephrased a more complete response in spanish so the students could hear the formal spanish discourse (“el está diciendo que en la idea de ela cuenta todas las cosas. entonces yeh math is more than numbers journal of urban mathematics education vol. 10, no. 2 125 alex dijo que cuenta cada palo, ¿verdad? cuenta de uno en uno voy a escribir eso. / okay, what you are saying, he is saying that ella’s idea is to count. alex said to count each of the lines, right? count one by one, i am going to write this.) semiotic resources other than spoken or written language, including gestures and physical movement, were often used. in the example above, alex pointed to another student example of a modeling one-to-one strategy to explain how the strategy required students to “count one by one.” by revoicing his explanation in spanish using the mathematical terms and writing down his observation on the poster sheet, laura showed that all forms of communication—verbal or nonverbal, english or spanish—were valued. in elise’s classroom, students also moved between language practices in their student-to-student interactions and during recess; however, hybrid and multimodal practices were seen less frequently than in laura’s case during classroom discussions. i contrast here laura’s use of students’ language practices during strategy shares with a vignette of elise’s class discussion of a strategy for subtraction with regrouping. the strategy is not the traditional u.s. subtraction wherein the algorithm subtracts from right to left, one place-value column at a time, regrouping as necessary. in this strategy, the students are subtracting numbers in expanded form, the subtrahend and minuend are written in expanded form in vertical order with the same place value arranged in columns and differences found for each place value: t: (points to the number 427 written in the problem 427–182.) ¿okay, que son las escenas de 427?  / ok, what is the expanded form for 427? s1: cuatro / four t: ¿cuatro, y como vamos hacerlo? / four, how are we going to do it? s2: cuatrocientos / 400 t: cuatrocientos / 400 (writes down 400, 20, 7 on the worksheet projected on the overhead, then points to the 20 and 7.) y que es esto? veinte y siete y ciento ochenta y dos (points to the 182) / and what is this? 20 and 7. and 182. ss: cien (writes down 100 below the 400 in the expanded form of 427) / 100 ss: 80 and 2 (writes down 100, 80, 2 in columns by place value below the 400, 20, and 7) t: ochenta y dos. ¿qué es siete menos dos? / eighty and two. what is seven minus two? ss: cinco (writes 5 down and points to “20-80”) / five yeh math is more than numbers journal of urban mathematics education vol. 10, no. 2 126 s3: sesenta / sixty t: espérese ¿cómo se puede restar 20 de 80? ¿qué le vas a decir? / hold on, how can you subtract twenty from eighty? what are you going to say? s4: puedes prestar de los 400. / you can borrow from the 400. t: ¿le prestamos 400 y ese se convierte¿ (points at the 400) / we borrow from 400 and what does this one convert into? s4: este se convierte en 300. (elise crosses out the 400 and writes 300 above.) / that one turns into 300. t: ¿por qué? / why? s4: porque le presaste uno prestado uno al 20. / because you lend one to the 20. t: ¿por qué le prestas? / why do you lend it? s5: porque 80 es más que 20. / because 80 is more than 20. in the interaction above, the class used verbal and nonverbal communication (gestures and the expanded form) to make sense of the regrouping process in subtraction. what follows is a brief episode of the discussion for regrouping in the next problem: (the class already decomposed 639 and 256 into place value in expanded form together, and this is shown on the paper on the overhead.) t: ¿entonces que estamos haciendo con nueve menos seis? / so what are we doing with 9-6? s1: tres / three t: ¿jay, que vamos hacer con treinta y cincuenta? / jay, what are we doing with thirty and fifty? s6: puedes prestar de seiscientos y poner quinientos. / you borrow from six hundred and you have five hundred. t: quinientos? (elise crosses out the 600 and writes 500 above) / five hundred? s6: quinientos / five hundred t: ¿pero porque le vas a dar una centena al treinta? / but why are you giving one hundred to the thirty? yeh math is more than numbers journal of urban mathematics education vol. 10, no. 2 127 s6: porque cincuenta es mast que treinta / because 50 is more than 30. the vignettes provide a window into the classroom discourse around regrouping for two problems; however, the same discursive pattern continued for the remaining 30 minutes of the lesson. as seen above, little language hybridity occurred during the strategy shares. the class discussion was only in spanish. this use can be expected, as students in elise’s classroom were a grade above laura’s and had more opportunities to develop their spanish proficiency. however, there was also less variance in what was said. note how student responses for both problems for the regrouping process (“porque le prestaste uno al 20.” / “because you lend one to the 20.”; “puedes prestar de seiscientos.” / “you borrow from 600.”), and the reason why regrouping was needed (“porque 80 es más que 20.” / “80 is more than 20.”; “porque cincuenta es más que treinta” / because 50 is more than 30.) were very similar. in elise’s class, spanish, mathematical representations and gestures were frequently used, but she modeled the representations and gestures first. student responses often consisted of a repetition of her words and phrases (“you borrow one”; “more than”). examining teachers’ opportunities to learn it is important to connect laura’s and elise’s teaching practices to their opportunities to learn to teach. what follows are teacher portraits for each teacher’s k–12 schooling, motivation for bilingual teaching, and their professional learning opportunities during teacher preparation and after graduation. laura. laura grew up in a bilingual household with a stepmother who only spoke spanish at home, while her father only spoke in english. laura took spanish in middle and high school, and majored in spanish in college. after college graduation, laura did not plan to pursue a career in teaching but struggled for a year to find work. with both parents as teachers, laura saw teaching, particularly a career as a bilingual teacher, as a stable career option. laura attended a teacher preparation program located in a city with a growing emergent bilingual student population. the program is designed to prepare teachers for the complexities faced in their teaching placements. the credential program focuses on developing pre-service teachers’ competencies in four areas: (a) developing an inquiry stance; (b) supporting second-language learners; (c) collaborating with faculty, peers, and mentors to continually improve practice; and (d) appreciating unique resources students bring to the classroom. during teacher preparation, laura took courses on language acquisition, multicultural education, child development, and mathematics methods. course readings and assignments focused on strengths-based approaches to learning, but yeh math is more than numbers journal of urban mathematics education vol. 10, no. 2 128 the context of teaching language learners explored theories and teaching practices for english language acquisition only. theoretical underpinnings, practical applications of bilingual education, and critique of english hegemony in u.s. schools were not included in the syllabi reviewed. as a teacher candidate working toward a bilingual authorization, laura was one of four students in a cohort of over 60 preparing to teach in bilingual settings. during the last interview of each year of the study, laura was asked to reflect upon the extent to which she relied on materials learned from her university courses. laura stated that she relied very little on her teacher preparation experiences. her coursework felt unrelated to “the realities of teaching spanish and english to students” and learning to teach emergant bilingual students for her was rooted in “professional developments outside of and after the credential program.” laura student-taught in a spanish/english bilingual kindergarten classroom at valadez elementary school when the bilingual education program was in its second year. at the time of student teaching, bilingual classes were only offered in kindergarten and first grade and only a small fraction of the school staff were bilingual. the student demographics were also linguistically diverse. the school had a predominantly latin@ student population (over 85%); half of the children in school were native spanish-speakers and the other half native english-speakers. instruction at her grade level was a 90/10 model, with curriculum instructed 90% in spanish and 10% in english. she described her student-teaching experience as vital in her development as a bilingual teacher: i spoke spanish, but the academic language needed to teach english, mathematics, and science required words i never used before. i never had to speak spanish all day; my own spanish improved so much during that time. then having to consider students’ spanish and english language development while planning lessons every week… laura attributed the learning of academic vocabulary, analysis of the basic elements of spanish and english language structures, and instructional strategies for dual-language development to being situated at a field site where both spanish and english were used for instruction. while the school offered a strong bilingual literacy program, laura noticed that the mathematics instruction at valadez consisted of “just numbers and procedures” and varied greatly from the inquiry-based instruction of the mathematics methods course. during the summer after graduation, laura asked her university field supervisor, julia, for additional professional development opportunities and attended a math professional development course and made site visits to observe cognitively guided instruction.5 5 cognitively guided instruction, often abbreviated as cgi, is an approach to mathematics teaching that builds on children’s problem-solving strategies (see carpenter, fennema, franke, levi, & empson, 1999). yeh math is more than numbers journal of urban mathematics education vol. 10, no. 2 129 at the time of laura’s hire, the state had just adopted new state standards and the principal at valadez requested additional professional training for the staff. a district mathematics coach, who happened to be julia, laura’s university field supervisor, came weekly for a year to work with the valadez teachers. laura worked with julia biweekly for the entire year. by the third year of the study, laura had developed a reputation for her strength in mathematics teaching and was selected to develop mathematics curricula for the district. teachers from other schools and districts interested in bilingual mathematics instruction responsive to students’ thinking frequently observed laura’s classroom. elise. elise wanted to be a teacher since kindergarten and was in a duallanguage program herself until california’s passage of proposition 227. the passage of the proposition led to the elimination of bilingual classes and elise’s transition to english-only settings in second grade. elise noted that she entered teaching with the intent to share critical knowledge with her latin@ students. deeply committed to supporting students’ self-identity, elise was actively involved in a community-mentoring program in which she received mentoring as a teenager and subsequently mentored latina youths. elise attended the same teacher preparation program as laura but a year later. similar to laura, elise was one of six bilingual teacher candidates in a cohort of over 60 preparing to teach in bilingual settings. when asked to reflect upon her teacher preparation, elise talked about the utility of the course readings from the university math and literacy methods courses and how these resources were still referenced occasionally, but she also described her loneliness and marginalization in the program: it’s hard. all the materials i have in the classroom (now) are translations of english textbooks to spanish. the spanish is off and the stories don’t connect to my kids. it (was) the same in the program. everything was about preparing us to work with students in english. how (a)bout for those of us that are not white and teach in english and spanish? elise noted that the prospective teachers and professoriate in the teacher education program were predominantly white and monolingual (in english); only one of her course instructor was bilingual and a person of color. both elise and laura had the same instructors for multicultural education, english language acquisition, child development, and the mathematics methods courses. preparation for teaching language was limited to one course devoted to “english language acquisition” in which she learned about specifically designed academic instruction for english (sdaie) strategies. these language methodologies explicitly target vocabulary development through teachers’ active modeling and teaching of conversational and academic language through the use of visuals, demonstrations, and hands-on learning (dutro & moran, 2003; krashen, 1989). yeh math is more than numbers journal of urban mathematics education vol. 10, no. 2 130 elise’s frontloading of language in which she emphasized key vocabulary and made the learning experience multi-sensory is representative. elise student taught in a second-grade classroom at valadez, where the dual language program was in its third year. with her master teacher, claudia, elise taught the first spanish immersion second-grade classroom at valadez. instruction in second grade was an 80/20 model, in which spanish was used 80% and english 20%. mathematics was taught in spanish, and english as a second language class was offered to students in the afternoon. after graduation, elise was hired to teach in the same grade, and claudia became her grade-level colleague. claudia and elise were the only bilingual second grade teachers at the school site, and valadez was the only school in the district to offer a dual language english and spanish immersion program. elise worked closely with claudia throughout the 3 years of the study. elise discussed curricular decisions with claudia daily and stated that she “basically did the same things” as claudia. claudia regularly shared lessons and student worksheets with her. given that there were limited instructional materials available in spanish, elise found the resources extremely helpful. during the 3 years of the study, elise attended two professional development workshops on cgi offered by the district and made three site visits to observe cgi at a local school. the school also received a 1-week workshop on their new math textbook during elise’s second year of professional teaching. elise regularly attended professional development workshops centered on discussion of language and literacy. however, none of these workshops combined discussions of language with mathematics subject-matter knowledge. elise talked about the school site as a place of struggle. the dominant presence of english diminished the place of spanish in the school-wide context. spanish was rarely heard on the playground or in the lunchroom. even in the classroom, students often spoke in english during group work. elise felt a responsibility to orient students to the importance of spanish. an avid proponent of students’ linguistic development, elise intentionally structured time at the start of math instruction to work on students’ disciplinary literacies in spanish. discussion the first part of the findings examined laura and elise’s mathematics teaching, and the second examined the teachers’ opportunities to learn. study findings confirm the importance of cultivating a linguistically diverse teaching force noted in previous research (e.g., achinstein & ogawa, 2011; sleeter et al., 2015; villegas & irvine, 2010). both laura and elise valued and honored students’ linguistic, cultural, and experiential knowledge. their mathematics lessons directly attended to the role of language in mathematics learning, promoted multimodal communica yeh math is more than numbers journal of urban mathematics education vol. 10, no. 2 131 tion, and encouraged students’ use of their native language. while both teachers used similar instructional strategies, how the strategies were used revealed differing orientations toward learning: learning as acquisition and learning as participation. a widely employed conceptual lens for understanding learning is the acquisition model, which views learning as the acquisition of specific knowledge and concepts. both teachers led lessons with the goal of supporting students to “acquire” knowledge of specific concepts (e.g., part-whole relations) and skills (e.g., subtraction regrouping). within this model, new knowledge is acquired and accumulates through a developmental progression, building from one concept to the next, through teacher facilitation and scaffolding. conceiving of learning as a developmental trajectory provides a useful model for building students’ cognitive academic language proficiency (calp), or formal academic language, from basic interpersonal communication skills (bics), or day-to-day language (cummins, 1984). the acquisition model encourages a trajectory from concrete, representational, to abstract stages for mathematical concepts. research on cgi demonstrates the benefits of teachers’ understanding of learning trajectories and pedagogical considerations responsive to students’ development (see, e.g., carpenter et al., 1999). as sfard (1989) has argued, however, an overemphasis on one metaphor for learning can be dangerous. conceiving learning exclusively within the acquisition model can lead to treatment of concepts as objects of learning and separate from the context in which they are learnt. a widely endorsed teaching practice is the frontloading of vocabulary, demonstrated by elise, in which language is taught independently of its application. the idea of concepts building upon each other has also led to viewing development as linear, to notions that certain vocabulary or mathematics concepts must be taught before students’ engagement in rich mathematical talk or more challenging tasks, and to the idea that knowledge must transfer from teacher to students. this linear transfer metaphor is widespread within western education, shapes curriculum and pedagogy, and shaped elise’s mathematics instruction. this linearity can be seen in how elise followed a lesson structure in which she gradually decreased the amount of “scaffolding” provided through repetition until students were able to replicate the learnt action on their own. just as dangerous is to view learning only as participation. while the acquisition model is entrenched in what happens in individual minds, the participation model views learning as situational, indexically bound to social context (e.g., brown, collins, & duguid, 1989; lave & wenger, 1991). the participation model can embed learning so completely within a given context that the individual is lost (sfard, 1998). therefore, a balanced model of learning is needed that considers both acquisition and participation in which learning emerges through interaction between the individual and the environment (engestrom, 2001). laura’s classroom reflects this balanced approach. laura strategically designed mathematical tasks and selected yeh math is more than numbers journal of urban mathematics education vol. 10, no. 2 132 student strategies to support the acquisition of specific concepts and skills based on her knowledge of students’ development. classroom interactions supported individual and collective understandings as laura and her students embraced and built upon the repertoire of practices available. by valuing diverse voices and perspectives, laura created an egalitarian space for learning in which students saw themselves in what was being taught. students who might have been dismissed as less capable in more traditional settings (e.g., developing in school-based discourse, solving problems using direct modeling, having only partial answers) were recognized by others and saw themselves as integral to the learning community. while both models, learning as participation and learning as acquisition, are recognized in research and practice, the acquisition model is dominant. schoolbased learning often follows the assumption that students should receive and learn what is clearly communicated and explicitly taught by the teacher (freire, 1970/2000; valenzuela, 2002). a balanced model reminds us that students’ learning as well as the teachers’ learning are bound by the elements available in the ecological system. this boundness brings to light the significance of elise and laura’s own learning to teach in bilingual settings. the study findings, like those by achinstein and ogawa (2011), challenge the cultural match assumption. as achinstein and ogawa argue, “just being a person of color does not guarantee effectiveness in teaching students of color, nor do new teachers of color necessarily have knowledge of pedagogy for diverse students” (p. 15). the cultural match assumption downplays the significance of teacher preparation and of the cultural, structural, and institutional racism that shape k–12 schooling. with few exceptions, most states do not require teacher preparation programs to offer coursework specific to teaching emergent bilinguals (lópez et al., 2013), and most teacher education programs require only one “diversity” course (aquinosterling et al., 2016; sleeter et al., 2015). the teacher preparation program attended by elise and laura provided more: a program mission for high-quality teaching in culturally and linguistically diverse communities, two mathematics methods courses, and a few “diversity” courses. still, their preparation was inadequate to classroom and programmatic needs. analysis of the program revealed that their teacher preparation was not designed to build upon the strengths and needs of bilingual teachers. the twosemester mathematics methods course focused on developing high-leverage instructional practices to support students’ development of central mathematical concepts; however, there was little emphasis on preparing teachers to support students’ development of language skills to engage meaningfully in disciplinary discourse. as well, there was an underlying assumption that these teaching practices were the “right” teaching methods and strategies for all students and all teachers. this onesize-fits-all model ignores the western, assimilationist perspective that dominates yeh math is more than numbers journal of urban mathematics education vol. 10, no. 2 133 u.s. schools (gutiérrez, 2002; khisty & willey, 2008; ovando, 2000; sleeter, 2001) and fails to leverage the different assets and needs bilingual prospective teachers bring to teacher preparation (arce, 2004; gomez et al., 2008; musanti et al., 2009; sleeter et al., 2015). language is an important medium through which equity and inequities are structured and sustained in k–12 classrooms and in teacher education (aguinosterling et al., 2016; gomez et al., 2008; khisty & willey, 2008; sleeter et al., 2015). the study findings, as well as others (e.g., moschkovich, 2012; remillard & cahnmann, 2005; setati, 2008; white et al., 2016), create an imperative for the development of linguistically and culturally relevant mathematics teacher education. this study is premised on the idea that bilingual teachers are “learners and innovators with cultural and linguistic repositories ... to challenge and inspire students toward academic achievement” (gist, flores, & claeys, 2015, p. 29). however, the study findings serve as a reminder that teacher educators (and researchers) must be careful not to romanticize bilingual teachers nor presume that bilingual fluency or ethnic identity alone are sufficient to challenge the subtractive underpinnings that permeate schools serving bilingual communities (cummins, 2000; 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(2010). teacher socialization. retrieved from https://pdfs.semanticscholar.org/113f/3c1f190e94e97aa9c181ef7f2e12224540ac.pdf yeh math is more than numbers journal of urban mathematics education vol. 10, no. 2 138 appendix a codes – organized mathematics learning for emergent bilingual students i. language support a. teacher direct modeling (e.g., repeated use of word, emphasizing pronunciation, sophisticated use of language, simplifying the language) b. revoicing c. student created products (e.g. word bank, definition, strategy) d. gestures i. student ii. teacher e. student analysis (e.g. compare words, strategies, word problem context) f. encouragement of l1 usage g. visual aids (e.g., word banks, student strategies, concept maps) h. games i. use of real-life context j. modify existing math textbook k. focus on meaning making, not just students’ production of “correct” spanish or english l. ask students to analyze (e.g. compare words, concepts or meanings) ii. knowledge authority a. teacher presents solution method b. teacher provides specific tool for problem solving c. teacher has final word about correct answer/solution d. multiple forms of student mathematical contributions are encouraged and valued i. incomplete ideas ii. incorrect answers iii. multiple approaches to problem solving iv. multiple representations e. teacher encourages students to have final word about correct answer/solution f. teacher intentional selection of student contribution to minimize status among students (and specific subgroups) g. position students to use one another as mathematical resource iii. language practices a. spoken words i. spanish ii. english yeh math is more than numbers journal of urban mathematics education vol. 10, no. 2 139 iii. hybridity b. written communication (e.g. writing on white boards, student written strategy, poster of student strategy) c. symbolic/equations d. physical i. math tools ii. manipulatives iii. realia iv. objects e. gestures i. teacher ii. student f. contextual i. teacher-generated ii. student-generated microsoft word 414-article text no abstract-2244-1-6-20210116 (proof 1).docx journal of urban mathematics education may 2021, vol. 14, no. 1 (special issue), pp. 24–44 ©jume. https://journals.tdl.org/jume jahneille a. cunningham is a postdoctoral scholar in the school of education and information studies at the university of california, los angeles, moore hall, 457 portola plaza, los angeles, ca, 90095; email: jah.cunningham5@gmail.com. her research interests include informal mathematics learning, critical approaches to parent engagement, mathematics identity, and teaching mathematics for social justice. “we made math!”: black parents as a guide for supporting black children’s mathematical identities jahneille a. cunningham university of california, los angeles black parents are often presumed to be uninvolved in their children’s education, especially in mathematics. these stereotypes are arguably sustained by white, middle-class expectations for parent engagement. this qualitative study challenges the dominant narrative by exploring the ways eight black parents support their elementary-aged children’s mathematical identities. although many scholars have examined the relationship between mathematics identity and academic outcomes, few have explored the role parents play in this identity development. drawing on martin’s (2000) mathematics identity framework and mccarthy foubert’s (2019) racial realist parent engagement framework, the author argues that black parents’ experiential knowledge of race and racism in mathematical spaces positions them to teach their children about the everyday importance and usefulness of mathematics. using parent interviews and family observations, the author’s findings suggest the parents supported their children’s mathematics identities using four approaches: 1) pragmatic (emphasizing financial literacy and basic life skills), 2) aspirational (promoting math-intensive careers), 3) affirmational (sharing words of encouragement), and 4) race-conscious (applying mathematical concepts to lessons in black history, culture, and anti-blackness). implications for educators are discussed, as parent identity support strategies may be useful for reform-oriented teachers seeking to foster positive mathematical identities in black children. keywords: black parents, elementary mathematics, informal mathematics learning, mathematics identity, parent engagement cunningham black parents as a guide journal of urban mathematics education vol. 14, no. 1 (special issue) 25 my dad, he was really good at math. he didn’t even graduate from high school, but he’s a musician, so i know that helped in his math somehow… he always had pride in his math. so, when he would teach me stuff, those were my best experiences. – lola, parent interview he excerpt above was taken from an interview with lola—a nontraditional college student and mother of three—who was reflecting on her childhood experiences learning mathematics. she proudly shared how her most memorable experiences in mathematics were not tied to her formal schooling but instead experienced with her father. as a black man who did not complete high school, many would deem him an unsuitable mathematics teacher. however, lola saw his career as a musician as providing sufficient qualifications. specifically, she acknowledged the relationship between his identities as a musician and as a “math person.” in listening to her story, i could not help but recall my childhood and how my father instilled a love for mathematics that brought me to this work. when i was in elementary school, my father would challenge me to determine change due at the cash register in the grocery store or to calculate a 15% tip at restaurants—a routine that bolstered my confidence in my mathematical abilities. both lola’s story and my own signify the essential role black parents, regardless of career or education level, play in shaping their children’s mathematical identities. background over the past two decades, critical scholars have pushed to center identity in mathematics education research. traditional notions of mathematics as a bias-free field have been challenged with evidence of racialized experiences, where race and racism define black students’ learning experiences (martin, 2006). these racialized experiences include structural impediments, from being denied access to upper-level mathematics courses (bryant, 2015) to low teacher expectations (copur-gencturk et al., 2020). additionally, jackson et al. (2020) found that poor school climate, including a lack of teaching materials and resources, partially mediated the relationship between black male high school students’ mathematics identities and mathematics outcomes. these findings suggest structural racism in schools impacts how black students see themselves as doers of mathematics as well as their ability to perform in mathematical settings. critical scholars have also challenged mainstream ideologies in mathematics education (civil, 2018; martin et al., 2010), including what counts as mathematics, who is good at mathematics, and when mathematics is useful. with an emphasis on student identities in the classroom, many researchers have argued that students’ mathematical identities do not develop in isolation but rather are co-constructed in the context of other salient identities, such as race, gender, and social class (aguirre, mayfield-ingram, & martin, 2013; nasir & de royston, 2013). for example, t cunningham black parents as a guide journal of urban mathematics education vol. 14, no. 1 (special issue) 26 english-clarke et al. (2012) interviewed 28 black youths and found that one-third of the participants shared racial-mathematical stories or messages they received from peers or adults. these stories included messages about racism and discrimination in mathematical settings, persisting in mathematics courses with few black students, and beliefs about asians being good at mathematics. this exemplifies how messages students receive about mathematics, and other social constructs such as race, carry implications for how they perceive their mathematical abilities relative to others. despite the growing body of literature on mathematics identity, many black students continue to be taught mathematics with little attention paid to their identity formation. mathematics classrooms in predominantly black schools are often teacher-centered (waxman et al., 2010) and test-driven (davis & martin, 2018), as teachers rely on rote memorization techniques to help students grasp mathematical concepts (ellis & berry, 2005). however, research suggests learning is optimized when students are active co-constructors of mathematical knowledge (boaler, 1998; boggan et al., 2010; franke & kazemi, 2001). as such, when black children learn mathematics in classrooms void of context and cultural relevance, their mathematical identities are stifled. consequently, students who learn mathematics in rigid, decontextualized environments lose the opportunity to draw mathematical connections to their everyday lives, including understanding how to use mathematics to critique and challenge social inequalities. with growing interest in identity in mathematics, many scholars have urged teachers to develop curriculum and pedagogy around the history and culture of black children (tate 1995; taylor, 2012) and other similarly marginalized students of color. the literature on funds of knowledge (gonzález et al., 2001), teaching mathematics for social justice (bartell, 2013; gutstein, 2012), and culturally relevant pedagogy (leonard et al., 2009; tate, 1995), for example, has underscored the importance of teaching mathematics with attention to social justice, student culture, and community-based/informal mathematical knowledge. arguably, these approaches can lead to improved mathematical outcomes because students are fully engaged when teachers value who they are outside of the classroom. yet some teachers report challenges with implementing these pedagogical approaches, as tensions arise when teachers attempt to balance social justice and/or cultural goals with mathematical goals (bartell, 2013; civil, 2018). additionally, teachers must be careful to incorporate student culture in curriculum without relying on essentialism or stereotypes that present black culture as monolithic (leonard et al., 2010). perhaps including parents in the decision-making process may help teachers learn about their students’ specific cultures and local communities, thus avoiding the pitfalls of essentialism. however, few scholars have explored parents as potential resources to inform such curriculum and practices (civil & bernier, 2006). in this paper, i argue that black parents possess a wealth of experiential knowledge—including knowledge of what it means to be a black student in a cunningham black parents as a guide journal of urban mathematics education vol. 14, no. 1 (special issue) 27 mathematics classroom as well as what it means to be black in america broadly— that makes them uniquely positioned to foster their children’s mathematical identities. i also argue that teachers can, and should, use black parents as a guide to model meaningful mathematics curriculum and pedagogical practices that support black children’s mathematical identities. decolonizing (black) parent engagement centering the knowledge and experience of black parents is important in critical scholarship given that they are often presumed to be uninvolved in their children’s education (cooper, 2009; latunde & clark-louque, 2016; powell & coles, 2021), especially in mathematics (martin, 2006). critical scholars have argued these stereotypes are sustained by white, middle-class values that dominate parent engagement expectations (howard, 2020; mcgee & spencer, 2015). this deficit narrative is supported by “traditional” expectations of parent engagement, which focus on schoolcentered activities such as providing homework assistance, volunteering at schools, and attending parent-teacher conferences. knowing that these activities often present substantial barriers to participation for black parents, particularly those from lowincome backgrounds who have rigid work schedules (cooper, 2009), educators and researchers have used interventions such as parent workshops to help “increase” parent engagement for black parents and other similarly marginalized parents. although these efforts may be well-intended, there remains an underlying assumption that black children’s academic success is dependent on their parents’ ability to engage with schools in these prescribed ways. however, some scholars have argued that these expectations limit the scope of what counts as parent engagement, reflecting white and middle-class values and expectations of parental involvement while black parents’ educational contributions go unnoticed (cooper, 2009; howard et al., 2019; jackson & remillard, 2005; mcgee & spencer, 2015). for example, jeynes (2010) found that black parents often use encouraging behaviors and affirmations to support their children’s academic selfconfidence and success. schnee and bose (2010) also found that while some black parents provide direct homework assistance, others chose not to help their children in order to instill the values of perseverance and independence. these examples underscore the ways black parents’ educational practices are often overlooked or misinterpreted as disengagement, highlighting the need for more asset-based research on black parent engagement. furthermore, many black parents’ educational contributions go unnoticed when we conflate schooling with learning. jackson and remillard (2005) distinguish between parental involvement in learning and parental involvement in schooling, as the authors found that the low-income black mothers in their study provided various learning opportunities for their children, including both planned and spontaneous informal mathematics activities. furthermore, eloff, maree, and miller (2006) found cunningham black parents as a guide journal of urban mathematics education vol. 14, no. 1 (special issue) 28 that black parents in south africa employed various strategies to facilitate their children’s mathematical learning in informal settings, including engaging in dialogue with their children about mathematics as it pertained to everyday activities. these examples suggest that black parents' mathematical support may be overlooked by schools when they are not directly tied to school-related activities. another component of black parent engagement that is often disregarded is racial socialization. in navigating their children’s education, black parents are also tasked with racial socialization, as they help their children develop an understanding of what it means to be black in society and in schools specifically (martin, 2006; white-johnson et al., 2010). with this comes the challenge of instilling dignity and pride in black children while also bringing their awareness to racial injustices. in interviewing black parents, martin (2006) found that many experienced racial discrimination while learning mathematics and consequently found it important to provide their children with role models in mathematics to overcome negative beliefs about the subject. given that mathematics classrooms remain highly racialized contexts, black parents’ lessons on race and racism in school provide their children with essential knowledge, yet these lessons are rarely captured by traditional notions of parent engagement. conceptual framework mathematics identity. here, i draw on martin’s (2000) conceptual framework for mathematics identity, which the author describes as “the dispositions and deeply held beliefs that individuals develop, within their overall self-concept, about their ability to participate and perform effectively in mathematical contexts and to use mathematics to change the conditions of their lives” (p. 206). in other words, mathematics identity consists of both beliefs about mathematical ability as well as understanding its usefulness in the real world. developed from an ethnography of black students specifically, this framework captures the ways racialized experiences in mathematics classrooms (e.g., low teacher expectations, stereotypes, etc.) can shape beliefs about one’s own mathematical abilities and the application of mathematics. given this, having a strong, positive mathematics identity is particularly important for black children, as it affords them access to power associated with mathematical competence and analytical skills to critique and change social injustices, such as racism, that impact their daily lives. racial realist parent engagement. this study also draws on racial realist parent engagement (rrpe; mccarthy foubert, 2019) as a theoretical framework that reveals the silenced narratives around black parents’ contributions to their children’s education broadly but can be applied to mathematics education specifically. the rrpe theoretical framework provides a critical lens to examine how black parents’ relationship with schools might impact the ways they support their children’s mathematical identities. drawing on derrick bell’s (1992) notion of racial realism, cunningham black parents as a guide journal of urban mathematics education vol. 14, no. 1 (special issue) 29 which contends that race and racism are a permanent part of the structure of american society, rrpe acknowledges the anti-black racism inherent in school policy and expectations of parent engagement. for too long, black children’s mathematics education has been hindered by discriminatory practices such as low teacher expectations (copur-gencturk et al. 2020) and tracking (mccardle, 2020). black parents, in turn, have been blamed for the outcomes (e.g., low test scores) resulting from these antiblack policies and practices. despite this, black parents “have been committed to and are engaged in their children’s education by surviving and resisting anti-black racism in schools” (mccarthy foubert, 2019, p. 15). in acknowledging black parents’ contributions to their children’s education, rrpe a) exculpates black parents who have been unjustly blamed for the so-called “achievement gap” and b) challenges anti-black school policies (e.g., tracking, discipline) and expectations for parent engagement that have impaired black parents’ relationships with schools. here, i argue that black parents are not uninvolved in their children’s mathematics education. rather, the ways they choose to navigate their children’s education broadly—and in mathematics particularly—are informed by their beliefs in the permanence of anti-black racism in schools. drawing on these frameworks, this study is guided by several assumptions about black parents and their role in their children’s mathematical identity development: a) black parents’ contributions to their children’s education are often disregarded, b) black parents’ experiential knowledge of anti-black racism in school and society inform the lessons they teach their children, c) black parents possess mathematical expertise in real-world settings, and d) black parents should be used as a model to support student identities in critical mathematics education. methods the study presented here is part of a larger study that examined black children’s out-of-school mathematical learning. in this qualitative, narrative inquiry (bhattacharya, 2017), i asked, how do black parents support their children’s mathematical identities? using a snowball-sampling method (noy, 2008), i recruited eight black families with children who attended public elementary schools in southern california (for details, see figure 1). although i initially sought socioeconomic diversity, the snowball sampling method resulted in a sample of college-educated parents (mostly mothers). i collected data through semi-structured parent interviews, family observations (including field notes) in the home, and artifacts and photographs gathered during observations. mothers were my main point of contact and were often the only caregiver present during data collection. i observed parents and their children during everyday activities (e.g., cooking, cleaning, homework, and leisure time). during interviews, i asked about their daily family routines, educational aspirations cunningham black parents as a guide journal of urban mathematics education vol. 14, no. 1 (special issue) 30 for their children, and the ways they supported their children mathematically. following up on our interviews, several of the parents offered evidence of their day-today activities by sharing artifacts such as mathematics workbooks, games, and cooking recipes, which i photographed for triangulation purposes. interviews were later transcribed and coded, along with field notes, in two coding cycles. i first employed an open coding technique (emerson et al., 2011), recording all emerging themes, and later reduced the data in a second, focused coding cycle. in second-cycle coding, i followed a codebook guided by my theoretical frameworks. table 1 study participants* participating parent (occupation) participating child(ren) (grade level) other household members data collected total home visits ashley (doctoral student) ada (3 rd grade) father (veteran) brother (14yo) parent interview family observation artifacts (photos) 3 ayo (hip-hop artist, community activist, doctoral student, teacher) ajani (2nd grade) brother (12yo) parent interview family observation artifacts (photos) 1 dana (doctoral candidate) devon (4 th grade) none parent interview family observation 1 eshe (nurse) ezra (phd candidate) elijah (kindergarten) sisters (1 and 3yo) parent interview family observation 1 jamel (research consultant, writer) jordan (2 nd grade) none parent interview family observation artifacts (photo) 1 kia (educator) keon (4 th grade) father (retired police officer) brother (6yo) parent interview family observation artifacts (photos) 1 lola (undergraduate student) leila (4th grade) liana (4th grade) stepfather (musician) sister (12yo) parent interview family observation artifacts (photos) 3 mona (chiropractor) mya (entering 1 st grade) father (assistant director) brother and sister (5 and 2yo) parent interview artifacts (photos) family observation 3 note. pseudonyms are used to protect identity of participants. cunningham black parents as a guide journal of urban mathematics education vol. 14, no. 1 (special issue) 31 findings collectively, the black parents in this study provided evidence of their mathematical identity support aligning with martin’s (2000) mathematics identity framework. however, there were some differences in their approaches to supporting their children’s mathematical identities: while some parents were more focused on building confidence, others were more concerned with showing their children the practical use of mathematics or promoting mathematics-intensive careers for their children. in the following section, i describe four themes in the approaches parents took to help foster positive mathematical identities in their children: a) an affirmational approach, b) a pragmatic approach, c) an aspirational approach, and d) a race-conscious approach. these themes are not mutually exclusive, as several parents used multiple approaches to support their children’s mathematical identities. the affirmational approach: “we got this, we can do this!” the affirmational approach—the most common form of mathematics identity support—could be seen in six of the parents (ayo, ashley, dana, jamel, lola, and mona) in this study. during interviews or informal conversations, these parents described the ways they supported their children’s mathematical identities, with a focus on building their child’s confidence in their mathematical ability. they verbally supported their child through words of praise when they were successful and words of encouragement when they were struggling in mathematics. jamel, the mother of 8year-old jordan, shared that she and his father “do a lot of praising of him, and just encouraging him and letting him know that he’s smart and he has within him what he needs to excel.” three parents (ayo, jamel, and lola) described specific routines they performed to support their children’s overall self-esteem, which they hoped would also translate to positive beliefs in their mathematical abilities. they all shared that they use “affirmations,” or chants that promote positive self-talk, as part of their daily routine with their children. the extant literature has well documented the benefits of parental encouragement as seen in the affirmational approach (jeynes, 2010), especially in improving persistence and interest among young girls in science and mathematics (alliman-brisset et al., 2004; gunderson et al., 2012; howard et al., 2019). in this study, i similarly found that parents of girls (mona, lola, and ashley) were especially concerned with their daughters’ confidence in mathematics. however, as the examples above show, parents of black boys have also relied on encouraging behaviors, as affirmational support has been documented as a common form of academic engagement in black parents (jeynes, 2010). cunningham black parents as a guide journal of urban mathematics education vol. 14, no. 1 (special issue) 32 the pragmatic approach: “when it comes to money, when it comes to money!” when using a pragmatic approach, parents would emphasize the instrumental importance of mathematics by reinforcing its everyday application to their children. specifically, four parents (ashley, ayo, dana, and lola) were concerned with their children’s financial literacy and used conversations and activities about spending and saving money to show the usefulness of mathematics outside of the classroom. ashley was particularly concerned with sharing the importance of everyday mathematics with her daughter, ada, who has a neurodiverse older brother. ashley: i worry about things, like will they be financially stable… i guess the everyday math-type things… the use of money, time, can you financially plan… we are hyper aware of that because… we have a 14-yearold [ada’s brother] who can’t tell the difference between a dime and a nickel, because he doesn’t have those discrimination skills. so those are part of the reason [ada] will have to care for him her whole life. ashley emphasized the importance of mathematical competence in everyday life to ada because she knew ada would eventually be her brother’s caregiver. thus, the consequence of her lacking these skills is greater because it would not only affect herself but also her brother. lola similarly shared the risk of being mathematically incompetent with her daughters by explaining to her daughter the importance of financial literacy. lola: leila mentioned before she wanna be in the wnba, so she even thought school’s not that important. i’m like ‘wait a minute!’ you don’t wanna have to pay someone to read contracts… or letting someone mismanage your money. all these factors play in no matter what you decide to do… you wanna be the main person saying ‘yes’ or ‘no,’ not having to pay someone cause you’re not good at math.” i noticed that when parents employed a pragmatic approach to sharing the instrumental importance of mathematics, they seemed to take a defensive stance. in both of these examples, there is a theme of protection, whether it be protecting assets (lola’s quote) or protecting family members (ashley’s quote). in other words, they shared the importance of mathematical knowledge by detailing the consequences of lacking said knowledge. teaching their children about financial literacy is important to black parents, who are well aware of racial wealth disparities (darity et al., 2018). as dana stated in the parent interview, she wants her son, deon, to be financially literate because, as black people, “…that’s a big thing in our community. we often don’t know how money works, and it is reflected in the generational wealth, oftentimes, we don’t have.” cunningham black parents as a guide journal of urban mathematics education vol. 14, no. 1 (special issue) 33 the aspirational approach: “i want him to be an engineer!” five parents (mona, jamel, kia, and eshe/ezra) expressed specific mathematical aspirations for their children pertaining to their future careers. like the pragmatic approach, parents were interested in helping their children see the utility of mathematics; however, here the focus was on future careers rather than the everyday use of mathematics. for example, jamel shared the types of extracurricular activities she would like her son jordan to become involved with. jahneille: where, if at all, do you see learning math fitting into the vision you have for [jordan] as an adult? jamel: i see it playing a really big part actually… i want him to get involved in—his school has a robotics program, they also have an engineering program—i want him to get involved in some extracurricular activities around math that will help him… his dad would like him to learn coding. in this example, jamel showed more interest in jordan learning mathematics to prepare for potential careers (e.g., robotics, engineering, or coding) rather than everyday mathematics or basic life skills (e.g., money management or grocery shopping). other parents expressed a similar desire for their children to pursue careers in science, technology, engineering, and mathematics (stem). eshe, an ethiopian immigrant, explained that mathematics and science expertise are highly valued in her culture. because of this, she bought toys for her fiveyear-old son, elijah, to foster an interest in these subject areas. during our interview, she explained that “[she] want[s] him to be [an] engineer, so [she bought] him legos, magna-tiles, blocks, anything for building.” mona, a mother of three, also expressed that it was important for her eldest daughter mya to understand mathematics well because “that’s gonna set her up well for [a] future in math and science, which will hopefully set her up well if she wants to go into engineering or [information technology].” as seen in these examples, parents who use the aspirational approach share the instrumental importance of mathematics through a career-focused perspective. given the dearth of black professionals in the stem fields (flynn, 2016; mcgee, 2016), these parents may be emphasizing these high-earning-potential careers to their children at an early age to encourage their matriculation into stem professions. as the literature on black stem college majors and professionals has suggested, parental encouragement and motivation in early mathematics play an important role in children’s successful stem trajectories (flowers, 2015; mcgee & spencer, 2015). cunningham black parents as a guide journal of urban mathematics education vol. 14, no. 1 (special issue) 34 the race-conscious approach: “we made math!” four parents (ayo, ashley, dana, and lola) used a race-conscious approach to support their children’s mathematical identities. this method was marked by its focus on collective black identity in the context of mathematics learning. in other words, parents would incorporate their knowledge of contemporary or historic black issues to support their children’s mathematical identities. this approach was often used in combination with one of the previously described methods. below, i describe how ayo, an educator and activist, used a race-conscious approach alongside an affirmational approach when her son was discouraged with mathematics. ayo: i went to a private school, all-black, pro-black, private school… so in my mind, we made math… so we gon’ do math at a whole ‘nother level… i cannot help but bring that to the table as a parent… that’s always in my tone when i’m talking to them about whatever the affirmation is. nah we got this, we can do this… break it down, slow it down… but all that foolishness and that nonsense, that don’t have no place up in this house. here, ayo references the collective black identity to encourage her children, reminding them that “we made math.” she mentions that this specific approach was motivated by her own experience learning mathematics at an afrocentric school where black history and identity were incorporated across the curriculum. she recalled having a classmate who would “do [math] problems in hieroglyphics” and that all of the students took advanced mathematics courses at an early age. ayo: if you challenge children, especially black children, to do certain things at an early age, you will be surprised how they could actually… adhere to that. so doing calculus in the 6th grade would sound crazy to somebody else, but at my school, they believed that they were all brilliant and capable of doing that. ayo comes from an educational background that simultaneously bolstered her mathematical and racial identity, a strategy she now employs as a parent. here, ayo shows confidence not only in her own ability or even her son’s but in all black children. it is this level of confidence that she hopes to instill in her children, which is why she is adamant that “all that foolishness and that nonsense [self-doubt] have no place…in [her] house.” her affirming behaviors serve as a form of resistance against the anti-blackness her children have experienced in schools that have set low expectations for black children, a sentiment that was echoed in other parent interviews and that has been well documented in the literature (cooper, 2009; delpit, 2012). ayo’s race-conscious approach to mathematics support is well aligned with the philosophies of the post-civil rights, afrocentric movement that motivated the start of afrocentric schools and interventions such as the algebra project (moses et al., 1989). cunningham black parents as a guide journal of urban mathematics education vol. 14, no. 1 (special issue) 35 lola, a non-traditional college student and african american studies major, similarly supported her daughters’ mathematical identities by centering their black identity; however, in doing so she highlights the instrumental importance of mathematics. in the excerpt below, lola combined a race-conscious approach with a pragmatic approach by using the board game monopoly to explain the harmful neighborhood effects associated with gentrification and rent increases. lola: so i like to tell them, like, our society is basically like a monopoly… property is everything… i show them the different neighborhoods and why they cost different prices… let’s say if this was like our neighborhood we grew up in… and somebody comes in and puts up these hotels, the rent increases and you’ll collect more. so when you’re deciding if you wanna win, you gotta buy up everything and raise the rent. but if you wanna be a good human being, you gotta… see what you wanna do.” here, lola showed her daughters the utility of mathematics by applying their knowledge of number sense to a real-world situation. however, this was no ordinary game of monopoly, as she utilized the game to explain to her daughters how gentrification relies on anti-black racism to dismantle communities like her childhood neighborhood. a native of the predominantly black neighborhood of south central los angeles, lola used her experiential knowledge of gentrification to raise a moral dilemma in the game: a) buy hotels, raise the rent, and win the game by forcing other players into bankruptcy or b) forgo buying hotels and be “a good human being” but risk losing the game. lola’s “mini-lesson” on gentrification is well aligned with ladson-billings’ (1995) recommendations for culturally relevant pedagogy, which included teaching practices that help students develop a “critical consciousness through which they challenge the current status quo of the social order” (p. 160). in both examples, the parents’ race-conscious approach appears to be motivated by their background—ayo’s experience of attending an afrocentric school and lola’s predominantly black childhood neighborhood. combined with their presentday experiences (ayo is an educator and activist and lola is an undergraduate student majoring in african american studies and sociology), their collective black identities and commitment to fighting against anti-black racism permeate their parenting concerning their children’s mathematics identities, similar to the parents in cooper’s (2009) study, whose parental involvement included political activism. however, other parents in this study were similarly race-conscious but did not necessarily incorporate this perspective in the ways they supported their children mathematically. jamel, for example, is a community organizer who works with black grassroots movements to support food-insecure and prison populations. yet, a clear connection between her community organizing and parenting around mathematics did not emerge from the data in the way that it did with ayo and lola. as such, what cunningham black parents as a guide journal of urban mathematics education vol. 14, no. 1 (special issue) 36 leads some parents to incorporate race-consciousness in supporting their children’s mathematical identities warrants further exploration. discussion despite the scholarly significance of these findings, this study has its limitations. first, the parents in this study were all college educated—ranging from one participant completing a bachelor’s degree to another with a doctoral degree. although i initially sought greater socioeconomic diversity, the snowball sampling method, which allowed participants to be active agents in the research process, resulted in participants that were very similar in education level. another limitation of this study was the limited participation of fathers. although both mothers and fathers were invited to participate in the study, many fathers were unavailable due to work schedules. perhaps fathers would have exhibited alternative forms of mathematical support. additionally, the amount of time spent during family observations varied across participants. although i only visited some families once, others i visited on multiple occasions. this decision was made out of respect for participants’ time because they were not compensated for their participation in this study. however, considering that i spent varying hours with participants, this gave me a deeper knowledge of some families compared to others. mathematical identity in black children the mathematics education literature is dominated by achievement gap research aimed at fixing black children, and other similarly marginalized children, with a focus on test scores and classroom performance. however, in the past two decades, scholars like gutiérrez (2008) and martin (2000), among others, have pushed the field to think past academic performance and to consider examining issues such as identity and power. here, i have highlighted the contributions black parents have made to their children’s mathematics education by supporting their identities as doers of mathematics. scholars who examine mathematical identities in black children specifically argue that a critical component of mathematical identity is the ability to use mathematical knowledge to critique the world around them and to transform their daily lives (davis & martin, 2018). the parents in this study, irrespective of their beliefs about their mathematical abilities, were concerned with supporting their children’s mathematical identities. they affirmed their children’s mathematical abilities in formal and informal ways and reminded them of the usefulness of mathematics—from becoming financially literate to gaining access to high-paying, math-intensive careers. some parents even employed a race-conscious approach in supporting their children mathematically, as they simultaneously tended to their children’s racial and mathematical identities by cunningham black parents as a guide journal of urban mathematics education vol. 14, no. 1 (special issue) 37 challenging anti-black racism in the context of mathematics learning. these findings support the growing body of literature aimed at challenging deficit views of black parent engagement in general (fenton et al., 2017; howard, 2020; howard & reynolds, 2008; jackson & remillard, 2005; mccarthy foubert, 2019) and provide an alternative perspective to view black children’s mathematics education, shifting the discussion away from the so-called achievement gap and toward mathematical identity, a crucial yet underexamined component of mathematics education. bridging home and school learning: implications for educators and parents the findings from this study suggest that educators can better support black students in mathematics by drawing connections to their everyday lives. mcgee and spencer (2015) asked what educators can learn from black parents while examining parental influence on college-aged stem high achievers. i raise the same question here for elementary educators who seek to bridge the gap between home and school mathematical learning for their black students. many of the parents in this study were “highly-involved” by traditional standards—they attended parent conferences, communicated with teachers, and were highly visible in their children’s schools. however, other parents in this study may have easily been mistaken for being uninvolved due to their alternative support strategies. their forms of support and involvement often go unnoticed, and in this study i have presented an alternative perspective on parents who employ subtle forms of academic support, as have other scholars (jeynes, 2010). most, if not all, of the parents in this study were concerned with their children’s safety and well-being in a highly racialized society. although teaching their children mathematics was not always their primary goal as parents, they provided their children with opportunities to simultaneously learn about race and mathematics. black parents’ awareness of racism and discrimination has been well documented in the literature on parental racial socialization. however, the dominant discourse around black parent engagement has focused on how to increase their involvement concerning school-based goals with the assumption that they lack the resources or interest to support their children’s education (fenton et al., 2017; howard & reynolds, 2008; jackson & remillard, 2005). educators must understand that for black parents what it means to be successful in school, and in mathematics in particular, is not simply a matter of their children’s academic and future career prospects. many black parents are preoccupied with their children’s safety; beyond the typical parental worries of their child’s well-being, black parents must also teach their children about racism and discrimination. as such, educators must consider the context of blackness when communicating with black parents about their children’s school success. the key to bridging the home-school connection may be for teachers to align their classroom goals with parental goals rather than expecting parents to passively comply with school expectations for parent engagement. cunningham black parents as a guide journal of urban mathematics education vol. 14, no. 1 (special issue) 38 considering that black parents, and other parents of color, already have the added burden of racism and discrimination, educators can build rapport with black parents by showing that they too are interested in their community-based knowledge, including issues that affect the black community in particular. in addition to helping students master mathematical concepts, educators can teach mathematics for social justice (leonard et al., 2010) and prepare lessons that explicitly address systemic social inequality that has disenfranchised black people. educators can also use as a guide the parents in the present study who employed race-conscious approaches to support their children’s mathematical identities in order to contextualize their students’ mathematical knowledge in ways that are meaningful to the black community. several parents in this study simultaneously supported their children’s mathematical and racial identities, using the game of monopoly to teach about gentrification or affirming their children’s mathematical abilities by reminding them about the legacy of black mathematicians. educators can show black children that mathematical knowledge is a tool that can be used to disrupt the racial hierarchy. just as many parents in this study reinforced their children’s mathematical knowledge and identities in the context of black history and contemporary social issues, teachers of black children should not be afraid to similarly affirm their students’ culture in the context of mathematics learning. as other scholars have noted, teachers can, and should, use social justice approaches (moses et al. 1989; wager & stinson, 2012) to teach mathematics. of course, incorporating a cultural and/or social justice lens into mathematics is a difficult task, as, generally, educators refrain from addressing issues of race because they can be difficult to implement effectively. scholars have acknowledged the tensions that may arise when social justice and cultural approaches are introduced to mathematics curriculum, as they might overshadow content knowledge (civil, 2018). aguirre, turner, et al. (2013) also demonstrated how superficial attempts to connect to children’s cultures can encourage stereotyping and promote harmful narratives about children and their communities. to avoid these scenarios, many educators rely on seemingly “race-neutral” teaching examples, such as pizza slices to teach about fractions, but these contexts do not represent authentic connections to students’ lives outside of school. if, as scholars have argued, mathematics learning is already a racialized experience, to ignore students’ racial identities in the context of mathematics learning is only reinforcing systemic inequality, as failure to address race in mathematics education by maintaining a color-blind curriculum only further alienates black children in mathematics classrooms. additionally, teachers might be concerned that discussions about race/racism may be inappropriate due to the age of the children. however, as seen in this study, parents are already having conversations about race with their children; perhaps black parents may serve as a resource for teachers in this regard. regardless of the racial composition of a particular school, social justice approaches to mathematics cunningham black parents as a guide journal of urban mathematics education vol. 14, no. 1 (special issue) 39 education can be employed successfully and can benefit all students, as the effects of racism and discrimination impact all members of society. as davis and martin (2018) have noted, teachers of black students can and should help students use mathematics as a tool to evaluate and critique historical and current events to increase understanding of their position in society, individually and collectively. in fact, elementary school classrooms may be an ideal setting for a race-conscious curriculum, as the multi-subject classroom settings create natural fluidity between content areas; teachers can connect mathematics to areas such as history and english/language arts, content areas where cultural connections may be drawn more easily. parents and teachers alike should also help protect children’s mathematical identities from harmful influences and be mindful of the ways mathematical identity is shaped by not only views of the self but also by the beliefs of others (nasir, 2002). in other words, school-based measures of mathematics success, such as grades, test scores, and teacher expectations, do not have to be the only factors that shape children’s mathematical identities. when children face challenges in school mathematics, parents and teachers can remind them of their mathematics successes outside of school by showing their children the many other ways they use mathematics, even when mathematical knowledge is not being assessed. for example, parents can talk to their children about measurements while cooking a family recipe together, reminding their child that a delicious meal that results from balancing flavors is evidence that the cook is good at mathematics. teachers can also emphasize mathematics as they pertain to other subjects; for example, teachers might highlight the rhythmic patterns in music to discuss ratios and proportions. such messages, although subtle, can help bolster children’s mathematical identities by increasing their confidence and understanding of the real-world importance of mathematics. this point is of particular importance now, as we are witnessing the bridging of home and school learning amid the novel coronavirus (covid-19) global pandemic (although data collection and analysis for the current study were completed prior to school closures in the united states). in a matter of weeks, parents were suddenly forced to homeschool their children in compliance with social distancing mandates. in a way, the pandemic has forced the bridge, as “home” and “school” have merged, even temporarily. this pandemic could potentially shift the structure of schools and further blur the boundaries between teachers and parents. i hope the findings presented here will encourage parents to trust that their life experiences have prepared them to teach their children mathematics and apply them to real-world settings. even if we operate under the optimistic assumption that schools will return to operation as usual post-pandemic, i hope that the time parents have spent in the dual role of parent/teacher will empower them in future interactions with teachers and school staff as they advocate for their children and support them as doers of mathematics. i also hope that teachers will begin to appreciate the contributions black cunningham black parents as a guide journal of urban mathematics education vol. 14, no. 1 (special issue) 40 parents make to their children’s education, even if they are not the contributions that are expected. conclusion reflecting on my identities as a black parent and mathematics education researcher, i am mindful of how the culmination of this study coincides with a moment in history where a global pandemic has forced many children out of physical classrooms. in this time, i have had the opportunity to spend more time with my sons, recalling my own childhood experiences in search of ways to create an educational but exciting environment to distract them from the chaos our global health crisis has ensued. however, i am also mindful of the surmounting pressure black parents are facing, juggling work and family all while supporting children with distance learning. in some ways, the call to acknowledge black parents’ contributions seems ironic, as distance learning has forced us to engage with schools in unprecedented ways. however, this does not mean that black children, or their parents, are any more protected from the anti-black racism they are subjected to in face-to-face classrooms. the biases and stereotypes about black children and their families are likely resurfacing in virtual classrooms. whether black parents’ contributions continue to go unnoticed is dependent on education researchers’ and teachers’ willingness to expand the scope of parent engagement. in this study, i have demonstrated the various ways black parents support their children mathematically. to date, few researchers have examined black parents’ contributions to their children’s mathematical learning, particularly to their identity as doers of mathematics. as we continue to push for black children to be seen in mathematics classrooms, i hope the findings from this study will also encourage education researchers and teachers to keep parents involved in mathematics reform efforts. we must remember that black parents are also teachers and will continue to teach their children well after they step foot out of the classroom. thus, if we make efforts to strengthen the relationship between schools and black families, we will have a better understanding of the pedagogical practices that are both mathematically rigorous and applicable to the everyday lives of black children. cunningham black parents as a guide journal of urban mathematics education vol. 14, no. 1 (special issue) 41 references aguirre, j., mayfield-ingram, k., & martin, d. 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(2010). parental racial socialization profiles: association with demographic factors, racial discrimination, childhood socialization, and racial identity. cultural diversity and ethnic minority psychology, 16(2), 237–257. copyright: © 2021 cunningham. this is an open access article distributed under the terms of a creative commons attribution-noncommercial-sharealike 4.0 international license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. journal of urban mathematics education december 2017, vol. 10, no. 2, pp. 25–38 ©jume. http://education.gsu.edu/jume paulo tan is an assistant professor in the department of education at the university of tulsa, 800 s. tucker ave, tulsa, ok, 74014; email: paulo-tan@utulsa.edu. his research interests include equity, teacher education, and students with dis/abilities in mathematics education. signe kastberg is a professor in the department of curriculum and instruction at purdue university, 100 north university st., west lafayette, in 47907; email: skastberg@purdue.edu. her research interests include relational practice, multiplicative thinking, and development of prospective elementary teachers’ models of children. commentary calling for research collaborations and the use of dis/ability studies in mathematics education paulo tan university of tulsa signe kastberg purdue university espite discussions of “mathematics for all,” opportunities that support the development of mathematical reasoning and understanding of mathematics as a human endeavor often do not exist for mathematics learners identified in schools as having dis/abilities.1 indeed, mathematics for all is consistently used to motivate the allocation of resources and attention to mathematics education in legislation, policy documents, and organizations’ vision and position statements. mathematics education researchers have served as advocates for marginalized students pointing out limitations in the mathematics for all rhetoric (martin, 2003). yet, students with dis/abilities are often left out of discussions regarding mathematics for all and equity research that has worked to contextualize and operationalize “achieving equity” and the process of “eliminating inequity” (tate, as cited in martin, 2003, p. 14). mathematics education researchers and organizations representing them use the term “equity” to refer to access and opportunities for all students. for example, the national council of teachers of mathematics’ (nctm) recent principles to actions: ensuring mathematical success for all called for systemic improvement in mathematics education for all (nctm, 2014). “access and equity” is identified as one of six guiding principles for school mathematics (p. 5). yet, the nctm’s (2014) position statement on access and equity leaves dis/ability out of the subgroups to which these goals apply. access and equity are identified as applying to racial, ethnic, linguistic, gender, and socioeconomic groups. furthermore, access and equity in standards-based mathematics education remains elusive for those stu 1 we use the term dis/ability to forefront power imbalances inherent in constructing and identifying dis/ability and the consequences of such imbalances in and out of school. the concept of dis/ability as socially constructed offers an entryway to reconstructing what mathematics education researchers mean when they use the term “disability” and to addressing inequities for individuals labeled with this educational and societal construct. the word “disability” is used when directly referencing works by other authors as they applied the term unless their work similarly use the term “dis/ability” (e.g., de freitas, 2015). d http://education.gsu.edu/jume mailto:paulo-tan@utulsa.edu mailto:skastberg@purdue.edu tan & kastberg commentary journal of urban mathematics education vol. 10, no. 2 26 dents with dis/abilities (browder, spooner, ahlgrim-delzell, harris, & wakeman, 2008; tan, 2014). in public schools, 13 disability categories are sanctioned for special education services. generally, only students in a few dis/ability categories, such as learning disability or speech/language impairments, spend a majority of their school day in the general education classroom. students identified as having “moderate” to “severe” disabilities (e.g., intellectual disabilities, autism, multiple disabilities) spend most of their school day in segregated special education classrooms or schools (u.s. department of education, 2013). jackson and neel (2006) compared mathematics instruction practiced in and outside general education classrooms. based on their classroom observations, the researchers coded time that teachers in four schools spent on conceptual instruction, algorithm instruction, and instruction or tasks unrelated to mathematics. jackson and neel operationalized algorithm instruction as “focused on imparting factual knowledge and mathematical procedures” (p. 597). a major finding was that, on average, approximately 30% of the time in general education classes was devoted to algorithm instruction compared to 75% of the time students in special education classes were subjected to this form of instruction. an additional concern is that accessibility for students with dis/abilities in general education mathematics classrooms remains a challenge. insufficient classroom supports lead to learning obstacles (baxter, woodward, & olson, 2001; bottge, heinrichs, mehta, & hung, 2002) and privileging narrow forms of mathematics expressions and sensory capacities (de freitas, 2015). as a result, access to the general education mathematics classroom and curriculum depends on the degree to which students with dis/abilities resemble conventional ways of operating such as communicating mathematical thinking via speech and writing. hence students with dis/abilities, even in general education classrooms, lack opportunities to build upon their existing ways of thinking mathematically. research represents a crucial component in the process of recognizing and addressing these inequities. ongoing commitment to equity and equity research in the mathematics education community has resulted in an increase in the number of equity focused research manuscripts submitted to journals (stinson, 2013). in the last 10 years (2006–2016), the journal for research in mathematics education (jrme) has included four reports of empirical studies focused on exploring teaching, learning, and curriculum of students with disabilities. the focus of the studies were impressions (lynch & star, 2014) and reasoning (lewis, 2014; xin, 2008) of students with learning disabilities. similarly, lewis and fisher’s (2016) review spanning 40 years of mathematics education research exclusively focused on students with mathematics learning disability.2 these articles begin to point the way 2 lewis and fisher (2016) operationalize mathematics learning disability (mld) in line with a dyscalculia diagnosis, which they described as a “biologically based difference in the brain” (p. 338). for the purposes of this commentary, we situate mld within the broader category of learn tan & kastberg commentary journal of urban mathematics education vol. 10, no. 2 27 toward research that illuminates mathematics reasoning and learning of students with disabilities. however, there is more to do to address dis/ability beyond those identified with learning disabilities to include other dis/abilities related to communication, emotion, behavior, attention, body, and health. indeed, outside of learning disabilities, no empirical study published in jrme has involved individuals with dis/abilities as doers and thinkers of mathematics. given the narrow focus of and few empirical studies on dis/abilities in mathematics education research, karp (2013) has termed this group the invisible 10%. we also find that dis/ability is underrepresented in mathematics education in the larger research community. although lubienski and bowen (2000) identified disability as a subgroup within the discussion of equity and mathematics for all, scholars have noted limited numbers of available educational research studies focused on mathematics education involving students with disabilities (karp, 2013; lambert & tan, 2016). lambert and tan (2016) analyzed 408 peer-reviewed journal articles with a mathematics education focus. they found that of the 42 articles explicitly including students with disabilities, two were published in mathematics education journals. the remaining 40 articles were published in special education or psychology journals whose fields are heavily influenced by behaviorist-based learning theories such as direct instruction (woodward, 2004). to be sure, lambert and tan’s (2016) research found that learners with disabilities and those without are conceptualized differently in mathematics educational research. whereas approximately 40% of studies subjected students with disabilities to learning traditions grounded in fixing or remediating deficits (e.g., medical or behaviorist models), only about 6% did so for students without disabilities. when it came to constructivists and social-constructivist traditions, 7% of the reviewed studies involved students with disabilities compared to 40% involving students without disabilities. similarly, lewis and fisher’s (2016) review of mathematics education research on students with mathematics learning disability found that most of the 164 studies “involved topics aligned with the kindergarten through third-grade standards and focused almost exclusively on basic arithmetic calculation” (p. 357). hence, the limited knowledge base in mathematics education research focused on students with disabilities points to alarming inequities. we attribute the construction of dis/ability as central to these inequities. broken minds as the origin of marginalization the construction of and response to dis/ability located within an individual is strongly ingrained and operated upon in society at large and consequently in ing disabilities distinguishing it from “moderate” to “severe” disability categorization such as intellectual disabilities, autism, and multiple disabilities. we return to the dyscalculia discussion later in the commentary. tan & kastberg commentary journal of urban mathematics education vol. 10, no. 2 28 schools. the field of special education emerged as a “rational” way for schools to organize and maintain order by viewing disability as a pathological condition (skrtic, 1991). skrtic argued that society and public education are grounded in theories of organizational rationality and human pathology. accordingly, school failure can be attributed to inefficient organizations and defective students. this rationality/pathology resulted in the emergence of special education “as a means to remove and contain the most recalcitrant students in the interest of maintaining order in the rationalized school plant” (p. 152). meanwhile, special education philosophy and research have historical roots in psychology and medicine guided by positivist orientations. these orientations encourage interventions for students with disabilities emphasizing performance (baroody, 1999; paul, french, & cranston-gingras, 2001). within the medical model, the concept of disability reflects organic deficiencies, “broken” bodies or minds as “something to fix, cure, accommodate, or perhaps endure” (andrews et al., 2000, p. 259). recently, for example, the study of developmental dyscalculia, a mathematics learning “disorder” included in the diagnostic and statistical manual of mental disorders (dsm-5; american psychiatric association, 2013), has sought to identify biomarkers of dis/ability within individuals by attempting to locate the cause of mathematical “disorders” in the brain (de freitas & sinclair, 2016). to identify these biomarkers, tests are used to determine the ability to closely estimate and compare quantities without counting. the outcomes of these tests are normed constituting “intact” or “deviant” forms of “number sense.” although mathematics education scholars have critiqued these tests (e.g., de freitas & sinclair, 2016) for over-emphasizing narrow dimensions of number sense, the results are taken at face value reinforcing the location of disability within an individual. indeed, scholars (e.g., paul, french, & cranston-gingras, 2001) have noted that despite advancements in social science perspectives, special education researchers have maintained a strong commitment to a positivist epistemology. as previously discussed, special education researchers conduct the majority of studies in mathematics education involving students with disabilities. hence the view of students with dis/abilities as mathematics doers and thinkers has remained narrow. at the same time, we should be careful not to cast all special education researchers as grounded in these historical traditions; there are important exceptions (e.g., cawley & parmar, 1992; lambert, 2015). taken together, the construction of and response to dis/ability becomes problematic for mathematics education researchers. similar to skrtic’s (1991) assertion that school organization delegates responsibility of the education of students with dis/abilities to special educators, we deduce that research practices fall along categorized lines of special education researchers and mathematics education researchers. the absence of dis/ability in mathematics education equity debates perhaps reflects how research involving students with dis/abilities falls within the purview of tan & kastberg commentary journal of urban mathematics education vol. 10, no. 2 29 special education researchers. we invite mathematics education researchers to challenge such research fragmentation while questioning positivist methods emphasizing narrow forms of mathematics performance (browder et al., 2008; tan, 2016). attending to these contradictions represents a constructive path toward building upon what we know about students as mathematics thinkers and doers. yet, martin (2003) asserts that the process of eliminating inequity will move beyond “rendering students eligible for opportunities that we assume and hope will exist for them” (p. 14) toward empowerment to use mathematics to “alter the power relations and structural barriers that continually work against their progress in life” (p. 15). measureable costs of inequity outcomes for individuals with dis/abilities in and out of school illustrate martin’s (2003) point. students with disabilities continue to be suppressed in segregated and self-contained environments where “life skills” curriculum fails to prepare them for real life (frattura & topinka, 2006; tomlinson, 2012). students with disabilities have the lowest high school graduation rates among all students, almost 20% lower than the national average (u.s. department of education, 2015). those who graduate from high school are less likely to enroll in and complete postsecondary educational programs than their counterpart students without disabilities (newman et al., 2011). moreover, restricted work opportunities after graduation result in large employment disparities between those with and without disabilities; 21% of working-age individuals with disabilities are employed either fullor part-time compared to 59% of those without disabilities (harris interactive, 2010). limited employment opportunities result in 28% of people with disabilities aged 18–64 living in poverty, as compared with 12.5% in the general population (denavas-walt, proctor, & smith, 2012). in addition, individuals with disabilities are less likely to report that they are very satisfied with life than those without disabilities—34% versus 61%, respectively (harris interactive, 2010). outside school, individuals with disabilities have taken leadership roles in self-advocating for civil rights despite having negative school experiences such as being bullied, labeled, and marginalized (caldwell, 2011). keith jones is one such advocate. mr. jones reflected on the low expectations that he experienced in and out of school as well as the work of gaining access to meaningful learning in mathematics: because of my being…i have to be secluded, stashed away…talked about in a way that there’s no expectation of me doing anything. and then you want me to succeed? you want me to be a product of this society…produce…put into this economy? but from the time i’m born to the time i die i’m being told i ain’t shit! ...teachers didn’t really push…it was more of ‘okay, here’s some manila papers and crayons, color’. tan & kastberg commentary journal of urban mathematics education vol. 10, no. 2 30 …i don’t care about pasting popsicle sticks, i want math…can i get some math? something! (habib, 2009) mr. jones’s reflection addresses martin’s (2003) assertion that achieving equity in mathematics is not bound by time in school. indeed, mathematics education researchers are positioned to examine and address inequities outside of school and beyond; consequently, historical, social, and economical analyses are all crucial in this work (king thorius & tan, 2015). unfortunately, mr. jones’ development as a self-advocate is an uncommon outcome for most individuals with dis/abilities. including individuals with dis/abilities in the discussions of mathematics for all is a starting place, but it is crucial to move to actions addressing the potential dreams and desires of such students. mathematics education researchers are central to this move. specifically, mathematics education researchers can utilize diverse research tools and frameworks for understanding mathematical thinking, knowing, and ways of being. mathematics education researchers also have diverse ways of talking and thinking about mathematics. collectively, such powerful research tools and knowledge of mathematics education are needed to counter blatant forms of dis/ability-based discrimination (i.e., ableism) in and out of schools. for example, ableism may manifest in discussions surrounding the “appropriateness” of engaging students with dis/abilities in constructivist and reform-oriented pedagogies. woodward and montague (2002) point out that special education scholars have resisted such pedagogies for students with disabilities. this resistance has resulted in the endorsement of teacher-directed practices for students with disabilities focused on explicit forms of instruction (national mathematics advisory panel, 2008) constituting practices that qualify as evidence-based (gersten et al., 2009). these practices are systematic procedures targeting measurable responses while providing reinforcement and error correction feedback (spooner, knight, browder, & smith, 2011). translating these systematic procedures to mathematics education practices, saunders, bethune, spooner, and browder (2013) provided the following description: the teacher identifies a skill to teach the student (e.g., how to identify obtuse, acute, or right angles) and finds an appropriate prompt to help the student get the right answer. this prompt is anything the student needs to get the right answer, and stays the same throughout the time it takes the student to learn the skill. it could be a verbal model (teacher presents an obtuse triangle and says “obtuse”), a gesture to the correct answer, or even a physical prompt by moving the student’s hand to the correct answer. first, the teacher presents the problem (e.g., “what kind of triangle is this?”) and immediately uses the prompt to help the student get the right answer. after doing this for a number of trials (sessions), the teacher fades that prompt by simply delaying it: the teacher presents the problem (e.g., “what kind of triangle is this?”) and waits 4-6 seconds before delivering the prompt. this gives the student time to answer independently, but also provides support. (p. 29) tan & kastberg commentary journal of urban mathematics education vol. 10, no. 2 31 how should mathematics education researchers respond knowing that students are being treated this way? we invite mathematics education researchers to rally together using their research expertise and voice to speak against blatant forms of ableism such as illustrated above. it is worth reiterating that these procedures are representative and widely supported by special education mathematics researchers hailing systematic instruction as one of the most significant advances in the field (spooner & browder, 2015). to support individuals with dis/abilities to achieve their goals as mathematicians and humans, they must be seen as both. opportunities to awaken the mathematicians within students with dis/abilities go beyond having access to “evidencebased” direct and systematic instruction. instead, a mix of pedagogies affording these mathematicians with opportunities to explore and create communicates the intent of mathematics for all and of all (tan, 2017). in doing so, mathematics education researchers work to address inequities and rise to the challenge of paulo freire’s critical question: “how many critical intelligences, how much curiosity, how many enquirers, how many capacities that were abstract in order to become concrete, have we lost?” (freire, d’ambrosio, & mendonça, 1997, p. 8). evidence of ability within students with dis/abilities mathematics education researchers have begun to support individuals with dis/abilities to achieve their goals as mathematicians and humans by building a body of knowledge identifying mathematics ability. some scholars maintain that the mathematics learning of students with dis/abilities, including those constructed as having a more “severe” disability, is similar to their non-dis/abled peers (baroody, 1999; tan, 2014; tan & alant, 2016; van den heuvel-panhuizen, 1996). these scholars posit that students with disabilities can benefit from reform-based practices (baroody, bajwa, & eiland, 2009). elaborating on such an assertion, we illustrate ways students with dis/abilities engage in, make meaning from, and express mathematics. van den heuvel-panhuizen’s (1996) study involving 61 fifthand sixth-grade students diagnosed with an intellectual disability found that participants utilized particular and, in some cases, sophisticated ratio problem-solving strategies (e.g., multiplicative reasoning, drawing concrete and abstract models) despite not being formally taught about ratios. similar findings have pointed to sophisticated problem-solving strategies in other mathematics domains such as subtraction (peltenburg, van den heuvel-panhuizen, & robitzsch, 2011) and combinatorics (peltenburg, van den heuvel-panhuizen, & robitzsch, 2013). in fact, baroody’s (1999) research synthesis determined that students with disabilities, including those with severe disabilities, do engage in mathematical practices such as inductive and deductive reasoning, and adapting/devising mathematical strategies. baroody con tan & kastberg commentary journal of urban mathematics education vol. 10, no. 2 32 cluded that because these students have the cognitive building blocks necessary to develop meaningful mathematics learning they can benefit from purposeful, meaningful, and inquiry-based approaches to mathematics learning. conviction about the learning potential of students with disabilities coupled with concerns about curricula lacking academic rigor were the impetus for development of a conceptually based mathematics curricula for these students (göransson, hellblom-thibblin, & axdorph, 2016). the authors examined mathematics lessons that teachers constructed in six classes involving students with disabilities. they found that teachers effectively created learning environments where students inquired, held mathematics conversations, shared their insights, and became interdependent. these studies confirm a core belief of mathematics education researchers–– that all humans are mathematics thinkers and doers. yet, narrow notions of what constitutes mathematics thinking and doing constrains this position. as such, these studies support the notion that dis/ability lies not within individuals but rather resides in limited opportunities and in rigid mathematics educational practices (e.g., narrow forms of assessment). dis/ability studies in mathematics education we join scholars who have challenged mathematics education researchers to build a process for exploring inequities central to our field (e.g., gutiérrez, 2013; karp, 2013; martin, 2003), inviting researchers to include students with dis/abilities in these efforts. our intention is for mathematics education researchers to partner with special education and dis/ability studies in education researchers to bring more diverse theoretical perspectives to research involving students with dis/abilities. in addition, we suggest the consideration of dis/ability studies in mathematics education (dsme) as a complimentary theoretical framework for research and advocacy. dsme synthesize elements of dis/ability studies in education (dse) and embodied mathematics. dis/ability studies in education dse is an emerging field that examines dis/ability as a social construction resulting in social exclusion and oppression (gabel, 2005). dse departs from the field of special education and groundings in positivist traditions (valle & connor, 2011; ware, 2005) depicting dis/abilities as deficits located within individuals. rather, dse aims to “fix” traditional practices leading to a disabling view of individuals. thus, in drawing from elements of dse, dsme focuses on reimagining mathematics education practices that enable and empower every student in ways that approach characteristics of students with dis/abilities (and all students) as repre tan & kastberg commentary journal of urban mathematics education vol. 10, no. 2 33 sentative of human diversity. rather than drawing lines of exclusion, dsme innovates for inclusive practices, expanding our capacity to support all learners in robust mathematics learning communities. another crucial element of dse is privileging individual voice and lived experiences. self-advocates, such as keith jones, provide critical insights to guide mathematics education researchers in their efforts. such insights foster understanding of dis/ability experiences, which offers considerations for directions of inquiry (baglieri, valle, connor, & gallagher, 2011). moreover, gutiérrez (2013) urged mathematics education researches to draw upon lived experiences of marginalized individuals and/or to apply and (re)write “theories and frameworks that give voice to others” (p. 57). as such, researchers can recognize “the embodied/aesthetic experiences of people whose lives/selves are made meaningful as disabled, as well as troubles the school and societal discourses that position such experiences as ‘othered’ to an assumed normate” (american educational research association, 2016). embodied mathematics central to embodied mathematics is that making sense of and expressing mathematics involves the body in ways that are not fully understood. we draw heavily on the works of elizabeth de freitas and nathalie sinclair (see, e.g., 2014, 2016) to situate embodied mathematics and reframe dis/ability in mathematics education research. we find problematic that “specific ways in which mathematics is represented, communicated, and explained tacitly privilege certain sensory capacities” (de freitas, 2015, p. 198). such privilege fosters restricted views of what constitutes engaging in mathematics. thus, de freitas directs us to explore the possibilities of “radically different sensing bodies” (p. 189). indeed, researchers have long known that by exploring other sensory capacities, students with disabilities often excel in creative productions (e.g., carter, richmond, & bundschuh, 1973). in particular, students with dis/abilities may engage in mathematics through various modes of interactions such as swaying, rhythmic movement, gesturing, tapping, feeling, facial expressions, or gaze (sinclair & heyd-metzuyanim, 2014). unfortunately, such embodied modes are not privileged in the school environment, leading to a diagnosis of learning dis/ability and/or the dismissal of some ways of operating as non-mathematical. thus, principles of embodied mathematics center on “the way that bodies are provisionally and temporarily enabled, directing our attention to the temporal contingency of dis/ability” (de freitas, 2015, p. 189) to counter notions of definitive potentialities and capacities. embodied mathematics then recognizes the potential of the human body and where “bodies can be seen as differently abled and differently organized rather than disabled or distracted” (de freitas & sinclair, 2014, p. 145). in citing eide and eide (2011), de freitas (2015) noted: tan & kastberg commentary journal of urban mathematics education vol. 10, no. 2 34 the curricular emphasis on alphanumeric aspects of mathematics, for instance, works against students with exceptional spatial skills. people diagnosed with dyslexia may struggle with procedural learning and rote memory tasks, but their memory of phenomenological details—details pertaining to physical aspects of an experience, such as tactile, motor, or spatial arrangements—exceeds that of non-dyslexics. (p. 199) other embodied arrangements such as music also provide access to mathematics engagement (edelson & johnson, 2003) through rhythmic movement. these movements possess spatial properties, sequencing, and patterning essential to mathematical concepts (geist, geist, & kuznik, 2012). students’ engagement in creating and moving to music provides opportunities for the development of insights about the structures of space and time in their creative activities. that is, embodied mathematics requires us to “simultaneously rethink the body in and of mathematics” (de freitas & sinclair, 2013, p. 454). importantly, like dse, embodied mathematics takes seriously the social-political dimensions and its “entanglements” in education (de freitas & sinclair, 2013). thus, for mathematics education researchers, the implication of embodied mathematics, dsme, and other sophisticated epistemologies is to better understand “the minute sensations that contribute to our students’ learning and invention of mathematical concepts” (de freitas, 2015, p. 192) situated within constructs of dis/abilities to address inequities in and out of schools. our role as mathematics educators is partnering with families, students, educators, and community members (nctm, 2008) to support, create, and advocate by addressing inequities through responsive mathematics education research. to facilitate this work, dis/abilities, as both a collective group and individual experiences, must be explicitly included in mathematics equity research and advocacy alongside other marginalized groups. importantly, this inclusion must cover the full-range of dis/abilities (e.g., autism, intellectual dis/abilities, emotional and behavioral “disorders”) not just mathematics learning disabilities. in doing so, we take on the responsibility of mathematics education research involving students with dis/abilities. we do this because we have long claimed that our work is about all students. we do this because we know the value of diversity and different perspectives in truly inclusive mathematics teaching and learning. we do this because we know that we must view every student beyond socially constructed labels and perceived limits. we do this because we are committed to honoring and understanding multiple ways of knowing, expressing, and engaging in mathematics. references american educational research association. 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(2004). mathematics education in the united states past to present. journal of learning disabilities, 37(1), 16–31. woodward, j., & montague, m. (2002). meeting the challenge of mathematics reform for students with ld. the journal of special education, 36(2), 89–101. xin, y. (2008). the effect of schema-based instruction in solving mathematics word problems: an emphasis on prealgebraic conceptualizations of multiplicative relations. journal for research in mathematics education, 39(5), 526–551. http://dspace.library.uu.nl/handle/1874/1705 journal of urban mathematics education july 2009, vol. 2, no. 1, pp. 22–51 ©jume. http://education.gsu.edu/jume lidia gonzalez is an assistant professor of mathematics at york college of the city university of new york (cuny), 94-20 guy r. brewer blvd, jamaica, ny, nyc 11451; e-mail: lgonzalez@york.cuny.edu. her primary research interest is in the teaching of mathematics for social justice. additionally, she is working on a study of alternatively certified mathematics teachers in nyc and interested, more broadly, in efforts at improving the educational experiences of urban youth. teaching mathematics for social justice: reflections on a community of practice for urban high school mathematics teachers lidia gonzalez york college of the city university of new york in this article, the author reports on a study that explored, in part, the developing identities of seven new york city public high school mathematics teachers as teachers of mathematics and agents of change. meeting regularly as a community of practice, the teachers and author/researcher discussed issues of teaching mathematics for social justice; explored activities and lessons around social justice; and created a unit of study that attempted to meet high school level mathematics standards, while addressing a social justice issue affecting the lives of urban students. the author reports on the mathematics teachers’ growing awareness of and concerns about infusing issues of social justice into their teaching as well as the teachers’ evolving conceptions of what it might mean to teach mathematics in an urban school, of the nature of mathematics itself, and of what their roles as educators might include. keywords: mathematics education, teacher development, teacher identity, teaching mathematics for social justice approaching mathematics through a social justice context has been proposed and used by some, including mathematics educators, as a way to address issues that confront urban youth from historically marginalized communities, while engaging them in the study of meaningful mathematics (see, e.g., frankenstein, 1983; gutstein, 2006, 2008). although the idea of education as a vehicle for social justice has been around for decades (see, e.g., freire, 1970/1993), it is only fairly recently that the idea has been applied to mathematics education: until recently, embedding mathematics pedagogy within social and political contexts was not a serious consideration in mathematics education. the act of counting was viewed as a neutral exercise, unconnected to politics or society. yet when do we ever count just for the sake of counting? only in school do we count without a social purpose of some kind. outside of school, mathematics is used to advance or block a particular agenda. (tate, 2005, p. 37) gonzalez mathematics for social justice journal of urban mathematics education vol. 2, no. 1 23 within recent years, there does appear to be a growing interest with respect to teaching mathematics for social justice, as evident by the recent published edited volumes that focus on mathematics and social justice (see burton, 2003; gutstein & peterson, 2005). how mathematics teachers might be prepared to teach mathematics for social justice, however, is an area still in need of exploration. gau (2005) argued: despite the potential teaching math for social justice has in addressing issues of equity in mathematics education, little research exists that examines mathematics teachers learning to teach for social justice, a necessary step in beginning to understand the entailments of teaching mathematics for social justice. (p. 3) it also has been argued that most of the existing examples of social justice units appear to rely on elementary mathematics rather than upper-level mathematics (brantlinger, 2007; brantlinger, gutstein, buenrostro, & turner, 2007), though this reliance is somewhat changing as more lessons and materials become available (see, e.g., gutstein & peterson, 2005; mukhopadhyay, powell, & frankenstein, 2009). thus, with these arguments in mind, i undertook a study with the explicit goals of illustrating how mathematics teachers might learn to teach mathematics for social justice and how teaching mathematics for social justice might be done within the context of the high school mathematics curriculum (see gonzalez, 2008). in my study, i reported on the formation of a community of practice consisting of seven new york city (nyc) public high school mathematics teachers and me, the researcher and a former nyc public high school teacher. as a community of practice, the teachers and i shared and developed ideas on the intersection between mathematics, mathematics education, and issues of social justice. together, we explored and generated knowledge around the idea of mathematics teachers as agents of social change and on the use of mathematics as a critical tool for understanding and working to improve social life, primarily those aspects most affecting the students served by the school at which the teachers of the study had taught. additionally, we developed a high school level curriculum unit around a social justice issue that the teachers saw as relevant in the lives of their urban students, while also attempting to successfully attend to the standards-based content of high school mathematics without compromising the nature of the mathematics learned. in this article, i report only a portion of my study, with an emphasis on teachers‘ developing identities as mathematics teachers and agents of change. in so doing, i focus on the teachers‘ beliefs about the teaching and learning of mathematics and their own roles as teachers of mathematics. because identity and awareness mediate both action and pedagogy (holland, lachiotte, skinner, & cain, 2003), the article focuses on the teachers developing identities, exploring gonzalez mathematics for social justice journal of urban mathematics education vol. 2, no. 1 24 shifts in their thinking and beliefs. in reporting partial findings, i draw heavily from the interviews and written reflections of the teachers who participated in the community of practice. through using the teachers‘ own words, it is my hope that their beliefs about their roles as both mathematics teachers and agents of change might become transparent. conceptual framework many education scholars, among them critical pedagogues, argue that mathematics (particularly algebra) stands as a gatekeeper to future success (apple, 1992; burton, 2003; gutstein, 2006; martin, 2000, 2003; moses & cobb, 2001). this gatekeeping status is especially evident for low-income students of color that, for the purposes of this article, will be referred to as students from marginalized communities. the inequities that exist between students of marginalized communities (such as those taught by the teachers in my study) and their ―mainstream‖ peers in terms of mathematics achievement, course-taking patterns, and enrollment in mathematics-related majors, are well documented (burton, 2003; gutstein, 2006; tate, 1995, 2005). addressing these inequities through the teaching of mathematics for social justice is viable and worthwhile to me, both as a mathematics education researcher and teacher of mathematics, especially given the extensive research highlighting the role that mathematics plays as a gatekeeper to future success (burton, 2003; moses & cobb, 2001; tate, 1995, 2005). defining teaching mathematics for social justice the phrase teaching mathematics for social justice is not uniformly defined within the research literature. there are numerous definitions ranging from equal access to upper-level mathematics courses to social reconfiguration spurred by the use of mathematics as an analytical tool to understand social life and the inequities that exist therein. the definition of mathematics for social justice that i rely on, and that guided my work with the teachers, draws from the work of several researchers and is comprised of four components. the first of these components is access to high quality mathematics instruction for all students. moses and cobb (2001) argued that mathematics is needed to be a full participant in society, and likened the struggle for access to high quality mathematics instruction for marginalized students to the civil rights movement and access to voting rights for african americans. another way to talk about this component is to say that all students deserve a strong grounding in what is usually referred to in the literature as dominant mathematics. gutiérrez (2007) defined dominant mathematics as that which ―reflects the status quo in society, that gets valued in high-stakes testing and credentialing, that privileges a static formalism in mathematics, and that is gonzalez mathematics for social justice journal of urban mathematics education vol. 2, no. 1 25 involved in making sense of a world that favors the views of a relatively elite group‖ (p. 39). a second component of the definition that i rely on is a re-centering of the curriculum around the experiences of students from marginalized communities. that is, teaching mathematics for social justice involves building upon the experiences of students from marginalized communities, while exploring issues of social justice through mathematics (gutstein, 2006). this component is supported by research that advocates for instruction to be centered on students‘ experiences in order for it to be meaningful to them (darling-hammond, french, & garcialopez, 2002; villegas & lucas, 2002). the third component is the use of mathematics as a critical tool for understanding social life; one‘s position in society; and issues of power, agency, and oppression. this component is often referred to as critical mathematics and often set in contrast to dominant mathematics. for instance, gutiérrez (2007) defined critical mathematics as ―mathematics that squarely acknowledges the positioning of students as members of a society rife with issues of power and domination…[and] takes students‘ cultural identities and builds mathematics around them in ways that address social and political issues, especially highlighting the perspectives of marginalized groups‖ (p. 40). in this way, mathematics becomes a tool used to examine social environments, increase awareness of social injustice, and serves as a valued language that can be used to further an agenda of social change towards a more just society. while increasing awareness is important, without a component that addresses change, the injustices that exist in society will continue to persist. in order to bring about social change, action and agency need to shape the perspectives with which we view mathematics for social justice. thus, the fourth component of teaching mathematics for social justice is the use of mathematics to radically reconfigure society so that it might be more just. mathematics for social justice units and lessons, according to gutstein (2006), should serve the purpose of ―liberation from oppression‖ (p. 22); he argued that schooling should be a vehicle for empowerment and social change. this component is consistent with the position of many educational scholars who argue that citizenship should not involve blindly following the rules of an inherently unjust society, but instead should involve being a critical observer taught to understand the world and work toward making it more just (aronowitz, 2004; burton, 2003; michelli & keiser, 2005). gutstein (2006) furthered this position specific to mathematics education, arguing ―a crucial aspect of mathematics for social justice is what students do with the mathematics‖ (p. 14). when mathematics for social justice is understood as a tool to further social change and the emancipation of oppressed communities, it is being viewed as an extension of paulo freire‘s scholarship (see, e.g., 1970/1993) and his pedagogy of liberation. frankenstein (1983) claimed, ―freire‘s theory compels mathematics gonzalez mathematics for social justice journal of urban mathematics education vol. 2, no. 1 26 teachers to probe the nonpositivist meaning of mathematical knowledge, the importance of quantitative reasoning in the development of critical consciousness…and the connections between our specific curriculum and the development of critical consciousness‖ (p. 318). understanding mathematics as a means to develop a critical consciousness makes clear that the end product is not confined to equal academic performance or to equal access, but to a complete rethinking and restructuring of the current society. defining socially just society given that the creation of a more socially just society is seen as a goal of teaching mathematics for social justice, it seems necessary to discuss and attempt to define what is meant by social justice and what a socially just society might look like. the work of zollers, albert, and cochran-smith (2000) looked specifically at the concept of social justice and its definition for a group of teacher educators. their study aimed to ―investigate individual understandings of the meaning of social justice and find the commonality necessary to ‗teach for social justice‘‖ (p. 1). the teacher educators in their study linked social justice to issues of fairness and equity, personal and institutional responsibility, and individual and collective action. michelli and keiser (2005) described a socially just society as one in which each individual can realize their potential and access all life‘s chances. furthermore, it is a society characterized by nonrepression and nondiscrimination in which no one individual or group oppresses another. a related way of understanding social justice is the principle of distributive justice; characterized by an equitable distribution of society‘s resources, including all that is both good and bad (rorty, 1979). this idea is to distribute both the benefits and burdens of society among its members, though issues arise when one attempts to define how such benefits and burdens can and/or should be distributed and is o ften accompanied by a discussion of wealth and access to opportunities. the scholarship noted above leads to a definition of social justice that includes access to opportunities and resources distributed in such a way as to not repress or discriminate against any one individual or group, whether for the good of another or not. a socially just society can therefore be characterized by equal opportunities, equal access, and the ability of all to reach their potential through access to all of life‘s opportunities (michelli & keiser, 2005). in addition to the scholarships noted, i rely on gutiérrez‘s (2007) benchmark for achieving equity in education. she argued that equity in education is ―being unable to predict student patterns (e.g., achievement, participation, ability to critically analyze data/society) based solely upon characteristics such as race, class, ethnicity, gender, beliefs, and proficiency in the dominant language‖ (p. 41). we can expand upon this idea from gutiérrez in order to define a socially just gonzalez mathematics for social justice journal of urban mathematics education vol. 2, no. 1 27 society. in so doing, i argue that a socially just society is one in which we are unable to predict success in life based upon characteristics including—but not limited to—race, ethnicity, gender, beliefs, citizenship status, and proficiency in the dominant language. proponents of teaching mathematics for social justice argue that a more socially just society is possible through the teaching of mathematics for social justice (see, e.g., gau, 2005; gutstein, 2006; gutstein & peterson, 2005). in order for teachers to teach mathematics for social justice, however, they must be prepared to do so. preparing teachers as agents of change professional development programs for inservice teachers as well as teacher education programs for preservice teachers are now beginning to address issues of social justice (darling-hammond, french, & garcía-lopez, 2002). sleeter (1997), in discussing a professional development opportunity for teachers to learn to teach in multicultural ways (a possible precursor to teaching for social justice), explained that the most common result of the training was that teachers became ―more aware of the differences among their students, student learning styles, racism in society, cooperative learning, curriculum, and school problems‖ (p. 688). gau (2005) also noted that the biggest change in the preservice mathematics teachers she worked with in her mathematics for social justice project was an increased awareness of differences. it is this awareness that teacher preparation programs should, i believe, strive for. by becoming aware of their students‘ backgrounds and of their own position in social life, both sleeter and gau argued that teachers often become ready to act on this knowledge for the betterment of students. teaching for social justice involves teachers and students becoming increasingly aware of their social realities and of one another‘s respective histories, cultures, and understandings. the development of teachers‘ identities as agents of change and discussion about the role of teachers as political agents is also necessary (villegas & lucas, 2002). as our behavior is mediated by our identities, changes in our behavior require changes in how we see ourselves (holland et al., 2003). identities affect agency and action, making identity development an essential element of teacher preparation. in defining identity, holland et al. stated: people tell others who they are, but even more important, they tell themselves and they try to act as though they are who they say they are. these self-understandings, especially those with strong emotional resonance for the teller, are what we refer to as identities. (p. 3) gonzalez mathematics for social justice journal of urban mathematics education vol. 2, no. 1 28 our identities therefore are ―something that arises from a transaction rather than being an inherent feature of a material body‖ (roth, 2005, p. 326). consequentially, teachers‘ identities, as those of all of us, are developed through social interactions. it is through interaction with others that we grow and develop in terms of how we see ourselves, forming and shifting our identities as we are pushed to entertain new ways of being (holland et al., 2003). entertaining these new ways of being often drives us to act in previously unexplored ways as we redefine who we are. considering new ways of being is the first step in changing one‘s pedagogy (florio-ruane, 2001). therefore, in order to affect changes in teachers‘ practice, preparing them to teach mathematics for social justice, we must, i believe, begin by affecting change in their identities. teachers need to come to see themselves as agents of social change if they are to implement mathematics for social justice in their teaching (gutstein, 2006). using communities of practice the idea that learning is a social process has led many teacher educators to use communities of practice as vehicles through which to prepare teachers (see, e.g., choi, 2006; florio-ruane, 2001). a community of practice, as defined by choi (2006), is a ―community that shares and creates real knowledge‖ (p. 143). it refers to groups of people; in this case, the participants and me, who are ―bound by their shared competence and mutual interest in a given practice‖ (p. 143)—the teaching of mathematics for social justice. according to wenger (1998), communities of practice contain the following three dimensions: mutual engagement, a joint enterprise, and a shared repertoire. when talking about mutual responsibility, however, it is important to note that wenger (1998) neither specified that the responsibility for the group be shared equally among its members, nor did he infer that equal sharing was possible. members have different knowledge, experiences, and positions within the group that they bring to the experience, allowing for collective work—with different contributions—on a joint enterprise. wenger (1998) also described communities of practice as communities in which there is prolonged engagement by the members as they work toward a joint enterprise. through using communities of practice as vehicles for professional development in the teaching of mathematics for social justice, the teachers work over prolonged periods of time (not the more common workshop model) to arrive at understandings about what it might mean to teach mathematics for social justice as well as how to prepare to do so. although it is not a necessary condition of a community of practice, the idea that power in the group should be shared, is supported by advocates of teaching mathematics for social justice, who argue that students and teachers together should be jointly responsible for what occurs within the classroom, including gonzalez mathematics for social justice journal of urban mathematics education vol. 2, no. 1 29 what is taught (gutstein, 2006; gutstein & peterson, 2005). while wenger (1998) argued that there are power dynamics at play in any community, the idea of sharing power with the teachers in a community of practice aimed at professional development is aligned with the goals of critical theorists who attempt to disrupt the power dynamics that presently exist in society, including those in situations such as the professional development of teachers (mclaren, 2000). given the social conception of learning and the characteristics of communities of practice as outlined, these characteristics become powerful ways through which teachers can develop as educators. communities of practice are touted by choi (2006) as ―the most suitable learning method not only for achievements of tacit knowledge based on participation and practice in real world contexts, but also for implicit knowledge, which is passed easily through represented and systematic forms by practice at a group level, not at a personal level‖ (p. 143). methods although the larger study aimed to answer four broad questions, this article focuses specifically on those questions pertaining to teachers‘ views and beliefs about the role of teachers and the nature of mathematics and mathematics teaching and learning. or, more broadly, the teachers‘ developing identities. as a result, the focus of the analysis presented here will be teachers‘ changing views of whom they are and what their practice does and should entail. to this end, i focus on two of they study‘s research questions: (1) how do these teachers view and understand the teaching of mathematics for social justice? (2) how, if at all, does exposure to ideas about social justice and mathematics affect teachers‘ beliefs about teaching mathematics, the nature of mathematics, and their roles as teachers and agents of change? recruitment of participants while working in an unrelated study, i assisted in the collection of data at several schools, one of which i will refer to as urban high school. this school is a large, comprehensive public high school in nyc that relies on a reform curriculum very unlike the ―traditional‖ mandated nyc curriculum. it was suggested to me that as a result of teachers‘ familiarity with a reform-based curriculum, the school might be a good fit for my study. this suggestion was based on the assumption that the teachers at urban high would be more open to trying activities around the teaching of mathematics for social justice as compared with teachers at other schools whose curriculum was more traditional and whose ideas about what constitutes mathematics and the teaching of mathematics might be more narrowly defined. gonzalez mathematics for social justice journal of urban mathematics education vol. 2, no. 1 30 i first approached the assistant principal of mathematics at urban high and later the principal to obtain support for the study to be conducted with urban high‘s mathematics teachers. after the appropriate permissions were obtained, the assistant principal of mathematics provided me with a list of email addresses for the teachers in the mathematics department. i forwarded a description of the research project to the teachers along with information about what participation in the study would include. participation included: attending group sessions, participating in three interviews, and writing reflections after each group session. potential participants were also told that in exchange for their participation, they would receive a copy of the edited volume rethinking mathematics: teaching social justice by the numbers (gutstein & peterson, 2005), copies of all of the articles and materials used during the study, and would be paid a monetary sum comparable to that which they would receive for participating in similar professional development opportunities through the nyc department of education. eight of the nearly 30 mathematics teachers at the school expressed interest in the study. of these, only one was declined due to scheduling issues, which is perhaps unfortunate as he would have been the only male participant. the participants, therefore, were seven female mathematics teachers who all worked in the same nyc public high school, urban high, during the 2006–2007 academic year, though by the time that the data were collected (during the 2007–2008 academic year) two of the teachers had moved to other schools; they elected to be a part of the study nonetheless. the school this section serves to describe the school and relies on statistics obtained from the nyc department of education‘s web site. in order to keep the name of the school confidential, the school‘s web site from which the data was obtained does not appear in the list of references, although the homepage of the nyc department of education does. urban high is a large, comprehensive, public high school in nyc, serving over 3,000 students in grades nine through twelve. the physical building is very large with wide, ample hallways. one particular floor contains multiple classrooms, many of which are devoted to mathematics. the rooms are wide, with dry-erase boards and trapezoidal tables arranged in groups of two forming hexagons. the school administration places emphasis on the use of group work; the tables facilitate this pedagogical approach. the vast majority of students at urban high (as of december 2007) were classified by the nyc department of education as black (55%) or hispanic (41%). furthermore, a commonly used—though often misleading—measure of a school‘s overall socioeconomic status (ses) is the percentage of students who qualify for free or reduced-priced lunch. at urban high, for example, in the gonzalez mathematics for social justice journal of urban mathematics education vol. 2, no. 1 31 2005–2006 academic year 38% of the students qualified for free or reducedpriced lunch. the percentage presented is the ratio of the ―number of approved lunch applicants‖ to the number of full-time students at the school (new york state school report card accountability and overview report, 2006). the participants in my study, however, explained that due to the large number of students at the school, that some of the cafeterias (there were more than one) were converted into classrooms and so as to not overcrowd the remaining cafeteria, students were encouraged to take their lunch period (a required part of students‘ education programs in nyc) at the end of the day. students who had lunch during the last period of the day usually elected to leave the school and eat lunch elsewhere. as such, the number of applicants for free or reduced-priced lunch was quite low because of the high number of students who do not eat lunch at the school. several members of the school‘s administration, when asked directly about this school policy, indicated that students may elect to take their lunch period last and that the overwhelming majority of those who do, leave the building at that time, as such they do not submit lunch applications. i asked several members of the administration, as well as all of the participants in my study, and several other teachers that i met on one of my visits, to estimate the percentage of students who might qualify for free or reduced-priced lunch (including those who might not apply), the various estimates i received hovered around 75%. this much higher percentage was representative of similar public schools in the city, as defined by the nyc department of education using data on its web site. that urban high is a large, comprehensive school in an urban area serving students who are primarily from historically marginalized communities made the school an attractive one for me to conduct my research study. furthermore, after a long struggle, urban high was able to select its own mathematics curriculum. the struggle to use a ―non-traditional‖ mathematics curriculum was challenging, as explained by the school‘s assistant principal of mathematics, but something that she felt was necessary. the teachers often made reference to the fact that the assistant principal of mathematics put her job ―on the line‖ to be able to use a non-traditional mathematics program. this non-traditional mathematics program is characterized by the use of exploration and discovery activities that highlight mathematics concepts which students discover as they work through the activities. the program places heavy emphasis on problem solving; problems are contextualized in units that revolve around certain themes. as with most teachers, the participants seemed to believe that the curriculum was strong in some areas, especially in getting students to break down problems, but that it was lacking in others. the teachers believed that the curriculum did not expose students to the types of questions they are likely to see on the standardized examinations and that it used imprecise language. while the curriculum encouraged students to engage in and struggle with non-traditional gonzalez mathematics for social justice journal of urban mathematics education vol. 2, no. 1 32 problems while learning to deconstruct the material, the teachers believed that students could not always relate to the contexts provided, and that supplementation was needed. in general, because the teachers worked with a non-traditional and contextualized mathematics curriculum, urban high was appealing to me. i imagined that the teachers might be more open to using mathematics for social justice lessons and activities, compared to those who worked at a school using a more traditional mathematics program. the participants the participants were seven female mathematics teachers who worked at urban high during the 2006–2007 academic year. each teacher had between 1 and 4 years of experience at the beginning of the study, and at least 1 year of experience using the school‘s non-traditional curriculum for mathematics. the familiarity with the school setting and curriculum assisted the teachers in ascertaining the supports and limitations of implementing different activities in the mathematics classroom. as a result of the self-selective nature of the group, the teachers were, in some ways, not representative of those in the school‘s mathematics department or the school in general, based on data obtained from the school‘s assistant principal of mathematics and from the participants themselves. in that, all of the participants were women, despite roughly one third of the mathematics department being comprised of men. additionally, as compared to others in their department and the school, the participants were more likely to be from racial and ethnic backgrounds similar to urban high‘s students. six of the seven teachers, approximately 86%, identified as black or hispanic, while in both the mathematics department and the school as a whole less than 33% of the teachers did so. according to data obtained by the national center for educational statistics (n.d.) 25.5% of public school teachers in urban areas such as nyc identify themselves as black or hispanic, significantly less than when considering the participants, approximately 86% of who identified themselves as black or hispanic (i.e., all but one, vanessa). moreover, this group of seven teachers tended to be less experienced than those in the school as a whole. while approximately 55% of the teachers at urban high had taught in the nyc public school system for over 3 years, only 43% of the teachers in my study had done so. additionally, while 25% of the teachers at urban high had been in the nyc public school system for over 5 years, no one in my study had been. the participants ranged from having 1.5 to 4.5 years of teaching experience, with the mean being 3.2 years. research highlights that teachers in urban settings tend to differ from their students with respect to characteristics such as race and ethnicity, family background, and socioeconomic status, making gonzalez mathematics for social justice journal of urban mathematics education vol. 2, no. 1 33 it difficult for them to relate to their students (darling-hammond, french, & garcía-lopez, 2002). yet, the teachers in this study were predominantly from the same racial backgrounds as their students. and all but two reported growing up in families of low-socioeconomic status, with four reporting that their family had been on public assistance when they were growing up. (table 1 provides background information about each teacher.) the mathematics for social justice group the mathematics for social justice group was a professional development opportunity for the teachers designed as a community of practice. it was not a college course, nor a course for which the teachers received credit or a certificate of any kind. it was not affiliated with any school or professional program. as the researcher, i was both a participant in the group and its facilitator. i did not ―grade‖ the teachers, nor report their ―progress‖ to anyone at the school or elsewhere. the group was a professional development opportunity for the teachers and formed part of my dissertation research (gonzalez, 2008). the group that the teachers participated in met weekly for a total of 10, 2hour sessions. meetings were held at urban high during the academic year. specifically, we met in vanessa‘s classroom on friday afternoons after classes had ended for the day and enough time had passed for those no longer at the school to arrive. our first meeting began with a discussion of what social justice is and how we might recognize a socially just society if we saw it. the idea was to allow an intellectual space for us to explore our own conceptions of social justice and, in later sessions, explore how our work as teachers of mathematics might serve as a vehicle for social change. our first five sessions involved reading texts related to the teaching of mathematics for social justice. articles, chapters from books, and other relevant materials from such authors as gutstein (2006), gutstein & peterson (2005), martin (2003), and tate (2005) were read and discussed with the aim of understanding how teaching mathematics for social justice is defined in the literature and how it might play out in the classroom. critical discussions about these readings and previous work that has been done formed the basis of our first few meetings. in addition, the participants and i engaged in activities and lessons that are currently available in the mathematics for social justice literature, allowing us an opportunity to explore some of the resources that exist and to discuss their perceived strengths and limitations. in the final sessions, the group developed a unit that linked high school mathematics (for first-year high school mathematics) with issues of social justice. gonzalez mathematics for social justice journal of urban mathematics education vol. 2, no. 1 34 table 1 summary of participants name (pseudonyms) age race/ ethnicity exp. (years) years at urban high college major teaching license articulated connections to school and students ellen mid 30s mixed (african american/white) 3.5 3; left to work at the suburban hs she had attended finance middle school math none, beyond her race and working at urban high jenna low 30s hispanic 3.5 3.5 information systems middle school math lives in neighborhood; similar in experiences and background (i.e., ses) mellissa mid 30s african american 3.5 3; left to work at small charter school economics/ accounting high school math none, beyond her race and working at urban high monica 29 african american 3.5 3.5 engineering high school math attended urban high; lives in neighborhood; similar in experiences (i.e., ses) nyo upper 30s african (nigerian) 4.5 4.5 engineering high school math attended urban high reina low 30s hispanic 2.5 .5; taught 2 years at a middle school engineering middle school math lives in neighborhood; similar in experiences and background (i.e., ses) vanessa mid 20s white 1.5 1.5 chemistry/ mathematics high school math similar in experiences and background (i.e., ses) it was my intention at the start of my study that the teachers take more responsibility with respect to how the group should run and what we would do as time went on. on several occasions, i told the teachers that if they had anything gonzalez mathematics for social justice journal of urban mathematics education vol. 2, no. 1 35 interesting for us to read and discuss that we could substitute their suggested readings for the readings i had originally planned; however, none of the teachers ever approached me with readings they wished to use. on other occasions, i set out some options in terms of how to proceed with the session at its start. despite these attempts to make the sessions more participant centered, for the first six sessions the teachers deferred to me in terms of how the session should proceed and none brought in a text to read or an activity that they wanted to share with the group. the shift in power and responsibility about how the group would run, and that i had longed for, did eventually become reality, however. in sessions 6 through 9, the teachers were the ones who determined what topic our unit would address, how to proceed with the development of the lessons, and who would be responsible for what. the last session again featured the teachers determining how we would proceed. in this session, each was an expert on work they put forth as they created it. if someone had walked in on this last session (any time after the first few minutes), it would have been virtually impossible to tell who the participants were and who the researcher was because each of us was taking turns as the presenter, as the leader of the group for the few minutes that we each took to speak about our part. everyone had something to offer and, at the same time, something to learn. our mathematics for social justice unit in their initial interviews (discussed later in the article), each of the teachers expressed a passion for education and many commented that the educational opportunities afforded to the students at urban high often leave them ill prepared for the future. this perspective, coupled with the desire that our unit be relevant to students and something they could all be ―on the same side of,‖ led the teachers to create a unit aimed at answering the question: how well does urban high school prepare its students for the future? the teachers saw this question as an issue of importance to their students and one that the students might have interest in. the teachers also felt that improving their school was an attainable goal their students might work toward. whereas, tackling a bigger social issue, such as how the unemployment rate or poverty levels are calculated, might lead to both good discussions and applications of mathematics, it might not result in any change or resolution. reina noted, ―i loved the unit we decided upon because it is something practical, that [students] would need immediately and could do something about‖ (session 9 reflection). it was important for the teachers that the students be able to instill change in meaningful ways. they wanted the students to feel aware and motivated, to be able to describe the situation using mathematics, and to take action at improving the situation they were to explore. additionally, keeping in mind the fact that they might not be able to use the unit as a whole giv gonzalez mathematics for social justice journal of urban mathematics education vol. 2, no. 1 36 en the scope, sequence, and pacing guide that drove their curriculum, the teachers hoped to create a collection of lessons that together could form a cohesive unit, but that could be used independent of one another so as to facilitate implementation. the overarching question of the unit was explored using two approaches. the first of these focused on how urban high was preparing their students as compared to other public high schools in nyc. vanessa, reina, and i worked on this part of the unit. relying on statistical datasets available on the nyc department of education web site, we developed lessons and problem sets that compared statistical data on urban high to two similar, large, open-admission schools and one specialized high school. the statistical data used for comparison included: graduation rates, standardized test scores, incidences of violence, number of advanced placement classes, and so forth. the focus of this part of the unit was the use of mathematics to understand the way in which urban high prepares its students as compared to other nyc public high schools and, from this analysis, to determine what changes might be needed to improve the school‘s ability to prepare its students for the future. exploring what might be extrapolated from these datasets involved mathematical analyses of why some of the statistics might be misleading, what can and cannot be answered by the statistics, and the implications of the statistics on student learning and preparation. this part of the unit was in line with the work of freire, as described by frankenstein (1983), who wrote, ―freire‘s concept of critical knowledge further directs us to explore not merely how statistics are non-neutral, but why, and in whose interest‖ (p. 324). the second approach used to determine how urban high prepared students was accomplished by comparing the opportunities students have at the school with the entrance requirements at various types of colleges and for various majors. by looking at how an urban high graduate might fare when applying to various colleges and analyzing how prepared they would be to pursue various courses of study (e.g., mathematics/science-related majors, liberal arts-related majors, performance majors), this part of the unit aimed to address the overarching question of the unit. ―my hopes are to get students thinking about the best fit for a college in terms of what they wish to study and how their grades help them fit into an appropriate area of study‖ (nyo, session 7 reflection). nyo and monica worked on this part of the unit, which also included lessons on understanding one‘s transcript and on the graduation standards. the teachers believed that despite having to meet these standards in order to graduate, many students were often unaware of them. following these two parts of the unit, the students would then prepare presentations to share their results with members of the school community, such as administrators, parents, teachers, and others. in so doing, they would not only demonstrate where the school was in terms of preparing students, but also they gonzalez mathematics for social justice journal of urban mathematics education vol. 2, no. 1 37 would advocate for changes that they saw as necessary for urban high‘s graduates to be prepared adequately for the futures they wish to pursue. the third part of the unit dealt with financial preparation. melissa and jenna undertook development of a sub-unit on financial mathematics with the belief that the school was not teaching students the skills required to be successful financially in the future. aimed at addressing this deficiency, this part of the unit included lessons on how to balance a checkbook and how interest is calculated on credit cards and other types of loans. as an element of this part of the unit, students would create a budget based on data obtained through the department of education web site about what the typical urban high student was planning to do after graduation. finally, ellen, who worked alone, explored ways in which the information students learned from the unit might be shared with others. she wanted to develop a forum for change where students could share their knowledge with incoming students so that new students could take full advantage of the opportunities available at the school. she also envisioned creating opportunities to inform parents and others about the school, the opportunities it provides, and also what students and parents need to do in conjunction with teachers and the school administration to ensure student success. her idea took the form of a ―success day‖ event that would have older students welcoming new students and their parents to the school in order to foster a culture of success at the school. she created an outline for the day‘s events that she proposed to the administration of her new school for use at the start of the 2009 school year. the teachers‘ engagement in the unit transcended our friday meetings. that is, their work on the unit was not confined to the 2 hours we met on fridays, but rather something that they did all throughout the week. for example, in her session 6 reflection, jenna noted, with respect to her group, ―we plan to spend this week doing a bit of research and bringing it into the next session.‖ similarly, others spent the time between sessions looking up information and reworking their parts of the unit. the teachers‘ work on the unit and the fact that they were spending much out-of-session time on it led us to postpone the last session. instead of meeting 1 week after session 9, we let 2 weeks go by before meeting in order that we would have more out-of-session time to work on the unit. the teachers responded positively to the unit they created and saw it as both relevant to and useful for students. vanessa explained, ―yeah, oh yeah, i think the kids would really be interested in it…this is stuff that‘s directly related to their life‖ (exit interview). others noted that students often complain about what they learn in mathematics, seeing no use for it in their daily lives. jenna wrote in her session 7 reflection: ―students have the habit of complaining that they aren't going to use most of what they learn in high school.‖ she added, however, that with respect to our unit, ―they're definitely going to need all of this.‖ gonzalez mathematics for social justice journal of urban mathematics education vol. 2, no. 1 38 interestingly, the teachers criticized the level of mathematics in some of the social justice activities we explored as part of the group sessions, but they did not raise this same concern with respect to the level of mathematics in our unit; even though some parts of it were informational, but devoid of any ―rigorous‖ mathematics. when asked about the lack of mathematical rigor in the finance and budgeting sections of the unit, melissa and jenna, the creators, both agreed that the mathematics in this part did lack rigor. on the other hand, they pointed to the fact that understanding the content of the lessons in this part is necessary for students as they move beyond high school and that, as it is not covered elsewhere in the curriculum, it is important that students be exposed to it. the most rigorous mathematics part of the unit was aligned with grade nine mathematics standards in nyc, addressing topics such as ratios, percents, and the use of graphs and tables to display data and probability. these are the same topics that are most often covered in the mathematics for social justice lessons which currently exist and those that the participants and i explored as members of the group. i think that the first part of the unit, the statistical comparison of the schools, was much more aligned with the ideals of teaching mathematics for social justice than the other parts. it used data that were readily obtainable to explore and compare various schools, thus exposing the differences in quality and scope of preparation offered to students. it highlighted the deficiencies that exist in some nyc public schools, specifically those that serve students in marginalized communities as compared to more ―successful‖ schools serving mainstream students. it prepared students to understand these inequities statistically with the hope that by so doing students will be motivated to advocate for changes within their school to assist in mitigate them. this advocacy was further supported by the framework of success day as well as by providing avenues for students to share their concerns and ideas for improving the school with other stakeholders such as school administrators, community leaders, and parent groups. data collection each teacher participated in two semi-structured interviews that i conducted with the goal of ascertaining the participants‘ initial and developing beliefs about their identities as mathematics teachers and agents of change. the initial interview was held prior to the start of the group sessions; it included in-depth questions that addressed the participants‘ views about their own activism, the teaching and learning of mathematics, and their identity as mathematics teachers and agents of change. also explored in the initial interview were the participants‘ beliefs about the role of teachers, their views of the students whom they taught, and their reasons for joining the study. to note developments in the teachers‘ thinking, the exit gonzalez mathematics for social justice journal of urban mathematics education vol. 2, no. 1 39 interview, also a semi-structured, in-depth interview, contained many questions similar in nature and content to those in the initial interview, allowing me to discern changes in the teachers‘ thinking about various issues. other topics driving the questions in the exit interview were teachers‘ views of their developing identities, their opinions about the community of practice, and their thoughts about the teaching of mathematics for social justice. (there were additional, ongoing interviews conducted; they are discussed later in the article.) researcher interview. i was interviewed by members of my dissertation committee at the start of my study as well as at its end. the interviews followed the protocol used for the initial and exit interviews of the participants, allowing me to ascertain my initial identity and beliefs and how these (might) have changed throughout the course of the study. this procedure also served as a way of somewhat gauging my beliefs and perceptions against those of the participants. teacher reflections. the teachers were asked to write a reflection at the conclusion of each group session, addressing the activities or discussion of that session. at times, open-ended questions were provided to the participants to guide their reflections. in all cases, however, teachers were reminded that they need not be bound by these questions and were encouraged to also address other issues or concerns that they might have. as the researcher, i, too, answered these guiding questions, when provided, in my own reflections. in addition to the teachers‘ reflections, i interviewed one teacher informally at the end of most group sessions in order to catch ―fresh‖ reactions, suggestions, and thoughts (these are the ongoing interviews previously mentioned). the teacher interviewed rotated so that each had a chance to be interviewed in this manner. this interview, an oral reflection, was done with the hope that i could probe teachers‘ reflections a bit more than was possible when they reflected on their own in writing in order to gather rich data about the participants‘ developing beliefs and identities. video data. while each of the group sessions was videotaped in its entirety, a thorough analysis of the video data has not been undertaken at this time. the analysis for this article is drawn from the interviews and reflections (as previously noted). nonetheless, an analysis of the video data, i believe, will provide further insights into the participants‘ developing identities, their beliefs and understandings of mathematics for social justice, and the use of communities of practice as vehicles for professional development. it is the focus of my future work. researcher journal. in attempting to understand my role in the group along with my identities as a researcher, mathematics teacher, and agent of change, i kept a researcher journal in which i reflected upon these topics after group sessions. these reflections involved formal reflections similar to those the participants completed at the end of each session, as well as informal writing about ideas and issues as they emerged. the journal served several purposes, one of which gonzalez mathematics for social justice journal of urban mathematics education vol. 2, no. 1 40 was to see how my own thinking and identity developed through time. a second purpose of the journal was to monitor my own subjectivity, attempting to understand and document how my own history and beliefs affected both data collection and analysis. finally, the journal served as the place where i reflected after group meetings, forming an initial, perhaps informal, way of understanding and analyzing the data. entries in the journal served the function of analytical, methodological, and personal memos (strauss & corbin, 1990). that is, in part they played an analytic role providing a place to make initial inferences about the data, raise quest ions, and note emerging themes. the entries also in part allowed me to consider how to approach the next session or phase of the research project; therefore, they served an important methodological role. data analysis data analysis was an open-ended process involving constant, continual reflection. in keeping with the recommendation of qualitative researchers, data analysis took place throughout the data collection process and not entirely at the end of the study (creswell 2005; strauss & corbin, 1990). this procedure enabled the refining of methods and future data collection. one example of this refinement was the addition of a written reflection by the teachers and me at the start of each session that was not part of the original data collection methods. these reflections were added later on in the study, both as a way of discerning participants‘ individual thoughts about warm-up activities and as a way of focusing the group at the start of each session. before my initial round of coding the data, i read through the text-based data several times in order to get an understanding of the whole of the transcribed discourse and, at that time, wrote some initial findings based on these readings that i then looked to for support when the data were later more systematically analyzed. given the nature of this work, my belief in the validity of teaching mathematics for social justice, and the goal of preparing teachers to teach in this manner, i came to my study with a fairly well-articulated (preconceived) agenda, complete with research questions, analytic categories (e.g., teacher identity, teacher understandings of mathematics for social justice), and the goal of preparing teachers to teach mathematics for social justice. although i was open to themes that might ―emerge‖ from the data, the fact that i had research questions i wished to address made it impossible to go into the coding process without any preconceived ideas. my interest in the teachers‘ developing identities and in their understandings of teaching mathematics for social justice necessitated the development of codes that addressed these issues. working definitions of the codes were constructed and refined as the coding scheme was applied to the data. these codes represented themes that were derived from my interaction with the data and gonzalez mathematics for social justice journal of urban mathematics education vol. 2, no. 1 41 included the teachers‘ developing understandings of teaching mathematics for social justice, their awareness of the literature, and changes in their ―style of talk‖ that i had not foreseen. the analysis of data was done in two ways. one was to look at the coded statements across time for each participant, allowing me to ―see‖ changes across time for each participant. a teacher‘s understandings of mathematics for social justice, for instance, could be traced using this method across time. a second method was to compare the coded statements across teachers by topic, noting agreement and disagreement between their beliefs and understandings. reliability of findings in an effort to present reliable findings, various procedures were undertaken. these included the use of multiple methods of data collection and multiple data sources, an essential component of trustworthy research (creswell, 2005; strauss & corbin, 1990). findings were triangulated by data source (participants and researcher) and data collection method (interviews and reflections). incongruous or conflicting information that surfaced was noted with the belief that negative cases strengthen research by contextualizing findings. ongoing interviews with the participants as well as their written reflections allowed me to learn about how they viewed their participation in the group, the nature of our meetings, and their understanding of various constructs, as well as their own developing ident ities. my own views and the patterns were checked against the views and patterns that the participants perceived and related back to me. additionally, the participants were presented with various preliminary findings through phone conversations and email exchanges and provided feedback with respect to these (i.e., member checking). findings my analysis of the data demonstrated that the teachers were acutely aware of the injustices that their students face; they were acutely aware of students‘ home lives, inadequate academic preparedness, and the lack of opportunity available to them and their families. as previously mentioned, five of the seven teachers are from similar backgrounds as their students and feel they share the experiences of these urban youth. those with similar backgrounds saw themselves as being able to succeed in society despite the lack of opportunity because of their reliance upon education. they saw education as the way to future success, although they were cynical about the education their students receive. in her initial interview, melissa, noted: gonzalez mathematics for social justice journal of urban mathematics education vol. 2, no. 1 42 i believe that sometimes the curriculum is set up towards the government. obviously, if they‘re picking the curriculum, they‘re picking what they want you to learn. they want to shape you in the way they want to shape you. the old school that i used to work with is just that—we just want to produce servants. we just want to produce someone who will be the serving class. it was not geared toward producing these high-level, educated, intelligent individuals. the teachers, as melissa‘s quote exemplifies, saw the public school at which they had taught, as well as similar schools, as producers of servants and not leaders. they feared that their students are being done a disservice and struggle with the fact that they are a part of the very system that is keeping these students from succeeding. melissa was one of two participants (the other being ellen) who have young sons of color and who vehemently opposed sending their sons to urban high or similar schools, as they believe that these schools are not adequately serving students—most notably, young men of color. they both talked about toying with the idea of starting a school specifically for this population of students. neither melissa nor ellen has their son enrolled in a nyc public school, nor do they have plans to do so. melissa explained: and i refuse—i told my husband, i will quit my job and be a home-school teacher instead of putting my child in this little zoo, any zoo that they got going here. i do not trust the system. i don‘t trust them, not with my black, male child. i know it sounds crazy, but i just don‘t because if you look at the, you know, what they have been producing, they haven‘t been producing much. (initial interview) her sentiments were echoed by ellen, who has her son enrolled in a suburban school, and by some of the other teachers who have labeled these schools as ―pipelines to prison,‖ especially for male students from marginalized communities. it is interesting to note that neither ellen nor melissa still work at the school, choosing instead to work in schools serving more mainstream students. the teachers‘ initial ideas about social injustice was that it is prevalent— something both they and their students deal with constantly—and that it could be addressed through school better than in school, as mathematics for social justice proponents aim to do. their love of mathematics and interest in social justice issues drove them to participate. it was their awareness of such issues and their eagerness to address them that led them to the group. although all but three of the teachers noted a lack of familiarity with the phrase teaching mathematics for social justice in the initial interviews, this lack of familiarity referred mainly to a lack of awareness of how the term is defined in the research literature. the teachers, as evidenced by their initial interviews and our first group discussions, did indeed have their own construction of what teaching mathematics for social justice might mean. this construct, to them, consisted of some aspects of the four components of the definition of mathematics for social gonzalez mathematics for social justice journal of urban mathematics education vol. 2, no. 1 43 justice previously discussed, as well as a view of how they, as teachers, might be agents of change. for example, two of the teachers, nyo and vanessa, admitted that they had not heard of the expression teaching mathematics for social justice, but explained their understanding of the topic. nyo‘s definition involved bringing real-life situations and contexts into the classroom. instead of relying on problems devoid of context, teaching mathematics for social justice for her, initially meant, ―incorporating social issues in sort of a word problem‖ (initial interview). nyo‘s initial understanding of teaching mathematics for social justice addressed the need to bring social issues and real-life contexts into mathematics education consistent with the re-centering component of the definition previously presented as well as with the use of mathematics as a way of examining and understanding issues in society. initially, vanessa described mathematics for social justice as, ―maybe like integrating certain things that students would relate…for them to have a better understanding about mathematical context using context, but something that‘s more familiar‖ (initial interview). vanessa also believed from the start that education should be a means for raising class consciousness and, though these are not her terms, teaching for liberation in the freirian sense: ―i wanna be able to raise some of these issues to my kids and be able to address them and discuss them and maybe to open up their eyes to what exists‖ (exit interview). melissa noted that she brings social issues into her teaching. mostly, this effort involved bringing up individuals of color who were noted mathematicians and scientists and asking the students to find examples of such individuals as well: i used to bring articles, and i used to—during black history month, i used to tell them that, ―you have to find a mathematician that was either african-caribbean, african-american, african-latino that you know, and read about it, and you get extra credit if you come up, and you present, and you talk about it.‖ and i would also, before the test, extra credit would be, ―i‘m gonna read you a passage of a person that created all these things, and they were black.‖ and i would read about it, and the kids would take notes, and they can use their notes for extra credit. (initial interview) while there is an element of critique or conscious raising that is consistent with teaching for social justice, melissa‘s comments are what many researchers call the ―heroes and holidays‖ approach to multiculturalism in education; this limiting approach was also common to the initial conceptions of teaching mathematics for social justice that some of gau‘s (2005) preservice teachers had at the start of her study. the definitions initially put forth by nyo, vanessa, and melissa include bringing the ―real world‖ into their classrooms, but are vague as to how to do so; again, similar to what gau found of her participants‘ initial views of teaching mathematics for social justice. gonzalez mathematics for social justice journal of urban mathematics education vol. 2, no. 1 44 as they were introduced to activities and lessons created around the idea of teaching mathematics for social justice, the teachers began to see the political nature of mathematics teaching and realized how mathematics might be used to highlight social injustice. the teachers quickly realized the power of mathematics for social justice activities to raise student awareness of the injustices prevalent in society. vanessa spoke of these lessons as a way of raising ―class consciousness‖ (exit interview), which is parallel to freire‘s (1970/1993) ―massified consciousness‖ (p. 17) and forms a key component of teaching for liberation (nasir, hand, & taylor, 2008). reina noted that mathematics for social justice lessons are ―a way to get the kids to be aware of what‘s happening around them‖ (exit interview). while the teachers disagreed as to how aware their students are of various social and political issues, they all commented that engaging students in mathematics for social justice lessons would result in increased awareness. when asked in their exit interviews about their roles as agents of change, all of the teachers pointed to the changes that they affect in their students within their own classrooms as evidence that they are agents of change. this response was consistent with the experiences of coti (2002) as he reflected upon a similar professional development opportunity he engaged in. vanessa, on the other hand, stressed her desire to raise class consciousness as a means of affecting broader change in society, noting that she needed to further consider how to best do so within her classroom. many of the teachers noted that they did not initially realize the power they had as teachers to affect change in the broader society and that this power was something they were now beginning to consider: so this group kind of made me more like, ―well, i have this intelligence. i need to use it for good.‖ yes. with much power comes much responsibility, so it just, it made me more aware that i need to be more socially active, that, you know, i need to be part of affecting change, because no one‘s gonna do it for me kind of thing, and it also made me feel like i have more of a sense of like the same thing i was saying about the kids, like ownership, like i have control over what could happen, you know, but i‘m choosing not to exert that control and that power. so these sessions kind of made me like, ―no, i have to. i have to, because i have that responsibility as someone who knows.‖ (reina, exit interview) the power to affect change in society through their students was also a new idea that many of the teachers were beginning to understand. ―i learned new ways students could change their environment while involving math,‖ wrote monica in her session 9 reflection, adding that she was excited at the possibility of helping students to do just that. the teachers realized that mathematics for social justice activities could lead to student empowerment and larger societal change: ―it would give [students] a voice if you realized that there was actually something that they could do or say about an issue‖ (ellen, exit interview), and ―definitely would make [students] gonzalez mathematics for social justice journal of urban mathematics education vol. 2, no. 1 45 more empowered‖ (reina, exit interview). empowerment, the teachers argued, could lead to change: ―you could change the community,‖ monica noted in her exit interview. similarly, in her exit interview, reina explained that through mathematics for social justice students would ―feel like they can affect change.‖ at some point or another, each of the teachers suggested that students often feel disempowered because of their situation/life experiences, noting their students ―see how hard the world can be‖ and ―feel like there‘s no hope‖ (monica, exit interview). having students realize their agency by working towards social change was seen as a way of combating this learned helplessness. the teachers saw this positive change in their students as a possible outcome of teaching mathematics for social justice, a belief consistent with that of mathematics for social justice advocates (gutstein, 2006; gutstein & peterson, 2005). as turner and font strawhun (2005) noted, ―we found that creating space for students to pose their own problems and to inject their interests and concerns into the curriculum was a powerful way of supporting student activism‖ (p. 87). the teachers began to consider ways that their teaching could be informed by the ideas and activities that we were discussing and using in our group. reina‘s written reflections are an indication of this awareness. she began to shift her writing toward ways she could incorporate the ideas and activities she was learning about in the group. she stated in her written reflection after the fourth session, ―i feel [my teaching] would look more like a way to use the math to make arguments about our point of view…possibly at the end of a math unit as a project where the students can now use the math topics we've learned to hold roundtable discussions on a specific social issue.‖ in another reflection, she noted, ―i would really love to work in a school where i could tie this into their social studies classes, where we do the investigations in mathematics and they talk about the social impacts in their social studies classes.‖ this second quote hints at the struggle that all but one (nyo) of the teachers expressed facing. these teachers dealt with their belief that engaging students in examining social injustices is a worthwhile endeavor while feeling simultaneously tied by a school culture that focuses on standardized exams the students are required to take as well as a curriculum implemented through an often rigid pacing guide. reina was so affected by the group and the identities of being a mathematics teacher and agent of change that she struggled with her role as a high school mathematics teacher at urban high. by her session 9 reflection, reina spoke of the frustration she was feeling as she noted that participation in the group ―made me very angry about how mathematics is currently taught.‖ as she did not feel she could teach mathematics in a relevant, meaningful way, reina explained in her exit interview that she was considering leaving the school or teaching in general. she explained why in her session 9 reflection: gonzalez mathematics for social justice journal of urban mathematics education vol. 2, no. 1 46 to make math relevant, to create students who are socially active in what occurs in their lives feels like a huge responsibility. i have always done what was required because it was required. to create for my students something i never needed for myself seems like too much of a stretch on top of everything else that is expected of a teacher. the thought is very overwhelming. by the time her exit interview was scheduled, reina had decided to continue teaching but to do so at the middle school level as she had done prior to her work at urban high. she felt that the middle school curriculum allowed her more freedom to address issues that her students were facing through mathematics. at the start of the study, the teachers feared that examining social issues would serve not to empower students but to paralyze them into inaction as they considered the many injustices that they must face. ellen noted in her reflection after the eighth session, ―as a participant in a research group on social justice, i often worried that making students aware of the injustices they are faced with would cause them to throw in the towel or take on an attitude of self-defeat.‖ throughout the course of our sessions, the teachers‘ concerns about this issue began to lessen. in that same reflection, ellen continued to say that now she realized, ―students are fully aware of the injustices they face each day, and all they need is some empowerment, backing, and the means to have their issues addressed.‖ many teachers echoed ellen‘s sentiment, noting that increasing awareness alone was not helpful to students and that opportunity for action, where students could exercise their agency, must accompany such work. the participants‘ desire for student empowerment and action as part of teaching mathematics for social justice is consistent with research that posits that activities around mathematics for social justice should include opportunities for action (gutstein, 2006; gutstein & peterson, 2005). as a result, many were drawn to an article by turner and font strawhun (2005), which described a project where students used mathematics to explore the space allotted to them as a small school housed in a larger building with other schools. the students in turner and font strawhun‘s study compared their space with that of the other schools and used their findings to support their argument that they were not given a fair amount of space in the building. what the teachers in the study were most drawn to in this project was that it ended with students presenting their findings to the school board in an attempt to change the situation and rectify the injustice being committed against them. although the teachers began to consider how to incorporate social issues and the activities we did into their classes, they were highly discriminating about what they would and would not be comfortable bringing into their classes. some, most notably nyo and vanessa, felt that students will be engaged in mathematics because of the draw of these social issues and that this is a way of hooking students. nyo wrote in reference to some of the activities we did, noting, ―i loved the gonzalez mathematics for social justice journal of urban mathematics education vol. 2, no. 1 47 use of the cartoons that illustrated political issues and how it tied into the data that were later handed out. [it was a] great introduction to the hardcore math topics.‖ she claimed that connecting mathematics to social issues might make the mathematics more meaningful and exciting to students. she added, in a reflection written after the ninth session, ―oddly enough i never liked social studies in school, but i feel if it were presented and related to math in a similar fashion i might have enjoyed it.‖ the findings discussed here highlight the fact that exposure to ways of incorporating social justice issues in mathematics can lead to teachers valuing such work and reconsidering the ways in which they teach their students, as well as the way in which they define what it means to be a teacher of mathematics. mathematics for social justice activities were met with interest, though not always with full support. specifically, the teachers worried if raising awareness about social issues would serve to paralyze rather than motivate students, highlighting the need to provide avenues for action along with such lessons as argued by the research literature (gutstein, 2006; gutstein & peterson, 2005). the teachers raised numerous other concerns as well, including the fact that teaching mathematics for social justice might not be supported by the school‘s administration or by parents. these concerns mirrored those described by the teachers in the study conducted by gau (2005). thus, in order that teachers are able to implement mathematics for social justice lessons into their teaching in a meaningful way, they must be supported and taught ways in which they could provide avenues for action to follow mathematics for social justice lessons if these lessons are to be used as a catalyst for social change and not merely a way to raise awareness that on its own might not be as beneficial to students. participation in the group also led to changes in how the participants (and me) saw themselves (and myself). consistent with the work of gau (2005), the teachers‘ conceptions of their roles as mathematics teachers expanded as a result of their exposure to the teaching of mathematics for social justice. they began to reconsider what it meant to teach mathematics and what counts as mathematics in the classroom. the study found that providing a forum to learn about the teaching of mathematics for social justice led to the teachers‘ growing understanding of teaching as a political act, as well as the power of mathematics to be used as a critical tool for analyzing social life. finally, as i considered issue of power and responsibility among the teachers and me in the group sessions, i noticed that the sessions during which we worked on the unit, unlike the others, were not planned out ahead of time. in the outline of group sessions that the teachers received at our first meeting, these sessions simply listed ―work on project‖ for the main activity to be done and i did not consciously think through or plan out how this work would be done. this omission might account for why the teachers were able to take responsibility for these latter sessions. taking responsibility for one of the earlier sessions would gonzalez mathematics for social justice journal of urban mathematics education vol. 2, no. 1 48 have meant disrupting the plan i had conceived and laid out for us in the session overview. that is much more challenging than taking responsibility for the working session because these were not planned prior and taking ownership of them would not necessitate a rejection of the plans i had already developed. these realities with respect to the level of participation in our group sessions highlight to me a need for professional development to be carefully structured to include avenues for participants to take both ownership of the work and responsibility for the development of the professional development experience. ownership is defined as, ―our ability to take responsibility for negotiating meaning‖ (wenger, 1998, p. 201). that is, opportunities are created for teachers to come to an understanding of the materials or methods, to bring in their points of view and experiences, and to impact what occurs as part of the professional development. given opportunities to truly engage in the material, teachers are more likely to value the professional development and more likely to use what they learn than if they are treated as mere receivers of information. implications, unanswered questions, and issues for further research this study was informed by, and hopefully adds to, the research literature in the teaching of mathematics for social justice and teacher development through communities of practice. the conclusions reached can inform future professional development programs, hopefully leading to improved experiences for teachers, and through them, for students as well. lessons learned about the developing identities of the participants can serve to inform future studies and also programs aimed at preand in-service teacher development. although my study added to the research in the areas specified above, it left unanswered questions that can serve to guide further research. answering such questions might provide valuable insight into a number of topics, including the teaching of mathematics for social justice, professional development through a community of practice, and teacher pedagogy. the focus here is on teachers‘ developing understandings of teaching mathematics for social justice and the effects of participation in the group on teachers‘ identities as agents of change. both of these are pre-cursors, or necessary conditions, for changes in teacher action leading to pedagogical shifts. following these teachers into their classrooms to determine how these changes in understandings and identity impact teacher‘s actual practice is a logical next step that might lead us to answer questions such as:  does participation in a community of practice centered on teaching mathematics for social justice change teachers‘ pedagogical practices? if so, in what ways? gonzalez mathematics for social justice journal of urban mathematics education vol. 2, no. 1 49 another area for further research involves studying the ways that teachers could be supported so that they move from understanding mathematics for social justice to implementing it in their classrooms. examining ways that teachers could be supported as they move from awareness to implementation seems an invaluable endeavor. any reform, in order to be successful, requires support from the school, and so while we see work on preparing teachers to teach mathematics for social justice, an example of which is this very study, i continue to wonder:  what resources does a school and/or administration staff need to provide in order that teachers might fully implement mathematics for social justice lessons into their teaching? another factor to further explore would be how the findings might have differed if the professional development group and the study as a whole were undertaken with teachers who were not necessarily all aligned with the goals of teaching mathematics for social justice from the start or with teachers whose political and social understandings were less congruous than those of the participants in this study.  how would the group and the experience in general for teachers have been different if their opinions on various social issues were not so closely aligned? these are very rich questions that i believe are valuable ways of focusing future research—my own included. before concluding, i would like to share a quote from a reflection written by nyo after the fourth session that i think speaks to the excitement that i, the teacher participants, and hopefully some of you reading this article feel about teaching mathematics for social justice: ―i absolutely liked the idea of mathematizing everything around us.‖ acknowledgments this article is based upon work partially supported by the national science foundation (nsf) 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(2000). in pursuit of social justice: collaborative research and practice in teacher education. action in teacher education, 22(2) 1–14. http://nces.ed.gov/surveys/sass http://schools.nyc.gov/ https://www.nystart.gov/publicweb/home.do?year=2007 microsoft word 462-article text no abstract-2710-1-18-20220330 (proof 1).docx journal of urban mathematics education december 2022, vol. 15, no. 2, pp. 8–40 ©jume. https://journals.tdl.org/jume gladys krause, ph.d., is an assistant professor at william & mary, p.o. box 8795, williamsburg, va 23187; email: ghkrause@wm.edu. her research centers on teacher knowledge and children’s mathematical thinking and how these two areas interact in classroom settings that involve multilingual and multicultural dynamics. melissa adams-corral, ph.d., is an assistant professor of mathematics education at the university of texas río grande valley, 1201 w. university dr., edinburg, tx 78539; email: melissa.adamscorral@utrgv.edu. her primary research interests involve the use of the theories and methods of community organizers to conduct research that actively supports teachers’ and young people’s efforts to improve their daily experiences at school. luz a. maldonado rodríguez, ph.d., is an associate professor of bilingual mathematics education at texas state university, 705 harwood dr., san marcos, tx 78666; email: l.maldonado@txstate.edu. her primary research interests follow the mathematical learning experiences of the bilingual learner, from elementary student to teacher candidate, in particular documenting empowering teaching and learning practices. developing awareness around language practices in the elementary bilingual mathematics classroom gladys h. krause william & mary m melissa adams-corral the university of texas río grande valley luz a. maldonado rodríguez texas state university this study contributes to efforts to characterize teaching that is responsive to children’s mathematical ideas and linguistic repertoire. building on translanguaging, defined in this article as a pedagogical practice that facilitates students’ expression of their understanding using their own language practices, and on the literature surrounding children’s mathematical thinking, we present an example of a one-onone interview and of the circulating portion of a mathematics class from a secondgrade classroom. we use these examples to foreground instructional practices, for researchers and practitioners, that highlight a shift from a simplified view of conveying mathematics as instruction in symbology and formal manipulation to a more academically ample discussion of perspectives that investigate critically both mathematical concepts and their modes of transmission, which involve language practices, that are crucial for educating bilingual children. keywords: bilingual education, elementary, mathematics, translanguaging krause, adams-corral, language practices in the elementary & maldonado rodríguez bilingual mathematics classroom journal of urban mathematics education vol. 15, no. 2 9 ual language programs have become popular across the united states as a promising bilingual alternative to english-only and transitional instructional models (martínez et al., 2015). however, this alternative has been shown to promote bilingualism through a policy of language separation that discourages students and teachers from using both of their languages during instruction (martin–beltrán, 2010; palmer & martínez, 2013). this language separation in the classroom occurs by content area, teacher, or time (e.g., on a weekly or daily basis; hamman-ortiz, 2019). in a review of the literature on the debate surrounding language separation, hammanortiz (2019) described how those in favor of language separation argue that “minoritized languages need a safe space to thrive and, thus, must be ‘protected’ from the infiltration of the majority language” (p. 388). in the same review, hamman-ortiz (2019) noted that language separation in the classroom perpetuates already existing societal language imbalances by encouraging learners to draw on features from the majority language during class time allocated to the minority language (see also ballinger et al., 2017). in addition, hamman-ortiz (2019) stated that students enrolled in programs promoting language separation often report a preference for the majority language. in many cases, students use the majority language more often when interacting with peers (hamman-ortiz, 2019). as a result, language separation often results in instruction that favors or promotes the use of the majority language. language separation is especially restrictive during mathematics instruction due to widespread perspectives on language and mathematics teaching and learning. a consensus among mathematicians is that, because the concepts of mathematics are universal, their language of expression should be irrelevant. as history has borne out, the concepts of mathematics aim to transcend differences of language, for they purport to be the distillation of certain laws of the human mind (lager, 2006; moschkovich, 2007; mosqueda, 2010; stillwell, 2010). as a consequence, the concepts of mathematics are independent of the language used for their expression, but a common practice among instructors is to spuriously conclude that the language of instruction is, as a consequence, irrelevant to the teaching of mathematics. bilingual contexts demand a rethinking of this assumption: for children learning a new language, requiring expression of mathematical concepts in the new language can impose a cognitive burden that undercuts the learning of the very mathematical concepts under study (bossé et al., 2019). additionally, as is often the case with emergent bilingual children who are policed by “symbolic language borders” in their schooling (valdés, 2017), language separation is touted as the means to achieve the academic language needed for mathematics achievement. our work counters these misconceptions and offers two examples from one dual-language classroom showcasing the work of an experienced bilingual teacher who recognizes what other researchers have proposed as a deeper and more powerful way of understanding the language practices of bilingual children, called d krause, adams-corral, language practices in the elementary & maldonado rodríguez bilingual mathematics classroom journal of urban mathematics education vol. 15, no. 2 10 translanguaging (garcía, 2009), defined here as a pedagogical practice that facilitates students’ expression of their understanding using their own language practices. in this classroom, translanguaging afforded learning opportunities that would not have arisen if the children had needed to limit their language practices to conform to an english-only model (maldonado rodríguez et al., 2020). importantly, this classroom’s mathematics instruction was based on children’s mathematical thinking (cmt; carpenter et al., 1989). children’s mathematical thinking-based instruction includes a body of teaching moves designed to elicit, as directly as possible, mathematical concepts as represented in a given child’s own mind. central to cmt-based instruction is that the teacher avoid imposing their own understanding of those same concepts, hint at correctness, or impart any other preconceptions that might unfaithfully render the child’s own understanding. teachers employing this methodology consequently design instruction by building on these elicitations of the child’s own understanding, rather than centering the teacher’s personal understanding in instruction. in this classroom, the children’s mathematical thinking was therefore expressed without limits or constraints towards an expected solution strategy or answer. children were encouraged to use a range of strategies to express their mathematical understanding, even when their ideas were incomplete or apparently not yet correct, because the underlying mathematical relationships within these strategies could be used as building blocks for extending everyone’s understanding (celedón-pattichis & turner, 2012; turner & celedón-pattichis, 2011). with these two tenets in mind—using the strategy of translanguaging and employing cmt-based instruction—we will draw on the literature and the examples from the classroom to propose a rethinking of the strict separation of languages during mathematics teaching. the two examples we present in this article seek to answer the question, “how might instructional practices displace the focus on monolingualism in the mathematics classroom to emphasize children’s mathematical ideas?” few examples exist as points of reference where proper use of standardized language is subordinated to the task of eliciting children’s unfiltered expressions of their mathematical understanding (valencia mazzanti & allexsaht-snider, 2018). by deemphasizing linguistic form, and even language choice, and allowing a freer bilingual mode of discourse, the resulting environment facilitates the teachers’ efforts to capture children’s ideas in their entirety before the strictures of “proper” language use have the chance to impede their expressions in mid-flow. for instance, in a study with a group of 5and 6-year-old spanish-english bilingual latinx children, valencia mazzanti and allexsaht-snider (2018) identified how the different representations (e.g., phonological and orthographic) used by the latinx children in the study provided avenues for learning that supported the children as they explored quantities through representations in different languages. through this work, the children developed an awareness of how numbers, words, and sequences are used in counting. krause, adams-corral, language practices in the elementary & maldonado rodríguez bilingual mathematics classroom journal of urban mathematics education vol. 15, no. 2 11 authors’ positionality an explanation to the readers of who we are will add context to the dialogues we analyzed, reflected on, and acted upon in this article. we are three spanish-english speaking latina educators. maldonado rodríguez and krause (henceforth mr&k) conducted all interviews for this study and collected all data from ms. adams’s classrooms. krause conducted the interview with hugo presented here. ms. adams, the teacher in this study, is second author. our work together goes back more than a decade, beginning when ms. adams was a novice teacher. since then, mr&k have learned with and from her. the three of us have also collaborated on different research projects with a focus on children’s mathematical thinking. our work together has always centered on foregrounding the voice and richness of bilingual children’s ideas in the mathematics classroom. much like the children that we worked with in this study, the three of us have existed as border crossers throughout our learning and academic experiences (anzaldúa, 2012). we view our research as relational (patel, 2016) and thus position the children not as lacking or missing something, but rather as people from whom we have something to learn. as a teacher, ms. adams has taught in defiance of language separation policies and emphasized bilingualism as the norm (garcía et al., 2016). in her classrooms, she offers opportunities to disrupt deficit narratives about emergent bilingual children’s mathematical capabilities. bilingualism and bilingual practices: review of terminology recent estimates suggest that at least half of the world’s population is bilingual (grosjean, 2010). bilingualism is so widespread throughout the world that it can arguably be considered more normal than monolingualism. in the united states alone, an estimated 60 million people (21% of the population) age 5 and over spoke a language other than english at home in 2011 (ryan, 2013). according to a report from the u.s. census bureau (2019), 61.4% of people who speak a language other than english in the united states speak spanish, amounting to roughly 42 million people. according to a report from the national center for education statistics (2018), in the fall of 2015, spanish was the home language of 3.7 million english language learners. this number represented 77.1% of all english language learners in k–12 classrooms. the historical context of u.s. education policy around language reveals a deficit-oriented stance towards this sizeable group of children, with a traditional overemphasis on mastery of english and no acknowledgement of the resources they possess (gándara & orfield, 2010; macdonald, 2004). in a recounting of historical events, hickey (2016) described policies from as early as 1754 where native krause, adams-corral, language practices in the elementary & maldonado rodríguez bilingual mathematics classroom journal of urban mathematics education vol. 15, no. 2 12 american youth were sent to schools “to learn english and to be stripped of their indigenous home languages” (p.16). in this same retelling of events, hickey noted that the bilingual education act of 1968 was a key element in recognizing the needs of children whose first language was not english. however, there is still a great deal of variation with regard to the meaning of bilingual education (hakuta, 1987). bilingual education was defined by the bilingual education act as follows: a program of instruction, designed for children of limited english proficiency in elementary or secondary schools, in which, with respect to the years of study to which the program is applicable . . . there is instruction given in, and study of, english, and, to the extent necessary to allow a child to achieve competence in the english language [emphasis added], the native language of the child of limited english proficiency, and such instruction is given with appreciation for the cultural heritage of such children, and of other children in american society, and with respect to elementary and secondary school instruction, such instruction shall, to the extent necessary, be in all courses or subjects of study which will allow a child to progress effectively through the educational system. (cubillos, 1988, p.10) the italicized phrase is critical. in essence, it says that other languages will be permitted only insofar as they support the learning of english. later, in 2002, when no child left behind (nclb) passed, it had a significant impact on the bilingual education act and bilingual education in the united states, mainly because of its emphasis on high-stakes testing. after nclb, the bilingual education act was renamed the english language acquisition, language enhancement, and academic achievement act (nclb, 2002). though the act still permits state and local educators to choose instructional methods and to employ bilingual methods, the accountability requirements further underline the fact that the primary objective of public educators continues to be english acquisition, and not bilingualism. see for instance the first purpose of the english language acquisition, language enhancement, and academic achievement act outlined under nclb (2002): (1) to help ensure that children who are limited english proficient, including immigrant children and youth, attain english proficiency, develop high levels of academic attainment in english, and meet the same challenging state academic content and student academic achievement standards as all children are expected to meet. (p. 115) in the 2016–2017 school year, texas, where this study was conducted, reported having 5,343,834 students, of which 18.8% were enrolled in bilingual and english language learning programs (swaby, 2017). at the same time, the texas education code requires the following for bilingual education and english as a second language program content and method of instruction: krause, adams-corral, language practices in the elementary & maldonado rodríguez bilingual mathematics classroom journal of urban mathematics education vol. 15, no. 2 13 (a) a bilingual education program established by a school district shall be a full-time program of dual-language instruction that provides for learning basic skills in the primary language of the students enrolled in the program and for carefully structured and sequenced mastery of english language skills. a program of instruction in english as a second language established by a school district shall be a program of intensive instruction in english from teachers trained in recognizing and dealing with language differences. (texas education agency, 2017, p. 112) it is important to point out that these goals for bilingual education do not mention content area learning. rather, the goals make language acquisition and basic skills the entire focus of bilingual children’s learning, inherently limiting their ability to develop critical content area skills. in addition to this, school districts nationwide face major challenges to implementing bilingual programs on a large scale; among the challenges are the current politics around bilingualism and the shortage of qualified bilingual teachers (gándara & escamilla, 2017). given the variety of definitions and terminology used in and around language and language practices in bilingual classrooms, we found it important that we define the terminology we use in the analysis of the work presented here. the terms bilingual and bilingualism in the present work, we use the following definition of what it means to be bilingual: “bilinguals are those who use two or more languages (or dialects) in their everyday lives” (grosjean, 2010, p. 4). it is worth noting that this definition (1) focuses on use, not fluency; (2) includes dialects; and (3) includes two or more named languages. we opted for this definition as it is ample, it allows for dialects to be included (which is important given the variations in the spanish and english language based on the culture of the speaker), and it deemphasizes the focus on fluency (which pertains to a point we will convey regarding mathematics instruction). we also use this definition because it allows us to talk more amply about the population of children we serve in the united states while also acknowledging that a more restrictive definition of bilingualism could apply to an appreciable portion of children in schools across the united states. the terms code-switching and borrowing two practices often observed in the speech of bilingual individuals are codeswitching and borrowing. code-switching is the “alternate use of two languages” (grosjean, 2010, p. 51), while borrowing is “the integration of one language into another” (grosjean, 2010, p. 58). there are two basic types of borrowing. the most common is when the speaker uses a loanword, that is, the speaker takes a word from one language and uses it with its original meaning in the context of speaking another krause, adams-corral, language practices in the elementary & maldonado rodríguez bilingual mathematics classroom journal of urban mathematics education vol. 15, no. 2 14 language, perhaps adapting its sounds and forms slightly to conform to the native style of the language being spoken. take for example the word “pocicle” in spanish (“popsicle” in english). the spanish word is used to represent exactly the same dessert as the english term, though the second “p” is lost in the spanish rendering because of the rarity of the sound combination “ps” in spanish. by contrast, the borrowing known as a loanshift takes a word in the language being spoken but uses it with a meaning that more properly approximates the sense of a clearly related word in another language. an example of loanshifting is the word “esmoquin” in spanish, which means “a tuxedo or dinner jacket.” this is a borrowing of the english word “smoking” into spanish. the sense of the borrowing derives from the english phrase “smoking jacket” (a jacket worn only while smoking) and refers to a jacket’s function. the loanshifting of esmoquin within spanish expanded the term to refer to jackets of a similar fashion so that it now refers to a particular jacket’s style rather than its function. when bilingual individuals code-switch, they are speaking in one externally identifiable language, say english, and for a moment shift to another language, such as spanish (e.g., “i went upstairs—sólo por un momentico—to check on the baby”). grosjean (2010) highlighted several negative attitudes encountered with respect to code-switching. for instance, some monolingual individuals reported that codeswitching can create “an unpleasant mixture of languages” or lead to a form of “semilingualism,” suggesting that a bilingual’s knowledge of a particular named language might be rendered somehow deficient by virtue of regular recourse to an alternate language (grosjean, 2010, p. 52). however, grosjean also explained some of the motivations for bilingual individuals to code-switch. in particular, a bilingual speaker may feel that some concepts or notions are better, more easily, or more economically expressed in another language. if the interlocutor also speaks that other language, then the bilingual speaker might employ that other language to express that particular concept in a way that more faithfully represents the intended sense. grosjean further pointed out how code-switching can fill a perceived linguistic need, for example, where the language currently spoken lacks certain technical terminology or a robust vocabulary for a particular body of understanding. for instance, the recent research project decolonise science is translating more than 180 scientific papers into african languages such as isizulu, northern sotho, hausa, yoruba, luganda and amharic (masakhane, n.d.; wild, 2021). in part, the motivation for the study derives from the paucity of native scientific terminology in these languages, which can often lack clear, specific, or conventional terms for originally imported concepts such as dinosaurs, viruses, bacteria, etc. these linguistic lacunae can have major consequences for teaching and learning in local languages within africa (masakhane, n.d.; wild, 2021). krause, adams-corral, language practices in the elementary & maldonado rodríguez bilingual mathematics classroom journal of urban mathematics education vol. 15, no. 2 15 both code-switching and borrowing rely on a perspective of bilingual language use that emphasizes practices that appear to cross the boundaries between named languages (garcía et al., 2016). however, the internal perspective of a bilingual speaker is just as important. as speakers are developing their use and understanding of language, they encounter gaps—be they of the language as such or of their current learning of it—and must navigate across fuzzy boundaries between named languages to meet the needs of particular communicative contexts. as they communicate, they are constantly drawing on all their linguistic resources in ways that navigate across the fuzzy boundaries between named languages while attending to the needs of particular communicative contexts. when learning language, sometimes these navigations lead to challenges and innovations. the term translanguaging translanguaging is the flexible use of linguistic resources across various everyday contexts (garcía & wei, 2014). otheguy et al. (2015) defined translanguaging as “the deployment of a speaker’s full linguistic repertoire without regard for watchful adherence to the socially and politically defined boundaries of named (and usually national and state) languages” (p. 283). naturally, this intersects with the notion of code-switching. as mentioned above, grosjean (2010) cited two common reasons for code-switching: 1) ease of expressing an idea in another language, or 2) meeting a linguistic need (i.e., a perceived deficit). it is worth noting, however, that grosjean’s description asserts no particular model for how language is represented in the bilingual mind. perhaps the bilingual holds in mind two languages kept completely distinct, or perhaps the bilingual holds in mind a unified linguistic apparatus of which named languages appear merely as facets or aspects. some researchers, like garcía and wei (2014), assert that when bilingual individuals engage in translanguaging, they are not alternating between two languages but rather are utilizing features from their single, encompassing linguistic system. in this paper, we work with a narrower concept of translanguaging: we view translanguaging as a pedagogical practice that allows code-switching when this facilitates a focus on, and an expression of, the concepts being learned in the classroom. that is, translanguaging emphasizes content while deemphasizing strict adherence to a target language, allowing learners to express their grasp of ideas by the linguistic means they find most suitable in the moment. translanguaging, from this perspective, need not assert a particular model of the cognitive representation of language within the bilingual mind (and thereby remain relevant even should neuroscience and linguistics refine or update their models of how language works in the brain). rather, it emerges as a powerful pedagogical technique that allows bilingual students to be bilingual when classroom discourse is focused on topics other than language. krause, adams-corral, language practices in the elementary & maldonado rodríguez bilingual mathematics classroom journal of urban mathematics education vol. 15, no. 2 16 the rising prevalence of the term translanguaging in the field of education seems to derive in part from a deemphasis on the particular systematics of the linguistic data characterizing certain bilingual practices. in fact, use of the term is a response to an increased emphasis on the social constructs that spurred these practices, as well as on the world view that these practices derive from and in turn facilitate or enable. translanguaging challenges and reframes. we, as educators, must attend to language as a social construct and the implications this social role can have in mathematics and in bilingual classrooms. situating in the literature teaching mathematics in bilingual settings research focusing on the teaching and learning of mathematics in the bilingual classroom has identified three common tensions for teaching: (1) the tension between using formal or informal language, (2) the tension between using children’s home languages and the language of the school, and (3) the tension between teaching mathematics and teaching language (adler, 2002). prior to adler’s (2002) study, moschkovich (1999) and khisty (1995) investigated teachers’ practices in promoting bilingual children’s participation and engagement during mathematical discussion. khisty (1995) found that those teachers who appeared more effective tended to pay attention to the interaction between language and mathematics content. when the educators in this study focused on language, they highlighted specific vocabulary that arises in both spanish and english that could cause ambiguity, hindering children’s understanding of mathematics concepts. in contrast, moschkovich (1999) found in her study that teachers who appeared more effective focused mostly on mathematics and placed less weight on correcting language or teaching vocabulary. most researchers now recognize that focusing on language when teaching mathematics does not simply mean starting with vocabulary. learning and doing mathematics includes mathematical ways of talking, arguing, and explaining (barwell, 2009); these are complicated topics in a single language, all the more so in bilingual settings. the contravening tendencies described in the studies above, therefore, might come as little surprise given the complexity of the teaching practices involved. although research studies on mathematics teaching, mathematical attainment, and bilingualism are fairly common and have reported some connection between proficiency in a second language and mathematical attainment (bialystok, 2018; henry et al., 2014; lager, 2010; martiniello, 2010; shannon & milian, 2002), they are far from straightforward. it remains unclear whether differences in mathematical attainment relate to language, culture, economic or social factors, or a combination of all of these (martiniello, 2010). krause, adams-corral, language practices in the elementary & maldonado rodríguez bilingual mathematics classroom journal of urban mathematics education vol. 15, no. 2 17 other studies hint at a positive relationship between bilingualism and mathematical attainment. for instance, clarkson (2007) has suggested that bilingualism allows children to think more efficiently when reasoning about mathematics. he has found evidence that young bilingual children, relative to their monolingual counterparts, show greater cognitive flexibility and creativity, as well as improved problemsolving abilities in mathematics. moschkovich (2007) has also suggested that bilingual children have an enhanced capacity to reason about mathematics problems. in particular, bilingual children can effectively identify the information relevant for solving problems and ignore less important information. research on translanguaging, and in particular research on translanguaging in the mathematics classroom, has further clarified the relationship between bilingualism and mathematical understanding. this derives from research investigating how rethinking language practices in the mathematics classroom can support students’ full language practices while also providing a foundation for understanding mathematics (dinapoli & morales, 2020; maldonado rodríguez et al., 2020). for instance, maldonado rodríguez et al. (2020) presented a study from a bilingual classroom where a child provided a wrong answer for a particular mathematics problem. importantly, the child provided the answer in spanish. the particular episode was used by the teacher to not only open the space for mathematical discussion to understand why the wrong answer would make sense mathematically, but also the teacher used this answer to build on mathematical language together. the authors of the study argued that allowing the child to share his idea in spanish provided a space not only for the child but for the entire class to use language as a tool through which mathematics was understood. despite the positive results of this relationship between bilingualism and mathematical attainment, teachers continue to use language practices in instruction that cultivate dominant language practices, defined as “standardized ways of speaking, listening, reading, and writing that are often referred to as ‘standard american english’ or ‘academic english’” (martínez & martínez, 2019, p. 234). these practices are often used even among linguistically diverse learners, which frequently excludes children’s cultural and linguistic knowledge in favor of monolingual english instruction (martínez-álvarez, 2017). such tendencies introduce the possibility of children seeing their own culture and language as incorrect, inappropriate, and in need of remediation (kohli, 2014; paris, 2012). in addition, these standardized practices in the mathematics classroom tend to obscure bilingual children’s actual grasp of mathematics (barwell, 2009). language practices directed toward a standard can implicitly or explicitly downgrade other forms of mathematical expressions (barwell, 2009). for instance, by focusing on “correctness” of vocabulary or grammar use, a teacher might foreground linguistic styles and understanding precisely when understanding a mathematical idea should take the foreground, letting idiosyncrasies of linguistic krause, adams-corral, language practices in the elementary & maldonado rodríguez bilingual mathematics classroom journal of urban mathematics education vol. 15, no. 2 18 expression take a temporary back seat. naturally, such idiosyncrasies of expression might mask underlying confusion, which only further probing can elucidate, but the change in mindset itself is meaningful. in particular, a focus on such idiosyncrasies can be detrimental to children, and in extreme cases potentially mix with racialization, that is, with “the extension of racial meaning to a previously racially unclassified relationship, social practice, or group” (omi & winant, 2014, p. 111). rosa (2016) in fact argued that dominant perceptions of language can serve to delegitimize the language of racialized groups while furthering their racialization. because of this, we argue that in order to truly center children’s mathematical thinking, we must also take care to counter ossified notions of what it means to speak mathematically. translanguaging in teaching mathematics our work in the mathematics classroom is rooted in the long-standing research on children’s mathematical thinking (carpenter et al., 1989; fennema et al., 1996; jacobs et al., 2007) and situated in the translanguaging work of garcía and wei (2014) and otheguy et al. (2015). in the context of teaching mathematics using children’s mathematical thinking, teachers must have a deep awareness of what their students know or can intuit in order to support them as they connect their current mathematical understanding to concepts of standardized mathematics. research has linked teachers’ understanding of children’s mathematical thinking to productive changes in teachers’ knowledge and beliefs, classroom practices, and student learning (carpenter et al., 1996; carpenter et al., 1989; fennema et al., 1996). in these studies, researchers have paid particular attention to the development of children’s problem-solving strategies, common misconceptions, as well as frameworks for understanding problem structures. other studies built upon and further established the effectiveness of instruction informed by knowledge of children’s mathematical thinking for children from diverse racial, socioeconomic, cultural, and linguistic backgrounds (adams, 2018; dominguez, 2011; dominguez & adams, 2013; turner & celedón-pattichis, 2011; villaseñor & kepner, 1993). we connect the translanguaging practices of building on children’s ways of knowing with this main tenet of research on children’s mathematical thinking because it can leverage children’s intuitive and informal ideas as the basis for instruction. during mathematics instruction, this means that children need to express their ideas in ways that make sense to them. when bilingual children are forced to accommodate their language to english-only instruction, they run a greater risk of misrepresenting their mathematical thoughts because they must reformulate their ideas in order to share them (krause & colegrove, 2020; moschkovich, 1999, 2007, 2012, 2015; turner & celedón-pattichis, 2011; turner et al., 2013). this could contribute to a teacher’s inability to access a true representation of what the child is thinking, which may in turn allow potentially brilliant ideas to be interpreted as incorrect or krause, adams-corral, language practices in the elementary & maldonado rodríguez bilingual mathematics classroom journal of urban mathematics education vol. 15, no. 2 19 incomplete. a critical aspect of being an effective mathematics teacher for linguistically diverse children is developing knowledge, dispositions, and practices that support building on children’s mathematical thinking, as well as on their cultural, linguistic, and community-based knowledge (adams, 2018; dominguez, 2011; dominguez & adams, 2013; turner & celedón-pattichis, 2011; villaseñor & kepner, 1993). representing ideas through language requires intellectual work. transforming ideas in order for them to be expressed in english only—when they were not initially conceived as such—forces a step that could potentially misrepresent the ideas themselves. through this lens of bilingual mathematics instruction, we see children as agents capable of expanding their sense of what they know and can do mathematically. children’s mathematical thinking ideally, when students learn mathematics, they learn ways of thinking that go beyond a collection of disconnected procedures for carrying out calculations. within this context, children learn how to generate mathematical ideas, how to express these ideas (in any way that makes sense to them), and how to explain these ideas and those of others (carpenter et al., 2015; franke et al., 2001). more than three decades of research on children’s mathematical thinking has shown that elementary school children are capable of engaging in this type of mathematical learning, but often they are not given the opportunity to do so (campbell et al., 1998; carpenter et al., 1996; carpenter et al., 2015; carpenter et al., 1989; empson, 2014; empson et al., 2020; empson et al., 2006; empson & levi, 2011; franke et al., 2001; jacobs et al., 2019; jacobs et al., 2007). when children are invited to solve problems on the basis of what makes sense to them, they use a variety of informal strategies driven by their experiences and the world around them. these strategies have been well documented by research (empson et al., 2020; empson & levi, 2011). for example, when children start learning fractions, they tend to partition items into pieces, repeatedly halving and then distributing the resulting pieces. as a result, halves and fourths tend to be familiar pieces for children. additionally, in these early strategies children tend to distribute wholes without considering the number of people sharing (hackenberg & lee, 2015). similar strategies for solving problems with whole numbers have been documented in the work of franke et al. (2001), jacobs and ambrose (2008), and jacobs et al. (2007), among others. children can at times also use procedures and conventions for solving problems. what is important is to ensure during mathematics instruction that children use such procedures and conventions because their understanding of mathematics has reached a level of fluency in which such operations are routine and they do not need to decompose their strategies into simpler computations. the notion of fluency here is important, both mathematically and linguistically. krause, adams-corral, language practices in the elementary & maldonado rodríguez bilingual mathematics classroom journal of urban mathematics education vol. 15, no. 2 20 methods our focus in this study was on describing how one bilingual elementary teacher emphasized bilingual children’s mathematical ideas by rejecting district policies that required mathematics to be taught entirely in english. by decentering the policies that sought to standardize her and her students’ language practices, all classroom community members were able to freely express their mathematical ideas. data were drawn from interviews with the students in this teacher’s classroom and observations of this teacher’s mathematics instruction. setting and participants the work we present in this paper comes from the second author’s classroom when she was teaching second grade. in the present work, we identify how she established a series of tasks and routines that created the space necessary for constant interaction among herself and her students. the bilingual classroom of ms. adams. the tasks and routines in ms. adams’s classroom entailed an expectation of collaboration and discussion that was centered on the children’s mathematical ideas. an example of these activities is the instructional practice of circulating, or monitoring children’s ideas in preparation for wholegroup instruction (stein et al., 2008). researchers such as jacobs and empson (2016) have suggested that teachers devote a greater share of their instructional time to circulating, because when they do so, they are provided opportunities to respond to their students’ mathematical thinking. furthermore, in order for children to share their mathematical ideas, they enact their linguistic repertoire (gutiérrez & rogoff, 2003). in ms. adams’s class, this repertoire was not restricted to the use of a single language. while circulating, ms. adams’s own linguistic repertoire had to respond flexibly to her students’ repertoire during conversations around problem solving. in ms. adams’s class, the children could expect that they would be permitted to share their mathematical ideas in whatever form and language they emerge, whether in english, in spanish, or in both. contrary to the conventional emphasis on the mastery of english and the general lack of interest in the particular knowledge and resources bilingual children possess (gándara et al., 2010), ms. adams embraced dynamic language practices in her classroom and was mindful to foreground the mathematical ideas of her students. we focused on the teaching of ms. adams in a bilingual class with 23 children. this classroom in a major texas city served children with a variety of language resources and latinx backgrounds. one child identified as biracial, while the other 22 identified as latinx with ties to mexico, puerto rico, el salvador, and honduras. bilingualism was the norm in this classroom, with most children coming from bilingual home environments, while one child spoke an additional language, otomí. krause, adams-corral, language practices in the elementary & maldonado rodríguez bilingual mathematics classroom journal of urban mathematics education vol. 15, no. 2 21 two of the authors, mr&k, followed ms. adams and the same group of children for two consecutive years, from second to third grade. ms. adams approached her role as an educator with the goal of empowering children to use their full potential as bilingual citizens of the world. at the time of the study, ms. adams had been an elementary school teacher for 7 years, teaching in bilingual classrooms from grades 2 to 5. during all her years of teaching, she has been connected with the exploration of children’s mathematical thinking. ms. adams possesses extensive knowledge of children’s mathematical thinking; focuses on children’s thinking in all her teaching, not just mathematics; and helps other teachers learn about children’s thinking and its role in instruction (adams, 2018; adams & busey, 2017). in her class, children’s thinking is valued and visible during problem solving. rather than demonstrating strategies herself, ms. adams encourages children to generate and use strategies that make sense to them, and she routinely elicits and builds on their ideas. children thus learn from one another, because they are expected to explain and justify each other’s thinking as a part of what it means to do and know mathematics in ms. adams’s classroom. data sources the work we present in this article comes from one-on-one interviews conducted by the first and third authors and an excerpt from a circulating portion of ms. adams’s class when children were working independently. the interviews were conducted using guidelines for clinical interviews in which the goal of the interview is not to guide a child to the correct answer, but to ensure that the interviewer understands the mathematical reasoning of the child’s strategy (ginsburg, 1997). questioning often focused on particular mathematical relationships noted in children’s strategies (jacobs & empson, 2016) or on extending questions designed to push children’s mathematical thinking further (jacobs & ambrose, 2008). both sources of data were collected near the end of the school year in the second year we worked with ms. adams. all interviews and lessons were video recorded, and field notes were taken. interviews. the interviews took place over the course of a single week. our plan was to interview every child in the classroom, but time constraints and occasional student absences prevented us from doing so. the first and third authors conducted 11 one-on-one interviews, each lasting about 30 minutes. we took turns when conducting the interviews. sometimes the first author conducted the interview, and the third author was responsible for the video camera; at other times the roles were reversed. each child solved at least one story problem and at least one equation. some of the children solved equal sharing problems, and some solved multiple groups problems. the equations included operations with whole and rational numbers. the krause, adams-corral, language practices in the elementary & maldonado rodríguez bilingual mathematics classroom journal of urban mathematics education vol. 15, no. 2 22 appendix includes a sample of some of the problems and equations the children were asked to solve during the interviews. the interview problem we describe in the present study is a multiple groups problem. a typical curriculum traditionally introduces equivalent fractions, followed by addition and subtraction of fractions (empson & levi, 2011). we depart from this sequence and follow the recommendations of empson and levi (2011), starting with a focus on children learning first how to create and name fractional quantities with equal sharing problems and then continuing with multiple groups problems. empson and levi defined multiple groups problems as word problems involving a whole number of equal groups of fractional amounts. this problem type allows children to engage in making connections foundational to understanding equivalency and operations (i.e., addition, subtraction, multiplication, and division) with fractions. individual problem solving—circulating. before the lessons we observed were taught by ms. adams, the three of us met and discussed the tasks ms. adams was going to use for each lesson. we came into ms. adams’s classroom understanding why she selected the tasks and what goals she had set for instruction. we simply followed her with the video camera and recorded the flow of the lessons. we focused on the circulating portion of the lesson because it offered a view into the individual and small group conversations that occurred before whole group discussion of strategies. for the circulating component of the lesson we analyze here, students were solving the following equation: 65 38 = ____. both of the mathematics problems presented in this article are considered highlevel cognitive demand tasks. in a high-level cognitive demand task, children’s attention is placed on making connections, analyzing information, and drawing conclusions (van de walle et al., 2013). high-level cognitive demand tasks are nonroutine tasks that engage children in productive struggle and challenge them to make connections to concepts and other relevant knowledge (van de walle et al., 2013). analysis our analysis of the two examples we present here involved not only a consideration of what the teacher and researchers said and did but also of the situation, including what the children said and did. we looked for conceptual breaks in the conversation to determine our unit of analysis (jacobs & morita, 2002). sometimes the unit consisted of a teacher’s single comment or question, and at other times it included a linked sequence of comments and questions because teachers often need to persist to support or extend children’s thinking. we started by transcribing the children’s interviews. then we developed a provisional list of codes. this list of codes came mainly from our proposed two tenets above, namely use of translanguaging and employing cmt-based instruction. for instance, an initial code was “language use.” here we were identifying all instances krause, adams-corral, language practices in the elementary & maldonado rodríguez bilingual mathematics classroom journal of urban mathematics education vol. 15, no. 2 23 where children used english or spanish to express their ideas. another was “use borrowing”. here we were trying to identify instances in which children decided to borrow a word from either english or spanish. in this list we also included other general aspects of how the child expressed his or her ideas, for example, “used notation to represent his or her mathematical thinking.” once we had completed the list of initial codes, we used a randomly selected sample of four child interviews to code. two of the authors, mr&k, double-coded this sample for reliability. after this round of coding, we shortened our list and identified a second list of possible codes, following the coding methodology described by saldaña (2015). then we randomly selected another set of interviews. this second sample was also coded by mr&k, who then met to discuss and resolve discrepancies. through this process, mr&k developed a list of codes that were related to one another in a coherent way and aligned with the research question. for example, we coded for translanguaging by individually identifying instances of code-switching and borrowing, because these can serve as evidence of the presence of translanguaging. we used observable instances of code-switching and borrowing to serve as externally visible evidence that children and teachers were free to deploy their linguistic repertoire strategically. we also coded as translanguaging instances where speakers might remain within the boundaries of one named language but showed an inventiveness to meet their communicative needs in one language though influenced by their knowledge of another language. this is evident in an example from the interview we analyze in this paper. these codes allow us to present to the reader examples from the mathematics classroom linked to what is being defined in the literature as the language practices of bilingual individuals. at the same time, we needed to study how these practices interplayed with the instructional decisions made by ms. adams when teaching mathematics. we completed all coding processes by hand. however, we used maxqda 2020, a software package for qualitative and mixed methods research analysis (verbi software, 2020), to watch the videos and add analytic memos during the first phase of coding. after finalizing the coding scheme, we randomly selected the interview with hugo, an 8-year-old mexican american boy in ms. adams’s classroom, for the purpose of the work we present here and separately coded it. through his interview, we were able to identify and distinguish the language practices of bilingual students (i.e., borrowing, code-switching, and translanguaging) described above at the same time that we were able to identify the mathematical ideas expressed by hugo. our purpose was to highlight the mathematical content in the conversations and the mathematical understanding represented through the connections the child makes when these practices occur. we additionally randomly selected and coded an interaction between ms. adams and gabriel, another bilingual student of ms. adams, during the circulating krause, adams-corral, language practices in the elementary & maldonado rodríguez bilingual mathematics classroom journal of urban mathematics education vol. 15, no. 2 24 portion of a lesson, identifying and analyzing similar behaviors and practices presented by hugo. in these conversations, we tracked the linguistic practices described above with links to specific examples so that the research team could discuss the appropriateness of these categorizations. these categorizations were brought to the teacher to check their appropriateness. findings the following section presents our analysis of the one-on-one interview with hugo and circulating time with ms. adams. in our analysis of hugo’s language choices during the one-on-one interview, we reveal the complex process of conveying mathematical ideas in bilingual contexts. at the same time, we see that there are at least two levers which we can use to manage the complexity effectively and efficiently in the moment: one is the lever of bilingualism itself, where a bilingual teacher will more likely follow a child’s bilingual expressions in real time without a great need for probing and rephrasing; the other is the willingness to table issues of language while trying to understand the content being expressed. in analyzing ms. adams’s circulating time in the classroom, there are several instances that require pausing and paying close attention to the mathematical ideas being discussed (e.g., the ideas behind regrouping or balancing an equation). below we attempt to unpack them. hugo’s interview hugo, a mexican american child from a bilingual home who is not labeled an english language learner, was given the multiple groups problem described in figure 1. the problem was written in both languages when presented to hugo. un chef está preparando 6 ensaladas de frutas. cada ensalada usa ½ manzana. ¿cuántas manzanas necesitará el chef para hacer las ensaladas de frutas? [a chef is making 6 fruit salads. each salad has ½ apple. how many apples would he need to make the fruit salads?] figure 1. multiple groups problem krause, adams-corral, language practices in the elementary & maldonado rodríguez bilingual mathematics classroom journal of urban mathematics education vol. 15, no. 2 25 figure 2. hugo’s work for 6 fruit salads each containing ½ an apple. hugo’s drawn solution is shown in figure 2. from our initial interpretation of hugo’s work, we regarded his notation of the solution as incorrect. when we asked him to explain what 3/3 meant in his solution, he responded as follows: hugo: este [apuntando al 3 en el numerador] son las manzanas. oh no! esto está mal. es dos-tres porque hay 3 manzanas y están esplitadas en doses. [this one [pointing at the 3 in the numerator] is the apples. oh no! this is incorrect. it is two-three because there are 3 apples and they are esplitadas in twos.] if we adopt hugo’s perspective, the notation makes sense. the 2 in the numerator represents the pieces into which each apple is split. the 3 in the denominator represents the total number of apples needed. in his explanation, we see that he is borrowing from english to say “e-split-adas” (literally “split-ed,” with “split” adopted from english and modified with the initial “e-” to fit the phonology of spanish), which makes sense to us in the context of his explanation. another noticeable feature of his explanation is his use of “doses” (the plural of “dos”). as hugo was providing his explanation, we noted these features in his speech and continued working with him: krause: yo vi que partiste las manzanas y colocaste un pedazo de manzana en cada ensalada. ¿por qué partiste las manzanas así? hugo: porque hay 6 de éstas [apuntando a las ensaladas] y nada más necesitas tres, porque las puedes cortar en un medio. y cada uno puede agarrar un medio de cada manzana. [krause: i saw you split the apples and add a piece of apple to each salad. why did you split the apples that way? hugo: because there are 6 of these [pointing at the salads] and you only need three, because you can cut them in a half. and each one can get a half of each apple.] krause, adams-corral, language practices in the elementary & maldonado rodríguez bilingual mathematics classroom journal of urban mathematics education vol. 15, no. 2 26 notice that the conversation with hugo focused on his strategy and not on his use of language. however, krause did not use hugo’s words, instead opting for “partir” to refer to the action hugo was referring to earlier as “esplitar.” later, hugo used “cortar” when referring to the same process. likewise, he employed “medio” instead of “doses.” in both cases, we can see that hugo’s use of language was not the result of lacking more vocabulary but rather a representation of his in-the-moment thinking. in this interaction between hugo and krause, translanguaging allowed krause to let go of expecting conventional fractional language and instead identify and understand hugo’s mathematical thinking. krause provided the space for hugo to freely express his thinking while simply responding by choosing different words in spanish. towards the end of the explanation, we notice that hugo decided to use “medio” rather than his initial choice of “doses.” both hugo and krause were interacting naturally, using the language that made sense to both of them for the topic at hand. as this went on, krause was able to see more and more of hugo’s understanding and hear more of what he knows. we chose to share the interview from hugo in this paper because in this article we invite both researchers and teachers to consider their own practices with bilingual students. students like hugo come from bilingual households but are not labeled english language learners. yet, as bilingual individuals, they accommodate their language choices to meet the context they are in (adams, 2015). that means our language choices can help shape the students’ own choices. as a result, this lends particular importance to how we respond to the features of their language repertoire. hugo knew the researcher interviewing him, watched her engage with his teacher, and had a sense of who she was, which influenced his choices to respond to her in spanish. he considered both that she spoke english and that she was speaking in spanish and so perhaps concluded that she preferred speaking in spanish. in other words, students will make language choices that accommodate us just as much as we make choices to accommodate them. gabriel’s individual problem solving (ms. adams’s circulating) children in ms. adams’s classroom were allowed to use any of the resources available in the classroom to solve the problems. in the example we share below, gabriel, a mexican american child from a bilingual home who is labeled an english language learner, was using a set of unifix cubes. the children were tasked with solving a subtraction problem using their own strategies. as the traditional subtraction algorithm—arranging differences vertically, subtracting in the ones position, borrowing from the tens, etc.—is not taught in this student-centered classroom, a variety of strategies was observed across the classroom. as he was solving the equation 65 38 = ___, ms. adams approached him and asked the following: krause, adams-corral, language practices in the elementary & maldonado rodríguez bilingual mathematics classroom journal of urban mathematics education vol. 15, no. 2 27 ms. adams: okay, gabriel, what’s up? gabriel: just making tens to make it to 60. ms. adams: okay, can i help you? how can i help you? can i make something for you? gabriel: you can make 20 tens and i can make some. you can make 10 tens and i can make 10 tens. … ms. adams: ¿tienes 65? [do you have 65?] gabriel: that makes no sense. ms. adams: what? gabriel: if it’s 65 minus 38, ‘cuz you’re taking away 38 and there’s only 5 in the 65, and then… and if you’re taking away 38 and then you, um, take away the 8 from the 30, then you take away 5, there’s gonna be 2 more in the 8. ms. adams: so, ¿lo que te está como atorando es la idea de quitarle 8 cuando sólo hay 5 en las unidades? [so, what is bothering you is the idea of taking away 8 when you only have 5 in the units?] gabriel: hmm… ms. adams: oh, okay. pues ¿cómo lo haríamos? ¿hay algún otro lugar donde le podríamos quitar? [oh, okay. so how can we do it? is there some other place where we can take it away from?] gabriel: [nods in agreement.] we note how ms. adams arrived with an attitude of support and allowed gabriel to lead the conversation. she was familiar with the children in the class and their ways of explaining their ideas, so she understood that when gabriel asked for “20 tens,” he was talking about making 20 with tens. as a result, ms. adams was able to infer gabriel’s thinking from the strategy he was using to solve the problem and did not focus on ensuring that he meant 20 and not 200. that is, ms. adams’s familiarity with gabriel’s linguistic habits allowed her to recognize “20 tens” as one of gabriel’s linguistic symbols for the correct mathematical concept at issue, regardless of the imprecision of the linguistic expression itself. this recognition allowed ms. adams to postpone linguistic interruptions (waiting until the concept is grasped before focusing on its linguistic form) in order not to derail gabriel’s mathematical train of thought. she was also able to use that inference to identify the part of the equation that he was having difficulty with. ms. adams intentionally created the space for gabriel to talk while she carefully listened to him. she articulated the issue at hand and provided scaffolds for his next steps in problem solving. she was also careful not to correct gabriel’s language around making tens early on in their interaction, an interruption that would have taken the focus away krause, adams-corral, language practices in the elementary & maldonado rodríguez bilingual mathematics classroom journal of urban mathematics education vol. 15, no. 2 28 from his strategy, and instead kept the focus on what he felt he needed in order to accomplish the mathematical task at hand. although she may embed conversations about what “20 tens” might really be during another day’s discussion, during this interaction her focus followed gabriel’s direction. another teaching move that we can highlight from this excerpt is the opportunity to change fluidly from one language to the other. while gabriel provided his explanation completely in english, ms. adams responded completely in spanish. taking on a more individualized approach, which shows knowledge of the children, their language preferences, and personalities, ms. adams decided how to respond based on what she knew about gabriel. during the coding process, ms. adams shared that she knew he would be able to understand if she questioned him in spanish, whereas other children in the class might have required a different approach. ms. adams made sure that gabriel was capable of comprehending the mathematical content they were discussing in both languages. she acknowledged and promoted gabriel’s bilingualism while providing a space that gave each child the right to choose a language to express their mathematical ideas. once gabriel recognized he could take 8 away from 10, ms. adams continued as follows: ms. adams: ah! okay, vamos a quitarle este 8. [okay, we’ll take away this 8.] gabriel: two, four, six, eight. ms. adams: okay, ¿qué nos quedó? [okay, what do we have left?] gabriel: ten, twenty, thirty, forty, fifty. fifty. fifty, fifty-five, fifty-six, fifty-seven. fifty-seven. ms. adams: okay, diez, veinte, treinta, cuarenta, cincuenta, cincuenta y uno, cincuenta y dos, cincuenta y tres, cincuenta y cuatro, cincuenta y cinco, cincuenta y seis, cincuenta y siete. so, lo que nosotros encontramos es que, para quitarle el ocho, tuvimos que entrarnos a uno de los dieces. ¿qué te faltó quitar? porque sólo le quitaste 8. [okay, 10, 20, 30, 40, 50, 51, 52, 53, 54, 55, 56, 57. so, what we found is that, to take away the 8, we had to enter one of the tens. what was left for you to take away? because you have only taken away 8.] gabriel: 30... m. adams: ¿ves una manera más fácil de quitar 30? [do you see an easy way to take away 30] gabriel: should i just take away these? [he points at a stack of 3 tens he had.] ms. adams: that sounds easier, right? gabriel: yeah. ms. adams: how many did you take away? gabriel: 30. the answer is twenty-seven. krause, adams-corral, language practices in the elementary & maldonado rodríguez bilingual mathematics classroom journal of urban mathematics education vol. 15, no. 2 29 ms. adams continued to support gabriel to make sure he understood how to subtract the units that were problematic for him. she also continued to provide the language support described up to this point. however, throughout the conversation gabriel was the one solving the problem, while ms. adams took advantage of her knowledge of gabriel’s work and abilities to ask questions that guided him through his own understanding. the mathematical details in gabriel’s strategy are important to note. he took away 8 from 10, and then took away 30 more from the 5 tens that remained. the flexibility in his thinking and his number sense (e.g., seeing 65 as a group of 6 tens and 5 ones) allowed him to see how to take away 8 from 10 rather than from 5, which he had reported as difficult for him. once this difficulty was noticed and addressed by ms. adams, she continued to talk to gabriel. in this excerpt, she proposed to go and look at the equation (62 38 = ___) they were solving earlier as a class. ms. adams: ¿hay alguna relación entre 24 y 27? so, tú me habías dicho: 62 más 3 es 65. ¿qué es 24, si le sumamos otros 3? [is there a relationship between 24 and 27? so, you had told me: 62 plus 3 is 65. what is 24, if we add 3?] gabriel: 27. ms. adams: 27. so, si hubiéramos ido con tu idea de hacer como una balanza entre lo que teníamos y lo que acabamos de hacer… tú dijiste, aquí le sumamos 3. pués aquí también le podemos sumar 3. y eso te dio la respuesta que sacaste, ¿no? [so, if we had gone with the idea of making a balance between what we had and what we just did… you said, here we add 3. and so here too we can add 3. and that gave you the answer you got, right?] ms. adams was able to identify an opportunity to go back to the initial ideas shared by gabriel. he had initially thought of 62 – 38 =___ and was thinking that if he added 3 to 62, he would get 65, but then he got stuck and could not continue. ms. adams saw the opportunity to extend his thinking, going back to his initial idea after he had solved the problem. in figure 3, we see how ms. adams used the idea of a scale to balance both sides of the equation gabriel was thinking about. ms. adams was attempting to help gabriel realize that the problem he actually solved, 62 – 38 =___, was nearly the problem he wanted to solve, 65 – 38 =___. this provides an example of how children can scaffold themselves into a solution to the original problem when given work they have done on a similar problem. figure 3. strategy shared by gabriel when solving the initial problem krause, adams-corral, language practices in the elementary & maldonado rodríguez bilingual mathematics classroom journal of urban mathematics education vol. 15, no. 2 30 as noted earlier, gabriel was labeled an english language learner. just as hugo, gabriel came from a bilingual home. ms. adams knew that his family chose a dual language program because they wanted gabriel to maintain and grow both languages and that he comfortably interacted in both throughout the day. translanguaging pedagogy (garcía & kleifgen, 2010) often involves alternating between input in one named language and output in another. ms. adams had been reading about and attempting to include these pedagogical practices in her bilingual teaching repertoire. so, when gabriel responded in english without hesitation and ms. adams continued to speak only in spanish, this resulted in a classic alternation between named languages across input and output. discussion this work seeks to foreground examples from a dual language classroom that allowed us to identify key aspects of how bilingual individuals interact in the context of teaching and learning mathematics in the elementary classroom. these examples not only allowed us specifically to identify practices of bilingual individuals described in bilingual research but also helped us notice two main aspects of how a bilingual teacher teaches mathematics in the bilingual classroom while minimizing the tensions described by adler (2002) and commonly found in previous research. specifically, we see here how the teacher foments bilingualism in a way that accepts expressions of mathematical understanding regardless of their language of expression. at the same time, she foregrounds mathematics in the instant a student expresses understanding, awaiting another moment to model other ways of expressing such understanding in one or the other of the languages of the classroom. although we cannot use these two aspects to generalize about bilingual classroom practices, we can use them as a point of departure for promoting future research that helps the field to make spaces for teaching and learning content without sacrificing bilingualism. below, we discuss how ms. adams allowed for a freer bilingual mode of discourse, language choice, and deemphasis of linguistic form as we attempt to describe a bilingual mathematics classroom that displaces monolingualism and emphasizes children’s mathematical ideas. children have uninhibited conversations with the teacher the present work focused on a part of a class lesson in which the teacher circulated and engaged in one-on-one conversations with children during problem solving. in our examples, ms. adams engaged in these conversations by responding to individual children’s mathematical thinking and used them as a platform for developing an even deeper understanding of the mathematical concepts. several points in the circulation portion of the class must be analyzed in slow motion. for instance, krause, adams-corral, language practices in the elementary & maldonado rodríguez bilingual mathematics classroom journal of urban mathematics education vol. 15, no. 2 31 ms. adams’s knowledge of the children in the class, the knowledge provided by her bilingualism, and her ability to listen carefully to the children’s ideas facilitated uninterrupted communication with gabriel, even when his initial response of “20 tens” was “mathematically” not what he needed to use for solving the problem. we cannot of course be certain what would have been the outcome of stopping gabriel, correcting him, and moving on with the conversation. however, we suspect that by not stopping him and by allowing the conversation to continue, ms. adams deemphasized linguistic form in order to create a space for a fluid sharing of mathematical ideas. ms. adams could then use these ideas to teach the target content of the lesson. we also surmise that this deemphasizing of linguistic form creates a freer bilingual mode of discourse. for instance, we can highlight the fluidity with which both teacher and child communicate in two languages as if they were only one, which aligns with what garcía and kleifgen (2010) have defined as dynamic bilingualism. the mathematical ideas were not inhibited and bilingualism was promoted. at the same time, we can identify in this excerpt that these interactions have the characteristics of a bilingual classroom, where the children will naturally switch languages and the teacher is able to attend to this constant variability in usage. creating these opportunities and making them widely available to children in the mathematics classroom have the potential to promote children’s acquisition of english while maintaining, and hopefully improving, their abilities in any other languages they bring to the classroom (garcía, 2009). at the same time, they continue to build on their mathematics understanding. translanguaging, in this case, allowed for children’s natural bilingualism to further their mathematical learning. the benefit of the approach demonstrated here lies in its ability to support their use of both their languages and to help instill in them an understanding that neither school nor mathematics need impose a strict adherence to monolingualism. in this way, as a nation, we could maximize the efforts to promote a more cohesive program for preparing bilingual learners and respond to language diversity not with a bias shaped by political, social, and economic forces but rather by a systematic idea about language itself. using one language in the interview we shared in this analysis, we can see how productive it was to avoid the rigid use of only one standardized language in order to maintain mathematical engagement and to further develop the children’s understanding. hugo’s engagement in a linguistic practice that is normal for bilingual students was made possible because krause afforded him the space to share his mathematical ideas in their purest form, that is, in the form in which they naturally emerged. because mr&k followed ms. adams’s class (with the same children) for two consecutive years, they knew ms. adams’s instructional practices and the children in the class well. importantly, mr&k knew they could afford, and in fact knew it was expected of them, krause, adams-corral, language practices in the elementary & maldonado rodríguez bilingual mathematics classroom journal of urban mathematics education vol. 15, no. 2 32 to give hugo the space to express his ideas in their purest form because that was the way in which ms. adams had taught mr&k. hugo solved the problem on his own, and it was not until he was asked to articulate his mathematical thinking (where he had been allowed to draw on his mathematical repertoires—those mathematical practices that he can apply and use when solving problems) that he had to draw upon his full linguistic repertoire in order to share his process aloud with the interviewer. sharing his thinking was not only to the benefit of the researcher, however. it was necessary for hugo to share his thinking because, at first glance, the symbolic notation he had written was incorrect, even though he had a valid strategy. the process of engaging in discussion around hugo’s thinking made clear the depth of his understanding of the problem and the logic of his invented notation. this ensures clearer pathways for future instruction and better recognition of children’s capacities. drawing on his full linguistic repertoire allowed hugo to make himself and his mathematics recognizable and legitimate. garcía and wei (2014) suggested that “translanguaging refers to new language practices that make visible the complexity of language exchanges among people with different histories, and releases histories and understanding that had been buried within fixed language identities constrained by nation-states (p. 21).” in this case, hugo and krause entered into their exchange with certain histories and institutional labels. translanguaging is what allowed for them to engage in an exchange that never sacrificed complexity and that resulted in new practices, including new mathematical practices. hugo was given the space to use complex language practices in order to communicate something new (his invented notation). that space, in part achieved by ceding the emphasis on standardized formalisms of both mathematical and linguistic expression while maintaining a focus on mathematical content, allowed his natural bilingual abilities to convey his understanding in a manner recognizable to the instructor. this is an example of how translanguaging in the mathematics classroom can serve a powerful mathematical purpose. krause in this case was able to identify what hugo knew and understood, which then could be used to make instructional decisions suited to his own understanding. this is consistent with what other researchers have found working in bilingual classrooms. for example, moschkovich (1999) found that bilingual teachers are more effective when they keep their focus on mathematical ideas, regardless of how they are expressed. imposing an unnatural formality of language may stifle children’s natural interest in mathematics. language separation policies carry with them the risk of marginalizing and denigrating bilingual children’s everyday translanguaging practices (martínez et al., 2015). mathematics classrooms are situated within wider sociolinguistic contexts, and language use becomes more than an instrument to teach mathematics (barwell, 2009). rather, it becomes a venue for promoting bilingualism as a norm and for equally valuing the use of a home language. krause, adams-corral, language practices in the elementary & maldonado rodríguez bilingual mathematics classroom journal of urban mathematics education vol. 15, no. 2 33 our goal with our work is to engage teachers and teacher educators in respectful—yet critical—dialogue around the complex nature of everyday bilingualism in the mathematics classroom. we recognize that teachers are often promoting policies that are imposed schoolwide or even districtwide and have not been given the space to question or challenge their utility. the present work provides evidence of an actual bilingual mathematics classroom where a teacher and her students engaged in translanguaging, a perfectly normal and natural mode of bilingualism (martínez et al., 2015). what happens in language instruction is not the main point we are trying to convey with our work. our focus is on how mathematics instruction can be used to better understand children’s ideas about mathematics and how children’s ability to hear their language as it is naturally expressed allows for more aware and conscious language teaching. in this article, we presented two examples of how a bilingual elementary teacher taught mathematics to her bilingual students. we set out to highlight these examples as a way to answer what seem to be common questions of practice: “how?” and, more importantly in the context of teaching mathematics in the bilingual classroom, “what does it look like?” were we to provide a “recipe” for teaching mathematics in bilingual classrooms, we would necessarily overlook numerous aspects of the complex interaction among language, culture, and learning. in both examples discussed in this article, we can see how institutional labels mask the complexity of children’s language repertoires. hugo, who is not labeled an english language learner, sustained a mathematical interaction in spanish with a bilingual researcher, sharing his in-the-moment mathematical thinking by flexibly deploying his linguistic repertoire. gabriel, who is labeled an english language learner, shared his thinking in english while his teacher responded in spanish, with their conversation easily transcending the boundaries of named languages (wei & ho, 2018). decision making on the part of ms. adams required knowing both children, their families, and their stories, details that cannot be assumed away or simplified for fast takeaways. these examples highlight how a deemphasis on the formalities of mathematical expression, both in their symbolic and linguistic form, can help teachers attain in the moment a truer glimpse into a child’s understanding of mathematical content. at the same time, such an environment encourages linguistic and social practices that serve to strengthen the child’s bilingual abilities and identities. earlier we mentioned the three common tensions of teaching in bilingual contexts: (1) between using formal or informal language, (2) between using children’s home language and the language of the school, and (3) between teaching mathematics and teaching language. in the examples we shared, language was ultimately not a cause of tension but rather a tool for foregrounding mathematical ideas. in the case of hugo, the “imperfection” of his language use was essentially inconsequential to the interaction, whose primary goal was to foreground the mathematical idea. hugo’s new way of expressing his krause, adams-corral, language practices in the elementary & maldonado rodríguez bilingual mathematics classroom journal of urban mathematics education vol. 15, no. 2 34 mathematical idea was a fantastically apt means for communicating his understanding of mathematics. his teacher’s ability to capture this understanding and build more complex mathematical ideas on top of it is exactly the kind of instructional skill that bilingual teachers need to develop in order to support bilingual children’s deep understanding of mathematics. the examples we presented here are meant to challenge some common assumptions regarding translanguaging. translanguaging does not require the abandonment of language goals; rather, it requires intentionality and thoughtfulness. if we want a performance in spanish, we should consider how we will respond to students’ lexical creativity. if we recognize students’ bilingualism regardless of labels, we may play with conversations where input and output vary across named languages. translanguaging and bilingualism are not monoliths or implemented simply; they are as complex and dynamic as people themselves. these examples represent moments in our data where language surprised us, as both researchers and practitioners. a final word in this article, we identified characteristics of a teaching practice for capturing and encouraging teaching that is responsive to children’s mathematical ideas and linguistic repertoire. however, we recognize that the caring and respectful stance evident throughout ms. adams’s teaching and her experience teaching is not enough. her knowledge of spanish and her own bilingualism are key components of the work she does in her classroom. as mathematics teacher educators, we have the responsibility to prepare more teachers with the same skills as ms. adams, and by promoting bilingualism in the schools we are developing a generation of bilingual citizens that may eventually become teachers themselves. research has also provided evidence that bilingualism has benefits that extend beyond the ability to communicate in multiple languages (kroll & dussias, 2017). for example, greater intercultural awareness and open-mindedness (byram, 1997) as well as increased access to post-secondary education (kroll & dussias, 2017) are a few examples of what we can accomplish if we focus our efforts on promoting and maintaining bilingualism. we agree that being bilingual is in fact an advantage, and we promote the development of a bilingual teaching practice in schools. however, as welch (2015) straightforwardly stated, “teachers need not be paralyzed by their own monolingualism” (p.93). within their own teaching practice, teachers have the option of assuming the role of both expert and learner (fránquiz & reyes, 1998). knowing their 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(30, 23) (86, 25) (127, 34) multiplication la clase de segundo grado se está organizando para una fiesta del fin de año. una de las mamás compra ___ cajas de peras para la fiesta. cada caja tiene ___ peras. ¿cuántas peras hay en total? the second-grade class is getting organized for an end-of-the-year party. one of the moms buys ___ boxes of pears for the party. each box has ___ pears. how many pears are there in all? (4, 12) (6, 18) (8, 24) separate change unknown hay ___ niños jugando afuera en el recreo. unos de los niños regresan a los salones. ahora hay ___ niños jugando afuera en el recreo. ¿cuántos niños se metieron a los salones? there are ___ children playing outside at recess. some of the children return to the classrooms. now there are ___ children playing outside at recess. how many children entered the classrooms? (43, 20) (51, 29) (125, 75) equations 1 = ½ + ____ ½ + ½ + ½ = ____ copyright: © 2022 krause, adams-corral, & maldonado rodríguez. this is an open access article distributed under the terms of a creative commons attribution-noncommercial-sharealike 4.0 international license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. journal of urban mathematics education december 2012, vol. 5, no. 2, pp. 112–132 ©jume. http://education.gsu.edu/jume laura mcleman is an assistant professor in the department of mathematics at the university of michigan flint, 303 e. kearsley street, flint, mi, 48502; email: lauramcl@umflint.edu. her research interests include preparing mathematics preservice teachers to work with underserved and underrepresented populations, and the preparation of mathematics teachers and mathematics teacher educators to integrate issues of equity and social justice in their instruction. anthony fernandes is an assistant professor in the department of mathematics and statistics at the university of north carolina charlotte, 9201 university city blvd., charlotte, nc, 28223; email: anthony.fernandes@uncc.edu. his research interests include preparing mathematics teachers to work with english learners and understanding the use of gestures in english learners’ mathematics communication. michelle mcnulty is a secondary mathematics and biology preservice teacher at the university of michigan flint, 303 e. kearsley street, flint, mi 48502; email: mimcnult@umflint.edu. her research interests focus on teaching methods for diverse populations of students, specifically english learners. regarding the mathematics education of english learners: clustering the conceptions of preservice teachers laura mcleman university of michigan flint anthony fernandes university of north carolina charlotte michelle mcnulty university of michigan flint in this article, using survey data, the authors examined conceptions about the mathematics education of english learners (els) from 292 preservice teachers (psts) in urban universities through cluster analysis to determine if certain background characteristics influenced the formation of homogeneous clusters. an analysis of the findings shows a two-cluster solution, where respondents in cluster 2 (n = 187) were more aligned with research on the mathematics teaching and learning of els than respondents in cluster 1 (n = 105). further, a chi-square test revealed that psts with three characteristics—exposure to issues related to els, field experience, and being female—were significantly higher in cluster 2 than cluster 1. the findings provide compelling evidence that exposure to el issues impact the conceptions that psts regarding the mathematics education of els. keywords: english learners, mathematics education, preservice teachers, mathematics teacher education, urban education eacher preparation programs play an important role in how and where teachers learn about practice (gay, 2009). yet, in the case of english learners (els 1 ), teacher preparation has not kept up with the high growth of els in the classroom. during the decade spanning 1998 to 2008, els accounted for nearly 50% of the growth in the overall pre-k–12 student population in the united states 1 we view english learners as those students who are still developing a proficiency in english and may, but do not always, consist of students who speak a language other than english at home. t mcleman et al. mathematics education of english learners journal of urban mathematics education vol. 5, no. 2 113 (national clearinghouse for english language acquisition, 2010), a majority of which were educated in mainstream urban classrooms (costa, mcphail, smith, & brisk, 2005). for example, 70% of elementary-aged els are educated within approximately 10% of the classrooms in the nation, a predominance of which is located within urban areas (consentino de cohen, deterding, & clewell, 2005). however, several researchers have reported that teachers do not feel that they are prepared to face the economic, demographic, and technological realities present in schools, including the education of els (see, e.g., durgunoğlu & hughes, 2010; levine, 2006; mayer & phillips, 2012). in consideration of the large percentage of els within urban schools, we were interested in understanding the conceptions of preservice teachers (psts 2 ), who attended universities located within urban settings. we conjectured that psts from these universities might have had multiple interactions with els, either as students or through field experiences, and therefore have developed specific conceptions of teaching mathematics to els. as such, in this study we sought to classify urban psts into groups or clusters based on their reported conceptions about the mathematics education of els. specifically, we researched the following question: 1. for psts who attend universities that are situated within an urban context, how do their conceptions about the mathematics education of els cluster? 2. what prior characteristics might account for the formation of these clusters? literature review numerous studies address how preand in-service teachers conceptualize cultural and linguistic diversity or the inclusion of els in mainstream classrooms (e.g., byrnes & kiger, 1994; flores & smith, 2008; hansen-thomas & cavagnetto, 2010; reeves, 2006). however, all these studies examined conceptions of diversity in contexts that were not specific to the mathematics teaching and learning of els. for example, byrnes, kiger, and manning (1997) used the 13 item language attitudes of teachers scale (lats; byrnes & kiger, 1994) to measure the attitudes that 191 teachers had about language diversity and linguistically diverse students in three states. youngs and youngs (2001) used two items adapted from lats to examine the nature of attitudes of 143 teachers towards els and predictors of these attitudes. they found that teachers who completed a foreign language or multicultural course, had english as a second language (esl) training, 2 for the purposes of this study, preservice teachers are those students enrolled in universitystructured teacher preparation programs. mcleman et al. mathematics education of english learners journal of urban mathematics education vol. 5, no. 2 114 experience abroad, worked with diverse or esl students, and were female had more positive attitudes towards els. as another example, pohan and aguilar (2001) developed the personal and professional belief scales that akiba (2011) used to examine the change in beliefs about diversity that took place after psts attended a diversity course and had a field experience. reeves (2006) also used part of pohan’s and aguilar’s scale to measure the attitudes that 279 subject-area teachers had towards including els in their classroom. as janzen (2008) noted there is a dearth of studies that document how psts conceptualize the instruction of els within the specific context of mathematics. according to cooney, shealy, and arvold (1998) and philipp (2007), beliefs and knowledge are tied to context and different contexts will elicit different conceptions. this study contributes to the literature by examining psts’ conceptions about els within the context of mathematics. psts generally believe that mathematics involves symbols and is less language intensive than other subjects (garrison & mora, 1999; walker, ranney, & fortune, 2005). however, numerous linguistic demands exist, including unpacking questions that contain complex phrases in the statements of problems, making arguments, justifying reasoning, and building on other’s arguments (bailey, 2007; barwell, 2005b; moschkovich, 1999, 2010; schleppegrell, 2010). teachers must understand linguistic complexity and make content comprehensible for els by providing linguistic and contextual support (echevarría, vogt, & short, 2008; gibbons, 2002), like modeling mathematical talk (khisty & chval, 2002) and scaffolding procedures (gibbons, 2002). we sought to understand the conceptions of psts from urban universities. additionally, we wanted to determine how the conceptions clustered and what characteristics might account for this clustering. conceptual framework conceptions, according to pratt (1992), are specific meanings of phenomena and impact how individuals view the world. we viewed the construct of conceptions similar to kitchen, roy, lee, and secada (2009), namely that conceptions constitute both knowledge and beliefs. for our study, the conceptions that research indicates psts would need in order to be effective mathematics teachers to els guided the development and interpretation of a survey we created. specifically, we framed the item design and data analysis through a non-deficit perspective of working with els. according to civil (2007) and moschkovich (2010), these perspectives assume that el students have valuable resources, including their culture and language, which can and should be used as an integral part of mathematics instruction. in this study, the use of an el’s native language was viewed as a resource in order to promote an el’s acquisition of the academic language of mathematics in english (garrison & mora, 1999). further, all parents from all mcleman et al. mathematics education of english learners journal of urban mathematics education vol. 5, no. 2 115 cultures were seen to value the academic growth of their children (civil, planas, & quintos, 2005), even if their ways of participating within the school structure did not fit within the traditional paradigm of parental involvement (e.g. attending parent-teacher conferences, volunteering in classrooms, etc.). thus in framing, and later scoring the items, we assumed that bilingualism and an el’s home culture were assets to the mathematical learning of an el. based on the framework of our survey, review of the literature, consultation with experts in the field, and our experience as mathematics educators, we created items for our survey that would assess psts’ conceptions in areas that would impact the mathematics education of els  interconnection of language and mathematics,  teaching mathematics to els,  language in the school context,  fairness, and  diverse cultures. these five areas guided the development of items for the survey as well as the analysis of the each of the participant's responses. throughout the findings, when appropriate, we frame the participants’ responses in terms of key findings from these areas of literature. methodology the survey instrument the survey consisted of 26 items that measured the strength of agreement or disagreement of psts’ conceptions about the mathematics education of els on a 5-point likert-type scale: strongly disagree (1), disagree (2), undecided (3), agree (4) and strongly agree (5). the 26 items on the survey were broken up into the five categories seen in table 1. additionally, participants were asked to provide demographic information, including gender, race, and knowledge of another language. additionally, we asked participants if they had been exposed to el issues through courses in their degree programs, and if they worked in classrooms as part of course-based field experiences. even though there are disadvantages to using a survey to measure psts’ conceptions (see ambrose, clement, philipp, & chauvot, 2004), we chose to do so for two primary reasons. first, we were concerned that psts might answer in a manner they thought was expected of them if we used an interview setting. second, there are no large-scale studies that can complement the smallscale qualitative studies about the conceptions that psts have. mcleman et al. mathematics education of english learners journal of urban mathematics education vol. 5, no. 2 116 table 1 breakdown of survey categories with number of items in each category category number of items language in the school context (lsc) 8 interconnection of language and mathematics (ilm) 3 diverse cultures (dc) 4 teaching mathematics to ells (tm) 8 fairness (f) 3 validity of the instrument a pilot version of the survey was developed and tested with psts at one university in the southeast united states. the findings of that survey were used to check psts’ interpretations of the items. after the items were refined, we ensured content validity by consulting 10 experts in the field, whose suggestions were incorporated into a further refinement of the survey items. face validity of the survey was addressed by asking psts at the end of the survey to answer three openended questions to determine the readability and clarity of the survey. data collection psts conceptions were measured through an online survey. data from 294 psts from universities situated within urban contexts from 12 different states around the united states were collected. for our purposes, a university that was located in an area with a highly dense population (based on classifications from the united states census bureau 3 ) was considered to be located within an urban context. we recruited the participants through personal requests to other mathematics teacher educators working with psts. all potential participants were provided a web link that directed them to the survey that was hosted on survey share (see http://www.surveyshare.com/). data analysis we analyzed the data using cluster analysis, a method that creates groups of respondents based on high within-cluster homogeneity and high between-cluster heterogeneity (hair & black, 2000). the data were prepared for analysis by reverse coding certain items based on our conceptual framework of non-deficit conceptions. a score of 1 represented a response that was least aligned with the research literature regarding els with a score of 5 representing a response that was most aligned. we conjectured, for example, that a pst who conceptualized that an el’s culture could negatively impact an el’s mathematical learning would be less open to seeing certain els’ home cultures as a resource in the classroom. in 3 visit http://www.census.gov/geo/www/ua/2010urbanruralclass.html for more information. http://www.surveyshare.com/ mcleman et al. mathematics education of english learners journal of urban mathematics education vol. 5, no. 2 117 total, 14 of the 26 items on the survey were reverse coded and are indicated with an r after the number. the next stage of preparation involved pre-screening the data for outliers based on the responses of the psts to the 26 items. outliers tend to distort the results of statistical tests and need to be removed at the outset (aron & aron, 1997); cluster analysis in particular is sensitive to outliers. the mahalanobis distance for multivariate data (p < .001) was used to determine the outliers (stevens, 1992; tabachnick & fidell, 1996). there were two outliers that were dropped from the subsequent analysis making the total number of responses examined 292. cluster analysis we used hierarchical cluster analysis using ward’s method to identify those respondents who had a high homogeneity of responses related to the 26 statements, with an end goal of identifying the characteristics of these groups. we examined the difference between the coefficients to determine the number of clusters. a new cluster was determined when the distance between a pair of adjacent coefficients was not relatively stable when compared to all other pairs of adjacent coefficients (milligan & cooper, 1985). upon examination of the re-formed agglomeration table, the approximate jump in coefficients of 640 between the last two stages of clustering and the relatively stable distance between all other pairs of adjacent coefficients (approximately 229, 200, and 155 for the next three pairs of adjacent coefficients, respectively) suggested a two-cluster solution. this jump is displayed in the distance on the right of the dendogram given in figure 1, which points to increasingly dissimilar clusters being combined. the two-cluster solution was validated by splitting the data into two equal sets and confirming the persistence of the same solution for the split data (hair & black, 2000). figure 1 dendogram showing two-cluster solution. mcleman et al. mathematics education of english learners journal of urban mathematics education vol. 5, no. 2 118 cluster description once the two-cluster solution was validated, demographic information was used to profile the two clusters. a chi-square test (p < 0.05) was performed on each demographic variable to seek its association to cluster membership. this profiling was extended to see how the psts in the two groups differed in their responses to the 26 items, as the goal was to identify the characteristics of each cluster. the differences between the means of cluster 1 (c1) and cluster 2 (c2) for each of the 26 items were calculated, and a two-tailed independent samples ttest (p < 0.05) on the mean scores with respect to each item was performed. we also performed independent samples t-tests using each of these variables as the grouping variable to determine the specific items where there was a significant difference between the means of c1 and c2. finally, given that the purpose of cluster analysis is to seek heterogeneous groups, significant differences between the means of the clusters are expected (hair & black, 2000). thus, the items where the differences between means were not significant were also documented. findings the first part of our findings describes the characteristics of the entire sample of 292 psts and the two clusters; the second examines items whose cluster means showed a significant difference and a difference in alignment to the research; and the third part examines items whose means did not differ between the two clusters. sample the sample of psts who responded to the survey was comprised of 86% females and 14% males. additionally, 85% were white, 7% were black, 3% were hispanic, and 2% were asians. most of the psts were interested in teaching k–5 (75%), with 14% and 11% interested in teaching middle and high school, respectively. the majority of psts had less than four years experience teaching, with 73% having no experience and 25% having between 0–4 years. a majority (78%) of the sample had some field experience, and 74% of the psts were exposed to issues related to els through their courses in their degree programs. though most of the psts had experience learning a second language (86%), there were only 8% who were actually fluent in another language. of those who self-reported that they were fluent in another language, 6 of 8 psts were hispanic, 4 of 6 psts were asian, and 9 of 239 psts were white. table 2 presents the number of respondents in each cluster, broken down by each demographic characteristic. mcleman et al. mathematics education of english learners journal of urban mathematics education vol. 5, no. 2 119 table 2 number of respondents per demographic variable cluster 1 (c1) 105 cluster 2 (c2) 187 total 292 gender male: 21 male: 19 male: 40 female: 84 female: 168 female: 252 teaching interest grades k-5: 76 grades k-5: 144 k-5: 220 grades 6-8:18 grades 6-8:23 grades 6-8: 41 grades 9-12:11 grades 9-10:20 grades 9-12:31 teaching experience none: 77 none: 135 none: 212 0-4 years: 24 0-4 years: 48 0-4 years: 72 5-10 years: 4 5-10 years: 4 5-10 years: 8 field experience yes: 72 yes: 155 yes: 227 no: 33 no: 32 no: 65 exposure to el issues yes: 67 yes: 149 yes: 216 no: 38 no: 38 no: 76 fluency in another language yes: 6 yes: 17 yes: 23 no: 99 no: 170 no: 269 experience learning a second language yes: 91 yes: 159 yes: 250 no: 14 no: 28 no: 42 as table 2 indicates, the ratio of individuals from c2 to those from c1 is about two-to-one (2:1) for almost every demographic component. in other words, there is about twice the number of individuals in c2 than in c1 across most demographic variables. for example, there are 144 participants in c2 that desire to teach grades k–5 as opposed to 76 in c1, a ratio of 1.89 to 1. however, there were significant associations for only three of the demographic variables and membership to c2: gender   2 15.507,p.05 , exposure to el issues   2 18.796,p.05 , and field experience   2 17.946,p.05 . this means, in the case of gender, the proportion of females in the two clusters is significantly different and females are more likely to be in c2 than c1. significant differences in the two clusters our analysis of the data revealed significant differences (p < 0.05) between the means in 21 out of the 26 items. however, here, we focus on the eight items where the two clusters were not only significantly different but also differed in their alignment to the research; that is, one cluster mean was more than three and the other was less than three. (even though there were significant differences be mcleman et al. mathematics education of english learners journal of urban mathematics education vol. 5, no. 2 120 tween the means for the other 13 items, both clusters on the whole agreed or disagreed with the item.) the eight (paraphrased) items are summarized as follows, with the cluster means and the percentage of undecided responses (3 on the likert scale) for each item given in table 3.  lsc13r: learning english is more important than native language.  lsc15r: speaking in a language other than english hampers the learning of english.  lsc16: state math tests should not be offered in different languages.  lsc18r: after one year, els are capable of rich math discussions.  f35r: the math work of els and non-els should be evaluated the same.  tm32r: els and non-els should be taught math in the same way.  ilm20r: conversational fluency implies capability to learn math like non-els.  dc25r: inherently, els from some ethnicities are better at math than others. table 3 items that were significantly different and had differing cluster alignment to research item cluster 1 (c1) cluster 2 (c2) mean % undecided mean % undecided lsc13r 2.6857 36.2 3.5668 20.86 lsc15r 2.8762 31.42 3.7326 23.53 lsc16 2.9619 24.76 3.8128 23.53 lsc18r 2.8952 51.42 3.2781 35.83 f35r 2.6476 32.38 3.3529 26.74 tm32r 2.7714 27.62 3.7059 24.06 ilm20r 2.5333 24.06 3.5187 23.81 dc25r 2.7619 39.05 3.2781 30.48 of the eight questions asked regarding language in the school context, the two clusters had differing alignment to the research for four of the items. about 45% of the respondents in c1 conceptualized that learning english is more important for els than maintaining their native language (lsc13r) compared to 17% in c2. on the other hand, only about 19% of respondents in c1 did not think that learning english was more important, where about 62% of the respondents in c2 did. about one-third of the respondents in c1 also conceptualized that the use of native language would hamper an el’s learning of english (lsc15r), with about only 8% of respondents in c2 sharing this conception. this pattern of responses was observed for item lsc16 as well. more than one-third (about 38%) of the psts in c1, based on their conceptions, felt that state tests should not be offered in different languages, where only 8% in c2 thought that this should be the case. for the last item, lsc18r, both c1 and c2 had a similar percentage of respondents that thought els could have rich mathematical discussions after being immersed in mcleman et al. mathematics education of english learners journal of urban mathematics education vol. 5, no. 2 121 english for a year: about 29% of the respondents from c1 and about 20% from c2. further, many of the psts in both clusters did not seem to have a definite view on this topic, with over half (51%) of the psts in c1 and a little over a third (36%) of the psts in c2 indicating that they were undecided. the lower means for c1 in the four items point to a common misconception described in the second language acquisition literature that more time spent learning english will allow for a faster acquisition of the language (gandára & contreras, 2009). however, this model helps els only acquire conversational language and not the academic language required for them to communicate their understanding of the content (cummins, 2000). cummins pointed out that it takes 5–7 years to acquire academic language proficiency, as opposed to 1–2 years for conversational fluency. to facilitate the comprehension of content, it is recommended that els can have richer discussions with each other or the teacher when they converse in their native language (domínguez, 2011; gutiérrez, 2002; moschkovich, 2010). furthermore, researchers (e.g., gandára & contreras, 2009) have discussed that the rate of acquisition in immersion and bilingual programs are about the same, a usual critique of bilingual education. clarkson (1992) and garrison and mora (1999) also pointed out that if students have learned content in their native language, as in the case of new immigrants, then after acquiring a certain threshold proficiency in both languages, they are able to transfer their mathematical knowledge from one to the other. so if native language benefits els’ comprehension of the content, allows for meaningful participation, and does not impact their rate of english acquisition, then it is advantageous for a teacher to be open to its use in the classroom. the misconception between conversational and academic language can be seen in the responses to ilm20r. psts in c1 were more likely to confuse fluency in conversational language with academic language fluency than psts in c2. in c1, 59% of the respondents either agreed or strongly agreed with the statement compared to 17% who either disagreed or strongly disagreed. in comparison, the percentages for c2 were reversed, with about 17% of respondents in c2 having either agreed or strongly agreed with the statement versus 59% who either disagreed or strongly disagreed. the high percentage of respondents in c1 that agreed may indicate those psts do not see mathematics and language as inseparable (barwell, 2005b), but rather as mutually exclusive constructs. researchers, such as schleppegrell (2007) and veel (1999), have pointed to the linguistic demands in mathematics that go beyond conversational fluency which would be required by the students to meaningfully participate in the classroom mathematics community. in items tm32r and f35r we observe the impact that some of the psts’ conceptions about language have on their conceptions of teaching els. about 50% of the respondents from c1 indicated that they would not differentiate how they mcleman et al. mathematics education of english learners journal of urban mathematics education vol. 5, no. 2 122 evaluated the mathematical work of els (f35r), with 18% agreeing that they would evaluate els differently. in contrast, these percentages for c2 were 22 and 51, respectively. in the context of fairness, about 44% of the respondents in c1 agreed that they would teach els in the same way that they taught non-els (tm32r), as opposed to 8% of the respondents in c2. research (e.g., bunch, 2010; campbell, adams, & davis, 2007; de jong & harper, 2005) has shown that there are extra cognitive demands on els as they try to understand new content in a language they are still learning. thus there is a need for els to have modifications and accommodations such as providing linguistic and contextual supports through scaffolding (gibbons, 2002) or by providing extended time for els to interact with peers and teachers about academic content (echevarría, vogt, & short, 2008; hanson & filibert, 2006). working within this paradigm, fairness does not mean sameness, something that the psts in c1 seemed to assume. finally, we examine psts’ conceptions about el students’ inherent ability to do mathematics (dc25r). about 41% of the respondents in c1 felt that some els were inherently better at mathematics, as compared to only 24% in c2. while for each cluster there was a high percentage of respondents who were undecided (about 39% for c1; 30% for c2), only 20% of the respondents in c1 disagreed that there was an inherent difference in mathematical ability for some els, compared to about 45% for c2. previous research (e.g., chval & pinnow, 2010; guttmann & bar-tal, 1982) supports these conceptions, specifically that psts can have stereotypical beliefs about students based on the language that the students speak. for example, they may believe that asian students are better at mathematics than latina/o students, even though they are both els (chval & pinnow, 2010). though some languages like chinese seem to offer advantages to speakers when formulating numbers, generalizing this to an overall superior mathematical ability is not a given (yee, 1992). it is possible that some of the psts considered only the difficulty of the discipline of mathematics. as noted earlier, some psts may have separated the ideas of language and mathematics. this separation might lead them to feel that an el’s grasp of mathematics is not dependent on knowing english but rather on the difficulty of mathematics itself. in other words, they might feel that els would have difficulties in a mathematics classroom regardless if they can converse in english since mathematics itself is inherently difficult (mcleman, 2012). non-significant differences in the two clusters significant differences between means of the two clusters are quite natural in cluster analysis, given that the process seeks to form heterogeneous groups. consequently, differences that are not significant can also provide mathematics teacher educators insights about conceptions that are held across psts. in our mcleman et al. mathematics education of english learners journal of urban mathematics education vol. 5, no. 2 123 analysis, 5 of the 26 items did not show a significant difference (p < 0.05) in means (see table 4). specifically, these (paraphrased) items are  ilm19r: math is an ideal subject for beginning els to learn english.  dc23r: some cultures negatively impact els’ learning of math.  dc24r: parents in some cultures place a higher value on education than parents from other cultures.  tm29r: limited math vocabulary helps els learn math.  tm33: els need discussion rich classrooms to learn math. table 4 items that were not significantly different item cluster 1 (c1) cluster 2 (c2) mean % undecided mean % undecided ilm19r 2.5905 30.48 2.6096 27.62 dc23r 2.7238 36.19 2.8289 33.69 dc24r 1.9333 9.52 1.8877 8.02 tm29r 3.4762 25.71 3.6096 23.53 tm33 3.8857 16.19 4.0321 17.11 given the conception among psts that mathematics is a universal language (garrison & mora, 1999; walker, shafer, & iiams, 2004) it was not surprising that both clusters had mean scores that were not aligned with the extensive research that describes the prevalence of language in the teaching and learning of mathematics. by extension both clusters also conceived that mathematics would be easier than other subjects for el students. these conceptions match with those expressed by middle school teachers who assumed that mathematics would be easier for el students because numbers are universal (hansen-thomas & cavagnetto, 2010). the majority of the psts had a definitive view about parents as indicated by the low percentage of undecided responses in each cluster. about 84% of the respondents from c1 and about 86% of the respondents from c2 agreed with the idea that there are parents from some cultures that value math education more than others (dc24r). further about 45% and 40% of the psts from c1 and c2, respectively, saw that some els’ home cultures would negatively impact their learning of mathematics. these conceptions are supported in the research where minority parents and the communities in which they live are considered to be primary reasons for els’ failure in schools (pappamiheil, 2007). finally, we see that psts in both clusters were aligned with research regarding the importance of discussion rich mathematics classrooms for els. in particu mcleman et al. mathematics education of english learners journal of urban mathematics education vol. 5, no. 2 124 lar, about 80% in both c1 and c2 agreed creating classrooms that promote rich language development is necessary in the mathematics instruction of els. discussion in this study, we were interested in examining how the conceptions about the mathematics education of english learners (els) from preservice teachers (psts) who attend universities situated within urban contexts cluster. further we were interested in what prior characteristics might account for this clustering. overall, cluster analysis on the 292 psts’ responses to the 26 items yielded a two-cluster solution with c2 (n = 187) more aligned with research than c1 (n = 105). close examination of the demographic distribution shows c2 was more likely to contain psts who were female, exposed to el issues in prior courses, and who had field experiences. these three characteristics of psts were similar to some of the predictors that previous research (e.g. byrnes et al., 1997; youngs & youngs, 2001) has found supports working with els in productive ways. youngs and youngs noted that teachers would be more positive about working with el students if they have been educated about working with this population. in regard to gender, given that the proportion of females is significantly more in c2, this would seem to confirm other research that females were more open or accepting of diversity issues within the classroom (e.g., akiba, 2011; pohan & aguilar, 2001; taylor, peplau, & sears, 1999; ottavi, pope-davis, & dings, 1994; youngs & youngs, 2001). with gender, exposure to el issues, and field experiences showing significant differences in group membership, it appears that providing field experiences in conjunction with readings concerning the education of els within teacher preparation can be a fruitful avenue to align psts’ conceptions to research. thus, further analysis was conducted on the eight items that had significant differences in means and differed in their alignment to the research given that these items spanned the three characteristics that were significant to membership in c2. findings from this secondary analysis showed that exposure to el issues, gender, and field experiences were significant for item tm32r (teaching both els and non-els in the same way). moreover, exposure to el issues and gender were also significant for items lsc13r (learning english is more important), ilm20r (conversational fluency implies academic fluency), and f35r (evaluation of math work should be the same). in other words, the means of the respondents with exposure to el issues and who were female were significantly different from the means of those without exposure and who were male. finally, in addition to the previously mentioned items, a significant difference was also found between the means for the respondents who had exposure to el issues and those who did not for items mcleman et al. mathematics education of english learners journal of urban mathematics education vol. 5, no. 2 125 lsc15r (speaking another language hampers the learning of english) and lsc16 (state math tests should be in different languages). as c2 was more likely to contain psts with these characteristics and to have items means that were aligned with the research, in comparison to c1, these findings mirror those of cho and decastro-ambrosetti (2006) in providing compelling evidence that exposure to el issues is one of the most important factors in helping psts have non-deficit views about the mathematics education of els. in particular, findings from this study suggest that when psts have had exposure to el issues they are more likely to understand the language demands of mathematics. for psts in c2, this idea translated to seeing that conversational language fluency is not equivalent to academic language fluency and an el’s need to speak in their native language would not hinder their development of learning english. through their conceptions, the psts in c2 also seemed to understand that an el’s linguistic needs must be acknowledged with accommodations to lesson planning, evaluation, and offering state-mandated assessments in languages other than english. moreover, akin to the findings seen by olmedo (1997), field experiences coupled with the knowledge gained about considering issues related to els may have provided psts avenues to begin to challenge the notion that teaching does not have be exactly the same in order to be fair. with less knowledge about and experience working with el students, the findings from this study show that psts hold deficit-based conceptions about the mathematics education of els. for example, like the teachers in reeves’s (2004) study, the psts in c1 felt that using the same standards to evaluate both els and non-els was fair perhaps because state and nationalized standardized do not alter testing for different populations of students. alternatively, the psts may have felt that els must be treated in the same exact way as non-els so as not to differentiate based on ethnicity and/or race, among other things, given that differentiating might be associated with discriminating. it is also possible that the psts believed that modifying standards or teaching differently would not best prepare els for the future. as one teacher in reeves’s study noted, “the real world” (i.e., future employers) will not make such accommodations. what is problematic about this view, and something that teacher educators must challenge psts about, is that it chooses to ignore the systemic inequalities underserved and underrepresented populations such as els face in the educational system (oakes, 2005). moreover, teacher educators need to make explicit to psts that, as research has shown (e.g., brimijoin, 2005), teachers can differentiate instructional/assessment strategies while still preparing all students to be successful on standardized assessments. non-significant differences cluster analysis was especially useful in isolating the conceptions that were held across the entire group of participants. in this study, ideas that cut across all mcleman et al. mathematics education of english learners journal of urban mathematics education vol. 5, no. 2 126 participants were related to psts’ conceptions about the universal nature of mathematics with minimal use of language and a deficit view of parents and communities of el students. these findings point to areas that can be targeted in teacher preparation through exposure to el issues in combination with appropriate field experiences. moreover, it is important to understand how psts consider the use of language in mathematics classes as well as how their interpretations might differ from that of researchers (e.g., bailey, 2007; barwell, 2005a; moschkovich, 1999). it is unclear if the psts alignment to the research on best practices of teaching mathematics to els shows familiarity with this research or if the responses stem from a desire to promote good teaching for all (de jong & harper, 2005). for the psts in c1, in particular, the latter seems likely considering their unalignment with the research on all other ideas regarding the mathematics instruction of els. indeed, based on recommendations from organizations such as the national council of teachers of mathematics (2000), teacher preparation programs have helped psts consider best practices to teach mathematics to all students, including the need to promote discussion within mathematics classrooms. it is important to note that while these types of recommendations are important, they are not synonymous with the construction of a discussion-rich classroom needed to facilitate the academic language and content knowledge of els such as the one detailed by khisty and chval (2002). limitations in general, each of the items discussed in this article had high percentages of respondents indicating that they were undecided. for c1, this percentage ranged from about 25 to 51 on the significantly different items. for c2, this range was slightly lower, from about 21% to 36%. with almost 98% of psts indicating they understood what the questions in the survey were asking and a little more than 93% of psts indicating that there were no ambiguous questions, the high percentages of psts who chose the undecided response seems to indicate that they were indeed undecided on whether or not they agreed or disagreed with a particular item. for the respondents in c1 (where a larger percentage of undecided responses were seen), a possible explanation for this may stem from the lack of exposure or experience in thinking about issues related to educating els, a problem in many teacher preparation programs (watson, miller, driver, rutledge, & mcallister, 2005). given that the proportion of individuals having exposure to issues regarding the mathematics education of els is significant to membership in c2, the individuals in c1 may have felt unprepared to indicate a view one way or another. on the other hand, psts may have chosen to mark undecided as a response because they felt that every situation is different and that there does not exist one mcleman et al. mathematics education of english learners journal of urban mathematics education vol. 5, no. 2 127 correct way to educate els. as one pst indicated, “i feel that many of these would be a case by case therefore it is difficult to pick either agree or disagree.” in general, these psts are correct that there is not one way to educate students. however, research does provide us information about which systemic, societal, and instructional practices in general support els’ achievement (e.g., cummins, 1981; echevarría et al., 2008; khisty & chval, 2002; moschkovich, 1999). thus, while the psts were correct in stating that not every situation is the same, their responses indicate that they may not be familiar with some of the systemic ideas regarding the mathematics education of els. implications for practice in the past decade, research has called for the integration of linguistic issues into teacher preparation programs (e.g., duff, 2001; fillmore & snow, 2002). this study supports those calls with findings revealing that issues related to the mathematics education of english learners (els) need to have a more prominent and integrated role within teacher preparation. for the psts in this study, all of whom attended universities situated within an urban context, exposure to el issues and gender were significant factors in non-deficit conceptions. however, there were a number of psts who still held deficit views, which shows that these issues still remain an “add on” within the profession of teaching. for example, the common core state standards do not address the mathematics education of els within the standards but rather attends to it within an addendum (see http://www.corestandards.org/assets/application-for-english-learners.pdf). addressing el issues in context can benefit the learning of instructional strategies that will support els’ mathematical learning while also helping psts to understand the linguistic complexity inherent in the teaching and learning of mathematics. in particular, psts need help in re-considering their perspectives about parental and family involvement in education to see that there are various ways for parents and families to value the education of their children, many of which are not visible (souto-manning & swick, 2006). moreover, there is a need to foster an awareness of the linguistic aspects that arise in mathematics beyond the syntax of symbolic manipulation (see bailey, 2007; barwell, 2005a; moschkovich, 1999). as nevárez-la torre, sanford-deshields, soundy, leonard, and woyshneral (2008) detail, however, this awareness will only be achieved through a redesign of teacher preparation curricula where specialized courses focused on els are required and knowledge and domains related to second language, language development, and culturally responsive teaching, among other things, are integrated into pedagogy courses. the findings of this study support this type of redesign and extend it by noting that an inclusion of linguistic issues must also occur within content courses. psts need to be provided experiences within the context of mathematics (i.e., while doing mathematics) to help them better underhttp://www.corestandards.org/assets/application-for-english-learners.pdf mcleman et al. mathematics education of english learners journal of urban mathematics education vol. 5, no. 2 128 stand the nuanced interconnection between language and mathematics (fernandes, 2012). these experiences would include, among other things, teacher educators in mathematics courses explicitly noting the linguistic features present in mathematics. implications for research this study goes beyond current ones that examine psts’ preparation regarding the education of diverse students. instead this study focused on clustering the conceptions of psts who attended universities within urban settings, a unique contribution to the field of mathematics education considering the lack of studies in this area (janzen, 2008). while this study confirmed many of the findings from previous research, speaking to the robustness of this research, patterns in some of the psts’ conceptions merit further investigation. to inform our perspectives for the improvement of teacher preparation of els, large-scale studies involving quantitative measures (such as the one reported here) and smaller-scale qualitative studies need to work in tandem. specifically, psts’ conceptions about the mathematics education of els should be investigated further through qualitative interviews in order to gain a deeper understanding. for example, since the psts in both clusters in this study reported conceptions about parents that were unaligned with the research with few undecided responses, future research will involve qualitative interviews with select psts to understand the deficit nature of this conception. additionally, more demographic information can be collected in order to provide a stronger picture of how different populations of psts conceive of the mathematics education of els. such information could include whether the psts themselves were classified as english learners in their k–12 education or what year study (e.g., freshman, sophomore) the psts are currently classified as at their institution. acknowledgments the study reported in this article was supported, in part, by the office of research at university of michigan flint. any opinions, findings, and conclusions or recommendations expressed here are 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(2001). predictors of mainstream teachers’ attitudes toward esl students. tesol quarterly, 35, 97–120. retrieved from http://onlinelibrary.wiley.com/doi/10.1002/tesq.2001.35.issue-1/issuetoc. http://www.mville.edu/images/stories/graduate_academics_education/changingsuburbs/ellstrategiesandresources/teacher_attitudes_in_classrooms.pdf http://www.mville.edu/images/stories/graduate_academics_education/changingsuburbs/ellstrategiesandresources/teacher_attitudes_in_classrooms.pdf http://www.mendeley.com/research/english-language-learner-representation-in-teacher-education-textbooks-a-null-curriculum/ http://www.mendeley.com/research/english-language-learner-representation-in-teacher-education-textbooks-a-null-curriculum/ http://onlinelibrary.wiley.com/doi/10.1002/tesq.2001.35.issue-1/issuetoc journal of urban mathematics education july 2012, vol. 5, no. 1, pp. 66–83 ©jume. http://education.gsu.edu/jume erica n. walker is an associate professor of matheamtics education at teachers college, columbia university, box 210, 525 west 120th street, new york, ny 10027; email: ewalker@tc.edu. her research focuses on social and cultural factors as well as educational policies and practices that facilitate mathematics engagement, learning, and performance, especially for underserved students. cultivating mathematics identities in and out of school and in between erica n. walker teachers college, columbia university n recent years, researchers have explored the question of literacy development both within and outside of school (hill & vasudevan, 2007; mahiri, 2004; morrell, 2007). many have focused on specific cultural practices that contribute to literacy acquisition and development. in addition, they have considered the notion of these sites and places as spaces (lefebvre, 1974; soja, 1989) in which literacy is developed and an identity related to one’s literacy experiences is acquired. these spaces encompass more than a physical location; they include “social and cultural ideas of place, the meaning humans attribute to place, and the cultural and social knowledge surrounding various locations” (cole, 2009, p. 22). in thinking about these directions in literacy research and how “literate persons” are developed through their backgrounds, experiences, and practices both within and outside of schools, it is worth thinking about the analogous question of how a “mathematical person” is developed. it is important to consider how people’s mathematics identities might be cultivated in spaces within schools, outside of schools, and in spaces in-between, and how these experiences might contribute to the development of a mathematical identity as well as the development and dissemination of mathematical knowledge. although some who have focused on the acquisition of mathematical knowledge have used out-of-school contexts to engage students in mathematics learning within schools (bonnoto, 2005; moses & cobb, 2001), others have demonstrated that both functional and rigorous mathematics can be done by those using methods obtained outside of school (saxe, 1991). further, researchers have also explored how mathematical conceptual understanding might be supported and developed out of school: for example, nasir (2000) explored how african american young men used percent and ratio when choosing players for their basketball teams. several have suggested that “mathematics learning and practice in and out of school can build on and complement each other” (masingila, davidenko, & prus-wisniowska, 1996, p. 177), and that formal and informal mathematics learning should not be experienced in schools or outside of schools as completely discrete entities (schoenfeld, 1991). the mathematics backgrounds, knowledge, and experiences that students bring with them to school can be effectively used to i walker cultivating mathematical identities bullock, e. c., alexander, n. n, & gholson, m. l. (eds.). (2012). proceedings of the 2010 philadelphia and 2011 atlanta benjamin banneker association conferences – beyond the numbers [special issue]. journal of urban mathematics education, 5(2). 67 develop mathematics understanding and knowledge and serve to engage students (bonnoto, 2005). it has been shown that these out-of-school contexts for enacting mathematics practices are informed by students’ cultural backgrounds and experiences (saxe, 1991). cobb and hodge (2002), martin (2000), nasir and saxe (2003) and others have attended to the role that culture, context, and community play in mathematics learning for young people. a broader discussion of cultural practices (gutiérrez & rogoff, 2003; nasir & saxe, 2003) that positively affect mathematics learning, practice, and socialization outside of school, particularly for young people of color in the united states, would be useful for developing in-school practices that support mathematics engagement as well as for creating intentional out-of-school spaces that do the same. for example, literacy educators and education researchers have begun to develop and explore literacy spaces outside of school that are sites for induction into literacy communities and contribute in intentional ways to individual and group socialization around literacy (e.g., kinloch, 2005). further, elements of these practices are used in informal and formal school settings. despite the compelling research exploring the role of culture and context in mathematics learning, we do not do enough to create meaningful spaces within cultural contexts for mathematics practice for young people outside of school (the young people’s project, which seeks to promote mathematics literacy among young people participating in the algebra project, founded by robert [bob] moses, is an exception), nor do we do enough to build on the mathematics experiences that they do have outside of school. what remains underexplored in mathematics education research is how the mathematically talented in the united states are socialized to do mathematics outside of school—how do they develop their mathematics skills, interests, and dispositions? in the past, i have explored the mathematical experiences of high achieving high school students and the networks that foster their mathematics success (walker, 2006). most recently, i have been conducting a study of african american mathematicians, exploring their formative, educational, and professional experiences in mathematics (walker, 2009, 2011). what has emerged as a key factor in the success of high achievers and mathematicians alike is the important role that out-of-school experiences and relationships, many rooted in specific cultural and social contexts, have played in their mathematics knowledge development and socialization. in this article, i discuss the mathematical spaces that mathematicians describe as important to their success. i identify mathematical spaces as sites where mathematics knowledge is developed, where induction into a particular community of mathematics doers occurs, and where relationships or interactions contribute to the development of a mathematics identity. these spaces may be physical locations like a school or classroom or locations to which the individual attaches a walker cultivating mathematical identities bullock, e. c., alexander, n. n, & gholson, m. l. (eds.). (2012). proceedings of the 2010 philadelphia and 2011 atlanta benjamin banneker association conferences – beyond the numbers [special issue]. journal of urban mathematics education, 5(2). 68 particular social, cultural, or mathematical meaning due to interactions and experiences she or he had there. here i focus on those mathematical spaces experienced by mathematicians during childhood and adolescence, with the goal of contributing to our thinking about how we might engage adolescents in mathematics, particularly those from underrepresented groups. in the conclusion, i suggest that we move from these sometimes “inadvertent” spaces that foster development for individuals to creating and examining “intentional” spaces that contribute in strong ways to mathematics socialization and talent development for larger groups, particularly for underserved students. efforts to craft purposeful mathematical spaces, i argue, should reflect the bridging of out-of-school and in-school networks, relationships, and practices. research context in a previous study exploring “academic communities” (interpersonal networks that supported mathematics success) of high achieving black and latina/o students at an urban public high school, it became apparent that these students were doing mathematics in various ways in spaces within school, outside of school, and in-between (walker, 2006). i began to wonder what kinds of mathematical spaces were experienced and created by mathematicians, arguably the highest achievers, and for black mathematicians specifically. what might we learn from their narratives about doing mathematics in and out of school? like many researchers, i suggest that engagement should be considered as a construct that simultaneously encompasses behavior, emotion, and cognition (fredricks, blumenfeld, & paris, 2004). as fredricks, blumenfeld, and paris (2004) describe: behavioral engagement draws on the idea of participation; it includes involvement in academic and social or extracurricular activities and is considered crucial for achieving positive academic outcomes and preventing dropping out. emotional engagement encompasses positive and negative reactions to teachers, classmates, academics, and school and is presumed to create ties to an institution and influence willingness to do the work. finally, cognitive engagement draws on the idea of investment; it incorporates thoughtfulness and willingness to exert the effort necessary to comprehend complex ideas and master difficult skills. (p. 60) considering these three components of engagement simultaneously allows us to deeply understand young people’s attitudes and actions around mathematics and develop a fuller picture of their mathematics identities and the socialization process that aids them in seeing themselves as doers of mathematics. while studies of mathematicians often describe their experiences in graduate school and within the profession (e.g., burton, 2004; herzig, 2004) and some research addresses the early socialization experiences of mathematicians, these studies do not critically examine the spaces in which these early socialization experiences walker cultivating mathematical identities bullock, e. c., alexander, n. n, & gholson, m. l. (eds.). (2012). proceedings of the 2010 philadelphia and 2011 atlanta benjamin banneker association conferences – beyond the numbers [special issue]. journal of urban mathematics education, 5(2). 69 occur. i argue that, for black mathematicians in particular, the locations that facilitate engagement are weighted with important historical, social, and cultural overtones that, for some, may be unique to their experiences as blacks pursuing mathematical excellence. related to these ideas about where mathematics is taking place is the notion of how those doing mathematics see themselves and are seen. the notion that one’s mathematical identity might have to be reconciled with one’s core identity—be it ethnic, gender, or otherwise—has also gained prominence in the literature (boaler & greeno, 2000; nasir & saxe, 2003). some of the research relating to ethnic identity and academic achievement suggests that students of color must negotiate multiple identities, at times compromising their ethnic identity in order to fully embrace their academic identity (e.g., fordham & ogbu, 1986). others have suggested that these identities overlap in positive ways (e.g., floresgonzalez, 1999; horvat & lewis, 2003). largely missing from these discussions, however, is how one’s mathematical identity might be formed and developed— and evolve—over time. in asking mathematicians about their formative experiences, i hope to contribute to our understanding of how one’s mathematics identity might shift and evolve over time, and how these shifts are related to one’s experiences within mathematical spaces. a particularly interesting facet of in-school learning versus out-of-school learning is the usual characterization of in-school learning as being focused on individual cognition, while out-of-school learning is seen as developed via shared cognition (resnick, 1987; masingila, davidenko, & prus-wisniowska, 1996). this “shared cognition” lends itself to martin’s (2000) formulation of identity and socialization being informed by community and interpersonal contexts. this study seeks to address these issues through an examination of the questions below: 1. what experiences contribute to mathematicians’ positive mathematics identity development and socialization? where do these experiences occur? 2. what are key characteristics of spaces that facilitate mathematics identity development and socialization? method the data presented here come from a larger ongoing study of african american mathematicians. the participants in this study are 27 african american mathematicians, all of whom were born in the united states, and whose phds in mathematics or a mathematical science were granted between 1941 and 2008. this is a purposeful sample; participants were identified using resources including the website created and developed by dr. scott williams (himself a black mathematician), mathematicians of the african diaspora (http://math.buffalo.edu/mad), as well as the text black mathematicians and their works (newell, gipson, rich, http://math.buffalo.edu/mad walker cultivating mathematical identities bullock, e. c., alexander, n. n, & gholson, m. l. (eds.). (2012). proceedings of the 2010 philadelphia and 2011 atlanta benjamin banneker association conferences – beyond the numbers [special issue]. journal of urban mathematics education, 5(2). 70 & stubblefield, 1980), published in 1980. in addition, once the initial pool of subjects was identified, snowball sampling (by which participants identified other mathematicians) was used to augment the sample. table 1 characteristics of the sample of black mathematicians male female phds 1940s–1970s 6 1 phds 1970s–1990s 7 2 phds 1990s–2000s 8 3 total 21 6 interviews were conducted using a semi-structured, open-ended interview protocol developed by me, and lasted between 45 minutes and 3 hours. most interviews lasted at least an hour. the interview questions focused on mathematicians’ early experiences with mathematics, in and out of school, as well as their later educational and professional experiences. interviews were recorded, transcribed, and coded by a research assistant and myself. we first coded interviews broadly for early episodes and experiences where a mathematician was describing doing or learning about mathematics. within these narratives, we examined the texts for incidences in childhood and adolescence that related to mathematics learning and categorized those as occurring within school (within or outside of the mathematics classroom) or outside of school (within some academic setting or not). after this initial coding, we examined these narratives within and across locations (in-school, out-of-school, and in-between spaces) for themes relating to aspects of mathematics identity and socialization, focusing on engagement (emotional, behavioral, and cognitive aspects) as well as racial, social, historical, and cultural themes relating to mathematics. after exploring these narratives for common as well as conflicting themes, i then purposefully selected 6 representative vignettes from 4 mathematicians’ narratives to describe and explore spaces in which mathematical identities are cultivated, both inside and outside of school, during childhood and adolescence. all mathematicians have been identified using pseudonyms. two of the mathematicians, eleanor gladwell (phd 1970s) and wayne leverett (phd 1960s), came of age in the 1950s, attended rural segregated schools in the south, were undergraduates at historically black colleges and universities, and were among the first african americans to integrate their previously allwhite graduate institutions. one, nathaniel long (phd 1980s), grew up in a multiracial, working class, urban neighborhood in the north and attended a predominantly white college and graduate school in the north. the remaining mathematician, craig thomas, earned his phd in the 1990s. he attended predominantly walker cultivating mathematical identities bullock, e. c., alexander, n. n, & gholson, m. l. (eds.). (2012). proceedings of the 2010 philadelphia and 2011 atlanta benjamin banneker association conferences – beyond the numbers [special issue]. journal of urban mathematics education, 5(2). 71 black elementary and secondary schools in a southern city, and attended a historically black college and predominantly white graduate school for the phd. “here it was in action”: cultivating mathematics identity in out-of-school spaces in vignette 1, nathaniel long talks about his experiences growing up in a multiracial working class neighborhood in a northern industrial city: i grew up in a—well, call it little italy—a mostly italian and some irish and german, but all catholic [neighborhood]. and then there were a smattering of black families. my mother actually grew up [on the block] the generation before, so they all knew each other. it was very close-knit. one of the kids, henry, was about five years older than me and he would play ball with us younger kids…i would play chess with him, and he started giving me these little puzzles. he would give me little problems to work on, little brain teasers and that sort of thing, which i was able to solve. henry was very interested in mathematics. he ended up majoring in mathematics and became a math teacher at a secondary school. at any rate, i think henry fletcher [pseudonym] was a very profound influence. even when we played sports, it was always correctness. a lot of kids just want to win, and there was always a sense of winning by the rules. it would be second or third down, and we would carefully reconstruct what had happened to make sure that we had the down right. he would go examine the sideline to make sure the ball was not out of bounds. there was always this sort of rigor to what actually happened. “were you tagged before? where was the ball when you were tagged? were you beyond the pole or not beyond the pole?” at any rate, i think that was a very positive influence on me in my early to late teens. nathaniel long’s framing of his neighborhood as one that supported intellectual engagement seems to contradict much of what has been popularly described about the lack of support for educational activities in predominantly black settings. long’s description helps to develop a “counterweight corpus of scholarship” (morris, 2004, p. 72) that challenges this notion (anderson, 1988; hilliard, 2003; morris, 2004; perry, 2003). in his narrative, long is careful to construct his experiences within the context of a “world within a world” that was predominantly black, describes the role of the “atmosphere” in contributing to intellectual development, and uses mathematical language to describe even the ways in which he and his peers played street football—down to the “rigor” of play and rules for use in determining outcomes. prominent in this narrative is the importance of one person, henry fletcher, an older peer who began “giving [long] these little puzzles” to solve. fletcher’s later becoming a mathematics teacher could almost have been predicted by long’s story. other mathematicians in the sample, like long, described the importance of older peers (siblings and cousins as well as classmates and friends) in creating walker cultivating mathematical identities bullock, e. c., alexander, n. n, & gholson, m. l. (eds.). (2012). proceedings of the 2010 philadelphia and 2011 atlanta benjamin banneker association conferences – beyond the numbers [special issue]. journal of urban mathematics education, 5(2). 72 environments that supported mathematics learning and engagement. these experiences took place at home, in school, on playgrounds, and in other settings. a few younger mathematicians, particularly those who lived in predominantly black neighborhoods and attended predominantly white schools, noted that they had a range of experiences with their “school friends” and “neighborhood friends”. they “just played” with their neighborhood friends and tended to have academic interactions with their “school friends”. in vignette 2, wayne leverett describes two key mathematics experiences that occurred outside of school and his realization that mathematics was a viable career option: one thing i remember is when i was in about the ninth grade, my uncle worked for a construction company. he saw the foreman using a slide rule. he just got curious about it, so the foreman said, “well, next time i place an order for equipment, i will order you one if you’d like.” so the slide rule came with a thick manual about trigonometric functions and those such things. it was way over my uncle’s head. he was a carpenter. on the gi bill he got trained to do carpentry. in the family, people thought that i was some sort of bookworm because i was always reading books. so he just gave it to me. i wanted to get to the basics of the thing. i wanted to understand it, so i actually read the manual. i knew enough algebra and trigonometry to figure out most of the scales. for me it became a hobby. one day, a good buddy of mine and i were idling time away walking down a country road headed home. we came upon a little white man who was surveying some land. he needed two strong fellows to help him pull some chains. he told us that he would pay us $.75 per hour to do this. this is a lot more than you could make working on the farm. you could earn two or three dollars a day by working on the farm, but here is a guy who is going to pay $.75 per hour. i thought that this was an enormous sum of money to pull these chains… when this guy started talking to us, he had a transit. he would set it up and sight through here and swing around through a certain angle and sight through there. he could compute the distance between two far away points. when he found out that i knew a little trigonometry, he started teaching me how to use this transit. he was so impressed with me and i was so amazed by how much money you could make using this trigonometry. so i said right away that i wanted to be an engineer because i thought that engineers made even more money than high school math teachers. i wanted to be a civil engineer. so this was a moving experience. i wish that students at the tenth grade level could see something like this, where “here is something i am learning in school that is being used to earn money.” meeting that engineer who was surveying land... he was friendly enough to teach me things about how he was actually measuring the distance, and in doing this without having to jump across that ditch over there to get to. now we had studied about triangles and all—if you know this side and you know this side and you know the angle between you can get the length of the third side and all that. but here it was in action. this was very powerful. walker cultivating mathematical identities bullock, e. c., alexander, n. n, & gholson, m. l. (eds.). (2012). proceedings of the 2010 philadelphia and 2011 atlanta benjamin banneker association conferences – beyond the numbers [special issue]. journal of urban mathematics education, 5(2). 73 embedded throughout wayne leverett’s interview are references to the importance of his family in facilitating his learning. another uncle marched him to a college registrar’s office and informed the registrar that leverett was a top student and should therefore be admitted to that particular college. he was. at that college, leverett benefited from a strong mathematics program. leverett—as described later in vignette 5—also describes the very important role of his high school teachers in his mathematics development. but in this vignette, leverett describes how two critical experiences with mathematics—both outside of school—contributed to his understanding of mathematics, and further, how mathematics was done in the real world. further, leverett expresses a wish that secondary school students could have this kind of real-world experience, where they can see the usefulness and importance of the mathematics they are learning in school to real problems in the world outside of school. his career—in private industry and as an academic—reflects these early experiences. several mathematicians of all ages speak to the importance of mathematics exposure outside of school, whether or not it was rooted in real-world contexts. one mathematician recalled she and her siblings embarking on home improvement projects with their father that related to mathematics; another described an experience attending a lecture in high school that discussed the still unsolved problems in mathematics that piqued his interest, and helped him to realize that mathematics was not “just a toolkit.” craig thomas’ experience with his grandfather below echoes these points. in vignette 3, craig thomas describes a mathematical experience he had with his grandfather in a southern city: my grandfather lived right around the corner from here [the college where thomas is now a professor]. i remember he would always have these mental challenges that he would give me all the time…i actually use one of them in particular [when i’m teaching]. we were on the front porch and he was asking me—he was saying, if he walked halfway to the end of the porch, and then halfway again, and then halfway again, and so on, how many steps would it take him to reach the end of the porch? and so, i may have guessed five or something, i don’t know. so then he actually proceeded to do it, halfway, and then halfway, and then halfway, but the idea was that he was converging—he didn’t use the term convergence, but he never actually reached it—but he got closer and closer and closer, and of course he didn’t say within epsilon… but anyway, i have fun when i’m teaching about convergence to really tap into it at this early level. one just because i have fun telling the story—but also to give my students an idea of the sorts of things they can do with their students, because some of them may go on to become teachers, or just with their grandchildren one day, whatever the case may be. these are the sorts of things that can really bring high level things in very early and just challenge the mind and make you think. walker cultivating mathematical identities bullock, e. c., alexander, n. n, & gholson, m. l. (eds.). (2012). proceedings of the 2010 philadelphia and 2011 atlanta benjamin banneker association conferences – beyond the numbers [special issue]. journal of urban mathematics education, 5(2). 74 this experience is so vivid in thomas’s memory that he shares it with students in his classes. he recognizes that this story is about more than mathematics: in sharing it, thomas underscores the importance of passing along mathematical ideas, that mathematics doesn’t just happen in school, and that his students can use accessible examples of mathematics when interacting with their own students or family members to illuminate complex mathematical ideas. these three vignettes reveal that, for these mathematicians, opportunities to engage in mathematics occurred within very disparate experiences that were all rooted in cultural contexts. for long, the peer culture that had developed in his neighborhood, perhaps as an offshoot of the close relationships among the mothers, supported intellectual pursuits within multiple contexts—promoting adherence to rigorous rules while playing sports but also engaging in games and puzzles. for leverett, his uncle’s curiosity and recognition that leverett could benefit from a book with trigonometric formulas, and further, the convergence of key events—his school learning, his own out-of-school book learning, and a chance meeting with someone who used mathematics in his career—all contributed to leverett’s understanding that there was more to do in life than work as a farm laborer, which was the most visible career opportunity for african americans in the rural south in the 1950s and 1960s. for thomas, the opportunity that his grandfather gave him to think about mathematics in a deep way while lounging on a porch one afternoon—not just focusing on drills and number sense, but in thinking about some complex mathematical concepts—has, according to him, had an impact on how he thinks about his own teaching and mathematical development. what is notable about all three vignettes is that these experiences contributed to the development of the three mathematicians’ mathematical selves. further, the people involved in these vignettes who have a great deal to do with how these mathematicians think about their early experiences with mathematics range from close family members (thomas’s grandfather and leverett’s uncle), to peers (long’s “mentor”), to individuals that are never seen or heard from again (leverett’s surveyor). within these out-of-school spaces—a porch, a field, a neighborhood street—there were opportunities to learn mathematics, to develop rigorous mathematical thinking, and to learn habits of mind that contributed to these mathematicians’ development. “that was it: i could do math”: cultivating mathematics identity within schools many successful adults can point to the critical role of teachers in their lives, and every mathematician that has participated in the larger study points to key experiences and relationships with dynamic and charismatic “teachers” both within and outside of school as being integral to their success. these teachers walker cultivating mathematical identities bullock, e. c., alexander, n. n, & gholson, m. l. (eds.). (2012). proceedings of the 2010 philadelphia and 2011 atlanta benjamin banneker association conferences – beyond the numbers [special issue]. journal of urban mathematics education, 5(2). 75 might have been peers as in the case of nathaniel long, traditional schoolteachers, or out-of-school adults like craig thomas’s grandfather, who were instrumental to these mathematicians’ development. but in this section i focus on in-school experiences, and how classroom teachers crafted mathematical spaces that were meaningful to these mathematicians. eleanor gladwell illustrates in vignette 4 how school relationships have an impact on one’s mathematics identity and describes the power of learning beyond the typical and traditional classroom teaching/learning dynamic: i had a high school math teacher who was a younger man that was recently out of college, so that means he had lots of energy and enthusiasm about mathematics. and that was about the time of sputnik. so they had all these institutes around the country to try to increase interest in math, and they had a lot of teacher’s institutes. and so he would go to those. i think it’s because of him that i really excelled in math in high school. for example, when i got to trigonometry, the county would not allow them to teach trigonometry because there were not enough students—you had to have enough for a big class. well, he only had five, six, or eight, you know. so he decided that we needed trigonometry to go to college. so he agreed that if our parents would bring us back in the evenings, he would teach trigonometry. and they did. that’s how we learned, that’s how we got our trig. when mr. holly said i could do math, that was it, i could do math. so i never thought that was strange at all. and the high school teachers, they all told us we could do whatever we wanted to, you know. so in a school with mostly black teachers, you got the message that you just needed to work hard and you could do whatever you wanted to do. what is notable about this story is mr. holly’s commitment to ensuring, despite policy constraints, that students he felt could benefit from having extra mathematics would get it-even outside of school hours. in addition, it underscores the parents’ commitment (siddle walker, 1996) to helping their children get the education they needed during an era of rigid racial segregation. like long, gladwell talks about the importance of the community in supporting intellectual endeavors—in this case the parents of the students making sure that they were able to take advantage of mr. holly’s after school instruction. but gladwell goes on to talk about the larger cultural context of her experience: the educational leaders that small, southern, predominantly black towns and communities have created. in addition, gladwell’s telling of this story about mr. holly reveals something about her mathematics identity. in a field where much of the discourse about women in the field focuses on their supposed lack of self efficacy, gladwell talks about how growing up in a segregated era posed clear challenges related to her race and gender. but she also describes how teachers who believed in her and demonstrated that belief in tangible ways, made her believe in herself and her walker cultivating mathematical identities bullock, e. c., alexander, n. n, & gholson, m. l. (eds.). (2012). proceedings of the 2010 philadelphia and 2011 atlanta benjamin banneker association conferences – beyond the numbers [special issue]. journal of urban mathematics education, 5(2). 76 mathematical talent, too. it was, therefore, not at all “strange” that she did well in mathematics. many mathematicians also describe teachers (of mathematics as well as of other subjects) who ensured that students were exposed to extracurricular mathematics opportunities outside of school—through enrichment programs, afterschool or before-school clubs, or summer activities. in vignette 5, wayne leverett continues his narrative, building on his story about the surveyor and the slide rule to one about the importance of his secondary school teachers in his mathematical development: but this slide rule was one of my first memories about experiences that got me hooked on math for sure. at school when the teachers discovered that i could use this thing, they were quite amazed. i remember maybe in the tenth grade algebra class, she [the teacher] gave me half the class [to teach]. my first memory of doing math [in school] was as a show off. i was having fun, but i think the fact that the teachers gave me praise really encouraged me to do a bit more. when we were taking algebra, mrs. barr gave me a college algebra book because i think she feared that i could keep up with the regular algebra easily. she gave me a college algebra book and would check off a couple of problems and say, “see if you can do these tonight.” i would go home determined to do them because i wanted to stay in her good graces. she thought i was smarter than i was and i wanted to keep it that way. so i would work on the problems, sometimes, half the night before i would figure out how to solve them, but i would come in the next day as if i had solved them in 15 minutes. “here is the solution, give me some more.” i managed to keep that going until i graduated…[a]t the end of the year when i tried to return the book, she said, “wayne, you keep that book. it will do you more good than it will do me.” i thought it was such a great treasure to have her book. there was another math teacher at school who did pretty much the same thing, except that he collected his books back at the end of the year. his name was mr. barr. i do feel that i had some sort of special treatment that at least two teachers at a very small school noticed that i had some abilities and they did it on their own. they didn’t get extra pay, but they were essentially giving me after school tutoring. nobody, not even the principal, [knew] that these things were going on. so i never have enough praise for those two teachers. the only thing that i have taken to everywhere i go is to remember what teachers did for me when i was in high school. because if burgess had ignored me, or if barr had ignored me, or [his college mathematics professor], i don’t know where i would be today. i certainly wouldn’t be here. so when i see a student who has some ability and is trying, i always try to pull them aside and do something special. i keep looking for students to befriend and yes, i try to find a good student to mentor and watch them and see how they grow. both gladwell and leverett talk about these particular high school experiences without mentioning much about the other students who were in their classes. but teachers are not the only actors within schools who have an influence on students. walker cultivating mathematical identities bullock, e. c., alexander, n. n, & gholson, m. l. (eds.). (2012). proceedings of the 2010 philadelphia and 2011 atlanta benjamin banneker association conferences – beyond the numbers [special issue]. journal of urban mathematics education, 5(2). 77 in vignette 6, craig thomas talks about his peers and the supportive culture for his mathematics work at his high school: one thing i often remember: i was in 8 th grade, i believe, and i was on a little local television show, a little game show. and the thing i remember about this, i [made it to the final round], and if i had gotten my [last] question right, i would have won. i remember getting to school the next day and i remember ladonna rogers [a pseudonym], she was one of those who would just make your life miserable. she was loud, she was just one of those who would make life hard, just because she could. and i remember i was so stunned—ladonna rogers came up to me and said, “we were rooting for you! i said, he’s in my homeroom!” i was so stunned—she liked to take digs at everybody and me, especially, and she’s up here rooting for me because i’m in her homeroom. and i remember that sticking out in my memory, that she was proud of me, i guess, or something, and that i guess i could infer from other experiences that there was more of that than i realized [at the time]. as thomas’ vignette 6 describes, the multiple roles of peers—in childhood and adolescence, but also continuing in adulthood—is a theme in many mathematicians’ interviews. some mathematicians describe separate peer groups for academic and social pursuits, others describe close knit peer groups that had a strong academic as well as social focus. peers, in many cases, are key sources of mathematics instruction and inspiration. as one mathematician revealed, his older classmate’s admonitions were rooted in the social context of the school, which had been recently desegregated: he said, “look, you have a responsibility.” i still remember to this day he says, “you’re better than any of us in terms of doing this stuff.” and he says, “you’re probably better that most of the white students.” he says, “you got to stay number one, and you also have an obligation to help, you know, to tutor and stuff like that.” so you know, anybody that was kind of interested i would help them. not because of him, i would have done that anyway. but i did feel this obligation because he would monitor what i was doing….he told me that that was my obligation, and i kind of believed him. there were times when i kind of didn’t feel like studying, i would kind of like hear his voice. which was really kind of interesting to me. and i wouldn’t remember his name or his face if he would walk up to me now at all. in a way i would like to thank him. however, gladwell, leverett, and thomas’s experiences also speak to the importance of mathematics teachers going beyond the prescribed curriculum to engage students’ mathematical interests and potential. in all three vignettes, as gladwell says, there is a “message”—“you just needed to work hard and you could do whatever you wanted to do”—about mathematics that is being sent to gladwell, leverett, thomas and their fellow students. in gladwell’s and leverett’s case, this message is directly rooted in the historical context of segregated schooling in the south. to counter this, gladwell’s teacher enlisted the communi walker cultivating mathematical identities bullock, e. c., alexander, n. n, & gholson, m. l. (eds.). (2012). proceedings of the 2010 philadelphia and 2011 atlanta benjamin banneker association conferences – beyond the numbers [special issue]. journal of urban mathematics education, 5(2). 78 ty’s aid in preparing students for college mathematics and leverett’s teachers gave him a very strong message about their beliefs in his ability, establishing him as co-teacher and giving him college-level work. thomas’s teachers, in an era where school segregation was unlawful but still present, sent a message not just to thomas, but to his peers: that his mathematical project is important enough to be presented to fellow students during instructional time. all of these teachers are using modes of instruction and induction into mathematical practice that are occurring inside as well as outside of the traditional mathematics classroom space. spaces in between: building bridges across multiple worlds, modes, and identities these six vignettes reveal that spaces and identities are not necessarily discrete. in-school and out-of-school spaces overlap, and the participants within these spaces take on different roles. for example, leverett’s learning of trigonometry took place in several settings and through several modalities: through his uncle, who gave him a slide rule manual with trigonometric functions; through school, where he learned mathematics in his classroom and from his teachers’ out-ofschool tutoring; and through the surveyor, who gave him practical experience but also reinforced mathematics content leverett had learned and fostered leverett’s understanding of possibilities of careers using mathematics. in addition, leverett in vignette 5 serves as student and apprentice teacher in his mathematics classroom, as does thomas in vignette 6 in his mathematics class and other classes. although gladwell’s trigonometry class occurred as a traditional teacher-students mathematics class arrangement, it occurred after school, when parents, teacher, and students all had to make special arrangements to participate. all of these characteristics facilitated mathematics learning and socialization within school, outside of school, and in “in-between” spaces. there was formal and informal mathematics learning, as well as important experiences that occurred outside of school that contributed to knowledge and understanding of school mathematics concepts. second, several vignettes echo previous findings in a study with high achieving high school students (walker, 2006) that show that persons and relationships from multiple worlds formed academic communities (comprising family members, peers, teachers, and others) who had an important impact on the mathematicians’ development. in addition, as i discovered in the high achievers study, the persons providing support or socialization opportunities for the mathematicians are not necessarily themselves mathematics teachers or mathematicians. in fact, some of these persons are those who would be considered “uneducated”, or “undereducated”, in the formal sense by many. walker cultivating mathematical identities bullock, e. c., alexander, n. n, & gholson, m. l. (eds.). (2012). proceedings of the 2010 philadelphia and 2011 atlanta benjamin banneker association conferences – beyond the numbers [special issue]. journal of urban mathematics education, 5(2). 79 third, the importance of opportunity in contribution to mathematics socialization for these mathematicians is key. much has been made of the concept of “opportunity to learn” and specifically, opportunity to learn rigorous mathematics. however, as i define the concept here, this idea is not limited to within school opportunities to learn, but opportunities in and outside of school that are both presented to and sought by mathematicians as adolescents. the mathematicians in this study—and the four whose vignettes are presented here—describe multiple opportunities and contexts in which to learn mathematics. mathematics learning occurred both outside of school (in family settings, with friends, through random interactions), within school (in mathematics classrooms, through school projects), and in in-between spaces (after school sessions and informal mathematics teaching and learning spaces in school buildings). conclusion these narratives suggest that we can be much more successful in improving mathematics outcomes and fostering interest in mathematics by rethinking how and where mathematics learning and practice occur, and where one’s mathematics identity is developed. to do this, this study of mathematicians suggests that we should build on out-of-school spaces that support mathematics socialization and also re-imagine the mathematics classroom to be a space that not only provides opportunities to learn meaningful mathematics, but supports mathematics identity development and positive socialization experiences. but we also have to think about how we insure that meaningful mathematics occurs beyond fleeting conversations, students’ individual experiences, and the spaces in which they happen to find themselves. our expectations of students’ abilities are key—if we think students have potential and if they are worthy of our attention in spaces that support mathematics learning, we become much more intentional and purposeful about creating these spaces. this is true in all four mathematicians’ vignettes—whether it is teachers, peers, or relatives contributing to early mathematics development. thus, opportunities to engage in meaningful mathematics have to have intentionality and purpose, and should not solely be haphazard or happenstance. for too many of our students, particularly our underserved black and latino/a students, these opportunities are limited. evidence shows that for black and latino/a high school students attending urban schools, even those with strong mathematics identities and positive socialization experiences, opportunities to learn mathematics may not be equivalent to those at other schools serving predominantly white students. opportunities to take advanced level mathematics classes, for example, are not equal across affluent and poor schools. elsewhere, i have written about how schools might maximize opportunity for underrepresented high school students (walker, 2007b) sug walker cultivating mathematical identities bullock, e. c., alexander, n. n, & gholson, m. l. (eds.). (2012). proceedings of the 2010 philadelphia and 2011 atlanta benjamin banneker association conferences – beyond the numbers [special issue]. journal of urban mathematics education, 5(2). 80 gesting that schools do the following to foster opportunity and belongingness: expand our thinking about who can do mathematics, build on students’ existing academic communities, learn from schools (particularly college and university programs) that promote mathematics excellence for underrepresented students, expand the options in school mathematics courses and enrichment opportunities, and reduce underrepresented students’ isolation in advanced mathematics settings. experiences that promote mathematics socialization, identity development, and learning are critically important, because in the lives of black mathematicians they have resonance. to wit, within these mathematical spaces, there are experiences that mathematicians had as young people that continue to resonate with them years later. these experiences contribute not only to their own construction of self as a mathematics doer, but also to their knowledge of mathematics content. further, these past experiences can contribute to how they think about their own practice as mathematicians. for example, thomas’s experience with his grandfather on the porch “shows up” in his classes when he is teaching the concept of convergence; leverett shares his surveyor experience as an example of “mathematics in action”. gladwell is renowned as a mentor to young mathematicians, and her experience with mr. holly may influence how she conceives of her work mentoring graduate students. in a program she has co-developed, the emphasis is on giving participants a “head start” on graduate level work, in much the same way mr. holly ensured that she and her peers would be prepared for college work. what appears to contribute greatly to a mathematical space’s resonance is the presence of key individuals and/or relationships. in all of these experiences, whether described by mathematicians or high achieving high school students, it is not just the space—the neighborhood, the school, the classroom—but it is also the significant relationships and experiences with family members, peers, teachers, mentors, and others in these spaces that are remembered. these relationships and experiences, built on high expectations, contribute to these individuals’ practices, and the larger practices, of the mathematical communities to which they belong (walker, 2007a, 2012). much of the research describing factors that contribute to the high achievement of underserved students in mathematics, in particular, points to the importance of relationships that are both personal and relate to the content (berry, 2008; moore, 2006). the process of crafting intentional spaces, rather than allowing for (or hoping for the possibility of) inadvertent spaces, in which young people learn and practice mathematics, develop a strong mathematics identity, and are inducted into a community of mathematics doers, i argue, must attend to these issues of opportunity and resonance. interviews with black mathematicians reveal that early on there were strong influences on their mathematical development and how they think about mathematics both inside and outside of school. what was surprising was that both out-of-school experiences and in-school experiences appear walker cultivating mathematical identities bullock, e. c., alexander, n. n, & gholson, m. l. (eds.). (2012). proceedings of the 2010 philadelphia and 2011 atlanta benjamin banneker association conferences – beyond the numbers [special issue]. journal of urban mathematics education, 5(2). 81 to be influential to talent and identity development, and at times, out-of-school experiences, however fleeting, had a seemingly lasting impact on black mathematicians’ conceptions of self, mathematics, and their mathematics ability. with these ideas in mind, schools can provide spaces for students that ensure that they feel that they rightfully belong to a community of mathematics doers, that they have opportunities to engage in meaningful mathematics, and that their experiences in these spaces are resonant and contribute to the development of their identities as mathematical persons. at the symposium during the symposium, i presented work from a study of black american high achievers—high school students and mathematicians—focusing on key mathematical spaces that fostered their mathematics excellence. during the discussion, we discussed links between mathematical spaces and young people's mathematics engagement, identities, and socialization, and how experiences in these spaces serve as counternarratives to the dominant discourse about high mathematics achievement that ignores mathematics excellence among underserved students. we had the opportunity to brainstorm about ways that schools, communities, and neighborhoods could develop formal and informal spaces that support mathematics learning, and discussed how more research in these settings could facilitate better understanding of how to improve mathematics teaching and learning for black students in particular. acknowledgments i am thankful to all of the mathematicians who are participating in this study for their generosity in sharing their narratives with me. i thank carol malloy, robyn brady ince, and lalitha vasudevan for their helpful comments on an earlier version of this paper. references anderson, j. 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(2012). building mathematics learning communities: improving outcomes in urban high schools. new york, ny: teachers college press. journal of urban mathematics education december 2009, vol. 2, no. 2, pp. 46–65 ©jume. http://education.gsu.edu/jume robert m. capraro is an associate professor of mathematics education and co-director of the aggie stem center at texas a&m university, department of teaching, learning, and culture, 4232 tamu, college station, tx 77843-4232; e-mail: rcapraro@tamu.edu. his research interests are centered on stem educational research initiatives, urban mathematics achievement and representational models, and quantitative methods. jamaal rashad young is a doctoral student in the department of teaching, learning, and culture at texas a&m university, 324 harrington tower, college station, tx 77843; email: jamaal-rashad-young@neo.tamu.edu. his research interests include technology integration and utilization in mathematics classrooms. chance w. lewis is the houston endowment, inc., endowed chair and associate professor of urban education in the department of teaching, learning, and culture in the college of education at texas a&m university, 4232 tamu, college station, tx 77843-4232; e-mail: chance.lewis@tamu.edu. his research interests are centered around the improvement of academic achievement for students of color, particularly african american students. zeyner ebrar yetkiner is a graduate student in the department of teaching, learning, and culture at texas a&m university, rudder tower 607, tx 77843-1360; email: zeyetkiner@hotmail.com. her research interests include quantitative research methods. melanie n. woods is a doctoral student in the department of teaching, learning, and culture at texas a&m university, 308 harrington tower, college station, tx 77843; email: mnwoods@tamu.edu. her research interests include teacher education reform and conceptual development in mathematics education. an examination of mathematics achievement and growth in a midwestern urban school district: implications for teachers and administrators robert m. capraro texas a&m university jamaal rashad young texas a&m university chance w. lewis texas a&m university zeyner ebrar yetkiner texas a&m university melanie n. woods texas a&m university in this article, the authors investigate the achievement gap in the context of a particular region and the factors associated with student learning in that region. data were collected over several years from recent administrations of the mathematics section of the measurement of academic progress in colorado. black and hispanic mathematics achievement and growth were compared to white student achievement and growth. the results indicate that gaps exist not only in mathematics achievement but also in mathematics growth. a statistically significant difference in mathematics growth rates between black and hispanic students from different economic backgrounds were found; however, a statistically significant difference in mathematics growth rates by gender was only found in black and hispanic third grade students. the authors provide explanations as well as implications of the factors associated with the results with the hope of influencing research and practice. keywords: achievement gap, gender differences, high-stakes testing, mathematics, reform curriculum, urban education capraro et al. mathematics achievement and growth journal of urban mathematics education vol. 2, no. 2 47 he no child left behind act of 2001 (nclb) 1 placed particular attention on the disaggregated results of student performance on state developed assessments. under nclb, the results of state assessments are analyzed along with er academic indicators, such as student attendance rates, student enrollment in advanced placement courses, and graduation rates to create an adequate yearly progress (ayp) profile that serves as a school ―report card‖ each academic year. the ayp academic indicators are designed to allow parents, community leaders, and school-district personnel to more objectively identify academic areas of strength, as well as academic content areas needing improvement (simpson, lacava, & graner, 2004). one of the primary components of ayp is the percentage of students being evaluated academically that meet the academic benchmark of proficient in each tested content area. data from the ayp profile are used to evaluate each local school site and the school districts’ ability to meet the academic needs of all subpopulations of students. each subpopulation of students is expected to improve by a certain percentage each year. based on the data provided from the various reports, this percentage of improvement is used to determine whether or not a school is consistently improving. typically, schools are measured on their ability to increase the percentages of particular subpopulations that perform at the proficient level (mccall, kingsbury, & olson, 2004). a ranking system, usually regulated by state education officials, is then used to determine whether a school’s accreditation should come into question by district and state education officials. a significant portion of school’s accountability structure is generated by this ranking system. this accountability structure requires that all educators and administrators critically evaluate the performance of all students; however, it is not uncommon that many of the school districts identified for improvement based on ayp are large urban districts that serve black and hispanic students (tracey, sunderman, & orfield, 2005). owens and sunderman (2006) found that the schools most likely to be identified as ―needing improvement‖ are highly segregated and enroll a disproportionate share of the state’s minority and low-income students. data from the national assessment of educational progress (naep) suggested that despite the efforts of nclb, the black–white and hispanic–white achievement gap in mathematics remains unchanged (lee, 2006). the naep was administered in grades 4 and 8 with slight fluctuations in student subgroup performance. for example, results of the naep indicated that white–black and white–hispanic gaps among 4th and 8th graders did not narrow meaningfully between 2003 and 2005 in mathematics (lee). the results of the naep also indicated that the racial gap change in mathematics between 2003 and 2005 was not statistically significant. however, a two-point reduction was found in the differ 1 no child left behind act of 2001, public law 107-110, 20 u.s.c., §390 et seq. t capraro et al. mathematics achievement and growth journal of urban mathematics education vol. 2, no. 2 48 ence between average mathematics achievement scores between white and hispanic students in grade 8 (lee). a feasible explanation for the current trends in mathematics achievement may be found in a thorough investigation of early mathematics achievement and mathematics growth. unfortunately, many black and hispanic students may enter school with several academic ―risk‖ factors that can inhibit their initial academic achievement and may translate to slower mathematics growth rates (rathbun, west, & walston, 2005). other factors such as previous exposure to mathematics and lack of adequate resources are usually outside the control of the student; however, these factors may influence the growth at which these students master mathematic skills and concepts. despite the differences in mathematics achievement and growth, nclb requires that students reach high academic standards and that all students progress at an acceptable rate. furthermore, the nclb act states that parents have the right to receive educational vouchers to transfer students to different schools at the expense of the current school district if the school district fails to improve the performance of all subpopulations to the degree specified. students should have an opportunity to be adequately educated by neighborhood schools. increasing the achievement of all students in mathematics begins with early recognition of mathematics deficiencies and evaluation of not only mathematics achievement but also mathematics growth. furthermore, educators, administrators, and researchers may learn valuable information about the achievement of black and hispanic students by investigating early trends in mathematics growth. as a result, the purpose of this study is to compare the mathematics achievement and mathematics growth of minority students and their white peers in an urban school district in colorado. the skills that students possess when they enter elementary school and their academic progress while in elementary school have a great impact on subsequent academic outcomes and experiences (national association for the education of young children [naeyc]; national council of teachers of mathematics [nctm], 2002). thus, this study seeks to explain student achievement across grade levels in regards to closing the achievement gap among constituents in a large urban school district, particularly black and hispanic students, who are usually impacted the most by standardized testing under nclb. factors impacting mathematics achievement and growth initial achievement and mathematics growth kindergarten students enter schools from various backgrounds and academic skills. initial academic differences may equate to differences in achievement and mathematics growth. some suggest that achievement trajectories may vary between different subgroups (jordan, kaplan, olah, & locuniak, 2006). initial capraro et al. mathematics achievement and growth journal of urban mathematics education vol. 2, no. 2 49 academic achievement differences in primary school are most pronounced between poor students and their more affluent counterparts and between minority and white students (benson, borman, & wisconsin center for education research, 2007). students who enter school with varying degrees of mathematical knowledge may gain mathematics skills differentially than their peers. for example, if one student enters kindergarten with a firm understanding of the concept of quantity, then he or she is at an advantage because any further enrichment adds to the student’s foundational understanding. several empirical studies indicate that initial performance predicts positive subsequent academic growth (aunola, leskinen, lerkkanen, & nurmi, 2004; bodovski & farkas, 2007; rescorla & rosenthal, 2004). the opposite was found for some students that entered school with lower initial mathematics achievement. fan (2001) suggested that some students are faced with ―double barreled‖ barriers of low initial performance and lower growth rates than their peers. yet, some students may enter school with low mathematics achievement but progress at nearly the same rate as their peers. ding and davison (2005) suggested that students can enter school with lower initial achievement and manage to progress at a rate that is not statistically significantly different than their peers. however, because of their lower level of initial achievement, the students were unable to reach the same academic levels as their peers. students identified as limited english proficient (lep) and students in special education have particular difficulties closing the initial gap in achievement (ding & davison). initial achievement differences do not account for all the subsequent variation in student academic progress and achievement; however, it puts the student at a disadvantage early in the educational pipeline. environmental factors affecting mathematics growth students enter the public school system with one or more factors that may contribute to lower academic achievement in mathematics (rathbun et al., 2005). specifically, coming from poverty, status as a racial or cultural minority, having parents who did not complete high school, and having parents who speak a language other than english in the home can negatively influence academic achievement and growth (croninger & lee 2001; natriello, mcdill, & pallas, 1990; rathbun & west, 2004). the aforementioned risk factors for lower academic achievement can possibly affect any student regardless of race or ethnicity. when considering the effects of language on mathematics performance students whose native language is not english had substantial difficulties on the mathematics portion of the naep (abedi, lord, & plummer, 1997). due to these factors, initial academic differences in some cases are more profound for some groups of students as opposed to others. capraro et al. mathematics achievement and growth journal of urban mathematics education vol. 2, no. 2 50 the underachievement in mathematics for african american students is widely discussed in extant literature due to the perceived gap (lubienski, 2002) in academic performance between african american students and their white counterparts. the factors frequently discussed in literature are related to the cultural, educational, and psychological barriers linked to the education of african americans in society. while there is widespread agreement about the role each of these plays in the schooling experiences of african american students, there are no conclusive findings about which factors have the greatest influence on their mathematics performance in school. there is an excess of research about the affect cultural differences have on students of color during their educational experiences. specifically, their cultural background sometimes determines to what extent these students have enough cultural capital (bourdieu, 1977) to navigate an educational system that may be foreign to them upon entering school. that is, roscigno and ainsworth-darnell (1999) discovered that lower ses blacks lack the resources to take family trips, purchase computers, and other resources needed to be successful in the classroom. in an effort to close the achievement gap, ladson-billings (1995) and tate (1995) discussed the need for more culturally relevant pedagogy for african american and latino students. arguably, students of color are often challenged by the instructional practices presented by white teachers unfamiliar with their students’ cultural backgrounds. consequently, the classroom becomes an environment where students of color are tracked into lower academic tracks (ladson-billings, 1997) and decline in taking upper-level mathematics courses in high school and college (davenport, davison, kuang, ding, kim, & kwak, 1998). as the dialogue continues regarding the widening mathematics achievement gap between black and white students, some researchers find this dialogue creates an internal psychological dilemma for students of color and how they perform in classroom environments. the dilemma, according to spencer, steele, and quinn (1998) is one to do with the perceptions held about groups of people (gender and race) and the targeted group not necessarily believing what is thought about them but simply having knowledge that these thoughts exist to the extent that it hampers performance. steele (1992, 1997) argued that minority students are more likely to experience what is known as stereotype threat because their intellectual ability continues to be compared to that of high-achieving white and asian students. moreover, osborne (2001) studied the effects of anxiety as a way to explain racial and gender differences in academic achievement of high school seniors and found that white students had less anxiety in mathematics as compared to their african american and latino counterparts and the difference was significant with respect to women learning mathematics. aside from being considered as racial or ethnic minorities, many hispanic students are considered language minorities (lm) as well. hispanic lm students capraro et al. mathematics achievement and growth journal of urban mathematics education vol. 2, no. 2 51 represent a highly diverse population in terms of socioeconomic status, linguistic and cultural background, level of english proficiency, amount of prior education, and instructional program experience (crawford, 2004). the aforementioned circumstance may have a significant effect on student mathematics achievement and growth. hispanic lm students enter kindergarten with fewer mathematics skills compared to other non-hispanic lm students, and these trends persist at least through the first grade (ready & tindal, 2006). this result suggests that lm status is an added challenge for hispanic students in the primary grades. saxe (1988) suggested that the effect of language on mathematics achievement could be direct (intrinsic) or indirect (extrinsic). among the extrinsic influences of language on mathematics achievement are: (a) entry mastery, (b) opportunities to learn, (c) motivational factors, and (d) measurement factors. according to saxe, the effects of their language status on classroom activities influence lm students’ mathematics achievement. for example, entry mastery is associated with the effects of different degrees of language competence on the influence of mathematics instruction for some students. many lm students receive mathematics instruction from a bilingual mathematics educator who may not be as competent in mathematics as other educators, which in turn affects the student’s opportunity to learn. hispanic lm students face a different set of challenges than other students whose home language is not english, due in part to the unfortunate reality that a larger percentage of hispanic lm students are affected by poverty (collins & shay, 1994; iceland, 2003; jargowsky, 1997; staveteig & wigton, 2009). the possible lack of financial resources may influence the hispanic lm students’ access to mathematics as well as language resources to enhance their academic performance. ruiz (1988) proposed that there are three basic orientations toward language diversity; these orientations were ―language as a right,‖ ―language as a problem,‖ and ―language as a resource.‖ furthermore, ruiz suggested that school programs in the united states have a history of embracing the language as a problem perspective. thus, instead of utilizing the student’s native language as a foundational resource, many educators as well as policymakers perceive that language is the problem. all children deserve an opportunity not only to learn but also to be successful regardless of their race, culture, socioeconomic status, or native language. examining the early trends in mathematics achievement and growth of the previously mentioned populations of interest may lead to a better understanding of the current and past trends in mathematics achievement, as well as vital instructional knowledge for educators and administrators. nonetheless, the education of all students despite initial achievement level or environmental risk factors is the responsibility of the instructional staff and school administration to overcome. thus, the factors that influence student mathematics achievement and growth at the school level are discussed in the following section. capraro et al. mathematics achievement and growth journal of urban mathematics education vol. 2, no. 2 52 research questions 1. what are the initial differences among different ethnic groups’ achievement in mathematics on measurement of academic progress (map) in grades 3, 4, and 5? 2. what are the differences among different ethnic groups’ growth in mathematics as measured by map in grades 3, 4, and 5? 3. what are the initial differences in mathematics achievement of non-english proficient (nep), limited english proficient (lep), fluent english proficient (fep), and native english speaker hispanic students on map in grades 3, 4, and 5? 4. what are the differences in mathematics growth of hispanic students with varying english language proficiencies as measured by map in grades 3, 4, and 5? methods participants measure of academic progress (map) growth scores were available for 2110 third graders (1010 female students and 1100 male students), 2209 fourth graders (1037 female students and 1172 male students), 2161 fifth graders (1056 female students and 1105 male students) in a large urban district during the 2005– 2006 academic year. the data collected were from two time periods to estimate learning trajectories for asian (4.1%), black (20%), hispanic (51.8%), and white (24.1%) students in grades 3, 4, and 5. the native american students, who comprised only a small proportion of the district (i.e., 0.8%), were not included in the analyses because of the imprecision in their parameter estimates. of all the students, 58.6% were eligible for free lunch with the highest percentage being within hispanic students (75.1%), and 8.5% were eligible for reduced lunch with the highest percentage again being among hispanic students (9.6%). special education students comprised 9.2% of all students in the dataset. these special education students were categorized as students with emotional (1.0%), perceptual (6.2%), and speech/language disabilities or disorders (2.0%). instrument the map—a multiple-choice, computer-based assessment administered to students in grades 2–10 (the northwest evaluation association [nwea], 2000)—is administered statewide in colorado. the nwea created the map for colorado aligned with the colorado state academic standards. it is different from conventional assessments because the map was developed to place student achievement and item difficulties on the same scale based on item response theory. the map is one measure for determining if a student has made one year’s growth in reading and mathematics. the test-retest reliability of the mathematics portion of map for grades 3 through 5 in spring 2002 changed from the mid .80s to the low .90s. capraro et al. mathematics achievement and growth journal of urban mathematics education vol. 2, no. 2 53 table 1 descriptive statistics on achievement and percent growth rates across grades 3, 4, and 5 by gender and ethnicity g r a d e ethnicity n pre-test mean pre-test sd post-test mean post-test sd % growth mean 3 asian female male black female male hispanic female male white female male 77 39 38 413 192 221 1133 543 590 487 236 251 187.12 187.51 186.71 179.68 181.59 178.03 178.52 179.28 177.82 186.86 186.23 187.45 12.98 11.03 14.86 11.85 11.28 11.59 11.39 10.76 11.91 12.85 11.91 13.67 200.70 199.74 201.68 192.30 193.93 190.87 191.60 192.07 191.16 199.48 198.32 200.57 12.79 10.44 14.91 12.99 12.01 13.93 12.58 11.73 13.31 13.03 12.16 13.74 109.19 98.73 119.92 92.40 92.23 92.54 93.69 92.62 94.68 100.73 96.17 105.01 4 asian female male black female male hispanic female male white female male 109 56 53 449 211 238 1126 532 594 525 238 287 198.94 198.34 199.57 190.32 190.65 190.03 190.27 190.15 190.38 198.02 197.39 198.55 13.50 12.29 14.77 12.87 13.25 12.54 12.67 11.93 13.31 12.58 11.76 13.21 209.48 208.90 210.19 199.47 200.18 198.85 199.99 199.86 200.10 207.50 207.41 207.58 15.79 14.78 16.90 14.17 14.28 14.07 13.31 12.31 14.16 13.77 13.45 14.06 110.9 112.3 109.42 88.27 92.23 84.77 93.52 93.62 93.43 98.50 105.23 92.92 5 asian female male black female male hispanic female male white female male 78 39 39 435 196 239 1100 537 563 548 284 264 206.65 207.97 205.33 198.89 199.46 198.41 198.82 198.90 198.74 206.41 206.74 206.06 13.61 13.31 13.94 12.51 11.65 13.18 12.09 11.71 12.46 12.56 10.41 14.53 216.51 219.59 213.44 207.82 208.79 207.03 207.36 207.70 207.03 216.22 216.34 216.08 15.79 13.19 17.65 13.79 13.32 14.14 12.95 12.36 13.49 13.60 11.72 15.39 109.08 127.57 90.61 94.78 100.09 90.42 90.46 93.3 87.75 107.56 106.18 109.05 capraro et al. mathematics achievement and growth journal of urban mathematics education vol. 2, no. 2 54 results differences in mathematics achievement and growth by ethnicity the preand post-test means and sds by gender within each ethnic group at each grade level are reported in table 1. standard deviations are an artifact of the number of items on a test. therefore, when scores are high so too are the standard deviations. for example, on a 5-point likert scale one would expect standard deviations less than 1, whereas on an i.q. test where the range is between 80 and about 130, one would expect a standard deviation in the tens place. to determine the differences between ethnic groups’ mean scores on the pretest in third grade, which is the first administration of map, confidence intervals were calculated. in general, across all analyses, asian and white students outperformed black and hispanic students. as shown in figure 1, black and hispanic students have statistically significantly lower mean scores as compared to their asian and white peers. in third grade, hispanics have a much smaller variance as compared to the other three groups but also a noticeably lower mean. figure 1. 95% cis for mean achievement scores in grade 3 by ethnicity on map pretest. when examining the analyses for fourth and fifth grade, the trend of asian (m4th grade = 198.94, sd = 13.50; m5th grade = 206.65, sd = 13.61) and white (m4th grade = 198.02, sd = 112.58; m5th grade = 206.41, sd = 12.56) students outperform capraro et al. mathematics achievement and growth journal of urban mathematics education vol. 2, no. 2 55 ing black (m4th grade = 190.32, sd = 12.87; m5th grade = 198.89, sd = 12.51) and hispanic (m4th grade = 190.27, sd = 12.67; m5th grade = 198.82, sd = 12.09) students held with no noticeable closing of the gap the confidence intervals change little and were not depicted. the fifth-grade analysis showed the width of the cis remained consistent across grades, and the relative performance also remained consistent without any noticeable difference in the achievement gap. the mean and the standard deviation of growth rates as measured in percentages across grades 3, 4, and 5 are presented in table 1. to compare the growth rate of students from different ethnicities across grades 3, 4, and 5, cis around the mean growth rates are provided in figure 2. these mean growth rates were calculated in percentages. the state established an expected growth for each student based on a set of normative tables that differentiate the average growth by grade level and starting point score. mean growth percentages were calculated based on what percent of the state established growth the student achieved. for example, if a student whose expected growth rate was 12 points had exactly 12 point increase would have a mean growth of 100%. one important characteristic of cis is they encourage meta-analytic thinking and contribute to cumulative knowledge (thompson, 2006). thus, cis help us to compare the growth rates of each ethnic group across grades 3, 4, and 5 and obtain a plausible range of the population parameters. a comparison of the first three cis in figure 2, which belong to asian students in grades 3, 4, and 5 respectively, provides evidence that their mathematics growth rate measured by map in the population may range from 96.75% to 121.62%. for white students this range is from 100.45 to 105.76 as seen in figure 2. when the cis in figure 2 for black and hispanic students are compared within themselves, the plausible range for population mathematics growth may range from 86.74 to 96.15 for blacks and from 90.27 to 95.29 for hispanics. it is clear in this analysis that black and hispanic students are not achieving their expected growth rates. differences in mathematics achievement and growth by gender within each ethnicity to determine the differences in mathematics achievement between genders within each ethnic group, cis for mean scores on the pretest in third, fourth, and fifth grades are provided in figures 3, 4, and 5, respectively. in third grade, gender difference was found within black and hispanic students. third-grade black and hispanic male students preformed statistically significantly lower than their female counterparts on map pretest. however, such a gender difference was not found in fourth or fifth grades. capraro et al. mathematics achievement and growth journal of urban mathematics education vol. 2, no. 2 56 figure 2. 95% cis for mean mathematical growth rates across grades 3, 4, and 5 by ethnicity as measured by map. figure 3. 95% cis for mean achievement scores in grade 3 by gender and ethnicity on map pretest. capraro et al. mathematics achievement and growth journal of urban mathematics education vol. 2, no. 2 57 figure 4. 95% cis for mean achievement scores in grade 4 by gender and ethnicity on map pretest. figure 5. 95% cis for mean achievement scores in grade 5 by gender and ethnicity on map pretest. capraro et al. mathematics achievement and growth journal of urban mathematics education vol. 2, no. 2 58 to compare the growth rates of female and male students from different ethnicities across grades 3, 4, and 5, cis around the mean growth rates are provided in figure 6. no statistically significant difference between genders was found regarding their mathematics growth as measured by map in third, fourth, and fifth grades within each ethnicity. figure 6. 95% cis for mean mathematical growth rates across grades 3, 4, and 5 by gender and ethnicity as measured by map growth. differences in mathematics achievement and growth by ses within each ethnicity the ses status was determined by students’ being eligible for free, reduced, or paid lunch. across third, fourth, and fifth grades, students from low ses families (i.e., students eligible for free lunch) achieved statistically significantly lower than their peers who were from higher ses families (i.e., students who get paid lunch) within each ethnicity (see figures 7, 8, and 9). white students from low ses families scored similar to their hispanic and black peers from high ses families across all grades. capraro et al. mathematics achievement and growth journal of urban mathematics education vol. 2, no. 2 59 figure 7. 95% cis for mean achievement scores in grade 3 by ses and ethnicity on map pretest. figure 8. 95% cis for mean achievement scores in grade 4 by ses and ethnicity on map pretest. capraro et al. mathematics achievement and growth journal of urban mathematics education vol. 2, no. 2 60 figure 9. 95% cis for mean achievement scores in grade 5 by ses and ethnicity on map pretest. a comparison of the mean growth rates of black and hispanic students from low ses families across grades suggested that their mathematics growth rate measured by map in the population might range from 75.00% to 95.58% and from 88.32% to 94.46%, respectively, which were both below the expected growth. for black and hispanic students from higher ses families, the plausible range for population mathematics growth might range from 80.58% to 108.55 and from 91.52% to 102.11%, respectively. differences in mathematics achievement and growth by english language proficiency status within each ethnicity to examine the initial differences in mathematics achievement of nonenglish proficient (nep), limited english proficient (lep), fluent english proficient (fep), and native english speaker students on map in grades 3, 4, and 5, cis are provided in figure 10. the narrower cis for hispanic students at all levels of english proficiency and for black and white native speakers reflected the precision of the parameter estimates. in other groups, less precision was obtained due capraro et al. mathematics achievement and growth journal of urban mathematics education vol. 2, no. 2 61 to the small sample sizes in each of these groups. non-english and limited english proficient students from every ethnicity were statistically significantly behind their fluent english proficient and native speaker peers on the map pretest. figure 10. 95% cis for mean achievement scores in grades 3, 4, and 5 by english language proficiency status and ethnicity on map pretest. conclusion the growth rate analysis suggests black and hispanic students start each grade throughout elementary school behind their asian and white peers in regards to their mathematics achievement. moreover, black and hispanic students have mathematics growth rates that are lower than their expected growth rates (i.e., less than 100%) as well as less than their asian and white peers making it virtually impossible for black and hispanic students in this colorado district to catch their asian and white counterparts. capraro et al. mathematics achievement and growth journal of urban mathematics education vol. 2, no. 2 62 it is worth noting that mathematics achievement and growth rate can be unstable; therefore, it may not truly reflect the performance of the students under investigation. linn and haug (2002) suggest that the achievement gain scores can be volatile and suggest that accuracy of results can be improved by combining results across different grades or years. in this study, achievement scores were combined across several grade levels; thus, the results are reasonably reliable. several factors may contribute to the achievement gap and persistent difference in mathematics growth presented in this study. in this discussion, initial achievement was presented as a possible factor contributing to the increasing mathematics achievement gap. the results of this study further suggest that ses has a dramatic affect on the mathematics achievement and growth of black students. this result coupled with cultural differences that are exasperated in many traditional classrooms may inhibit the ability of many black students from mathematics excellence. students in ell programs face different challenges that may contribute to their gaps in mathematics achievement and growth. language is typically presented as a major contributing factor to the lack of growth in language arts as well as mathematics for non-native english speakers. escamilla, mahon, rileybernal, and rutledge (2003) claimed that the hispanic achievement gap could not be attributed to language issues alone. the authors suggested that the structure of the assessment systems might inhibit hispanic students’ ability to meet the academic standards. in particular, the exemption process may prevent hispanic students from receiving the same quality of instruction as other students in the same institution. thus, many of these students are not given the opportunity to improve their skills because their learning is systematically constrained. one suggestion is to adjust the current educational policy that influences the systematic constraints on hispanic students’ achievement. native language assessments, portfolios of academic progress, or language simplified test in english may be reasonable policy reform suggestions (mahon, 2006). the hispanic ell’s in this study may be confronted with very specific factors, but other factors can influence both black and hispanic student populations. educators, administrators, and parents should remain cognizant of the many factors that influence student mathematics achievement and growth. the results presented here indicate that a gap in the mathematics achievement of black and hispanic student begins early in their academic career. the results also suggest that if left unattended, this trend can continue because the mathematics growth rates of black and hispanic students are lower than their asian and white peers. furthermore, the results are far more detrimental to black and hispanic male students. proper identification and remediation may help to better prepare these students for subsequent mathematics assessments. furthermore, educational policy reform may enhance the learning opportunities of hispanic students. capraro et al. mathematics achievement and growth journal of urban mathematics education vol. 2, no. 2 63 references abedi, j., lord, c., & plummer, j. r. 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(2005). changing nclb district accountability standards: implications for racial equity. cambridge, ma: the civil rights project at harvard university. commentary: the new political economy of urban education and mathematics educaiton journal of urban mathematics education december 2012, vol. 5, no. 2, pp. 6–17 ©jume. http://education.gsu.edu/jume pauline lipman is professor of educational policy studies and director of the collaborative for equity and justice in education, university of illinois at chicago. her multidisciplinary research focuses on race and class inequality, globalization, and political economy of urban education, particularly the inter-relationship of education policy, urban restructuring, and the politics of race. her newest book is the new political economy of urban education: neoliberalism, race, and the right to the city (routledge, 2011). commentary neoliberal urbanism, race, and equity in mathematics education pauline lipman university of illinois at chicago n a germinal book on urban education, gerald grace (1984) argued against the prevailing instrumental “policy science” approach to the study of urban education problems in the united states and united kingdom. drawing on sociologist c. wright mill’s critique of “abstracted empiricism,” grace rejected policy science’s “technical and immediately realizable” within-the-system solutions to urban education problems abstracted from the urban context (p. 32). he proposed a “critical policy scholarship” that situates urban education in the social, economic, political, and cultural contexts shaping the city. critical policy scholarship illuminates the material and cultural struggles in which schooling is located and is generative of social action toward social justice (p. 41). an underlying assumption is that policy is an expression of values arising out of specific interests and relations of power. grace notes critical policy scholarship requires a multidisciplinary approach that draws on urban studies as well as urban sociology. in a somewhat similar vein, rury and mirel (1997) argue, “educational researchers [in the united states] too often accept the urban environment as a given natural setting, rather than one that has itself been determined by larger economic and political processes” (p. 85). what is needed, they argue, is a political economy of urban education that examines the contested dynamics of power and wealth that shape the urban context, its historical dimensions, and how it is articulated spatially. rury and mirel propose that we place questions of power, particularly the role of capital and race in structuring urban space, at the center of the education research agenda. in june 2009, i attended a national conference of grass roots education activists, youth, parents, and teachers. we traded similar stories from dallas, new york, new orleans, philadelphia, detroit, and chicago. school districts disinvesting in schools in black and latina/o neighborhoods and then closing them, students transferred across the city, increases in violence, charter school expansion, attacks on teacher unions, no real public participation, gentrification of african american and latina/o working class neighborhoods and families pushed farther out of the city. our conversations coalesced around similar questions: why is this i lipman commentary journal of urban mathematics education vol. 5, no. 2 7 happening? what is the relationship of school closings and the gentrification and economic polarization of our cities? how can we collectively fight for truly quality education for low-income children of color in this new environment and what would that education look like? it was these questions that compelled me to write the new political economy of urban education: neoliberalism, race, and the right to the city (lipman, 2011). today, at the end of 2012 and four years into the deepest economic crisis since the great depression, these questions are even more pressing—and more generalized to cities of all sizes. the city of detroit is “clear-cutting” whole neighborhoods, closing dozens of neighborhood schools and expanding charter schools; philadelphia’s school district is being transformed into a marketplace of school operators; and african american and latina/o students in cities large and small face school closings and disinvestment in their communities while processes of gentrification and privatization of public services are the norm. without a deeper understanding of the political and economic forces influencing schools, we can neither uncover the genesis of these education policies nor effectively advocate for more equitable and just policies and practices. in this short commentary, following from the book, i want to argue that it is necessary to examine the intersection of education policy and urban restructuring in order to understand what is happening in urban schools, and by implication, to assess the possibilities for more equitable and socially and culturally relevant mathematics education. neoliberal urbanism, race, and education neoliberalism is the defining social paradigm of the past 30 years—a free market strategy to manage the structural crises of capitalism by providing new opportunities for capital accumulation (harvey, 2005). put simply, neoliberalism is an ensemble of economic and social policies, forms of governance, discourses and ideologies that promote free markets, individual self-interest, unrestricted flows of capital, deep reductions in the cost of labor (by lowering wages and benefits and dismantling unions), and sharp cuts in government responsibility for social welfare. neoliberals champion privatization of social goods and retrenchment of government from provision of housing, health care, and education on the premise that competitive markets are more effective and efficient. but neoliberalism is not just “out there” as a set of policies and explicit ideologies; it has generated a new common sense about social institutions and social relations. it has reshaped identities—who we are as academics, teachers, students, parents, school administrators. many readers of jume know that over the past 30 years, public education in the united states has been radically restructured in accord with a bi-partisan neoliberal agenda. the pillars of that agenda are top-down accountability and lipman commentary journal of urban mathematics education vol. 5, no. 2 8 standards, education markets, and business practices and ideologies. it features mayoral control of school districts, closing public schools or handing them over to corporate-style “turnaround” organizations, expanding school “choice” and charter schools, instituting teacher incentive pay based on test scores, and diminishing the power of teacher unions. the most comprehensive manifestations of this agenda are the bush administration’s no child left behind act and obama administration’s $4.35 billion federal education stimulus package known as race to the top. this radical restructuring of public education is a rendition of a global project to gear education to “economic competitiveness” and to impose market discipline on all aspects of schooling (compton & weiner, 2008; gutstein, 2010; rizvi & lingard, 2009). these policies and the contestations over them shape the debate on urban education today. although much has been written about this issue, not enough attention has been paid to ways in which education policies, and their contestations, are intertwined with the radical economic, political, and spatial transformations of cities. building on the work of critical urban political economy and geography and critical analyses of race, i argue that education is both shaped by and deeply implicated in the processes that are reshaping cities and that have produced and intensified inequality and marginalization. these processes coalesce in the neoliberal restructuring of the city, or neoliberal urbanism, and its racialized dynamics (brenner & theodore, 2002). in the new political economy of urban education, i look particularly to chicago as a laboratory for the articulation of education policy and the contested political and economic dynamics of neoliberal urbanism. neoliberal urbanism and race [c]ities (including their suburban peripheries) have become increasingly important geographical targets and institutional laboratories for a variety of neoliberal policy experiments, from place-marketing and local boosterism, enterprise zones, tax abatements, urban development corporations, and public-private partnerships to workfare policies, property redevelopment schemes, new strategies of social control, policing and surveillance and a host of other institutional modifications within the local state apparatus. the overarching goal of such experiments is to mobilize city space as an arena both for market-oriented economic growth and for elite consumption practices. the manifestations of destructively creative neoliberalization are evident across the urban landscape: the razing of lower income neighborhoods to make way for speculative development; the extension of market rents and housing vouchers; the increased reliance by municipalities on instruments of private finance; the privatization of schools; the administration of workfare programs; the mobilization of entrepreneurial discourses emphasizing reinvestment and rejuvenation; and so forth. (peck, brenner & theodore, 2008) lipman commentary journal of urban mathematics education vol. 5, no. 2 9 the neoliberal city is an entrepreneurial city driven by market ideologies and the regulatory power of global finance. this is so not only for “global cities,” such as new york, london, and chicago, but for economically devastated urban centers such as post-katrina new orleans and detroit and smaller cities across the united states. as peck, brenner, and theodore (2008) describe, cities are laboratories for neoliberal economic and social experiments and their attendant polarities of wealth and poverty, centrality and marginality that characterize u.s. society as a whole. these contrasts, especially sharp since the deep recession of 2008, are on display in wealthy gated communities and policed low-income african american and latina/o areas, glittering downtowns and disinvested working class neighborhoods, and elite consumers juxtaposed with a new low-wage workforce (primarily women, people of color, and immigrants) and the permanently unemployed (particularly african american males and displaced older workers). cities are key sites for flexible, unregulated labor, privatization of public infrastructure (e.g., roads, bridges, parks, schools, hospitals), cuts in social welfare, and neoliberal forms of governance. neoliberal city governments make policy decisions based on satisfying investors and real estate developers and promoting growth in corporate investment and elite consumption (high-end housing, retail, and leisure). they rely on property and real estate taxes and debt financing. bond rating agencies such as standard & poor’s, the gatekeepers of global capital markets, have become the institutional regulators of city governments as municipal debt (in the form of bonds and municipal securities) is traded in the global financial markets (hackworth, 2007). the new logic of urban government is: anything that hurts investment is “bad” for bond ratings and thus “bad” urban policy. the enduring history of racism has been pivotal to the neoliberal agenda (wilson, 2007). conditions for neoliberal urban restructuring were set by racial segregation of cities and post world war ii policies that led to white flight to suburbs, disinvestment in “inner cities,” and urban decline. in turn, neoliberal policy has intensified structural inequality based on race. while some people of color gained greater access to education and employment, and a few amassed wealth in the boom years of the 1990s, the vast majority bore the brunt of deindustrialization, cuts in social welfare, attacks on unions, and intensified policing. racial inequality in income is greater today than 40 years ago, and criminalization and mass incarceration of african americans and latinas/os afflicts low-income urban neighborhoods, especially children. ideologically, racism is the subtext for insistence on “personal responsibility” and ending “dependency” on the state. constructing people of color as the undeserving poor (lazy, pathological, and welfare dependent) provides a rationale to restructure or eliminate government-funded social programs and to discredit social welfare (katz, 1989). the cultural politics of race provide the ideological lipman commentary journal of urban mathematics education vol. 5, no. 2 10 framework to privatize urban public institutions, including schools. in particular, public institutions identified with the “inner city”—housing, schools, hospitals— are pathologized by a racially coded discourse of failure and dysfunctionality that legitimates their dismantling and/or privatization. this racialized logic has justified the wholesale displacement and relocation of thousands of public housing residents and closing their schools (bennett, smith & wright, 2006; lipman, 2011; greenbaum, 2006). based on the logic of “deoncentrating poverty,” people have been dispersed and relocated, without self-determination, while their neighborhoods are redeveloped for a new class of residents (imbroscio, 2008). it is these urban conditions that have given rise to demands for affordable housing, anti-eviction campaigns, and campaigns to defend public clinics. neoliberal urbanism, school closings, and gentrification i argue that urban education is shaped by, and implicated in, the contested economic and social dynamics that are reshaping cities. gentrification and the policy to close “failing” neighborhood schools is a clear case (greenlee, hudspeth, lipman, smith, & smith, 2008). gentrification (replacement of working class housing with housing for the more affluent) is a pivotal sector in urban economies (smith, 2002). reliance on property and real estate taxes to fund public services and to collateralize municipal bonds makes cities dependent on, and active subsidizers of, the real estate market. gentrification also drives displacement, shortage of affordable housing, homelessness, and exclusion of workingclass and low-income people from the city itself (hackworth, 2007). policies to close schools are implicated in the gentrification process. schools are community anchors, particularly in disinvested communities facing loss of affordable housing, foreclosures, lack of jobs and public services, and overall destabilization. policies that destabilize schools undermine this important role, particularly as the economic crisis further threatens working-class and low-income students and families. closing neighborhood schools is a key lever pushing out low-income people. the schools that replace them are refurbished and re-branded to attract a new clientele—stripped of their associations with the low-income communities that were displaced. in chicago, philadelphia, new orleans, detroit, and other cities strategies to close neighborhood schools are integral to plans for mega-real estate development projects (buras, randels, salaam, & students at the center, 2010; cucciara, 2008; pedroni, 2011). chicago’s renaissance 2010 plan to close 60–70 neighborhood schools is an example. planners (city officials in partnership with corporate partners) targeted the first stage of renaissance 2010 to the african american bronzeville area where thousands of units of public housing were demolished to make way for a government-subsidized private development of primarily middleclass and high-end housing (lipman, 2011). new schools were an explicit ele lipman commentary journal of urban mathematics education vol. 5, no. 2 11 ment of redeveloping the area. rationales to gentrify neighborhoods and close schools are also linked discursively. the pathologization of disinvested lowincome african american and latina/o communities as “gang infested, bad neighborhoods” echoes the pathologization of their schools as “dysfunctional” and having a “culture of failure.” this is paired with the regenerative discourse of gentrification as “rebirth”—minus the people who have lived there. demonization and deficit discourses attached to young people of color are part of the coercive culture of too many urban schools and low-level academic tracks and are deeply intertwined with the demonization of their communities. just as being uprooted from one’s home and neighborhood tears apart the web of human connections that constitute communities and ground lives, so does closing schools that have anchored communities. these schools might have been provided the resources and support to improve. some have symbolized african american communities’ strength, endurance, and intellectual and cultural achievements. the decision to close schools without community participation is plainly coercive, and racist, and has evoked a storm of protest in many cities. the psychic trauma and insecurity that is the “collateral damage” of these policies (lipman, person, & kenwood oakland community organization, 2007) reverberates in children’s educational experiences and life chances. in urban school districts today, any effort to address educational inequity is frustrated by this reality. this is why opposition to school closings and demands for community voice in school improvement are emerging as a new civil rights struggle, as reflected in the journey for justice, a national campaign against school closings led by parents and students from 19 cities (see www.youtube.com/watch?v=lioqulbfreo). from the standpoint of equity, the wave of school closings has been devastating, leading parents and community organizations to propose models of community-driven school transformation as an alternative to closing schools. since 2001, the chicago board of education has closed, consolidated, or turned around (fired all adults and turned over schools to an outside operator) 105 schools. eighty eight percent of the affected students are african american, and schools with more than 99% students of color have been the primary target (caref, hainds, hilgendorf, jankov, & russell, 2012). many experienced years of disinvestment, loss of staff and programs, test-driven narrowed curriculum, and a revolving door of failed top-down mandates, programs, and supervisors (chicago teachers union, 2012; gutierrez & lipman, 2012; lipman, 2004). in chicago, school closings have led to spikes in violence and 80% of elementary students from closed schools were transferred to schools no better than the ones they left (gwynne & de la torre, 2009). during this period, racial gaps in achievement on the national assessment of educational progress have widened citywide. school closings and turnarounds also result in disproportionate loss of experienced african american teachers (caref et al., 2012) even as increasing the proportion of http://www.youtube.com/watch?v=lioqulbfreo lipman commentary journal of urban mathematics education vol. 5, no. 2 12 teachers of color is known to be an important aspect of furthering equitable and culturally relevant education. neoliberal urbanism, education markets, and corporate school reform turning over public goods and public services to the market is the dogma of neoliberal urban governance. on the assumption that the private sector is more efficient and productive than the public sector, city governments outsource public services to private operators, often eliminating public sector unions in the process. selling off public assets is a key source of city revenue, and privatization has escalated with the economic recession and budget shortfalls faced by over-leveraged city governments. under mayor richard m. daley, chicago privatized bridges, parking meters, public parking garages, schools, hospitals, and public housing. the city also outsourced trash pick-up service and some police functions to private companies and privatized parking at o’hare airport, city parking enforcement, street resurfacing, engineering, purchasing, vehicle towing, and delinquent tax collections. education is a key arena of urban privatization with charter schools, vouchers, and privatization of education services constituting a huge investment opportunity (burch, 2009; saltman, 2010). backed by gates, walton, and other venture philanthropies, charter schools have become the central vehicle to open up public education to the market and to weaken teachers unions. although in the early 1990s charter schools were developed as laboratories for educational innovation and teacher and community control, the charter school strategy has been exploited as the pathway to education markets and rearticulated to the interests of education entrepreneurs, venture philanthropists, investors, and corporate-style charter school chains. while charter expansion is national, the focus is urban school districts where disinvested, inequitable, and unresponsive public schools are fertile territory for education markets. in chicago, many elementary students have limited access to physical education, arts, library/media instruction, science laboratories, computer science, and world language classes; 160 chicago public elementary schools do not have libraries (chicago teachers union, 2012)—most are in low-income neighborhoods of color. at the same time, charter schools are subsidized by corporate philanthropies and, in some instances, the state. under chicago’s market-based renaissance 2010, chicago public schools (cps) closed over 100 public schools and opened 100 charter schools plus a number of selective and magnet schools. the goal is to make one-third of the district charter. in the urban change/regeneration discourse there is no alternative to marketdriven restructuring of schools, housing, neighborhoods, and downtowns. as renaissance 2010 rolled out in 2004, chicago’s mayor-appointed school board president (a real estate developer) characterized opposing parents as people “who don’t want change.” president obama evoked this trope when he contended criti lipman commentary journal of urban mathematics education vol. 5, no. 2 13 cism of race to the top “reflects a general resistance to change. we get comfortable with the status quo” (obama, 2010). in the face of an historically compounded “education debt” owed african american students (ladson-billings, 2006), neoliberal policies become the only option to “fix” urban schools, co-opting the discourse of “change” and “reform” to contain debate about the kind of change needed and who should participate. however, research indicates charter schools tend to exclude “difficult” students (miron, urschel & saxton, 2011), reinforce inequality (frankenberg, siegel-hawley & wang, 2010), are mostly non-union, and have high teacher turnover and more inexperienced teachers (caref et al., 2012). the experience of this reality is beginning to take the shine off charter schools, yet a public/private “portfolio” of schools is taken for granted in urban districts. the contention over education markets and other aspects of corporate education reform is about more than schools. it is about reshaping identities of students, parents, and teachers and how we envision society and our participation in it. the post-world war ii period of (limited) social welfare framed people as “citizens” with civil rights and the state as responsible for a level of social wellbeing. although women, people of color, non-english speakers, gays and lesbians, people with disabilities, and so on had to fight for inclusion, there were grounds to extend civil rights and economic justice. claims could legitimately be made on the state to better our social condition. in the neoliberal social imaginary, rather than “citizens” with rights, people are “empowered” as individual consumers in the marketplace of schools, healthcare, individual retirement accounts, and more. one improves one’s situation by becoming an “entrepreneur of oneself,” cultivating the image and resume that enhances one’s competitive position in the marketplace of “human capital.” schooling is organized around productivity (high test scores) and preparing a globally competitive workforce, not human development and social responsibility. schools are to be run like businesses, teachers treated as employees, education as a product, and school leaders as managers (increasingly recruited from the business world by the broad foundation’s corporate school leadership institute). the contradictions generated by closing public schools and degrading teaching and learning have given birth to resistance nationally and to a new antineoliberal, equity-centered “social movement unionism” in chicago. it was these contradictions that coalesced in the chicago teachers’ strike of 2012, which was about much more than wages and benefits (the only demands the union could officially strike over). teachers had had enough of disrespect, test-prep curriculum, school closings, and education inequity. and so had parents and students. picket lines at schools and mass rallies of over 30,000 that took over downtown chicago were joined by parents and students demanding art, music, school counselors, psychologists, no school closings, and an end to charter school expansion. the lipman commentary journal of urban mathematics education vol. 5, no. 2 14 connections between an inequitable and privatized city and an inequitable, privatized school system are quite clear. neoliberal urbanism and equitable mathematics education the contested space of neoliberal urbanism is the context within which efforts to enact equitable and socially just mathematics unfolds. curricular and pedagogical reforms have to contend with not only broader education policies but also the raced, classed, and gendered economic, social, and spatial effects of actions to remake the city with which they are intertwined. i am not a mathematics educator, so i defer to mathematics education researchers and practitioners to elaborate the implications for mathematics education in urban schools. but i would like to end with a couple suggestions. first, as the city becomes more inequitable, more unliveable for those at the bottom of the economic and racial hierarchy, the conditions of poverty and insecurity and violence faced by so many students acutely affect urban schools, broadening the need to pursue equity beyond curriculum and pedagogy to issues of poverty and housing and healthcare and justice. the intertwining of education and urban policy requires us to take a “critical policy scholarship” approach, to think beyond our fields, and to “connect the dots” between what is happening in mathematics reform and larger economic and social agendas. this calls for mathematics education researchers, faculty, and practitioners to engage the literature on the broader urban context and to examine how processes of neoliberal urbanism are unfolding in specific cities and their implications for public education, and mathematics education specifically. second, we have to ask, what is the relationship between specific mathematics education policies and “reforms” and broader structural and social changes in the city. is access for some students to more challenging curricula and more funding for mathematics education actually a move toward greater equity? to take an example discussed by gutstein (2010) in this journal, the emphasis on stem (science, technology, engineering, and mathematics) education and the common core state standards initiative (cssi) is driven by u.s. economic competitiveness goals. urban school districts support this agenda as part of marketing the city, through, for example, specialized math-science academies. but gutstein points to inherent inequities in this agenda in the context of a highly stratified and racialized labor force in which low-wage, low-skilled labor is the largest growing sector. this labor stratification and the growth of low-wage jobs is a central aspect of the economic and social inequalities of the city. moreover, he argues the emphasis on stem and cssi runs counter to a liberatory education that is mathematically rich and rigorous and helps students use mathematics to think critically about the issues of inequality and injustice that plague the city. in my opinion, it is this sort of approach—connecting urban conditions, broad social and educa lipman commentary journal of urban mathematics education vol. 5, no. 2 15 tional policy, and what is actually taught in schools—that is needed in mathematics education research, teacher preparation, and practice. finally, as public schools are closed, urban school systems are turned into “portfolio” districts, and “productivity” is the end goal of schooling, the space to discuss equity in public education is narrowed. parents are positioned as shoppers in an educational marketplace rather than members of society who deserve a quality, relevant education in their neighborhood and need to work collectively to realize it. teachers are subjected to the competitive pressures of performance pay for productivity gains (raising achievement), fear of school closings, and the expansion of non-union charter schools. on the other hand, the contradictions of neoliberalism and the emergent social movements they are engendering, create an opportunity for a more robust, broader, more radical (going to the root) struggle for educational equity—one that joins socially and culturally relevant, rich mathematics with the defence and transformation of public education itself. this suggests that mathematics educators committed to equity can not afford to stand outside the debate over education markets, marketization of public schools, and movements of parents and teachers to transform, rather than close them. moreover, it is not possible to separate the kind of urban schools—and the kind of mathematics education—we have from the kind of city we live in. this is an opportunity as well as a challenge for mathematics educators at all levels. references bennett, l., smith, j. a., & wright, p. a. 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(1997). the political economy of urban education. review of research in education, 22, 49–110. saltman, k. (2010). the gift of education: public education and venture philanthropy. new york, ny: palgrave macmillan. smith, n. (2002). new globalism, new urbanism: gentrification as global urban strategy. antipode, 34, 427–450. wilson, d. (2006). cities and race: america’s new black ghetto. london, united kingdom: routledge. journal of urban mathematics education december 2016, vol. 9, no. 2, pp. 81–121 ©jume. http://education.gsu.edu/jume luis a. leyva is an assistant professor in the department of teaching and learning at vanderbilt university, peabody college of education and human development, pmb 230, gpc, 230 appleton place, nashville, tn, 37212; email: luis.a.leyva@vanderbilt.edu. his research interests include issues of gender and sexuality in stem (science, technology, engineering, and mathematics) education, marginalized student populations’ experiences in undergraduate mathematics education, and gender-affirming and culturally responsive mathematics teaching and stem support programs. an intersectional analysis of latin@ college women’s counter-stories in mathematics luis a. leyva vanderbilt university in this article, the author discusses the intersectionality of mathematics experiences for two latin@ college women pursuing mathematics-intensive stem (science, technology, engineering, and mathematics) majors at a large, predominantly white university. the author employs intersectionality and poststructural theories to explore and make meaning of their experiences in relation to discourses of mathematics ability and pursuits of stem higher education. a cross-case analysis of two latin@ college women’s counter-stories details the development of successoriented beliefs and strategies in navigating the discourses that they encountered institutionally and interpersonally in their mathematics experiences. implications are raised for p–16 mathematics and stem education to broaden equitable learning opportunities for latin@ women and other marginalized groups’ construction of positive mathematics identities at intersections of gender and other social identities. keywords: gender, higher education, intersectionality, latin@ women, mathematics identity, undergraduate mathematics “she pursued a math career, culture caught up with her” – tracey, latin@ college woman xtant research has often adopted either gender or race for its lens of analysis in understanding experiences of marginalization among women and students of color in mathematics (see, e.g., berry, 2008; boaler, 2002; damarin, 2000; fennema, carpenter, jacobs, franke, & levi, 1998; martin, 2000; mendick, 2006; stinson, 2008; terry, 2011). research that focuses on a single dimension of identity, however, risks homogenizing group experiences and overlooking within-group differences for negotiating discourses1 in mathematics and society at large. 1 in this context, i draw on stinson’s (2008) definition of discourses as the “language and institutions as well as complex signs and practices that order and sustain sociocultural and sociohistorical constructed forms of social existence” (p. 977). e http://education.gsu.edu/jume mailto:luis.a.leyva@vanderbilt.edu leyva latin@ college women and mathematics journal of urban mathematics education vol. 9, no. 2 82 at the undergraduate level, scholars have largely focused on examining statistical trends of negative influences among women, african americans, and latin@s2 with limited analysis of factors for their retention and academic success in stem (science, technology, engineering, and mathematics; chapa & de la rosa, 2006; cole & espinoza, 2008; simpson, 2001). in efforts to disrupt discourses of stem underachievement and underrepresentation associated with these populations, it is also important for research to qualitatively unpack these student populations’ strategies of resilience and persistence in stem higher education at intersections of gender, race, and other dimensions of their social identities. researchers exploring equity issues in stem higher education have begun to leverage intersectional analyses to obtain nuanced understandings of social influences on retention and student experiences among underrepresented groups, particularly, historically marginalized women of color and members of the lgbtq+ (lesbian, gay, bisexual, trans*, queer/questioning, and other) community (camacho & lord, 2013; cech & waidzunas, 2011; espinosa, 2011; lord et al., 2009; reyes, 2011). mathematics education scholars have also called for such intersectional considerations of marginalized students’ strategies in negotiating mathematics experiences with gendered and racial discourses (esmonde, brodie, dookie, & takeuchi, 2009; lim, 2008; martin, 2009; oppland-cordell, 2014; zavala, 2015). i argue that intersectional analyses can nuance our understandings of mathematics as a gendered and racialized space across the p–16 school pipeline as well as inform ways to better support and broaden opportunities for minoritized populations in mathematics. coupled with a conceptualization of gender as socially constructed and discursively produced differently across contexts and individuals (butler, 1990), intersectional analyses allow for explorations of how the racialized masculinization of mathematics structures inequitable access and opportunities among african american and latin@ women—two marginalized populations whose experiences are largely unexplored in mathematics education research. in this article, i present case studies of two latin@ college women in their second semester of pursuing mathematics-intensive majors3 at a large, predominantly white institution in the northeastern united states. using poststructural theory (e.g., st. pierre, 2000) and intersectionality (crenshaw, 1989, 1991) as well as critical race theory (crt) methodology (e.g., solórzano & yosso, 2002), i closely examine these latin@ women’s counter-stories focusing on negotiations of their 2 drawing on gutiérrez (2013), the term latin@ decenters the patriarchal nature of the spanish language that traditionally groups latin american women and men into a single descriptor latino, denoting only men. the @ symbol allows for gender inclusivity among latin americans compared to the either–or form latina/o, implying a gender binary. 3 in the context of this study, mathematics-intensive is defined as stem majors requiring at least two semesters of calculus based on the university’s curriculum. such majors include but are not limited to astrophysics, chemistry, computer science, engineering, mathematics, and physics. leyva latin@ college women and mathematics journal of urban mathematics education vol. 9, no. 2 83 experiences in high school and first year of college with discourses in mathematics, stem higher education, and society more broadly. this analysis proposes the following questions to explore mathematics as a socially exclusionary space: 1. what are the dominant discourses of mathematics ability and stem higher education raised in the two latin@ college women’s counter-stories? 2. to what extent did they encounter these discourses in school and classroom structures as well as interpersonal relationships during high school and in college? 3. what strategies did they employ in making meaning of their experiences and navigating these discourses at intersections of their gender, race, and other identities? literature review in this section, i present a review of relevant literature starting with an overview of intersectionality theory (crenshaw, 1989, 1991). with intersectionality as a tenet of crt, i then examine the extent to which intersectionality has been explored in prior mathematics education research that adopted crt methodology (solórzano & yosso, 2002). i then review research outside of crt work to document how intersectionality has been leveraged in exploring issues of gender in mathematics education. i particularly focus on research about gender as it is the other dimension of social identity besides race that has been most widely studied in mathematics education as well as comprised the original intersection (race/gender) that motivated the theorization of intersectionality in black feminist thought. to conclude the review, i examine intersectional analyses across equity work in stem higher education to argue for the use of intersectionality in detailing mathematics as a socially exclusionary space and informing ways to better support marginalized populations, including women of color, during their transition into undergraduate mathematics education. intersectionality and black feminist thought intersectionality is a concept based in black feminist thought that was coined and adopted by crenshaw (1989, 1991) to detail intersectional forms of marginalization legally and politically experienced by historically marginalized women of color in the united states. crenshaw (1991) highlighted the importance of attending to the “intersecting patterns of racism and sexism” (p. 1243) that are often not reflected in feminist and antiracist discourses shaping legal and political structures. leyva latin@ college women and mathematics journal of urban mathematics education vol. 9, no. 2 84 although crenshaw’s legal analyses focus on black women’s lived experiences at intersections of race and gender, she acknowledged that the complexity of intersectional oppression can be further elucidated by attending to other dimensions of individuals’ social experiences including class, immigration, and sexuality (crenshaw, 1991). intersectionality then refers to unique forms of intersecting oppression that emerge from gender, race, class, sexuality, and other social categories “function[ing] as parallel and interlocking systems” (collins, 1993, p. 29) of domination and subordination. at intersections of gender and race, for example, hooks (1981) detailed the american black female experience during times of slavery, the civil rights movement, and feminist movements by attending to “both the politics of racism and sexism from a feminist perspective” (p. 13) rather than solely sexism. w. e. b. du bois similarly pursued an intersectional analysis of the black political economy with race, class, and nation as intersecting “social hierarchies that shaped african american access to status, property, and power” (collins, 2000b, p. 42). collins (2000b) critiqued du bois’ analysis as “progressive yet paternalistic” (p. 43). while du bois aligned with black feminist thought in his description of black women’s unique societal oppression, he viewed gender more as a “personal identity category” (p. 42) than as a system of power and thus was left unexamined in his intersectional analysis. crenshaw’s (1991) and collins’ (2000b) assertions for the analytical significance of centering intersections to obtain more holistic understandings of gendered as well as racialized lived experiences have been taken up in relation to class (davis, 1981; feagin & sikes, 1994), nation and immigration (anthias & yuval-davis, 1992; yuval-davis, 1997), and sexuality (lorde, 1982; moraga & anzaldúa, 1981). intersectionality in mathematics education research in mathematics education research, studies have illustrated how systems of power result in gendered and racialized struggles among marginalized groups to access high-quality educational opportunities and to prove their academic legitimacy (barnes, 2000; berry, 2008; boaler, 2002; mcgee & martin, 2011; mendick, 2006). these analyses advanced understandings of social influences that impact mathematics achievement and identities with varying degrees of attention to the intersectionality of experience among sampled populations and study participants. much research that adopted crt methodology foregrounds race in its analyses while intersections of race and other social forms of oppression (e.g., gender, class) are left implicit beyond the sampling of participants (berry, 2008; berry, thunder, & mcclain, 2011; mcgee & martin, 2011; terry, 2010). in their study of leyva latin@ college women and mathematics journal of urban mathematics education vol. 9, no. 2 85 six academically resilient, black4 mathematics and engineering college students’ life stories, for example, mcgee and martin (2011) examined how these students co-constructed their mathematics and racial identities through stereotype management in response to racial stereotypes of blacks’ limited mathematics ability and non-academic behaviors. mcgee and martin acknowledged room for further analysis in their study about the intersectionality of the black students’ racialized experiences and argued that such an analysis can only be considered by first unpacking the racialization of mathematics: our focus on race does not imply that race, class, and gender intersections are not important. however, more nuanced understandings of race…must be developed among mathematics and science educators if these intersections are to be seriously considered. (p. 1349) i agree with mcgee and martin as well as other critical race scholars about the importance of documenting variation of racialized experiences to challenge deficitbased racial discourses in mathematics. however, i argue that holding off on intersectional analyses is unnecessary as intersectionality serves as an analytical tool for capturing such variation in racialized forms of mathematics achievement and experiences to disrupt deficit discourses about students of color. in his participatory action research (par) study, terry (2010) used crt methodology to document seven high school black male youth’s construction of mathematics counter-stories about imprisonment and college enrollment. he pointed to the lack of generalizability of his findings across different intersectional subgroups including black women who must navigate uniquely racialized and gendered discourses (e.g., crack mothers, welfare queens). he also acknowledged the generative opportunities of framing future par research in “broader theoretical discussion of constructed academic identities vis-à-vis black masculinity” (p. 96). i argue that this opportunity can be extended to mathematics education research more broadly to provide insight into the gendered variation among women and men of color in making meaning of racialized experiences in mathematics and society. this call for analytical attention to notions of black masculinity captures the need for not only more intersectional understandings of mathematics as “racialized forms of experience” (martin, 2006, p. 198), but also theorizations of gender as a non-binary 4 here, i use the term black rather than african american to be consistent with the cited authors’ language choice. mcgee and martin (2011) acknowledged the variation in racial identity within the group descriptor black in relation to cultural background and nation of origin. black refers to a group of multigenerational individuals that can include those born in the united states and those who immigrated to the united states (clark, 2010; swarns, 2004). the term african american refers to a subgroup of black individuals tied to a history of slavery and struggles for civil rights in the united states (joseph, hailu, & boston, in press). i use the term african american throughout the remainder of the article to remain consistent with the participants’ racial self-identifications. leyva latin@ college women and mathematics journal of urban mathematics education vol. 9, no. 2 86 social construct and thus shaped by other vectors of identity. this intersectional analysis would be an advancement of the sex-based conceptualizations of gender across much crt work in mathematics education that leaves implicit how participants’ mathematics experiences are shaped by the dynamic interplay of race, gender, and other social identities (leyva, in press). intersectionality has also been minimally examined outside of crt work in mathematics education to better understand achievement and experiences among marginalized populations in mathematics. scholars, in addition to issues of race, have examined mathematics as a gendered space in light of women’s underachievement and underrepresentation as well as the valuing of heteronormatively masculinized norms of mathematical engagement (barnes, 2000; boaler, 2002; fennema et al., 1998; mendick, 2006). elsewhere, i reviewed research on gender in mathematics education to document how the majority of this extant work problematically conflated gender with biological sex as well as minimally attended to gendered variation in mathematics achievement and participation at different intersections of identity (leyva, in press). it is important to take note, however, of research studies that were exceptions to this lack of intersectional considerations for sex and gender, including works which highlighted influences of race and ethnicity (birenbaum & nasser, 2006; brandon, newton, & hammond, 1987; riegle-crumb & humphries, 2012; stanic & hart, 1995), culture (hanna, 1989; oppland-cordell, 2014), class (lubienski, 2002; mcgraw, lubienski, & strutchens, 2006), and sexuality (esmonde et al., 2009). such intersectional insights not only challenged the long-standing discourse of male superiority in mathematics, but also captured the affordances of coupling quantitative and qualitative insights to detail how the social construction of gender gives rise to variation in mathematics achievement and participation among marginalized groups. intersectionality and women of color in undergraduate stem education intersectionality has been largely taken up in research that explores equity issues in stem higher education with a strong focus on the experiences of women of color. the summer 2011 issue of the harvard educational review (malcom & malcom, 2011) for example, featured an assortment of works about women of color in stem higher education including a quantitative analysis of stem persistence predictors compared to white women (espinosa, 2011) and an interview study detailing challenges in transferring from community colleges to 4-year universities as stem majors (reyes, 2011). these works raised implications for higher education institutions in carving opportunities that attend to the intersectionality of being a woman of color in stem, including active recruitment in undergraduate research programs, building community with other minoritized women in and out of stem leyva latin@ college women and mathematics journal of urban mathematics education vol. 9, no. 2 87 classrooms, and professional development for stem faculty to better inform teaching based on the needs and experiences of women of color. camacho and lord (2013) pursued a more focused intersectional analysis specifically on the gendered and racialized experiences of latinas5 navigating the exclusionary culture of undergraduate engineering education. in reflecting on their findings about latinas managing microaggressions6 of academic ability and establishing counter-spaces, camacho and lord (2013) echoed espinosa’s (2011) and reyes’ (2011) calls for change in higher education that attends to the intersectionality of latinas’ experiences particularly through rethinking recruitment, mentorship, and co-curricular program design. looking across these intersectional analyses of student experiences in undergraduate stem education, i argue that there remains analytical space to leverage intersectionality as a tool to detail the culture of undergraduate mathematics and document marginalized students’ strategies in navigating this space as well as the role of institutional agents for student support. such intersectional analyses are particularly important to consider for underrepresented students during their first year of undergraduate studies in light of the academic, social, and affective challenges that they encounter during their transition into stem majors (crisp, nora, & taggart, 2009; hurtado, newman, tran, & chang, 2010; reyes, 2011). more specifically, an intersectional analysis of the mathematics experiences among women of color, including latin@ women, during their first year of undergraduate stem studies will add insights to the field of mathematics education—a space with minimal work on the intersectionality of experience among women of color and a recent call for equity research at the undergraduate level (rasmussen & wawro, in press). theoretical perspectives in this section, i elaborate the theoretical perspectives that informed data collection and analysis. race is conceptualized as a social construct that intersects with property rights giving rise to systemic inequalities (including education) in the united states among people of color (ladson-billings & tate, 1995). gender is theorized as a social construct discursively produced or performed differently across individuals and contexts (butler, 1990). the theoretical perspective of intersectionality was adopted to detail the latin@ women’s strategies in navigating institutional structures and interper 5 i use the term latina here to remain consistent with camacho and lord (2013)’s adopted language in their study. 6 microaggressions are defined as everyday communicative actions or verbal expressions that may or may not intentionally slight target or marginalized individuals (sue, 2010). macroaggresions are broader communicative acts toward target or marginalized individuals on systemic rather than individual levels (sue et al., 2007). leyva latin@ college women and mathematics journal of urban mathematics education vol. 9, no. 2 88 sonal relationships that shape mathematics as a socially exclusionary space at different intersections of social identities. poststructural theory guided the detailing of discourses raised in participants’ negotiations of their mathematics experiences and identities as latin@ women pursuing mathematics-intensive stem majors before and during their first college year at a large, predominantly white university. intersectionality and critical theories as previously discussed, intersectionality (crenshaw, 1989, 1991) is a theoretical perspective from black feminist thought that considers how intersections of race, gender, class, and other social identities shape marginalized individuals’ narratives of oppression and resistance. it serves as a tenet of crt, latin@ critical race theory (latcrit), critical race feminism (crf), and various other subfields of critical theory. crt applied to education addresses the “centrality of race and racism and their intersectionality with other forms of subordination” (solórzano, ceja, & yosso, 2000, p. 63) such as sexism and classism in schools and classrooms. intersectionality in educational research framed by crt, therefore, “challenges the separate discourses on race, gender, and class by showing how these three elements intersect to affect the experiences of students of color” (solórzano & yosso, 2002, p. 24). race, class, and gender are three of the many intersecting social categories that can be considered in detailing intersectionality of experiences in education and society (collins, 2000a, 2000b; crenshaw, 1991). latcrit, a “theoretical cousin” to crt, addresses the intersectionality particularly among latin@s with analytical considerations of experience at intersections of race, sex, gender, class, and other social dimensions (solórzano & delgado bernal, 2001). latcrit, therefore, complements crt by considering issues of culture, immigration, and language among latin@s that often go unexplored in crt (delgado bernal, 2002). similar to latcrit, crf is another theoretical offshoot from crt that foregrounds the intersectional experiences of marginalization and empowerment particularly among women of color (wing, 2000). crf serves as a “feminist intervention within crt” (wing, 2000, p. 7) that uses an anti-essentialist lens to examine the experiences of women of color not only as distinct from those of men of color, but also widely variable at different intersections of identities including race, gender, class, and sexuality. the use of internationality theory in this study, therefore, allowed me to leverage the complementary nature of crt, latcrit, and crf to center the latin@ women’s voices as well as detail the intersectional variation in their mathematics experiences at institutional, interpersonal, and ideological levels of analysis. poststructural theory poststructural theory’s conceptualizations of discourse and power were used to frame the analysis of latin@ women’s counter-stories. a discourse is a “historically, leyva latin@ college women and mathematics journal of urban mathematics education vol. 9, no. 2 89 socially, and institutionally specific structure of statements, terms, categories, and beliefs” (scott, 1988, p. 35). discourses structure behavior and language in ways that highlight the “surface linkages between power, knowledge, institutions, [and] intellectuals… as these intersect in the functions of systems of thought” (p. a. bové, as quoted in st. pierre, 1990, p. 54). in this study, discourses refer to norms and behaviors of mathematics ability and stem higher education structured by racism, sexism, and other intersecting systems of oppression that shape the mathematics experiences of marginalized student populations including latin@ women. power is theorized as multiple systems of relations in constant flux across individuals and contexts (foucault, 1997/1984). halperin (1995) writes, “power is thus a dynamic situation, whether personal, social, or institutional” (p. 17, emphasis in original). discourses are, therefore, contextual manifestations of these varying power relations that inform one’s positioning across the intersecting systems at different moments. foucauldian thought asserts that power comes with resistance in which strategic moves are adopted in reaction to sociocultural discourses of opportunity and oppression. such agency in these reactions, however, is within limits maintained by these power relations that perpetuate the status quo in society (st. pierre, 2000). drawing on martin’s (2009) and mendick’s (2006) scholarship, this study theorizes mathematics as a source of power that structures a hierarchy of ability aligned with society’s inequitable opportunities for dominant and marginalized groups. in poststructural theory, individuals are conceptualized as discursive subjects whose identities are socially constructed through their negotiations of sociocultural discourses and power relations (st. pierre, 2000; walkerdine, 1990). individuals’ identities, therefore, are in a perpetual state of flux and produced as discursive responses to dynamically changing power relations in everyday society. poststructural theory goes further to describe how identities and systems of meaning are mutually produced with such “meaning[s]… strategically reinterpreted, reworked, and deferred since there is no referent for the subject” (st. pierre, 2000, p. 503). using a sociocultural lens, martin (2006) defines mathematics identity as— dispositions and deeply held beliefs that individuals develop, within their overall selfconcept, about their ability to participate and perform effectively in mathematical contexts…. a mathematics identity encompasses a person’s self-understanding of himself or herself in the context of doing mathematics. (p. 206) by applying a poststructural lens to martin’s (2006) definition, mathematics identities are social constructions constantly negotiated across different contexts in response to discourses shaped by intersecting systems of oppression such as racism, sexism, classism, and heteronormativity (see, e.g., berry, thunder, & mcclain, 2011; boaler & greeno, 2000; esmonde, 2009; mendick, 2006; stinson, 2008). individuals as multidimensional beings construct their mathematics identities at junc leyva latin@ college women and mathematics journal of urban mathematics education vol. 9, no. 2 90 tures of multiple yet often contradictory discourses upheld by institutions (e.g., schools, mathematics classrooms), other individuals (e.g., peers, teachers), and society (barnes, 2000; stinson, 2008). the nature of individuals’ negotiations of these discourses varies when attending to different intersections of identity dimensions including class, culture, gender, race, and sexuality. mendick (2006) and stinson (2008), for example, detailed how students are constantly negotiating gendered and racial discursive binaries of mathematics ability including “masculine/feminine” and “white/black,” respectively, with masculinity and whiteness mapping onto mathematics ability. stinson (2008) went further to highlight the complexity of such discursive negotiations by describing how “[discursive] subjects live at intersections of these binaries” (p. 992, emphasis added) with the saliency of binary dimensions varying across contexts and individuals. these negotiating practices inform the study’s intersectional analysis used to examine latin@ college women’s experiences and thus detail how they co-constructed their social and mathematics identities as responses to discourses in mathematics, stem higher education, and society. overall, these theoretical perspectives collectively informed the study’s methodology that addresses the need for scholarship in mathematics education exploring the intersectionality of mathematics experience among historically marginalized student populations. more specifically, these perspectives were applied to closely examine two latin@ women’s co-constructions of their mathematics and social identities by making meaning of their mathematics experiences in relation to discourses that are gendered, racialized, classed, and so on. this study, therefore, examined how in engaging with these discourses, the two latin@ women differentially negotiated their positions along the hierarchy of mathematics ability across contexts and at different intersections of their social identities. such discursive negotiations offered insight into the latin@ women’s success-oriented beliefs and strategies in navigating these discourses encountered institutionally and interpersonally throughout their mathematics experiences. methods crt informed the study design including the analytical construction of the two latin@ women’s counter-stories. the counter-storytelling methodology is an analytical approach to “telling the stories of people whose experiences are often not told (i.e., those on the margins of society)” (solórzano & yosso, 2002, p. 32). while counter-stories can be narratives that challenge dominant discourses of marginalized groups in society, they can also be what solórzano and yosso (2002) call “unheard counter-stories” (p. 32) that may not necessarily push back on these discourses yet still offer insight into individuals’ strategies of survival and resistance in navigating sociopolitical contexts. the counter-story analysis presented here attends leyva latin@ college women and mathematics journal of urban mathematics education vol. 9, no. 2 91 to both kinds of narratives of experience shared by the latin@ women that either did or did not challenge discourses in mathematics and stem higher education. solórzano and yosso (2002) identified three types of counter-stories: (a) personal stories or narratives, (b) other people’s stories or narratives, and (c) composite stories or narratives. counter-stories presented in this analysis are based on other people’s stories or narratives—namely, the two latin@ college women’s reflections on navigating mathematics as a socially exclusionary space. the construction of their counter-stories was aligned with solórzano and yosso’s outline of the four functions that counter-storytelling methodology serves: (a) building community among members of marginalized groups, (b) challenging dominant discourses to transform established societal beliefs, (c) presenting different realities of marginalization to individuals with shared forms of oppression, and (d) constructing another world richer than the counter-story and lived experience alone can provide. with a poststructural analysis and sociocultural view of mathematics identities, i examined the latin@ women’s counter-stories as discursive productions mapping onto instances of disconnect and marginalization as well as affirmation and empowerment that impacted the co-construction of their mathematics and social identities (martin, 2009). the coupling of poststructural theory and crt’s counter-storytelling, therefore, guided the detailing of racial, gendered, and other discourses across the latin@ women’s counter-stories that were raised to make meaning of their mathematics experiences. the analysis, therefore, attended to the varying influences of these discourses across institutional contexts and interpersonal relationships as the latin@ women made meaning of their experiences. intersectionality, as a tenet of crt and other critical theories that frame the analysis, allowed for highlighting the variation across the latin@ women’s discursive moves in these identity constructions including beliefs and strategies that they adopted to navigate them at intersections of gender, race, and other social identities. a qualitative case study methodology (miles & huberman, 1994; yin, 2003) was employed such that the latin@ women’s counter-stories were the “cases,” or units of analysis, used in detailing the extent to which mathematics was a socially exclusionary experience for them. in efforts to gain holistic accounts of the latin@ women’s mathematics experiences, their counter-stories were constructed by looking across multiple data sources including mathematics autobiographies, individual interviews, and focus group discussions. these counter-stories addressed the research questions by exploring what were the dominant discourses raised, when and where the latin@ women encountered them, and how they similarly and uniquely navigated them in co-constructing their mathematics and social identities. research context i began conducting the study in spring 2013 at a large, public 4-year university located in the northeastern united states. according to the university’s 2013 in leyva latin@ college women and mathematics journal of urban mathematics education vol. 9, no. 2 92 stitutional profile, african americans and latin@s comprised less than a quarter of the fall 2012 undergraduate student population. latin@s made up about 13% of full-time students in fall 2012. men and women in general enrolled at the university at comparable rates. of the undergraduate population in their first year during fall 2006, latin@s graduated from the university within 4 years at a rate of 34%. african americans and latin@s earned baccalaureate degrees at rates ranging between 10 and 15% by the end of the 2011–2012 school year. considering only about 10% of the university’s conferred degrees were in stem, an even smaller percentage of these degrees were conferred to latin@s. african americans and latin@s comprised about 6% of the university’s full-time faculty in fall 2012. latin@ women were least represented among full-time faculty members, a total of 26 with and without tenure in fall 2012. participants eight first-year college students pursuing mathematics-intensive majors at the same university were recruited for the research study. the term first-year is used as a descriptor of the participants’ first year of university enrollment. this included two african american women, two african american men, two latin@ women, and two latin@ men. participants were drawn from a stem support program at the university. the program’s support services focused on providing underrepresented college students with co-curricular activities and networking opportunities to advance their academic and professional development in stem. by the start of data collection, participants had taken at least one college mathematics course (e.g., pre-calculus, calculus) during their first semester. this article focuses on findings related to the two latin@ women participants, lauren and tracey (pseudonyms). to address the research questions, lauren’s and tracey’s counter-stories are presented as cases with each offering a rich and unique account of the intersectionality of experience for a latin@ woman in mathematics and stem higher education. the cross-case analysis, presented later, was conducted with the intent to detail similarities as well as differences between lauren’s and tracey’s individual experiences. it should be noted that my analytical focus on latin@ women’s mathematics experiences is not intended to perpetuate notions of intersectionality as being solely about issues pertaining to women of color. rather, it addresses an intersectionality of experience minimally discussed in the mathematics education literature while attending to crenshaw’s (1991) discussion of how other social identities besides race and gender (e.g., class, culture, immigration) shaped the two latin@ women’s oppression and empowerment in mathematics. the following table presents profiles for lauren and tracey. this table outlines participants’ pursued stem majors, high school student demographics, most leyva latin@ college women and mathematics journal of urban mathematics education vol. 9, no. 2 93 recently completed college mathematics course, and career goals. participants’ initial stem major interests continued throughout the study. the variation across the two participants’ high school demographics is noteworthy especially when considering their respective experiences of transitioning into a predominantly white university space. along with taking calculus during her first semester, tracey had prior calculus experience in high school and a summer bridge program for incoming stem students at the university. table 1 participant profiles name race/gender stem major high school demographics completed college mathematics career goals lauren latin@ woman computer science predominantly white advanced pre-calculus undecided tracey latin@ woman mathematics predominantly latin@ calculus i for stem majors high school mathematics teacher data collection three data sources were used to construct the first-year latin@ college women’s counter-stories: (a) mathematics autobiographies, (b) individual interviews, and (c) focus group discussions. in this section, i discuss the nature of each data source and how it contributed to answering the research questions on mathematics as a socially exclusionary experience for the latin@ college women participants. to layer the data collection, excerpts from participants’ mathematics autobiographies were incorporated into individual interviews and focus group discussions to clarify and probe meanings of key statements. this sequential data collection reinforced and provided more nuance of findings throughout the study. mathematics autobiographies. participants wrote twoto three-paragraph autobiographies on their mathematics experiences during high school and college. they reflected on their favorite and least favorite mathematics courses with details on the nature of their participation, relationships with teachers, and classroom structures and interactions. participants provided similar reflections on their most recently completed college mathematics courses. the mathematics autobiographies were submitted prior to participants’ individual interviews. stimulus excerpts were used during interviews to probe participants about connections between their mathematics experiences and being a latin@ woman. in addition, high school and college mathematics reflections were used to probe participants on what being successful looked like across these con leyva latin@ college women and mathematics journal of urban mathematics education vol. 9, no. 2 94 texts, how these messages of mathematics success were communicated, and what strategies they employed to meet these standards of success. individual interviews. each participant completed a 90-minute, semistructured individual interview focused on four themes about their mathematics experiences, including high school, college courses, stem support program participation, and views on women and racially minoritized groups in mathematics and stem at large. interviews were audiotaped and transcribed. excerpts from the mathematics autobiographies were used to examine gendered, racialized, and other social influences on high school and college mathematics experiences. some questions used to probe these dynamics included: “how do you feel as though latin@ women are encouraged or discouraged from pursuing mathematics?” and “why do you think so many fellow latin@ women like yourself do not make it in mathematics?” focus group discussions. after the interview, participants completed a focus group discussion with three other participants to motivate peer discussions on african american and latin@ women and men in mathematics and stem at large. each focus group participant was paired with another participant of the same intersectional identity. these pairings were intended to establish welcoming discussion environments so participants would not feel tokened and could possibly relate to at least one other participant’s shared perspectives while also highlighting differences between their experiences. finally, by having two different intersectional identities represented, this focused the analysis on similarities and differences between intersectional groups as well. focus group discussions were audiotaped and transcribed. during each focus group discussion, participants were presented with five mathematics student narrative excerpts from four mathematics education articles on marginalized students’ mathematics experiences (berry et al., 2011; lombardi, 2011; mendick, 2005; stinson, 2008). participants were asked to read these five excerpts and select one or two with which they either strongly associated or disassociated based on their experiences. these excerpts, drawing on stinson’s (2008) methodology, provided participants with language to engage in expressive, critical conversations surrounding social issues in mathematics. participants were probed for gendered, racialized, and other social significance in their reactions to the stimulus narrative excerpts. some probing questions included: “to what extent does diversity play a role in your college mathematics experiences at the university?” and “what are some examples from your high school and college experience when you felt your mathematics ability was judged based on your gender and/or race?” in addition, participants were asked to discuss issues regarding african american and latin@ women’s and men’s current underrepresentation as well as these groups’ projected future participation and success in stem. latin@ college women participants were presented with questions such as “how has your stem support program involvement, if at all, influenced how you see yourself as a latin@ leyva latin@ college women and mathematics journal of urban mathematics education vol. 9, no. 2 95 college woman in stem?’ and “as a latin@ college woman pursuing a mathematics-intensive stem degree, how do you see the future of women and underrepresented minorities in stem including mathematics? what ideas or experiences bring you to raise this claim about the future diversity of stem?” data analysis connections across these three data sources were used to write counter-stories of the latin@ women’s mathematics experiences. intersectionality guided a crosscase theme analysis to examine patterns in the first-year latin@ college women’s constructions of mathematics identities as responses to discourses related to intersections of gender, race, and other identities (miles & huberman, 1994). using poststructural and intersectional lenses of analysis, the counter-stories were openly coded for discourses specific to mathematics ability and stem higher education (bowleg, 2008). axial codes were used to identify institutional and interpersonal contexts in which participants encountered these discourses as well as the strategies adopted for navigating them (strauss & corbin, 1998). (see leyva [2016] for full details of coding.) one of the key aspects of intersectional analysis in qualitative research is making participants’ implicit experiences of intersectionality explicit, including when participants do not report them (bowleg, 2008). counter-stories, therefore, were examined for “subtexts” of how participants discursively constructed meanings of their mathematics experiences particularly at the intersections of gender, race, and other social identities (banning, 1999). open and axial coding of counterstories and their subtexts served to illuminate the intersectionality across participants’ discursive negotiations of mathematics as a socially exclusionary experience. analytical memos were written throughout the data collection and analysis processes. memos were dated to trace the development of data interpretations including possible themes, areas of needed clarification, and key connections to the research literature. memos and annotated transcripts were used to develop the open and axial coding schemes and strengthen the rigor of the findings. a colleague independently coded excerpts from the transcribed data to confirm the accuracy of the open and axial coding schemes. this colleague was provided a detailed codebook containing descriptions and sample instances of each open and axial code. an assortment of excerpts corresponding to different open and axial codes was selected for this independent coding task. any coding disagreements were discussed during code reconciliation meetings until they were resolved and necessary coding scheme adjustments were made. member checking the accuracy of the transcripts, coding scheme, and data interpretations was also completed (creswell & miller, 2000). two member checks were held with participants in spring 2013. i structured the member checks as informal group conversations where participants, as “authors of their own experienc leyva latin@ college women and mathematics journal of urban mathematics education vol. 9, no. 2 96 es,” were positioned as experts whose critiques and recommendations to strengthen the rigor of my analysis were welcomed. participants were presented with coded excerpts across data sources and asked to reflect on the accuracy in terms of transcription and interpretations. some questions used to structure the member checks included: “did the applied codes accurately reflect the nature of influences that shaped the mathematics experiences shared in these excerpts?” “are the strategies that i claimed you adopted in being a mathematics student accurate?” and “is there anything that you think should be added to the coding scheme or my interpretations that is a critical part of your experience as a latin@ women in mathematics?” the member checks informed necessary revisions to the coding scheme and preliminary findings. disagreements during member checks about codes and interpretations were discussed until resolved. if the disagreement stemmed from a participant wanting something from her experience to be captured, i incorporated these comments into the final write up of her counter-story. if any coding or interpretation disagreement was unresolved, it was not included in the analysis. researcher identity, positionality, and trustworthiness. as a latin@ man who graduated as a mathematics major and researches mathematics experiences of underrepresented populations in stem, i brought an understanding of my positionality in pursuing data analysis and interpretations with strong subjectivity to develop nuanced understandings of mathematics as a socially exclusionary experience. i was aware that my latin@ identity allowed me to relate to feelings of underrepresentation, academic disadvantage, and struggle that participants as students of color may have experienced with mathematics and stem higher education. mutual identification with participants as a person of color who also pursued a mathematics-intensive stem major allowed for the establishment of intersubjectivity that built positive rapport and trustworthiness (glesne & peshkin, 1999; lincoln & guba, 1999). as a 4-year employee in the university office overseeing the stem support program, i approached the study with strong familiarity of the program and ongoing visibility to student members during monthly meetings and events. such participant–researcher connection captures the use of my “multiple identities as an interaction quality” (berry, 2008, p. 472) to create welcoming spaces for the first-year latin@ college women to share and reflect on their mathematics experiences. despite these mutual understandings of experience, i also acknowledge my gendered privilege and varying social distance from the latin@ women participants in experiencing mathematics as a man. awareness of my shifting positionality throughout the study played an important role in identifying moments “where self and study were intertwined” (stinson, 2008, p. 987). thus, consciousness of my positionality as a latin@ man in mathematics allowed me to connect in different ways with the latin@ women as well as be willing and open to learn from them. leyva latin@ college women and mathematics journal of urban mathematics education vol. 9, no. 2 97 findings i organize this section by first presenting the two cases, lauren’s and tracey’s counter-stories. each case begins with an overview of each latin@ college women participant’s connections to mathematics, including reflections on her favorite and least favorite mathematics courses. the two counter-stories focus on the intersectionality of lauren’s and tracey’s experiences as latin@ women studying mathematics in high school and at the university. they were constructed to detail institutional and interpersonal contexts in which lauren and tracey encountered discourses of mathematics ability and stem higher education. the counter-stories also describe the success-oriented beliefs and strategies that they adopted in negotiating their identities as latin@ women in mathematics with these discourses. i then present a cross-case theme analysis of lauren’s and tracey’s counterstories. i organized the analysis by four identified discourses across the counterstories. the theoretical and methodological principles of intersectionality guide this cross-case analysis by highlighting both common and distinct experiences of marginalization and empowerment as latin@ women in mathematics at intersections of gender, race, culture, class, and immigration. lauren’s counter-story lauren is a first-generation, el salvadoran-american woman pursuing a computer science major. her interest in computer science began when her high school accounting teacher took note of her mathematical problem-solving skills and connected her with the school’s computer science teacher. she saw herself as “always good at math.” while a shy student across classroom spaces, lauren’s mathematics ability brought her to feel more comfortable in high school mathematics courses where it was “okay for [her] to talk.” lauren described how classmates would comment on her being less engaged during mathematics classes, which she saw as being related to her “natural” mathematics ability: a lot of the times students would tell me or classmates would say like, “you barely pay attention but you get good grades”; and that’d be true like i really would be on my phone for the whole class and i would pay attention and i’d do my homework and all that, but i would get good grades ‘cause it came naturally to me. (individual interview) lauren positioned herself as not needing to invest as much effort in paying attention during class as her classmates because of her innate mathematics ability. she also distinguished herself from peers whose lack of natural mathematics ability may have brought them to struggle with understanding the subject and thus not like mathematics. lauren commented: “i feel like it’s not hard to understand math, but leyva latin@ college women and mathematics journal of urban mathematics education vol. 9, no. 2 98 maybe that’s just me because i get it. maybe if i didn’t get it, i wouldn’t be saying the same thing.” favorite and least favorite mathematics courses. a supportive relationship with teachers was the defining quality of lauren’s more favorable mathematics course experiences. algebra ii honors, for example, was lauren’s favorite high school mathematics course in light of her “friendly relationship” with her teacher. lauren described how her teacher genuinely cared for his students. he also “did not give up on any student” even when they expressed struggles with the course material. along with establishing a positive connection with her teacher, lauren reflected on working well with algebra ii classmates who enjoyed mathematics like her as opposed to those who seemed to not care about mathematics as much: some of them didn’t want to learn math or didn’t much care much for math so it was like hard with them. but others—they enjoyed math as much as me or as i did so it was easy [with them]…. some of the students didn’t really engage. like they didn’t just want to be there. maybe because they didn’t like math. (individual interview) here we see how lauren separates fellow students into two groups—namely, those who like mathematics (including herself) and then those who do not. lauren acknowledged the important role that one-on-one teacher attention can play in helping students understand mathematics and thus influence their liking of the subject. according to lauren, “if [a] person who said they don’t like math, if they had an individual person teaching them instead of being in a big class, they would understand it.” this statement captures how, in spite of viewing mathematics ability as innate and possibly tied to one’s affect toward the subject, lauren perceived teachers as being able to facilitate students’ understandings of the content and appreciation of mathematics. a notable tension, therefore, resides in lauren’s reflections on mathematics ability—namely, between views of it being innate and natural and views of it being developed through teaching and student learning. pre-calculus was lauren’s least favorite high school mathematics course. she shared how her pre-calculus teacher lacked a “consistent method of teaching” characterized by inaccurate assumptions of what students already knew and a “fast pace” without re-visiting earlier concepts. the teacher’s instructional approach, therefore, contrasted with that of lauren’s algebra ii honors teacher who “would stop and ask if everyone understood what he was doing…so rarely did anyone not understand his [teaching] method.” lauren, therefore, also valued mathematics teaching that focused on student understanding during high school. although lauren saw herself in the same position as her pre-calculus classmates as not learning much, she set herself apart as being less affected by her teacher’s poor instruction than other “students [for] who…it takes a long time for them to get math.” it is noteworthy how lauren draws on her leyva latin@ college women and mathematics journal of urban mathematics education vol. 9, no. 2 99 natural mathematics ability to distance herself from struggling peers across two mathematics courses with contrasting instructional approaches. this distinction between her and her peers’ mathematics classroom experiences is also observed in lauren’s discussion of her relationship with the precalculus teacher. although the pre-calculus teacher seemed “really nice” in her teaching and cultivated a “friendly environment” in the classroom, lauren described how she was not approachable in the context of seeking extra help by negatively judging students who struggled. nonetheless, lauren saw herself having a relational advantage over classmates because she felt as though mathematics teachers established stronger connections with students who were “good at math” like her. lauren remarked, “when you’re good at math, teachers find that they enjoy talking to you more because you have that connection like you can relate.” she saw the pre-calculus teacher as seeming to have “really liked the fact that [she] liked math” yet was firmer with her than other students with lower levels of mathematics ability. lauren, therefore, perceived her mathematics ability as affording her status in pre-calculus that allowed her to receive higher-quality and more rigorous support from teachers. the relational spaces of mathematics classrooms (including quality of instruction and student support) coupled with views of mathematics ability as innate or developed, thus, largely shaped lauren’s reflections of her favorite and least favorite mathematics courses. peers and family. during high school, lauren was subjected to peer comments particularly from latin@ boys about latin@ women not being good at mathematics or expected to go to college. she looked back on how these boys would often make remarks to latin@ girls who were also young mothers such as “you should be at home taking care of the kids” when they saw them in class or thinking about applying to colleges. below lauren reflects on latin@ boys’ remarks about latin@ girls and mathematics: like all of the hispanics like my friends they would say things like even to us—their friends—like “oh you shouldn’t be going to college or trying to be good at math” cause they just think that it’s not normal for a girl to be good at math especially a latina girl. (focus group discussion) lauren explained how these comments seemed to originate from parents’ and older generations’ gendered division of labor in the latin@ household. for example, she described how her older family members upheld values where “the men worked and the women stayed at home” based on times of their upbringing. despite hearing high school peers’ disparaging remarks and living in a household with gendered family roles, lauren shared how this did not impact her academic pursuits because her family still encouraged her to “break that tradition” and pursue a college degree. lauren distinguished her family’s academic encouragement with many other lat leyva latin@ college women and mathematics journal of urban mathematics education vol. 9, no. 2 100 in@ family situations that perpetuate traditional cultural expectations of women becoming mothers and wives rather than pursuing a college education. upon graduating from high school, lauren did not see fellow latin@ graduates trying hard to advance academically or professionally. she described how it was “really disappointing” that most of her latin@ high school peers seemed complacent with receiving a high school diploma and “not really doing anything” like seeking higher education. she alluded to also observing such forms of complacency among her latin@ family members, particularly, her older cousins: a lot of my older cousins are living their life with a job and not having much of school besides a high school diploma…. i feel like they will begin to see that you can live a life with a degree and more comfortably. and my aunts and uncles, that’s what they would like for their children and i feel like they will end up pursuing school because they know it leads to a better future. so that’s what i’m hoping: that latinos will start getting involved more [in stem]. (individual interview) here we see how lauren’s older cousins receive forms of academic encouragement from her aunts and uncles about pursuing higher education similar to that which she receives from family to disrupt ideas of latin@ girls not receiving college degrees. lauren, furthermore, seems to use this example about her older cousins to illustrate a way in which latin@s can increase their participation in stem. she saw her pursuits of a stem degree as a first-generation college student serving as an opportunity to be a “role model” for younger family members: i am the first generation…to go to college in my family so obviously the older people like they are doing their own thing. they are not going to college so that doesn’t influence or impact them. but the younger ones like my nephews, my nieces, my cousins, they still have time so i feel like i would just like to be their role model so they can feel like they can do it as well…. hopefully they’ll go into stem. (individual interview) in this excerpt, lauren discussed how being a role model in her family would allow younger relatives to see another latin@ pursuing stem and thus be encouraged to feel like they can do the same. she perceives such academic encouragement in her family for pursuing stem higher education as a way to increase latin@s’ representation in stem and thus “become a majority in these fields.” family, therefore, plays an important role for lauren in managing peers’ disparaging comments about latin@ girls as well as perceiving the value of higher education for broadening latin@s’ participation in stem. first year of college. at the university, lauren enjoyed her advanced precalculus course with a professor and teaching assistant who “wanted everyone to do well.” she reflected on how the professor’s “friendly” nature and one-on-one support opportunities may have largely motivated students’ office hour participation, which she viewed as uncommon across her university courses. it is noteworthy how leyva latin@ college women and mathematics journal of urban mathematics education vol. 9, no. 2 101 lauren’s positive experience with college pre-calculus aligned with her preferred aspects of high school mathematics—namely, teachers with a “friendly” nature as well as classroom instruction structured for student learning and support. it was during college calculus, however, when lauren reconsidered her pursuits of a computer science major. lauren mainly attributed her reconsideration to her struggles in learning calculus due to the quality of her professor’s teaching. she described how she was doing “not as good as [she] should be” in calculus because she was unable to connect with the calculus professor’s teaching and felt less engaged in class compared to her high school years: that’s partly because i rely a lot on the teacher and if i feel like the teacher isn’t teaching then well i kind of slack…. i don’t answer questions or ask questions either…. and it’s not because i’m not comfortable with the people around me, but it’s like i’m not comfortable with the teacher ‘cause i know that if i ask him, he’ll explain it to me and i still won’t understand it. (individual interview) despite lauren’s claim of mathematics “coming naturally” to her, it is interesting to consider how she saw herself struggling in calculus because she depends on teachers to be successful. her calculus professor’s teaching, therefore, brought lauren to resort to teaching herself the material by reading the textbook and seeking help from other students in her first-year, all-women residence hall. these actions made lauren feel “overwhelmed with math” as well as fearful that she will always be teaching herself and, in turn, stop liking mathematics: i don’t want to have to be in too many classes that it’s too much for me to handle or that i don’t enjoy to do it anymore. it kind of scares me. i don’t want to ever feel like i’ll be bad at math. right now in calc, there is a possibility that i’ll learn it and that i understand it but i don’t want to have to like continue to having to teach myself like that. (individual interview) lauren’s calculus experience brought her to feel as though she would face similar struggles throughout the remainder of her college mathematics coursework to the point of no longer liking and feeling like she was bad at mathematics. although lauren previously experienced disagreeable instruction and an unapproachable mathematics teacher in high school, these high school experiences did not cause her to think that she would stop being good at or enjoying mathematics as much as it did during college. when asked to identify an either positive or negative turning point in her experience as a mathematics student, lauren reflected on how college calculus brought her to fear that she would have to possibly compensate for her future mathematics professors’ limited teaching abilities: leyva latin@ college women and mathematics journal of urban mathematics education vol. 9, no. 2 102 the professors—they may be great at math or like really good at math but i don’t feel like teaching is their thing.… it kinda scares me that i’ll forever be like teaching myself and i don’t want to have to like have to teach myself even though a lot of students do it. (individual interview) lauren’s idea of being naturally good at mathematics coupled with how teaching herself was not resulting in better grades led her to question continuing the computer science major. it is important to note how lauren distinguished becoming “less interested in mathematics” and the process of “teach[ing] herself.” this distinction is important because it captures how lauren did not disassociate with mathematics, but rather the idea of learning mathematics without receiving teacher support—a defining quality of her more favorable mathematics courses. lauren’s reconsiderations, therefore, do not arise from seeing herself as mathematically incapable, but rather from exercising her agency to avoid situations like teaching herself that jeopardize her confidence and enjoyment of mathematics. another observed difference between lauren’s high school and college mathematics experiences was the salience of her gender identity as a woman in stem classrooms. lauren reflected on often being the only or one of the only women in her college courses. for example, she saw student enrollment in her computer science courses as racially diverse but being less balanced between boys and girls with her often being the “only girl.” lauren then went on to contrast this with her high school mathematics experience: yeah, i guess i experienced it differently than like in my high school. the math department was really split up. like there was both female teachers and male teachers. i don’t feel like it was a gender thing. i mean like it probably would come out if there were more male teachers, but because there was female teachers, i don’t think the gender thing was that much of a thing or a big problem. (focus group discussion) lauren perceived the presence of women as mathematics teachers in her high school serving as a way of showing students, particularly the boys, that “a female can do it [mathematics] too” and thus reduced possible “tension…with females” in mathematics. thus, the noted contrast of gender representation lauren to feel as though issues of gender in mathematics and stem were more salient in college than in high school. during her first college year, lauren took a women’s leadership course where she came across statistics indicating that latin@ women were the most underrepresented group in stem. lauren attributed these statistics to latin@ women “lack[ing] the encouragement” and how they “weren’t mentored enough” to pursue stem including mathematics. in contrast to remarks of latin@ women not being good at mathematics that lauren encountered during high school, she rationalizes latin@ women’s underrepresentation in terms of not receiving adequate forms of support allowing them to feel that they can do it. for example, lauren commented leyva latin@ college women and mathematics journal of urban mathematics education vol. 9, no. 2 103 on how other latin@ girls may not consider doing mathematics due to lack of encouragement from their families: i don’t really think that they [latin@ girls] find like it’s an option for them. maybe they feel they should be more like in the liberal arts or…doing something else. like they don’t really consider math. but then maybe they weren’t taught that they could do math. like their parents might say like, “you can’t do math. what are you gonna do with that?” (individual interview) it is important to consider how lauren’s explanation of latin@ girls’ underrepresentation in stem aligns with her experience of receiving academic encouragement from family to pursue stem higher education as a latin@ woman. tracey’s counter-story tracey is a cuban-american woman pursuing a double major in mathematics and theater arts to become a high school mathematics teacher. she saw mathematics as “what [she] was good at” and “made [her] confident. tracey reflected on how helping peers in high school affirmed her ability and confidence in mathematics. her confidence, however, was challenged as a high school sophomore when she received a c in her algebra ii honors course that brought her to perceive success in mathematics as a matter of hard work rather than natural ability. tracey recalled thinking to herself: “i’m not going to get good at this if i don’t take time out of my day [to study]. i can’t just rely on the fact that ‘oh, i’m good.’ i have to work on being good.” the algebra ii honors course grade, therefore, served as a turning point in tracey’s mathematics experience: a “heartbreaking” and a “very memorable moment” that guided her approaches to future mathematics courses of investing and time and effort to do well. when looking back on the algebra ii honors course, tracey recalled the challenge of balancing academics and “girl things”: it was a lot of girl things that were going on…. i was in a lot of clubs. i was having my quincenera so that was stressful. i was president of like the choir and i was one of the leads in the musical and in a performing arts academy. and i had a lot of things on my plate so it was really hard to balance everything. (individual interview) she clarified that in addition to extracurricular activities, “girl things” included interest in boys and managing friendships, which she saw as distracting her from doing well in mathematics. tracey described, “i think they affected them [her algebra ii course experiences] greatly because i couldn’t only focus on math because i was focusing on social life and real-life problems.” favorite and least favorite mathematics courses. tracey’s passion for mathematics was largely influenced by her high school education that began early when leyva latin@ college women and mathematics journal of urban mathematics education vol. 9, no. 2 104 she was recommended to take algebra i honors in eighth grade. although tracey chose algebra i honors as her least favorite high school mathematics course, this disfavor was based more on the “overwhelming” academic transition into high school rather than aspects specific to her teacher and the classroom. tracey viewed eighth-grade mathematics as a building block for algebra so she saw herself struggling because it was “really hard to have to balance two math classes” with the high school course going at a faster pace. advanced placement [ap] calculus was tracey’s favorite high school mathematics course. the teacher, mr. sosa (pseudonym), was available for extra help during and after school hours as well as approached his calculus teaching in ways that allowed her to make connections with previous topics. tracey reflected on how mr. sosa communicated his prioritization of the students’ mathematics success to the class: mr. sosa from day one told us, “no one in this class is gonna fail…. i am putting you up on this pedestal. you’re gonna reach it. you have no other choice,” in a nicer way than the way i’m saying it right now. and he just gave us so many opportunities. (individual interview) mr. sosa’s coupling of academic and relational support played an important role in shaping ap calculus as tracey’s favorite mathematics course. to do this, he reminded students of his high academic expectations for them and openness to supporting them throughout the course. tracey kept in contact with him after high school graduation as it was “thanks to him that [she] enjoy[s] math.” another favorable aspect of the ap calculus course experience was her classmates who she previously knew from growing up in their “small, well-knit town.” the familiarity of her ap calculus classmates facilitated the formation of informal peer study groups outside of class. tracey saw study group meetings as comfortable, supportive spaces that were “really, really fun” and allowed her to collaborate in learning mathematics with her ap calculus classmates. tracey commented on how mutually identifying with mathematics teachers in terms of social and cultural backgrounds could be beneficial in having shared understandings of how gendered, racial, and other marginalizing stereotypes shape educational experiences: it helps i guess if the teacher is the same race and/or gender…. because i mean if like okay let’s go with race, if they are of the same race, they understand more the cultural side and the cultural generalizations that have been made about you, are made about you, and will be made about you. so they understand your background and what you’re dealing with at home…. if i have a girl teacher…she’s gonna understand the competition with the men in the class so it would be nice to have someone to relate to. (focus group discussion) leyva latin@ college women and mathematics journal of urban mathematics education vol. 9, no. 2 105 tracey’s remarks illustrate how shared social identifications enable teachers to build relationships with students that leverage cultural insights and gender awareness to disrupt discourses that may limit educational opportunities. to illustrate this dynamic, tracey discussed how most of her high school’s faculty was comprised of latin@ men including mr. sosa who would often “stick up” for her during moments when classmates criticized her “basic small questions” in mathematics: everybody in class would be like, “we learned that 5 years ago. what are you doing here?” it’s just like those little snarky comments…i get comments from the students and then like teacher [mr. sosa] would stick up for me and say: “guys, it’s a question. you probably don’t know the answer. do you know the answer? no.” so, he was on my back. (focus group discussion) tracey presents a classroom moment of how she saw mr. sosa managing status of mathematics ability often tied to gender and race by responding to peers’ criticisms of her questions perceived as reflecting lower mathematics ability. such teacher– student connections responsive to diversity in mathematics classrooms highlights the importance of teachers being critically aware of and actively challenging discourses about mathematics ability as well as the need for increased diversity of mathematics teachers in the united states. tracey’s high school mathematics experience motivated her pursuits in mathematics so she can return as a mathematics teacher in her hometown and support latin@ girls with managing gendered cultural stereotypes. she stated: “i wanna be a math major…. because i love my high school and my town so much, i wanted to be a high school teacher for math and go back and show all the little girls that you can do it. beat the culture!” peers and family. tracey took note of girls’ underrepresentation as being one of only four girls in the class. such underrepresentation gave rise to what tracey called “gender battles” between girl and boy classmates framed by notions of boys being better at mathematics and smarter than girls: there was four girls in the class…. so, it was really nice to have at least my two very best friends in the class with me ‘cause we were the only girls…. the guys would try to give us gender battles like it’s ‘cause guys are smarter. and i was number two in calculus just proving to the guys. (individual interview) tracey described how such “gender battles” created a competitive learning environment in the classroom that brought her to feel as though she had to constantly prove being smarter than the boys. such gendered competitiveness brought some boys to get upset about “getting beaten by a girl,” which motivated tracey to do well so she didn’t “lose again to these guys… [who] think they are better.” furthermore, tracey discussed how the “gender battles” motivated her and fellow “really smart” girl classmates who were also latin@ to support each other in leyva latin@ college women and mathematics journal of urban mathematics education vol. 9, no. 2 106 bringing boys to feel “threatened by [their] girly greatness.” she referenced such collectivism among the latin@ girls to make a case for its important role in the mathematics success of latin@ women underrepresented in stem fields. tracey commented: “there’s not many of us [latinas] out there. in my [ap calculus] class, there were only four of us: one girl didn’t do so well and the other two did i guess because we stuck together.” tracey’s reflections capture how the potentially marginalizing discourses of girls’ mathematics ability that fueled the “gender battles” of the ap calculus course came to be challenged and engaged constructively among tracey and her fellow latin@ girl classmates. tracey described how such support from her fellow latin@ girl classmates was meaningful due to their shared understandings of gendered cultural pressures: all of my latina friends from high school, we were either first-generation american or were immigrants that were born out of the country and came here. so it’s really nice to have the support from them ‘cause they all had what i had at home which was, “have a child young, get married.” we all said: “we are not gonna do that. we are gonna go to college.” (focus group discussion) this reflection illustrates the important role that peer connections with other latin@ girls played in tracey’s experience of not only challenging notions of girls not being good at mathematics, but also negotiating latin@ family values with her pursuits of higher education. she described her sense of satisfaction and mutual support with having several of her hometown friends—fellow latin@ college women—enroll in the same university so they could “stick through [it] together.” one of the girls from the ap calculus course, for example, enrolled in the university’s engineering program so tracey and her continued studying together for mathematics after high school. tracey commented on how it was helpful studying with her because she was someone familiar—namely, her “best friend for like six years.” by sustaining this peer network from high school, tracey and her peers continued supporting each other in navigating college academics while also negotiating family expectations of maintaining the cultural status quo of latin@ women. moreover, tracey discussed how teenage pregnancy was common, or a “casual thing,” for latin@ girls in her hometown. she recalled hearing comments from high school mathematics classmates made toward latin@ girls such as “go back to the kitchen” and “you need to be married.” tracey described how others at the university were “shocked and…appalled” learning about her friends’ early pregnancies rather than the support expressed by high school peers. such contrasting reactions to teenage pregnancies at the high school and university capture peers’ different levels of awareness between the two spaces about familial pressures that latin@ girls experience about becoming young mothers and wives. by asserting their shared pursuits of a college education instead of marrying and having children, tracey and her latin@ girl peers held mutual understandings of this gendered cul leyva latin@ college women and mathematics journal of urban mathematics education vol. 9, no. 2 107 tural expectation and supported one another in “beating the stereotypes of race and…the social norms of gender” by pursuing higher education. family, therefore, also played a role in shaping tracey’s connection to mathematics. she alluded to family influences of immigration, financial struggle, and gendered cultural narratives in describing why she saw herself as “a perfect example of why the latinas don’t make it in mathematics.” tracey reflected on how being a first-generation college student in the united states entailed managing elderly family members’ “very old-fashioned” views of a gendered division of household labor taken up in cuba—namely, “the men work, the women nurture…. the men provide the money, the women clean the house.” tracey, for example, discussed how her grandmother who recently immigrated to the united states criticized her for “slacking” because she was attending college at eighteen years old instead of “get[ing] married [and] start[ing] a life” so she can provide her with great grandchildren. such family expectations of motherhood and marriage shaped tracey’s views of latin@ women throughout her upbringing: “latina women—you’re a mom. you are to breed and work for, tend to your husband. at least that’s what i grew up thinking.” this view of latin@ women’s role in the family stood in opposition to tracey’s intended pursuits of “further[ing] [her] education” and obtaining a college degree in mathematics. tracey saw her mother as a frame of reference of what can happen to a college-bound latin@ woman when “culture catches up” to her in the united states. in particular, her mother was unable to complete a college degree in mathematics education after she was pregnant with tracey and began financially supporting the family as the only working adult in the household: i grew up and still live in poverty. my mom works a horrible job that she hates. my dad can’t work…. he’s disabled…. my grandmother can’t work. she’s old…. we struggle…. i saw what happened to my mom. she pursued a math career, culture caught up with her, and now she’s in that situation. if i go down that path, i am not going to move forward…. she doesn’t want what happened to [her] to happen to [me]. (individual interview) tracey uses her mother’s story as motivation to major in mathematics and become a mathematics teacher. furthermore, tracey acknowledged her pursuits in mathematics allow her to avoid being in the same position as her mother after falling victim to gendered cultural pressures faced by latin@ women. in addition, tracey reflected on hope for change in the prevalence of stereotypes about latin@ women as young mothers and not being college-educated so opportunities for latin@ women like her in stem are increased. she alluded to how such change would allow her to “move forward and…carry on [her] mom’s legacy” of becoming a mathematics teacher. thus, tracey’s reflections illustrate how knowledge of her mother’s culturally-influenced experiences as well as shared leyva latin@ college women and mathematics journal of urban mathematics education vol. 9, no. 2 108 experiences with latin@ girl classmates navigating stereotypes of mathematics ability and higher education shaped her academic and professional pursuits to challenge discourses that marginalize latin@ women. first year of college. tracey remarked on her appreciation of having personable college mathematics instructors who connected with their students. she described how her first-semester calculus teaching assistant, vince, was the “funniest man ever” who the students regularly approached during weekly problem-solving workshops with questions. in addition, tracey appreciated vince’s student-centered facilitation of the calculus workshops such as soliciting questions from students as well as circulating the classroom to offer guidance and support to problem-solving groups. vince, thus, provided tracey with the opportunity to learn from a mathematics teacher in college who prioritized student learning and support much like mr. sosa did. given the common use of teacher-centered lecture formats for mathematics teaching at the undergraduate level, vince’s teaching approach allowed tracey to continue receiving the academic and relational support from teachers that played a central role in her positive high school mathematics experiences. with a smaller latin@ representation at the university compared to high school, tracey felt as though latin@s were lumped together as one homogeneous racial group as opposed to the appreciation of latin@s’ cultural diversity in high school. she described how instead of valuing how “hispanic culture had many branches,” her first year at the predominantly white university brought her to feel as though latin@s as an entire group were positioned as: “you’re all together. you’re all here.” this lack of acknowledged within-group variation in college, as a result, contributed to her feelings of being subjected to racial judgments framed by the discourse of latin@s not being good at mathematics. for example, tracey contrasted this with her experience of attending a predominantly latin@ high school where she felt judged more for being a girl and less for being latin@. she described: “there wasn’t any really degrading on my intellectual skills based on my race because we were all the same race. it was more based on ‘cause i was a girl.” to illustrate, tracey reflected on her university peers’ surprised reactions upon learning that she was a mathematics major and then proceeding to ask about her ethnicity and performance in mathematics: when i came here [to the university], it was really weird when people would ask me, “what major are you?” and i’d say, “oh, i’m thinking of math.” “oh! math?! really? wow! i wouldn’t have expected that.” a good amount of people ask me, following up the question of “what major are you?” with “where are you from?”…. and i’d say, “oh i’m cuban.” “oh you’re cuban. oh, alright. that’s cool. alright, so how are you doing in math?” alright, i guess they just wanted to know where i was from? (individual interview) leyva latin@ college women and mathematics journal of urban mathematics education vol. 9, no. 2 109 tracey was perplexed by the logic behind others’ “awkward follow-up question” regarding her ethnicity after sharing that she was a mathematics major. however, she also acknowledged being more aware with time of how racialized ideas of mathematics ability including latin@s not being good at mathematics contributed to such interpersonal exchanges at the university. tracey’s reflections on others’ reactions to her being a mathematics major, thus, capture her heightened awareness of her high school and university as spaces racialized in different ways that in turn shaped others’ potentially negative judgments of her mathematics ability in college and lack thereof in high school. tracey, however, found herself questioning if she wanted to continue pursuing a mathematics major after her first exam in second-semester calculus. she described how her efforts such as attending lectures, contacting instructors for extra help, and completing homework assignments did not translate into the grades that she expected. during this discouraging experience, tracey reached out to advanced peers pursuing a mathematics major for advice on handling this situation: i spoke to different people who i’ve met along the way, who are also math majors, and have gone through this and they all say that…calc 2 is the hardest math. “you just have to stick through, hold your head high, get through it, and after calc 2, you’ll make it.” and it really helped to have peers say: “you know, every single person in the universe struggles with calc 2. it’s hard, it’s really hard. once you pass that milestone, it gets better.” (individual interview) although this peer support was not content-specific such as recommended studying or test-taking strategies, it provided tracey with closure in not feeling alone in her situation with second-semester calculus. tracey’s connections with more advanced peers at the university, thus, not only allowed her to maintain a peer study network for mathematics in college, but also provided emotional support and affirmation of the struggles that come with being a mathematics major. cross-case analysis of counter-stories looking across these two counter-stories, we see the variation in how lauren and tracey invoked and negotiated discourses related to mathematics ability and stem higher education to make meaning of the intersectionality of their experiences as latin@ women in mathematics. i organize the following cross-case analysis by the most dominant discourses present in their counter-stories: (a) mathematics ability is innate, (b) women and latin@s are not good at mathematics, (c) latin@ women are underrepresented in stem, and (d) latin@ women become young mothers and wives instead of college students. insights into how the two latin@ women encountered them institutionally and interpersonally are also discussed. finally, in the analysis, i explore strategies that lauren and tracey adopted to navigate these discourses and advance their respective mathematical pursuits. leyva latin@ college women and mathematics journal of urban mathematics education vol. 9, no. 2 110 mathematics ability is innate. lauren and tracey, in different ways, engaged the discourse of mathematics ability as innate. for example, lauren saw herself as being naturally talented in mathematics that allowed her to pay less attention during high school mathematics courses, feel as though the subject came easier to her than others, build stronger relationships with teachers, and work well with peers at similar levels of ability. such innateness of ability, however, ran counter to lauren’s acknowledgement that her mathematics success was largely contingent on receiving high-quality instruction and support from her teachers in high school and college. in contrast, tracey challenged the innateness discourse after a turning point in her mathematics experience—namely, receiving her first low grade of c in high school. mathematics ability was what allowed tracey to feel “intellectually smart” and confident, but her overconfidence in algebra ii honors brought her to appreciate that mathematics success was less about natural talent and more about hard work and effort. this discourse continued to inform tracey’s approach to future mathematics courses including ap calculus where she and fellow underrepresented latin@ girls engaged in “gender battles” to collectively disprove notions of men being better at mathematics than women. lauren’s and tracey’s respective ways of engaging the discourse of innate mathematics ability captures the pressures of academic success that can be placed on students especially latin@ women and other populations underrepresented and marginalized in mathematics. in lauren’s counter-story, we observe the impact that high-quality teaching can have in disrupting such notions of innate ability often attributed to men, whites, and asian american students who hold higher rates of achievement and representation in mathematics within the united states. tracey’s reflections on her low grade in algebra ii honors and underrepresentation in ap calculus point to systemic issues of underachievement and underrepresentation among women and racially minoritized groups in mathematics. when innateness of mathematics ability—a discourse framed by colorblind and gender-blind ideologies—is coupled with these systemic issues, women’s and racially minoritized groups’ underachievement and underrepresentation come to be explained as these groups being inherently deficient of potential for mathematics success (battey & leyva, 2016, this issue; martin, 2009, 2013; mendick, 2006). thus, a gendered and racialized hierarchy of mathematics ability is produced. innateness of ability operates as a colorblind and gender-blind way of discussing this racialized and gendered hierarchy. it can bring fellow members of marginalized groups to position each other as being more or less latin@ and more or less feminine based on perceived mathematics ability (mendick, 2006; stinson, 2008). this positioning is particularly oppressive for latin@ women as an underrepresented group in mathematics in navigating gendered and racialized judgments of ability that bring their success to be deemed unexpected and transgressive. an example of this positioning was observed when lauren discussed how latin@ boys in high leyva latin@ college women and mathematics journal of urban mathematics education vol. 9, no. 2 111 school engaged in gendered peer policing premised on the notion that it is not normal for a girl and “especially a latina girl” to be good at mathematics. women and latin@s are not good at mathematics. lauren and tracey also engaged discourses of women and latin@s not being good at mathematics. it is important to note that the ways in which lauren and tracey perceived schools, classrooms, and other institutional spaces as gendered and racialized shaped the level of saliency of these discourses throughout their experiences (moore, 2008; stinson, 2008). lauren, for instance, described feeling more conscious of the “gender thing” and women’s underrepresentation in stem at the university than in high school. she saw women’s representation in her high school mathematics faculty as an implicit way of communicating to students, especially boys, that “a female can do it [mathematics] too” and thus challenging gendered discourses in mathematics. tracey felt her mathematics ability subject to scrutiny in relation to her gender identity at her predominantly latin@ high school, but judged more so racially at the predominantly white university. the discourse of girls being less smart and good at mathematics than boys shaped the competitive atmosphere of tracey’s ap calculus classroom with the outnumbered latin@ girls engaged in “gender battles” over top spots in the class with boy classmates. at the university, tracey felt her shared pursuits of a mathematics major to be received with skepticism and surprise including an “awkward follow-up question” about her racial background. this contrasted with tracey’s high school experience where “there wasn’t really a discouragement…on race and mathematics” in light of being in a predominantly latin@ town. lauren’s and tracey’s counter-stories, thus, capture how consciousness of their social identities varied across contexts and thus led to different positionings of themselves and others along the hierarchy of mathematics ability over time. an evolving theme across lauren’s and tracey’s engagements with discourses of women and latin@s not being good at mathematics is how these gendered and racial discourses shaped the nature of instruction and teacher-student relationships in high school and college mathematics (battey, 2013; battey, neal, leyva, & adams-wiggins, 2016). both latin@ women described how their mathematics success in high school was largely attributed to establishing positive, supportive relationships with teachers who held high academic expectations of them and fellow students in alignment with notions of culturally responsive pedagogy in urban schools (gay, 2010; ladson-billings, 1995). high school mathematics teaching, therefore, provided them with opportunities to be academically challenged and supported in ways that were affirming of their identities as latin@ women. during their first year at the university, lauren and tracey reconsidered their continued pursuits of a computer science and mathematics major respectively. they raised concerns about feeling disconnected from their mathematics professors’ teaching approaches and seeing their efforts not translate into better grades. these reflections raise questions about the shift in mathematics teaching and learning ex leyva latin@ college women and mathematics journal of urban mathematics education vol. 9, no. 2 112 periences between high school and college. lauren’s and tracey’s counter-stories warrant our analytical considerations of the extent to which undergraduate mathematics education in predominantly white universities and other higher education institutions sustains forms of cultural and gender responsiveness that the latin@ women received during high school (jett, 2013; rodd & bartholomew, 2006; stinson, 2016). such forms of support in mathematics teaching better attend to the intersectionality of experiences for underrepresented and marginalized populations including latin@ women and thus broaden students’ opportunities to co-construct positive academic and social identities across the p–16 mathematics pipeline. latin@ women are an underrepresented group in stem. lauren and tracey raised the discourse of latin@ women being an underrepresented group in stem. lauren interpreted statistics of latin@ women as the most underrepresented group in stem as a reflection of the minimal encouragement and mentorship that they receive to pursue stem education and careers—a departure from deficit notions of latin@ women inherently not being good at mathematics. she engaged the discourse of latin@ women’s underrepresentation to make meaning of why she went on to pursue higher education in stem. lauren distinguished her situation of being supported by her family to major in a mathematics-intensive field like computer science from that of other latin@ women whose families may not have “taught [them] that they can do math” or that stem was a viable career pathway for them. her stem pursuits also allowed lauren to serve as a role model for her younger relatives to sustain such family encouragement. tracey similarly invoked the discourse of latin@ women’s underrepresentation in stem when reflecting on her experience as one of the four latin@ girls in her ap calculus course. she explained that because they “stuck together,” she and her fellow latin@ girl classmates were successful in the course and their first year at the university. this sense of collectivism was powerful because tracey and her peers understood each other’s similar situations as not only being underrepresented in the ap calculus classroom, but also dealing with gendered cultural pressures of becoming young mothers and wives rather being good in mathematics and pursuing higher education. this supportive peer network motivated tracey and her latin@ girl classmates to challenge discourses that marginalize latin@ women’s opportunities in stem throughout the ap calculus course and into college. lauren’s and tracey’s counter-stories, thus, illustrate how they respectively took up the discourse of latin@ women’s underrepresentation to mobilize change through their stem higher education. in addition, they saw their pursuits of mathematics-intensive majors serving as opportunities to sustain the interpersonal encouragement from family (in lauren’s case) and peers (in tracey’s case) that largely influenced their academic and professional endeavors in stem. the potentially marginalizing influence of the underrepresentation discourse, therefore, instead stimulated the latin@ women’s adoption of success-oriented beliefs of stem edu leyva latin@ college women and mathematics journal of urban mathematics education vol. 9, no. 2 113 cation as an opportunity to become role models for future generations of latin@ women as well as strategies like building peer networks for navigating the socially exclusionary landscape of mathematics and stem education more broadly. latin@ women become young mothers and wives instead of college students. the discourse of latin@ women expected to become young mothers and wives rather than being college-bound emerged across lauren’s and tracey’s counterstories. the latin@ women were both subjected to the discourse through peer remarks in high school about taking care of children, returning to the kitchen, and getting married rather than doing mathematics and pursuing a college degree. lauren and tracey saw such peer policing as a reflection of traditional cultural values of a gendered division of labor taken up in latin@ households. while lauren was aware of such gendered cultural expectations that latin@ women “stayed at home” instead of working or studying, she reflected on how her family did not perpetuate such values and instead encouraged her to “break that tradition” by pursuing a college education. although lauren was a first-generation college student, she was pushed less toward marriage and child rearing than tracey whose mother would have been the first family member to complete a college degree in the united states. tracey, however, discussed how her grandmother frequently criticized her for not thinking about marriage and raising children instead of applying to and attending college. her mother also served as a frame of reference of what can happen to a latin@ woman when “culture catches up” to her and derails her stem career pursuits. as a result, tracey viewed returning as a high school mathematics teacher in her hometown as an opportunity to pick up where her mother left off in her stem career development. this returning was also a way for her to encourage younger generations of latin@ girls to “beat the culture” and not fall victim to discourses that steer them away from applying their mathematics ability and interests. these different experiences raise considerations of the influence of immigration in the latin@ women’s negotiations of family expectations and stem higher education with the presence of recently immigrated family members contributing to a stronger salience of gendered cultural discourses at home. despite the varying influence of the discourse about latin@ women as young mothers and wives, it is important to note how lauren and tracey both leveraged family-oriented motivations of pursuing mathematics-intensive majors to directly challenge the discourse. what is a potentially marginalizing discourse that stems from traditional latin@ family values, therefore, was met with the latin@ women’s empowering strategies of resistance for the advancement of their family situations. this resistance illustrates how the two latin@ women’s familismo (marín & marín, 1991; suárez-orozco & suárez-orozco, 1995), or sense of loyalty and responsibility to the latin@ family unit, played a critical role in their motivations to excel in mathematics while negotiating stem higher education with family expectations. leyva latin@ college women and mathematics journal of urban mathematics education vol. 9, no. 2 114 social class also shaped lauren’s and tracey’s perspectives on the importance of pursuing mathematics-intensive majors and stem careers as latin@ women. lauren set herself apart from the “really disappointing” complacency academically and professionally among her latin@ older cousins who did not pursue higher education. in doing so, she described how serving as a role model for her cousins as a college graduate in stem would help them realize how they can also “live a life with a degree and more comfortably.” tracey alluded to her family’s financial struggle with her mother taking an unenjoyable, low-paying job as the sole household contributor. she perceived such financial struggle to be a consequence of her mother falling victim to the gendered cultural expectations of becoming a young mother as a latin@ woman. tracey draws on this awareness to stimulate her mathematical pursuits as a pathway for avoiding a similar situation and “carry[ing] on [her] mom’s legacy” as a future high school mathematics teacher. immigration and class, therefore, are closely tied to the development of latin@ family values such as maintaining cultural integrity and pursuing a college education for aims of social mobility in the united states. the interplay of such family influences with the discourses of mathematics ability and stem representation results in varying intersectionalities of experience for the latin@ women like lauren and tracey. discussion and implications in this study, i presented cases of two latin@ college women’s mathematics experiences with analytical attention to how they navigated discourses of mathematics ability and stem higher education encountered institutionally and interpersonally. the intersectional analysis of lauren’s and tracey’s counter-stories details the complexity of how they made meaning of their experiences as mathematics students and negotiated their identities as latin@ women with mathematics success and pursuits of stem higher education (esmonde et al., 2009; martin, 2009). more specifically, the analytical construction of their counter-stories allowed for detailing the variation between two latin@ women’s mathematics experiences and adopted strategies in managing discourses of ability and underrepresentation mapping onto empowerment, resilience, and success (bowleg, 2008). by exploring gender as socially constructed, this work departs from sexbased, binary analyses comparing achievement and participation differences between women and men in mathematics and thus allowed for capturing within-group variation of experience particularly among latin@ women (leyva, in press; mendick, 2006). the analytical foregrounding of lauren’s and tracey’s gendered experiences shaped by other social influences including race, class, and immigration advances work in mathematics education that, by and large, has left intersectionality of experience, especially among women of color, implicit (leyva, in press). the poststructural analysis documents the emergent ways in which the two leyva latin@ college women and mathematics journal of urban mathematics education vol. 9, no. 2 115 latin@ women were discursively gendered as mathematics students while negotiating marginalizing perspectives of mathematics ability and sense of familismo. the analysis generated nuanced understandings of what it means to be a latin@ woman studying mathematics as well as how educational structures and stakeholders can more effectively advance educational equity for latin@ women and other marginalized groups in p–16 mathematics. for example, the high school-tocollege transition was a challenging experience that came with pedagogical and relational shifts in mathematics teaching that resulted in both lauren’s and tracey’s reconsiderations of mathematics-intensive stem majors. both lauren and tracey alluded to gender-affirming and culturally responsive moments in high school mathematics that played a major role in their pre-college interests and academic success. such social support and teacher–student connections, however, were minimally experienced at the university with the latin@ women finding themselves in positions of needing to “teach themselves” and re-building peer networks on their own. this corroborates findings from extant research in stem higher education that identified institutional selectivity, or attending research-intensive colleges or universities that hold teaching at a lower priority for faculty, as a negative statistical predictor of stem persistence for women of color (espinosa, 2011). furthermore, findings from this study echo resounding calls for mathematics teaching responsive to gender, racial, and other forms of diversity at the undergraduate level (jett, 2013; rodd & bartholomew, 2006; stinson, 2016). lauren’s and tracey’s counter-stories, therefore, call into question the extent to which the coupling of academic and social support critical to their construction of positive mathematics identities as latin@ women in high school is sustained and valued in undergraduate mathematics education—a space largely shaped by gendered and racial discourses of mathematics ability—especially at predominantly white institutions (mcgee & martin, 2011; mendick, 2006; shah, in press). thus, p–16 mathematics teachers play an important role in being aware of deficit discourses and supporting marginalized student populations, including latin@ women, in successfully navigating them to broaden opportunities for mathematics success. this support is especially important at the undergraduate level with entry-level mathematics courses like calculus documented as a filter resulting in attrition of stem majors (chen, 2013; rasmussen, marrongelle, & borba, 2014). this filtering was evidenced in lauren’s and tracey’s reconsiderations of their mathematics-intensive majors by firstand second-semester calculus respectively. despite these discourses’ marginalizing influences and minimal social support available to navigate them in undergraduate mathematics, lauren’s and tracey’s counter-stories illustrate the two latin@ women’s development of success-oriented beliefs and strategies in negotiating these discourses with their social identities. it is important to note how the two latin@ women’s familyand community-related motivations to seek higher education in mathematics-intensive stem areas were leyva latin@ college women and mathematics journal of urban mathematics education vol. 9, no. 2 116 largely influenced by mathematics teaching, particularly in high school, that brought them to see themselves as being “good at math.” therefore, i argue that p– 12 stem education should be a critical area of focus in contemporary calls and initiatives for broadening participation in stem among underrepresented populations. a mathematics or other stem degree is often a prerequisite in being certified as a p–12 mathematics teacher. increasing retention rates in undergraduate mathematics and other mathematics-intensive stem areas, thus, would further diversify the p– 12 teaching force and connect underrepresented students with teachers who look like them and who can relate to their personal experiences in dealing with deficit discourses in mathematics and stem broadly. this diversifying is important as lauren and tracey discussed the influence of seeking support from as well as building networks with fellow women and latin@s, but they were mostly family members and peers as opposed to mathematics or stem educators. findings from this study corroborate those from extant work in urban mathematics education on the importance of peer networks in the mathematics success of students of color (oppland-cordell, 2014; treisman, 1992; walker, 2006). lauren and tracey either sought academic support from peers at the university or sustained peer networks from their home communities that served as resources for their success in high school and college mathematics. tracey, in particular, commented on how maintaining peer connections from high school were both academically supportive and socially empowering. however, such ties to peer networks are often severed for women and students of color once they start their first year of college, thus leaving them with the responsibility of re-building such networks on their own for their continued success in mathematics. lauren’s and tracey’s abilities to seek, establish, and sustain peer networks in mathematics challenge notions that the ability to form such networks is inherently missing among students of color (treisman, 1992). more importantly, this highlights the responsibility that colleges and universities, especially predominantly white institutions, have in supporting marginalized students’ peer network development for academic and social support in stem. such support includes the management of deficit discourses and feelings of underrepresentation and as well as navigating institutional spaces for undergraduate stem success (mcgee & martin, 2011). a limitation of this study is the lack of observations of the two latin@ women across undergraduate mathematics classrooms. ethnographic observations and field notes would complement their reflections of mathematics as a socially exclusionary space in terms of their engagement with content, peers, and instructors at the university. there is analytical space for future research that examines the instructional and relational spaces of undergraduate stem education including mathematics classrooms to document how opportunities for mathematics learning are promoted or hindered for marginalized populations including latin@ women. although there exist significant issues related to latin@ students’ experiences in p–12 leyva latin@ college women and mathematics journal of urban mathematics education vol. 9, no. 2 117 mathematics, these issues are only recently becoming part of the undergraduate mathematics conversation. insights from this study and those that follow can inform professional development for undergraduate mathematics departments and mathematics educators as they work to create academic and social supports that sustain opportunities for students of color like lauren and tracey to continue to be successful in their higher stem education. references anthias, f., & yuval-davis, n. 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email: sue.wilson@canterbury.ac.nz. her research interests include mathematical practices in school contexts, initial teacher education, and mathematical tools and representations. liz brown is the kaiārahi matua (senior māori adviser) for te rāngai ako me to hauora – the college of education, health and human development – te rāngai pūtaiao – the college of science – and te rāngai toi tangata – college of arts at te whare wānanga o waitaha – university of canterbury, 20 kirkwood avenue, ilam 8041, christchurch, new zealand; email: liz.brown@canterbury.ac.nz. her research interests include biculturally responsive pedagogy and practice, preservice teacher education, leadership and change management. cultural competencies and planning for teaching mathematics: preservice teachers responding to expectations, opportunities, and resources susanna wilson university of canterbury new zealand jane mcchesney university of canterbury new zealand liz brown university of canterbury new zealand in this article, the authors report on a small-scale study set in a context of a firstyear mathematics education course for preservice primary teachers. professional documentation from three different sources were analysed in relation to the national document tātaiako: cultural competencies for teachers of māori learners, which was used as a key course resource in a year-one mathematics education course for preservice teachers. the authors found evidence that the preservice teachers used the resource to identify important learning and teaching practices, and as a source of language and examples. a further tentative finding was how relational aspects of teaching mathematics were adopted as indicators of culturally connected practice. keywords: cultually responsive teaching, preservice teacher’s expectations, teacher education ver many years, there has been a growing international focus on equitable access to mathematics education for students from diverse cultural backgrounds. yet students from minority cultures continue to be overrepresented in the o wilson et al. cultural compentenices journal of urban mathematics education vol. 10, no. 1 96 lower end of mathematics assessment data (boaler, 2002; civil, 2007; nasir & cobb, 2007). this overrepresentation is also the case in new zealand where the school mathematics assessment data continue to be concerning for māori (the indigenous) students (ministry of education, 2015). national data in mathematics achievement show that 35% of māori students achieved below the “national standards” at years 1 to 8 (5 to 12 years old) compared with 20% of pākehā/ european students (ministry of education, 2015). a minority of māori students are enrolled in kura kaupapa māori education (māori language immersion schools). in 2011, there were 6,132 students attending kura kaupapa māori, compared with 165,664 learners who identified as māori, enrolled in mainstream schools (ministry of education, 2013). the majority of students of māori descent therefore are attending mainstream (mostly state) primary schools that are required to provide teaching and learning programmes to meet and satisfy the educational needs of māori students. concerns over equity are not new, nor confined to new zealand, and teachers have been developing and trialling multiple approaches for addressing student underachievement for decades. beginning teachers are not exempt from this equity focus and their initial teacher education (ite) programmes can prepare them to teach students from diverse cultural backgrounds (downey & cobbs, 2007; gay, 2002; kitchen, 2005; white, murray, & brunard-vega, 2012). increasingly, expectations that teachers address issues of diversity and culture in their teaching are mandated at national policy level in standards for teachers. in new zealand, these are explicit in the graduating teacher standards (new zealand teachers council, 2013a) and the registered teacher criteria (new zealand teachers council, 2013b). a teacher graduating from an ite programme is expected to, for example, “promote a learning culture which engages diverse learners effectively” (2013a), and on completion of registration (after 2 years of teaching in schools), “respond effectively to the diverse language and cultural experiences, and the varied strengths, interests and needs of individuals and groups of (learners)” (2013b). one approach that aims to engage students from diverse backgrounds is known as culturally responsive teaching, an approach that has been around for a number of years (subsequently discussed). creating opportunities and designing curriculum innovations to address issue of diversity, within an ite setting, can be a complex task, even when direction and guidance are provided in national policies (downey & cobbs, 2007). mathematics curriculum courses, however, are legitimate sites for preservice teachers (psts) to learn how to implement culturally responsive mathematics teaching. so how might psts learn how to plan, teach, and adapt practices that illustrate culturally responsive teaching? one professional learning process for developing expertise is that of teacher “noticing” (jacobs, lamb, & philip, 2010; van es & sherin, 2002). in a particular curriculum context, mason’s (2002) notion wilson et al. cultural compentenices journal of urban mathematics education vol. 10, no. 1 97 of disciplined noticing is “to make a distinction, to create foreground and background to distinguish some things from its surroundings” (p. 31). all teachers, particularly psts, face the complexity of implementing mathematics teaching practices as well as culturally responsive practices, and teacher educators play an important role in supporting psts to develop the ability to professionally notice within the discipline of mathematics. psts, unlike more experienced colleagues and without a repertoire of knowledge to draw from, “cannot be aware of or respond to everything that is occurring” (jacobs et al., 2010, p. 170). within ite courses, teacher educators can deliberately include strategies such as making explicit relevant aspects of practice, selecting specific resources, and scaffolding course context for close attention. in this article, we report our findings of a small-scale study of psts based around a first-year primary (elementary) mathematics education ite course. we collaborated with colleagues who have expert cultural and pedagogical knowledge to redesign the course in order to embed a focus on culturally resposive teaching. we chose to include a national document as a key resource for the psts. this national document—tātaiako: cultural competencies for teachers of māori learners—sets out cultural competencies as important features of teacher practice in early childhood, primary, and secondary settings. the focus of tātaiako is for teachers to support “māori students to enjoy education success as māori” (ministry of education, 2011, p. 4). our research investigation was to determine what psts notice and record about tātaiako within their professional documentation in three different contexts. we discuss how we incorporated a focus on cultural competencies within course requirements as an indicator of valued knowledge for psts. drawn from an ongoing research study, where the main focus is how psts plan for mathematics teaching, we report on the ways that a small group of psts identified cultural competencies within their planning. we also report ways they included tātaiako in their written planning during a 4-week practicum that followed the completion of the mathematics education course. we believe the findings from this small-scale study have a number of implications for our work as mathematics teacher educators. in the brief final section, we discuss those findings and our proposed next steps. culturally responsive teaching a growing number of teachers aim to engage students from diverse backgrounds by adopting teaching practices that are attuned to and connect with students’ cultural heritages and educational experiences (cochran-smith et al., 2015). such practices fall under the umbrella of culturally responsive teaching (crt), an approach that has been around for a number of years. gay (2002) describes culturally responsive teaching as using “the cultural characteristics, experiences, and per wilson et al. cultural compentenices journal of urban mathematics education vol. 10, no. 1 98 spectives of ethnically, diverse students as conduits” for more effective teaching (p. 106). the underlying pedagogical theory of crt, according to gay, is that when academic knowledge and skills are situated within the lived experiences and frames of reference of students, they are more personally meaningful, have higher interest appeal, and are learned more easily and thoroughly. (p. 106) in these ways, crt not only provides students with greater access to learning opportunities but also these more meaningful experiences make a difference to their academic achievement (gay, 2002; villegas & lucas, 2002). crt has been taken up in a range of educational contexts from early childhood schooling to tertiary and, in particular, with teacher professional learning, including ite (kitchen, 2005; nasir, 2016; villegas & lucas, 2002). ite programmes can begin preparing psts to teach students from diverse cultural backgrounds by implementing various types of courses designed for psts to learn how to teach in culturally responsive ways. some of these multicultural education courses are offered as discrete courses without connections to other courses or as optional courses for psts, which means some psts may graduate with minimal knowledge of how to teach students from a variety of cultural backgrounds. while these courses play an important role in preparing psts for teaching in culturally responsive ways, more in-depth learning can occur when teacher educators commit to including crt practices within the context of their subject courses. the inclusion of crt relies on teacher educators having knowledge not only of their subject areas but also of cultural aspects including knowing a range of crt practices suitable for their ite setting (villegas & lucas, 2002). in their recent review of teacher education programmes and the preparation of psts for teaching in diverse classrooms, cochran-smith and colleagues (2015) contend that psts need to experience more than one course dedicated to crt, and suggest they need several opportunities within their entire programme to develop such practices. in addition, psts can be agents of change when they influence what happens in a classroom (kitchen, 2005), and although they are not solely responsible for “transforming the education system,” psts do have an important role to play (villegas & lucas, 2002, p. xix). teacher educators, therefore, need to design and implement ite course experiences that help psts develop a repertoire of crt practices (cochran-smith et al., 2015). when designing courses, gay (2009) advises teacher educators to avoid following “the course of least resistance” by either leaving course content as-is, or merely making cosmetic changes (p. 194). furthermore, psts need to do more in coursework than focus on superficial elements of different cultures, such as food, holidays, and festivals, which are described as multicultural “tourism” (l. derman-sparks, as cited in lenski, crumpler, stallworth, & crawford, 2005, p. 87). wilson et al. cultural compentenices journal of urban mathematics education vol. 10, no. 1 99 some teacher educators have designed and researched programmes that more effectively address culture and cultural differences. lenski and colleagues (2005) studied their beyond awareness research project and identified the importance for psts to consider their own beliefs and assumptions about different cultural aspects, particularly the “values and practices of families and cultures different to their own” (p. 85). it is also important for psts during ite course experiences to develop positive “affirming” and “asset oriented” views, as opposed to long-held deficit views of diversity (cochran-smith et al., 2015, p. 114). these educators have also noted that there have been many studies about how ite has influenced psts’ beliefs and attitudes but far fewer studies of influences on psts’ practice in school settings, that is, whether changes in beliefs lead to changes in teacher actions. culturally responsive mathematics practices in ite nasir (2016) claims that while mathematics has been traditionally viewed as a subject unrelated to culture, like all subject areas, mathematics is a rich source of cultural knowledge and practices. when connections between culture and mathematics are recognised within an ite setting, worthwhile learning can occur when both are taught together, and ite mathematics courses are “a viable place to begin deep level changes” (gay, 2009, p. 191). in a study of secondary mathematics psts, kitchen (2005) found that psts needed help to notice explicit connections between cultural and mathematics practices. he designed and implemented course experiences such as exploring the cultural origins of important mathematics ideas and situating mathematics learning in a cultural context of an american indian reservation. he also included a focus on equity practices by critiquing the effects of ability tracking on students from non-dominant cultures. his aim, within the context of the mathematics education course, was to support the psts to incorporate equitable teaching practices in implicit and explicit ways so that they could effectively teach students from diverse cultural backgrounds. in another study, white, murray, and brunard-vega (2012) found that psts’ dispositions toward their students’ cultural backgrounds influence their awareness and sensitivity toward diverse learners, and consequently shape their selection of classroom mathematical teaching practices. if psts do not understand the interconnections between multiple layers of culture and their classroom practices, then psts risk continuing to “create classroom cultures and engage in classroom practices that perpetuate limited opportunities and barriers for students to learn and do mathematics” (p. 41). mathematics teacher educators therefore have an important role in their curriculum (methods) courses to teach both mathematics and cultural practice simultaneously (cochran-smith et al., 2015; gay, 2009; kitchen, 2005; nasir & cobb, 2007). there are two complementary goals: to prepare prospective mathematics teachers to implement the curriculum and to teach diverse learners. psts can then begin to develop the skills to “be change agents in the lives of their wilson et al. cultural compentenices journal of urban mathematics education vol. 10, no. 1 100 students, mediating the educational equalities and experiences by their students by promoting challenging mathematics curriculum and instruction in the classroom” (kitchen, 2005, p. 37). culturally responsive mathematics teaching in new zealand similarly, in new zealand “while mathematics can be seen by some as being culture free—it can provide powerful contexts for developing knowledge and understanding of one’s cultures and cultural values” (averill, taiwhati, & te maro, 2010, p. 167). unfortunately, this misconception that mathematics is “culture free” leads some teachers to “abrogate their responsibility to be culturally responsive” (averill, te maro, taiwhati, & anderson, 2009, p. 27). ideally, in a small country with a bicultural focus, psts could enter an ite programme ready to implement culturally responsive mathematical practices, but there is still much for psts to learn, and mathematics teacher educators have an important role in explicitly teaching both cultural and mathematical practices to ensure the success of diverse learners. psts are unaware of strategies to include bicultural perspectives in their teaching, with a consequence that māori students perceive that “mathematics is not a subject for them” (averill & te maro, 2003, p. 89). one approach for teacher educators is to present mathematics as being “of and from our everyday human realms, sitting right there in our culture” because “we can talk it, argue it, and describe it in more than one language, and in many contexts” (averill et al., 2010, p. 176). if psts are to teach māori students in culturally responsive ways, they need to know about te ao māori (the māori world) and understand how to incorporate this knowledge appropriately when teaching mathematics (averill et al., 2009). averill and colleagues also stress the importance of cultural knowledge and practices being taught in ways that avoid superficial and tokenistic interpretations of cultures because teaching cultural aspects this way can have negative effects on both the learning and the achievement of māori students and can “strengthen their feelings that the only valuable aspects in education are those which come from european viewpoints and knowledge” (averill et al., 2014, p. 35). psts need to understand key cultural concepts and not just “simple translations from one language to another” (averill et al., 2014, p. 35). examples include te reo māori (the māori language), which is valued as a way to pass on knowledge and traditions; māori pedagogies such as learning through participation, song, storytelling, metaphor, and observation; and the concept of ako, where teachers and learners are intertwined (averill et al., 2009). one framework has been developed to help teachers understand key māori concepts that are relevant for mathematics teaching (averill, te maro, taiwhati, & anderson, 2009). the framework includes four elements of māori conceptual understanding: wilson et al. cultural compentenices journal of urban mathematics education vol. 10, no. 1 101 • knowing and understanding each other as people, • knowing and understanding each other as learners, • knowing and understanding each other’s cultures, and • enhancing feelings of cultural identity. (p. 31) this framework helps psts to focus on broader and less superficial aspects of culture and prompts them to consider these elements when planning for mathematics teaching. in an ite mathematics education course designed to include bicultural perspectives, there was a focus on active partnerships between learners, māori language (te reo māori), as well as māori pedagogies, contexts, beliefs, philosophies, protocols, and values. these aspects were selected to “model ways that these can be acknowledged and reflected in the student’s own teaching” (averill & te maro, 2003, p. 88). at the end of the course, the educators found that the psts could identify a wide range of bicultural perspectives within both course content and structure. they also found that they needed to be explicit in their use of bicultural perspectives and practices so that psts could “recognise, acknowledge and draw from all perspectives of the course” (p. 94). later studies (see, e.g., averill et al., 2009) generated six conditions necessary for psts to teach mathematics in culturally responsive ways. these conditions included: deep mathematical understanding, effective and open teacher–student relationships, cultural knowledges, opportunities for flexible approaches and for implementing change, accessible and non-threatening mathematics learning contexts, responsive learning communities, and cross-cultural partnerships (p. 180). some of these conditions relate to overarching relational practices and professional selfknowledge, while some relate specifically to mathematics. in particular, deep mathematical understanding may be connected to an ability to recognise culturally related contexts for mathematical learning. one aspect still to be explored was the psts’ willingness and ability to implement these culturally responsive practices when teaching in classrooms during practicum. tātaiako: cultural competencies for teachers of māori learners tātaiako highlights and illustrates aspects of māori culture appropriate for educational settings and emphasises the importance of teachers’ relationships and engagement with māori learners, their whanau (family) and iwi (tribal grouping). its main aim is for māori students to reach their full potential and “map[s] out a path” for teachers to support students to do so by emphasising how education can be delivered in the context of “vibrant contemporary māori values and norms, reflecting the cultural milieu in which māori students live” (ministry of education, 2011, p. 3). wilson et al. cultural compentenices journal of urban mathematics education vol. 10, no. 1 102 tātaiako sets out five cultural competencies: • wānanga: communication, problem solving, and innovation (participating with learners and communities in robust dialogue for the benefit of māori learners’ achievement). • whanaungatanga: relationships (students, schools, communities) with high expectations. actively engaging in respectful working relationships with māori learners, parents and whānau, hapū (sub tribe grouping), iwi, and the māori community. • manaakitanga: values integrity, trust, sincerity, equity, showing integrity, sincerity, and respect toward māori beliefs, language, and culture. • tangata whenuatanga: place-based, sociocultural awareness and knowledge affirming māori learners as māori. providing contexts for learning where the language, identity, and culture of māori learners and their whānau is affirmed. • ako: practice in the classroom and beyond taking responsibility for their own learning and that of māori learners. (ministry of education, 2011, p. 4) we included the competencies in full to illustrate how important māori concepts and practices are explained in ways that link to recognisable aspects of relationships, respect, and values within educational contexts such as schooling. each competency is explained in terms of “behavioural indicators” (term used in the document) that teachers would demonstrate at different stages of their teaching careers. as an example, for manaakitanga, psts on entry to an ite programme are expected to “value cultural difference,” and at the point of graduation from the programme demonstrate “respect for hapū, iwi, and māori culture in curriculum design and delivery processes,” and, finally, experienced teachers are expected to demonstrate “integrity, sincerity, and respect towards māori beliefs, language and culture” (ministry of educaton, 2011, p. 8). desired outcomes for each competency are also described and are written from both a learner’s and a whānau (family) perspective. a learner’s perspective for manaakitanga is “my teacher uses te reo māori in class and encourages us to speak māori if we want” (p. 9); and from a whānau perspective, the teachers “care about our children and always talk positively about them” (p. 9). the purposes of these exemplar outcomes are to describe each competency and to outline possible teacher actions to support māori learners in educational settings. we selected tātaiako as a conceptual framework for the course because we recognised it as a potentially useful document written to support psts and teachers, and we also saw the potential for psts to develop their understandings of the competencies by linking them to mathematics practices and resources. additionally, the psts had already been introduced to tātaiako in an earlier professional education wilson et al. cultural compentenices journal of urban mathematics education vol. 10, no. 1 103 course. the mathematics education course is in a 3-year, full-time ite programme that prioritises bicultural perspectives and practices, and we wanted to provide another opportunity for psts to continue their learning of bicultural practices. researching psts noticing of tātaiako context the compulsory mathematics education course was in the first year, second semester and had 48 taught hours and 100 independent student hours during 10 consecutive weeks. there were twelve 2-hour lectures for the whole student cohort and eight 3-hour workshops in groups of approximately 35 students. course content includeed pedagogical approaches for teaching school mathematics with a specific emphasis on exploring new zealand mathematics curriculum content and related resources and preparing students for planning and teaching mathematics in a 4week practicum that followed the course. there were two assignments for the course plus a course professional workbook that contained core resources and an organising template for recording notes for each of the lectures and workshops. tātaiako was the focus of a second week lecture where liz, the third author, (in her leadership role of kaiārahi māori; i.e., māori strategy manager in the college of education) began the lecture by revisiting tātaiako and discussing the meanings of each of the five competencies. liz provided some general examples of each competency in relation to primary schooling and sought further ideas from the psts. in the second half of the lecture, sue (the first author) focussed on examples from school mathematical practices and resources, and again drew on ideas from the psts. for example, wānanga (communication, problem solving and innovation) was linked to practices related to mathematical talk (askew, 2012), such as prompts in te reo māori for example, kōrero ki to hoa (tell your partner) and me pēhea koe ka mahi (how did you do that?). tangata whenuatanga (place-based, sociocultural awareness and knowledge) was illustrated by linking mathematical learning to local contexts, such as investigating geometric patterns in traditional māori art and buildings. a variety of photographs were presented to draw attention to existing resources within local communities. sue then showed examples of readily available new zealand mathematics learning resources containing a wide range of māori contexts, and these were used to illustrate existing resources that psts could use and adapt for their future teaching. finally, sue encouraged psts to think critically and to care about the selection of resources in relation to both cultural integrity and mathematical authenticity. for subsequent lectures and workshops, the psts were expected to make links with the competencies of tātaiako in a similar way to the process that was modelled in the lecture. links were initially co-constructed between lecturers and wilson et al. cultural compentenices journal of urban mathematics education vol. 10, no. 1 104 psts who recorded their notes within the tātaiako section on their workbook template, and over time lecturer support was withdrawn and the psts were expected to do this linking independently. psts were also required in their second course assignment to make links between the tātaiako competencies and mathematics practices and resources, and then connect the competencies to their detailed written lesson plans, one each for a geometry and a measurement lesson. we chose geometry and measurement as the focus for the assignment because examples were provided during lectures and workshops, such as whanaungatanga – students will be able to work with their friends to make the reflective symmetrical patterns. manaakitanga – students will be introduced to the māori names for the two dimensional shapes and practice labelling and saying these names as they classify these shapes. when planning these lessons, the psts drew on previously selected and analyzed student learning tasks. they had to transform these tasks for teaching and then design written lesson plans, using a template required for their professional education course. these tasks and lesson plans contributed to preparation for their 4-week professional practicum where psts were required to plan and teach three mathematics lessons and to identify and include a cultural focus in their mathematics lesson plans. participants and data collection the aim of this study was to determine which aspects of tātaiako were noticed and recorded by psts and the links they made to mathematics teaching. generally speaking, the study is an example of the coursework category b-2 from cochran-smith and villegas (2015), examining “the impact of opportunities provided to teacher candidates through courses, with or without field assignments” (p. 13). investigating teacher “noticing” is complicated, and we prioritised psts’ generated writing as indicators of their thinking that were important enough to record, within the power dynamics of required coursework. we adopted a sociocultural lens in our interpretation of the psts’ writing, with a particular focus on meanings about practices and relationships (averill, 2012). we collected documentation data for three main reasons; each data source provided chronologically different data, served different coursework purposes, and lastly, due to living in a post-earthquake environment, we were mindful of any extra pressure of time commitments that might be placed on our pst participants (mcchesney & wilson, 2016). wilson et al. cultural compentenices journal of urban mathematics education vol. 10, no. 1 105 thirteen psts from sue’s 2014 workshop class volunteered to participate in the larger study, providing informed consent for their contributions to be anonymous. the psts had been in different schools during practicum and once all course and practicum assessment requirements were completed, they provided their mathematics education documentation as sources of data. three psts volunteered their workbooks, 11 volunteered their assignment lesson plans, and seven provided practicum lessons plans. the workbooks were not an assessment component of the course, the second assignments had been assessed and returned to the psts at the end of the course, and the practicum lesson plans were assessed and returned to them before being used as a data source. returning student work in this manner provided a gap between course activity and data collection, which protected the students’ academic outcomes because all course requirements were completed. the workbooks, assignment plans, and lesson plans were analysed using content analysis based on the language of the five competencies of tātaiako. data analysis involved reading each document carefully, identifying references to the competencies, and organising them into different categories (which were the five competencies). the data analysis was then checked between the first two authors, eliminating some data before finalising each category (cohen, manion, & morrison, 2011). within each category (of tātaiako competency) we analysed for specific references, use of terms, and examples either copied or written in their own words from tātaiako, as well as references and notes related to mathematical practices and resources. the data analysis of each category is presented in the next section, beginning with the three competencies with the most data, followed by the summary reporting of the two remaining competencies. analysing psts’ documentation wānanga: communication, problem solving, and innovation the psts linked wānanga to mathematics teaching and learning in three ways. the first was students communicating while learning mathematics. the data sources were mostly within the assignment and the practicum lesson plans where psts planned for students to learn mathematics by talking and discussing their learning with each other during the lessons. there were 13 references to communicating in the assignment plans, where a range of verbs (communicating, talking, discussing, sharing, describing, and explaining) were used to describe how the concept wānanga could be embedded into their lessons. for a measurement lesson, which required students to draw a monster by measuring different lengths and shapes, one pst wrote an extended description: students will be encouraged to talk about mathematical solutions, problems and questions with classmates and the teacher. the students will be able to talk with their wilson et al. cultural compentenices journal of urban mathematics education vol. 10, no. 1 106 classmates to attempt the activities which will allow the students to use topic appropriate language. being able to communicate any problems with classmates could also help the students with any difficulties and means they can solve issues without needing to ask the teacher. seeking help from the teachers though will still be hugely encouraged. for this pst, communication meant interactions between students and between students and teacher. this excerpt was similar to data from other psts. for example, in her measurement plan, another pst described wānanga occurring at the beginning of the lesson “where the students and i will be able to share our ideas about what we think the task requires.” in the practicum lesson plans, wānanga references were also about talking, discussing, and sharing ideas during the lessons. one pst who planned and taught three lessons about time wrote “students will have discussions and learning interactions with other students,” and her lesson plans indicated opportunities for student interactions. the second connection to wānanga related to the psts’ plans for arranging how students work together. in the course workbook data, three entries described “working in pairs” or “doing the activities in groups.” after completing the fractions workshop where psts had experienced practical activities and different representatons for fractions, one pst wrote that it was important for students to “be allowed to work together” so that they could manipulate equipment and resources while learning. this sentiment was a common thread in the assignment lesson plans, where 12 references were about arrangements for learning, such as students working in pairs, with partners, or in groups for some or all of the lessons. in a measurement lesson that involved students estimating and then measuring lengths in centimetres and metres, one pst wrote “students will work together in a group of ten with the teacher, and in pairs, switching between each activities,” and these tātaiako notes matched her lesson plan. the third connection to wānanga related to aspects of “problem solving,” which was noted both in the workbooks and in the assignment lesson plans. there were 13 references to problem solving that included statements related to the problem-solving nature of the mathematics tasks for each lesson, students solving problems themselves and in their pairs or groups, and creating problems for others to solve. the use of problem-solving strategies was also mentioned, for example, in a geometry lesson that required students to follow instructions to create a path. one pst wrote, “students will be using their innovation and problem-solving strategies within the lesson,” another wrote, “students will use their problem-solving strategies to investigate why temperatures vary throughout the country.” others were more specific and described how students would be encouraged to use “trial and error strategies” when designing a path for a specified length. less formal problem-solving strategies were also recorded, for example, “students will have to figure out ….” in the practicum lesson plans, a typical example of a reference to wilson et al. cultural compentenices journal of urban mathematics education vol. 10, no. 1 107 problem solving was from a series of three fractions lessons, “students will need to use their problem-solving strategies within the lesson.” ako: practice in the classroom and beyond – reciprocal teaching and learning there were six entries in the workbooks that connected ako and mathematical learning with the concept of reciprocal discussed during the lecture and workshop sessions. several psts wrote this connection as “teaching = learning.” in the assignment lesson plans, 10 students made links to the competency of ako. these were clustered into main ideas; the first continues the practice of students working together, and with their teachers, to learn mathematics. one pst described ako as “the students and teacher developing their knowledge of area and perimeter together,” another as “the students will be learning and teaching through discussion about the three dimensional objects they have made,” and another as “sharing experiences.” although these descriptions appear to be closely related to arrangements for learning in the previous category, our analysis has identified this shared student activity as more closely linked with social processes of learning and as relational participation between students. the psts’ role as a teacher was next identified in the ako category. the psts wrote that they needed to value and encourage learners, to provide them with the guidance, support, and resources they needed to participate during the lessons. it was important that teachers acknowledge that students had prior mathematical knowledge, and “students will appreciate that i am interested in finding out what they already know, so that we can build on that.” another pst wrote on her geometry plan: “respecting each other’s ideas and working with student’s strengths to achieve and acquire new knowledge.” these comments were more related to students as learners, what they bring to each new task, and how teachers respond to student’s knowledge. the final meaning of ako related to using examples from students’ worlds as contexts for mathematical learning, for example, “asking children about their own experiences with looking in mirrors and seeing reflections,” and using “everyday life in the classroom and beyond e.g., can relate temperature to other objects as well as climate.” manaakitanga: values – integrity, trust, sincerity, and equity. manaakitanga was the category that had more specific māori examples related to language and cultural contexts. the data from workbooks recorded examples of mathematical terms in english and te reo māori, and this was more pronounced in the workshops where measurement and geomentry were the focus mathematics topic, for example, “students will be introduced to māori names for two dimensional shapes.” this practice was continued by one student in the assignment plans: “students will be encouraged to use the māori words when wilson et al. cultural compentenices journal of urban mathematics education vol. 10, no. 1 108 counting. the square mats will have both english and māori representations.” another pst wrote, “manaakitanga means showing integrity, sincerity and respect towards māori beliefs, language and culture—māori place names have been incorporated into the work.” she had included māori place names alongside english place names in the task and had also adopted the terms used for manaakitanga. lastly, psts noted that a teacher’s role was to provide support for learners, as well as a respectful and safe learning environment. an example of an entry in an assignment lesson plan is “treating all students equally, and respecting their input and ideas. i will create a trusting environment in which the students learn.” whanaungatanga: relationships (students, schools, and communities) tangata whenuatanga: place-based, sociocultural awareness, and knowledge with whanaungatanga being explicitly linked with relationships, psts focused on the relational ways students could work together when learning mathematics. during a geometry workshop, one pst wrote in her workbook “students will be able to work with their friends to make symmetrical patterns.” similarly, in the assignment lesson plans typical entries were “students will build relationships with their peers as they work together as a group and in pairs,” and another, “students will be able to work in pairs to solve each question and discuss their buildings” (related to building and drawing three dimensional models with cubes). the final comment written by another pst who planned for a similar geometry task: “students will be able to work with their classmates … and means the students can support each other to achieve the activity.” for tangata whenuatanga there was one workbook entry that related to using local contexts for carrying out statistical investigations and “using statistical activities for the community.” this idea continued in the assignment plans where one pst wrote about using “local information within this (measurement) activity” that required students to investigate different temperatures around new zealand. this meaning of this competency was about situating mathematical tasks within local contexts. collectively, the analysis of the workbook entries, the assignment lesson plans, and the professional practice lesson plans provided a window into what was important enough for these 13 psts to write as a record for their professional learning. they often used the names of the tātaiako competencies to label their entries, some provided brief explanations of how these could be embedded in mathematics teaching, while others provided more detail. wilson et al. cultural compentenices journal of urban mathematics education vol. 10, no. 1 109 discussion and conclusions in this small-scale study, we set out to investigate which aspects of tātaiako the psts noticed and the links they made between the cultural competencies and practices for teaching mathematics. the psts had recorded each competency from tātaiako in some way; some had explicit links to mathematics, some were less explicit but still aligned with current teaching practices for mathematics, and some linked to examples of “the māori world” that could be embedded in mathematics. in this final section, we discuss these findings, and then focus on implications for initial teacher education. the psts explicitly connected the competency of wānanga to mathematics practices as shown by the psts linking “communication” and “problem solving” as familiar terms. these terms, along with associated mathematics practices, had been often discussed in the course sessions as important aspects of effective mathematics teaching and learning for all students (anthony & walshaw, 2009). “problem solving” described mathematics practices such as “figuring out” or “working out problems,” and some psts referred to students using specific problem-solving strategies. we found that communication was a broad term that encompassed a range of actions related to student mathematical learning. verbs such as talking, discussing, debating, sharing, and explaining their thinking were commonly found in all three data sources. we claim that the use of these verbs illustrates that the psts prioritised a social dimension of learning, where talk was not only a communication tool but also an essential ingredient of learning. in addition, the psts noted specific strategies in their planning that would promote opportunities for social interaction and shared mathematical learning. in both wānanga and whanaungatanga, the psts made connections to how students are organised for mathematical activity, particularly the multiple references to students working “in pairs” or “in groups.” related to the importance of social interactions for learning mathematics, we found that all psts noted the importance of relationships between students and between teachers and students. whanaungatanga was the competency that was described as being about relationships with students and while not specifc to mathematics, this was connected to teaching practices such as “building relationships with pairs.” similarly, the competency of manaakitanga was about relationships, with an emphasis on the teacher–student relationships, highlighting the roles of a teacher in “scaffolding, supporting, respecting” and “treating all students equally.” these aspects of both cultural and mathematics practices were also emphasised for ako, where there were several entries about teaching and learning being connected. this connection is seen in comments such as “the students and teachers developing their knowledge of area and perimeter together” and the prevalent use of the equation “teaching = learning.” we suggest this equation was recorded on wilson et al. cultural compentenices journal of urban mathematics education vol. 10, no. 1 110 documentation because this abbreviation had been used often during coursework to represent the reciprocal nature of learning between student and teachers, and the term “reciprocal learning” was included in tātaiako. this connection also illustrates the psts awareness of mathematics teaching and learning being a collaborative endeavour between students and teachers and recognises the important role psts have in enacting practices that allow such collaboration to occur (averill et al., 2009). while we found less specific inclusion of aspects related to māori culture, the competency of manaakitanga was where the psts linked to te reo māori (māori language) as seen by the examples of māori words used for learning geometry and measurement (averill et al., 2010). these translations of mathematical terms into te reo māori had been modelled during course sessions, particularly during geometry and measurement sessions. another link to the māori world was the inclusion of specific māori contexts, such as traditional māori symmetrical patterns, which was relevant as a context for learning about geometrical transformations. in conclusion, the psts used the resource of tātaiako in different ways. some psts copied tātaiako words and phrases directly onto planning documentation, others used these as prompts to record in their own words. irrespective of the detail, both were important for helping the psts to adapt and adopt cultural practices for mathematics teaching. we suggest that frameworks such as tātaiako have an important role in clarifying for psts expectations for embedding culturally responsive practices when teaching, and in our future work we will look for similar sources of language prompts and exemplars. we also want to investigate further opportunties for psts to mediate between the cultural competencies and practices for effective mathematics teaching (kitchen, 2005). our findings also show that while the psts were able to work with tātaiako, the mathematics course experiences were essential opportunities for them to interpet the competencies and make links to mathematics. we found that far from being “culture free” (nasir, 2016), the mathematics course provided both implicit and explicit opportunities for the psts to connect cultural and mathematics practice. although our study was small, it highlighted that psts can be supported to begin to develop culturally responsive mathematics teaching (villegas & lucas, 2002). we have only begun our work in this area and plan to continue with other mathematics education courses. we plan to further investigate how to include opportunities in our mathematics education courses where psts can delve more deeply into what it means to support “māori learners to achieve as māori in mathematics” (p. 3). in closing, we acknowledge that aspects of the cultural competencies in tātaiako and the practices identified by the psts in our study are relevant for all learners, irrespective of their cultural background. averill and colleagues (2014), however, remind us that māori students and whānau (family) wilson et al. cultural compentenices journal of urban mathematics education vol. 10, no. 1 111 believe it is more: “māori students having connections with te reo me tikanga māori (language and ways of doing things), having pride in māori identity, feeling valued and comfortable to be themselves at school, and being able to walk comfortably in māori and pākehā (non-māori) worlds” (p. 33). we believe psts can support māori learners to achieve in mathematics and can be leaders in this area as they move beyond the ite setting (averill, 2012). references anthony, g., & walshaw, m. 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(2013a). graduating teacher standards. retrieved from https://educationcouncil.org.nz/sites/default/files/gts-poster.pdf new zealand teachers council. (2013b). registered teacher criteria. retrieved from http://www.nzei.org.nz/documents/help/faq-docs/rtc%20%20frt%20ptca%20draft%20april%202011-1.pdf van es, e., & sherin, m. (2002). learning to notice: scaffolding new teachers’ interpretations of classroom interaction. journal of technology and technology education, 10(4), 571–596. villegas, a. m., & lucas, t. (2002). educating culturally responsive teachers: a coherent approach. albany, ny: state university of new york press. white, d., murray e., & brunard-vega, v. (2012). discovering multicultural mathematics dispositions. journal of urban mathematics education, 5(1), 31–43. retrieved from http://edosprey.gsu.edu/ojs/index.php/jume/article/view/161/103 journal of urban mathematics education december 2012, vol. 5, no. 2, pp. 18–27 ©jume. http://education.gsu.edu/jume jacqueline leonard is the director of the science and mathematics teaching center at the university of wyoming and professor of mathematics education, 1000 e. university avenue, dept. 3992, laramie, wy 82071; email: jleona12@uwyo.edu. her research interests include access and opportunity in mathematics education and critical pedagogy, such as teaching for cultural relevance and social justice in mathematics classrooms. commentary er’body talkin’ ‘bout social justice ain’t goin’ there jacqueline leonard university of wyoming he title of this commentary 1 is inspired by the negro spiritual everybody talkin’ ‘bout heaven. the song is most often sung a cappella; the lyrics are as follows: everybody talkin’ ‘bout heaven ain’t goin’ there everybody talkin’ ‘bout heaven ain’t goin’ everybody talkin’ ‘bout heaven ain’t goin’ there oh my lord well i read about the streets of gold and i read about the throne not everybody callin’ “lord, lord” is gonna see that heavenly home everybody talkin’ ‘bout heaven ain’t goin’ there everybody talkin’ ‘bout heaven ain’t goin’ everybody talkin’ ‘bout heaven ain’t goin’ there oh my lord the spirituals were born out of an oppressive condition, which we know as chattel slavery. thus, the songs are often referred to as the sorrow songs. yet as w. e. b. dubois (1903/1995) reminds us, “through all the sorrow of the sorrow songs there breathes a hope—a faith in the ultimate justice of things” (p. 274). here, i replace the word heaven with the words social justice as i focus on the black experience in america and the experiences of black children in our nation’s schools. while linking social justice with religion is not new, the use of the term has become prevalent in education, in general, and mathematics education, 1 a revised talk delivered at the research wednesdays speakers series, college of education, georgia state university, on october 17, 2012. t leonard commentary journal of urban mathematics education vol. 5, no. 2 19 specifically. while there are several definitions of teaching for social justice (wager & stinson, 2012), i define teaching mathematics for social justice as the following: holding specific social-justice-related perspectives and actions that provide all students with opportunities to learn rigorous mathematics in culturally specific and meaningful ways that seek to improve the economic and social conditions of marginalized individuals and groups, and that work toward the reduction (if not the complete elimination) of deficit-oriented beliefs and dispositions. (leonard & evans, 2012, p. 100) ideally, “the bottom line is…enhancing students’ learning and their life chances by challenging inequities of school and society” in order to “redistribute educational opportunity” (enterline, cochran-smith, ludlow & mitescu, 2008, p. 270). the operationalization of this definition implies a k–16 commitment that results in a long-term investment which has the potential to redistribute economic wealth for poor students and students of color. thus, teaching for social justice is teaching for empowerment and liberation. if not, then it’s merely talk. in other words, er’body talkin’ ‘bout social justice ain’t goin’ there. through the spirituals, blacks indicted the hypocrisy of the day. likewise, i intend to highlight the educational dilemma for black children in this decade. what educational policies intend to do and what they actually do are in conflict. teacher education programs that claim to have a social justice mission and to focus on teacher dispositions need to do more than list their mission and goals on the program website. to further illustrate this principle, i share an experience that i had in the western united states about a week ago. i was invited to attend a luncheon and a ribbon-cutting ceremony. my mother was visiting me at the time, and we were the only african americans attending the luncheon. as we mingled, something happened that reinforced for me what it is like to be black in america. before the luncheon, i was introduced as the director of the science and mathematics teaching center at the university of wyoming to a retired u.s. senator. in response to the introduction, the beloved ex-senator responded: “you people have come a long way since you were 3/5ths of a person.” needless to say, i was speechless. intentional or unintentional, these words invoked prejudice, bigotry, and racism. how can a former u.s. senator who took an oath to uphold the constitution make such a statement? “you people” implies that blacks are alien and outside of what is considered normal. despite changing demographics in the united states, whiteness remains the norm. “you... have come a long way” gives some credit to effort. blacks have come a long way, and there is a growing black middle class. however, this part of the statement seemed to have an element of sur leonard commentary journal of urban mathematics education vol. 5, no. 2 20 prise in terms of expectations. in other words, i did not expect you to get this far. finally, the crux of the statement, “since you were 3/5ths of a person,” employed mathematical terms to remind me of slavery and to place me in a category that was less than human. in this case, mathematics was used to disempower me, rob me of my satisfaction, and minimize my accomplishments—that is, if i let it. that experience did not have to overshadow the occasion or detract from the good will of others who greeted me warmly. the reoccurrence of daily situations such as this one in the lives of black folks was the purpose of the spirituals. they were written to uplift the souls of black folk who understood that what someone called you did not define you. benjamin banneker, renowned mathematician and scientist, illustrated this point in his letter to secretary of state thomas jefferson: one universal father hath given being to us all, and that he hath not only made us all of one flesh, but that he hath also, without partiality, afforded us all the same sensations, and endued us all with the same faculties; and that, however variable we may be in society and religion, however diversified in situation and color, we are all of the same family, and stand in the same relation to him. 2 therefore, i choose to use my experience to “flip the script” and use the encounter with the senator as a springboard to talk about social justice from a mathematical perspective that empowers rather than disempowers and liberates rather than demoralizes black children and other children of color. to do this, i use the 3/5ths rule as a metaphor to discuss black students’ mathematics experiences in american schools. i conclude with a second experience that reveals the power of one and how the unit of one in both mathematical and social justice terms can empower and liberate black americans to reach their full potential. the 3/5ths rule to understand the black experience in america, it is important to understand the context of the ex-senator’s statement about the 3/5ths rule, and how this law has impacted educational policy in relation to black children. the articles and provisions in the constitution as it relates to the 3/5ths rule states: article i, section 2 [slaves count as 3/5 persons] representatives and direct taxes shall be apportioned among the several states which may be included within this union, according to their respective numbers, which shall be determined by adding to the whole number of free persons, including 2 as quoted in benjamin banneker’s pennsylvania, delaware, maryland and virginia almanack and ephemeris for the year of our lord 1792. baltimore: william goddard and james angel, 1792. leonard commentary journal of urban mathematics education vol. 5, no. 2 21 those bound to service for a term of years, and excluding indians not taxed, three fifths of all other persons [i.e., slaves]. article i, section 9 – clause 1 [no power to ban slavery until 1808 and tax levied on the import of slaves] the migration or importation of such persons as any of the states now existing shall think proper to admit, shall not be prohibited by the congress prior to the year one thousand eight hundred and eight, but a tax or duty may be imposed on such importation, not exceeding ten dollars for each person. article iv, section 2 [free states cannot protect slaves] no person held to service or labour in one state, under the laws thereof, escaping into another, shall, in consequence of any law or regulation therein, be discharged from such service or labour, but shall be delivered up on claim of the party to whom such service or labour may be due. article v [no constitutional amendment to ban slavery until 1808] …no amendment which may be made prior to the year one thousand eight hundred and eight shall in any manner affect the first and fourth clauses in the ninth section of the first article. while some believe the constitution is the greatest document ever written, these articles reveal that the founding fathers actually “wrote both the institution and the benefits of slavery into [the] constitution” (harding, 1981, p. 46). but before the revolutionary war (1775–1783), political leaders met in philadelphia in 1774 for the first continental congress. there, they proclaimed: we will neither import nor purchase any slave imported after the first day in december next, after which time we will wholly discontinue the slave trade and will neither be concerned in it ourselves nor will we hire our vessels nor sell our commodities or manufactures to those who are concerned in it. (harding, 1981, p. 45) nonetheless, what they proclaimed and what they did were very different. as shown in the articles, the founding fathers crafted laws in 1787 that prohibited the ban of slavery until 1808 and levied taxes on the import of slaves. such laws not only instituted slavery into american life but also provided revenue for the federal government. ten dollars per slave is a great deal of income when millions of slaves are being auctioned on the block. historians estimate that 10 to 12 million slaves were sold by europeans during the antebellum period (johnson, smith, & wgbh team, 1998). furthermore, counting blacks as 3/5ths of a person for the benefit of having more southern votes exploited oppressed people to skew the results to ensure that slavery continued. in other words, a candidate running for president of the united states would receive more votes from the south. to add insult to injury, if a slave were to escape to a free state, federal laws were in place (see article iv, section 2) that guaranteed the slave would be returned thus pro leonard commentary journal of urban mathematics education vol. 5, no. 2 22 tecting the right to own slaves in perpetuity. this guarantee was a grave contradiction in terms of social justice given that liberty and justice was not for all. er’body taking ‘bout social justice ain’t going there. in such a system of oppression and institutionalized racism, public education was born. the plessey vs. ferguson case in 1896 established an educational policy of separate and (un)equal that would remain constitutional for the next 58 years. exactly another 58 years has passed after the brown vs. board of education ruling in 1954, but schools are still segregated by race and class thus maintaining a two-tiered system of education (leonard, 2008, 2009). sometimes this twotiered system is manifested as a school-within-a-school—one for blacks and one for whites. this two-tiered system is often operated under the guise of a magnet school. blacks and other students of color are in the “regular” school while most of the white students are in the magnet school. on paper, this type of school looks diverse, but often white students in the magnet school never interact with black students in the regular school. in some cases, they do not even have lunch or physical education at the same time. tracking operates in the same manner, except in this system, whites, in many cases regardless of ability, are tracked into honors and advanced courses while poor students and students of color are tracked in remedial and special education courses (blanchett, 2006). these kinds of schools are not models of social justice. rather they fail to level the playing field and operate to limit the educational opportunity of black and other underrepresented minority children. beyond 3/5ths education while the quality of life for blacks in america has improved, blacks continue to experience higher school dropout rates and higher unemployment rates compared to white americans (lang, 2011). while there are many factors that lead to these results, none is more evident than low teacher expectations, deficitoriented pedagogy, and a two-tiered system of education. to illustrate the fact that low expectations continue despite social justice mission statements in colleges and schools of education, consider the following statement made by a white female who was enrolled in a mathematics methods course that i taught a few years ago: the reason children in urban schools need to learn basic skills and children in suburban schools need to learn problem solving is because inner-city children will grow up to work in the fast food industry or in factories and children living in the suburbs will grow up to be managers and business leaders. needless to say, i could have heard a pin drop in the classroom after this statement was made. however, this white student, who was embarking upon a career leonard commentary journal of urban mathematics education vol. 5, no. 2 23 in the teaching profession, verbalized what is reality in terms of black education and black employment. black children, for the most part, are still receiving a sharecropper, skill-based mathematics education (moses & cobb, 2001). results of standardized tests such as the national assessment of educational progress (naep, 2011) continue to show performance disparities by race. while everyone continues to be lifted, comparisons by race reveal blacks are underperforming. however, when data are examined differently by type of school (lubienski & lubienski, 2006), one finds public schools significantly out-perform catholic schools. when charters and non-charters are compared, charter schools score significantly lower than non-charter schools. among private schools, lutheran schools have the highest scores, and conservative christian schools have the lowest scores (lubienski & lubienski, 2006). these data suggest there is more than one way to slice the data to determine how well our children are doing in mathematics. thus, economic variables and race may not be the most salient factors when comparing mathematics achievement. to create additional learning opportunities, i advocate for culturally relevant and social justice pedagogy. there should be less stress on computation where underrepresented minorities continue to score high on assessments like naep and more emphasis on number theory, data and statistics, measurement, algebra and geometry (leonard, 2008). for example, one of the more difficult questions that appeared on the eighth-grade version of the mathematics naep test in 2011 was as follows: which of the following is an equation of a line that passes through the point (0, 5) and has a negative slope? a. y = 5x b. y = 5x – 5 c. y = 5x + 5 d. y = -5x – 5 e. y = -5x + 5 to solve this problem in a culturally relevant way, the teacher could use geographic information systems (gis) and overlay a coordinate grid on a map of the students’ neighborhood to help them understand the concept of slope. for example, take my old neighborhood in the st. louis area. the location of my former home is labeled a, which can serve as the origin (see figure 1); tables can be created to obtain the coordinates and then plot the points (see table 1). leonard commentary journal of urban mathematics education vol. 5, no. 2 24 table 1 coordinate values c. y = 5x + 5 x y 0 5 1 10 2 15 3 20 e. y = -5x + 5 x y 0 5 1 0 2 -5 3 -10 in two instances, the first condition of the line passing through (0, 5) is met (c & e); and in two instances, the second condition for a negative slope (y = mx + b), where m is the slope (-5), is met (d & e). e is the only equation that meets both conditions so by process of elimination and direct proof the answer must be e. a negative slope goes (tilts to the left) toward goodfellow, and a positive slope (tilts to the right) toward hodiamont. contextualizing the problem with the students’ neighborhood is culturally relevant and will anchor the instruction so students might not so easily forget the mathematics. figure 1 google map of st. louis, mo 63112. (0, 5) e c leonard commentary journal of urban mathematics education vol. 5, no. 2 25 this example suggests the context in which students live imbue culture and can provide examples for teaching mathematics. moreover, cultural relevance has been linked to stem (science, technology, mathematics, and engineering) fields through robotics clubs and computer programming classes (bracey, in press). providing opportunities for underrepresented minority students to link culture to stem fields has led to greater retention of mathematics concepts (bracey, in press; leonard & hill, 2008). in bracey’s study (in press), the use of mentors and role models from diverse backgrounds were important in terms of motivation and retention in mathematics. while teachers of any background and gender can develop dispositions that relate to social justice (villegas, 2007), it is important that black students experience the tutelage of black teachers, especially black males, who can serve as role models. according to barr, sadovnik, and visconti (2006), black children had higher performance in mathematics when they had black teachers. gloria ladson-billings (2005), in her book beyond the big house reported that black teachers held 40% of black professional jobs from 1890 to 1910. however, according to the schools and staffing survey (national center for education statistics [nces], 2004), black teachers were only 7.9% of the public teacher workforce in 2003–2004. similarly, fewer blacks are employed as professors and teacher educators. in 2009, 6% of all faculty in higher education in the united states were black (nces, 2011). how do we restore teaching as an honorable professional among black college students? how do we ensure that blacks are significantly represented among the 100,000 mathematics and science teachers that president obama is calling for? we have to go out and get them. i am pleased to learn that institutions like georgia state university (gsu) surpass the national norm in terms of diverse student enrollment. latest enrollment figures show the student body is 33% black, 15% asian, and 8% hispanic. in addition to undergraduate education, graduate programs have made tremendous strides at gsu, particularly in mathematics education. currently, there are 46 phd students in mathematics education, and 32 (nearly 70%) of these students are black/african american. i know of no other graduate program in mathematics education that has such a record. institutions like gsu are making a difference in the state of georgia and the nation. however, once we recruit diverse students, we must be serious about our social justice stance in order to retain them. furthermore, additional effort is needed to support blacks and other underrepresented students if they are to fill the ranks of the professoriate. such aspirations begin in our nation’s classrooms with our youngest students, including my grandson, christopher, who began kindergarten at a magnet school in september. will he experience the regular school or the magnet school? his leonard commentary journal of urban mathematics education vol. 5, no. 2 26 cousin, terrance, who also began kindergarten this fall, attends a private religious school. will he be tracked in the redbird, bluebird, or the blackbird reading group? these young african american boys represent the class of 2025. as my progeny, will they grow up in an america that looks backward and perpetuates a 3/5ths mentality or one that looks forward to 100% participation in terms of equal educational access? closing remarks in closing, the same weekend that i was rendered speechless by a former u.s. senator who referenced the 3/5ths rule in terms of my success, i also had the awesome privilege of meeting and shaking the hand of congressman john lewis (u.s. representative for georgia’s 5th congressional district). congressman lewis was visiting the city of denver to encourage people to vote. he reminded the black church of the power of one. every person is considered a whole person and is entitled to one vote. however, rules and laws that have the potential to suppress the right to vote have been instituted in 2012. with less than 20 days to go before the november 6th election, the intent of voter-approved ids is to limit the number of votes cast by the poor and people of color. while wisconsin and texas have taken such laws off the books and judges have blocked the law in pennsylvania and south carolina, voter id laws remain in effect in kansas, indiana, new hampshire, tennessee, and georgia (bronner, 2012). present day voter-approved id laws perpetuate a 3/5ths mentality. yet congressman lewis came to challenge that mentality insisting that “regardless of socioeconomic status, race, or gender, everyone has one vote.” he reduced it down to the least common denominator of one. if everyone has the right to cast one vote without discrimination, then we can all participate equally in the democratic process. repressing the vote and demanding identification to vote, are attempts to undo that equality. nevertheless, i was so encouraged by this giant of civil rights. i was reminded of how he suffered on bloody sunday. how many blacks died trying to get the right to vote? i remembered freedom rides and how the bus he rode was bombed. congressman lewis is only one person, but he left a lifelong impression upon me. while the struggle is not over, i refuse to give up hope that as a nation we will truly experience the power of one: one vote, one hundred percent access to high-quality schools, one nation under god with liberty and justice for all. social justice is a verb and not a noun. er’body talkin’ ‘bout social justice ain’t going there. what are you prepared to do? leonard commentary journal of urban mathematics education vol. 5, no. 2 27 references barr, j. m., sadovnik, a. r., & visconti, l. (2006). charter schools and urban education improvement: a comparison of newark’s district and charter schools. the urban review, 38, 291–311. blanchett, w. (2006). disproportionate representation of african american students in special education: acknowledging the role of white privilege and racism. educational researcher, 35, 24– 28. bracey, j. (in press). black student engagement and cognition in math. in j. leonard & d. b. martin (eds.) the brilliance of black children in mathematics: beyond the numbers and toward new discourse. charlotte, nc: information age. bronner, e. (2012, october 3). voter id rules fail court tests across county: pennsylvania is latest. the new york times, 162(55,913), p. a1, a17. dubois, w. e. b. (1995). the souls of black folk (intro. r. kenan). new york, ny: penguin books. (original work published 1903) enterline, s., cochran-smith, m., ludlow, l. h., & mitescu, e. (2008). learning to teach for social justice: measuring change in the beliefs of teacher candidates. the new educator, 4, 267–290. harding, v. (1981). there is a river: the black struggle for freedom in america. new york, ny: harcourt brace. johnson, c., smith, p., & wgbh series research team. (1998). africans in america: america’s journey through slavery. new york, ny: harcourt brace. ladson-billings, g. (2005). beyond the big house: african american educators and teacher education. new york, ny: teachers college press. lang, c. (2011, august 28). race, class, and obama. the chronicle review. retrieved from http://chronicle.com/article/race-classobama/128787/. leonard, j. (2008). culturally specific pedagogy in the mathematics classroom: strategies for teachers and students. new york, ny: routledge. leonard, j. (2009). “still not saved”: the power of mathematics to liberate the oppressed. in d. b. martin (ed.), mathematics teaching, learning, and liberation in the lives of black children, (pp. 304– 330). new york, ny: routledge. leonard, j., & evans, b. (2012). challenging beliefs and dispositions: learning to teach mathematics for social justice. in a. wager & d. stinson (eds.), teaching mathematics for social justice: conversations with mathematics educators (pp. 99–111). reston, va: national council of teachers of mathematics. leonard, j., & hill, m. l. (2008). using multimedia to engage african-american children in classroom discourse. journal of black studies, 39(1), 22–42. lubienski, c., & lubienski, s. t. (2006). charter, private, public schools and academic achievement: new evidence from naep data. new york, ny: national center for the study of privatization of education. moses, r. p., & cobb, c. e., jr. (2001). radical equations: math literacy and civil rights. boston, ma: beacon press. national assessment of educational progress. (2011). the nation’s report card. retrieved from http://nationsreportcard.gov/math_2011/gr8_national.asp. national center for education statistics. (2004). schools and staffing survey, public school teacher data file, table 18. retrieved from http://nces.ed.gov/surveys/sass/tables/state_2004_18.asp. national center for education statistics. (2011). digest of education statistics, 2010 (nces 2011015), table 256. retrieved from http://nces.ed.gov/fastfacts/display.asp?id=61. villegas, a. m. (2007). dispositions in teacher education: a look at social justice, journal of teacher education, 58, 370–380 wager, a. a., & stinson, d. w. (2012). (eds.). teaching mathematics for social justice: conversations with educators. reston, va: national council of teachers of mathematics. http://chronicle.com/article/race-classobama/128787/ http://nationsreportcard.gov/math_2011/gr8_national.asp http://nces.ed.gov/surveys/sass/tables/state_2004_18.asp http://nces.ed.gov/programs/digest/d10/tables/dt10_256.asp http://nces.ed.gov/fastfacts/display.asp?id=61 teaching collegiate-level mathematics courses for social justice: considering the mathematical journey of secondary mathematics preservice teachers journal of urban mathematics education july 2013, vol. 6, no. 1, pp. 71–80 ©jume. http://education.gsu.edu/jume laura mcleman is an assistant professor in the department of mathematics at the university of michigan flint, 303 e. kearsley street, flint, mi, 48502; email: lauramcl@umflint.edu. her research interests include preparing mathematics preservice teachers to work with underserved and underrepresented populations, and the preparation of mathematics teachers and mathematics teacher educators to integrate issues of equity and social justice in their instruction. joyce piert is a lecturer in the department of mathematics at the university of michigan flint, 303 e. kearsley street, flint, mi 48502; email: piertjoy@umflint.edu. her research interests are urban education, mathematics and identity, and social justice. considering the social justice mathematical journey of secondary mathematics preservice teachers laura mcleman university of michigan-flint joyce piert university of michigan-flint in this essay, the authors share some of their journey as they seek to make sense of what it might mean to prepare secondary mathematics preservice teachers to teach mathematics for social justice. the focus on how to prepare mathematics teachers to critically consider the world around them and to further develop the dispositions to become agents of change has been discussed in the research literature. what it might “look like” to enact this type of programmatic-level teaching at a college or university, however, has rarely been examined. through the sharing of their thoughts and reflections, the authors hope others might draw inspiration to reconsider the teaching of mathematics courses for social justice at the program level. keywords: mathematics, mathematics teaching, preservice teachers, social justice even of us sat around the room in a circle formation. we had gathered for the first meeting of our teaching circle that was funded by the university’s center for teaching and learning as a venue for professional development. all the members of the teaching circle were instructors within the mathematics department in some capacity: four mathematicians, two mathematics educators, and one retired high school teacher. during this meeting, we brainstormed which qualities we wanted our secondary mathematics preservice teachers (psts) to possess upon graduation from our program. “we want teachers who can teach to a diverse population of students,” was one professor’s response. as participants in that teaching circle, we held the philosophy (and still do) that effective teaching of mathematics to a diverse population of students must allow students to use mathematics to “examine one’s own lives and other’s lives in relationship to sociopolitical and cultural-historical contexts” (gutstein, 2006, p. 5). the focus on how to prepare mathematics teachers to critically consider the s mcleman & piert social justice mathematical journey stinson, d. w., & spencer, j. a. (eds.). (2013). privilege and oppression in the mathematics preparation of teacher educators [special issue]. journal of urban mathematics education, 6(1). 72 world around them and to further develop their dispositions to become agents of change is one that has been discussed in research (e.g., de freitas & zolkower, 2009), this is what we refer to as teaching mathematics for social justice. yet, until these teaching circle meetings, neither of us (laura and joyce) had really considered what it might “look like” to enact this type of programmatic-level teaching at the university level. as the conversations within the teaching circle continued, we wondered such questions as: what would it look like to teach collegiatelevel mathematics content courses through a lens of social justice? do we as mathematics instructors possess the necessary knowledge to teach mathematics in this manner? the purpose of this essay is to share some of our journey as we seek to make sense of what it might mean to prepare secondary mathematics psts to teach mathematics for social justice within our institution. our hope is that through the sharing of our honest thoughts and reflections, other individuals can begin to clarify their own thinking or draw inspiration to reconsider the teaching of mathematics courses within their own teacher preparation programs. our journeys it is important to situate our backgrounds within our work. as foote and bartell (2011) argue, our life experiences shape what we attend to in our work, including the questions we ask and the interpretations we draw. laura: growing up, i didn’t know that i led a privileged life, as i realize that i wore blinders towards the consideration of inequities that existed within society. i lived in a middle class, white community where racial, ethnic, and socioeconomic diversity were practically non-existent. while both my mom and dad worked to support us, my family never went without. i grew up in a nice home, and we always had food on the table. from my viewpoint, people looked like me and shared the same values and norms as my family. joyce: i grew up during the turbulent 60s, a period when the issues of social injustice and inequalities were constantly in the news and on the forefront of many people’s mind. these events shaped my young mind while living with my family in a working class urban community. early in school, i discovered that i had a love for mathematics, and i often found myself in classes where using arithmetic and algorithms were expected. i was one of the first girls to be permitted in a drafting class, and i honed my skills with fractions in home economics. as an african american, i was intimately aware of the disparities of high unemployment, low quality schools, and high crime that existed within my community. this con mcleman & piert social justice mathematical journey stinson, d. w., & spencer, j. a. (eds.). (2013). privilege and oppression in the mathematics preparation of teacher educators [special issue]. journal of urban mathematics education, 6(1). 73 sistent awareness facilitated my evolution as a teacher and my drive to promote equality within society for all people. laura and joyce: we both excelled as mathematics students, at the k–12 and collegiate levels alike, having learned how to navigate the school environment. through past experiences, we learned how to create a teacher-centered environment within a “banking concept of education” (freire, 1970/2000, p. 72): lecture, require students to take notes, maintain a limited amount of engagement, and assess learning. these experiences, however, were challenged for each of us as we took high school teaching positions in low-income urban areas with students whose lived realities were very different from our own. we saw that our teachercentered lectures, even coupled with some cooperative learning activities, did not help our students succeed in a traditional sense. we realized that the problem lay not with the students, but with how we were approaching mathematics teaching, yet were unsure of how to change. with a need to know, we each moved on from teaching secondary school mathematics to pursuing doctoral degrees in education. laura: i was fortunate to be a fellow in the center for the mathematics education of latinos/as (cemela 1 ) throughout my doctoral program. it was through cemela that i was exposed to theories and frameworks surrounding issues of social justice, specifically from a freirian paradigm of critical pedagogy (cf. freire, 1970/2000). initially, i was resistant, as i felt overwhelmed considering this new paradigm, as well as defensive about the subject at which i had so excelled in school. over time, however, i began to connect what i was learning with the experiences i had had as a high school mathematics teacher and the quest i had to help all students be successful in mathematics. after completing my doctoral program and taking a tenure-track position at a university, i was invited to participate in the privilege and oppression in the mathematics preparation of teacher educators (prompte 2 ) conference. throughout the conference, i explored and grappled with issues of privilege and oppression within my own life. further, i reflected upon these constructs within my own teaching and how i work with future secondary mathematics teachers. 1 cemela is a center for learning and teaching supported by the national science foundation, grant number esi-0424983. any opinions, findings, and conclusions or recommendations expressed herein are those of the authors and do not necessarily reflect the views of the national science foundation. 2 privilege and oppression in the mathematics preparation of teacher educators (prompte) conference (funded by create for stem institute through the lappan-phillips-fitzgerald cmp 2 innovation grant program), michigan state university, battle creek, mi, october 2012. any opinions, findings, and conclusions or recommendations expressed herein are those of the authors and do not necessarily reflect the views of the funding agency. mcleman & piert social justice mathematical journey stinson, d. w., & spencer, j. a. (eds.). (2013). privilege and oppression in the mathematics preparation of teacher educators [special issue]. journal of urban mathematics education, 6(1). 74 joyce: in my graduate experience, i continued to cultivate and nurture the “mission” for social justice that was already growing inside of me. i was a child and adolescent during the civil rights era. it was during these years that i was exposed to the philosophies of such activists as septima clark, ella baker, and kwame toure (stokely carmichael). the voices of these activists influenced my viewpoint and convinced me that it is within the power of the grassroots (common people) to make the world a more humane place for all people. while in graduate school, i was able to merge the philosophies of the activists of my youth with the philosophies of such critical pedagogues and scholars as paulo freire, henry giroux, peter mclaren, and bell hooks. it is this marriage of philosophies that shape my current thoughts and pedagogy around mathematics education. moving forward each of our journeys has led us to our current location: flint, michigan. flint is a community that has attracted the nation’s attention for various reasons over the last few decades. the city is struggling with emergency financial takeover by the state, high unemployment rates, high crime rates, and closing schools. it is a city that is ripe for community transformation, and this transformation could be facilitated through the vehicle of education. it is in this place where we engage university students as faculty members of a mathematics department (laura, a tenure-track assistant professor; joyce, a lecturer). here, we wish to prepare secondary mathematics psts to integrate issues of equity and social justice throughout their instruction. through our ongoing efforts in our one-semester mathematics methods course, we are making strides to challenge perceptions of what it means to teach and learn mathematics. we realize that our efforts will most likely be the first opportunity for most, if not all, of our psts to question their own views regarding mathematics and its potential as a vehicle to critically challenge the world. ideally, we would like our course to be a space in which psts see themselves as agents of change within the educational system and as such develop and acquire strategies that they can put into action in their own mathematics instruction. in order to achieve this ideal, though, we posit that psts need to experience social justice as a cornerstone of mathematics teaching throughout their mathematical career. with this in mind, logistical questions arise. some of these questions include:  should all college-level mathematics courses be taught through a lens of social justice? if not, which ones should be?  what would a curriculum that uses social justice as its basis look like in a college-level mathematics course? what would a program of study that mcleman & piert social justice mathematical journey stinson, d. w., & spencer, j. a. (eds.). (2013). privilege and oppression in the mathematics preparation of teacher educators [special issue]. journal of urban mathematics education, 6(1). 75 uses social justice as a basis look like (i.e., sequencing, textbooks, course goals, assessments, etc.)?  what knowledge would be necessary for mathematics teacher educators in order to instruct in this manner? do we possess this knowledge, and if not, how do we obtain it?  how do we prepare both students and colleagues for this paradigm shift in teaching and learning? how do we address resistance from colleagues and/or teachers?  do we have the courage to teach these courses outside the scripted norm? are we secure enough in our own constitution (and positions) to undertake this journey? a dialogue in this section, we recount some portions of our ongoing conversation 3 in an effort to grapple with one of these big questions, namely, should all college-level mathematics courses be taught through a lens of social justice? if not, which ones should be? through our dialogue, it is possible to see how our thinking diverged, but ultimately converged when we began to focus on a specific entry point for our psts to begin a mathematical journey focused on social justice. laura: should all mathematics classes that our psts take be taught through a lens of social justice? that’s a difficult question. for one, i’m not sure i have the necessary knowledge to teach, for example, an abstract algebra course in this manner, as my schooling experience had me focus on acquiring a classical knowledge of mathematics (gutstein, 2006). specifically, i would have a hard time relating the content to issues of social justice. for example, how would i teach concepts such as unique factorization domains through a lens of social justice? and more importantly, should we? we need to consider what the goals are of such a course: why are psts taking this course? what purpose does it serve for their preparation? then we need to see how those goals align with the goals of a course that has social justice as its focus. perhaps it doesn’t make sense for our psts to have all of their mathematics courses be taught through a lens of social justice, especially if the goals of the class are mutually exclusive from the goals of a class with social justice as a focus? joyce: if students experience the foundations of mathematics (such as functions) through a social justice lens throughout their mathematics career, then they would 3 the dialogue presented does not represent a transcribed conversation as grammatical edits have been made since the conversation took place. mcleman & piert social justice mathematical journey stinson, d. w., & spencer, j. a. (eds.). (2013). privilege and oppression in the mathematics preparation of teacher educators [special issue]. journal of urban mathematics education, 6(1). 76 infuse these “higher-level” mathematics courses with the social justice perspectives themselves. the proverb “if you give a man a fish, he will eat for a day, but if you teach him how to fish, he will eat for a lifetime” comes to mind. if i can expose students to social justice perspectives of mathematics early in their mathematical careers, then i believe that they will have the tools to shape mathematics for themselves into instruments of social justice whenever they encounter mathematics. laura: it seems that you might be saying that the students will integrate issues of social justice regardless of how we teach college mathematics courses, as long as we teach earlier mathematics courses through a social justice lens. if so, i’m not sure i agree. some of the mathematics courses at the college level (e.g., abstract algebra, complex analysis) are very different than the ones they experience earlier on in their mathematical careers (e.g., algebra i, geometry) in their direct application to phenomena within the world. so, while psts can know how to view certain ideas in mathematics through the lens of social justice (such as factoring real numbers), they might not be able to transfer it to more abstract mathematical concepts (such as factoring over fields other than the real numbers). but i do agree that early exposure is an important and essential component in students’ mathematical lives. with this exposure, psts may think to question how the higher-level abstract mathematical concepts in their later mathematics courses can be used to see, read, and write the world (gutstein, 2006), and thus, hopefully be able to guide their future secondary students to do the same. i still question what the goals are of some of the collegiate mathematics courses. in conversations with some mathematicians, they seem to be of the mindset that mathematical concepts do not always have to be directly applicable to context—that there is value in studying mathematics for the inherent beauty of the discipline. if these are the goals of some of the mathematics courses that psts are taking and if teaching mathematics through a lens of social justice means that social justice is an integral and authentic part of the mathematics course, perhaps we will do more harm than good if we try to “force” issues of social justice into a mathematics course when the issues are less authentic to the nature of the subject. let me be clear… i am not saying that there is no place for issues of social justice in higher-level mathematics courses. we just need to integrate the issues in such a way that is authentic to the content of the course. joyce: let me back up… i need to revisit this big question: should all mathematics courses be taught through a social justice lens? let me use art as an analogy. there are aspects of art that i learn because i am interested in learning to improve my creativity. but then there are aspects of art that are beautiful and i would like to learn simply for the appreciation of art. so, speaking of mathematics now, the mcleman & piert social justice mathematical journey stinson, d. w., & spencer, j. a. (eds.). (2013). privilege and oppression in the mathematics preparation of teacher educators [special issue]. journal of urban mathematics education, 6(1). 77 question for me becomes whether all mathematics is utilitarian? or are there parts of mathematics that are for pure beauty and appreciation? i think each question has its own answer. we know that there are mathematics courses that we teach for utility, in other words, to enhance students’ abilities to solve problems. we refer to those courses as applied mathematics, while the courses that we teach for appreciation and beauty, of course, we refer to as pure mathematics. with these definitions in mind, shaping a course around the issues of social justice in a collegiate-level mathematics curriculum definitely creates a conundrum. as we think about issues of social justice, we may find it easier to pose problems of inequity or societal challenges that could easily align with the applied aspects of collegiate mathematics. it’s only when we attempt to imagine a framework to hang issues of social justice on from the pure aspects of mathematics that we begin to feel like we are bumping our heads against the wall. for me, issues of social justice are issues of the heart. it can reasonably be assumed that it is not simply the act of “examining” their lives and the lives of others that we want our students to experience, but rather an attempt to facilitate a raising of the consciousness of their life experiences in relationship to others. so if it is essentially affecting the heart that we are looking to achieve, then it would seem that pure science and social justice have much in common given that both attempt to provide an experience of the heart—one in appreciation and the other in valuing others. laura: i would argue that mathematics is historically taught free from context, and that because of this lack of contextualization, mathematics tends to attract a type of student who has a particular disposition towards not considering mathematics as a venue for issues of social justice. i agree, then, that we are affecting the heart, where we want our students to develop a disposition towards the world to view mathematics in light of issues of social justice. i believe that integration of social justice issues into the curriculum is key, as it may seem, upon first glance, that only certain courses can be taught through a lens of social justice. with that said, one could argue that a more natural fit of teaching through a lens of social justice would be the earlier level courses (earlier in the psts’ mathematics journey i mean), that is, the calculus sequence and linear algebra. i say this because these courses seem to have content that directly aligns with scenarios which individuals can encounter in their lives (e.g., rates of change in calculus; engineering concepts in linear algebra). this content also directly aligns to what the psts may teach in secondary schools. joyce: well, i believe that we must wrestle with the question of where to begin our psts’ experience of mathematics from a social justice lens. as a lecturer, my experience has been teaching mathematics to students who are novices in their mathematical journey. in fact, these courses are similar to the level of mathematics that the majority of psts will teach in schools. i feel that it is at this level of mcleman & piert social justice mathematical journey stinson, d. w., & spencer, j. a. (eds.). (2013). privilege and oppression in the mathematics preparation of teacher educators [special issue]. journal of urban mathematics education, 6(1). 78 mathematics education (intermediate algebra, college algebra, pre-calculus, and, perhaps, calculus i) that students need to have the experience of learning mathematics from a social justice lens. for the psts who start their mathematical careers in these earlier classes, they would have as a point of reference those experiences in the mathematics courses that are taught from a social justice lens. these earlier experiences could have a positive impact on mathematics methods courses, in that less time would need to be spent on the introduction of this ideology and more time could be devoted to the methodology of creating lessons for social justice prior to a student teaching field experience. yet, thinking about how to instruct abstract algebra from a social justice lens leaves me perplexed. however, i do feel that drawing from the tenets of ethnomathematics (d’ambrosio, 1985) and showing how mathematics of this caliber was/has been discovered and used in non-western societies and that elements of these mathematical concepts had their genesis with these diverse groups would benefit psts greatly. laura: i agree that we need to teach the courses you mention through a lens of social justice, but we still need to consider what mathematics courses our psts would take. i’m not sure that a lot of our psts would start at the levels you mentioned … well, maybe calculus i. so perhaps we could just focus on what curriculum would look like for a calculus i course. and then maybe, as you said earlier, if we provide our psts social justice mathematics opportunities early in their program, they could begin to develop the disposition to question and challenge how social justice fits within their the mathematics classes later in their mathematical journey. joyce: reminiscing on my experience as a preservice teacher taking mathematics courses, i actually began my coursework with pre-calculus; that is why i suggested beginning with those specific earlier courses. however, i think that precalculus and/or calculus i could be a good place to start our work. another option might be to undertake a mission of sorts, to have the students of higher-level mathematics courses (as part of course requirements) create their own projects for social justice that utilizes mathematical concepts. at the very least, we could create an upper-level mathematics course that could be offered as an elective, in which the primary focus would consist of developing mathematical models for social justice. doing so would “kill two birds with one stone:” students could discover for themselves how to link social justice to the mathematical concepts taught, and we could collect a repertoire of instructional materials to incorporate in future courses or even refine for other courses. i am all in favor of developing a social justice curriculum for calculus i. another thought: what about creating an interdisciplinary course, partnering with colleagues in the sociology department for example, that would be eligible for mcleman & piert social justice mathematical journey stinson, d. w., & spencer, j. a. (eds.). (2013). privilege and oppression in the mathematics preparation of teacher educators [special issue]. journal of urban mathematics education, 6(1). 79 general education credit? the purpose of this course could be to look at critical social problems and current challenges (be it, political, social, economic…national and global) and to create mathematical models and develop sociological paths towards solutions to these problems from a grassroots perspective. reflection as professionals who believe that our job is to change the paradigm of teaching and learning mathematics, it is clear throughout our conversation that we are grappling with what courses should be included in our psts’ mathematical journey to help them become agents of change, and further where to begin this journey. we started the conversation from different perspectives, each taking into account the level of mathematics that we teach. upon reflection, though, this difference is not surprising given that the merging of a variety of viewpoints is, at its core, a communication one. however, we find numerous strengths in our communication. first, we were not afraid to start a dialogue around a difficult question, although we didn’t know where it would lead. second, all the thoughts and contributions put forth were not only valued but also respectfully challenged. finally, the conversation continued to evolve and develop until both of us were standing on (or somewhere close to) mutual ground. while our conversation regarding the instruction of all college-level mathematics courses is still ongoing, the process of sharing our thoughts helped us identify a starting point for how we could begin to integrate social justice into the mathematical journey of our psts. more importantly, our dialogue served as a springboard to move us forward from theory to reality. using others’ work as a starting point (e.g., staples, 2005), we are now working on the design of a calculus i course that focuses primarily on issues of social justice. this beginning is an important implication as this is new territory for us; one we have not travelled before and do not have a prescribed template from which to work. however, we now understand that we cannot have answers to all our questions before we jump in to this uncertain domain. although the fear of not possessing the “right” knowledge is very much a reality for us, in order to help our psts rethink mathematics—and in order for us to rethink mathematics ourselves—we realize that we must take a leap. we hope that others who are also considering challenging the purpose of collegiate mathematics teaching and learning will take the leap with us, and that we can learn from each other. references d’ambrosio, u. (1985). ethnomathematics and its place in the history and pedagogy of mathematics. for the learning of mathematics, 5, 44–48. mcleman & piert social justice mathematical journey stinson, d. w., & spencer, j. a. (eds.). (2013). privilege and oppression in the mathematics preparation of teacher educators [special issue]. journal of urban mathematics education, 6(1). 80 de freitas, e., & zolkower, b. (2009). using social semiotics to prepare mathematics teachers to teach for social justice. journal of mathematics teacher education, 12, 187–203. foote, m. q., & bartell, t. g. (2011). pathways to equity in mathematics education: how life experiences impact researcher positionality. educational studies in mathematics, 78, 45–68. freire, p. (2000). pedagogy of the oppressed (m. b. ramos, trans.; 30th anniv. ed.). new york, ny: continuum. (original work published 1970). gutstein, e. (2006). reading and writing the world with mathematics: toward a pedagogy for social justice. new york, ny: routledge. staples, m. (2005). integrals and equity. in e. gutstein & b. peterson (eds.), rethinking mathematics: teaching social justice by the numbers (pp. 103–106). milwaukee, wi: rethinking schools. template pme28 journal of urban mathematics education december 2012, vol. 5, no. 2, pp. 157–181 ©jume. http://education.gsu.edu/jume robin averill is a senior lecturer of mathematics education in the school of education policy and implementation at victoria university of wellington, new zealand; email: robin.averill@vuw.ac.nz. her research interests include culturally responsive education and leadership, teacher-student relationships, resource development, and initial teacher education. reflecting heritage cultures in mathematics learning: the views of teachers and students robin averill victoria university of wellington, new zealand te whare wānanga o te ūpoko o te ika a māui in this article, the author explores the views of six teachers and 136 indigenous (māori), pacific nations, and new zealand european students on reflecting the heritage cultures of māori and pacific peoples in mathematics learning. findings show that teachers responded to students’ cultures through their classroom interactions but not through contexts used in mathematical tasks—participants largely viewed māori and pacific nations cultures and mathematics learning as distinct. findings indicate that substantive incorporation of heritage cultures in mathematics instruction requires changes in teacher and student beliefs regarding the place of heritage cultures within mathematics learning and sector and school-based support for teachers to develop their cultural knowledge and understanding of how to utilize such knowledge in their teaching. keywords: culturally responsive teaching, diversity, indigenous populations, mathematics teaching and learning, student voice nternational research literature evidences increased attention on addressing the impact of discontinuities between minoritized cultural groups and dominant school cultures on student affect, engagement, and achievement (e.g., gay, 2010; kidman, yen, & abrams, 2012; nieto & bode, 2008; presmeg, 2007; sleeter, 2011; tyler et al., 2008). education is a complex system. schools—and teachers’ work within them—exist within broader societal contexts in which systemic power imbalances in relation to colonialism, decision-making, resource distribution, and school sector policy and structures (often intrinsically linked to the ideologies of dominant cultures) all affect how issues of diversity in education can be addressed in classrooms (brayboy, 2005; samu, 2011). within these contexts, culturally relevant or culturally responsive pedagogies, which acknowledge, draw from, and strongly link with students as culturally located individuals, are advocated as pathways toward enhancing student outcomes by reducing discontinuities between students’ homes and schools where they exist (gay, 2010; ladson-billings, 1994; wlodkowski & ginsberg, 1995). these pedagogies employ strong ties between learning contexts and the cultural backgrounds of students which, along with effective teacher-student relationships and student-centred practices, help create classi averill heritage cultures in learning journal of urban mathematics education vol. 5, no. 2 158 room programmes that are inclusive of the culturally located behaviours, ways of learning, expertise, knowledge, interests, and experiences that students bring to their learning (e.g., averill et al., 2009; fletcher, parkhill, fa’afoi, taleni, & o’regan, 2009; kanu, 2011; tyler et al., 2008). similarly, bishop (2008) emphasizes the importance of teachers providing opportunities for indigenous and minoritized students to learn within and through contexts consistent with and linked to their ancestral cultures, referred to in this article as their heritage cultures. students are part of many cultural groups, each with their own set of behaviours, practices, ways of interacting, and values (banks, 2004). for example, students may belong to church groups, sports clubs, school-based groups, social groups, and family groups. in this article, heritage culture is used to refer to the languages, practices, knowledge, ways of being, and behaviours linked to the heritage ethnicities of students as nurtured within them by their family and others in their extended heritage cultural group. also included within the term are current and historical issues, events, and other information (e.g., legends, celebrations, customs) clearly related to a heritage culture. differences between students’ and teachers’ cultural backgrounds and teachers’ limited knowledge of the students’ cultures present challenges for teachers to plan and teach in ways that capitalise on their students’ background experiences (villegas & lucas, 2002; zumwalt & craig, 2008). here, i describe a recent study designed to examine the potential for links to be made between students’ heritage cultures and mathematics instruction in new zealand. in the study, i investigated the perceptions of mathematics teachers and students of teachers’ knowledge of the students’ māori and pacific nations heritage cultures, mathematics learning linked to māori and pacific nations heritage cultures, and beliefs about the role of the students’ heritage cultures in their mathematics learning. due to the fundamental importance of effective teacher-student relationships for the learning of māori and pacific nations students (bishop, berryman, tiakiwai, & richardson, 2003; fletcher et al., 2009; hill & hawk, 2000), the perspectives of teachers and students of these relationships in their own mathematics classroom were included in the investigation. i begin by describing the national and international context and theoretical background, and then outline the study and present teachers’ and students’ views on teacher-student relationships and incorporation of cultural heritages within mathematics learning. finally, i discuss the study findings and their implications for educators, policy makers, and future research. the new zealand context there are approximately 760,000 new zealand school students of new zealand european, māori, and pacific nations heritages (with approximately 59%, 22%, and 9% in each group, respectively). the proportions of māori and pacific nations students are increasing relative to their new zealand european counterparts. the ethnicities of new zealand’s māori and pacific nations averill heritage cultures in learning journal of urban mathematics education vol. 5, no. 2 159 secondary school students and those of their teachers are often poorly matched, particularly in mathematics. new zealand māori, pacific nations, and european students’ cultural heritages vary greatly across and within ethnicities. these limited ethnic classifications do not convey the marked diversity that exists within and across groups in relation to the depth of individuals’ engagement with their heritage culture/s, the combinations of individuals’ heritage cultures, and the number of generations in their family who have lived in new zealand. mathematics achievement also varies within and across ethnic groups; new zealand european students tending to perform better on traditional achievement measures than their māori and pacific nations peers (crooks, smith, & flockton, 2010). most new zealand european students are in the ethnic majority in their classes and schools; however, those represented in this study are not. new zealand education policy encourages culturally responsive teaching to improve the academic success of māori and pacific nations students, reduce disparities in achievement, acknowledge and strengthen students’ cultural identities, and strengthen the understandings of all students of the cultural diversity of new zealand society (ministry of education, 2007a, 2011). according to these policies, it is essential for teachers to acknowledge, reflect, and value student identity and culture in their practice for māori students to achieve “as māori” (ministry of education, 2008, p. 4). teachers are encouraged to ensure māori learners can see themselves in their education, realize their “cultural distinctiveness and potential” (p. 18), and participate in and contribute to te ao māori (the māori world). similar policy goals exist regarding pacific nations students (ministry of education, 2006). however, teaching that reflects such goals has yet to reach widespread classroom implementation. for example, despite observing 100 secondary mathematics lessons, many of which exhibited examples of culturally responsive practices such as student-centred classroom discourse, i found few examples in which the teacher acknowledged the heritage cultures of their students, and none of the observed instructional activities reflected links with students’ māori and pacific nations heritage cultures (averill, 2012). traditional māori pedagogies include experiential learning; context-based and integrated learning; tuakana-teina (learning from more experienced peers); learning through stories, metaphor, and song; and strategies that develop shared ownership of and responsibility for learning (hemara, 2000). schools successful with māori students encourage parental and community involvement and incorporate “substantial elements of traditional and contemporary māori language, culture and knowledge into the curriculum” (education review office, 2002, p. 1). strong mathematics achievement gains by māori students have been linked to school-wide commitments to culturally responsive teaching such as by incorporating taonga (māori treasures), māori language and protocols, and tikanga māori (māori practices) such as making personal connections and sharing responsibility, into classroom practice (te maro, higgins, & averill, 2008). for pacific nations students, relevant teaching averill heritage cultures in learning journal of urban mathematics education vol. 5, no. 2 160 strategies include using humor to encourage engagement, using pacific sources as curriculum materials (ministry of education, 2007b), and telling stories to explain concepts and locate information within the lives and experiences of students. a recurrent theme in the literature regarding the learning of both māori and pacific nations students is the potential for growth in teacher recognition of the students’ experiences, background knowledge, needs, and skills of how to capitalize on these in their teaching (e.g., amituanai-toloa, mcnaughton, kuin lai, & airini, 2009; bishop & berryman, 2006; ministry of education, 2007b; tuuta, bradnam, hynds, higgins, & broughton, 2004). to develop their understanding of the diversity of māori and pacific nation peoples, new zealand teachers are encouraged to learn about linguistic, cultural, and culturally linked pedagogical differences across groups, and between these groups and teachers’ own cultures (fletcher et al., 2009; hill & hawk, 2000; macfarlane, 2004). however, many teachers in loorparg, tait, yates, and meyer’s (2006) new zealand study felt that adopting culturally responsive practices is challenging in part because it is left for individual teachers to manage, and they called for a range of supports such as increased access to suitable resources to help them respond to cultural diversity in their teaching. similarly, in a study which included interviewing 32 māori year 5 and 6 students in one urban māori language immersion school, kidman, yen, and abrams’ (2012) found that elementary science teachers also face challenges related to positioning indigenous culture and knowledge in the curriculum. they contend that the resulting peripheral inclusion of māori culture and knowledge in school science conveys tacit and hidden messages for students about the nature of science education and the place of indigenous culture and knowledge within it, contributing to indigenous student disengagement with science learning. the crucial importance of positive teacher-student relationships—students having a feeling of connectedness with the teacher and teachers incorporating the perspectives students have about their learning in their teaching decisions—for students’ engagement, motivation, and learning has been identified and discussed by many in national and international contexts (see, e.g., anthony & walshaw, 2007; eccles, 2004). strong academic relationships are considered essential for māori and pacific nations student learning in particular because they enable students to feel comfortable and safe in the classroom and to know that the teacher understands and is focussed on their learning and best interests (bishop et al., 2003; fletcher et al., 2009; hill & hawk, 2000; ministry of education, 2007b, 2008). new zealand teachers are encouraged to establish and maintain effective teacher-student relationships by knowing, respecting, and valuing their students and by building new learning on the experiences, knowledge, and ways of being that students bring to the classroom. averill heritage cultures in learning journal of urban mathematics education vol. 5, no. 2 161 the cultural capital of students and mathematics learning two recent new zealand studies highlight the potential for linking mathematics instruction to the cultural capital (forms of knowledge, attitudes, skills, and advantages in relation to society in general) that students who are typically underserved acquire through their upbringing (bourdieu, 1986). hunter and anthony (2011) explored teaching practices aligned with pacific nations’ cultural practices within inquiry-based mathematical instruction. in their case study of a pacific nations teacher working with pacific nations students, the teacher referenced core cultural beliefs of reciprocity, collectivism, and communalism as the basis for classroom grouping arrangements and expected behaviours. teaching styles affected students’ mathematical agency and their accountability for their own and others’ mathematical learning, demonstrating a relationship between positive student outcomes and the use of instructional methods reflecting group participation values from students’ heritage cultures. in a recent study (averill, 2012), i reported the extent to which dimensions of teacher care could be referenced to cultural perspectives of indigenous māori and pacific peoples through analysing teaching practices observed in multi-ethnic secondary mathematics classrooms using durie’s (1998) indigenous model of health and well-being. durie’s (1998) whare tapa wha (the four-sided house) model uses the house as a metaphor for showing the interdependence of four dimensions of personal health and wellbeing: physical, social, spiritual, and cognitive and emotional dimensions. the model has proven useful for examining and developing culturally responsive mathematics teaching practice (averill, 2012; tertiary education commission, 2010). for example, my study linked caring mathematics teacher practices (such as using inclusive language, providing encouragement, being explicit about teaching decisions) with durie’s cognitive and emotional, social, spiritual, and physical dimensions of health and well-being, indicating the holistic nature of teacher care for students of the target ethnicities. the two studies highlighted (i.e., averill [2012] and hunter & anthony [2011]) indicate the potential of drawing from the cultural capital of indigenous and other marginalised groups for enhancing equity of access to mathematics achievement. nasir, hand, and taylor (2008) and bartell (2011) are among many who have advocated that teachers develop cultural competence to enable them to build on the existing cultural capital of their students. bartell examined the literature regarding theories of care and culturally relevant and responsive pedagogy and provided the following description of a caring teacher: teachers that care with awareness know their students well mathematically, racially, culturally, and politically. they work to understand and make connections with students’ cultures and communities; help students develop positive racial, cultural, and political identities; reflect critically on their own assumptions and practices about students’ cultures and communities, including rejecting and con averill heritage cultures in learning journal of urban mathematics education vol. 5, no. 2 162 fronting deficit and colorblind perspectives; and labor to neutralize status differences within and beyond the classroom walls. (p. 65) to teach in ways consistent with bartell’s definition, teachers need to know and understand their students as individuals and to care strongly about many aspects of their students’ lives. they must recognize students’ cultural capital and be able to link learning content and pedagogical practices to it. strong teacher-student relationships are known to be vital for creating learning environments necessary for students to be comfortable with being known so well by their teachers and with culturally linked classroom activities being used in their learning (bishop et al., 2003; gay, 2010; hattie, 2009). in their professional development research project, mcculloch and marshall (2011) focused on teachers’ orientation towards, and competence with, connecting the out-of-school experiences of students to in-school mathematics learning. across their year of involvement in the professional development, teachers participated in eight full days of interactive activities that explored the mathematical content knowledge of teachers and their understanding of how students learn mathematics, and considered the influences of cultural factors on teaching and learning. the researchers found that despite an increase in the number of out-of-school/in-school connections made by teachers in their teaching, these efforts remained infrequent and superficial, and few were “mathematically meaningful” (p. 60). the impact of the professional development included increased teacher use of students’ informal mathematics and language. in addition, there was an increase in teacher awareness of attending to possible effects of cultural difference, for example, through teachers clarifying contexts that students may have found unfamiliar. teachers’ lack of mathematical knowledge and the “simplistic connections [teachers] drew between race and culture” (p. 62) were thought to be factors that limited their ability to further reflect students’ out-of-school experiences in their practice. enhancing teachers’ cultural competence is also the focus of an extensive new zealand professional development project, te kotahitanga (berryman, 2011; lawrence, 2011). the project, which involved classroom-based professional development for year 9 and 10 classroom teachers in using an “effective teaching profile” (bishop et al., 2003, pp. 95–116), has resulted in enhanced teacher-student relationships, improved mathematics outcomes for māori students, and shifts in the understanding of teachers of their role and of incorporating māori culture into classroom teaching across curriculum areas (bishop, berryman, wearmouth, peter, & clapham, 2012; hynds et al., 2011). teachers and students reported deepened teacher understanding of māori educational concepts and their relevance to the classroom and sensitivity to the perspectives of students (savage et al., 2011). māori students were positive about changes in their learning through the project and specific examples of te kotahitanga teachers using learning tasks drawn from the māori culture have been reported in some curriculum areas (e.g., savage et al., 2011). although many mathematics lessons were found to feature māori cultural knowledge or language and averill heritage cultures in learning journal of urban mathematics education vol. 5, no. 2 163 opportunities for student co-construction, no specific examples of mathematics learning tasks drawn from the māori culture have yet been reported from the project. mathematics was also the curriculum area with the highest proportion of observed lessons that did not exhibit these elements. many studies have shown ways that the cultural competence of mathematics teachers can be developed and discontinuities between the heritage cultures of students and their mathematics learning reduced. examples include teachers working alongside educators with strong cultural knowledge (e.g., celedon-pattichis, 2004; howard, perry, & butcher, 2006), drawing from community knowledges to build mathematics learning opportunities (e.g., civil, 2007; civil & andrade, 2002), teaching mathematics for social justice (e.g., gutstein, 2003), teachers getting to know individual students and their families to develop their classroom practice (e.g., foote, 2010), and curriculum development such as that carried out with the indigenous yup’ik people (e.g., lipka et al., 2005). the views of teachers are important to understand in order to inform policy and teacher education. two recent studies exemplify that student voice is also vital for informing mathematics teaching practice to reduce culturally linked differences in educational opportunities. the stories of latina/o students in gutiérrez, willey, and khisty’s (2011) study indicated a prevalence of ideologies of the dominant cultural group in their instruction, schools’ neglect of the culturally linked learning tools and resources of students, and student resistance to particular schooling practices. in siope’s (2011) new zealand study, secondary school pacific nations students described how effective teacher-student relationships enhanced their learning and how they coped with school and home demands by keeping their school and home lives separate. in summary, researchers and policymakers have called for classroom teaching in general and mathematics teaching in particular to acknowledge and strongly link with the cultural backgrounds of students, and many projects have made valuable contributions towards understanding ways of doing so and of the associated supports and challenges for teachers. the views of teachers and students of such teaching have been less frequently explored but are vital for informing teacher practice and curriculum and policy development. these views were the focus of this study. examining the views of teachers and students: the study the sites for this research study—part of a larger research project (averill, 2009)—included three urban, state co-educational, mid-low socio-economic schools, each with roughly equal numbers of māori, pacific nations, and new zealand european students to ensure data could be collected from students of the three target ethnic groups. none of the study schools were involved in the te kotahitanga professional development project (bishop et al., 2003). the study participants were 136 year 10 students from the classes of six teachers, with roughly equal numbers of students of the target ethnicities, according to averill heritage cultures in learning journal of urban mathematics education vol. 5, no. 2 164 official data. this article does not discuss differences in the responses of students by ethnicity because students’ self-reported ethnicities indicated much greater complexity exists than is shown by the official data (e.g., 42% of students claimed more than one ethnicity). the three heads of department volunteered to participate and each nominated one other teacher. four of the teachers (three new zealand european: mr. a, ms. b, and mr. c; and one māori/new zealand european: ms. d) were very experienced in teaching students of the target ethnicities. two teachers (both first generation new zealanders: mr. e and mr. f) were in their second year of teaching. (full ethical approval was gained for the study.) to capture the desires of teachers and students for mathematics learning at the beginning of the school year as well as their later experiences of mathematics teaching and learning, data were gathered over 9 months including early, mid, and late in the academic year using three cycles of questionnaires and semi-structured interviews. all student questionnaires were available in english, the māori language, and in samoan, the language of the largest pacific nations participant group to enable students to respond in their heritage language. questionnaires included both open questions to capture the range of participant opinions and likert scales to enable measurement of the strength of views. i assumed a sociocultural epistemological stance for the study and used both qualitative and quantitative methods. māori and pacific nations cultural advisors were consulted on multiple occasions in order that their advice would inform every stage of the study design, implementation, and data analysis. strategies to maximise data validity included extensive development and trialling processes for creating the data gathering tools, using a sole researcher to gather all data, data triangulation (three research sites and collecting data multiple times at each site), methodological triangulation (more than one method of collecting data), and peer examination feedback from cultural and other advisors collected across the duration of the study (averill, 2009). the dataset included 18 teacher questionnaires and 96, 107, and 136 student questionnaires respectively for the first, second, and third data gathering rounds. both a sample of students of the target ethnicities and all teachers were interviewed in each data gathering round to follow up responses and themes arising from the questionnaires (138 student interviews and 18 teacher interviews in total). students had the option of being interviewed alone or with a peer for their comfort and to maximise the quality of the dataset. statistical package for the social sciences (spss) was the statistical analysis tool used for quantitative questionnaire data. in analysing the qualitative data, statements were deemed to indicate links with the māori or pacific cultural heritages of students if they included any direct reference to the languages, practices, or accepted behaviours of māori or pacific nations heritage cultures or to current or historical issues, knowledge, events, or other information clearly related to māori or pacific nations peoples. in the section that follows, i discuss study results in relation to teacher-student relationships and averill heritage cultures in learning journal of urban mathematics education vol. 5, no. 2 165 students’ heritage cultures, beginning with teacher perspectives followed by those of students. the voices of teachers: “it’s more than just the maths.” teacher-student relationships all of the teachers believed it important to develop a positive relationship with each student and all but one prioritized learning students’ names. all study teachers believed that both knowing their students as people and making some things about themselves open for students to know were important. for example, one said: [knowing about the students] is useful for interacting with them, and also for getting them to know little bits and pieces about us as individuals. these kids are very inquisitive in that sense. some staff think of that as being rude or nosy but the kids often want to know about the people around them: where you come from, who your wife is, what kids you’ve got, and that sort of stuff. and for them, they are always pleased when we show an interest in something that they’re doing. knowing about each other just helps. at times it can help you through an awkward situation in teaching. (mr. a) all teachers believed it important to incorporate aspects relevant to students’ in-school and out-of-school lives in their teaching to help show relevance of the mathematics to students and enhance student motivation. teachers liked knowing about their students including being aware of their “successes” (all teachers), “sports” (5 teachers), “personalities” (5), “families” (4), “family commitments” (4), “cultural activities” (4), and “progress in other subjects” (3). teachers used formal and informal ways of getting to know their students. two teachers asked their students to write letters about themselves at the start of the year, one telling the students about herself before setting the task: i always tell them something about myself, how long i’ve been teaching, my children, that kind of thing, and if they want to ask me anything, that’s the time. always on the first night for homework i get them to write me a letter to tell me everything that they think i should know. i have particular themes: how their last year went in maths; what they want to do this year; their family; and anything [they think] i need to know. (ms. d) aside from this letter-writing task, there was no evidence of systematic ways of learning information about students. the informal methods teachers used to get to know their students as people included having conversations with students over time (3 teachers) and with families (1 teacher). strategies teachers reported using to get to know their students as learners included diagnostic testing, working one-to-one with students, “interacting with as many students as possible each lesson,” and “brainstorming” when introducing new learning. teachers described a range of ways that they believed helped their students know them as a teacher, including “establishing a averill heritage cultures in learning journal of urban mathematics education vol. 5, no. 2 166 worthwhile programme,” “encouraging students,” “having patience and listening,” “having an organised [classroom] environment,” and “setting clear boundaries, routines, and expectations.” teachers reported personal and pedagogical strategies for establishing and maintaining teacher-student rapport: “knowing students’ names,” “using correct māori and pacific nations pronunciation,” “creating a friendly environment,” “using a sense of humour,” “being patient,” “giving students choices,” talking with, and listening to, students about their life, their progress, out-of-school activities, and problems, and “helping students in subjects other than mathematics.” four teachers believed that mathematics teachers faced greater challenges than teachers of other subjects in developing teacher-student rapport and knowledge of one another, arguing that it can be difficult to interest students in mathematics content and that missed lessons are more problematic in mathematics than in other subjects due to new learning often building on previous work. national assessment methods (“reducing students’ interest in mathematics”) and students’ negative experiences of, and feelings towards, learning mathematics were also identified: often [mathematics teachers] have to work harder [at developing rapport]. kids’ relationships with you are affected by their view of the subject and if they don’t feel switched on by the subject it’s probably going to be difficult to get a good relationship with them. so somehow we’ve got to get them to come in and feel comfortable, so being successful in maths to start with is important. that means on the first test [i make sure] everyone gets more than eighty percent, so they feel good about it. (mr. a) students’ heritage cultures no teachers mentioned the heritage cultures of students in questionnaire responses regarding how they get to know and develop rapport with their students. apart from the focus on pronunciation, teachers did not state any deliberate links between their pedagogical choices and students’ heritage cultures. however, when specifically asked about students’ heritage cultures, four of the six teachers believed that it was important for students’ mathematics learning that they as teachers knew and cared about aspects of the cultural heritages of their students. views varied regarding the relative importance of showing care for students’ heritage cultures and other factors for students’ learning: “caring about students’ cultures shows students that you value their background; it is all part of sharing an interest in each other so that the learning environment is one that all cultures can work in.” (mr. c) mr. a, who used his knowledge of students’ heritage cultures to create connections with them, believed that students’ learning was more strongly influenced by factors other than showing care for students’ heritage cultures: teacher a: i’ve worked in the pacific for 12 years. what we’ve brought back here is knowledge that helps. you look at their faces and look at their names. i said to one of the kids a couple of years ago, averill heritage cultures in learning journal of urban mathematics education vol. 5, no. 2 167 “you’re a cook islander” and she said “how do you know?” and i told her the story about a student that i had in the cook islands that had the same surname and i said this girl came from mangaia, one of the outer islands and so does she, so it’s from those kinds of clues that you can establish some personal link… it is about caring for kids and showing a bit of interest in them. it’s more than just the maths. interviewer: do you think it makes a difference to those students in terms of their learning? teacher a: it can do. there are other things that affect their learning more, but it’s just that they’re more likely to talk to you more…it just helps that [teacher-student] relationship. teachers stated that they showed that the heritage cultures of their students are important to them by developing their own cultural knowledge and skills (e.g., by learning some māori, samoan, or fijian language or knowing how to interpret the body language of different groups), by “talking with students about similarities and differences between our cultures” (mr. c) and by “giving students opportunities to work in the māori language” (mr. a). most teachers were positive about developing their cultural knowledge; “anything that helps with rapport, which is culturally important, i will try to learn about” (mr e). however, teachers mostly did not identify specific pedagogical practices that they intentionally used in order to teach in culturally responsive ways: “if students of different cultures feel comfortable coming into my classroom, i think in ways i am addressing the differences of the cultures in the room” (mr. f). one teacher stated that many of her students want to keep parts of their home and school worlds separate, but she tries to reduce the separation: many [students] go to great lengths to keep their lives in compartments, for example lots have “school” names [in english] because they believe teachers will not be able to pronounce their real names. i hear other students using their samoan names so i try to know their real names. (ms. b) although five of the six teachers showed some interest in knowing about māori and pacific cultures, they were largely neutral or negative about using mathematics examples linked to using such knowledge in their teaching: “i’m not good at this, that’s for sure. overall i don’t make a difference between new zealand european or māori or something. it’s the same, it is in our school. i can’t see a difference between them” (mr. f). two teachers felt that they should be reflecting heritage cultures in their teaching, for example the teacher with māori heritage said: “i know that in my classroom i really should have māori words up or the kowhaiwhai patterns (māori rafter patterns)…and i haven’t got those…. i should be doing more to promote biculturalism in mathematics” (ms. d). averill heritage cultures in learning journal of urban mathematics education vol. 5, no. 2 168 all teachers reported finding it difficult to include māori and pacific nations contexts in their teaching due to lack of resources (3 teachers; “textbooks only pay lip service to [incorporating māori and pacific nations cultures]” [mr. a]) and lack of personal knowledge (3 teachers; “i’m not sure how i can help somebody who’s from another culture to learn maths better” [mr. f], “i don’t know enough about other cultures and how they use maths” [mr. c], “identifying realistic situations to use is difficult” [mr. a], “to link the culture to the maths you have to have a really good understanding of the culture, and i don‘t think i have” [mr. e]). teachers’ practice in relation to reflecting the māori and pacific nations heritage cultures of their students in mathematics learning seemed also to be limited by their beliefs about the nature of the personal engagement of their students with their heritage cultures (e.g., “most of these students don’t have strong connections with their heritage culture” [mr. a], “most of the [pacific nations] students are new zealand born and urban so their contexts are different to those born outside of new zealand and/or rural” [mr. a]) and concerns about possible student discomfort (e.g., “students might take it the wrong way and not like it” [mr. e]). the voices of students: “i see them as different, there’s your culture and there’s maths” teacher-student relationships teachers and students being able to be themselves in the mathematics classroom was important to the students in this study because it enabled them to get to know the teacher, feel the teacher was interpreting their interactions appropriately, and engage with the learning: interviewer: what makes a good maths teacher? student: personality and that he works [well] with people. interviewer: how does that help your maths learning? student: it makes you get more into it. many students felt it important that they and the teacher knew one another for effective communication, mutual trust, teacher-student rapport, and for their learning. comments from two of mr. c’s students help illustrate students’ thinking: “teachers need to know the [students], so if you’re not getting something, they can use examples from things you like…so they are using something you are interested in.” because it kind of breaks down the wall, they can relate to you as a person. you normally think it’s going to be a scary thing talking to your teacher but if you know a bit about them and they know about you, you feel confident in talking with them. if they get to know you, you know you can trust that teacher and you know you’re going to get a good lesson, but if they just write something on the board and get you to copy it, you’re not really learning anything. averill heritage cultures in learning journal of urban mathematics education vol. 5, no. 2 169 overall, students valued knowing some things about their teacher and their teacher knowing about them. they held differing preferences regarding what they wanted to know about their teacher, what they wanted their teachers to know about them, and how they wanted their teachers to find those things out. the first set of questionnaire responses showed that “my successes,” “my personality,” and “how well i learn in other subjects” were what students most wanted their teachers to know about them, with over 40% of students in each heritage culture group selecting each of these. almost one fifth of students wanted their teacher to know something about their “family,” “sports,” and “cultural activities.” four of the 136 students indicated that they did not want their teacher to know anything about them. most students who had been asked to write letters to their teachers were in favour of that way of sharing information about themselves: “because if you were too shy to speak about it in class, or didn’t want to tell anyone, you just write down in the letter and she’d just read it.”; “[writing the letter] was a really good idea because it gave us a chance to explain about ourselves. it’s giving us a chance to write what we want.” however, one student stated that she would not be comfortable sharing information about herself with her teacher until later in the year when she would know her teacher better. students who had not been asked to write letters about themselves liked their teachers to get to know them through teacher-student interactions over time, completing a questionnaire, or by talking with their previous teachers. some felt comfortable telling the teacher what they would like the teacher to know but stated that it was harder to do in mathematics than in other subjects because fewer spontaneous opportunities for this existed: student a: i’d just tell them straight up and then they’ll understand. student b: i reckon that’s pretty hard in maths though, ‘cause, you just work with numbers, you don’t have conversations about reading a book and then you can’t talk about applying it to your personal life or something. students’ heritage cultures many students believed their mathematics learning and their heritage culture were not linked (“culture doesn’t really have anything to do with maths”) and although some students (22%) believed that mathematics teachers being responsive to their heritage culture may positively affect their learning opportunities, most either stated they did not know (between 47% and 60% of each group) or that it would not be important for their learning (24%). pacific nations students were more likely than māori or europeans to believe their culture was both relevant at school (59% compared with 37% and 28% respectively) and to their mathematics learning (25% compared with 11% and 12% respectively). the recurring theme of students believing that mathematics learning and their heritage cultures are two separate things suggests they had averill heritage cultures in learning journal of urban mathematics education vol. 5, no. 2 170 experienced or noticed few or no links between them in their current or previous mathematics learning: interviewer: you’ve been learning maths from year one all the way through to now, so is that something that you’re used to happening in maths, teachers bringing in examples from different cultures? student: no, just one usually, european, which is all right ‘cause i kind of understand it better. student: it would just be the same as normal maths if they added on a bit of culture. interviewer: but do you think it would help you with learning maths? student: i think it would be better if they keep it how it was. moreover, students often showed surprise at the inclusion of questions regarding their heritage cultures, further evidence that they did not expect such links to be made. however, one interview indicated that despite initially viewing mathematics and heritage cultures separately, students could quickly think up an example linked to their heritage nation: student c: i see them as different, there’s your culture and there’s maths, so maths is with all these numbers and the teacher, and the culture is with your parents and family, so it’s quite different. interviewer: so you see them as two separate things? student d: i reckon you could use culture for, like statistics, like how many people live in samoa and things like that, or how many children were born there in certain years. interviewer: and you’d like examples like that to be used? student c: yeah, like to have true statements that were taken from the [samoan] census would be good. some students felt uncomfortable about mathematics learning being linked to their heritage culture, being concerned for themselves (such as avoiding feeling shame in relation to their own lack of knowledge of their culture or language), or for others: interviewer: do you see your heritage culture as something that is separate to your mathematics learning or something that is important in your mathematics learning? student: i kind of want it to be separate. interviewer: can you explain why? student: well, i really don’t know my own culture, or how to speak it. student: if you have examples and they’re all to do with, like pacific type questions, and then people would think, “oh, why is she just doing that culture? why not anything else?” but if you do a range of them then that would be all right, but kind of different. responses gathered in the second and third questionnaires showed that most students did not know whether or not their mathematics teacher knew about their heritage culture (more than 35% for all groups, and 61% of māori students). pacific nations students were more likely than other groups to be averill heritage cultures in learning journal of urban mathematics education vol. 5, no. 2 171 lieve their teacher knew about their heritage culture (51% compared with 28% and 38% for māori and european respectively). higher proportions of māori and pacific nations students wanted their teacher to know about the māori and pacific worlds (20% and 45% respectively) than to use examples that drew from this knowledge in their mathematics lessons (9% and 28% respectively). later questionnaire data showed that despite learning for many months with the same mathematics teacher, most students did not know whether or not their teachers knew about the māori and pacific worlds (over 64% and over 59% across all groups respectively), suggesting either that the teacher had not shown such knowledge or used learning examples linked to māori or pacific nations cultures, or that if they had, it had not been noticed by their students. some comments suggested students thought their teacher needed only limited knowledge about their heritage cultures: i think it’s good for [teachers] to know what your cultural identity is and what one you belong to but for them to fully understand it, that’s not really necessary ‘cause as long as they know who you are, they don’t need to know everything about you. some students believed that subject areas such as english, art, languages, and social science were more suitable than mathematics for connecting learning with their heritage culture: “we’re there for maths and not for discussion about cultural things, that’s for a different classroom”; “focus on the maths and leave other things to other teachers.” while many students did not believe teachers needed to know much about their heritage cultures, they did believe it very important that teachers showed respect for them: it just depends on how they use my culture…if they’re going to use an example in the language they may as well speak it properly because if they’re going to describe something in maths, they should get it right. according to students, teachers showed the importance they placed on students’ heritage cultures by showing “respect” for them, “giving help,” trying to learn words in the students’ heritage languages, and treating students of different cultures in the “same” way or in different ways. this apparent contradiction was explored further in the interviews. students believed both to be important with the way in which it was shown dependent on the context. for example, students felt teachers should treat everyone the same in relation to respecting everyone’s cultures equally, while believing that teachers should treat students differently by responding appropriately to cultural difference: student: teachers need to respect our cultures. interviewer: should they treat everybody the same or treat everybody differently? student: the same but respect everyone’s culture. yeah, the teacher might say something that would offend someone from a dif averill heritage cultures in learning journal of urban mathematics education vol. 5, no. 2 172 ferent culture, so if she knew that was their culture she wouldn’t say it. students’ expressed that their teachers’ concern for their heritage culture was important for their mathematics learning. reasons for this belief included themes of inclusion, fairness, workload, and suitability of linking heritage culture and mathematics learning: important: so that the teacher would be inclusive, so they could reflect class cultures in maths activities, to enhance students’ feelings of cultural identity, because we might learn in different ways. not important: teachers should treat everyone the same, teachers are already too busy without having to learn about students’ cultures, culture and maths are two separate things, our cultures do not affect our learning, it is none of the teacher’s business. (paraphrased from student questionnaire responses) one pacific nations student suggested “teachers could have a good ethnic buddy,” a practical suggestion consistent with celedon-pattichis’s (2004) and howard and colleagues’ (2006) research which includes incorporating cultural experts as partners to enhance teachers’ cultural knowledge and expertise. such practice has the potential to develop teachers’ knowledge of, and sensitivity towards, students’ heritage cultures and their pedagogical approaches to reflecting these in their mathematics teaching. discussion and conclusion teacher-student relationships overall, the study results are consistent with those of previous studies and the literature on teacher-student relationships (e.g., bishop et al., 2003; eccles, 2004; fletcher et al., 2009; hill & hawk, 2000) in that both students and teachers believed in the fundamental value for effective learning of positive teacher-student relationships and knowing about one another. this study adds to the literature by providing evidence of the connection between effective teacher-student relationships and teachers knowing students well as individuals, including identifying what students may want their teacher to know, and how teachers can sensitively learn these things without detracting from curriculum delivery time. the findings indicate the value of enabling student choice about what and when to share, as demonstrated by the range of students’ preferences regarding what they would like their teachers to know about them and how and when they would feel comfortable for the information to be gathered. the findings indicate that bartell’s (2011) ideal of “teachers who care with awareness” (p. 65) is likely to be difficult for mathematics teachers to achieve. to teach in ways consistent with bartell’s definition teachers need to know, understand, and strongly care about many aspects of their students’ lives. teachers who work exclusively with one group of students (such as many el averill heritage cultures in learning journal of urban mathematics education vol. 5, no. 2 173 ementary school teachers) from one or a small number of different cultural backgrounds (as in hunter & anthony’s [2011] study) must be better positioned to demonstrate cultural competence and achieve elements of bartell’s description of caring teacher practice than teachers who work across many class groups with students of a wide range of cultural backgrounds. this study shows that some students may not want to be known by their mathematics teacher as well as would be required for all aspects of the definition to be adopted. furthermore, this study indicates that teachers and students may need convincing that knowledge of heritage cultures will assist mathematics learning and can be used with sufficient respect and sensitivity. a teacher’s knowledge about their students’ heritage cultures was not amongst the main factors directly reported by teachers and students as fostering or maintaining effective teacher-student relationships or rapport. however, participants’ responses regarding the importance of teachers respecting students’ cultures and responding to students in culturally appropriate ways indicate that participants believe teachers’ knowledge of students’ heritage cultures is essential in relation to interpersonal interactions. this study indicates that teachers were teaching in culturally responsive ways in some respects (e.g., placing importance on knowing their learners as individuals, acknowledging cultural differences, developing positive teacher-student relationships, and managing student-centred classrooms), but had few systematic ways of knowing they were being culturally responsive in their teaching or of developing their culturally responsive practice. for example, the study teachers were interested to know about some aspects of their students’ lives but such knowledge was mostly gathered informally through ad hoc individual interactions with some students during class lessons. students’ heritage cultures the study findings provide further evidence of discontinuities between schools and students’ cultural heritages and contribute to understandings of how and why such discontinuities occur and how they can be reduced. in contrast to participants in the te kotahitanga project (berryman, 2011), neither the study teachers nor the students readily identified how students’ heritage cultures were linked with mathematics learning. no evidence emerged in the study of schools, teachers, or students drawing from the heritage cultural knowledge of parents, whānau (family groups), or the wider school community to inform mathematics teaching and learning. the study results could indicate that teachers and students may not have strong personal cultural identity or if they do, they may see it as something they prefer to keep apart from the classroom. no schoolor discipline-based discussions focused on responding to cultural diversity in learning programmes were mentioned by participants, suggesting either such conversations and development are not taking place in the study schools, or if they are, mathematics teachers do not see them as relevant, important, or easily implementable. the student voices outlined the dichotomy averill heritage cultures in learning journal of urban mathematics education vol. 5, no. 2 174 many students encounter in identifying with their own cultural heritage while being part of the other cultural groups in their lives; for example, their class, school, sports, and social groups and the mainstream cultures within which these sit. in addition, the students in this study were in their adolescent years, a time when many can desire or feel compelled to minimise differences between themselves and others (meece & daniels, 2008). it may be that the opinions and practices of teachers are a deliberate or subconscious response to their sense of the complexities of students’ beliefs, perceptions, and prior learning experiences in relation to culturally responsive mathematics teaching and variation in the cultural identity of individual students. both the teacher and student voices in the study indicate that many conflicts and challenges exist for teachers in developing and using culturally linked mathematics learning tasks. consistent with the findings of amituanai-toloa and colleagues (2009), loorparg’s (2006) team, and others, and claims of researchers such as villegas and lucas (2002), the teacher voices collected in this study illustrate that development of teachers’ knowledge of students’ heritage cultures, how to reflect these in their mathematics teaching, and why this is important to achieve are necessary and urgent. whilst many new zealand mathematics textbooks do not yet incorporate māori and pacific nations cultural contexts in a substantial way, a few suitable resources have been available (e.g., averill, phillips, & french, 2003; heays, copson, & mahon, 1994). the absence of mathematics tasks linked to contexts drawn from māori or pacific nations cultures in study students’ current and previous mathematics learning was strongly apparent, despite the teachers’ access to mathematical tasks with links to māori and pacific nations cultures, and study schools being situated within multiethnic communities. given this finding, responses to questions regarding integration of aspects of cultural heritages within mathematics learning must have been based largely on participants’ beliefs of what such practice might be like, rather than from experience. culturally responsive teaching the study included six mathematics teachers and their students from one year group across three urban schools in the same city and caution should be used in interpreting the findings more widely. limitations notwithstanding, this study adds to what is known about culturally responsive teaching in several ways. first, students may not recognize some elements of teachers’ responsiveness to students’ heritage cultures. teachers’ knowledge of cultures and cultural difference informs their classroom behaviours and decisions, how they conduct one-to-one teacher-student interactions, and their body language, any of which may be deemed by their students as suitable, comfortable, and caring. the absence of such culturally responsive behaviours, however, is perhaps more likely to be noticed by students than their presence (banks, 2004). for example, although culturally responsive methods were the teacher’s intention and practice in hunter and anthony’s (2011) study, it is unlikely that the stu averill heritage cultures in learning journal of urban mathematics education vol. 5, no. 2 175 dents recognised that the ways in which their learning was being managed were consistent with practices used in their heritage cultures, unless this was explained. findings from hunter and anthony’s (2011) and averill’s (2012) studies suggest that mathematics learning is enhanced when classroom practices are compatible with important aspects of students’ heritage cultures. however, unless the nature of those practices that are intentionally or unintentionally linked to classroom practice is discussed with students, it is unlikely that they will also contribute to policy and research goals of strengthening students’ cultural identities. secondly, many students viewed mathematics and culture as separate and, in spite of multiple opportunities to consider the place of heritage cultures in relation to mathematics learning over the study, most did not readily identify benefits from integrating cultural knowledge and mathematics instruction. the students’ varied and often vague responses and their surprise at the inclusion of questions linking cultural heritage and mathematics learning may also indicate that some had not previously considered the place of deliberate acknowledgement of students’ heritage cultures in their learning or that they do not prioritise such practice. while teachers did not seem as surprised as their students to be asked questions linking heritage cultures and mathematics learning, they generally appeared not to have developed a firm philosophical stance nor to have deeply considered how they would or could incorporate students’ heritage cultures into their mathematics teaching. this is particularly surprising given four of the teachers had extensive experience teaching māori and pacific nations students and held departmental leadership positions. despite the reasons for their practice, as found by kidman and colleagues (2012) and cautioned by sleeter (2011), teachers not linking mathematics learning to their students’ heritage cultures is likely to portray to their students that such links are not able to be made or are not desirable. there is growing evidence that teachers need substantial assistance to enable such practice. finally, the findings indicate that practices consistent with educational research and policy regarding reducing discontinuities between students’ homes and schools in relation to cultural heritage are not yet in place across mathematics classrooms, indicating likely limitations in relation to the learning and career opportunities of students not of the dominant culture (gay, 2010; kidman et al., 2012). these results, from multiethnic schools educating indigenous students in a multiethnic community and country, are of concern. the findings support those of others (e.g., fletcher et al., 2009; macfarlane, 2004) in indicating that increased teacher knowledge of students’ cultural heritages, of te ao māori, and pacific nations is urgently needed. it is hard to see how teacher-student connectedness can be maximised and teaching be truly culturally responsive while a mismatch between teachers’ and students’ ethnicities remains, teachers’ cultural knowledge base is either limited or invisible to students, and students’ understandings of links between heritage culture and learning are tenuous. averill heritage cultures in learning journal of urban mathematics education vol. 5, no. 2 176 implications there are implications from this study for school communities, professional development and teacher education, and policy development and implementation. this study’s findings, alongside those of others (e.g., bishop et al., 2003; gorinski, ferguson, samu, & mara, 2008; siope, 2011), indicate the challenges for new zealand mathematics teachers to substantially reflect māori and pacific nations cultures in their work are so great that substantial resources, professional development, and community involvement are required for the ministry of education’s (2006, 2008) goals, and calls from the literature, to be fully realised. teachers are but one part of the education system and of wider society. historical and current societal contexts within which schools sit amplify the challenges teachers face in developing culturally responsive practice because the ideologies of dominant cultures are often not only entrenched, but they and their effects can be invisible to teachers, parents, and students (brayboy, 2005; samu, 2011). issues of power and influence within education as a whole, what counts as knowledge and achievement, and whose cultural heritages and languages are valued in schools and in society are vital to attend to in order to establish the conditions necessary for inequalities in relation to diversity in education to be addressed in classrooms. all share responsibility for addressing these issues. for example, school communities are needed as partners in working together to support policy implementation by voicing high expectations of cultural responsiveness within schools, teaching, and learning, and sharing their cultural knowledge, understanding, and expertise, and how they believe this should be reflected in the learning of their students. evidence suggests that school-wide projects in which teachers develop close links with individuals of different cultural groups and their families (as in foote’s [2010] project), the use of accessible cultural models on which to base classroom practices (as in hunter & anthony’s [2011] study), and mathematics teachers having easy, frequent, and ideally, classroom-based access to educational cultural advisors (as in howard et al.’s [2006] project) offer positive ways to develop culturally responsive mathematics teaching practice. this study provides a glimpse into the beliefs and understandings of teachers and students regarding culturally responsive practices that could inform such developments. the extent to which school communities are involved in professional development, mathematics teacher behaviours and the contexts used in mathematical tasks are visibly and sensitively linked to students’ heritage cultures, and teacher and student beliefs regarding the relevance of heritage culture to mathematics learning are altered, could provide useful measures of the success of such professional development in advancing culturally responsive teaching and providing environments that can enhance students’ cultural identities. in response to the challenges identified for teachers above and education policy requirements, initial teacher education programs must also prepare students well to understand and implement, and even to be leaders in, culturally averill heritage cultures in learning journal of urban mathematics education vol. 5, no. 2 177 responsive practices. aspiring teachers without knowledge of the heritage cultures of new zealand students have much knowledge and cultural understanding to acquire through their initial teacher education and will need guidance, support, practice, and critical reflection on their practice that school-based associate teachers may not be able to provide. such development must be expected and supported by initial teacher education providers and qualifications. the development and implementation of education policy are complex processes affected by many factors. it is reasonable to argue that to enable social change, education policy must be idealistic, inspirational, and aspirational. however, it must also be grounded by indications research provides regarding what is possible to achieve within societal constraints (marshall, coxon, jenkins, & jones, 2000). furthermore, policy must be supported by in-depth processes and resources to maximise its implementation and the impact it will have within schools, teaching, and students. international contexts will vary. this study indicates the value of uncovering teacher and student beliefs towards understanding the challenges for teachers and education systems and society in ensuring culturally responsive practices are the norm rather than the exception. students being able to draw from, and link to, their heritage cultures through their mathematics learning is a vital and challenging goal to pursue towards enhancing students’ cultural identities, continuity between homes and schools, inter-cultural knowledge and understanding, and equity of access to mathematics achievement. teachers’ and students’ perspectives are essential to consider in developing practice. internationally, this study offers glimpses of the complexities that can exist in relation to incorporating pedagogies and learning experiences linked to heritage cultures in mathematics instruction that can help to inform teacher education, curriculum and resource development, school practice, and policy implementation. acknowledgements the author sincerely thanks megan clark, fuapepe rimoni, and herewini easton for their careful and wise advice through the study, the new zealand institute of mathematics and its applications (nzima) for funding which contributed to the completion of this study, and the study participants for their very thoughtful contributions and very valuable time. references amituanai-toloa, m., mcnaughton, s., kuin lai, m., & airini. 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(2008). who is teaching? does it matter? in m. cochrane-smith, s. feiman-nemser, d. j. mcintyre, & k. e demers (eds.), handbook of research on teacher education: enduring questions in changing contexts (3rd ed., pp. 134–156). new york, ny: routledge/taylor & francis group & association of teacher educators. http://www2.nzmaths.co.nz/numeracy/references/comp07/tpt07_temaro_higgins_averill.pdf http://www2.nzmaths.co.nz/numeracy/references/comp07/tpt07_temaro_higgins_averill.pdf http://www.educationcounts.govt.nz/__data/assets/pdf_file/0011/6968/te-kauhua.pdf the complex challenge of a being black high school student and exposed to out-of-school stem learning journal of urban mathematics education december 2013, vol. 6, no. 2, pp. 20–41 ©jume. http://education.gsu.edu/jume ebony o. mcgee is an assistant professor of diversity and urban schooling in the peabody college of education and human development, vanderbilt university, 230 appleton place, nashville, tn 37203; email: ebony.mcgee@vanderbilt.edu. she is also a member of scientific careers research and development group at northwestern university. her research focuses on the role of racialized biases in educational and career attainment, resiliency, mathematics identity and identity development in high-achieving marginalized students of color in stem fields. high-achieving black students, biculturalism, and out-of-school stem learning experiences: exploring some unintended consequences ebony o. mcgee vanderbilt university in this article, the author discusses the complex challenges of high-achieving black students who are successful in becoming immersed in predominately white stem (science, technology, engineering, and mathematics) spaces and how such immersion can exacerbate their experiences of racial stereotyping and other forms of racial bias. the author illustrates these complex racialized experiences through the story of maurice, a high-achieving high school mathematics student who successfully negotiated the white spaces he occupied yet did not indicate interest in pursuing a stem-related career. although maurice developed sophisticated bicultural competencies that allowed stem professionals and educators to view him positively, he decided that compromising his own racial, cultural, and individual identities to pursue a stem college major and career was too costly. the discussion in general highlights how racial and ethnic stereotyping which is endemic throughout stem education and careers can push mathematically competent black students out of the stem pipeline prematurely. keywords: bicultural competencies, high-achieving black students, mathematics education, out-of-school stem experiences, racial stereotypes, stem just because i know how to play the game doesn’t mean i like it. —maurice, an african american junior high school student overall gpa 3.7 – mathematics gpa 4.0 1 aurice is clever and charismatic. as the president of his junior class and his high school’s international ambassador, he negotiated successfully among a cadre of racially and culturally diverse in-school and out-of-school authority figures (e.g., school administrators, counselors, teachers, maintenance staff, community members, employers). maurice was the “go to” person for other black students at 1 all proper names throughout are pseudonyms. m http://education.gsu.edu/jume mailto:ebony.mcgee@vanderbilt.edu mcgee biculturalism and stem journal of urban mathematics education vol. 6, no. 2 21 new beginnings charter high school who wanted to learn how to “play white.” his peers called it playing white because they observed that maurice had “not lost any of his blackness” in order to achieve in mathematics and science. his talent for code switching 2 made him popular with both his teachers and his fellow students. many of the other 23 students in the larger study 3 expressed envy as they discussed the various ways maurice could switch his demeanor to emulate mainstream white ways of speaking and behaving and then in an instant flip back to “being black.” i informally interviewed and spoke often with maurice’s mathematics teachers, mathematics coach, and the school principal who all considered him to be one of the “best and brightest” students in the school. maurice certainly knew “how to play the game,” which he defined as being able to operate in white spaces with the adoption of white cultural and social values (e.g., speech and dialect, mannerisms, dress, charisma, etc.). nevertheless, he had many negative experiences during his out-of-school stem (science, technology, engineering, and mathematics) learning experiences that he described as “uncomfortable experiences revolving around my race.” these negative experiences led maurice to conclude that, despite his interest in and proclivity for stem-related activities, he was not interested in pursuing a career in stem. in this article, i explore how maurice, a high-achieving high school mathematics student who successfully negotiated the white spaces of stem, could make the decision not to pursue a stem related career. first, i offer a review of literature related to black students in stem and how black students negotiate white spaces. i then provide details of a phenomenological case study of maurice that illustrates the complexities of adopting biculturalism as a means to negotiate white spaces. biculturalism, or bicultural identity, has been defined as extending one’s ethnic identity and sense of belonging to two or more different cultures without losing one’s original cultural identity (lafromboise et al., 1993). i conclude the article with a discussion that highlights some probable unintended consequences of biculturalism, suggesting that, in some cases, it might be a failed strategy. black students in stem black students’ recruitment and retention in stem the president’s council of advisors on science and technology (holdren, 2 code switching refers to the ability to shift one’s behavior or the practice of moving between variations of languages in different contexts to fit the norms of more than one group (jones et al., 2012) 3 this article represents a single case that was part of a larger study. see mcgee (2013a, 2013b) and terry & mcgee (2012) for reports from the larger dataset. mcgee biculturalism and stem journal of urban mathematics education vol. 6, no. 2 22 lander, & varmus, 2011) brought national attention to the low recruitment and retention rates among undergraduates in stem fields. for many years, the united states has relied on a relatively homogenous stem workforce comprised, for the most part, of white and asian men. in 2009, asian and white students had the highest percentage of undergraduate degree attainment within the stem fields, approximately 16 and 9 percent, respectively (aud, fox, & kewalramani, 2010). however, the percentage of whites in specific stem occupational groups, such as the life sciences, has decreased significantly from 2001 to 2009, and much of that employment has shifted to asians (2010). recruitment and retention in stem college majors is a problem that disproportionately affects students born in the united states; however, the situation is most acute for african americans. in 2009, blacks received just 6% of all stem bachelor’s degrees, 4% of master’s degrees, and 2% of phds (national center for education statistics, 2010), despite constituting 12% of the u.s. population. at all levels of postsecondary education (community college through postdoctoral), the percentage of stem degrees blacks received in 2009 was 7.5%, down from 8.1% in 2001. in mathematics and the physical sciences, the numbers are even more striking. in 2009, african americans received 4% of degrees granted in mathematics and statistics in the united states. in the same year, out of 5,048 phds awarded in the physical sciences (e.g., physics, chemistry, astronomy), a mere 89 degrees, or less than 2%, went to african americans. education researchers have begun investigating strategies for recruiting and retaining american students in general and african american students in particular to stem fields. in 2009, blacks were only 3% and latinas/os only 2% of all stem workers. stem occupations are projected to grow by 17% from 2008 to 2018, compared to 9.8% growth for non-stem occupations (beede et al., 2011). rising above the gathering the storm (committee on prospering in the global economy of the 21st century, 2007), a report on the stem fields, documented the fact that underrepresented minorities’ completion rates for stem degrees would at least have to triple to reach a goal of 10% of 24-year-olds receiving their undergraduate degrees in science or engineering. these figures are compounded by the fact that the u.s. stem faculty is 78% white and 14% asian (di fabio, brandi, & frehill, 2008). explanations for the small number of historically underrepresented students pursuing academic and career trajectories in stem fields revolve around their lack of interest in and academic preparation for pursuing college majors in these disciplines (leggon, 2006). those who study structural inequalities in education argue that african american students generally do not have adequate opportunity to take the most challenging mathematics and science courses in high school (martin, 2009, 2012; tate, 1997). it should be noted, however, that black students who do take advanced placement, international baccalaureate, or higher-level mathematics and science courses are as likely as white students to pursue stem mcgee biculturalism and stem journal of urban mathematics education vol. 6, no. 2 23 degrees (tyson, lee, borman, & hanson, 2007). gutiérrez (2008) argues that issues of access (i.e., having adequate resources to engage with quality mathematics) and achievement measures (i.e., standardized test scores, participation rates, and the mathematics pipeline) are two critical elements that impede the achievement of black students. she and others (e.g., martin, 2009) call for a healthier understanding of identity (i.e., maintaining cultural and racial connections) and power (i.e., agency to create change in schools) when addressing mathematics achievement and equity issues for marginalized students (martin, 2009). responding specifically to the low number of black and latina/o students who have successfully negotiated mathematics and science success over the last 20 years, an increasing number of programs have been developed to improve student recruitment and retention rates. considerable attention has been given to the mathematics and science outcomes of middle and secondary school students, which provides a critical preparation and foundation for enrolling in a college stem major. for example, the algebra project, which is rooted in the civil rights movement, provides culturally sensitive instruction to increase algebra literacy among african american middle and high school students (moses & cobb, 2001). project syncere’s (supporting youth’s needs with core engineering research experiments) goal is to increase the number of historically underserved and female students pursuing careers in the stem fields. the program provides a robust project-based curriculum for grades 3 through 12 (talbert, 2012). the latino stem alliance partners with schools, private industry, community groups, and academia to bring stem to underserved youth who would otherwise not have such an opportunity. latino stem alliance identifies and conducts proven stem enrichment programs in partnership with schools and community groups (jones, 2012). these programs enable underserved youth to experience hands-on stem enrichment activities with the goal of inspiring them to consider stem-related careers. in spite of the existence of programs such as the algebra project, project syncere, and latino stem alliance, which primarily focus on black and latina/o stem students, only modest gains have resulted (national center for education statistics, 2010). nevertheless, there are historically underrepresented students who do achieve success in stem and who do so in the midst of multiple inequities that test their abilities to be stem-resilient. black students’ success in stem investigating black student success is critically important in addressing the experiences they endure and their abilities to persevere against a host of obstacles, and adds another layer of description and understanding of their academic and educational decision-making (berry, thunder, & mcclain, 2011; stinson, 2013). carter andrews (2009) investigated the construction of black students’ racial and mcgee biculturalism and stem journal of urban mathematics education vol. 6, no. 2 24 achievement self-concepts in a predominantly white high school as an entry point to understanding their black achiever identity. the students’ narratives showed they discussed achieving in the context of being black. thus, these black high achievers provided a critical insight into how race and racism have operated to potentially constrain their success. her follow-up study (carter andrews, 2012) examined how high-achieving black students at a predominantly white high school respond to experiencing racial microaggressions by integrated forms of hypervisibity and invisibility that she terms racial spotlighting and ignoring. the students’ ability to resist microaggressions helped to enable them to maintain academic and social success despite experiencing racism in the school context. these studies add to the growing research base focused on black student success (conchas, oseguera, & vigil, 2012; toldson, 2008). investigation of black high school student success in stem has also revealed students’ determination to cope with academic-related stereotypes and to use their mathematics (and general academic) success as a form of resistance leading to increased resilience (mcgee, 2013a, 2013b). walker (2006, 2012) explored the role of peer influence in cultivating urban 4 high school students’ academic success in mathematics, providing a counter narrative of the importance of support from peers, families, and communities in reinforcing mathematics success. munro’s (2009) and sullins’s (2010) research studies both reveal a strong association between high school students’ interest and self-confidence in science and mathematics and their continuing stem studies through college, beyond enrollment and achievement outcomes. other studies showed that training for teachers in stem subjects had positive associations with students persisting in stem fields (maltese & tai, 2011). out-of-school stem experiences have been studied as well. for example, wright (2011) investigated community-based practices for 11thand 12th-grade african american students and described the ways academically successful african american male adolescents interpret their social and academic lives so they are able to be successful in school while maintaining a healthy racial identity. taken as a whole, these studies demonstrate that there are broader measures than disaggregated “achievement data” to provide an understanding of the learning, social, emotional, and cultural resources that facilitate african american students’ learning and academic achievement in the stem fields. 4 the descriptor urban does not merely describe the population density of a school’s surrounding community. among other things, urban describes schools with many students of color, schools for which many contemporary policies are designed, and usually refers to certain unspoken about and thus qualities considered undesirable of the students and community who belong in that space (chazan, brantlinger, clark, & edwards, 2013). mcgee biculturalism and stem journal of urban mathematics education vol. 6, no. 2 25 black students negotiating white spaces educators have long suggested that acculturation—the process of learning or adapting to a new culture—might lead to a partial rejection one’s own ethnicity or race and adopting the dominant culture (nguyen & benet-martínez, 2007). the historical view of black students who are academically successfully is often based on the notion that successful black students must extend their cultural identity— imitate at one end of the spectrum and internalize mainstream white identity and behavior at the other end—to achieve the best learning outcomes, essentially, they need to become bicultural or even raceless (fordham, 1988, 1996). researchers who study biculturalism have long argued that when blacks and other marginalized students develop bicultural identities and/or competencies they are more successful academically, or, at the very least, they master the dominant culture, which allows them to negotiate the school experience more successfully (anzaldúa, 1999; lafromboise et al., 1993). this process requires knowledge of the language, personality characteristics, and patterns of social behavior of at least two distinct cultural groups (scherman, 2010), and the ability to operate and interact in both cultures without relating to either in a hierarchical manner (lafromboise, berman, & sohi, 1994). in sum, being bicultural is said to allow a person to negotiate two cultures and to know which culture is better to embrace in particular contexts. research on biculturalism frequently presupposes that bicultural individuals internalize and make use of their two cultures seamlessly, uniformly, and with little internal conflict (benet-martínez, lee, & leu, 2006; phinney, 2003; tadmor & tetlock, 2006). african american students are presented as behaving in ways that are predominantly identified with being black (e.g., speaking ebonics, playing basketball, wearing pants that sag) and are “schooled” to adopt behaviors and competencies deemed acceptable within white culture as a key to gaining or maintaining academic achievement (diemer, 2007; oyserman, brickman, & rhodes, 2007). however, there appears to be little room within the u.s. education system for black students to exhibit their own culture without being subject to misrecognition, misunderstanding, and disciplinary sanctions (good, dweck, & aronson, 2007; maryshow, hurley, allen, tyler, & boykin, 2005). students’ cultural assets are frequently misrecognized (george, 2012; walshaw, 2011), causing black students to be misidentified as deficient in their level of learning, language, ideologies, and practices (hand & taylor, 2008; malloy & malloy, 1998; martin, 2007, 2009). misunderstandings occur when authority figures interpret black children’s behavior as anti-school and self-defeating—at worst, they are seen as “prisoners in waiting” (alexander, 2010; noguera, 2003; smith, 2009). managing the classroom has become more important than focusing on learning, and black students are sanctioned more often than their white counterparts for many types mcgee biculturalism and stem journal of urban mathematics education vol. 6, no. 2 26 of behavior (gregory, skiba, & noguera, 2010; thomas, coard, stevenson, bentley, & zamel, 2009). these multiple forms of educational and social oppression have a deleterious impact on the learning and classroom participation of black students in the mathematics classroom (martin, 2012). having a bicultural identity or adopting bicultural competencies in the education context is based on the notion that african american students fare best by learning how to operate “properly” within the dominant white cultural milieu. bicultural competence has been described as being able to balance multiple experiences (dominant and non-dominant forms of cultural capital) and to negotiate different physical and racial borders without compromising the essence of one’s home/cultural identity. carter (2006) defines the term cultural straddlers as african american students who operate biculturally in high school. carter differentiates cultural straddlers from cultural mainstreamers; the former group embraces their own cultural and racial identity, whereas the latter group appears to be fully assimilated into white culture and ideology. cultural straddlers are also different from non-compliant believers, who reject the premise of straddling two cultures and prefer to operate almost exclusively within their own cultural and racial identity. black students who adopt a bicultural identity and become cultural straddlers are said to embrace and value of both cultures and to negotiate across them with ease. they use both non-dominant and dominant forms of cultural capital to negotiate the education system. therefore, cultural straddlers are considered to be best positioned to maintain academic success. phinney and devich-navarro (1997) suggest that biculturalism is a way for blacks to manage racism, achieve academically, and maintain a strong sense of group identity. carter (2006) reasons that a bicultural identity enables black students to overcome racially hostile environments. grounded in research conducted over a hundred years ago (e.g., du bois, 1903/2003; woodson, 1933/2006), contemporary researchers have vacillated between the notion that marginalized students overwhelmingly practice biculturalism and that to the degree that they experience identity confusion, racism, classism, sexism, and other forms of marginalization, they are forced to conform to a dominant u.s. identity, sometimes at the expense of their original ethnic identity (harris & marsh, 2010). an opposing view of biculturalism dismisses the notion that it is a healthy fusion of two identities—the ethnic/home cultural identity and the dominant mainstream identity—and highlights the fact that some students sacrifice their ethnic identities in order to achieve academically and progress through the education system. for example, some researchers (e.g., rudmin, 2003; vivero & jenkins, 1999) have reported that biculturalism can be maladaptive, leading to stress, isolation, and anxiety due to constant pressure to choose between being more like the dominant culture or true to one’s own ethnicity. rowley and moore (2002) note that biculturalism could lead to identity confusion mcgee biculturalism and stem journal of urban mathematics education vol. 6, no. 2 27 and a sense of resentment toward feeling obliged to operate in two cultures, with the dominant mainstream culture viewed as the ideal. moreover, researchers who study african american racial identity recognize the contextualized ways identity and race often operate, which may complicate notions of adopting white mainstream culture as a path to achieving academic and career success (rowley, & sellers, 1998; sellers, morgan, & brown, 2001; sellers, smith, shelton, rowley, & chavous, 1998). in their research on the biculturalism and academic achievement of african american high school students, rust, jackson, ponterotto, and blumberg (2011) determined that cultural identity and academic self-esteem are important factors for academic achievement, but that biculturalism is not. similarly, black high school students who perceive a conflict between their second-class racialized status and their high academic achievement may experience internal strife (mcgee, 2013a). however, as schwartz and unger (2010) suggest, in some situations and contexts it may make the most sense for students to behave and think in ways that are consistent with the dominant culture. other theorists have investigated the relevance of racial/cultural identity in unpacking the academic achievement of african american students (e.g., du bois, 1903/2003; arroyo & ziegler, 1995; murrell, 2009; nasir, mclaughlin, & jones 2009). mcgee and martin (2011) provides evidence that black stem college students can be high achievers in their respective fields but often at a high psychological cost, due to racial stereotyping and other forms of bias. therefore, adopting a bicultural identity may not be enough to enable high-achieving students to fend off negative racial stereotypes fueled by inequitable academic, environmental, and social conditions as they attempt to survive in an education system that perpetuates an ideology of racial inferiority (warmington, 2009). the literature on the mathematics education of african american high school students is still largely silent about what happens to students who succeed academically in mathematics and mathematics related fields using a bicultural strategy to negotiate their success. exploring biculturalism the larger study from which this case is extracted explored the college and career aspirations of 24 mathematically talented black youth who attended “urban” high schools in chicago. the focus of the work was to gain a healthier and more grounded understanding of the strategies and competencies these students used to negotiate their environments and to achieve academically. motivated by 12 students’ narratives, focusing in particular on maurice’s counter-narrative, additional questions were explored: mcgee biculturalism and stem journal of urban mathematics education vol. 6, no. 2 28 1. how did biculturalism and/or the adoption of bicultural competencies operate in mathematically high achieving high school students who pursue stem internship/employment opportunities that are predominately white? 2. what role if any does early exposure to stem subjects and high achievement in mathematics play in students’ college decision-making? 3. how do the students make sense of their future college and career goals? this study adds yet another layer to the pros and cons of altering one’s character or appearing to be transformed in order to climb the career (and academic) ladder. the study also seeks to explicitly differentiate the internalization of american middle-class values from adoption of these values without internalizing them, and thus using them, as maurice’s interview revealed that his mother instructed him, “to beat them [whites] at their own game.” acknowledging students’ abilities to operate in ways considered markers of success (e.g., the ability to socialize with white employers, high achievement in stem classes) while suppressing critical components of their cultural/racial identity (e.g., having to lie about their mothers’ employment status to avoid reinforcing stereotypes about black women, smiling and agreeing with derogatory and racist/gendered comments about black boys) could contribute to a more structural, systemic, and comprehensive explanation of the educational challenges faced by african american high school students (murrell, 2009), who are academically poised for a future in the stem fields. the participant in preparing to interview mathematically high-achieving black high school students who reside primarily in chicago’s lowand mixed-income neighborhoods, i gathered recommendations from mathematics teachers, who identified their top five to seven students, including those who scored well on traditional measures of academic and mathematics success. i conducted interviews with 24 of these students; all had a gpa of 3.0 or higher in mathematics courses. the interviews averaged 65 minutes each and were semi-structured. because these students were operating in predominantly black environments, neighborhoods, and schools, my assumption was that there would be minimal discussion of the role racial stereotypes and other forms of racialized bias play in their lives. however, the coding and analysis showed that the exposure to stem summer internships at predominantly white businesses gave students their first entry into managing white spaces, as their prior career and academic experiences required functioning in mostly black educational spaces. thus these new more racialized situations provided insight into the hierarchical racial and cultural ideologies that exist within these structures. in those discussions, a subset of 12 students shared the experiences they had encountered within predominately white spaces, which were primarily stem internships or visits to stem businesses. mcgee biculturalism and stem journal of urban mathematics education vol. 6, no. 2 29 however, 11 of the 12 students in this study were not able to offer much perspective on their ability to function within predominantly white contexts because of their limited opportunity to operate in such spaces. according to the high school’s mathematics teachers, mathematics coach, and principal, maurice was the student who appeared most familiar with negotiating both black and white domains, having four stem summer and afterschool internships since middle school. however, the other 11 students’ experiences in those spaces strongly suggest that maurice’s narrative was similar to what theirs would have been, had they participated in greater numbers of stem internships, visits, meetings, and speaking engagements in predominantly white environments. the methods a phenomenological frame was applied in conjunction with an augmented form of life-story interview (mcadams, 2008, 2009). the philosophy underpinning phenomenology—in-depth description of particular phenomena or the appearance of these things as lived experience (milner, 2007)—helped in designing a research strategy that flowed directly from the research questions and goals of this study (patton, 2002; van manen, 1997). the students’ lived experience, influenced by their internal perceptions and identities and the external context, helps give meaning to students’ perceptions of a particular phenomenon. life-story interviews were ideal for gaining an understanding of these students’ mathematics and life experiences at various critical points in their development (mcadams, 2008, 2009). life-story narrative allowed for chronology, and enabled me to conduct an analysis focused on how elements were sequenced, how the past shaped perceptions of the present, how the present shaped perceptions of the past, and how both shaped perceptions of the future (reissman, 1993). my initial research design included 12 of the 24 students who discussed their racialized experiences in relation to stem as a result of being in predominantly white contexts. i sought to provide space for these students to articulate their viewpoints individually and within the larger paradigm of educational achievement. the aim was to better understand these students as mathematically high-achieving african american high school students attending urban schools in the same city. similar findings across all participants had the potential to extend or challenge previous biculturaland achievement-related theories. the first reading of the transcripts revealed one striking component: the divide between the participants’ stellar mathematics achievements and their intended college majors. i first coded for the rationale behind their chosen majors and careers, and then went into the body of each interview and coded for situations that helped to explain their rationale for majoring in non-stem college majors. after a second and third reading of the transcripts, however, i realized that only one of these 12 students, maurice, had discussed in depth a host of experiences mcgee biculturalism and stem journal of urban mathematics education vol. 6, no. 2 30 where he accomplished success in dominant spaces by, as he expressed it, “playing white.” i chose to focus on maurice in this iteration of the study because of his expressed ability to negotiate a number of environments dominated by whites and while remaining authentic and successful in black contexts. he dealt with racist language, mannerisms, questions, and dialogue, but, nonetheless, was admired by both peers and white and black educational and occupational stakeholders for being successful. for example, another participant in the larger study said that maurice was “just as much admired by super duper white folk as [by] his own boys…and he’s still black.” thus, i continued my data analysis by examining maurice’s videoand audio-tapes, the written transcripts, and his demographic questionnaire. the analysis my interpretation of the research data was ongoing, and the phenomenon of biculturalism and bicultural competencies appeared throughout all iterations of my data analysis. during the coding process, i extracted statements in which maurice addressed his past, current and anticipated future career trajectories, and then coded for experiences during which he said his identity was compromised. for example, he described a situation where he felt hurried into taking a leadership position; feeling as though he was being used as he mumbled, “a token and showpiece” for his high school. i coded for those types of reflections and subsequent decisions he made based on his perceptions of those situations. i applied additional sets of codes that focused on his choice of college major. various forms of racial bias were evident in his narrative. i coded for these influences, in particular to highlight how he made sense of these racialized experiences. once the interview was coded, coordinated, and rearranged, i conducted a thematic analysis (braun & clarke, 2006), the aim being to understand maurice’s social realities in a subjective yet scientific manner. after multiple iterations of identifying core consistencies and meanings, the categories and subcategories revealed several themes related to the aforementioned research questions. this analysis helped to organize the dataset and describe it in rich detail, and to interpret various aspects of the research study (2006). each theme captured important components of the data relative to the research questions and represented some level of patterned response across time periods of maurice’s life. as part of my assumption that a student’s beliefs and responses change as she or he develops, matures, and accumulates life experiences, i organized the results into three overarching academic periods of maurice’s life, resulting in three overlapping frames—k–8, high school, and future college outlook—taking into account changes as he developed and leaving room for new themes that might emerge. the future college outlook frame is where perceptions about his future identities and college aspirations presented a glimpse into maurice’s likely mcgee biculturalism and stem journal of urban mathematics education vol. 6, no. 2 31 future career choices. this analysis emphasized the importance of unpacking the human development of this mathematically high-achieving african american student in an urban high school who is negotiating predominantly white stem spaces. the findings maurice took issue with having to “play the game” by adopting policies and ideologies that were centered on white middle-class culture, although he seemed to have mastered that “game.” maurice had sufficient experience to be considered an effective cultural straddler (carter, 2006). he practiced a variety of bicultural competencies (e.g., code switching, appearing comfortable and proficient in both dominant and non-dominant settings, and watching tv shows and reading newspapers that made him knowledgeable about both cultural contexts). meeting the challenges of appeasing both groups, however, caused him great anxiety and apprehension. despite being somewhat of an expert at adapting and imitating white mainstream behaviors and mannerisms, maurice, in fact, made a considerable effort to resist embracing a bicultural identity. in some ways, his story counters the notion that a bicultural identity and competencies provide an optimal path toward postsecondary academic success, particularly in the stem fields. the results of this study are presented concisely through maurice’s interview; however, his story is not told in isolation. of the 24 mathematically highachieving high school students who participated in this study, only two indicated a desire to pursue a stem-based college major. maurice is just one of the many students in this study who was a consistent high achiever in mathematics and demonstrated the presence of bicultural competencies. however, as discussed below, he had already decided to opt out of a stem path in college and career. maurice was a senior in high school at the time of the interview. he was said to have a “dynamic personality” and an “ability to connect with people of very different backgrounds,” two comments taken directly from a mathematics teacher and the mathematics coach at his high school, respectively. maurice was known for having innovative bicultural competencies. he did not “totally” deplete himself of black dialect, style, behaviors, and mannerisms, thus maurice learned when to use his black cultural capital to his advantage (carter, 2003). he was able to learn from and communicate effectively with a variety of people without compromising his affinity for his own black culture. maurice began the interview with vivid memories of his childhood and early schooling. he was grateful that his parents, married over 20 years, transferred him from a “bad” neighborhood school to a predominantly black catholic school. during maurice’s k–5 years, his family struggled financially, but they later achieved what he defined as middle-class status: “two cars, nice house, summer camps, and family vacations.” maurice briefly discussed his responsibility as a mcgee biculturalism and stem journal of urban mathematics education vol. 6, no. 2 32 role model for his younger brother and spoke extensively about his middle school and high school years, praising his teachers, parents, and mentors for recognizing his talent in mathematics and his leadership skills. he commented that he refined his english language and speaking skills in middle school when he attended a culturally affirming poetry and rap afterschool program. he believed this program helped him to learn how to code switch, a term he was already familiar with and described as, “masking my blackness while wowing them with their own tools.” maurice attended several stem-based afterschool programs and summer camps and secured a paid internship as a civil engineer in training, working with a predominantly white civil engineering company that provided environmentally friendly building and bridge design services. i concluded the interview by asking maurice a few questions about his college and career goals. based on my impressions of his talents and opportunities in the civil engineering field, i was convinced that maurice would choose engineering as a major (even though i was not supposed to draw such conclusions). to my surprise, when i asked him what type of career he would pursue, he replied, “i want to be a football player.” all my training on appearing objective and non-biased evaporated and my jaw dropped. maurice, being keen on interpreting the behavior of traditionally educated persons, acknowledged the change in my behavior with this poignant comment: you see, doc, it’s really not just about the football. if i become a football player, i can get 8 tattoos, have 3 baby mommas, and walk around [with] an entourage of niggas, and throw big house parties in my mansion. i can do all of that and no one is going to question my abilities and talents on the football field. further, they are going to allow me to engage in all of that behavior, maybe even encourage it, as long as it does not affect my game. now, if i went into civil engineering and i got my phd, let’s say i even got two phds, every day, someone somewhere is going to question and challenge my ability to do my job. sometimes it will be subtle, others times it will be in my face, but it will always be there. that i know. maurice explained that he had been thrust into situations where he became a victim of racial stereotyping and other forms of racial bias. for example, he recalled being treated liked an anomaly when a member of a prestigious scholarship committee said to him, “your race should be proud of such a well-behaved, wellmannered young man like yourself.” in another instance, maurice was assumed to fit the stereotyped of black male underachievement and deception when a substitute mathematics teacher accused him of cheating because he scored 100% on a quiz. after the principal vouched for maurice’s intellectual ability and integrity, the substitute teacher admitted that she had never taught a black male who “was able to get a perfect score.” although maurice attended a predominantly black urban high school, he was constantly being told that he had to work twice as hard academically and three times as hard in engineering because his future was going to be full of low expectations about his abilities in that field. his two summer in mcgee biculturalism and stem journal of urban mathematics education vol. 6, no. 2 33 ternships at an engineering firm, along with his participation in afterschool stem programs that were sponsored and conducted through a fortune 500 company, illustrated the spoken and unspoken racial dynamics in a stem workplace. the companies maurice visited and interned with had similar racial employment dynamics: non-u.s.-born engineers, white managers, black female secretaries, and, according to maurice, “if i’m lucky, a black [male] janitor.” maurice went on to say that his frequent and seemingly constructive experiences in stem-related workplaces actually did him more harm than good. in his words, “i’ve seen too much, i know too much.” although maurice clearly was skilled at exhibiting bicultural behaviors, had exceptional mathematical ability, and was seen as a prized engineering intern, he did not care for what he experienced in the context of his internships. furthermore, he did not want to spend his career in a field where proving himself or faking his identity would be a normalized daily task. as maurice made sense of his stem experiences, he decided that the multitude of racial stereotypes and other forms of bias were “too much” and that he wanted to carve out a more racially affirming career path. it was not that he did not like engineering, but rather, as he concluded, “i just don’t like [engineering] enough to put up with the bs.” maurice this racialized narrative to discuss his rationale for not accepting a third summer internship at a local engineering company: the indian and some of the white engineers in my group section rarely spoke to me, unless my manager was around. then they [the engineers] treated me extra nice, which disgusted me even more. most of the engineers hate the managers too. all the managers are white males, except for one white lady. now you would think that we [the one white female manager] would get along because they treat her kinda like they treat me [invisible], but she treats me like i’m a nuisance. the white managers treat me like an affirmative action, token negro…all they talk about is my potential. i could design a bridge system that could save the company a hundred thousand dollars and they would still be talking about my potential. later in the interview, maurice admitted that it was not his desire to throw big parties and have lots of “baby mommas,” or tattoos, although he did enjoy hanging out “with [his] boys.” maurice wanted to follow in his father’s footsteps and be “married forever” and have two children. he used the stereotypical black male pro football player example merely to make a point about finding a career in which racial and gender stereotypes would not impede other people’s perceptions of his abilities. at this point in his development, maurice appears unwilling to deal with the trauma that he faced and the distress he expects to occur regularly in the engineering workplace, where being both black and proficient in engineering is deemed a suspicious exception rather than the rule. many of the other participants in the study expressed interest in careers where they saw examples of successful blacks, such as doctors, rappers, or music producers, or in fields they per mcgee biculturalism and stem journal of urban mathematics education vol. 6, no. 2 34 ceived as serving black communities, such as social workers, psychologists, and civil service workers. these choices reflect the results of beasley’s (2011) study, which documented the experiences of black students who graduated from elite colleges, finding that their career trajectories were positioned toward less prestigious employment in fields known to directly address social and racial inequities, such as education, social work, and community and nonprofit organizing. biculturalism: unintended consequences and flawed strategy okay, instead of asking us [black students] to get good at playing the game, ya’ll should be questioning, why do we have to play the game in the first place? —maurice maurice practiced biculturalism in ways that both created opportunities (e.g., building stem competencies, having increased stem exposure, and receiving financial compensation) and created a culturally repressive set of identity constraints (e.g., feeling the need to master white ways of being). members of historically underrepresented groups often grappled with how to succeed in fields that are riddled with real and perceived racial, ethnic, and gender stereotypes. because of this struggle, maurice decided against a career in engineering, which he felt would obligate him to yield more to the dominant way of being than he was willing to give. he described this dominant ideology as, “white ways of thinking, behaving, and doing. …i’ll probably have to even start dreaming in white if i want to keep up the façade.” although he had developed the skill set needed to be competent in the field if he so desired, maurice was unwilling to risk compromising his more culturally encouraging identity. it would be difficult to conclude from this data that biculturalism is a necessary skill for negotiating early stem career opportunities; however, it would be safe to say that being bicultural, for maurice, was an essential but identity-compromising competency that troubled his aspiration to choose a stem field as a viable career option. (of course, it is important to note that maurice’s feelings documented here captures but a brief moment in his life, much can happen that might change his future career aspirations.) although maurice was quite savvy about moving in and out of mainstream culture, he did so with silent resistance and quiet trauma as he gravitated toward a life in which his own culture would be validated. maurice seemed to conform to mainstream behaviors and ideologies only as a strategy for achieving educational and employment success. his choice of football as a career was undoubtedly risky, as it is more likely that maurice could become a civil engineer than a professional football player. in 2011, there were roughly 1,119 black professional football players versus 128,042 blacks employed in four types of engineering ca mcgee biculturalism and stem journal of urban mathematics education vol. 6, no. 2 35 reers—in other words, there are approximately 114 times as many black engineers as black professional football players (national action council for minorities in engineering, 2011; plunkett, 2012). even though he excelled in mathematics and negotiating white stem spaces while in high school, both of which were viewed as dominant cultural markers and forms of cultural capital, maurice’s identity was primarily embedded in his ethnic and racial cultures, making football, in his eyes, a mentally healthier option to preserve his “good sense.” maurice’s experiences would be hard to quantify on a measure or survey. it would be fairly easy to attribute his career decision-making to a lack of appropriate college and career counseling, if it were not for the extensive career and college counseling his high school provided. maurice’s parents and high school were very skillful in negotiating educational and scholarship opportunities, particularly for students like maurice who exhibited talent in mathematics and science. maurice’s early and repeated exposure to the stem fields and courses was initiated and nurtured by his parents and enhanced by multiple forms of support from the high school. maurice’s narrative should lead us to wonder how many other black or historically marginalized high school students who are exposed to and, more importantly, who succeed in the world of stem early in their education are turned off by the multitude of racial put-downs and bias they encounter. this issue applies particularly to students like maurice, who are able to function and thrive in stem settings through their skill in stem disciplines and their bicultural competencies but may be troubled by what they experience. his narrative troubles the definition of biculturalism, which often is framed as belonging to two or more cultures (mok & morris, 2012), as opposed to belonging to non-dominant culture and feeling obligated to pretend to belong in a dominant culture (acevedopolakovich, quirk, cousineau, saxena, & gerhart, 2014). this understanding of biculturalism implies that black students, through part-time assimilation, can sustain educational ideologies and behaviors that perpetuate the dominant power structure in our society (boyles, carusi, & attick, 2009). for black students who are persuaded to adopt bicultural competencies and behaviors, assumptions are made about what they know about negotiating white educational spaces, as if adopting a bicultural identity simply means having mainstream cultural values deposited into black students’ culturally empty brains (freire, 2005). biculturalism as a strategy for black students adopt to achieve success requires closer examination in terms of the particular ways the dominant culture stigmatizes historically underrepresented groups, which puts them in danger of experiencing multiple forms of disadvantage and marginalization. du bois (1903/2003) described the trouble associated with black people experiencing double-consciousness as two warring souls, one black and one white, where the white soul remains privileged and normalized as ideal, which problematizes the mcgee biculturalism and stem journal of urban mathematics education vol. 6, no. 2 36 freedom black students have in resisting biculturality. biculturalism mandates that black students co-opt educational ideologies that stress individualism, competition, and a host of culturally mainstream policies and practices considered more compatible with the values that are valued by white mainstream culture (bonilla-silva, 2001). as a result, some black students are left with the impression that their original cultural identities are deficient and pathologic and they learn to think of their race and culture as inferior (boyles et al., 2009). those who challenge this narrow framing may also resist careers that are strongly associated with maintaining dominant cultural capital, such as stem careers. this was, in fact, an unintended consequence of introducing maurice to an early stem career trajectory. concluding thoughts some high-achieving black students may reject biculturalism and other forms of assimilation and thereby reject an educational system that ignores or dismisses their cultural identities. denouncing biculturalism, however, may block those students from opportunities to learn and succeed in college. maurice’s resistance to being stereotyped and marginalized, in part, determined his projected post-secondary career choice as a nfl professional football player, where he has .215% chance of success. as a result of his racialized experiences with his stem summer and employment opportunities—which were deemed successful from the perspective of his school principal, mathematics teachers, mathematics coach, and the stem employer—maurice very well may be one of the many mathematically high achieving students opting out of stem. maurice acted out bicultural competencies without embracing biculturalism as an identity. i believe experiencing stem through these biased contexts forced him to become more aware of the racial inequities in this country that create and perpetuate the normalized racial abuse faced by many marginalized yet high achieving students. mathematically high-achieving black high school students who disengage from pursuing stem college and employment opportunities represent a loss of potential and talent for the stem workforce, which can leave these students dodging from prosperous and innovative career pathways, despite being fittingly qualified for them (beasley, 2011). maurice’s story speaks to a larger demand echoed by some critical educators that stem education and careers be altered in ways that respect and appreciate the intellectual and humanistic qualities of all individuals (akins, 2013; strayhorn, 2013). thus, i argue that that black students like maurice will continue to resist working in fields where they must perform in ways that honor the mainstream and subjugate their own culture and identities, which could manifest by foregoing racially insensitive or hostile college majors and career fields. my hope mcgee biculturalism and stem journal of urban mathematics education vol. 6, no. 2 37 is that maurice’s story will serve as a reminder that great and seemingly honorable efforts to make stem experiences equitable for marginalized students are insufficient. addressing the crisis of blacks and underrepresented students in the stem fields must extend beyond the racialized nature of academia and progress to the stem workplace. references acevedo-polakovich, i. d., quirk, k. m., cousineau, j. r., saxena, s. r., & gerhart, j. i. 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(2011). k–16 and beyond: african american male student engagement in stem disciplines. journal of african american males in education, 2(1), 1–5. journal of urban mathematics education december 2013, vol. 6, no. 2, p. 86 ©jume. http://education.gsu.edu/jume journal of urban mathematics education vol. 6, no. 2 86 reviewer acknowledgment january 2012–december 2013* nathan alexander, teachers college, columbia university dan battey, rutgers university joanne becker, san jose state university robert berry, university of virginia lecretia buckley, jackson state university carrie chiappetta, stamford public schools haiwen chu, wested teresa dunleavy, university of san diego indigo esmonde, university of toronto gheorghita faitar, d’youville college lidia gonzalez, york college, cuny jessica hale, georgia state university jacqueline hennings, griffin regional educational service agency jennifer jones, rutgers university brian lawler, california state university, san marcos maxine mckinney de royston, university of california berkley james telese, university of texas, brownsville la mont terry, occidental college anita wager, university of wisconsin, madison morgin jones williams, georgia state university candace williams, dekalb public schools desha williams, kennesaw state university khoon wong, national institute of education, singapore special issue guest editors and open reviewers volume 5, number 1 spring/summer 2012 proceedings of the 2010 philadelphia and 2011 atlanta benjamin banneker association conferences – beyond the numbers erika bullock, university of memphis nathan alexander, teachers college, columbia maisie gholson, university of illinois, chicago volume 6, number 1 spring/summer 2013 privilege and oppression in the mathematics preparation of teacher educators (prompte) david stinson, georgia state university joi spencer, university of san diego * note: during a website update that took place over the summer months of 2013, several newer members’ user ids were inadvertently eliminated. if you were a reviewer for a manuscript during january 2012– december 2013 and your name and affiliation is not listed, please contact david stinson at dstinson@gsu.edu (the update rolled over the review sent but not the reviewer’s id). we apologize in advance for the omission; a correction to the list will be made. it is our sincere intent to recognize all those who generously give of their time and expertise to the continued success of jume. http://education.gsu.edu/jume mailto:dstinson@gsu.edu i am from… journal of urban mathematics education december 2012, vol. 5, no. 2, pp. 53–54 ©jume. http://education.gsu.edu/jume tamika n. ball is a mathematics teacher at therrell educational complex, school of health sciences and research, atlanta, ga and a first-year doctoral student at georgia state university. she is the 2012–2013 atlanta public schools district teacher of the year; her research interest includes the implementation of instructional technology in the urban mathematics classroom. public stories of mathematics educators i am from… tamika n. ball therrell educational complex school of health sciences and research i am from the ear piercing southern bell pearl wearing south i am from curtsey practicing debutant balling church going any given sunday suburban world i am from village raising lady like displaying surroundings i am from toe-touching school spirit loving high volume talking atmospheres i am from single parent living midnight studying straight “a” making higher learning households i am from nightly homemade biscuit making interior decorating walls i am from day dreaming hopeful can’t stop wanting until i get it lifestyles i am from learning more when you listen and treating others the way you want to be treated living i am from live by example someone’s always watching you, so be on your p’s and q’s environment i am my environment, i live my environment, i am from here can’t you see well sure mrs. ball we see, the pearls, the “tude,” you reaching for the stars, but do you see me? i am from the hood wearing, tight leggings, walking out my mother’s house without a head turning home i am from late waking, no book carrying, asking for a pencil, yes a pencil, 180 days out of the year bright star ball public stories journal of urban mathematics education vol. 5, no. 2 54 i am from calm your neck phrasing, curse sailing, hallway skipping except your class “cause” you’re cool school i am my environment, i live my environment, i am from here can’t you see well sure bright star i see, the style, the “tude,” you acting like you don’t really want it but do you see “we”? we are from afternoon tutoring, cornell note writing, math i, ii, and iii passing reality we are from repeated practicing, white board relay winning, tic-tac-toe “matho” champion class we are from reward card winning, homework pass receiving, online assignment doing before mrs. ball calls my momma again order we make our environment, we live our environment, we are from here can’t you see journal of urban mathematics education december 2015, vol. 8, no. 2, p. 127 ©jume. http://education.gsu.edu/jume journal of urban mathematics education vol. 8, no. 2 reviewer acknowledgment january 2014–december 2015 * nathan alexander, san francisco state university dan battey, rutgers university joanne becker, san jose state university clare bell, university of missouri–kansas city robert berry, university of virginia angela brown, piedmont college joan bruner-timmons, miami-dade county public schools lecretia buckley, jackson state university patricia campbell, university of maryland college park susan cannon, georgia state university robert capraro, texas a&m university carrie chiappetta, stamford public schools ervin china, georgia state university marta civil, university of arizona teresa dunleavy, vanderbilt university indigo esmonde, university of toronto gheorghita faitar, d’youville college mary foote, queens college, cuny susan gregson, university of cincinnati jessica hale, georgia state university victoria hand, university of colorado boulder shandy hauk, wested crystal hill, indiana university-purdue university indianapolis keith howard, chapman university jennifer jones, rutgers university rick kitchen, university of denver brian lawler, california state university, san marcos percival matthews, university of wisconsin–madison maxine mckinney de royston, university of pittsburgh eduardo mosqueda, university of california, santa cruz angiline powell , university of memphis mary raygoza, university of california, los angeles laurie rubel, brooklyn college, cuny james telese, university of texas at brownsville luz valoyes chávez, university of missouri anita wager, university of wisconsin– madison erica walker, teachers college columbia university candace williams, dekalb county school district desha williams, kennesaw state university morgin jones williams, georgia state university khoon wong, national institute of education, nanyang technological university, singapore maria zavala, san francisco state university * note: during a website update that took place over the summer months of 2013, several newer members’ user ids were inadvertently eliminated. if you were a reviewer for a manuscript during january 2014– december 2015 and your name and affiliation is not listed, please contact david stinson at dstinson@gsu.edu (the update rolled over the review sent but not the reviewer’s id). we apologize in advance for the omission; a correction to the list will be made. it is our sincere intent to recognize all those who generously give of their time and expertise to the continued success of jume. http://education.gsu.edu/jume mailto:dstinson@gsu.edu microsoft word 4 final turner et al vol 9 no 1.doc journal of urban mathematics education july 2016, vol. 9, no. 1, pp. 48–78 ©jume. http://education.gsu.edu/jume erin e. turner is an associate professor in the department of teaching, learning and sociocultural studies in the college of education, at the university of arizona, 1430 e. second street, tucson; email: etruner@email.arizona.edu. her research interests include issues of equity and social justice in mathematics education, and preparing teachers to teach mathematics with students from culturally and linguistically diverse backgrounds. mary q. foote is an associate professor in the department of elementary and early childhood education at queens college, cuny, powdermaker hall 054w, 6530 kissena blvd., flushing, ny 11367; email: mary.foote@qc.cuny.edu. her research interests fall broadly within issues of teacher knowledge. more specifically, her interests are in teachers’ understanding of students’ cultural and community knowledge and practices, and how they might inform equitable mathematics teaching practice. kathleen jablon stoehr is an assistant professor in the department of education at santa clara university, 500 el camino real, santa clara, ca, 95053; email: kstoehr@scu.edu. her research interests include issues that relate to preservice and early career teachers’ processes and understandings of learning to teach related to equity and social justice including language, race, culture, and gender. learning to leverage children’s multiple mathematical knowledge bases in mathematics instruction erin e. turner university of arizona mary q. foote queens college, cuny kathleen jablon stoehr santa clara university amy roth mcduffie washington state university tri-cities julia maria aguirre university of washington-tacoma tonya gau bartell michigan state university corey drake michigan state university in this article, the authors explore prospective elementary teachers’ engagement with and reflection on activities they conducted to learn about a single child from their practicum classroom. through these activities, prospective teachers learned about their child’s mathematical thinking and the interests, competencies, and resources she or he brought to the mathematics classroom, and then wrote reports that included instructional suggestions as to next steps to further the child’s growth in mathematics. the authors’ analyses of these reports indicate that there were a variety of ways which prospective teachers made connections to one or more of their child’s knowledge bases. in a high percentage of cases, prospective teachers attended to one of these knowledge bases, indicating that they were attending to particularities about their child and developing the dispositions to continue to do so. implications for research and practice are discussed. keywords: children’s funds of knowledge, children’s mathematical thinking, mathematical tasks, prospective teacher education turner et al. children’s multiple mathematics journal of urban mathematics education vol. 9, no. 1 49 quipping prospective teachers (psts) with the necessary strategies and tools to meet the mathematics learning needs of today’s diverse student populations is critical (leonard, 2008; turner et al., 2012). research suggests, for example, that to support student learning, teachers, both prospective and practicing, need to build connections with their students, families, and communities, and to draw on these connections in their mathematics teaching (civil, 2007; ensign, 2005; ewing, 2012; gay, 2010; lipka et al., 2005; matthews, 2003; meaney & evans, 2013; turner, celedón-pattichis, & marshall, 2008; vomvoridi-ivanović, 2012). this building includes drawing on knowledge of children’s mathematical thinking (carpenter, fennema, peterson, chang, & loef, 1989) along with understandings about children’s interests and competencies, incorporating their cultural, home and community-based knowledges and experiences (civil, 2002; ladson-billings, 2009; gonzález, moll, & amanti, 2005). we refer to these multiple understandings and experiences that have the potential to shape and support students’ mathematics learning as children’s multiple mathematical knowledge bases (mmkb; turner et al., 2012). each of these areas (i.e., children’s mathematical thinking and children’s cultural and community-based knowledge and experiences) has received individual attention in research, but research in children’s mathematical thinking has rarely considered the familial and cultural funds of knowledge children bring to thinking about mathematics; conversely, research in children’s funds of knowledge has typically not focused in detail on children’s mathematical thinking. furthermore, research on how teacher preparation programs can support psts’ understandings and practices related to children’s mmkb is limited. this article describes how one research program, amy roth mcduffie is associate dean for research and external funding and a professor of mathematics education in the college of education, at washington state university, 2710 crimson way, richland, wa 99354; email: mcduffie@tricity.wsu.edu. she researches preservice and practicing teachers’ professional learning and development in and from practice. specifically, she focuses on mathematics teachers’ developing equitable instructional practices and their work with curriculum. julia maria aguirre is an associate professor of education at the university of washington tacoma, wcg 319 box 358435, 1900 commerce street, tacoma, wa 98402; email: jaguirre@uw.edu. her research interests include equity and social justice in mathematics teaching and learning, teacher education, and culturally responsive mathematics pedagogy. tonya gau bartell is an associate professor in the department of teacher education at michigan state university, 620 farm lane, east lansing, mi, 48824; email: tbartell@msu.edu. her research interests include equity in mathematics education and teacher education for social justice. corey drake is an associate professor and director of teacher preparation in the department of teacher education at michigan state university, 620 farm lane, room 116m, east lansing, mi, 48824; email: cdrake@msu.edu. her research interests include teacher learning from and about mathematics curriculum materials and supporting prospective elementary teachers in learning to teach mathematics to diverse groups of students. e turner et al. children’s multiple mathematics journal of urban mathematics education vol. 9, no. 1 50 teachers empowered to advance change in mathematics (teach math), engaged psts in mathematics learning case studies to support the psts in learning about and connecting to children’s mmkb in their plans for mathematics instruction. conceptual framework in this section, we outline research on psts’ orientations towards children and families from marginalized communities (e.g., immigrant communities, poor/ working class communities, communities of color), and discuss psts’ knowledge and practices related to connecting to children’s mmkb in mathematics instruction.1 psts’ (re)orientations toward children and families psts bring limited experiences with children and families from cultural, racial, and linguistic backgrounds different from their own (bleicher, 2011; silverman, 2010; taylor & sobel, 2001). furthermore, psts’ limiting beliefs and assumptions about children from marginalized backgrounds can undermine student learning (sleeter, 2001). some psts hold deficit-based notions of what students from diverse cultural and linguistic groups are capable of learning and should learn (artiles & mcclafferty, 1998; kidd, sánchez, & thorp, 2008) and have fears related to working with marginalized students and their families (bleicher, 2011). psts also tend to be unaware of social and educational inequities associated with race, class, and ethnicity, and this lack of awareness may lead psts to faulty conclusions related to students’ successes or struggles at school, particularly in mathematics (ensign, 2005; kidd et al., 2008). these orientations are widespread; they have been specifically noted about psts and practicing teachers working in urban contexts and are evident in teachers in a variety of teaching contexts both in the united states and internationally (chong, 2005; planas & civil, 2002), and can be resistant to change (rodriguez & kitchen, 2005). scaffolded learning experiences in teacher education programs, however, can support psts in developing more positive, resource-based orientations toward children from marginalized communities (aguirre, zavala, & katanyoutanant, 2012; darling-hammond & mcdonald, 2000; kidd et al., 2008; turner et al., 2014). for example, research outside of mathematics education has found that conducting case studies of individual children can provide psts with opportunities to critically examine their own biases, and to “learn how to look closely at children, to see them as growing individuals, and to find ways to foster their learning” (darling-hammond & mcdonald, 2000, p. 42). by examining a specific student’s learning across 1 a more extensive review of prior research in this area is found in turner and drake (2015). turner et al. children’s multiple mathematics journal of urban mathematics education vol. 9, no. 1 51 home, school, and community contexts (i.e., through observations, interviews, and student work), case studies provide psts with opportunities to identify children’s strengths, progress, and learning needs (horowitz, darling-hammond, & bransford, 2005). in our own work, we have documented how mathematics methods course activities, including interviews with individual children, helped to (re)orient psts toward students from marginalized groups by focusing psts’ attention on the knowledge, skills, and competencies that these children bring to the classroom (bartell et al., 2013). collectively, this research suggests that teacher education programs may play an important role in supporting psts to explore their students’ home and community experiences in order to support mathematics learning. connecting to children’s mmkb in mathematics instruction learning related to children’s mathematical thinking. an increasingly prominent line of research in mathematics teacher education has examined teachers’ understandings and practices related to children’s mathematical thinking (i.e., children’s problem-solving strategies, connections between strategies and problem structures, common confusions). this work, which often draws on the cognitively guided instruction (cgi) research program (e.g., carpenter et al., 1989; fennema et al., 1996), has linked teachers’ knowledge of children’s mathematical thinking to productive changes in teachers’ classroom practices and student learning. for instance, fennema and colleagues (1996) found that as teachers learned about the development of children’s problem-solving strategies in specific content domains, they began to use this knowledge to inform instructional decisions (e.g., lesson planning, problem selection). in turn, students demonstrated significantly higher levels of achievement on problemsolving tasks (carpenter et al., 1989; fennema et al., 1996). jacobs, lamb, and philipp (2010) found that prior to coursework focused on mathematics teaching and learning, psts have a limited capacity for attending, interpreting, and responding to children’s mathematical thinking. yet, scaffolded learning experiences such as conducting and analyzing problem-solving interviews with children have been shown to further develop psts’ competencies (mcdonough, clarke, & clarke, 2002; philipp, thanheiser, & clement, 2002; philipp et al., 2007; sleep & boerst, 2012). more specifically, ambrose (2004) found that psts benefited from repeated opportunities to interview and interact with children about their reasoning, as psts’ beliefs about children’s problem-solving capacity are often resistant to change. yet for psts, knowledge about children’s thinking does not always transfer to instructional practices. vacc and bright (1999) found that although psts experienced significant shifts in their knowledge of children’s thinking across methods courses and student teaching experiences, there was little change in how they used this knowledge for instructional planning or teaching. in a study of how psts adapted mathematics tasks based on knowledge of students, nicol and crespo (2006) found few instances where psts’ adaptations were aimed at further exploring or connecting to children’s mathe turner et al. children’s multiple mathematics journal of urban mathematics education vol. 9, no. 1 52 matical thinking. research with practicing teachers has also indicated that connecting to children’s mathematical thinking in instruction is a complex teaching practice that takes time to develop (carpenter et al., 1989; fennema et al., 1996). learning related to children’s home, cultural, and community-based knowledge and experiences. research has also begun to explore teachers’ (and to a lesser extent psts’) understandings and practices related to children’s home, cultural, and community-based experiences and practices, or their funds of knowledge (civil, 2002, 2007; gonzalez, moll, & amanti, 2005). this work is supported by studies which have shown that historically underrepresented groups benefit from instruction that draws on their cultural, linguistic, and community-based knowledge (ladson-billings, 2009; lipka et al., 2005; turner, celedón-pattichis, & marshall, 2008). for example, civil (2007) illustrated that practicing elementary school teachers drew on children’s and family experiences with gardening to deepen students’ understanding of mathematical concepts related to measurement, area, and perimeter. turner and colleagues (2008) documented how bilingual kindergarten teachers used familiar storytelling-like conversations about family trips to the supermarket, classroom activities, or upcoming cultural celebrations to support students in successfully solving a range of basic word problems. taylor (2012) and wager (2012) conducted and studied a yearlong professional development focused on supporting elementary teachers’ efforts to connect school mathematics lessons to the mathematics that children used outside of school. initially, teachers connected lessons to students’ interests, or familiar out-of-school activities (e.g., finding the area of a soccer field because children play soccer), but not to the ways that children used mathematics outside of school. taylor (2012) argues that teachers’ tendency to connect first to familiar contexts, and only later (and with support) to ways that children and families use mathematics in these contexts suggests a possible trajectory in teachers’ practice. much less is known about how psts learn to connect to children’s out-of-school experiences in mathematics instruction (see turner & drake, 2015). in our prior work, we studied problem-solving-based mathematics lessons that psts created grounded on learning about mathematics in children’s communities (aguirre et al., 2013; turner et al., 2014). we found that psts often began lessons with traditional word problems that reflected familiar names and places from children’s neighborhoods, and that most psts found it challenging to make “consistent and substantive connections to [their students’] cultural funds of knowledge” (turner et al., 2014, p. 45). also relevant when considering how psts might connect to children’s experiences outside of school, is how psts conceptualize real-world connections in mathematics. this conceptualization is important as teachers’ beliefs and understandings about connecting school mathematics to situations, contexts, or activities outside of school influence how and whether they choose to make such connections in their teaching (lee, 2012; meaney, trinick, & fairhill, 2013). for example, in lee’s (2012) turner et al. children’s multiple mathematics journal of urban mathematics education vol. 9, no. 1 53 study of 71 k–8 psts, participants collected, created, and evaluated contextualized problems that they (the psts) believed reflected exemplary real-life connections. lee found that while psts thought that real-world connections could enhance student engagement and motivation, their vision for how teachers might include such connections in their mathematics instruction was limited to posing textbook-like problems that involved calculations with money or time. in other words, lee argued what psts think about real-life connections and what they do may not always coincide. one could conjecture that this gap may be even more pronounced in psts’ attempts to pose problems that connect not just to the real world but to specific contexts in students’ homes and communities. in summary, prior research has established that psts (a) can increase their understanding of children’s mathematical thinking by conducting problem-solving interviews with students; and (b) have an emerging capacity to learn about children’s interests, families, and communities, and to (re)orient themselves, generally, to the competencies that children from marginalized groups bring to the classroom. however, there remains much to be known about how psts leverage and integrate their emerging understandings about children’s mmkb as they plan mathematics instruction. to address this gap in the literature, we focus here on the participation of psts in a purposefully designed set of experiences with a single case study child—the mathematics learning case study—aimed at introducing psts to the practice of connecting to children’s mmkb in their mathematics teaching. in this analysis, we examine how psts used what they learned about their case study child’s mmkb (i.e., the child’s mathematical thinking, and her or his interests and home and communitybased experiences) to make suggestions for future mathematics instruction. we thus address the following research question: in what ways do psts draw on knowledge of children’s mmkb as they make instructional suggestions for their case study child? methods participants and context participants were 79 psts who were enrolled in a mathematics methods course as part of their teacher preparation program at one of five universities. 2 the five sites 2 the larger teach math project includes six university sites. data for this study were drawn from five of those sites (sites a, b, d, e, and f). for consistency among papers written about the project, we use those designations when referring to participants. the sixth site (site c) is not included here because data at that site were collected at a later time. turner et al. children’s multiple mathematics journal of urban mathematics education vol. 9, no. 1 54 reflected diverse geographic contexts (e.g., urban, suburban, borderland,3 and a mix of urban and suburban) and programs (e.g., variance in field placements and prior coursework). fifty-five of the participants identified themselves as white/european descent, 10 as hispanic/latin@, six as mixed ethnicity, five as asian american, and two as african american/black. one participant did not identify her or his race or ethnicity. the findings here focus centrally on work done by 59 of the psts who constructed contextualized problems for their case study children. of these 59 psts, 56 identified as women and three as men. forty-four of these psts identified as white/european descent, one as african american/black, three as asian american, six as hispanic/latin@, and four as mixed ethnicity. one pst provided no data about racial and ethnic background. (see table 1 for a listing of the racial and ethnic background of psts by site.) psts that identified their racial and ethnic backgrounds as other than white/european descent were more likely to speak a language in addition to english (71%4: 10 of 14 spoke another language), as compared to 41% (11 of 44) of the white/european descent psts. table 1 overview of psts and children by site number of psts racial and ethnic background site a (n = 8) site b (n = 17) site d (n = 9) site e (n = 13) site f (n = 12) white/european descent 4 15 3 11 11 hispanic/latin@ 2 0 3 0 1 african american/black 0 0 1 0 0 asian american 1 1 1 0 0 mixed ethnicity 1 0 1 2 0 no data 1 number of children child grade level site a site b site d site e site f k–1 2 8 2 5 2 2–3 2 7 4 2 4 4–5 4 2 3 2 5 6–8 4 1 3 by borderlands, we are referring to that zone in the southwestern united states that shares a border with mexico. students who live along that border often travel back and forth between mexico and the united states as frequently as once a week to visit family, make purchases, and so forth. they bring particular funds of knowledge to u.s. classrooms and so we identify that area specifically. although there are northern states that share a border with canada and so might be thought of as borderlands, it is not as typical for students who live along that border to travel back and forth between canada and the united states as previously described. 4 all percentages are rounded to the nearest whole number. turner et al. children’s multiple mathematics journal of urban mathematics education vol. 9, no. 1 55 the case study children with whom these 59 psts worked ranged from kindergarten to grade 8 (table 1 also lists children’s grade levels by site). while specific demographic information about case study children is not directly available, according to written reports from the psts, there were 25 hispanic/latin@ children, 13 white/european descent children, 11 african american/black children, seven mixed ethnicity children, two asian american children, and one native american/indigenous child. of these children, psts reported that 24 spoke english only, 21 spoke at least some spanish, five spoke another non-english language (e.g., hebrew, vietnamese), and the language background was unknown for nine of the children. overall, our participants were slightly more diverse in terms of race and ethnicity (but not gender) than national trends in the elementary school teacher population would predict (hollins & guzman, 2005). data sources data sources included three written reports that psts completed as part of the mathematics learning case study. psts were asked to focus throughout the semester on one child in their practicum classroom who was different from them in one or more sociocultural aspects (e.g., race, socioeconomic status, home language, etc.). psts were also encouraged to choose a case study child who struggled at least somewhat with mathematics. with the goal of supporting psts in learning about mmkb, psts interacted with and observed their case study child and then wrote reports based on these experiences. more specifically, the mathematics learning case study activities included a “getting to know you/funds of knowledge” interview in which psts talked with their case study child about their interests, home and community activities, and their experiences in school mathematics, and problem-solving interviews with the case study child in the areas of operations with whole numbers, fractions, or baseten concepts. psts produced written reports for both of these activities. psts also observed their case study child across multiple weeks, during mathematics lessons and at other times of the day, with the goal of learning about the child’s strengths and resources. (see appendix a or the project website https://teachmath.info for a more detailed overview of the mathematics learning case study components.) the mathematics learning case study culminated with a final report in which psts analyzed and reflected on interactions with their case study child across the semester and proposed appropriate next steps in mathematics instruction. data analysis the first two authors conducted data analyses with the assistance of the third author, a graduate research assistant. our goal was to see the extent to which psts turner et al. children’s multiple mathematics journal of urban mathematics education vol. 9, no. 1 56 learned about and drew on their case study child’s mmkb to make instructional suggestions in the final report. in the first round of analysis, we read through the datasets for the 79 participants and identified a total of 144 instructional suggestions and associated justifications. all psts included at least one instructional suggestion in their report, and some included multiple suggestions. instructional suggestions included: (a) particular participation structures that might support the child’s success, such as seating an english learner with another child who spoke their home language; (b) additional work in specific topics such as multiplication or measurement; (c) activities parents could engage in with their children outside of school; and (d) specific problems, both contextualized and bare number problems. given our research question, we were particularly interested in instructional suggestions that were embedded within a context (i.e., a word problem). these contextualized tasks presented opportunities to explore how psts attended to children’s mathematical thinking (e.g., via number choices or problem structures), as well as how they drew on children’s interests and their cultural, home and community-based knowledge and experiences. 5 of the 144 suggestions, 96, representing the work of 59 psts, were contextualized tasks. in the second round of analysis, we focused on the 96 contextualized tasks. using a process of analytical induction (bogdan & biklen, 2003), we coded each task and associated justifications along multiple dimensions that connected to key ideas in the literature.6 we attended to (a) orientations towards children’s strengths or learning needs (e.g., problems that built on children’s competencies) (foote et al., 2013; kidd et al., 2008); and (b) ways that psts justified task contexts, problem structures, and number choices (vacc & bright, 1999; land, drake, sweeney, johnson, & franke, 2015). we also coded instances when psts relayed specific knowledge about the case study child that they gathered as part of the mathematics learning case study. this included knowledge of the child’s (a) mathematical thinking (problem types or number ranges with which the child had been successful or unsuccessful), (b) interests, and (c) home or community activities (e.g., family budgeting and cooking practices, afterschool activities). we also identified instances where psts described ideas about children in general, such as ideas about objects, places, or activities that psts thought would be relevant or of interest to all children, or general knowledge about mathematical concepts or skills that were ap 5 although attention to issues of language strengths and needs of some students were noted in several reports, this was not the focus of any of the instructional suggestions involving contextualized problems and thus does not figure in our analysis. 6 it is important to note that our analysis was limited to the written products that psts produced as part of the mathematics learning case study. it is probable that psts had additional knowledge about their case study students that was not reported in written assignments. while psts may have considered a range of factors in their suggestions for future instruction, our analysis is limited to the explanations and justifications for those suggestions contained in their written reports. turner et al. children’s multiple mathematics journal of urban mathematics education vol. 9, no. 1 57 propriate for children of a particular grade level. we coded these instances as general ideas about children, to distinguish them from the knowledge that was explicitly linked to the case study child. we collaboratively coded a subset of the 96 contextualized tasks to establish consistency and to refine coding categories and definitions. we then each coded a small number of the tasks individually, and met to discuss discrepancies. we repeated this process, reconciling differences in coding and further refining code definitions until 85% intercoder reliability was achieved. we then each coded a third of the remaining tasks individually. in the third round of analysis, we looked across coding categories for themes (creswell, 2007), or patterns in how psts drew on specific knowledge about their case study child’s mmkbs as they planned contextualized tasks for future instruction. the initial sections of our findings are organized according to these themes. in the fourth and final round of analysis we examined disaggregated data for potential differences among sites. we compared categories of tasks posed by psts at each site and investigated how those differences related to other pst or child characteristics. we also disaggregated data by pst/child pairings and examined potential differences in tasks posted by white/european descent psts paired with white/european descent children (n = 8), white/european descent psts paired with non-white children (n = 36), non-white psts paired with white/european descent children (n = 4), and non-white psts paired with non-white children (n = 10). the final section of our findings reports the results of these analyses. findings we begin the findings by discussing briefly how psts integrated knowledge of children’s mathematical thinking as they developed contextualized tasks. we next discuss in more depth how psts integrated knowledge of their case study child’s experiences, interests, and practices into these instructional suggestions. we conclude with a discussion of similarities and differences in the tasks posed across sites and among psts and children of different racial and ethnic backgrounds. integrating knowledge of children’s mathematical thinking we found that as psts generated contextualized tasks for their case study child, a focus on children’s mathematical thinking was prominent (evident in 81 of the 96 examples: 84%). moreover, we found that psts leveraged their understandings about children’s mathematical thinking in ways that mirror what has been reported in prior research, with both prospective and practicing teachers (fennema et al., 1996; jacobs, lamb, & philipp, 2010; vacc & bright, 1999). for example, psts carefully selected numbers and problem structures based on the kinds of strategies and reasoning that children used during the interviews. psts also made deci turner et al. children’s multiple mathematics journal of urban mathematics education vol. 9, no. 1 58 sions about the kinds of tools or manipulatives they would make available to the child based on what they had learned about the child’s strategies and understanding of particular number relationships. for example, one pst [b322] began with a multi-step problem from the problem-solving interview about boxes of chocolates (i.e., sara has 3 boxes of chocolate with 4 pieces of chocolate in each box. then she eats 5 pieces of chocolate. how many pieces of chocolate does she have left?). she then modified the problem so that it was framed in terms of her second-grade case study child (michael) and his mother. additionally, as the pst was interested in supporting the child’s understanding of equal size groups and skip counting (something that she learned was difficult for the child during the problem-solving interview), she adjusted the problem structure so that it focused only on combining boxes of equal size. the new problem read: “michael has 5 boxes of chocolates and each box contains 3 pieces of chocolate. michael’s mother gives him 2 more boxes of the same chocolates. how many pieces of chocolates does michael have now?” these findings confirm what has been noted in prior research related to how scaffolded learning interactions with individual children around mathematical tasks can support teachers’ understandings and practices related to children’s mathematical thinking (jacobs et al., 2010; sleep & boerst, 2012). for this reason, we do not describe these findings in detail. instead, we focus on patterns related to how psts integrated understandings about other aspects of children’s mmkb (e.g., children’s interests, home and community experiences, etc.) as they planned contextualized tasks for future instruction. integrating knowledge of children’s interests, experiences, and funds of knowledge we found that psts drew on knowledge related to children’s interests, and their home and community-based experiences in four different ways as they generated contextualized tasks. psts based tasks on (a) assumptions about familiar or relevant contexts, not necessarily linked to the case study child; (b) specific knowledge of objects or activities that were familiar to the case study child; (c) mathematizations of family practices; and (d) ways that the case study child engaged in mathematics in home or community activities (see appendix b). we discuss these patterns in the next four sub-sections. category 1: assumptions about familiar or relevant contexts. psts frequently drew on assumptions about places, objects, and activities they thought would be relatable to children, including, but not necessarily specific to, their case study child. thirty-seven of the 96 contextualized tasks (representing the work of 32 psts) were categorized in this way, making it the most prominent category in our analyses. these tasks often resembled textbook-like word problems (e.g., one or two sentences that provided information and a question to be solved). frequently (in 20 of the 37 problems), psts began with a basic word problem structure and then replaced potentially unfamiliar or less relevant details (such as winter sports turner et al. children’s multiple mathematics journal of urban mathematics education vol. 9, no. 1 59 for children in a desert context) with objects or activities that the pst thought “all children would like” or “be familiar with” such as jellybeans, swings, pennies, and classroom activities and supplies (see problem (a) in appendix b). for example, a pst [a306] justified a problem (for a grade 2 student) about buying gumballs with pennies by explaining, “this problem is relatable to children since they have pennies and buy things occasionally, so this is a topic that they could really imagine happening.” also common were instances when psts adapted problems from the problem-solving interviews or from the textbook used in the child’s classroom (17 of the 37 problems), often with the justification that the contexts in these problems were already familiar to the child (see sample problem (b) in appendix b). often these adaptations were limited to changing the names of the characters in the problem. as one pst [f404] explained (about her problem for a grade 4 student), “my problem sets would be very similar in format to those … in the interview packet, adjusting the questions to make them more personal to her.” in one example, a pst [b313] adapted a multiplication problem (for a grade 1 student) about placing three stickers in each of four pockets so that it included the child’s name and easier numbers, moves which the pst felt would support the first grader’s understanding of the problem structure. other psts explicitly stated that they adapted problems they had seen in children’s mathematics textbooks because these problems were not only familiar to children but also reflected what children “will continue to see, and if [case study child] is not able to master it now, he [a grade 1 student] will continue to have trouble with math” [d404]. in summary, the most prevalent findings category contained tasks that did not necessarily reflect the experiences of the particular case study child, but instead were based on assumptions and general ideas about contexts that would be familiar or useful for all children. category 2: knowledge of familiar objects or activities. in other instances, psts drew on specific knowledge about their case study child’s interests or preferences (i.e., “favorites” such as bugs [a307], for a kindergarten student; softballs [f404], for a grade 4 student; or toys at the flea market [f408], for a grade 5 student) to generate problem contexts. thirty-one of the 96 contextualized tasks (representing the work of 25 psts) reflected these kinds of connections. most often, the problems resembled textbook-like word problems, similar to those previously discussed. in these cases, however, psts included contexts that they “knew” were of high interest to their case study child because they believed that connecting to children’s interests would enhance their engagement and motivation (e.g., “my child “lights up” every time i mention …” [b306], for a grade 2 student; “[this connection] helps her to focus on the problem more” [b323], for a grade 4 student; see sample problem (c) in appendix b). in another example, this pst generated a multiplication task for her fourth-grade case study child that involved calculating the number of boxes of macaroni and cheese in 14 containers that each held 8 boxes. turner et al. children’s multiple mathematics journal of urban mathematics education vol. 9, no. 1 60 the pst explained, “the object of macaroni and cheese will help her focus since that is one of her favorite foods” [b323]. while most problems in this category reflected attempts to connect to objects or people that were of high interest to the case study child, in a few instances, psts posed tasks that connected to familiar activities that arguably may include mathematical activity (e.g., collecting cans to recycle [f417], for a grade 5 student; playing video games [f414], for a grade 3 student; shopping at the mall with friends [a300], for a grade 2 student; and doing homework [b324], for a grade 2 student; see sample problem (d) in appendix b). for example, one pst [f402] knew that her first-grade child enjoyed watching and playing football, and argued: “by using football, he will better understand how to work [problems] out. … i am activating [child]’s schema and building on his prior knowledge of football to teach him math problems that he struggles with.” in general, these tasks reflected artificial scenarios that seemed forced on a context relevant to children (e.g., how many footballs would you have left if you had 11 and lost 3?) and did not connect to the mathematics that children might engage in as part of the activity (e.g., keeping track of scores, etc.). however, psts justified these problems by explaining that connections to familiar activities would help students “to picture what is going on in the word problem a little better” [a300] and thereby support students’ understanding. to further support their decisions, psts often drew on specific moments when they had witnessed increased interest and understanding from their case study child in response to these moves. this category contrasts with the instructional suggestions coded as falling into category 1, wherein psts made suggestions based on assumptions they made about the relevance of problem contexts. in the case of category 2 suggestions, psts began to draw on specific knowledge of the case study child they had gained through interactions with the child. category 3: mathematizing family practices. in other instances, psts constructed problems connected to activities in which the family of the case study child engaged (e.g., eating dinner, grocery shopping, doing laundry) and then considered how people might use mathematics as part of this activity. twenty-one of the 96 contextualized tasks (representing the work of 17 psts) were categorized in this way. a few tasks in this category focused on ways that parents could engage their child with mathematics as part of family activities (see sample problem (e) in appendix b). in a more elaborated and more realistic example, a pst [e405] suggested the following multi-operation task for a fifth-grade case study child: she shared with me that her mother is planning a quinceañera for a family friend. … perhaps she could become involved in the preparations for food, figuring out how much food would be needed to feed all of the party guests. she could create a shopping list of ingredients (including quantities) and determine how much it would cost. turner et al. children’s multiple mathematics journal of urban mathematics education vol. 9, no. 1 61 although the child did not specify the kinds of mathematics that her family might use as they prepared for this event, the pst identified mathematical concepts and practices that might be involved (e.g., scaling quantities in recipes for a specific number of people, operations involving rates and decimals to calculate total cost). in other examples, psts constructed tasks that teachers could implement that connected to family activities or practices (see sample problem (f) in appendix b). in this case, knowing that her fourth-grade case study child enjoyed eating out with his family, the pst [d412] constructed a division task involving equally sharing a pizza partitioned in eighths among four family members. although the child mentioned to the pst that he saw mathematics being used when he went out to dinner with his parents, it was unclear whether he was referring to the mathematics involved in partitioning and sharing food. that said, families may engage in this mathematical activity, and in this way this task reflects another attempt, as in the quinceañera example, to pose problems that connect to how case study children’s families might use mathematics in real-world situations. another pst [f415] drew on knowledge that the child’s mother operated a nail salon to construct a series of problems that involved using multiple operations to calculate the cost of various salon services. according to the pst, “[the child, a fifth grader] goes occasionally to help her mother at work, and she really helped a lot for the grand opening of the salon.” an example of a problem the pst thought would be familiar to the child is the following: there are 5 girl friends who want to go to the salon to get looking all pretty for their slumber party. one pedicure is $30, one manicure is $20, $5 for just paint, and massage is $20. two girls want to get one pedicure and manicure each. one girl wants to get a pedicure, manicure, and a massage and the other two girls want to get a pedicure and a massage each. how much money total will the salon make when these girls go? the pst explained that this problem would encourage the case study child to use multiple strategies, including mental calculations, versus always relying on a standard algorithm. the pst noted that mental math strategies are important for working in the salon, as “[the child’s mother] can’t always depend on the calculator because nail salons can get pretty busy, so she should be able to tell her customers right away the final price.” in summary, in this category psts focused both on suggestions for how parents and families could connect to mathematics in their daily activities and on attempts to mathematize family practices outside of school for use in the classroom. across these examples the psts’ goal seemed to be “layering” mathematics onto known family activities. in this way, the examples in this category, as with category 2, contrast with tasks in categories 1 where psts simply inserted contexts that were assumed to be familiar or of interest. turner et al. children’s multiple mathematics journal of urban mathematics education vol. 9, no. 1 62 category 4: identifying mathematics in activities in which the child engages. in a small number of examples (7 of the 96 tasks representing the work of 6 psts), psts selected task contexts that related to their case study child’s activities and identified ways that the child engaged in mathematics as part of the activity. although psts sometimes made inferences about children as they constructed these tasks (e.g., inferences about the specific ways that the child used mathematics), these tasks (more so than any others included in our analysis) reflected attempts to connect to mathematical funds of knowledge that children might bring from their experiences outside of school. for instance, in sample task (g) in appendix b, the third-grade case study child had described to the pst [d400] that he received an allowance each week and was working to keep a record of his earnings. although the child did not explain the specific mathematical practices he engaged in while saving, tracking, and spending his allowance, the pst knew he might bring mathematical knowledge from this practice (e.g., skills related to estimation, or organizing data) to the problem-solving situation. in another example, a pst [d416] learned from her fifth-grade case study child that his father was a construction worker and that the child helped the father with painting and mixing cement on weekends. the pst drew on this knowledge, and the case study child’s struggles with multiplication word problems during the interviews, to generate a task that involved calculating the number of gallons of paint that the case study child and his father would need to paint eight rooms if it takes two gallons of paint per room. the pst intended to connect to the child’s funds of knowledge, including the mathematics that the child and father might engage in when purchasing supplies. additionally, the task used a familiar context to help the child recognize how multiplication and rates can be used in real-world situations. another pst [e403] knew that her eighth-grade case study child was an avid basketball player with extensive knowledge and experience with the game. she also knew that he struggled to make sense of basic word problems and needed opportunities to explore alternate methods (beyond the standard u.s. algorithm) and to reason about the results of his calculations. the pst constructed the following problem about calculating scores to help the child generate mental strategies for operations involving equal groups: “how many shots would i have to make within the three point line [each shot would be worth 2-points] to get 27 points if i have already made 5 free throws?” the pst explained that the child would be able to draw on his experiences playing the game with siblings to solve the problem, which she felt would be particularly beneficial as the child “didn’t exhibit any confidence in his math ability.” in summary, tasks in this category reflected the clearest attempts to link to ways that children used (or might use) mathematics in their out-of-school activities. examples in this category are once again similar to those in categories 2 and 3 in turner et al. children’s multiple mathematics journal of urban mathematics education vol. 9, no. 1 63 drawing on specific knowledge of the case study child. in addition, like category 3, category 4 tasks focused not only on familiar contexts but also on connections to the practices and activities of children and families. what distinguished the tasks in this category is that psts went beyond “layering” mathematics onto family activities (i.e., category 3) and instead connected to the mathematical practices that students already engaged in outside of school. patterns across sites, psts, and children patterns by site. analysis of tasks generated at each of the five research sites revealed notable patterns. at both site b and site f, approximately two-thirds of psts generated at least one category 1 task that reflected assumptions about activities, contexts, or objects that would be relevant to case study children (65%: 11 of 17 at site b; 67%: 8 of 12 at site f). in fact, more than one-third of psts at these sites generated only category 1 tasks (35%: 6 of 17 at site b; 42%: 5 of 12 at site f). however, while tasks based on assumptions (category 1) were common among psts at sites b and f, approximately 60% of psts also posted at least one task that drew on specific knowledge of the child’s interests, activities, or practices (i.e., tasks at categories 2, 3, or 4; 65% at site b and 58% at site f). one notable difference was that only one pst at site f posed a task at category 3 or 4, while five psts at site b posted category 3 or 4 tasks. thus while psts at sites b and f posed tasks based on assumptions (category 1) versus based on specific knowledge of the case study child (categories 2, 3 or 4) with similar frequency, psts at site b were more likely to generate category 3 and 4 tasks. as noted in table 1, psts at sites b and f were similar demographically; at both sites approximately 90% of psts identified as white/european descent. there were also similarities in the tasks posed by psts at sites d and site e. only one-fourth to one-third of the psts at sites d and e posed category 1 tasks (33%: 3 of 9 of the psts at site d; 23%: 3 of 13 at site e), while a majority posed at least one task that drew on specific knowledge of the case study child (67%: 6 of 9 of psts at site d; 100% at site e generated tasks at category 2 or higher). moreover, psts at these two sites were more likely to generate category 3 or 4 tasks that connected to practices of children and their families (56%: 5 of 9 of psts at site d; 77%: 10 of 13 of pst at site e). thus unlike sites b and f, psts at sites d and e infrequently posed tasks based on assumptions about children (category 1) and instead generated tasks that drew on knowledge of the case study child, and in particular, knowledge of family practices (category 3 and 4). as noted in table 1, at site e, 85% (11 of 13) of psts identified as white/european descent, while psts at site d reflected greater diversity in racial and ethnic background. of the nine site d psts, three identified as white/european descent, three as hispanic/latin@, one as asian american, one as african american/black and one noted mixed ethnicity. turner et al. children’s multiple mathematics journal of urban mathematics education vol. 9, no. 1 64 at site a, psts were the least likely to generate tasks that reflected specific knowledge of their case study child’s experiences, activities, or interests. only 38% (3 of 8) of site a psts posed tasks at category 2 or higher. more often, psts at site a generated tasks based on assumptions about contexts that would be relatable or familiar to the children they worked with. eight-eight percent (7 of 8) of psts at site a wrote at least one category 1 task, and 63% (5 of 8) generated only category 1 tasks. interestingly, psts at site a reflected more racial and ethnic diversity than those at sites b, e, and f. half of the psts (4 of 8) identified as white/european descent, and the remaining psts identified as hispanic/latin@ (2), asian american (1), or mixed ethnicity (1). other patterns by psts and student groups. analysis of potential relationships between the pairing of psts and children with the category of tasks generated suggest that differenced may be negligible. for example, we examined potential differences between white/european descent psts paired with white/european descent students (n = 8), white/european descent psts paired with non-white students (n = 36), non-white psts paired with white/european descent students (n = 4), and non-white psts paired with non-white students (n = 10). across three of the four subgroups, approximately 40% of tasks posed by psts reflected assumptions about the case study child (category 1), and approximately 60% of tasks drew on actual knowledge about the child (categories 2–4). similar patterns were found with tasks that connected to practices of children and families (approximately 20– 30% of tasks were in category 3 or 4, across the different subgroups). the one exception occurred with the small (n = 4) group of non-white psts who worked with white/european descent case study children, where tasks based on assumptions about the child were less evident. in short, results suggest that the specific pairing of psts and case study children did not substantially influence the categories in which psts posed tasks. given that, by design, psts did not share backgrounds with their case study child, it is reasonable that none of the pst-case study child pairings produced higher instances of tasks that drew in meaningful ways on knowledge of the case study child. finally, we compared the tasks posed for case study children of different racial and ethnic backgrounds. as noted in table 1, 13 case study children were white/european descent, and 46 students were identified as non-white, including african american/black (11), hispanic/latin@ (25), asian american (2), native american/indigenous (1) or mixed ethnicity (7).7 we found that children identified 7 while we recognize the substantial diversity of students within any given racial and ethnic group, and acknowledge that it can be problematic to sort students according to specific demographic features, for the purposes of this analysis we use these labels to examine possible differences between tasks posed for students of different backgrounds. we consider patterns in tasks posed for white versus non-white students, as well as patterns in tasks posed for students from each racial and ethnic group. turner et al. children’s multiple mathematics journal of urban mathematics education vol. 9, no. 1 65 as non-white were more likely to receive at least one task based on assumptions (59%: 27 of 46 of non-white children received category 1 tasks as compared to 38%: 5 of 13 of white/european descent). this difference was particularly pronounced for hispanic/latin@ children, 68% (17 of 25) of whom received category 1 tasks. similar trends were noted for category 2–4 tasks. for instance, non-white children were slightly less likely to receive category 2 tasks that drew on specific knowledge of their interests and experiences (65% as compared to 77% of white/european descent children), or category 3 or 4 tasks based on practices that they or their families engaged in (35% of non-white children received category 3 or 4 tasks, as compared to 46% of white/european descent). once again, these trends were particularly prevalent for hispanic/latin@ children, only 28% (7 of 25) of whom received category 3 or 4 tasks. these moderately higher level of tasks received by the 13 white/european descent case study children is not explained by the pairing of psts and child (as previously discussed) or by the racial and ethnic background of psts (as white/european descent children worked with psts from various backgrounds). what seemed more salient were general differences among sites. white/european descent children at site f received category 1 and 2 tasks from white/european descent psts, which follows the overall trends at that site. in contrast, white/european descent children at site b, all of whom worked with white/european descent psts, received a similar, but slightly broader range of tasks (categories 1, 2, and 3), which again follows the overall trends among sites. white/european descent children at site d received many category 3 and 4 tasks, similar to other students at this site. also, there were no white/european descent students at site a, where psts were most likely to generate lower-level tasks. these findings suggest that white/european descent children were underrepresented at sites that produced higher percentages of category 1 and 2 tasks (site a in particular) and over represented at sites that produced higher-level problems (site d in particular, where almost half of the case study children were white/european descent). the slightly lower level of tasks received by the 25 hispanic/latin@ children also seems related to general differences across sites. for instance, hispanic/latin@ children were overrepresented at sites a and f, where psts were more likely to pose tasks at categories 1 and 2. whereas 34% (20 of 59) of all case study children worked with psts at sites a and f, 44% (11 of 25) of hispanic/latin@ children were from these two sites. when considering hispanic/latin@ children that received category 1 tasks in particular, the differences are even more pronounced. as previously noted, while 68% (17 of 25) of all hispanic/latin@ children received at least one task based on assumptions (category 1), 91% (10 of 11) of the hispanic/latin@ children at sites a and f received tasks this level. in con turner et al. children’s multiple mathematics journal of urban mathematics education vol. 9, no. 1 66 trast, at sites d and e, where psts were most likely to generate tasks at higher levels, only 38% of hispanic/latin@ children received category 1 tasks. in summary, the differences in the category of tasks received by children from different racial and ethnic backgrounds seem less related to the pairing of psts and children, or the racial and ethnic background of the pst, and more related to the overall differences between sites. we conjecture potential reasons for differences among sites in the next section. discussion and implications categorization of tasks as previously noted, in 81 of the 96 contextualized tasks (84%) analyzed in this study, psts drew on children’s mathematical thinking. furthermore in 59 of the contextualized tasks (61%), psts drew on specific knowledge about the case study child’s interests, activities, or practices. these 59 tasks were discussed in findings categories 2, 3, and 4. additionally, in 47 of those 59 tasks (80%), psts were able to draw on both knowledge of children’s mathematical thinking and specific knowledge about the case study child’s interests, activities, or practices. we find this result notable because it suggests that connecting to children’s mmkb in plans for future instruction, a practice that can be challenging even for experienced teachers (taylor, 2012; wager, 2012), is accessible to psts, at least in an emerging form. this result may reflect the affordances of the mathematics learning case study, specifically the scaffolded learning opportunities to focus closely on a particular child, and to learn about the child’s mmkb through multiple interactions, over time. we suspect that the high incidence of connections to children’s mathematical thinking may have been an artifact of the structure of mathematics learning case study assignments. the problem-solving interviews, coupled with requests to generate specific follow-up problems for the child to solve, may have made it easier for psts to draw directly on knowledge gained about children’s mathematical thinking, as compared to other types of knowledge about the child as they developed tasks for future instruction. category 1 and 2 tasks. in terms of the different ways that psts made connections to children’s interests, experiences, or funds of knowledge, the most prevalent single category in our findings were tasks based on assumptions about objects or activities that would be familiar to all children (category 1, approximately 40%). yet, if we look at the tasks that drew on the specific knowledge about the case study child (categories 2, 3, and 4), we see that this occurred in approximately 61% of the tasks. but tasks in categories 1 and 2 both reflected only slight adaptations, such as inserting an object or setting that was known to be of interest, to what otherwise would be standard textbook-like word problems. considering that psts turner et al. children’s multiple mathematics journal of urban mathematics education vol. 9, no. 1 67 conducted “getting to know you/funds of knowledge interviews,” had numerous informal interactions with their case study child, and were encouraged to draw on what they learned about children’s activities and family practices when generating instructional suggestions, these results underscore the documented difficulty that even practicing teachers have had in connecting deeply to children’s experiences in mathematics tasks (hedges, cullen, & jordan, 2011; taylor, 2012; wager, 2012). part of the challenge may be related to vacc and bright’s (1999) finding that, although psts demonstrate gains in their knowledge about children as a result of activities in methods courses, they tend not to use this knowledge in planning instruction or in teaching. lee’s (2012) study offers another possible interpretation, suggesting that the low incidence of tasks that went beyond superficial connections to children’s experiences may be related to the gap between psts’ vision for making connections in mathematics teaching, and their ability to enact this vision in instructional plans and practice. differences in the prevalence of category 1 and 2 tasks across sites suggest other possible explanations; we elaborate on these points later in the discussion. despite the superficial nature of the tasks in these two categories, we nonetheless argue that they constitute emerging and potentially useful attempts to connect mathematics instruction to children’s interests and experiences. psts explained that including relevant and familiar contexts enhanced student engagement, referencing instances when children participated more actively in problem solving when highinterest objects, activities, or people were used in tasks. this interpretation that the use of relatable, high-interest contexts contributed to student success may have motivated psts to emulate this practice in their plans for future instruction. furthermore, some recent research offers support for psts’ interpretations, suggesting that connections to children’s interests in mathematics word problems can help children make sense of tasks and can enhance achievement, particularly for lower-achieving children and for challenging tasks (renninger, ewen, & lasher, 2002; walkington, petrosino, & sherman, 2013). additionally, hedges and colleagues (2011) argue that connections to children’s interests can serve as entry points to children’s funds of knowledge because “children’s interests are stimulated by the experiences that they engage in with their families, communities and cultures” (p. 187). in this way, connections to children’s interests can be productive, particularly if teachers also attend to the mathematical knowledge and practices that children may bring related to those interests (civil, 2002, 2007; gonzalez, moll, & amanti, 2005). in fact, it may be useful to conceptualize the tasks in category 1 and 2 tasks as an initial and potentially productive step along a path toward more meaningful connections. category 3 and 4 tasks. the tasks in category 3 (20%) reflected psts’ attempts to connect to understandings about family practices. yet lacking specific knowledge about how families use (or do not use) mathematics as part of their practices, psts imagined the mathematics that might be involved, or how one could turner et al. children’s multiple mathematics journal of urban mathematics education vol. 9, no. 1 68 layer mathematics onto the activity. wager (2012) identified a similar category of connections in her work with practicing teachers. she explains, “the activity was identified [by teachers] first and then a school mathematical practice was matched to that activity” (p. 15). the small number of tasks in category 4 (approximately 7%) reflected attempts to connect to the mathematical knowledge and practices that children brought from experiences outside of school. we conjecture that the low incidence of tasks that evidenced these connections may reflect a tension that arises when teachers attempt to connect to out-of-school practices (e.g., cooking, gardening, shopping) in school mathematics lessons (taylor, 2012; wager, 2012). when teachers try to make the mathematics in these activities more explicit, the connections to what actually happens outside of school can be lost so that the problem generated for classroom use no longer resembles the out-of-school practice (gonzález, andrade, civil, & moll, 2001; masingila, davidenko, & prus-wisniowska, 1996). we see glimpses of this tension in each of the examples in categories 3 and 4. in the case of the basketball task for example, it was reasonable for the pst [e403] to assume that the child would be knowledgeable about scoring and might even have strategies for quickly calculating scores that he could draw on to solve the problem posed. yet, while some basketball players may calculate how many shots would be needed to score a given number of points, for others these types of calculations may not be an authentic part of their play. a related tension in constructing problems is attention to the appropriateness of concepts and number choice considering both the grade level and the mathematical competencies of the child. in this case, although the numbers may seem inappropriate for an eighth grader, the pst appears to be responding to specific needs (understanding basic word problems, exploring alternative calculation methods) she had identified for the child. while it is important to acknowledge these tensions, we argue that it is productive for psts to analyze and attempt to connect to mathematical practices in out-of-school activities. posing these kinds of problems, even if they do not always mirror the specific ways that children and families engage in mathematical reasoning outside of school, opens a space for children to talk about their out-of-school mathematics practices. teachers can use these tasks to position student’s home and family activities as mathematical, validating children’s funds of knowledge. we see this effort as a critically important part of challenging deficit-based narratives about children and youth from marginalized communities. also important is that children may learn mathematics more deeply when problem contexts are familiar and build on children’s knowledge. walkington and colleagues (2013) found this to be true when working with older children. we conjecture that for younger children as well, when tasks connect to familiar contexts, children need to expend less effort under turner et al. children’s multiple mathematics journal of urban mathematics education vol. 9, no. 1 69 standing the context, making available more resources for understanding the mathematics. there may be additional reasons why it is difficult for psts to generate tasks that connect to the ways children and families engage in mathematics outside of school. in comparison to practicing teachers, psts are just entering the profession and have not had the same experiences with elementary school mathematics and mathematics classroom practices or curricula. it may also be that the activities within the mathematics learning case study did not always provide the range and depth of opportunities required for psts to learn about how children and families used mathematics in everyday activity. prior research has documented that it takes time for teachers to build relationships with children and families and to move beyond serendipitous or piecemeal connections to children’s funds of knowledge (hedges et al., 2011). while the mathematics learning case study was designed to support psts in attending closely to children’s knowledge, experiences, and resources, psts may need additional opportunities to learn about children in different contexts and over time. differences in the prevalence of category 3 and 4 tasks across sites support this conjecture, as we elaborate below. differences across sites as outlined in our findings, analysis across sites revealed notable contrasts between the categories of tasks posed by each group of psts. in this section we conjecture and discuss possible explanations for these differences. to begin, the differences in tasks posed cannot be attributed to the racial and ethnic background of the psts. at some sites, white/european descent psts wrote many category 3 and 4 tasks that connected to practices of children and families (site e), while other sites with mostly white/european descent psts evidenced a higher proportion of category 1 and 2 tasks, including tasks based mainly on assumptions, rather than specific knowledge about case study children (sites b and f). conversely, in one instance a group of psts from diverse racial and ethnic backgrounds proposed numerous category 3 and 4 tasks (site d), while at another site where psts reflected racial and ethnic diversity, all tasks were in category 1 or 2 (site a). a more viable explanation for the notable contrasts between sites is differences in how the mathematics learning case study module was implemented. for example, at site a, where psts were most likely to write category 1 tasks, the case study assignment was completed during a 4-week period at the beginning of the semester-long course. for 3 weeks, psts conducted weekly interviews and observations of their case study children, and then during the fourth week, wrote the final report. in contrast, at several other sites where psts produced a greater range of tasks and where a majority of tasks drew on specific knowledge of the case study child, psts worked on the case study assignment across a significant portion of the course (site d, e, and f). for example, at site d, psts met their child at the begin turner et al. children’s multiple mathematics journal of urban mathematics education vol. 9, no. 1 70 ning of the semester, conducted initial getting to know you interviews in week 4, and problem-solving interviews in weeks 5 and 6 and then continued to observe and interact with the case study child several times a week for the balance of the semester. final case study reports were not submitted until the last week of the semester. sites e and f followed a similar schedule. at site b, where tasks generated resembled those produced by psts at site f, psts completed the case study assignment during an intensive weeklong immersion in the field. during this week, psts focused intently on learning about the case study child through multiple interviews, observations, and interactions. these differences in how the case study assignment was implemented suggest that psts may benefit from prolonged or intensive interactions with their case study child that allow extended opportunities (beyond those included in the structured interviews) to learn about their child’s experiences, interests, competencies, and practices. another possible explanation for the differences across sites includes the age and prior life experiences of psts. for instance, at site e, where psts were most likely to pose category 3 and 4 tasks that drew on specific knowledge of children’s and families’ practices, psts were graduate students completing a combined master’s degree and teaching certification program. while some of these psts entered the graduate program immediately following their undergraduate degree, others returned to school after working in other fields. similarly, at site d where psts also generated many category 3 and 4 tasks, psts reflected greater diversity in age and prior life experience than those at other sites. psts at site d were all undergraduate students, yet some were parents, and others immigrated to the united states as children and thus had experiences with different cultures, languages, and school systems. at the other three sites (site a, b, and f), psts were almost all traditional undergraduate students who entered college immediately following high school. while the numbers of psts at each site are small and do not support broad generalizations, the differences noted across sites suggest that both extended opportunities to interact with case study children coupled with a broader range of life experiences (due to age, work experience, and family background) may support psts in learning about children’s experiences and connecting to this knowledge in their mathematics teaching. these findings related to how different factors (i.e., pst and child background, pst-child pairings, site implementation) seemed to influence the tasks that psts posed are an important contribution to the literature, and one that outlines promising direction for future research. conclusion despite the challenges, we see a hopeful story emerging. as noted earlier, nearly half of the 96 contextualized tasks (46%) attended to both children’s mathematical thinking and specific knowledge of the case study child’s out-of-school in turner et al. children’s multiple mathematics journal of urban mathematics education vol. 9, no. 1 71 terests and activities, and all but three of the 96 tasks (97%) connected to either the child’s mathematical reasoning or the child’s interests and activities. the fact that almost every pst focused on some specific knowledge about their case study child indicates that psts were attending to particularities about children, and potentially developing the disposition to continue to do so (leonard, 2008). our findings also suggest that psts were learning to (re)orient to children from diverse cultural, linguistic, and racial backgrounds in ways that reflected a resource-based (rather than deficit-based) perspective. through their scaffolded learning assignments in the mathematics learning case study, psts learned specific things about children’s knowledge and experiences and positioned that knowledge as a resource to support school mathematics learning. in this way, our study demonstrates the value of case study experiences for learning to orient children as mathematical learners in ways that recognize their families, communities, and out-of-school interests and activities as resources that can support mathematics learning. finally, this study challenges the notion that teaching that connects deeply to children’s mmkb is out of the reach of psts and thus should not be a focus in teacher preparation programs. our findings suggest that with specific kinds of support, such as extended or intensive opportunities to interact with children both in and out of the classroom, psts can begin to leverage knowledge about children’s and families’ out-of-school activities in their plans for instruction. implications for research and practice in addition to the importance of extended interactions with children discussed here, psts may benefit from additional support as they learn to connect to children’s varied knowledge bases in their planning. we found that psts often learned far more about their case study child’s interests, experiences, and strengths than they incorporated in their instructional suggestions. this finding suggests that additional scaffolds are needed to help psts utilize that knowledge as they plan for instruction. more attention, for example, could be focused on adapting problem contexts to align with the information about their case study child accessed in the “getting to know you/funds of knowledge” interviews, or in informal conversations and observations. psts also participated in a community mathematics walk during the methods course in which they learned about the mathematical resources in the community surrounding the elementary school. yet, we saw little attention to the knowledge gained during this activity in the psts’ instructional suggestions. linking this experience more explicitly to the mathematics learning case study may provide psts with additional entry points into making connections to children’s mmkb. psts may also benefit from examples of how experienced teachers draw on children’s mmkb in their teaching. methods course instructors might invite mentor teachers (teachers who work with psts in the field) to share this aspect of turner et al. children’s multiple mathematics journal of urban mathematics education vol. 9, no. 1 72 their practice, and/or help mentor teachers to explicitly mark these teaching moves when psts are in their classrooms. additionally, our findings draw attention to a type of knowledge that psts should develop to be effective teachers of mathematics that has not been given significant attention in mathematics teacher preparation. in order for teachers to support the learning of all students, particularly those from under-represented populations, it is important that mathematics teacher educators support psts in delving deeply into the knowledge and experiences that children bring to school so that they can be leveraged in the service of their mathematical learning. the mathematics learning case study shows promise in moving teachers toward making these connections; particularly in contexts where extended time during the semester is devoted to interactions with and observations of the child. research on the effectiveness of this project and other attempts to support psts’ ability to connect to children’s experiences and funds of knowledge would be a fruitful direction for future research. at a minimum, these results point to the need for more coordinated, multi-site research in teacher education to better understand how differences in implementation and program context impact pst learning and practice. whether it is possible to improve results through additional activities or scaffolds in methods courses warrants further research. acknowledgments the work reported here is based on research supported by the national science foundation under grant #1228034. any opinions, findings, conclusions, or 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(2013). supporting algebraic reasoning through personalized story scenarios: how situational understanding mediates performance. mathematical thinking and learning, 15(2), 89–120. turner et al. children’s multiple mathematics journal of urban mathematics education vol. 9, no. 1 76 appendix a overview: mathematics learning case study module this module focuses on the mathematics learning and dispositions of a case study child. prospective teachers (psts) work with a case study child over the course of a semester, and consider how to use what they learn in mathematics instruction. the activities within this module include (a) conducting a mathematics “getting to know you/funds of knowledge” interview, (b) conducting one or more problem-solving interview assessments, (c) conducting informal observations of the child during mathematics lessons and other times of the day, and (d) engaging in written analysis and reflection. activity 1: mathematics “getting to know you/funds of knowledge” interview psts conduct an interview with one child in their practicum classroom in an effort to become more familiar with the child’s activities and interests, the child’s home and community knowledge base, and home and community resources. activity goals are: • to find out more about the child including her or his interests, activities she or he engages in outside of school with family and friends, and what she or he identifies as activities at which she or he excels (i.e., does she play soccer at a local park, does he go to a community center, where does she or he shop, etc.). • to identify places, locations, and activities in the community that are familiar to the child, and to find out what the child knows about potential mathematical activity in those settings. these could include locations in the neighborhood immediately surrounding the school, locations/settings in the neighborhood in which the child lives, as well as locations/settings in the broader community with which the child is familiar. • to find out more about the child’s dispositions towards mathematics. activity 2: problem solving interviews psts conduct one or more problem-solving interviews with their case study child. these interviews provide an opportunity to practice eliciting, interpreting, and assessing children’s thinking about mathematics, with a particular focus on their understanding of number concepts. psts are provided with a set of sample problem-solving tasks, but are also encouraged to adapt questions as needed for their case study child. psts take detailed notes during the interview and collect all student work. whole number interview protocols and guidelines were adapted from the work of tom carpenter and the cognitively guided instruction (cgi) group (carpenter, fennema, franke, levi, & empson, 1999), as well as work of susan empson and colleagues (empson, turner, & junk, 2006). the fraction interview protocol was adapted from work done with edd taylor. activity 3: synthesizing and connecting across activities this assignment is designed to cut across the previous activities in this module. in this written report completed outside of class, the pst reflects across the multiple interviews and observations that she or he completed with her or his case study child, and considers how to use this knowledge in mathematics instruction. turner et al. children’s multiple mathematics journal of urban mathematics education vol. 9, no. 1 77 appendix b example tasks posed by psts findings category sub-category examples category 1: generating tasks based on assumptions about familiar or relevant contexts 37 of 96 tasks (representing 32 psts) tasks based on assumptions about objects and activities that would be relatable to children. tasks resembled typical textbook-like word problems (e.g., information followed by a question to be solved) (a) amanda went to the store and bought 6 stickers. she already had 18 stickers. how many stickers does amanda have in all? tasks adapted from the problem-solving interviews or from the curriculum used in the case study child’s classroom (b) there are 9 birds flying in the air, some of them landed on a tree. now there are only 4 birds left flying in the air. how many of the birds landed on tree? category 2: generating tasks based on knowledge of objects or activities familiar to case study child 31 of 96 tasks (representing 25 psts) tasks based on specific knowledge about the case study child’s interests or preferences. tasks resembled textbook-like word problems but included objects and people known to be of high-interest to the case study child (c) mary has some frogs. kevin gave mary 3 more frogs. mary now has 8 frogs. how many frogs did mary have at the beginning? tasks connected to activities known to be of high interest to the case study child, in this case football. however, tasks did not connect to the math that children might do as part of that activity. (d) jason has 9 footballs. he loses 5 of them. how many footballs does jason have now? category 3: generating tasks by mathematizing case child’s family practices 21 of 96 tasks (representing 17 psts) tasks based on an activity or practice in which the family of the case study child engaged. psts considered how people might use mathematics as part of this activity. tasks focused on home activities in which parents could engage the case study child in mathematics. (e) the next time you [the parent] go to the grocery store with [child’s name], you might ask him to count how many items were in your cart as you check out. tasks focused on connections teachers would make in classroom mathematics lessons to home or family activities. (f) john goes out to eat with his parents on friday night. his mom orders a large pie for herself, john, john’s sister, and john’s dad. this large pizza comes with eight slices. if each person in john’s family wants to eat the same amount of slices, how many slices will each person get? turner et al. children’s multiple mathematics journal of urban mathematics education vol. 9, no. 1 78 category 4: generating tasks by identifying mathematics in activities in which the case study child engages 7 of 96 tasks (representing 6 psts) task that related to activities in which psts knew their case study child participated, and also identified mathematics that the child engaged in as part of the activity note: no sub-categories within category 4. (g) if you receive 5 dollars a week for allowance, how much would you have after a month? if after one month, you spent 13 dollars on a new soccer ball, how much money do you have left? caught in a web: linkages between racial narratives about mathematical ability journal of urban mathematics education december 2016, vol. 9, no. 2, pp. 49–80 ©jume. http://education.gsu.edu/jume dan battey is an associate professor in the department learning and teaching at rutgers, the state university of new jersey, graduate school of education, 10 seminary place, new brunswick, nj, 08901; email: dan.battey@gse.rutgers.edu. his research interests include understanding mathematics education as a racialized space, classroom relational interactions, and professional development that integrates mathematics and equity. luis a. leyva is an assistant professor in the department of teaching and learning at vanderbilt university, peabody college of education and human development, pmb 230, gpc, 230 appleton place, nashville, tn, 37212; email: luis.a.leyva@vanderbilt.edu. his research interests include issues of gender and sexuality in stem (science, technology, engineering, and mathematics) education, marginalized student populations’ experiences in undergraduate mathematics education, and gender-affirming and culturally responsive mathematics teaching and stem support programs. a framework for understanding whiteness in mathematics education dan battey rutgers, the state university of new jersey luis a. leyva vanderbilt university in this article, the authors provide a framework for understanding whiteness in mathematics education. while whiteness is receiving more attention in the broader education literature, only a handful of scholars address whiteness in mathematics education in any form. this lack of attention to whiteness leaves it invisible and neutral in documenting mathematics as a racialized space. naming white institutional spaces, as well as the mechanisms that oppress students, can provide those who work in the field of mathematics education with specific ideas about combatting these racist structures. the framework developed and presented here illustrates three dimensions of white institutional space—institutional, labor, and identity— that are intended to support mathematics educators in two ways: (a) systematically documenting how whiteness subjugates historically marginalized students of color and their agency in resisting this oppression, and (b) making visible the ways in which whiteness impacts white students to reproduce racial privilege. keywords: mathematics education, race, racism, whiteness, white supremacy hiteness is a widespread ideology in society (see, e.g., leonardo, 2004; lewis, 2004). while the effects of whiteness are receiving more attention in the broader education literature, mathematics educators have generally been immune to researching its impact on students (battey, 2013a). only a handful of scholars address whiteness in mathematics education in any form (see, e.g., battey, 2013a; brewley-kennedy, 2005; martin, 2007, 2008, 2009, 2013; stinson, 2008, 2011). this lack of attention to whiteness leaves it invisible and neutral in documenting mathematics as a racialized space. racial ideologies, however, shape the expectations, interactions, and kinds of mathematics that students experience. martin (2009) calls for the de-silencing of race in mathematics through ideologies of colorblindness and whiteness by actively acknowledging students’ co-constructed academic and racial identities as well as providing opportunities to engage with mathematics as a tool for social change. this call, for us, means documenting the w http://education.gsu.edu/jume mailto:dan.battey@gse.rutgers.edu mailto:luis.a.leyva@vanderbilt.edu battey & leyva whiteness in mathematics education journal of urban mathematics education vol. 9, no. 2 50 ways in which whiteness subjugates historically marginalized students of color (e.g., different forms of microand macro-aggressions1) and their agency in resisting this oppression, as well as to make visible the ways in which whiteness impacts white students to reproduce racial privilege. the goal for us is to support the development of a mathematics space that builds collective consciousness of racism to prevent students of color from internalizing deficit ideologies (feagin, 2006; moore, 2008). this collective consciousness, in turn, could open more space for student identities that challenge existing racial hierarchies in mathematics. scholars posit that such perpetuation of racist structures in white institutional spaces like mathematics classrooms can be halted through purposeful analyses of whiteness with the voices and experiences of those marginalized brought to the fore (andersen, 2003; martin, 2009; moore, 2008). in addition to these voices, however, researchers must also deconstruct the ways in which institutional spaces privilege whites (martin, 2008). naming white institutional spaces, as well as identifying the mechanisms that oppress and privilege students, can give those who work in the field of mathematics education specific ideas of how to better combat racist structures. as martin (2013) states, few “white scholars, have turned their analyses inward to examine the internal structure of mathematics education as a politically oriented project in order to expose its own enactments and validations of racial hierarchies and inequities” (p. 322). along the same lines, we offer this (developing) framework to support mathematics education scholars in general, and white scholars specifically, in examining the racist internal structure of mathematics education. but before reviewing existing literature, we wish to clarify some key concepts, including white supremacy, white privilege, whiteness, and racism. evidently, these terms are interwoven. it is crucial to note, however, that in developing a framework for whiteness in mathematics education, our goal is not to re-center whiteness, but rather to document white supremacy as opposed to white privilege. leonardo (2004) makes the case that for white privilege to take shape it must be accompanied with a process of racial domination. in other words, while white privilege refers to benefits from racism in favor of whites, white supremacy is the systemic maintenance of the dominant position that produces white privilege. therefore, white supremacy takes on power more centrally than privilege alone and focuses “around direct processes that secure domination” (leonardo, 2004, p. 137). whiteness is the ideology that maintains white supremacy, valuing one racial group over others. thus, the foundational ideology of whiteness maintains a system of white supremacy, which produces privilege. finally, in relating racism and whiteness, kivel (2011) states: 1 microaggressions are subtle, possibly unconscious, automatic insults to individuals from historically marginalized groups (solórzano, 1998). macroaggressions are broader acts against marginalized groups on systemic levels (sue et al., 2007). battey & leyva whiteness in mathematics education journal of urban mathematics education vol. 9, no. 2 51 racism is based on the concept of whiteness—a powerful fiction enforced by power and violence. whiteness is a constantly shifting boundary separating those who are entitled to have certain privileges from those whose exploitation and vulnerability to violence is justified by their not being white. (p. 17) thus, whiteness is a foundational concept for racism. whiteness creates an ideal race, with which to devalue and subsequently oppress other racial groups. understood in this way, whiteness has a dual nature: privileging whites and oppressing those outside the boundary of white. while this boundary is not static, it can be viewed through both lenses to determine its presence in an institution. whiteness then is the fictive ideology from which racism is established. the goal of the framework presented here is not merely to name white privilege in mathematics education but rather to document the institutional ways in which white supremacy in mathematics education acts to reproduce subordination and advantage. we begin by briefly reviewing some of the approaches that scholars have taken to document whiteness in mathematics education. we follow with a review of work on whiteness in literatures across law, sociology, history, and education as an introduction to central ideas. because so few scholars have taken up martin’s (e.g., 2009, 2013) calls around interrogating whiteness, we intend the review to be a broad introduction for mathematics educators. we first address whiteness as a construct that shifts over time, but oppresses those outside its boundary. from there, we examine work on how white supremacy currently functions through dialectical mechanisms: symbolic (ideologies) and material (resources). we then move to examples of whiteness functioning through colorblind policies in housing, taxes, and education. throughout the review on whiteness, we draw specific connections to what the concepts can mean for work within mathematics education. after the review, we introduce our theoretical framework aimed at documenting mechanisms operating in white institutional spaces within mathematics education. in the framework, we illustrate three dimensions in documenting white institutional space: institutional, labor, and identity. we hope this framework serves as a tool to detail the ways in which whiteness reproduces advantage and disadvantage in mathematics to consciously find ways to confront and challenge its effects. the question that the framework aims to address: how does whiteness operate in mathematics education? whiteness in mathematics education while there is limited work on whiteness in mathematics education, the approaches taken to examine whiteness have been quite varied. researchers have examined whiteness as hegemonic discourse, property, identity, and privilege. stinson (2008, 2011), for example, explores how academically and mathematically successful african american male students negotiate discourses of whiteness. in particular, battey & leyva whiteness in mathematics education journal of urban mathematics education vol. 9, no. 2 52 the african american male participants of his study responded to fordham and ogbu’s (1986) burden of acting white theory (among others) to show how they accommodated, reconfigured, or resisted the discourse. stinson claimed that the burden of acting white was most often an oversimplification or misinterpretation of their experiences; in that, the participants did not place academic success as only within whiteness or somehow outside blackness. instead, they experienced whiteness more generally as pressure to mold themselves into a white ideal rather than being called out as acting white by black peers. the young african american men in stinson’s study, however, were able to navigate the expectations of whiteness successfully. this work represents a unique approach in having african american students, through counter-storytelling, deconstruct the hegemonic narratives of whiteness. taking a different approach to whiteness, battey (2013a) provides an example of whiteness at an institutional level. in his project, battey calculated the investment in whiteness due to mathematics course taking in the united states. in this sense, he examined access to mathematics education as property. given that there is no reason to expect different “races” to do better or worse in terms of mathematics—other than historical and institutional inequities producing differential opportunities and access—the differences in course taking can be tied to racist structures. battey examined data from three time points, 1982, 1992, and 2004, using mathematics coursework as a proxy for property, to predict differential racial investments upwards of $1.5 trillion advantaging whites over latin@s,2 african americans, and native americans. while 25% of this total can be attributable solely to race (see rose & betts, 2001), a net advantage for whites of over $400 billion still remains. these calculations illustrate how mathematics education reproduces racial income and wealth differences that perpetuate an ideology of whiteness. at an individual level, brewley-kennedy (2005) explores how one mathematics teacher educator’s white identity influenced her instructional practices in a mathematics methods course. from interviews and observations, brewley-kennedy found that the teacher educator encountered several struggles when attempting to implement an “equity” agenda. first, the teacher educator worried about creating an emotionally heated space if she explicitly challenged preservice teachers’ beliefs about race and poverty. rather than an intellectually challenging space, she wished to maintain a “safe space.” in addition to worrying about her preparedness with pedagogical practices and having simply “book knowledge,” she also avoided exploring equity more broadly due to the pre-service teachers’ resistance about dis 2 we use the @ sign to indicate both an “a” and “o” ending (latina and latino). in alignment with gutiérrez (2012), we see this use as a way to de-center the patriarchal nature of the spanish language. it is customary for groups of males (latinos) and females (latinas) to be written in the form that denotes only males (latinos) and we see the @ symbol as better than denoting an either/or (latino/a) form that promotes a gender binary. battey & leyva whiteness in mathematics education journal of urban mathematics education vol. 9, no. 2 53 cussing it. while the teacher educator was comfortable with discussing gender and special education because they were not, from her perspective, emotionally loaded, she was worried about “politically correct” terms and overly generalized issues of race. brewley-kennedy’s research illustrates how an identity of whiteness serves to constrain a teacher educator in discussing equity with future teachers. her research exemplifies what diangelo (2011) calls “white fragility,” or whites’ discomfort with race resulting in behaviors of silence, fear, guilt, or avoiding discussions about race altogether, which serves to reproduce a status quo of white supremacy. the most extensive work on whiteness in mathematics education comes from martin (see, e.g., 2007, 2008, 2009, 2013), who has discussed various concepts such as racialized mathematics identity, white privilege, and white institutional space in his scholarship. because we review martin’s work on white institutional space later, here we focus on his research on racialized mathematics identity and white privilege. in his article “beyond missionaries or cannibals: who should teach mathematics to african american children?” martin (2007), drawing from the work of bonilla-silva (e.g., 2003), discussed four ideological frames—liberal individualism, naturalism, cultural racism, and minimization of racism—that can constrain white teachers’ expectations and teaching of african american children and youth. these frames serve to position african american students as deficient and as needing to “live up,” so to speak, to white norms of behavior and achievement. the flip side to these frames is the assumed privilege that they bestow to white students. assumptions of whites’ high ability in mathematics and avoidance of pathologizing whites serve to privilege them as a group. for instance, despite mathematics achievement tests showing asian american students outperforming white students, society resists pathologizing whites’ underachievement (martin, 2009). whiteness here serves as a means to resist attaching deficient frames to white students. across this work, scholars discuss various aspects of mathematics education and whiteness. we take a broad view of mathematics education that includes policy, ideologies, research, curriculum, instruction, identities, and the people who populate the field. as noted in the introduction, we aim to introduce the research base for mathematics education scholars and practitioners to better understand critical facets of whiteness in the existing literature. we begin with a focus on its ideological construction. whiteness in law, sociology, history, and education lipsitz (1995) states that the fictive concept whiteness appeared in law as an abstraction, and it became actualized in everyday life. much like “black” is a cultural construction based on skin color, not biology, “white” developed out of the reality of slavery and segregation, giving groups unequal access to citizenship, im battey & leyva whiteness in mathematics education journal of urban mathematics education vol. 9, no. 2 54 migration, and property. by giving whites a privileged position in relation to the “other,” european americans united into a fictitious community. whiteness is a constantly shifting boundary separating those who are more valued from those whose exploitation is justified by not being white. the boundaries of this social construction show how the definition has shifted over time. while many in the united states conceptualize race as a black/white binary, groups such as jews, native americans, asians, and latin@s have proved more difficult to classify in the racial hierarchy within the u.s. context (haney-lopez, 2006). in the 1840s and 1850s, for example, citizens of california had debates about the status of mexicans and chinese. there were some mexicans with considerable wealth who partnered with whites, while the chinese were exploited for work on railroads and in fields, which impacted who could become citizens, own land, and marry whites (almaguer, 1994). to complicate things further, though mexican americans were considered white legally, they were denied rights and privileges that whiteness bestowed (foley, 2002). despite being ruled as white in california courts, the u.s. government added a category of mexican in the 1930 census, counting only 4% of mexicans as white. this action prompted the league of united latin american citizens (lulac) to turn its back on civil rights battles of the 1940s and 1950s with statements such as “tell these negroes that we are not going to permit our manhood and womanhood to mingle with them on an equal social basis” (b. marquez, as quoted in foley, 2002, p. 56). in contrast to lulac, the chican@ movement of the 1960s rejected the assimilationist strategies. they rejected whiteness and all it had come to mean, including the rejection of ancestry and cultural heritage as well as the adoption of “american” values. the response from whites was “why do you insist on being different? why do you have to be mexican or chicano? why can’t you just be american?” (foley, 2002, p. 56) such questioning failed to recognize that differential treatment and institutional racism did not afford chican@s the benefit of being american or white. thus, the lure of whiteness and all that it entails has been a contested boundary for those in the latin@ community, some seek it out and others reject it (bonilla-silva, 2002; haney-lopez, 2006). muddying the definitional space further, from 1878 to 1909, courts in the united states heard twelve naturalization cases of persons seeking citizenship. eleven of those cases were barred from citizenship including persons from china, japan, hawaii, as well as two mixed race applicants (foley, 2002; haney-lopez, 2006). across the cases, neither white skin nor being caucasian guaranteed one’s rights to citizenship and thus, whiteness. there is extensive work examining the shifting definitions of white historically, from jews, irish, and russians to eastern and southern europeans (e.g., haney-lopez, 2006). over the last few decades, although there is still prejudice against these groups, they are generally considered white in the united states (brodkin, 1998). many ethnic groups have sought out battey & leyva whiteness in mathematics education journal of urban mathematics education vol. 9, no. 2 55 equalization through citizenship, but when african american citizens still had to sit at the back of the bus and could not vote, assimilation became the goal for some. when the 1940 census stopped distinguishing foreign-born versus native-born whites, official assimilation as white became a possibility. as “not-yet-white” ethnic immigrants strove to assimilate as a way to attain whiteness (roediger, 2002), “immigrants of color always attempt to distance themselves from dark identities (blackness) when they enter the united states” (bonilla-silva, 2003, p. 271). rather than laws, concrete definitions, or biology determining citizenship, an ideology of whiteness wedded to the idea that blacks were culturally and biologically inferior to whites (morrison, 1993), determining access and power. what all of this means within mathematics education is that an ideology of whiteness operates to devalue, oppress, and discriminate those perceived as “less” or not white. in conjunction with this devaluing, the ideology maintains whites in an objective and neutral position of power to divvy up access. an ideology of whiteness would then serve to position white people, white ideas, and white behaviors as more valued institutionally and in classrooms, which may not always be visible in terms of curriculum designers and policy developers. moreover, whiteness oppresses blackness through deficit ideas, poor treatment, and lower quality of instruction. the creation of a racial ideology of whiteness then brings with it very real consequences. we next detail the function of this racial ideology and how it interacts with colorblindness to produce material racism. the function of racial ideologies of whiteness and colorblindness ideologies provide a framework for making sense of the world and they gain power based on legitimizing the present state of things. racial ideologies, then, work best when they offer invisible, commonsense explanations to keep the status quo (hall, 1990). these forever-present but invisible ideologies retain power and endure to the extent that they enable an understanding of the stratification of society, securing the positions of both the dominated and the dominant (lewis, 2004). thus, the functioning of racial ideologies like colorblindness and whiteness is complex. whiteness is supported by a colorblind ideology, a form of maintaining the social order institutionally, and with the appearance of not being racial. bonillasilva (2003) connects colorblindness with the resistance to framing, defining, or pathologizing whiteness and the ways that race plays out in the united states since the civil rights movement. while racism often calls forth overt practices such as slavery, the jim crow era, and lynchings, the more recent avoidance of explicit racial discourses signifies colorblind racism, the dominant racial ideology since the civil rights movement (bonilla-silva & forman, 2000). this racial ideology fits with martin’s (2009) discussion about the framing of white achievement versus that of historically marginalized students of color along two lines. first, it shows the battey & leyva whiteness in mathematics education journal of urban mathematics education vol. 9, no. 2 56 denial to recognize how institutional inequality bestows unearned advantages to whites. this denial allows the dominant ideology to locate racism today in a few prejudiced individuals. second, it fits with framing lower achievement by students of color as due to cultural deficiency. this reasoning essentially blames african americans and latin@s for their lower status (bonilla-silva & forman, 2000). an unwillingness to question how institutions benefit whites, coupled with statistics showing lower achievement scores for african american and latin@s shifts the blame to students, families, communities, and culture and away from whiteness. colorblindness shifts explicit racial arguments about genetics to supposedly non-racial arguments or proxies of student failure, uncaring parents, and devaluing of education, which leaves whiteness invisible and allows those who assert it to defend their views in apparent nonracial ways (bobo & hutchings, 1996; bobo & smith, 1994; bonilla-silva, 2003; bonilla-silva & forman, 2000; carmines & merriman, 1993; jackman, 1994). for instance, colorblindness as a racial ideology provides an explanation for the framing of disparate achievement as “gaps” when white students (the dominant group) score better than students of color (the dominated group), yet not when whites are scoring lower, in the case of international comparisons (martin, 2009). under whiteness, it does not matter whether whites are racially conscious or not. whites benefit from an external reading of themselves as white (lewis, 2004), whether or not they identify as white. in other words, a felt identity or groupness is not a prerequisite to reap unearned privileges. lewis discusses this situation as a passive collective; whites are unified by their actions around certain objects (passive collective), rather than a self-conscious choice to be a member of a group (identity). lewis writes— although numerous all-white groups are not explicitly racial, their racial composition is not an accident but is a result of whites’ status as members of a passive social collectivity whose lives are shaped at least in part by the racialized social system in which they live and operate. (p. 627) the concept of a passive collective allows for the enactment of whiteness through institutional racism, including unearned advantages, without the intentionality of whites. all whites experience race daily, living and working within racial structures, though race and racism are not necessarily explicit for them. for instance, white identities can operate more explicitly in the form of exclusive policies (e.g., country clubs that do not allow blacks to join), but in a colorblind society, these are less acceptable. other settings function as a result of an outcome of exclusive policies (e.g., housing segregation affects who shops at particular grocery stores and attends local schools). here, there is no felt identity of whiteness in these settings, although housing segregation whitewashes particular stores, schools, and communities. a third type of setting also functions as a passive collective based on long his battey & leyva whiteness in mathematics education journal of urban mathematics education vol. 9, no. 2 57 tories of racial exclusion. an example of this type is when educational and work experience is used for promotions and better jobs, based on past racial discrimination of the labor market, allowing more whites to serve on boards, attend partner meetings, or serve on personnel committees (lewis, 2004). these settings illustrate the different ways that whiteness can work as an ideology, without it necessarily being adopted as a white identity. whiteness functions within structures, deciding how resources, labor, and space will be distributed by means of housing segregation as well as educational and financial stratification. these structures are in place to benefit future generations. the point is not that all whites benefit the same in all of these settings or by all policies, as this would be essentializing a highly diverse group of people, but rather that a person’s racial position is constructed in relation to a racial history, which has distributed space, resources and labor as well as generated racist language and discourses (young, 1994). this distinction is important in understanding whiteness as an ideology rather than as an identity and thus shifts how one might study whiteness in mathematics education. as opposed to approaching whiteness through interviews and posthoc analyses of individual experiences, which has worked so well for example with african american identity, the formation of a passive collective will make study of the construct more difficult. because most whites passively identify as white, directly asking them how whiteness affects them will glean limited insights. instead, a researcher must study the proxies used for race, moment-to-moment interactions, and the institutional settings in which whites participate. doing so places more emphasis then on observation and examining multiple levels of mathematics education including policy, curriculum, and teaching, in addition to identity construction as reflected in the framework presented here. symbolic and material racism in policies sewell (1992) and lewis (2004) discuss racism both ideologically and concretely through considering its dialectical nature: symbolic (ideological) and material (structural resources). race is more dynamic than having racial ideologies create material differences; racial ideologies are also reproduced in relation to material circumstances. more specifically, sewell (1992) explains how race is socially constructed by the dialectic relationship between symbolic and material resources as follows: must be true that schemas are the effects of resources, just as resources are the effect of schemas…. if resources are instantiations or embodiments of schemas, they therefore inculcate and justify the schemas as well…. if schemas are to be sustained or reproduced over time…they must be validated by the accumulation of resources that their enactment engenders. (p. 12) battey & leyva whiteness in mathematics education journal of urban mathematics education vol. 9, no. 2 58 racial groups, therefore, are divided because of real material conditions; which, in turn, reproduce deficit ideologies about intelligence, effort, and values. at the same time, racial ideologies are employed in the continual production of this material stratification (west & zimmerman, 1987). the dialectic relationship between ideologies creating more racial disparity, and racial disparity producing these ideologies is critical in understanding the consequences of policy and educational institutions including in mathematics education. the production of whiteness then can be felt in material form. lipsitz (1995) coined the phrase possessive investment in whiteness over two decades ago. in his article, he discussed federal policies in the united states around housing, taxes, and education among other areas, which have reproduced an ideology of whiteness. many policies seem neutral (i.e., colorblind), yet their material effect is anything but that. as one example, the federal housing administration used a confidential city survey as well as destroyed housing in city centers, which affected twice the percentage of african americans compared to whites in the 1950s and 1960s. they have also shifted loan money and therefore future investment in real estate away from communities of color and toward whites since 1934 (lipsitz, 1995; logan & molotch, 1987). more recently, studies have shown that african americans are 60% more likely than whites to be turned down for loans in boston (controlling for credit qualifications), disqualified for loans almost three times as much in houston, and are four times less likely to receive conventional financing in atlanta (campen, 1991; logan & molotch, 1987; massey, 1996; ong & grigsby, 1988; orfield & ashkinaze, 1991; zuckoff, 1992). there is an extensive literature showing how seemingly colorblind policies have produced material stratification in resources and, in turn, reproduced whiteness. furthermore, changes in federal tax policies during the 1980s made taxation on goods and services higher than it was for profits from investments (lipsitz, 1995). this colorblind policy has allowed whites to increase their wealth in comparison to blacks (oliver & shapiro, 2006). to illustrate this point, whites held $20,000 more wealth in 1984 than blacks in the united states and increased to $95,000 in 2007 (an over 40% increase, controlling for inflation). a policy aimed at lowering investment taxes on capital gains benefited those owning their own homes and profiting from raised home values, transforming a supposedly neutral policy that advantaged whites who benefited from more home ownership and increased property values due to previous racist policies. similarly, proposition 13 in california granted tax relief to property owners and reduced funds by $13 billion a year for public education and other social services (mcclatchy news service, 1991). with 69% of whites owning homes in california versus 46% of blacks (u.s. census bureau, 2004), this tax relief served to return more wealth to whites. these policies again reproduced advantages for whites. battey & leyva whiteness in mathematics education journal of urban mathematics education vol. 9, no. 2 59 educationally, funding in schools is one way that whites maintain privileges. while the history of brown v. board of education is well known, the fact that we are now at similar levels of segregation in schools as the 1960s (see orfield, losen, wald, & swanson, 2004) means the problems of resource differentials are just as big an issue today as they were 40 years ago (fairclough, 2007; walker, 1996). policies of school funding tied mostly to local property taxes have maintained differential funding for suburban schools at levels twice that for urban schools (kozol, 1991). this well-documented difference impacts teacher quality, curricula, building conditions as well as numerous other educational issues. these differences then play out in instructional resources, quality of instruction, and achievement test scores, further reifying whiteness in terms of “achievement gaps” over historically marginalized students of color. despite these investments in whites—generated through slavery, segregation, and colorblind social reforms—a poll noted that 70% of whites believe that african americans “have the same opportunities to live a middle-class life” (orfield & ashkinaze, 1991). these policies purport colorblindness while advantaging a racial group, whites; however, these policies serve to increase racial stratification rather than ameliorate it. colorblind policies then maintain the guise of neutrality while reproducing whiteness by increasing material benefits due to historic advantages. mathematics education has similar policies such as “algebra for all” and the “common core” that espouse colorblindness, but reproduce or even increase material differences for students (martin, 2003, 2013). the production of racial stratification (material) in course taking or achievement differences then provides evidence validating the higher value of whites, reproducing whiteness (symbolic). bringing a lens of whiteness to policy can support analyses that deconstruct rather than accepts the claimed neutrality. such a dialectical relationship between symbolic and material racism is also at play within mathematics education. the belief in a racial hierarchy of ability in mathematics—namely, whites and asian americans at the top—produce real benefits for these groups. perceptions are then made real as far as how african americans and latin@s are treated in mathematics classrooms, the forms of instruction available to them, and what courses (advanced placement [ap] or not) schools provide; which, in turn, lead to different testing outcomes (gaps). institutions make these ideologies concrete when they provide african americans and latin@s impoverished forms of instruction through tracking and reduced funding in the form of property taxes. this then serves to legitimize the ideology that african americans and latin@s are innately worse at mathematics rather than deconstructing the role of institutions or noting the efforts of educators and communities to combat these racist structures daily. the framework presented here examines multiple levels of mathematics education to document mechanisms that reproduce whiteness both symbolically and materially. battey & leyva whiteness in mathematics education journal of urban mathematics education vol. 9, no. 2 60 mathematics education as a white institutional space in laying out the framework, we discuss three dimensions to consider in documenting how white institutional space operates in mathematics education: institutional, labor, and identity (see table 1). each of these dimensions interacts with the other, but they provide three lenses to capture the operation of whiteness. we draw on a number of scholars in building our framework, but none more so than martin. more specifically, martin (2009, 2013) calls for research on whiteness operating in mathematics education to address forms of racism in relation to achievement, participation, and student learning. sociological work (e.g., feagin, vera, & imani, 1996; moore, 2008) informs martin’s conceptualization of mathematics education as a white institutional space based on four tenets: (a) numerical domination by whites and the exclusion of people of color from positions of power in institutional contexts, (b) the development of a white frame that organizes the logic of the institution or discipline, (c) the historical construction of curricular models based upon the thinking of white elites, and (d) the assertion of knowledge production as neutral and impartial, unconnected to power relations. (martin, 2013, p. 323) not surprisingly, these four tenets specifically informed the institutional level of the framework. for example, martin’s fourth tenet (d) connects to the maintenance of whiteness as neutral and objective, which relates to ideological narratives about whiteness. meanwhile, tenets (a) and (b) speak to the organizational logic of an institution, which includes how power is distributed across demographics. finally, tenet (c) is specifically about history of curricula, although we broaden it to include the history of schools and communities and how they speak to economic and racial segregation. despite our close attention to martin’s tenets, other scholars were also central in building each dimension of the framework. in addition to martin’s influence at the institutional level, moore’s and acker’s respective work is critical. moore (2008) specifically laid out the physical space of institutions including images displayed, history illustrated, and signs of recognition showing the values of specific institutions. therefore, her work is cited under the element of physical space as well as the elements that martin (2013) drew on in his work. in line with martin’s notion of the logic that organizes an institution, acker (2000) used work on gender to examine how organizational logic impacts intersections with race and class through processes, actions, and meanings, maintaining inequities within broader society. disparities in decision-making, control over resources, distribution of work, job security, and opportunities for promotion and recognition are ways to control and leverage power in organizations (acker, 2006). using these mechanisms, institutions can distribute power in a seemingly neutral and objective way while reproducing whiteness. across these four elements battey & leyva whiteness in mathematics education journal of urban mathematics education vol. 9, no. 2 61 are key points of expressing and carrying out an ideology of whiteness at the institutional level. the labor level draws on both acker’s and moore’s work as well. acker (1990) discussed how gendered forms of interaction place more of an emotional burden on women that they must then bear in the workplace. in extending this work, moore (2008) connected this emotional labor to the burden that african americans bear due to whiteness. this emotional labor comes in the form of regulating dissatisfaction, frustration, and anger due to being subjected to deficit views, racial slights, and forced compliance. such regulation is required when these emotions are deemed unacceptable in schools, with expectations of students being unemotional and placid in mathematics, placing the emotional burden of dealing with racism and discrimination on those oppressed. behavioral regulation works similarly in schools, especially through fear of black boys and youth, who the ideology of whiteness degrades as aggressive and violent and therefore in need of being controlled, while white boys and youth are seen as fooling around and playful (gregory, skiba, & noguera, 2010). finally, steele and aronson’s (1995) work on stereotype threat and dovidio’s (2001) work on implicit racism raised the cognitive effects of dealing with racism. together, they have found that deficit framing of historically marginalized groups depresses test scores and hinders group problem solving. whiteness, therefore, serves as a dividing line between those implicitly asked to perform additional cognitive, emotional, and behavioral labor within mathematics education. the last level of the framework, identity, draws most heavily on martin’s work once again. martin (2009) argues that colorblind ideologies and practices marginalize students of color and prevent their positive co-construction of racial and mathematics identities. we want to stress that the conception of identity we utilize is one that is interpersonally constructed and thus negotiated between the individual and the multiple contexts in which she or he participates, which is what we mean by the co-construction of meaning. lewis (2004) discusses schools as key sites of identity formation through racial-ascription processes that distinguish whites from non-whites using markers of otherness (e.g., culture, language, skin color, socioeconomic status). while boundaries of whiteness shift, the markers for being non-white are still signifiers of lower status. for example, lower status is something that white students do not have to deal with as they are assumed to be legitimate participants in mathematics. while not all whites are attributed this legitimacy equally (e.g., female and low ses students more times than not are not afforded the same guarantees in mathematics classrooms as their middles-class, male peers), they still benefit from being seen as white. meanwhile african americans, latin@s, and native americans are delegitimized mathematically, raising the need to prove themselves (see mcgee & martin, 2011). battey & leyva whiteness in mathematics education journal of urban mathematics education vol. 9, no. 2 62 despite such racial de-legitimization in mathematics, it is important to also consider students’ agency in negotiating their racial identities and mathematics success. although martin (2009) uses african americans’ experiences to illustrate these racial struggles in mathematics, his discussion can be extended to other racially oppressed student populations as they “negotiate and resist the racialization processes that attempt to position and confine [them] within an existing racial hierarchy” (p. 325). it is in this light that we consider identity, because as racism acts on students through institutions and interpersonal interactions, students also act back on these dimensions to resist racism in mathematics. table 1 framework of whiteness in mathematics education dimensions elements links to literature institutional ideological discourses martin, 2013 (d); moore, 2008 physical space moore, 2008 history martin, 2013 (c); moore, 2008 organizational logic acker, 2000, 2006; martin, 2013 (a, b) labor cognition dovidio, 2001; steele and aronson, 1995 emotion acker, 1990; moore, 2008 behavior gregory, skiba, and noguera, 2010 identity academic (de)legitimization martin, 2009 co-construction of meaning lewis, 2004; martin, 2009 agency and resistance mcgee and martin, 2011; moore, 2008 before examining each dimension in more detail, it is important to communicate that the dimensions are not independent of each other. instead, they provide three lenses with which to detail and examine the construction of white institutional space in mathematics education. for example, the identities that students develop are negotiated with respect to the ideologies they must navigate as well as the emotional and behavioral norms established with others. therefore, to establish that whiteness is framing a context where one lens or dimension may be primary, the others must be taken into account as well. additionally, we reference the dual nature of whiteness once again. in order to document whiteness, the dual nature of privilege and oppression needs to be considered. as whites are advantaged by the ideology, whiteness positions people of color as culturally deficient, intellectually inferior, and behaviorally aberrant. we detail ways in which each of these dimensions operates using various elements as well as providing indicators for each element. institutional institutional spaces constrain or afford differential access to people, resources, and work. in distributing this access, institutions legitimize certain ideologies battey & leyva whiteness in mathematics education journal of urban mathematics education vol. 9, no. 2 63 through the physical space, positioning of different groups in terms of power, and presentation of history (see table 2). just as with the examples mentioned earlier related to housing, when institutional decisions are made through a colorblind lens, it is easy to predict their impact on historically marginalized communities of color. if a freeway is going to be built in an urban area that is predominantly african american, then we know whose homes will be destroyed. this decision is not racially neutral. much the way that testing and standards shift over time, we can predict the ways in which these policies that distribute resources will impact the organizational logic and physical space of schools and districts. in this way, the institutional dimension of our framework shapes the labor and identity dimensions given that it is responsible for the organization of labor and determines the ideologies and people with whom individuals will develop relationships. this shaping, however, is not to say that people cannot be agents of change in mathematics, but rather that institutions establish what those agents are acting against. table 2 institutional elements with indicators dimension element indicator institutional ideological discourses  racial hierarchy of mathematics ability  innateness of mathematics ability  mathematics as neutral  abstract individualism  meritocracy physical space  concrete representations  school messages  visibility of students  control of physical expression history  histories of schools  patterns of inclusion and exclusion  curricular perspectives organizational logic  distribution of power and work  organizational structure  positioning of stakeholders (e.g., parents) ideological discourses. broad discourses such as colorblindness, meritocracy, and abstract individualism often accompany whiteness in institutional spaces. these examples of symbolic racism, as discussed previously, are helpful in examining the presence of whiteness as has been detailed elsewhere (bonilla-silva, 2003; martin, 2009; moore, 2008; ullucci & battey, 2011). here, however, we focus on specific ideologies common in mathematics education. within mathematics education, whiteness takes the form of racial hierarchies of mathematics ability (martin, 2009), the maintenance of mathematics as a cultureless or neutral domain, as well as the battey & leyva whiteness in mathematics education journal of urban mathematics education vol. 9, no. 2 64 innateness of mathematics ability (ernest, 1991). each of these ideologies plays off the others positioning african americans, latin@s, and native americans as less engaged, intelligent, and mathematically capable (shah, in press). in particular, the innateness of mathematics intelligence aligns with colorblind ideologies in mathematics to produce advantages for whites. the dialectic discussed earlier in terms of symbolic and material racism is important here. for example, if one holds the belief that mathematics ability is solely innate, then a teacher has less responsibility and control in generating student learning, and interestingly, this perspective has been shown to make teachers more susceptible to unconscious stereotyping (jordan, glenn, & mcghie-richmond, 2010; levy, stroessner, & dweck, 1998). this ideology (symbolic), coupled with racial achievement differences (material) in mathematics that are constantly reported in the news and academia, produces the racial hierarchy of ability. the achievement differences are material evidence that innate mathematics ability is not equitably distributed by race. therefore, a belief in innate mathematics ability serves as a colorblind way of unconsciously believing in the racial hierarchy of ability shaped by whiteness. the racial hierarchy of mathematics ability benefits the identities that white and asian american students can construct with the domain while the accompanying discourse of innateness of mathematics ability makes the racial hierarchy stable. evidence of these discourses can be observed in teachers’ and schools’ stable notions of “high” and “low” mathematics students that are then institutionalized in forms of tracking and subsequent differential access to cognitive demand. pervasive discourses in schools about fixed levels of low and high students as well as “honors” or remedial labels construct ways to discuss innateness that link to racial discourses (dime, 2007). in terms of privilege, the discourses are evidenced by the automatic attribution of asian americans and whites as being good at mathematics and surprise when these students struggle. even young students understand the value of different races in predominantly white schools (lewis, 2001). ideologies play out daily, becoming part of classroom routines to the point that students internalize positive associations with whiteness and negative ones with blackness. identifying the ideologies at play in contexts provides a way to note when whiteness is present in mathematics education. physical space. physical manifestations of ideologies also connote power through representations such as office size and placement of different participants. rousseau anderson (2014) urges urban mathematics educators to seriously consider space. bullock and larnell (2015) build on this idea to remove the veil of what is considered “urban” by taking seriously the physical urban space in detailing race and racism in mathematics education. physical representations can come at many levels. for instance, it can come in the form of images, charts, symbols, and objects serving as concrete representations that communicate values and other central aspects of institutions. moore (2008) uses these concrete representations to paint a battey & leyva whiteness in mathematics education journal of urban mathematics education vol. 9, no. 2 65 picture of whiteness in two law schools. pictures that designated notable people, student recognition, and school history passed on messages about who was accepted and welcomed as well as who excelled academically. images, histories, and perspectives of african american and latin@ students can be invisible at times (moore, 2008). this invisibility can contrast with the hypervisibility (higginbotham, 2001) when students feel as though they are asked to speak for their race or when teachers hyper-focus on the “misbehavior” of students of color. aligned with visibility, charts about acceptable behavior can be racialized ways of controlling students. for instance, behavioral norms that promote militaristic rules of order, zero-tolerance policies, and student “uniforms” are clear messages that the school sees students as needing to be controlled. repeated school slogans in schools such as “i’m smart! i know that i’m smart” found in kozol’s (2005, p. 36) work communicate just the opposite. if students were assumed to be smart, there would be no need to repeat these types of mantras. similarly, the lack of these messages in predominantly white contexts is an implicit transmission that students are expected to be intelligent and under control or that these students do not need to see representations of current and historical figures that do not look like them. such transmissions are also a way to perpetuate whiteness; communicating that there is such a limited number of significant african americans or latin@s that white students do not need to know about them. rubel, lim, hall-wieckert, and sullivan’s (2016) research also considers physical space from the perspective of mapping communities and place-based education. building on soja’s (2010) concept of spatial justice, rubel and colleagues connect this work to discuss the ways “unjust geographies” can be researched in mathematics education: injustice leads to the production of “unjust geographies” which can manifest at a microlevel as intersections of the body by unequal politics (e.g., police stop and frisk policies) or at a broader spatial scale as inequitable distributions of resources (soja, 2010). (p. 6) this claim illustrates how ideologies impact space at multiple levels in mathematics education such as treatment of physical bodies, physical representations inside and outside of schools, distribution of resources, the positioning of various educational participants in physical space, and the positioning of communities in a broader geographical sense. these physical representations are material manifestations of specific ideologies and serve to reify racism. history. schools have histories that are inseparable from issues of exclusion, segregation, and differential resources in the united states (e.g., ignoring tax policies on capital gains discussed earlier). with a history of racially constructed access to jobs and wealth in the united states, raising sales taxes while lowering capital gains taxes have no other option but to oppress and advantage various groups. battey & leyva whiteness in mathematics education journal of urban mathematics education vol. 9, no. 2 66 these historical issues contribute to current educational inequality. for instance, a school may have originally been segregated followed by the bussing of african american students, only to see white flight result in home prices dropping and the tax base that determines school funding collapsing. history of inclusion or exclusion, therefore, has an impact on teacher retention, school demographics, and school funding. for example, rousseau anderson and powell (2009) examine the interrelationship between a rural and urban school district. they explore how history influences demographics, economic development, and changes in zoning ordinances to transform the landscape of education in one metropolitan context. in effect, they document how race was central in this history to effect opportunities to learn. the historical context of mathematics education also impacts curricula. curricula present who and which groups have been involved in constructing history. the inclusion or exclusion of groups within curricula communicates to students whose perspectives matter and who is important. in predominantly white settings, teachers and students may feel that multiculturalism does not apply to them and thus result in colorblind curricula (lewis, 2001). the perspective within curricula communicates notions of exclusion, assimilation, resistance, or valuing regarding different cultures and values. martin (2008), for example, describes how the national mathematics advisory panel’s curricular recommendations focused on algebra and other mathematical content to advance white elites’ agenda of international competitiveness. and to connect history with ideologies, one can examine curricula for presentations of mathematics as neutral or cultureless as well. therefore, taking a historical perspective on mathematics education affords many insights into how whiteness preserves privilege, distributes resources, and maintains the status quo. organizational logic. acker (2000) states: “organizational hierarchies constitute and replicate dominance–subordination relations that are characteristic of class” (p. 196). she goes on to make the case around race as well, namely that structures of slavery, segregation, and the exclusion of historically marginalized people of color from certain jobs are still present today in the organizational logic that constructs hierarchies within institutions. these hierarchies can be seen in the distribution of power in a school; for example, who is administrating, teaching, as well as whether parents are seen as influential participants. schools are organizations that situate people in different ways and distribute power and decision-making accordingly. how that power is distributed and to whom it is distributed matters. for instance, parents who are viewed as over-involved can influence or determine curriculum. this influence positions them as having power in contrast to those framed as oppositional in defending their children, uninvolved, or not caring. in these differing logics, parents are granted varied power in schools. the same can be true for teachers and students. organizational logic is what determines who has power, who does what work, and who evaluates whom (acker, 1990). in this distri battey & leyva whiteness in mathematics education journal of urban mathematics education vol. 9, no. 2 67 bution of power, there is the potential to have different races in more privileged and more subservient roles leading to inequitable racial representation in positions of power. this distribution determines different forms of labor, including the labor that is required of students and the extent to which this labor prepares them for future success. along these lines, organizational logic can be seen through the rules that guide behavior, what counts as appropriate emotions, and sanctioned responses. for example, lewis (2004) discusses how predominantly white schools frame anger as inappropriate for responding to discrimination. the sanctioning of “appropriate” responses then goes on to produce more emotional labor for students of color. the organizational logic, therefore, constructs the hierarchies and assumed ways of being that distribute differential forms of labor. labor how labor is divided can reflect the influence of whiteness. as noted earlier, we draw heavily on acker’s (1990) and moore’s (2008) work in documenting the different labor required of whites versus those positioned as less powerful in systems. moore’s (2008) research particularly documents how law schools, operating as white institutional spaces, function to require more emotional labor of students of color as they manage microaggressions doled out by administrators, professors, and fellow students. it is this notion of whiteness producing interpersonal dynamics that requires different kinds of labor among students that we want to highlight here. normative expectations of emotional and behavioral work can restrict students to being certain types of students—controlling them to fit unquestioned cultural expectations. administrators and teachers may form a passive collective, not consciously or daily identifying as white, which only further maintains these expectations as “neutral.” when forms of labor are restricted in such a way that the contributions and behaviors of students of color are not seen as valid, and thus have to put in additional work in managing their emotions and behavior, it can be a sign that whiteness is operating in this context. this then is a way to expose the nonneutrality of the labor in the classroom and other spaces within mathematics education. we use three elements of labor to detail how whiteness can operate: cognitive, emotional, and behavioral (see table 3). cognitive. cognition is interpersonal in the sense that the kinds of mathematical work students are asked to do communicates expectations and messages about what students are capable of doing. the literature on african american and latin@ students is replete with classroom settings that only ask students to replicate procedures, follow worksheets page by page, and that lack the opportunity to engage in cognitive depth (ladson-billings, 1997; lubienski, 2002). teachers and schools that frame these pedagogical choices as the only cognitive work necessary for students subjugates students with respect to a white ideal. additionally, how authority is distributed, both for classroom procedures and the mathematics, also speaks to battey & leyva whiteness in mathematics education journal of urban mathematics education vol. 9, no. 2 68 whether a teacher holds expectations that students can self-monitor their behavior and gain command of the mathematics (nasir, hand, & taylor, 2008). if these ways of parsing classroom cognition are coupled with ideologies of a racialized hierarchy of mathematics ability, then they are signs that whiteness is operating. but it is also more complex than this. for instance, even in a mostly african american classroom, some students may have more access to content and authority than others. if the students who are seen as more capable fit norms of white participation, then whiteness is still central. patterns as to which students have access to cognitively demanding tasks can be quite telling. table 3 labor elements with indicators dimension element indicator labor cognitive  differential cognitive demand  distribution of mathematics authority  academic expectations  stereotype threat emotional  management of emotional experiences  regulation of emotion  range of emotional experiences allowed behavioral  discipline  management of behavior  language norms  teacher praise/acknowledgment steele and aronson’s (1995) work on stereotype threat can also be seen as the result of added cognitive labor due to being in situations where stereotypes are applied. some of the work around stereotype threat examines when racial stereotypes are primed, or explicitly accessed, and the effect can be seen on test scores. but their work also means that even when discourses such as a racial hierarchy of mathematics ability are passively acting in an environment, latin@s, african americans, and native americans must manage additional cognitive load while engaged in completing mathematical tasks. steele (1997) also found that this additional load may cause disidentification with school altogether (discussed later in the identity dimension of the framework). monitoring the cognitive demand provided by the teacher and the cognitive load managed by students provides evidence about the extent to which whiteness is constricting the classroom environment. teachers who are enacting microaggressions or providing tasks of low cognitive demand are reproducing the racial hierarchy of mathematics ability, and thus positioning underserved students at the bottom. emotional. coping with discrimination and racism in everyday experience requires significant emotional labor in terms of sadness, frustration, and anger battey & leyva whiteness in mathematics education journal of urban mathematics education vol. 9, no. 2 69 (moore, 2008). this emotional labor is intertwined with undue cognitive burden on students as well. dovidio (2001) found that when solving mathematics problems, african american students working with white students whose racial bias was implicit performed slower on the tasks than those working with white students who were unbiased (the fastest groups), and even slower than african americans in groups with white students whose racial bias was explicit (second fastest groups). these findings demonstrate the impact of emotionally processing racial interactions, particularly those that are ambiguous, but even explicitly racial interactions. in addition, this emotional labor is connected to slowing students’ cognitive mathematical engagement as well. schools and classrooms often do not provide the time, space, or support for students to process these racial experiences and emotions. when students do process or exhibit these emotions, they can be seen as angry, aggressive, or violent rather than struggling with a complex and unfair world. moore (2008) discusses how law schools ignore and undervalue this emotional labor: coping with everyday racism in the law school frequently produces frustration, anger, or sadness, but the institutional logic of the law school does not recognize expressions of these emotions as legitimate. students are thus forced to manage their emotion in order to avoid further marginalization…. this demands that students of color perform invisible and emotional labor that their white counterparts are not required to perform. both in the law school and in the profession of law, this labor is expected of law students of color, yet it goes unrecognized and unrewarded. (p. 31) additionally, students must manage the ways in which they express emotions to avoid deficit discourses about being perceived as argumentative, angry, aggressive, and a multitude of other negative associations. when students of color are expected to handle experiences that they consider unfair in a calm, dispassionate, and disconnected way, whiteness is restricting acceptable ways of grappling with the emotions of discrimination and racism (moore, 2008). mcgee and martin’s (2011) work is also illustrative here. they discuss the impact of dealing with daily hassles and stereotype management in mathematics education. even for students achieving mathematical success, students still found the management of hostile environments to be emotionally debilitating. whether it was the effort to prove a stereotype wrong or “fronting” to project conformity, the students experienced emotional distress and exhaustion. therefore, more racially hostile environments produce more emotional labor from racism. again, we want to highlight the complex ways in which emotional labor impacts cognitive functioning and how overlapping these elements are across this research. behavioral. one way in which labor is handled is by deeming certain student behaviors appropriate and others inappropriate. this mishandling has immense consequences in classrooms as harsh and recurrent discipline has been found to fre battey & leyva whiteness in mathematics education journal of urban mathematics education vol. 9, no. 2 70 quently lead to missed instructional time and expulsion from school for male african americans and latin@s in particular (gregory, skiba, & noguera, 2010). african american students, in particular, are two to four times more likely to be referred to the office for disciplinary reasons than their white peers as well as being punished more severely for their behavior (skiba et al., 2011). additionally, due in part to “behavioral reasons,” african americans are over-identified for special education and placed in comparatively more restrictive learning environments, excluding them from access to mainstream instruction (skiba, poloni-staudinger, gallini, simmons, & feggins-azziz, 2006). within mathematics, this can take the form of deeming certain ways of language use as inappropriate for mathematical argumentation or by requiring students to sit still in seats in regimented ways (battey, 2013b). furthermore, whiteness can function by valuing behaviors of white students over others in subtle ways, resulting in the implicit communication that white students’ language and behavior are deemed more appropriate within the mathematics classroom. when students align with white ideals of classroom behavior, their actions will more likely be praised. additionally, when white boys and youth, for instance, act out, it is often seen as instances of immaturity and playfulness, but certainly not aggression or violence, leading to less severe punishment and discipline (ferguson, 2000). when african american and latin@ students do not align with white norms of behavior, maybe through being too argumentative, too quiet, too excited, or abrasive, we would expect to see behavior being called out, positive behaviors going unnoticed, and a hyper-focus on misbehavior, leading to increased discipline and eventually suspensions and expulsions (skiba et al., 2011). in classrooms that employ such behavioral control, despite substantive mathematical contributions from students of color (see battey, 2013b), it is evidence that a broader racial ideology is at play. the ideology is enacted through the positioning of students cognitively, emotionally, and behaviorally in classrooms and thus influences their identities—namely, the extent to which they see themselves as legitimate mathematical participants. identity martin (2009) defines mathematics identities as “dispositions and deeply held beliefs that individuals develop about their ability to participate and perform effectively in mathematical contexts and to use mathematics to change the condition of their lives” (p. 326). the construction of mathematics identities, however, is not a strictly personal, internal process as it is constantly negotiated with ideologies, institutions, and interpersonal encounters. the organizing white frame too often relegates african americans, latin@s, and other non-whites as incapable and thus grants unquestioned legitimacy to whites in educational spaces (moore, 2008). this organizing frame aligns with martin’s (2009) concept mathematics as a racialized form of experience. the social construction of whiteness as a privileged battey & leyva whiteness in mathematics education journal of urban mathematics education vol. 9, no. 2 71 identity in everyday society is maintained in classrooms and other mathematics spaces through inequitable learning opportunities as well as feelings and experiences of academic de-legitimization experienced by historically underserved students of color. as a result, a racialized hierarchy of ability is constructed in mathematics education (as discussed in the institutional section) that shapes students’ identities in seeing themselves as doers of mathematics. table 4 identity elements with indicators dimension element indicator identity academic (de)legitimization  identification with mathematics  legitimacy of intellectual ability co-construction of meaning  hierarchy of mathematics ability  peer perceptions of each other  hypervisibility/invisibility agency and resistance  relationship with deficit discourses  forms of (dis)engagement  association with peer group academic (de)legititmization. mathematics contexts that function as white institutional spaces require students to negotiate academic legitimacy with a racialized hierarchy of mathematics ability based on white norms and values. understanding mathematics identities, therefore, can only be attained by detailing processes of negotiation with racialized discourses as opposed to traditional analyses of achievement gaps between different races (martin, 2009). with whites and asian americans—considered “honorary whites” (see bonilla-silva, 2003)—at the top of the hierarchy of mathematics ability, whiteness in mathematics classrooms operates in ways that whites are assumed or assume themselves to be innately intelligent in mathematics. conversely, the legitimacy of historically marginalized students of color is always under question so that they may feel the need to prove themselves mathematically capable with respect to white views of success, which structure the academic spaces. even with past success, their ability will still most likely be questioned (leyva, 2016, this issue). deficit perspectives of students of color and their mathematics ability stem from ideological discourses; in turn, these discourses position students as illegitimate members of mathematics classrooms resulting in poor relationships with teachers, lower-quality instructional experiences, and at times disidentification with mathematics (oppland-cordell, 2014; spencer, 2009). therefore, whiteness constructs spaces where some are assumed to be legitimate participants and others are delegitimized. likewise, when the behaviors, language, and other presentations of self among students of color are aligned with the ways that whites and honorary whites do mathematics, we would expect them to be more welcomed into the mathematics battey & leyva whiteness in mathematics education journal of urban mathematics education vol. 9, no. 2 72 classroom, although this could still be contested at any moment. this uncertainty can pit african american and latin@ students against one another such that students might position each other as less latin@ or less african american if they do succeed mathematically (leyva, 2016). it is important to note that it is the white institutional space constructing this positioning that can play out between peers. the alternative is a classroom that promotes multiple ways of being successful mathematically and does not pit certain students as more or less legitimate in the way that they engage with mathematics (featherstone et al., 2011). co-construction of meaning. students construct identities through relations with other people and the institutions in which they participate. moore (2008) discusses peer perceptions of academic support programs in law schools where some white students thought that students of color were admitted to the school solely based on race rather than earning it by merit. this anti-historical view ignores the reasons for programs that remedy institutional racism. it also perpetuates whiteness by not recognizing the material racism that produced and continues to produce differential access to educational quality. however, as institutions leave these ideologies and material racism unaddressed, they participate in limiting spaces for students of color to construct identities that counter the racial hierarchies contained by whiteness. the ways in which ideologies are enacted have direct consequences for schools. the explicit and implicit ways in which people and institutions pass on such messages are critical for how students develop their mathematics identities. for instance, ability grouping or tracking along racial lines creates a material manifestation of racism that sends messages to students about the racial hierarchy of ability (lewis, 2004). as students are placed in these structures, they act in relation to the position to which they are assigned. similarly, teacher comments about low and high students or needing to learn the “basics” pass on messages more overtly (battey & franke, 2015). whether tracking, ability grouping, or overt comments from teachers, schools construct what being “good” at mathematics means and place students along a continuum of ability. recent work has explored how students are positioned with peers and in terms of the whole class to better understand how advantaged and disadvantaged roles are constructed (engle, langerosuna, & mckinney de royston, 2014; esmonde, 2014; langer-osuna, 2015). for example, white students may query an african american student if they are in the right place when they attend an ap mathematics class, relaying the message that they do not expect success from african americans in mathematics. students, however, are not naïve to the ways in which they are positioned institutionally and interpersonally. understanding how students react to being constructed as certain kinds of mathematical doers speaks to the ways in which students are positioned. they develop identities in relation to these positions and decide how to reposition themselves in response. battey & leyva whiteness in mathematics education journal of urban mathematics education vol. 9, no. 2 73 agency and resistance. such racialized experiences in mathematics include the positioning of historically marginalized students of color as both invisible (feagin & sikes, 1994) and contradictorily hypervisible (higginbotham, 2001). students are invisible because their perspectives, experiences, and history are not represented, but hypervisible because there may be so few students of color in academic spaces, which runs the risk of tokenism, essentialism, and having to represent their race (moore, 2008). examining both the invisibility and hypervisbility of students of color in mathematics spaces with respect to being successful in highertracked mathematics courses, for example, shows whiteness systemically acting within educational institutions. what is just as important to note are students’ responses to these forms of positioning in mathematics. for instance, students can respond by objectifying or rejecting their racial selves, thus removing their experiences and histories from discourse (moore, 2008). responses can also come in the form of seeing oneself as an exception to racial hierarchies (leyva, 2016). students can be strategic with peers in downplaying their academic success or purposefully disengaging in academic settings (moore, 2008). unfortunately, these responses tend to reproduce white institutional spaces and accompanying ideologies. however, it is in noting the need for these responses to broad discourses that we as a field can document how whiteness is operating. another option in response to whiteness is to fight ongoing battles with white racial norms. mcgee and martin (2011) call this option stereotype management. in their analysis, they found that successful black mathematics and engineering students were constantly aware that their racial selves were undervalued and moved from proving stereotypes wrong to more internal motivation in achieving academic success. this “proving” can be exhausting and is often done more collectively. racial groupings, thus, provide a space for emotional coping and support in navigating white spaces (tatum, 1997). these racial groupings, in turn, allow students to build collective consciousness and resist the internalization of an ideology of whiteness (feagin, 2006). confronting white institutional spaces collectively allows students to build a shared narrative to view their individual experiences as a broader structural problem (moore, 2008). moore (2008) also states, “however, it also adds to the pain of racism experienced by these students because they become aware of how frequent and common racism is in the law school when they learn of the stories of their peers” (p. 131). while building collective consciousness can be a racial support, the fact that it is needed at all signifies that racialized discourses are being perpetuated in the context. unchallenged racial discourses keep individual experiences of race internal for both whites and students of color. however, for students of color, doing so can be detrimental. when racial discourses are unchallenged, student of color may disassociate from their race, community, and history to succeed mathematically, or internalize the discourse. battey & leyva whiteness in mathematics education journal of urban mathematics education vol. 9, no. 2 74 discussion we wish to illustrate two ways of using this framework in different contexts. the first is in the context of a predominantly white school. naming race as operating in white contexts can be difficult given that whites often avoid the mention of race. nevertheless, markers of whiteness can be key in these settings in terms of privileging some and devaluing others. markers that link even white students to less status in terms of sex, gender, ses, or geographical accents (e.g., rural vs. urban) serve to differentiate privileges. this differentiation is built on whiteness setting particular norms of what it means to be white based on middleand upperclass, male standards. institutionally, examining the distribution of students across the school and the ways in which teachers and administrators engage with lowerstatus white students or the few students of color attending the school can be quite enlightening. for instance, tracking might place these students in lower mathematics courses showing the organizational logic of the school. additionally, the lack of attending to race and the invisibility of perspectives and histories of people of color may be evident in ways such as parents of color not being on the parent–teacher association (organizational logic), no images other than martin luther king, jr. present on the school walls (physical space), and curricula that do not represent anything but white problem contexts (history). specifically, with respect to the labor of mathematics in classrooms, whiteness would force teachers to hyper-focus on the behavior of lower-status white students or students of color, while their mathematical thinking remains invisible. additionally, those lower-status white students who are doing well would need to constrain themselves to white norms of behavior such as sitting quietly and only talking when called on as well as exhibiting speed and accuracy to be perceived as mathematically intelligent. the same would be true of students of color in this context. teachers and white peers may frame students of color as not belonging in mathematics classrooms (academic delegitimization), or are surprised when they are present. this framing prompts students of color to respond by forming collectives (agency), feeling the need to prove themselves (emotional labor), disidentifying from their race (identity), or rejecting mathematics (resistance). the point of this framework is to identify these behaviors not individually among specific students, but as responses to whiteness and institutional racism operating in mathematics education. we use this example to illustrate the need for mathematics education researchers to document the presence of race within predominantly white spaces. additionally, whiteness can be viewed within predominantly african american and latin@ contexts. in this case, we consider a historically white immigrant community; white immigrants who moved out when historically marginalized people of color began moving into the area. institutionally, with the influx of african americans, housing prices have been reduced, which has limited school resources battey & leyva whiteness in mathematics education journal of urban mathematics education vol. 9, no. 2 75 as well. simply looking at the demographics of the teachers and administrators versus the students is one sign of who has the power in the school (organizational logic). there is an african american principal, for example, but half of the teachers are white in a school in which the student body is 100% african american and latin@. students are required to wear uniforms, an act of controlling physical expression (physical space). similar to the predominantly white context, regardless of the race of the administrators and teachers, if students are succeeding by aligning with white norms of intelligence and behavior (labor), then whiteness is present here as well. while a racial match between the teacher and student can be beneficial to students (gershenson, holt, & papageorge, 2016), teachers of color can also perpetuate the same white norms as well. discourses of “acting white” (fordham & ogbu, 1986; stinson, 2011), which are intended to narrow the diverse ways in which african americans and latin@s can act, are a form of whiteness constraining ways of being (co-construction of meaning). the predominant mathematics pedagogy is “back to the basics” (ideology), where students must know the basics before moving onto more complex mathematics and problem solving in meaningful contexts. as a result, the students rarely if ever get access to meaningful mathematics (cognitive labor). finally, the majority of teachers perceive parents of color as either absent or aggressively fighting the school (ideology), and are too often excluded from participating (organizational logic). the point here is that if whiteness is systemic, it does not depend on white actors or villains. it can be internalized and reproduced by even those who do not intend to perpetuate racism. while these are two brief examples to illustrate some approaches to using the framework, we want to highlight that entering an analysis through any of the framework dimensions would suffice. certainly, interview studies can show the discourses that students are navigating and the labor that they have to employ to succeed in mathematics classrooms. likewise, researching at the classroom level can show not only the interpersonal interactions but also the types of identities evident as well as the physical space and discourses that are accessed by teachers and students. or a researcher could look at documents, policies, and the community in detailing whiteness at play in a particular context. there are many ways to do this work, but we need to begin attending to the dynamics of racism by foregrounding the operation of whiteness across contexts within mathematics education. conclusion at the beginning of this article, we cited martin (2009) calling for desilencing race in mathematics. for us, this entails destabilizing the racial neutrality of whiteness, something that has received little attention to this point. doing so, however, does come with some cautions. in many ways, whites are already centered in conversations around race as the racial norm or standard-bearer. this cen battey & leyva whiteness in mathematics education journal of urban mathematics education vol. 9, no. 2 76 tering is implicit in talking about other groups having deficits or gaps in comparison to white standards. in making some of these comparisons explicit, there is a danger in re-centering whiteness. that is why discussions of whiteness must go beyond simply discussions about white privilege so as to name and to counteract the mechanisms and institutional ways in which white supremacy in mathematics education reproduces subordination and advantage. the framework presented here is intended to support researchers and teachers to document the operation of institutional white supremacy in mathematics, through ideologies, physical space, interactions, and students’ agency and resistance. our hope is that as this work continues, this framework will become more detailed and targeted toward the ways that privilege and oppression get reproduced in mathematics education. additionally, as we gain a better understanding of whiteness in mathematics education, this understanding could serve the field in counteracting its effects among historically marginalized students of color. we think moore’s (2008) call in law schools is pertinent for mathematics educators as well: “deconstructing the white institutional space will require that we discard this constraining white frame and center the experiences and voices of students of color in the project of identifying and eliminating the structural remnants of our white racist past” (p. 163). the framework detailed here is an attempt to support mathematics educators in deconstructing and discarding the white frame of mathematics education. references acker, j. 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(1992, october 9). study shows racial bias in lending. boston globe, 1, 77–78. http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/34/12 journal of urban mathematics education december 2016, vol. 9, no. 2, pp. 11–28 ©jume. http://education.gsu.edu/jume brian r. lawler is an assistant professor in the department of secondary and middle grades education at kennesaw state university, 1000 chastain rd., kennesaw, ga, 30144; email: blaw@kennesaw.edu. his research interests include power and privilege in mathematics education, and system change toward humanizing and equitable mathematics practices in schools. response commentary* to rectify the moral turpitude1 of mathematics education brian r. lawler kennesaw state university first, i must confess that over the past few years i have been gravely disappointed with the white moderate. i have almost reached the regrettable conclusion that the negro’s great stumbling block in the stride toward freedom is not the white citizen’s counciler or the ku klux klanner, but the white moderate who is more devoted to “order” than to justice; who prefers a negative peace which is the absence of tension to a positive peace which is the presence of justice; who constantly says: “i agree with you in the goal you seek, but i can’t agree with your methods of direct action;” who paternalistically believes he can set the time-table for another man’s freedom; who lives by the myth of time and who constantly advises the negro to wait until a “more convenient season.” shallow understanding from people of goodwill is more frustrating than absolute misunderstanding from people of ill will. lukewarm acceptance is much more bewildering than outright rejection. – dr. martin luther king, jr., 1963 rofessor danny bernard martin delivered a different talk at the national council of teachers of mathematics (nctm) research conference in april 2015 (published by jume later that year); he established an important concern that few in the room would soon forget. martin, like king (1963), was not interested in waiting for a more convenient season to achieve a positive peace in mathematics educa 1 the notion of “moral turpitude” is a legal concept, referring to conduct that is thought to be against community standards of justice or morality. i do not claim here that any person or formal organization ought to be brought to justice; however, it is certain that the current practices of mathematics education are unjust and fail even its own moral standards. *editor’s note: in the spring/summer 2015 issue (vol. 8, no. 1) jume published, as a commentary, dr. danny bernard martin’s invited plenary address delivered at the nctm research conference april 2015 in boston, massachusetts (martin, 2015). in the fall/winter 2015 issue (vol. 8, no. 2), jume published a response commentary, authored by drs. diane j. briars, matt larson, marilyn e. strutchens, and david barnes (briars et al., 2015). the response commentary here continues this important discussion; we invited others to keep things going while they are still stirring (see “contributing a commentary to jume: keeping things going while they are still stirring”). p http://education.gsu.edu/jume mailto:mailto:blaw@kennesaw.edu http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/308/193 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/308/193 lawler response commentary journal of urban mathematics education vol. 9, no. 2 12 tion. speaking truth to power (rustin, 2012), he questioned whether nctm could be the organization to create the reform needed to change the conditions of the collective black (bonilla-silva, 2003) in mathematics education. or is nctm, the “public voice of mathematics education” (nctm, 2014, p. ii), this white moderate, seeking an absence of tension, more dedicated to order than to justice. in his invited talk, martin (2015) expressed his disappointment with the “white moderate” position that nctm has taken. he recognized that a nearly thirty-year history of documents, policies, and position statements from nctm reflect a nearly static concern with standards for mathematical content, instruction, and assessment (berry, ellis, & hughes, 2013), yet there has been little if no reduction of dehumanizing practices or positive impact on inequities in student achievement. our profession, mathematics education, seems to lack the will to shift from an orientation toward mathematics and mathematics education that systematically oppresses, or fails to reduce the inequities of the larger society. representatives of the nctm leadership (briars et al., 2015) authored a response to martin on behalf of nctm, set as a call to action. in that response, they restated the many efforts toward more equitable mathematics education that have been part of its recent history, and recognized there is more work to do—as they had done previously: “over the past twenty-five years, we have learned that standards alone will not realize the goal of high levels of mathematical understanding by all students. more is needed than standards” (nctm, 2014, p. vii). in this call to action, nctm (i.e., briars et al., 2015) invited the mathematics education community to participate in the conversation to consider what more is needed, committing to using its role to strive for improvement. i commend the organization for this stance; it is my intent to use this forum to take them up on the invitation to consider action. while there is great complexity in the challenge for nctm to end its participation in the stratification of society, i focus in this paper on three core issues: what counts as mathematics, explicit pedagogical attention to how mathematics works in society, and revaluing mathematics as human activity. these three issues could be considered ontological, epistemological, and axiological concerns; fundamental structures or underlying worldviews of the mathematics education community that in their present state obstruct the pursuit of a more just mathematics education. to forge a liberatory and re-humanized mathematics education, i suggest each of these core issues must be addressed. the changes i propose to each issue, while seemingly radical, are insights recognized for decades within our own field. prior to the discussion of these three issues, i elaborate concerns raised by professor martin (2015) then respond, reflecting on my 25-year career in mathematics education. following elaboration on the three issues, i argue that a new organization must emerge to speak back to nctm, wholly convinced that the nature of nctm as an organization could never direct nor condone the work that would lawler response commentary journal of urban mathematics education vol. 9, no. 2 13 radically change the collective conditions of african american, latin@, indigenous, poor, queer, and otherwise underserved students. a consistent history of the core ideas in martin’s (2015) invited talk, several resonated strongly with me. first, the various standards for content and practice, teaching and professionalism, and assessment are far more similar than different over a nearly thirtyyear history. this lack of advancements in ways of thinking about content, pedagogy, and assessment suggests an inadequate interest for a positive peace. second, standards and similar documents reflect a particular ideology of what counts as mathematics and mathematical ways of thinking. this ideology is presented as both non-negotiable as well as necessary for a successful life after schooling. yet many adults find happiness and comfort in life, but do not have a level of mathematical literacy called for in these standards (ernest, 2000). and third, given the lack of significant progress on equity targets in mathematics education, it strikes me that very different ideas and very different voices are essential to make change. the points raised by martin (2015) are not entirely new (cf., bishop, 1990; frankenstein, 1983); mathematics education functions in a manner to preserve itself. foucault (1981) recognized, “any system of education is a political way of maintaining or modifying the appropriation of discourses, along with the power and knowledge they carry” (p. 64); mathematics education is no different. for example, martin’s remarks during the talk provide an analysis of principles to actions: ensuring mathematical success for all as an exemplar of “race, racial projects, and mathematics education” (martin, 2013), an essay published in the nctm periodical journal for research in mathematics education. as martin notes in that essay, questions such as whose interests are served by mathematics education? persist, having been considered by critical mathematics educators for several decades. it is curious why now nctm responds. more importantly, an honest analysis of this condition would likely be important to understanding how impactful changes can be achieved, what alternate voices might be heard. nctm’s mission charges it to be the public voice of mathematics education. while nctm is to be commended for publications that reflect many voices representing the diverse perspectives of mathematics educators, its stance as the public voice remains problematic. as martin (2011, 2013, 2015; see also martin, gholson, & leonard, 2010) argues, there is a need for nctm as this public voice to recognize and then interrogate its position as an institutional space of whiteness serving a majority white audience. this status is likely not by design, rather a historical product of a deeply racist society and educational system, a product of both physical and psychological colonization by european settlers and western thought. the danger of this status is that nctm is unlikely to recognize possibility that lies out lawler response commentary journal of urban mathematics education vol. 9, no. 2 14 side its paradigm. furthermore, because of its responsibility to its members (recall, “the voice of mathematics education”), nctm is unlikely to take a stance that would not be readily embraced by the (predominantly white) mathematics education community. because of its status, nctm is unlikely to advocate for children over mathematics. because of its incorporation, survival of the business is the first priority. ultimately, nctm operates in service to the white majority: its economic desires, its rationalities and sensibilities, its benevolence. martin (2015) claims that given this frame, gains for the collective black can only come hand in hand with benefits for whites. his is not a singular perspective; modern sociology recognizes these conditions as the present operations of racism in modern day america, where not only the criminal justice system but also the educational system is creating a new jim crow (alexander, 2010; thomas, 2013), a uniquely american approach to ensuring a caste social system. the need for introspection nctm’s actions since the late 1980s occurred within a larger neoliberal economic, political, and social context, generally a turn toward free market ideology and individualism.2 for example, nctm’s policies typically came with a message that there are many ways a teacher can support students to achieve the standards; we will not prescribe any particular pedagogical strategy. furthermore, the context has allowed for schools to view the student as “being solely responsible for the consequences of the choices and decisions they freely make” (thorsen, 2006, p. 208). these actions fail to interrupt (if they do not create) a bifurcated educational experience that works more insidiously than to create simple discrepancies such as achievement gaps. in its present form, mathematics education operates as a centaur state (wacquant, 2014), liberal at the top and punitive at the bottom. the enterprise of mathematics education flouts egalitarian ideals, evidenced by cries for mathematics for all and strong support for equity—as seen routinely in conference programs as an “equity strand.” attention to mathematical processes, habits, or practices value problem solving, discourse, and the conceptual-oriented thinking of mathematicians. yet for students of the collective black, they are more likely to experience mathematics classrooms that focus on procedures and memorizing (davis & martin, 2008), skills for employability or servitude. all the while, classrooms that include children of society’s more privileged focus on higher order reasoning, problem solving, and discourse (anyon, 1980; ladson-billings, 1997; lubienski, 2002). 2 liberalism is generally a strategy toward the prevention of social conflict. neoliberalism has emerged as economic policy grounded in ideas of free market capitalism and the glorification of individualism. lawler response commentary journal of urban mathematics education vol. 9, no. 2 15 sociologist loïc wacquant (2014) argues that neoliberal policies for addressing inequalities experienced by marginalized populations, as we see in mathematics education, leads to the “criminalization” of poverty (i.e., the creation of the collective black). an example of this sort of educational criminalization is the deficit orientation to the learner, an ideology baked into our discourse. further examples of neoliberalism in mathematics education include brutal assessment policies, which work to disproportionately hold children of the collective black further and further behind;3 policies that emerged from the federal elementary and secondary education act of 2002 create shutdowns or takeovers of a disproportionate number of urban schools (sunderman & payne, 2009); research programs yield pedagogical strategies focused on deficit, such as student misconceptions (e.g., smith & stein, 2011); and the notion that certain families do not value education (aguirre, mayfield-ingram, & martin, 2013) allows for lessened expectations. this criminalization of children leads to heighted punitive results such as increased suspensions (advancement project, 2007) and more years repeating math classes (e.g., fong, jaquet, & finkelstein, 2014), leading to our grave awareness of the efficiency of the school to prison pipeline (amurao, 2013) for children of the collective black. the centaur state of mathematics education creates liberal ideals with minimal oversight for those at the top, and repressive practices with strict controls for those at the bottom. this mathematics education is systematically failing all our children—whether by oppression or by creating a false sense of righteousness. martin (2015) argues well that the negative outcomes experienced in mathematics education by the collective black serve to support larger economic, social, and political agendas. in nctm’s role as the public voice of mathematics education, i expect the recognition of and calls to action against such injustices. these calls, however, are not enough; martin has challenged nctm with introspection, to consider how it may be complicit. a liberatory mathematics education the present centaur state of mathematics education serves to sediment the neoliberal (and neocolonial) standardization agenda. the particular mathematics knowledge and ways of knowing put forth by nctm is to format (skovsmose, 1994), or fabricate (lawler, 2012), the child. here i suggest shifts in three principles foundational to mathematics education, necessary, yet not sufficient for a radi 3 for example, “the achievement gap between children from highand low-income families is roughly 30 to 40 percent larger among children born in 2001 than among those born 25 years earlier” (reardon, 2011, p. 91). lawler response commentary journal of urban mathematics education vol. 9, no. 2 16 cal4 change that may allow for a liberatory and re-humanized mathematics education. the first is ontological—the status of mathematical knowledge; the second, epistemological—an interrogation of how mathematics education is complicit in racism; and the third, axiological—what a re-humanized mathematics education should value.5 i point to these underlying philosophical principles that have invisibly guided mathematics education to its present state of moral turpitude. rethink the ontological status of mathematics the present ontological status given to mathematics reflects western logic; a way of thinking that arose during the age of reason. this current status reflects the realist (or platonist) tradition, which emerged from rationalists of 17th century europe. they claimed that all knowledge could be acquired on the power of reason alone; mathematics is the model for such knowledge. furthermore, this mathematical knowledge is considered to exist independent of human experience. the realist orientation to knowledge allows for metaphors such as discovery for learning, and guidance or facilitation for teaching (davis, sumara, & luce-kapler, 2015)— metaphors common to present mathematics education. this ontological perspective persists for several reasons. among them is that many working mathematicians are mathematical realists. they see their work as discovery of naturally occurring objects, not the products of their humanity or constructions of their mind. davis and hersh (1981) suggested, “the typical mathematician is a platonist during the week and a formalist on sundays” (p. 321). that is to say, mathematical objects are real, they exist “outside the space and time of physical existence” and “quite independent of our knowledge of them” (p. 318). a second force maintaining this ontology is how it has so successfully sorted and ranked our current society by defining levels of intelligence (lakoff & núnez, 2000; lawler, 2005); mathematicians are often thought of as brilliant, having attained the highest form of knowing. an example of this societal belief can be found in our print media: “one reason why people who learn more mathematics earn more is because doing maths makes you smarter and more productive” (schrager, 2009, ¶3). as a result, those who possess knowledge in the domain (i.e., the discipline of mathematics) are “more aligned with communities of practice that hold more power” (nasir, hand, & taylor, 2008). this sort of status bestowed upon the nobles precluded them from alternate considerations of the ontological status of mathematics that have the potential to remove them from the court. 4 a shift in these foundational principles may reflect the notion of violence summoned by martin (2015). 5 for the purposes of this article, i bend the epistemological argument to attend to how the noetic emphasis of a realist ontology sharply limits what counts as mathematics, and thusly can be used to sediment hegemony. lawler response commentary journal of urban mathematics education vol. 9, no. 2 17 this present, persistent ontology afforded to mathematics produces innumerable iniquities, one being that it allows for deficit language in mathematics; we focus on misunderstandings—not possessing the correct, enlightened way of knowing. foucault (1965) writes that this sort of deficit orientation to forms of discourse, knowledge, or ways of knowing pathologizes not only the discourse but also the person. mathematics educators are so steeped in the dominant, realist, “standardized” mathematics (gutiérrez, 2007) that we can no longer imagine, let alone have the ears to hear, our students’ subjugated knowledges (foucault, 1980)— mathematical ways of thinking and knowing that are left out, opposed, or ignored by the dominant culture. we can only not hear the brilliance of the collective black. what is needed is a counter-discourse, oppositional to the standard system of mathematical knowledge and knowing. this counter-discourse must reveal how the accepted knowledge is built on exclusion and confinement. it is within these hidden forms of knowledge that we may recognize the limits of the knowledges that disqualify them. this sort of disruption of western ontologies already exists within mathematics education. one example is in the ethnomathematics tradition; the dominant view of mathematics is recognized as only one of many (see powell & frankenstein, 1997). similar perspectives emerge in relational (belenky, clinchy, goldberger, & tarule, 1986; thayer-bacon, 2003) and indigenous epistemologies (battiste, 2013; sarra, 2011). in some of these perspectives, the learner is characterized as a mathematical author (povey, burton, angier, & boylan, 1999). povey and colleagues argue that this shift in perspective on “author/ity amongst teachers and learners will support a renegotiation of the relations of dominance embedded within current conceptions of the nature of mathematical knowledge” (p. 244), a disruption that has the potential to revalue whose mathematics counts. there is a second, ignored tradition in mathematics education that interrupts the realist ontology of the dominant orientation to mathematics—the constructivist orientation to knowing and learning that arose in mathematics education just prior to nctm’s first standards. smock and von glaserfeld (1974) posited that knowledge is not passively received through the senses or by communication; rather the cognizing subject actively builds it up. in other words, knowledge is the result of a self-organizing process, a human construction generated in interaction: “coming to know is a process of dynamic adaptation towards viable interpretations of experience. the knower does not necessarily construct knowledge of a ‘real’ world” (von glaserfeld, 1990). constructivism suggests an ontology in which “knowledge is not the commodity the tradition of western philosophy would have us believe” (von glaserfeld, 1988, p. 83).6 unfortunately, the radical ontological 6 quoting montaigne, “la peste de l’homme, c’est l’opinion de savoir” or “mankind’s plague is the conceit of knowing” (von glaserfeld, 1988, p. 83). lawler response commentary journal of urban mathematics education vol. 9, no. 2 18 notions of constructivist epistemologies that briefly enamored the profession (davis, maher, & noddings, 1990) failed to take hold.7 presently, many applications of constructivist theory emphasize the activity of the learner, but fail to shift the ontology (lawler, 2014). since constructivism’s explosion and quick departure in mathematics education, there have been others whose work disrupts the ontological status of mathematics from a sociocultural and sociopolitical orientations (see, e.g., gutiérrez, 2013; nasir et al., 2008) rather than psychological. yet these, too, have yet to find traction. the implications for the proposed ontological shift from one to many mathematics are systemic. fundamentally, it demands a re-humanization of both mathematics and mathematics education. could the institution and pedagogies of mathematics education recognize many mathematics and many ways of being mathematical? such an ontological shift suggests a pedagogy focused on listening8 rather than telling, a move away from authority and control, evoking new relations of power. teachers listen to form conjectures of student conceptions, serving to generate for the teacher a tree of potential new ways of knowing for the child. the teacher’s role becomes to design an environment that may occasion the emergence of new mathematical ways of knowing (davis, 1996). the child leads, the teacher follows. confront the role of racism in mathematics (and mathematics in racism) the second foundational shift i propose is for mathematics education to be explicit about the consequences of and its complicity in racism and similar oppression, an epistemological emphasis on how mathematics functions. at present, the field is nearly silent on the roles of racism in mathematics (cf., dime, 2007; powell, 2002; spencer & hand, 2015) and mathematics in racism (bishop, 1990; frankenstein, 19839). much that i have documented above demonstrates the ways in which the racist american society propagates itself through a particular mathematics—that racism flourishes is in part through this mathematics’ ability to create a centaur state. others have demonstrated mathematics’ functions in racism. for example, through the use of counting and emphasis on hierarchal relationships, mathematics provides a colonizing power to administer and govern (foucault, 2009). davis and 7 a notable exception is the idea of enactivism (davis, 1996), still present in some research programs outside the united states, canada in particular. 8 davis (1997) proposes a hermeneutic listening, which “is intended to imply an attentiveness to the historical and contextual situations of one’s actions and interactions” (p. 370). 9 “traditional mathematics education supports the hegemonic ideologies of society” (frankenstein, 1983, p. 328). lawler response commentary journal of urban mathematics education vol. 9, no. 2 19 martin (2008) argue how the subordination of blacks is built on scientific methods of mathematical measures to support and validate racist beliefs. frankenstein (1983) suggests, “the hegemonic ideology of ‘aptitudes’—the belief, in relationship to mathematics, that only some people have a ‘mathematical mind’—needs to be analyzed” (p. 329). furthermore, popkewitz (2004) argues that modern pedagogies in mathematics serve to “divide, demarcate, and exclude particular children from participation” (p. 4). mathematics, as a colonizing force historically and presently, is thought to represent the highest forms of western thinking, and is assumed better than any indigenous mathematical systems or ways of knowing. this second recommendation is that not only must our profession recognize and embrace the colonizing and formatting power of both mathematics and mathematics education, but also the study of these issues must become a required element of the k–12 mathematics curriculum. again, the ethnomathematics tradition may offer a start toward this alternative epistemology. powell (2002) notes that ethnomathematics “departs from a [western] binary mode of thought and a universal conception of mathematical knowledge that privileges european, male, heterosexual, racist, and capitalistic interests and values” (p. 17); it disrupts the notion of a singular mathematics. the mathematics classroom could include study of others’ mathematics, and include questions like where do degrees come from? and who decided we will measure angles with degrees? given that answers to questions like these are along the lines of “because some people determined it should be that way,” such a pedagogy can allow students to begin to recognize mathematics has a cultural history (bishop, 1990), another re-humanizing move. similarly, the traditions of criticalmathematics (frankenstein, 1983; powell, 2002) and mathematics for social justice (gutstein, 2005; leonard, brooks, barnesjohnson, & berry, 2010) offer frameworks for examining as well as teaching a different mathematics, such as gutstein’s (2005) call to embrace classical, community, and critical mathematics. furthermore, there is a wealth of strategies and materials available for use in the classroom, and a broad community of mathematics educators embracing this approach (gutstein & peterson, 2005; wager & stinson, 2012). critical and social justice mathematics offer different pedagogies, yet as martin (2013) has noted, the perspectives have not been consistent in explicitly attending to issues of racism. culturally relevant pedagogy (ladson-billings, 1995; leonard et al., 2010) offers this missing element, calling for a critical pedagogy of cultural critique, to attend to “political underpinnings of the students’ community and social world” (ladson-billings, 1995, p. 477). often a critical or social justice approach to mathematics teaching is rejected, arguing such topics lie outside the discipline (beck, 2014; ravitch, 2005). nctm could elect to impact this rationale; in fact, it already has a rich history of embracing interdisciplinary approaches to mathematics education—often connections to lawler response commentary journal of urban mathematics education vol. 9, no. 2 20 scientific or economic topics. seeing mathematics as a product of human activity offers a seemingly natural interdisciplinary fit with humanities and social studies. mathematics has been used for great evils including the creation of ranked social structures and the validation of racist beliefs. it would be incomplete to not recognize it as also been used for great good, and can be used to disrupt injustices (gutstein, 2005). a critical social and historical understanding of the role of mathematics in people’s lives would create better understanding of the roles mathematics has contributed socially and politically historically, the way it operates presently, and how it can be harnessed to fight injustices and create a better world (d’ambrosio, 2007). the recognition that mathematics works in particular ways, including to stratify society and perpetuate racism, and that how it works can be changed, leads to a need for consideration or the moral code underlying the field of mathematics education. highlight the humanity of mathematics a third foundational element of mathematics education in need of examination is its moral code, an axiological consideration. mathematics is not neutral (nasir et al., 2008) nor value free (frankenstein, 1983); mathematics and mathematics education are the products of human activity (davis & hersh, 1981; kilpatrick, 2012), “math needs people” (gutiérrez, 2013, p. 48). what is the kind of behavior we aspire to? why do we teach mathematics, to what end? several years ago i interviewed laurie reyes hart and george m. a. stanic (lawler, 2005), two mathematics educators whose careers focused on issues related to equity. the interview was grounded in their review of research focused on differential achievement in mathematics based on race, sex, or socioeconomic status (see reyes & static, 1988). i asked both why should we teach mathematics?—a question stanic (1984) traced historically for his doctoral dissertation; a question usually assumed to be so self-evident that we as a field forget to ask. yet that mathematics seems complicit in both benefits and iniquities10 compels contemplation of the justification question. in their response, both hart and stanic (see lawler, 2005) questioned arguments about the utility of mathematics; they frequently ignore that most people function well in their daily lives without a profound level of mathematics.11 fur 10 i intentionally use iniquity, a term to indicate a gross injustice, or wickedness. my use was prompted by stanic in my interview, in which he expressed his notion of equity “as the opposite of iniquity, as the opposite of something evil. so that it’s more than the kind of gentle word than we think of it as.... when you start thinking of it as that which is the opposite of iniquity, suddenly you seem to have more responsibility” (lawler, 2005, p. 36). 11 there is a “simultaneous objective relevance and subjective irrelevance of mathematics in society” (m. niss, as cited in ernest, 2000, p. 3). lawler response commentary journal of urban mathematics education vol. 9, no. 2 21 thermore, both dismissed the justification that we must teach mathematics so future citizens can contribute to the economy. hart surmised that mathematics should be taught for access to power and resources, as well as awareness of mathematics as a tool of oppression. stanic added that mathematics seems to be “this interesting phenomena that has arisen among human beings, and thus worthy of study because it’s such an important part of human life, historically” (p. 35). stanic (see lawler, 2005) suggested that not only is humanity defined in part by our mathematics but also the study of mathematics is in part a study of our humanity. beyond the shift to recognize mathematical ideas are generated by people and in cultures, there is an onus to examine ethics of mathematics, and mathematical ways of knowing. what are the responsibilities that come with learning mathematics? what are the responsibilities that come with knowing a particular mathematics? what are the responsibilities for using mathematics? furthermore, what might it mean to develop a mathematics education with a project to erase inequities between people, mathematics, and the globe (gutiérrez, 2007)? gutiérrez (2013) argues a key aspect of equity is for the mathematics education community to “become experts at supporting learners to maintain a sense of wholeness while doing mathematics” (p. 61). classroom and professional foci such as those indicate in the questions above would contribute not only to the development of robust children’s mathematical identities (aguirre et al., 2013), but also to those of our professional community. similar to a study of the sociohistorical productive and destructive uses and impacts of mathematics as previously encouraged, a broader study of the humanity of mathematics would begin to achieve an ethical imperative to humanize the discipline. the stories of mathematicians, wide-ranging biographies, may become part of the curriculum. some stories may be of a particular problem that puzzled this person, serving to pose a problem for classroom study. the effort could extend to the study of peoples, in which once again the ethnomathematics tradition offers many examples. over these past 30 years, nctm has asserted standards for mathematical practices and identified habits of a mathematician. these human aspects of doing mathematics and the development of mathematical identity (aguirre et al., 2013), however, seem to be largely ignored. during the same time period, epistemologies that embrace a critical orientation to the intersection of mathematics and society, and postmodern shifts in ontology have also been ignored in the practices of mathematics education, as well as in the policy statements and other actions of the nctm. the three recommendations suggested above are radical, in the sense they call for difficult shifts in mathematics education. of course there are recommendations offered by others that i have not recognized here, such as a modernized and germane content, culturally relevant tasks, a diversity of teachers that match that of students, and an end to the accountability rhetoric (suspitsyna, 2010) and the testing lawler response commentary journal of urban mathematics education vol. 9, no. 2 22 regime. it is my sense that each of those is secondary to the changes i have argued. that is, making those shifts are predicated on the three i suggest, and are not sufficient to change the experiences of students to be something other than schooling as a colonizing experience into the white, upper-middle class value system and ways of knowing. what nctm did not (could not?) hear i have suggested three recommendations fundamental to a liberatory and rehumanized mathematics education. these recommendations are not new; they are already represented in numerous traditions and research programs within the larger mathematics education community. yet they remain unfulfilled. important questions to ask: why has the field of mathematics lacked political will to enact rehumanizing recommendations? where is the moral outrage? (spencer, 2015) martin (2015) suggests that the current state of affairs affords not only nctm as an organization but also its leadership and its members social, economic, and political privileges. it is certain that the mathematics education community is a conservative force (kilpatrick, 2012), lacking the will to disrupt this current state of affairs. as it operates now, the mathematics education project writ large benefits from the status quo. there are large sums of money to be garnered for research programs that rely on seeing the collective black as deviant (berry et al., 2013), and potentially greater dollars to be made peddling equity-themed products and professional development—all in the guise to improve the lives of the collective black. furthermore, as mathematicians and mathematics educators, our status is conferred by a particular view of mathematics; it is not in our interest to disrupt this enlightened, ordained form of knowledge. kilpatrick (2012) notes: “teachers of mathematics may derive considerable status from presiding over a subject that others find difficult or even impenetrable. why should they lower it from its elite pedestal?” (p. x). these changes are messy, imbued with power and ego, requiring mathematics educators to “reject a view of their subject that may have been a mainstay of their scholarly identity” (p. x). from my perspective, the nctm response (see briars et al., 2015) ignores significant aspects of martin’s (2015) critique. this non-response was predicted by kilpatrick (2012); it is the product of the organization’s conservative drive to maintain the present state of affairs in mathematics—the decades-long, persistent cry for equity. nctm’s response seems to not recognize two critical concerns voiced by martin: (a) to redefine the goals of the mathematics education enterprise to escape a colonizing orientation, and (b) to wonder if nctm is capable of leading such change. martin (2015) states that the mathematics education community, in general, and nctm, in particular, has not made significant progress in addressing the op lawler response commentary journal of urban mathematics education vol. 9, no. 2 23 portunity gap; that is, it has not made significant progress in changing “the conditions of african american, latin@, indigenous, and poor students in mathematics education” (martin, 2015, p. 22). nctm either refused to hear, could not hear, or ignored martin’s intent with this passage. nctm remains muddled in the notion of an opportunity gap, but martin has no interest in this framing of an opportunity gap; it leaves unquestioned the eurocentric and colonizing form of mathematics education indisputed, undisturbed. at present, mathematics education functions to perpetuate the ideology of the dominant view of mathematics. the emphasis on an opportunity gap for nctm is to improve the opportunities for the collective black so as to assimilate to the dominant view, in effect to become white washed.12 as martin (2015) puts it, nctm seems unable to get beyond an orientation to equity for the collective black “to enjoy contingent benefits of the system, [a system] that is not set up for them or by them” (p. 22). martin (2015) characterizes the change necessary to the status quo of mathematics education as violent, a change that would put the last first. this language is strong, yet “decolonization is always a violent phenomenon” (fanon, 1968, p. 35). as a white heterosexual man, quite successful in a system set up for me,13 i can empathize with the anxieties this sort of language brings. if the collective black were to become those who succeed, people like me could be repositioned as the recurrent failures. it is clear that such a result does not sit well for neither me nor others; but the fierceness in which we refuse such a solution seems not to be matched in our duty to correct the contemporary reverse injustices. the present iniquitable outcomes of mathematics education are evidence that the eurocentric mathematics of schools serves as a colonizing force to the minds of children, possibly all children, but certainly those whose cultures do not align to the dominant american (u.s.) culture. the last 27 years suggest that relying on the white benevolence of nctm will not achieve the kinds of equity called for.14 for correction to be other than an incremental change, the history of mathematics education suggests that some form of impassioned, vigorous, violent—in the sense of total—action must be taken. this is not the violence of bloodshed, but the overhaul of a colonizing institution and belief system about what counts as mathematics, and what is valued in a mathematics education. it is to be a violence on the level of the 12 to understand the trauma associated with this sort of experiencing of the eurocentric mathematics school culture, consider, for example, stinson (2006). 13 i owe “the fact of [my] very existence… to the colonial system” (fanon, 1968, p. 36). 14 gutiérrez (2007) proposes a redefinition for equity goals in mathematics education. she sets the target for equity to be three-fold: (a) to be unable to predict achievement and participation based solely on student characteristics; (b) to be unable to predict ability to analyze, reason about, and critique the knowledge and events in the world based solely on student characteristics; and (c) to erase inequities between people, mathematics, and the globe. lawler response commentary journal of urban mathematics education vol. 9, no. 2 24 routine psychological, social, and institutional violence perpetrated on the collective black. mathematics education incurs a psychological, not physical violence; it is a psychological colonization (fanon, 1968). the decolonization project must liberate the colonized mind from the effects of alienation and dehumanization. it is certain to be an intense, sociopolitical contention in which race and class-based struggle play a key role. in this struggle, nctm is a political organization representative of the colonial bourgeoisie. it is not in their interest to oversee a removal of the system that has created their dominance: “is nctm the kind of organization that is capable of facilitating the kind of violent reform necessary to change the conditions of african american, latin@, indigenous, and poor students in mathematics education?” (martin, 2015, p. 22). i suspect the institution, the voice of mathematics education, can only imagine a reform entailing tweaks and modifications. nctm will promote compromise, a non-violence. the question i see unasked in martin’s comment is what violence is necessary? i suggest the violence necessary is embedded in three shifts i have recommended here, actions that reclaim and humanize the ontological, epistemological, and axiological roots of the present mathematics education. it is my hope that these recommendations could result in a liberation of the consciousness, reverse the effects of alienation and dehumanization for both students and educators, build solidarity in the struggle for liberation, and reconstitute the structures and social institutions of the present mathematics education. to close, i call for a movement to rise, possibly from within mathematics education, that not only strives to improve the condition of the collective black but also conceives further to disrupt the status quo in such a way that moves the last to first, 15 to decolonize mathematics education. this movement cannot become a pawn or affiliate of the nctm, “it is the colonist who fabricated and continues to fabricate the colonized subject” (fanon, 1968, p. 2). based on nearly thirty years of equity rhetoric and little action, nctm’s content adherence to “myth of time” (king, 1963, p. 10), that “the very flow of time will inevitably cure all ills” (p. 11), suggests that nctm “paternalistically believes [it] can set the timetable for another man’s freedom” (p. 10). the colonized subject must break from nctm and create a counter-organization to express its own voice. this new community must emerge to counter the conservative institutional force of nctm. we cannot afford the mathematics education that presently exists under our watch. 15 it is my sense that in a new ontology, all can be first—an absence of hierarchy, a postcolonial heterarchy. lawler response commentary journal of urban mathematics education vol. 9, no. 2 25 references advancement project. 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(2012). teaching mathematics for social justice: conversations with educators. reston, va: national council of teachers of mathematics. http://files.eric.ed.gov/fulltext/ed543514.pdf http://www.truth-out.org/opinion/item/16406-education-reform-in-the-new-jim-crow-era 1 journal of urban mathematics education july 2009, vol. 2, no. 1, pp. 81–105 ©jume. http://education.gsu.edu/jume roland pourdavood is a professor of mathematics education at cleveland state university, 2300 chester avenue, chester building 266, cleveland, oh 44115-2214; email: r.pourdavood@csuohio.edu. his research interests are mathematics teachers‟ dialogue and reflection for transformation and school reform. nicole carignan is an associate professor of intercultural education at university of quebec at montreal, c. p. 8888, succursale centre-ville, montreal, quebec, canada, h3c; email: carignan.nicole@uqam.ca. her research interests are cultural diversity, sociocultural aspects of education, and emancipatory action research for personal and social praxis. lonnie c. king is a project manager at nelson mandela metropolitan university, nmmu missionvale campus, p.o. box 77000, port elizabeth, 6013, south africa; email: lonnie.king@nmmu.ac.za. his research interests are preservice and inservice teachers‟ understanding of geometry and teacher professional development. transforming mathematical discourse: a daunting task for south africa’s townships roland g. pourdavood cleveland state university nicole carignan university of quebec at montreal lonnie c. king nelson mandela metropolitan university in this study, the authors describe the voices and practices of four mathematics teachers in a k–7 “coloured” township school in the context of the eastern cape of south africa. the authors begin with an explanation of the context of the township in terms of its history, languages, diversity, educational system, and mathematics education. through observing and describing four mathematics classroom discourses, two from the foundation phase and two from the intermediate phase, the authors illustrate the relationship between the participating teachers’ voices and practices within their socio-cultural and socio-historical context. additionally, the authors describe the complexity of transforming mathematics education and suggest that implementing the revised national curriculum statement requires an epistemological shift in perspective regarding teaching and learning that promotes mathematical understanding. the authors’ analyses show the importance of transforming mathematics education and the responsibility of higher education institutions in the preparation of future teachers compatible with the new societal demand. keywords: access and equity in mathematics education, beliefs and practices, mathematics classroom discourse, mathematics teacher professional development, mathematics teacher transformation the exponential growth of communication technology has created a world in which even the most remote classroom and school system can now be part of pourdavood et al. mathematical discourse journal of urban mathematics education vol. 2, no. 1 82 the world discussion about social justice, access and equity, and teacher preparation. our time is fortunate to witness the demise of educational systems that intentionally oppress the weakest members in society in preference of the strongest. in the field of education, those abuses created systems of education that were inherently prejudicial and seriously undermined the ability of certain groups of people to advance and succeed in their communities. due to technology, however, the world can now see what was once hidden. technology has made it possible to act upon what is seen and there is now a global consciousness developing that demands that there be standards developed regarding access and equity in education, that social justice norms be applied to all students in a school system, and that professional preparation programs be developed which will equip teachers with the experiences needed to teach in a manner compatible with twenty-first century expectations. in this research study, we address these new expectations and the rise of global consciousness to the importance of education by observing a previously underrepresented and underserved urban township school system in a country once governed by one of the world‟s most oppressive governments. south africa, once famous for its class system known as apartheid, is now a democratic state attempting to remake itself into a model of equity and fairness for all of its cit izens, regardless of race. in this study, we consider the ramifications that apartheid had on its educational system and its lingering impact today. we discuss with teachers in four different classrooms the effect their inadequate teacher preparation has had on their students who leave school with alarmingly high rates of illiteracy in mathematics. we also observe the imprints created by apartheid in its unequal funding of south africa‟s four racial school systems and the large class sizes found in black and coloured schools especially. south africa is now addressing these failures and has made great strides toward overcoming their past historical struggles. by observing and describing mathematics teaching and learning in one school system and its attempt to overcome such a history of educationally inadequate and harmful practice, we hope the research community will recognize the importance of this study for (urban) education worldwide. situation of south africa before 1994 the history of south africa (sa) and its present struggles with educational reform in general, and mathematics education reform in particular, cannot be understood unless one understands the history of apartheid itself, and the colonialism from which it grew. under apartheid, white south africans (about 9.2%) dominated the other three racial groups: the blacks (or africans) 79.5%; the coloureds 8.9%; and the indians/asians, the smallest group, 2.5%. the system they created was an attempt at social engineering; its official goal was to deliver to pourdavood et al. mathematical discourse journal of urban mathematics education vol. 2, no. 1 83 each separate culture what it deemed necessary for that culture. a “convenient myth” (morrow, 1990, p. 175) was perpetuated, which held that each group‟s racial differences rendered them incompatible with each other and therefore unable to coexist within the same society. in reality, the apartheid system was a thinly disguised but extreme form of racial discrimination designed to separate and control the “lesser” races of people. apartheid created a language barrier as well for the education of the nonwhite racial groups. in 2001, it was reported that approximately 75% of the population of sa spoke one of nine indigenous languages: ndebele, pedi, sotho, swazi, tsonga, tswana, venda, xhosa, and zoulou (gilmartin, 2004). before 1994, there were only two official languages, namely english and afrikaans (which is an adapted dutch language) and respectively spoken by 13.5% and 8.5% of the population. no language represents the majority in sa. although most of the population is either black or coloured, and speak a variety of other indigenous languages, all education in sa was delivered in these two official and foreign tongues even though less than a quarter of the entire population spoke either of these two languages at home (gilmartin, 2004). the educational system established under apartheid was incredibly disparate and inequitable across these racial groups. four distinct school systems were developed, each serving their own separate groups. purporting to offer education along the lines of the usa south‟s “separate but equal” methodology, these four systems were anything but equal. edusource data news reported that funding per student (in rands) at the time of liberation (1994) for the african student was 2,184, while the white student received 5,403, the indian student 4,687, and the coloured student 3,691 (edusource data news, as cited in gilmour, 2001). the national education policy investigation (nepi) reported that core syllabi, not entire curriculum, were made available to each school in each system, but were so diverse and numerous (over 1400 different ones) that they, combined with different examination systems, created even more separation and isolation among groups (nepi, 1993, as cited in gilmour, 2001). teacher preparation also presented a glaring inequality in sa schools. when compared to a standard measure of matriculation plus 3 years teacher education; 46% of african teachers, 29 % of coloured teachers, 7% of indian teachers, and 1 % of white teachers were found to be under qualified to teach in their respective schools (gilmour, 2001). in apartheid sa, it was usual practice for white teachers to attend school up to the twelfth grade and then receive 3 additional years of teacher education to qualify them as teachers. in contrast, the coloured person, who is neither black nor white, but descendant from white afrikaans and blacks, attended school up to the eighth grade and then received 2 additional years of teacher education. the black person received the least amount of education; she or he attended school only to the sixth grade and then received 2 additional years pourdavood et al. mathematical discourse journal of urban mathematics education vol. 2, no. 1 84 of teacher education to qualify her or him for teaching. it is easy to see that these separate systems for preparing teachers for the classroom were incredibly unequal and could only result in disproportionate success and failure across racial groups. class size is also an indicator of the inequitable educational system in place during apartheid. for example, in the eastern cape of sa, the student-teacher ratio in 1994 was 56:1 for the black, 39:1 for the indian, 28:1 for the coloured, and 21:1 for the white populations (gilmour, 2001). given that funding to each of these four school systems has been inequitable, that language has been made a barrier to learning, that non-white teachers have been insufficiently prepared to teach in terms of both content knowledge and pedagogy; the added burden of large class sizes only further exasperates the possibility of learner success for the non-white. situation of south africa after 1994 the new government of sa has made great strides in their effort to re-create their country into a model of democracy. a new constitution was drafted that clearly aimed at equality for all racial groups in sa and laid the foundation for the future by guaranteeing fair and accessible education for all south africans. the constitution makes redress to the language barrier created by apartheid and attempts to honor all of the languages of sa by declaring 11 of them official languages. this declaration by sa government makes access to education possible for all citizens and provides opportunities for learning to occur in the language most natural to the student. the new sa is still struggling with severe unemployment that affects 44% of black peoples, among others 50% of young men co mpared to 7% for the whites (l„encyclopédie de l‟agora, dossier : afrique du sud, 2005). the government has also established the equitable shares formula (esf), which is intended to make just the division of national revenue among the nine separate provinces created in sa after the elections of 1994 (equity in education expenditure, n.d.). this process is designed to make funds available to provinces on the basis of need in order to bolster the poorest of areas first before the funding of the wealthiest areas. this redistribution will no doubt help but problems still exist in the making of this redress a reality. currently, due to lack of money in local schools, fees are being applied to students desiring to attend certain schools. although schools are now free and open to all students, local fees are making enrollment prohibitive and in some cases are being applied in order to keep certain students barred from attendance (spreen & vally, 2006). likewise, an understanding of the difficulties confronting educators in sa as they work to reform their educational system, and specifically mathematics education, must include an explanation of the unique circumstances created by the pourdavood et al. mathematical discourse journal of urban mathematics education vol. 2, no. 1 85 apartheid regime; which, up to 1994 when the country‟s first democratic elections occurred, dominated and severely restricted the development and growth of the country‟s peoples and cultures. in the context of school reform, history books, mathematics schoolbooks, english books, afrikaans books, social studies books, and others have been revised in order to celebrate the cultural diversity and the contribution of all south africans from any racial background. in the eastern cape of sa, where our study was conducted, the language spoken in most black homes is xhosa. the language spoken in most coloured home is afrikaans. the tendency in south african black and coloured communities, however, is to want to learn english and be able to communicate fluently in english. according to the 1993 constitutional law, each citizen up to the age of 16 has the right to be educated in her or his native language. but, black parents, who speak limited english, as well as coloured parents, believe that their children will be better educated if they use english as the language of teaching and learning (ltl). some black parents believe that their children will be better prepared for future careers if they attend white public schools and are taught by white teachers. it has been left up to the individual schools to decide on the ltl for the particular community that they serve. for example, in the eastern cape of sa they have chosen english, afrikaans (main language of coloured people), and xhosa (main language of black people). the english language however is “becoming the lingua franca of the new south africa” (gilmartin, 2004, p. 415). because of this reality, most black and coloured people tend to use english as the ltl in their schools. in black township schools where 100% of students are black, the ltl is mainly xhosa for early childhood grades (grade k–3), gradually changing to english in middle childhood grades (grade 4–7), and then completely to english at the secondary level. black and coloured parents and their communities believe that the english language is the language of power and opportunity for success in sa; therefore, they strongly encourage their children to master english while they are at school. in our study, which was conducted in a coloured township school, class sizes were, on average, 38 pupils to one teacher, even higher than the 1994 figure for coloured schools. although research has shown that there is no direct correlation between class size and student performance (o‟sullivan, 2006); it has also been shown that available resources and teacher qualification have significant impact on the effectiveness of instruction (howie, 2003). with this historic inadequacy and inequity regarding teacher education programs, particularly for black xhosa speaking and coloured afrikaans speaking teachers in the eastern cape of sa, transforming mathematics education in the classroom remains a daunting task. in keeping with making funding and language accessible for all of the provinces is the hope that teacher preparation will follow. pourdavood et al. mathematical discourse journal of urban mathematics education vol. 2, no. 1 86 as our study suggests, this is not yet the case. the teachers that we observed and interviewed expressed concern for their chances of success in an environment that is, while hopeful, not yet perfect. they remain under resourced and under prepared. in the next section we will define the context of the current study. context the 2002 revised national curriculum statement (rncs) sets forth impressive objectives for the future of education in sa. its scope covers a breadth of content areas and clearly objectifies what mathematics content is to be taught, learned, and assessed for grades k–9. the rncs intends to expand school mathematics beyond arithmetic to include mathematical meaning consistent with the needs of the new society. in addition, the rncs aims to establish equitable instructional and assessment practices compatible with current research on teaching and learning mathematics. these goals are interesting and encouraging. however, there is little research on how educators implement the rncs in their day-to-day classroom mathematical discourses and whether such goals are actually being met. likewise, at the current stage of transformation, there is limited curriculum material consistent with the rncs agendas. this lack of research and supplemental learning materials continues to place teachers in a vulnerable and precarious situation. in this study, we describe the voices and practices of four mathematics teachers in a grade k–7 coloured township school in the eastern cape of sa. the four teachers‟ ages ranged from late 30s to early 50s. all four teachers and their students were coloured and from the same township community. in this sense, the teachers were serving their own community. the four teachers‟ teaching experience ranged from 10 to 25 years. we describe four mathematics classroom discourses; two from the foundation phase (k–3) and two from the intermediate phase (4–7). our descriptions illustrate the relationship between the participating teachers‟ voices and practices within their socio-cultural and socio-historical context. theoretical framework here, we define mathematical discourses as mathematical discussions and social interactions occurring in a mathematics classroom between the teacher and students and among students as they attempt to solve problems and communicate their thinking and reasoning. to understand the complex interplay between a teacher‟s pedagogy and practice, we incorporated hufferd-ackles, fuson, and sherin‟s (2004) model for observing and describing four participating teachers‟ classroom practices. hufferd-ackles et al. (2004) observed and identified four pourdavood et al. mathematical discourse journal of urban mathematics education vol. 2, no. 1 87 levels of discourses that exist in a math-talk learning community that describe the relationships between teachers‟ beliefs and practices. through a professional development project, they observed several primary school teachers‟ practices in a bilingual setting. hufferd-ackles et al. identified four transitional stages in mathematical discourse ranging from a traditional teacher-centered classroom (level 0) toward a classroom in which the students and teacher participated in and contributed to the activities of the community (level 3): level 0 in the framework represents a traditional, teacher-directed classroom. in the level 1 classroom, the teacher in the study began to pursue students‟ mathematical thinking, but still played the leading role in the math-talk learning community. in level 2, the teacher began to stimulate students to take on important roles in the learning community and backed away from the central role in the math talk. in level 3, the teacher coached and assisted her students as they took on leading roles in the math-talk learning community. (p. 91) one must recognize that these four levels of classroom discourses are not linear and sequential. for example, at each particular classroom activity, a teacher may incorporate two or more of these levels depending upon the classroom dynamics and the classroom learning situations. however, based on the teacher‟s content knowledge, beliefs, and pedagogical values, one of these four levels may play dominant role in the classroom mathematical discourses. an example of each level is presented below. consider the following problem: kim bought six movie tickets for her six friends. she spent 48 dollars total. how much did each ticket cost? in a level 0 classroom, the focus of classroom discourse is on the teacher‟s question, the student‟s short numerical response, and the teacher‟s evaluation of the right answer. the classroom teacher is the sole authority for asking questions and validating students‟ responses. the social interactions occur only between the teacher and individual students. if a student responds “the answer is eight dollars,” then the teacher validates the correct response and continues posing similar routine problems. in a level 1 classroom, in addition to focusing on the right answer, the teacher shows an interest in her or his students thinking and reasoning. she or he may ask, “how did you figure out the answer?” or “is there any other way to solve this problem?” nevertheless, she or he still plays a dominant role relative to the classroom mathematics discourse. in a level 2 classroom, the teacher may pose the question to the classroom and provide students with opportunities to work in their small groups for problem solving and communication. then she or he may invite students to defend their ideas in front of the class. students‟ solutions are subject to change by other students or by the teacher through classroom dialogues. the teacher facilitates classroom mathematical discourse via her or his questioning strategies. problem solv pourdavood et al. mathematical discourse journal of urban mathematics education vol. 2, no. 1 88 ing, reasoning and proofs, communications, mathematical connections, and mathematical representations are a central part of the classroom discourse. multiple solution strategies are discussed. the students are treated as members of the classroom community. for example, one student may solve the problem algebraically by showing 6 × x = $48, x = $8. another student may solve the problem geometrically by drawing a 6 by 8 rectangular figure, and yet another student may use her or his manipulatives materials such as base-10 blocks for mathematical modeling and visual representation. in a level 3 classroom, the teacher may invite students to pose similar problems for their peers and challenge them to find solutions. in this kind of classroom mathematical discourse, co-teaching and co-learning occurs among members of the classroom community. the classroom teacher acts as a learner with her or his students. she or he still plays the role of a facilitator and guide. she or he, however, provides the students more leadership opportunities. methods this observational and descriptive research was grounded in constructivist inquiry (lincoln & guba, 1985, 1994). the study is context specific (i.e. four classrooms from a coloured township school). in this sense, the study intends to share the ideas and experiences of the four participating teachers and hopes that others will identify with the research context and apply the findings to their own particular settings. our data collection started in mid-january 2004 and ended mid-april 2004. it consisted of preliminary and active phases. the purpose of the preliminary phase was to establish a research framework and to discuss and decide the following issues: (a) defining the research goals, (b) establishing a timeline for research activities, (c) contacting the school community and sharing our intentions for the research, (d) targeting potential classrooms for observation (i. e., foundation and intermediate phases), and (e) clarifying the role of the research team. the active phase included conducting 14 mathematics classroom observations and 6 interviews with four participating teachers. the four teachers volunteered to participate in the study; two were women and two were men. the two women‟s ages ranged from mid 40s to early 50s. they taught secondand thirdgrade classrooms respectively. the two men‟s ages ranged from late 30s to mid 40s. they taught in sixthand seventh-grade classrooms. mathematics classroom observations occurred once a week, every monday between 8:00 a.m. and 2:00 p.m. during the observations, the researchers focused on the teacher-students and student-student mathematics discourses; field notes were taken during the mathematics classroom observations. the subsequent interviews were audio-taped and transcribed verbatim. the criteria for the selection of teachers included: (a) their willingness to participate, (b) their ability to articulate who they are, (c) the diver pourdavood et al. mathematical discourse journal of urban mathematics education vol. 2, no. 1 89 sity of their background (i. e., gender, age, home language), (c) their diverse mathematical content knowledge and qualification, and (d) their teaching experiences. data collection and data analysis occurred simultaneously during the course of the study. constant comparative data analysis (lincoln & guba, 1985, 1994; mccracken, 1988) was used for making sense of data. data was reviewed and analyzed independently by each member of the research team. this method was a form of triangulation among the researchers (denzin, 1984). triangulation also occurred when the research report was shared with the participants (i.e. the principal, deputy-principal, and teachers). the compatibility between the researchers‟ interpretation and the participants‟ stated beliefs and actions evinced the trustworthiness of the data analysis. consistent with the hufferd-ackles et al. (2004) model, we described classroom mathematics teaching and learning as well as teachers‟ voices towards mathematics education. we used the term teaching strategies 1–4 instead of levels 0–3 to demonstrate mathematics classroom discourses. this change in terminology occurred during our research report to the participating teachers. the teachers mentioned that the word “level” was too rigid. they also mentioned that the term level 0 can be interpreted as the teacher‟s incompetence in teaching. we agreed with the participating teachers and modified the terminology. teachers’ background, voices, and classroom practices to demonstrate the kind of mathematics instructions occurring at this school in both the foundation and intermediate phases, we selected four classrooms: a second grade, a third grade, a sixth grade, and a seventh grade. based on analysis of the 14 classroom observations, these four representations of foundation and intermediate phases as episodes were selected. the physical structure of the classrooms in both foundation and intermediate phases was very similar. students were sitting around their tables in pairs. two, or sometimes three, tables were joined together to make a cluster of four to six learners sitting together as a group. the number of pupils in each classroom was between 36 and 40. in the following section, we describe each teacher‟s background, voice, and her or his mathematics classroom practices in an effort to highlight the particular difficulties these teachers face and to explain the effect these difficulties have on students‟ learning and classroom discourses. in the following section, we describe each teacher‟s background, beliefs, and classroom mathematics practices. we aim to explain the challenges that these teachers faced in teaching mathematics, the decisions they made during their classroom mathematical discourses, the strategies they used, and the (probable) consequences of these strategies on students‟ learning and classroom discourses. pourdavood et al. mathematical discourse journal of urban mathematics education vol. 2, no. 1 90 teacher 1: second-grade teacher’s background, voice, and classroom practices the second-grade teacher was from the foundation phase (k–3) background. she was in her mid 40s, coloured, with 15 years of teaching experience. she obtained a diploma up to the eighth grade plus 2 years of teacher preparation in 1989, before liberation. she spoke afrikaans as her mother tongue and english as a second official language. during her first 10 years of teaching, she used afrikaans as the ltl. for the last 5 or 6 years, however, because of parental demand, she used english as the ltl in her class. the teacher stated that she suffered from scarce funding for professional development. her pupils learned mathematics and other subjects in english although they were speaking afrikaans as their mother tongue at home. the teacher lived in the same community in which her students lived. in this sense, she was serving her own community‟s pupils. her voice the second-grade teacher mentioned that the three most important things for her were: (a) to see progression in her learners‟ academic growth and to acknowledge their achievement, (b) to provide her learners ample opportunities to excel, and (c) to assist struggling learners to reach an average level and to become confident in their own abilities. she asserted that for learners to understand and have a better appreciation of mathematics, she would integrate the subject with other learning areas: “if you look at maths, it‟s all-encompassing, it‟s everywhere. if you look at the child as a whole then there is a connection.” to make her teaching interesting, she would give her learners apparatus such as a counting-chart or a hundred-chart. the teacher mentioned that using the children‟s environments for number operation and mathematical patterns and relationship was one of her instructional goals. her biggest challenge was her class size and dealing with learners who were qualitatively slower: “when the class is so big, 39 children, you can‟t focus on your class as a whole and on those you really must focus on.” in order to cope with this challenge, she used a heterogeneous, small-group, cooperative learning strategy: we have cooperative learning in the class. this year i am taking my weaker learners and focusing more on them. my first group, the clever ones, i can give more work for them to carry on with, like enrichment activities in maths. according to her, using this type of cooperative learning strategy, all children would benefit. the higher achievers learn by teaching and assisting lower achievers. the higher achievers also do the peer-assessment and group assessment: “we had an open session where they gave their views of the learners‟ work and also encouraged them to do better.” she also mentioned the need for profes pourdavood et al. mathematical discourse journal of urban mathematics education vol. 2, no. 1 91 sional development through ongoing school-based workshops. she believed that by empowering herself she could make her mathematics classroom more enjoyable and meaningful for her students: i would like to learn more too, by attending more workshops so that i can equip myself to assist them in the classroom. sometimes you sit there and become stereotyped doing the same thing over and over, when there are other little things you can do, when you go to the workshops, your spirit is just uplifted…there is a big need for maths workshops. there are not many workshops that are held for the foundation phase teachers. she acknowledged that working with her colleagues in the foundation phase was very meaningful for her professional development: “sharing our portfolios help us a lot, getting information from the first grade and then moving on. so, we keep reports of the children and pass it on.” we found the second-grade teacher‟s attitudes towards relearning mathematics interesting and encouraging. although she mentioned the limitation her poor teacher preparation program created, she was hopeful that through her dedication and commitment towards her own professional growth, she could provide her learners better educational opportunities. the second-grade teacher‟s beliefs correspond to two important equity issues: (a) the issue of her students‟ achievement in mathematics; and (b) the issue of access for her underrepresented students to do significant mathematics. her practices [note: in subsequent sections, t = teacher, l = learner, and ls = learners.] t: how many fingers do we have? how could we make sure that we have 10 fingers? [she called a learner.] l1: [he counted 1 through 10 using his fingers. the teacher had her 100-chart on the board and asked learners to read various numbers on the chart.] t: [she called a learner.] show me the number 10 on the board. l2: [she went to the board and showed number 10.] t: which number comes before 10? [learners raised their hands for response. the teacher called a learner.] l3: [he showed number nine on the board.] t: which number comes after 10? [learners raised their hands for response. the teacher called another learner.] l4: eleven. the counting activity continued with the teacher questioning the students to recognize the number before a given number, after a given number, below or above a given number. learners would raise their hands for responses and the teacher would call a learner for a response (i.e., strategy 1 or level 0). pourdavood et al. mathematical discourse journal of urban mathematics education vol. 2, no. 1 92 t: how can i make 6 with my fingers? can somebody come and write me six on the board? [the learners raised their hands. she called a learner.] the student wrote the word “six” on the board. the teacher called another student to draw or model number six. the student drew six small circles, side-by-side on the board. the teacher called a different student to group the six circles by 2s. students were listening and participating in the activity. the teacher attempted to combine literacy and numeracy in her teaching by asking children to write numbers in english. in order to make counting by 2s more relevant to students‟ prior experience she used the students‟ bodies (i.e., counting six pairs of eyes). the activity was consistent with her statement about using children‟s natural environment for teaching mathematics. she encouraged her students to write and model mathematical symbols (i.e., drawing). the mathematical discourse, however, was dominated by teacher questioning, students‟ responses, and teacher evaluation (strategies 1 and 2, or level 0 and 1). although providing her students with opportunities to make sense of numbers and operations was one of her priorities, in practice she focused only on some routine problems for students to answer. the level of mathematics discourse was at or below first grade. the teacher acknowledged that the way she taught was relative to the way she was “trained.” she mentioned that she was the product of an educational system that historically treated teachers inequitably, thus she is ill-prepared in terms of content and pedagogy for teaching mathematics with understanding. however, she wanted to better prepare herself through ongoing school-based professional development activities. in our other classroom observations, the teacher followed the similar patterns of teaching and classroom discourses throughout the course of the study. teacher 2: third-grade teacher’s background, voice, and classroom practices the third-grade teacher was from the foundation phase (k–3) background. she was in her early 50s, coloured, with 25 years of teaching experience. similar to the second-grade teacher, she obtained a diploma up to eighth grade plus 2 years of teacher preparation in 1978, long before liberation. she spoke afrikaans as her mother tongue and english as a second official language. during her first 20 years of teaching, she used afrikaans as the ltl. for the last 5 or 6 years, however, because of parents demand, she used english as ltl in her class. similar to the second-grade teacher, this teacher stated that she suffered from scarce funding for professional development. the teacher lived in the same community in which her students lived. in this sense, she was serving her own community‟s pupils. pourdavood et al. mathematical discourse journal of urban mathematics education vol. 2, no. 1 93 her voice she said it was important for her to know her learners. according to her, by the end of the first week of school, she would have already assessed her learners in order to see where the problems were in terms of learning mathematics, reading, and phonics. she stated that she wanted to get to know her learners not only by name but also by the way they worked: “i do a lot of observation, i observe five or six children at a time.” she would observe the same children again the following day just to see if her observation and assessment were adequate. she asserted that one of her strategies was to challenge all children including “at-risk” learners: “i push them very hard during the first two-three weeks of the year to find out what their limitations and boundaries are, and then i can get my pace.” for her, planning was very important. when she did her planning, she paid particular attention to the integration of mathematical contents and themes. she also focused on how her learners understood the mathematical ideas presented to them: for example, when i do time and then fractions, obviously fractions must also be quarters, halves, three-quarters because we talk about time which is about quarters and halves. so, i cannot do my planning for 10 days in advance. the child sets the pace and determines the next day‟s work. as a result, she constantly adjusts her planning and her “pace” based on where her learners are: every year is different, especially with the new revised curriculum statement, with no guidelines for implementation from the department of education, no support from them; i am constantly changing because children are exposed to many things now, obviously methods change, and the subject or content changes. living in the same community where her learners lived provided her with ample opportunities to get to know her students‟ backgrounds and to build relationships with them and their parents. one of her biggest challenges was the implementation of the rncs into her day-to-day mathematics instruction. she said that she had not figured it out yet. her practices [note: the focus of mathematics instruction was time measurement.] t: how many minutes are in an hour? ls: 60 t: 60 minutes. can you tell me what the purpose of long hand is? ls: it tells the hours. pourdavood et al. mathematical discourse journal of urban mathematics education vol. 2, no. 1 94 t: hours or minutes? ls: minutes t: how about short hand? ls: it tells the hours. t: [she had a manual clock in front of her and she modeled the movement of the long hand for one complete revolution.] what do we call that? ls: clockwise. [then she modeled counter-clockwise.] t: now i want all of you to put your clock on the time that school starts. what time is the break? [she waited and observed.] ls: 10 o‟clock. [they modeled it.] t: when do we go home? ls: 12 o‟clock. t: what time do you come back? ls: one o‟clock. [the teacher and learners continued their discussion about various time of day and events. they also would write the words such as morning/light, afternoon, evening, dusk, night/dark, a.m. and p.m. the teacher encouraged multiple perspectives.] the students were expected to communicate their thinking and reasoning. the teacher asked the students to work on similar problems from the book. she circulated around the room, encouraging students to work together cooperatively and responsibly. concurrently, she gave additional time to lower-achieving students (i.e., strategies 1, 2, and 3; with strategy 2 being dominant during the classroom discourses). the classroom observation revealed several important issues such as: the teacher‟s pedagogical content knowledge, her questioning strategies for understanding and assessing students‟ understanding, her wait-time for students‟ responses, and her knowledge of her students and targeting of those children who were qualitatively slow. the teacher put mathematics content into context that made relevant sense to all students. the questioning strategies and wait-time afforded all students an opportunity to participate in and contribute to the activities of the classroom community. nevertheless, the teacher mentioned the lack of adequate resources for teaching mathematics in a culturally responsive manner. teacher 3: sixth-grade teacher’s background, voice, and classroom practices the sixth-grade teacher was from the intermediate phase (4–7) background. he was in his early 40s, coloured, with 12 years of teaching experience. similar to the second-grade and third-grade teachers, he obtained a diploma up to eighth grade plus 2 years of teacher preparation before the liberation. he spoke afrikaans as his mother tongue and english as a second official language. during his first 6 years of teaching, he used afrikaans as ltl. for the last 5 or 6 years, however, because of parental demand, he used english as the ltl in his class. the pourdavood et al. mathematical discourse journal of urban mathematics education vol. 2, no. 1 95 teacher lived in the same community in which his students lived. in this sense, he was serving his own community‟s pupils. his voice similar to the second-grade teacher, the sixth-grade teacher mentioned that teachers teach the way they were “trained.” he said his biggest concern was for children to see the connection between what they do in the classroom and real life. as a mathematics teacher, however, he wants to help his learners to make the connection between school mathematics and its application in the real world. he believes that if teachers put mathematics in context then children will learn more easily. it was important for him to find a way that teachers would be able to recognize that a child has shown more progress when he is compared to previous years because he was challenged within her or his ability: “as mathematics teachers, if i and the other teachers look at mathematics teaching and learning this way, we will hopefully improve in the long term, and the children‟s level of maths will grow.” his second concern was with the school‟s professional development activities. he stated that, although many teachers have been teaching mathematics for many years, not much has been done to upgrade their skills and methods. he suggested that one way to encourage teachers to empower themselves was through ongoing school-based professional development and workshops: “getting people in the school may just touch that spark a bit, so that they can rejuvenate that interest again.” he believed that even if teachers receive new methods of what to teach and assess, they are not necessarily ready to implement these ideas in their day-today classrooms. his practices on the board, the teacher wrote various units of metric measurements such as millimetre, centimetre, metre, and kilometre. the teacher started his instruction in a calm and peaceful voice. t: today we are going to focus on the introduction of measurement and conversion. how many millimetres are in a half of a metre? l1: 500 millimetres, teacher. t: 500 millimetres. it is written? nought [zero] comma [decimal] 500 m. [he wrote on the board 0,500m.] how about half of a kilometre? how do we show that in terms of metres? what is the relationship between a quarter of metre and half of a metre? [he wrote on the board 1/4 m and 1/2 m.] l2: it is a half of a half, teacher. t: it is a half of a half metre. okay, if i have a quarter of a metre, how do i show that? l3: it is nought, comma 250 m. [the teacher wrote on the board 0,250 m.] pourdavood et al. mathematical discourse journal of urban mathematics education vol. 2, no. 1 96 t: now, pick up your notebook. [learners had their rulers, pens or pencils, notebooks, and textbooks in front of them. then the teacher went from table to table giving different assignments to different groups of learners. the task was for each group to measure certain objects such as the length of a pencil, the length of a table, the length of an open hand, the length of a pencil case.] you will decide in your group whether you want to use millimetre, centimetre, or metre. i don‟t want one person doing all the measuring. i want one person be the reporter. [classroom was noisy; however, the learners were on task.] the activity on measurement was interesting and engaging. students wanted to find the solutions for various objects. the teacher used strategies 1, 2 and 3 (i.e., levels 0, 1 and 2) for mathematical discourses. our systematic observations of classroom discourses, however, suggest that the strategy 2 (level 1) was the dominant one in most cases. there were several interesting points that emerged from the observation of this classroom discourse. first, the teacher‟s calm and peaceful manner created a safe environment, conducive to learning. second, although the teacher‟s instructional approach was mostly conventional (i.e., teacher-directed instruction, teacher questions, students‟ short answers, and teacher evaluation), the latter portion of his instructional strategy afforded his students opportunities to engage in meaningful mathematical discourse among themselves. in this sense, his instructional approach was less teacher-directed and more student-centered. third, the activity was contextualised and it had a real life mathematical application. teacher 4: seventh-grade teacher’s background, voice, and classroom practices the seventh-grade teacher was from the intermediate phase (4–7) background. he was in his late 30s, coloured, with 10 years of teaching experience. similar to the other teachers in this study, he obtained a diploma up to eighth grade plus 2 years of teacher preparation before the liberation. he spoke afrikaans as his mother tongue and english as a second official language. during his first 4 years of teaching, he used afrikaans as the ltl. for the last 5 or 6 years, however, because of parental demand, he used english as the ltl in his class. the teacher lived in the same community in which his students lived. in this sense, he was serving his own community‟s pupils. his voice reflecting on his practices, the seventh-grade teacher believed that his way of teaching has to be transformed: “the frame of mind i am in will have to change as far as integrating is concerned.” he stated, “this is definitely the area that i must improve in because it is something that i have been struggling with since the start of obe [outcome-based education].” from his perspective, teachers were pourdavood et al. mathematical discourse journal of urban mathematics education vol. 2, no. 1 97 taught to teach mathematics in three steps: (1) teacher teaching and learners following, (2) asking some learners to repeat what was taught, and (3) individual seatwork and homework: “they have to follow the procedure or the teaching methods. teachers first do this, and then they do that. but it has changed. because of that change, i feel that i personally need more guidance.” his personal reflection and the new national requirements for effective teaching afforded him an opportunity to recognize the limitations of his method of direct instruction. the teacher asserted that one of his personal responsibilities was to make sure that once the learners leave grade seven they are confident and competent to compete with other learners from more privileged schools. he wanted his learners to see the application of mathematics in the real world and be empowered mathematically: “i would like them to stand up and be counted amongst others as someone who has really progressed as far as maths is concerned.” the seventh-grade teacher was sensitive to the notions of access and equity in mathematics education. he wanted his students to be able to use technological tools for learning mathematics, the tools that he was denied having when he was at school. his practices after greeting, the teacher started his instruction on number patterns, number relations, and squaring numbers. on the board he wrote “number patterns.” below that title, he wrote two rows of numbers. 1 2 3 4 5 6 7 8 0 2 6 12 20 30 42 t: can we establish a rule? [pause] can we establish a rule in order to determine the relationship between the two rows of the numbers? what would be the next number? [he was referring to the number below 8.] l1: number 8 will be 56, teacher. t: how did you get it? [the teacher paused. l1 is thinking while l2 raised his hand.] l2: the number of the next term multiplied by the previous one? [he was referring to the numbers in the first row and multiplying two consecutive numbers 7 and 8 in order to get his answer 56. the learners observed the patterns on previous numbers such as 1 × 2 = 2; 2 × 3 = 6; 3 × 4 = 12; 4 × 5 = 20; 5 × 6 = 30; 6 × 7 = 42; and 7 × 8 = 56.] t: right. now, i have these two rows of numbers. [he wrote on the board.] what is the rule? [he paused.] 1 2 3 4 5 6 7 8 9 3 5 7 9 11 13 15 17 19 l3: the rule is n × 2 + 1 = pourdavood et al. mathematical discourse journal of urban mathematics education vol. 2, no. 1 98 t: if i continue the first row up to 15, what would be the corresponding number for 15? l4: 31, teacher. t: how did you get it? l4: 15 × 2 + 1 = t: square numbers. who can give me a square number? ls: [several voices.] 16, 9, 4 … t: how do i get a square number? i get a square number by multiplying a number by itself. [then he called different learners to give square numbers.] l5: [one girl who was sitting quietly all of a sudden raised her hand and said:] i can make a square with a picture. if each side of a square is 4 centimetres then we get an area of 16 centimetres. [the class became very quiet. learners were thinking about the girl‟s reasoning. however, the teacher did not follow up the learner‟s observation. instead, he told her that she was answering a different question.] t: but, what you are saying is geometry. we are not talking about geometry. we are talking about square numbers. okay. we are talking about numbers not geometry. [the learner quietly accepted the teacher‟s response to her observation.] what number pattern do you see on these numbers? [referring to 1 = 1 × 1 = 1 2 ; 4 = 2 × 2 = 2 2 ; 9 = 3 × 3 = 3 2 ; 16 = 4 × 4 = 4 2 .] l6: teacher, i see a pattern there. i see 1, 2, 3, 4, [referring to the bases for 1 2 ; 2 2 ; 3 2 ; 4 2 .] t: yes. but, what is the square of that? [the student responded 1, 4, 9, and 16.] the mathematics discourse continued with the teacher‟s questioning, students‟ responses, and the teacher‟s validations. there are several important observations from this classroom episode. first, although the mathematics instruction was dominated by the teacher‟s talk, mathematical content was focused on conceptual understanding of number patterns (i.e. strategies 1 and 2). second, in most cases, the teacher‟s wait-time for students‟ responses was very helpful. there were, however, two critical moments for mathematical learning that the teacher missed (a) the connection between geometry, measurement, and square numbers, and (b) the relationships between the square root of a number and its square. one might wonder, perhaps, if the teacher‟s uncertainty to see the connection prevented him from taking advantage of this learning opportunity. asking for elaboration would encourage active listening to multiple perspectives for conceptual understanding of mathematics and modeling a “teacher as learner” in his classroom. in our one-on-one interview with the teacher we asked him about these critical incidents: r: in your mathematics lesson of squaring numbers, a student observed a squared geometric figure. your response to the student‟s observation was that, “we are talking about number patterns not geometry.” why did you respond this way? t: i was just focusing at that stage on my presentation, but i did reflect afterwards. i need to be guided more because i tend to focus only on whatever aspect i am doing. i did not pay much attention to what they were saying at that moment. i am just grateful for the observation about the squared numbers; it has opened my mind to be aware of other perspectives and other instances where the same principle can be applied. when i teach pourdavood et al. mathematical discourse journal of urban mathematics education vol. 2, no. 1 99 maths, i must be aware of what else can be interlinked, so that when pupils come up with something, i can focus on that as well to empower myself and the students. through the process of dialogue with the researchers and his reflection on his classroom mathematical discourses, the teacher began to understand the limitations of his direct instruction. from our perspective, this type of confrontation is a positive step toward teacher transformation in terms of content knowledge and pedagogical content knowledge. the seventh-grade teacher mentioned that he needed to empower himself in terms of mathematical content and pedagogy so that he could help his students. he also realized the importance of actively listening to students and facilitating classroom dialogues through redirecting, probing, and prompting questions. discussion in our study of these four teachers, we observed that mathematics instruction varied from mostly teacher-centered (strategy 1 or level 0) and teacherdirected combined with questioning for answers (strategy 2 or level 1), to smallgroup, cooperative learning and learner-centered (strategy 3 or level 2) in both foundation and intermediate phases. strategy 2 (level 1) was dominant in most classroom discourses. strategy 4 was not observed in any classroom. communication about mathematics ideas in all classes occurred mostly between the teacher and the learners. the nature of those interactions focused mainly on how to calculate and produce a correct answer. the reason for calculating and interpreting a given answer was observed in some of the classes in both phases. our classroom observations suggest that the main focus of mathematics instruction was not to afford students with a conceptual understanding of the lesson presented by teacher but purposed only to impart the mechanical procedures necessary to achieve the correct answer. in the aforementioned four classrooms episodes, we attempted to illustrate the dialectic relationships between the participating teachers‟ stated beliefs and practices. these reflexive relationships must be understood within the sociocultural and socio-historical context in which these teachers were educated long before the 1994 election. the nature of these teachers‟ enactments is inherently socio-historical and experiential. they expressed the challenges they face teaching mathematics compatible with the new obe of the rncs. for these teachers to be successful in administering the rncs, they will have to improve their content knowledge and their pedagogical approaches for teaching mathematics, thereby equipping their students with necessary knowledge, skills, and dispossessions. these teachers will need professional development support if they are to become the kind of teachers that the rncs envisions; teachers who are “qualified, competent, dedicated and caring,” and who are “mediators of learning, inter pourdavood et al. mathematical discourse journal of urban mathematics education vol. 2, no. 1 100 preters and designers of learning programmes and materials, leaders, administrators and managers, scholars, researchers and lifelong learners, community members, citizens and pastors, assessors and learning area/phase specialists” (rncs, 2002, p. 8). the rncs does not provide any specific methods for achieving these stated outcomes. rather, it lays down a broad vision for the future of education and attempts to delineate what the new teacher, the new learner, and the future citizen of a united sa is desired to become and possess. the rncs states that the “south african version of outcomes-based education is aimed at stimulating the minds of young people so that they are able to participate fully in econo mic and social life” (rncs, 2002, p. 12). how the teacher in the classroom accomplishes these goals must become the responsibility of teacher preparation programs. our participating teachers themselves suggested common sense ways that they may begin to achieve these goals. they suggested that collaborative planning among their school colleagues and school-based professional development would be beneficial and useful. these teachers face the daunting task of implementing the new obe in their day-to-day classrooms. in the east cape of sa, rural schools are facing large student-teacher ratios, administrative failure, lack of financial and material resources, and the lack of adequate teacher preparation necessary to meet the goals of the new standards. blanton and harmon (2005) have shown that these obstacles can be overcome through the provision of organizational support of teachers at the school level. this support will help teachers to assume leadership responsibilities in their schools and facilitate the creation of curricula that are aligned with rncs standards. teachers can be educated how to design and implement professional development activities for their colleagues, and to revise lesson planning and delivery to better achieve the desired outcomes for learning. professional partnership and development along these lines is possible for sa teachers and schools. in a review of literature concerning teachers‟ values and presuppositions about mathematics instruction it has been concluded that teachers generally approach the teaching of mathematics as rule driven and formula based, and as a content area that is best taught through text-book problems and memorization (zevenburgen, 2005). this literature also shows that these sorts of views are difficult to change without proper professional support (zevenburgen, 2005). the teachers in our study, however, expressed their desire to do just that. they expressed the desire to connect their teaching with their students‟ socio-cultural backgrounds and to make their teaching contextually relevant to their students‟ lives. a first step in doing so would be to create a classroom environment where dialogue is treasured and students‟ voices are encouraged to be heard. this recommendation is consistent with the math-talk community climate discussed by hufferd-ackles et al. (2004) where the conducive learning milieu is one where the student contributes largely to the discussion of mathematics and is allowed to pourdavood et al. mathematical discourse journal of urban mathematics education vol. 2, no. 1 101 frame an understanding of specific content and concepts in her or his own terminology. traditional mathematics instruction intends only to impart the skills required to solve mathematical problems; the new outcomes desired to be met by our south african teachers intend for students to understand mathematical concepts and the myriad of ways of representing them. participation in the larger global society, another one of the outcomes desired by the rncs, requires that students not only be able to understand but also to learn to construct new knowledge, to explain, conjecture, organize, predict, and to collaborate with others toward a common goal (english & watters, 2004). traditional approaches to teaching mathematics have focused only on following procedures for arriving at an answer (english & watters, p. 336). english and watters note that the problem with this approach is that a child does not then have to understand or interpret the mathematics problem involved, she or he has only to be able to follow the proper sequence of steps towards a specific goal. this sequence of steps is problematic because this approach does “not address adequately the mathematical knowledge, processes, representational fluency, and social skills that our children need for the 21st century” (english & watters, p.336) nor does it equip the student with any real sense of understanding of the language of mathematics. it is through the dialectic interplay between student–teacher and student–student that clarification and consolidation of conceptual understandings of mathematical ideas is fostered (national council of teachers of mathematics, 2000). additionally, the participating teachers reflected on their teaching methods in mathematics, about the need to get to know all students, and about putting mathematics content in relevant contexts that connects with students‟ prior knowledge and their socio-cultural experiences. students are more successful in their learning when their socio-cultural heritage is visible and reflected in the classroom. teachers can create learning opportunities for all students by establishing socio-mathematical norms in the classroom relative to their students‟ cultural backgrounds (e. yackel & p. cobb, as cited in frempong, 2005). moreover, the teachers we interviewed discussed the importance of active listening to students‟ voices, which is vital if students are to feel that they have a secure and welcoming place within the classroom and an opportunity to succeed. students hold distinctive perceptions of learning and possess individual knowledge bases that are available for teachers to access if teachers learn to take the time required to foster these teacher-student relationships (a. cook-sather, as cited in boyer & bishop, 2004). the teachers also talked about the need to commit themselves to allowing extra time for targeting attention on their lower achieving students, to affording all students sufficient opportunity to work together cooperatively and responsibly, and to using multiple strategies for assessing students‟ understanding of mathe pourdavood et al. mathematical discourse journal of urban mathematics education vol. 2, no. 1 102 matical procedures. these ideas and strategies are compatible with the rncs requirements but are understandably difficult given the current inequitable funding of schools and large class sizes. concluding remarks the intention of this study was not to question the professional integrity of these teachers. these educators took their teaching responsibilities very seriously. with inadequate resources and limited practical guidelines from the department of education, these educators tried hard to provide their students with the best possible educational opportunities. we also recognize that what these teachers enacted in their classrooms was/is connected to the larger socio-political and socio-historical movement that discriminated against them and intentionally prepared them poorly so that they would function minimally within their local settings. nevertheless, we must express our constructive criticism regarding the inadequacy of most of the classroom practices we observed and the consequences of such practices for educating pupils. in this study, we described the complexity of transforming mathematics education. resulting from this observational and descriptive process we would suggest that the implementation of the rncs requires an epistemological shift in teaching, learning, and assessing learning. most of the participating teachers taught the way they were “trained” to teach. they clearly stated how their teacher education programs were structured. they also reflected on the last 10 years of education in sa. they were supportive of the rncs in terms of what to teach, learn, and assess in mathematics. they were, however, unclear and uncertain regarding how to implement these national requirements in their day-to-day classrooms: “because of that change, i feel that i personally need more guidance” (seventh-grade teacher). our study highlights similar concerns expressed elsewhere in studies of this sort. research has shown that teachers often prefer “a more favourable learning environment than the one perceived to be present” (aldridge, laugksch, & fraser, 2006, p. 137). research has also shown that teacher preparation in outcome-based education has been significantly small when compared to the expectation of the educational reforms in sa (aldridge, laugksch, & fraser, 2006). teachers desire to have better resources, more preparation, better relations with colleagues, and stronger parental support (aldridge, laugksch, & fraser, 2006). these desires echo the desires of our four teachers. in addition to these concerns the large class sizes found in these provinces combined with a lack of professional support makes it “difficult to use progressive teaching methods in the classroom” (aldridge, laugksch, & fraser, 2006, p. 138). pourdavood et al. mathematical discourse journal of urban mathematics education vol. 2, no. 1 103 for teachers to be competent in teaching mathematics in sa, they must be prepared more thoroughly in terms of mathematical content and pedagogy in grade appropriate contexts. the teachers we interviewed clearly expressed that they were aware of their own professional deficiencies and they worried about the consequences of these deficiencies for their students. currently, sa has done much to further their educational reforms and to provide this preparation to their teachers equitably. the department of education has developed a number of resources in addition to the rncs. the publication entitled “toward effective school management” (tesm) exists as a series of manuals that provide a framework for the department of education to oversee their new educational reforms. each manual covers a variety of issues related to the management of the diverse schools existing in sa and are designed to give individual school management teams the necessary guidance needed to manage their schools effectively (tesm, n.d.). in addition to this series the department of education has put together a web-portal that allows educators to access a wide variety of professional development resources, informational resources, and curricular resources (http://www.thutong.org.za). notwithstanding these efforts, the system is still struggling to prepare its teachers effectively due to the slowness of these materials and funding to reach the poorest areas of sa. it is vitally important for these teachers to be prepared to teach mathematics in a way that is culturally and contextually relevant in order to meet the needs of their students. teachers must be “provided with evidence of best practice that is based on solid research” so they can create the best learning environments possible for their students (webster & fisher, 2003, p. 324). in 1999, sa mathematics achievement for eighth grade was the lowest of all participating countries, as reported in the third international mathematics and science study (timss; highlights from the timss, 1999). this poor showing did not change after 4 years when students were tested again; in 2003 sa placed last of the 45 participating countries (timss, 2003). south african educators continue to face heavy challenges as they struggle to overcome the tough realities facing them as teachers. after 15 years of reform, the funding of schools is still insufficient and inequitable, class sizes are still cumbersome, preparation remains inadequate, and language barriers still exist. the legacy of the apartheid system of education and its deep impact on denying equitable education for all citizens, keeps transforming mathematics education an uncertainty in the new multilingual classrooms of sa. democratic education may not be realized if it does not reach the next stage of connectivity and relationship with the existing day-to-day classroom situation. at the present time, there is limited research-based curriculum material in mathematics. therefore, more research is needed in this area. http://www.thutong.org.za/ pourdavood et al. mathematical discourse journal of urban mathematics education vol. 2, no. 1 104 it is important to transformative mathematics education that higher education institutions of learning take hold of their responsibility to provide effective teacher preparation programs. future mathematics teachers need to be prepared to face the challenges of a new multilingual society. will they be prepared for the new societal and educational demands that await them? acknowledgments this research was supported partly by the u.s. fulbright scholar exchange program 2004 to south africa and partly by nelson mandela university, department of science, mathematics, and technology education. opinions presented in this manuscript belong to the authors. references aldridge, j. m., laugksch, r. c., & fraser, b. j. (2006). school-level environment and outcome based education in south africa. learning environments research, 9(2), 123–147. blanton, r. e., & harmon, h. l. (2005). building capacity for continuous improvement of math and science education in rural schools. the rural educator, 26(2), 6–11. boyer, s. j., & bishop, p. j. (2004). young adolescent voices: students‟ perceptions of interdisciplinary teaming. research in middle level education online, 28(1), retrieved january 30, 2009, from http://www.nmsa.org/publications/rmleonline/tabid/101/default.aspx denzin, n. (1984). the research act. englewood cliffs, nj: prentice hall. english, l. d., & watters, j. j. (2004). mathematical modeling with young children. proceedings of the 28th international conference of the international group for the psychology of mathematics education, bergen, no, 2, 335–342. equity in education expenditure. (n.d.) retrieved january 22, 2009, from http://www.info.gov.za/aboutsa/education.htm frempong, g. (2005). exploring excellence and equity in canadian mathematics classrooms. proceedings of the 29 th international conference of the international group for the psychology of mathematics education, melbourne, au, 2, 337–344. gilmartin, m. (2004) language education and the new south africa. tijdschrift voor economische en sociale geografie, 95(4), 405–418. gilmour, j. d. (2001). intention or in tension? recent education reforms in south africa. international journal of educational development, 21(1), 5–19. hufferd-ackles, k., fuson, k. c., & sherin, m. g. (2004). describing levels and components of a math-talk learning community. journal for research in mathematics education, 35(2), 81– 116. highlights of the third international mathematics and science study. (1999). retrieved january 22, 2009, from http://nces.ed.gov/timss/results99_1.asp howie, s. j. (2003). conditions of schooling in south africa and the effects on mathematics achievement. studies in educational evaluation, 29(3), 227–241. l‟encyclopédie de l‟agora, dossier : afrique du sud. (2005). retrieved january 15, 2007, from http://www.agora.qc.ca/mot.nsf/dossiers/afrique_du_sud. lincoln, y. s., & guba, e. g. (1985). naturalistic inquiry. beverly hills, ca: sage. lincoln, y. s., & guba, e. g. (1994). comparing paradigm in qualitative research. in n. denzin & y. lincoln (eds.), handbook of qualitative research. thousand oaks, ca: sage. mccracken, g. (1988). the long interview. newbury park, ca: sage. http://www.nmsa.org/publications/rmleonline/tabid/101/default.aspx http://www.info.gov.za/aboutsa/education.htm http://nces.ed.gov/timss/results99_1.asp http://www.agora.qc.ca/mot.nsf/dossiers/afrique_du_sud pourdavood et al. mathematical discourse journal of urban mathematics education vol. 2, no. 1 105 morrow, w. e. (1990). aims of education in south africa. international review of education, 36(2), 171–181. national council of teachers of mathematics. (2000). principles and standards for school mathematics. reston, va: national council of teachers of mathematics. o‟sullivan, m. c. (2006). teaching large classes: the international evidence and a discussion of some good practice in ugandan primary schools. international journal of educational development, 26(1), 24–37. revised national curriculum statement grades r–9. (2002). retrieved january 29, 2009, from http://www.info.gov.za/view/downloadfileaction?id=70257 spreen, c. a., & vally, s. (2006). education rights, education policies and inequality in south africa. international journal of educational development, 26(4), 352–362. third international mathematics and science study report (2003). retrieved january 22, 2009, from http://nces.ed.gov/timss/results03.asp toward effective school management. (n.d.). retrieved january 22, 2009, from http://www.thutong.doe.gov.za/ webster, b. j., & fisher, d. l. (2003). school-level environment and student outcomes in mathematics. learning environments and research, 6(3), 309–326. zevenburgen, r. (2005). primary preservice teachers understanding of volume: the impact of course and practicum experiences. mathematics education research journal, 17(1), 3–23. http://www.info.gov.za/view/downloadfileaction?id=70257 http://nces.ed.gov/timss/results03.asp http://www.thutong.doe.gov.za/ journal of urban mathematics education july 2013, vol. 6, no. 1, pp. 1–5 ©jume. http://education.gsu.edu/jume david w. stinson is an associate professor of mathematics education in the department of middle and secondary education in the college of education, at georgia state university, p.o. box 3978, atlanta, ga, 30303; e-mail: dstinson@gsu.edu. his research interests include exploring socio-cultural, -historical, and -political aspects of mathematics and mathematics teaching and learning from a critical postmodern theoretical (and methodological) perspective. he is a co-founder and current editor-inchief of the journal of urban mathematics education. joi a. spencer is an associate professor of mathematics education in the department of learning and teaching in the school of leadership and education sciences, at the university of san diego, 5998 alcala park, san diego, ca 92110; e-mail: joi.spencer@sandiego.edu. her research examines african american student mathematics identities and the mathematics instructional practices of those who teach black students. guest editorial conversations about privilege and oppression in mathematics education david w. stinson georgia state university joi a. spencer university of san diego …i think the african american community is also not naïve in understanding that, statistically, somebody like trayvon martin was statistically more likely to be shot by a peer than he was by somebody else. so folks understand the challenges that exist for african american boys. but they get frustrated, i think, if they feel that there’s no context for it and that context is being denied. and that all contributes i think to a sense that if a white male teen was involved in the same kind of scenario, that, from top to bottom, both the outcome and the aftermath might have been different. …now, the question for me at least, and i think for a lot of folks, is where do we take this? how do we learn some lessons from this and move in a positive direction? i think it’s going to be important for all of us to do some soul-searching. …[i]n families and churches and workplaces, there’s the possibility that people are a little bit more honest, and at least you ask yourself your own questions about, am i wringing as much bias out of myself as i can? am i judging people as much as i can, based on not the color of their skin, but the content of their character? that would, i think, be an appropriate exercise in the wake of this tragedy. …and so we have to be vigilant and we have to work on these issues. and those of us in authority should be doing everything we can to encourage the better angels of our nature. – barack hussein obama ii  44th president of the united states of america he purposefully selected quotes (above) from president obama’s remarks on trayvon martin delivered in the white house press briefing room on july 19, 2013 1 effectively frame the intended spirit of this jume special issue (co-guest edited with joi spencer). the special issue was conceived of in october 2012 as we 1 for president obama’s complete “remarks by the president on trayvon martin,” see http://www.whitehouse.gov/the-press-office/2013/07/19/remarks-president-trayvon-martin. t http://www.whitehouse.gov/the-press-office/2013/07/19/remarks-president-trayvon-martin stinson & spencer guest editorial stinson, d. w., & spencer, j. a. (eds.). (2013). privilege and oppression in the mathematics preparation of teacher educators [special issue]. journal of urban mathematics education, 6(1). 2 (david and joi), along with approximately 40 other mathematics educators from approximately 25 universities, attended the privilege and oppression in the mathematics preparation of teacher educators (prompte 2 ) conference held in battle creek, michigan. mathematics educators from michigan state university (beth herbeleisenmann, tonya bartell, kristen bieda, sandra crespo, higinio dominguez, and corey drake) and bucknell university (m. lynn breyfogle) convened the conference. the electronic-mail invitation to the conference stated: we would like to invite you to participate in a small conference titled privilege and oppression in the mathematics preparation of teacher educators (prompte), where we will engage in conversations about systems of privilege and oppression (e.g., racism, classism, sexism, heterosexism, ableism) in our work as mathematics teacher educators (mtes). although mtes have begun to talk about these issues in relation to the preparation of mathematics teachers (mts) and mathematics teaching, we rarely talk about them with respect to our own preparation and the preparation of future mtes. our hypothesis is that concentrated attention to thoughtful discussion and action related to identifying, understanding, and confronting systems of privilege and oppression can improve our work as mtes and, ultimately, will impact mts’ and students’ learning experiences in mathematics classrooms, especially students who have been historically underserved in schools. this conference will provide a venue in which to plan and take thoughtful action. by “thoughtful action,” we mean action that can allow us to change our own interactions related to systems of privilege and oppression, develop strategies for working on these systems amongst ourselves and with our graduate and undergraduate students, and enable us to invite others into such conversations. (b. herbel-eisenmann, personal communication, august 8, 2012) as part of the “thoughtful action” called for in the invitation and throughout the 3-day conference, the intention of this jume special issue is to invite all mathematics educators (and others) into conversations about systems of privilege and oppression. the request for manuscripts for possible inclusion was sent to all prompte conference participants. manuscripts were limited to 3,500 words or less (exclusive of references, tables, figures, etc.), although, longer manuscripts were considered; revised through an open, peer-review process; and accepted in a variety of formats (e.g., narratives, reflections, dialogues, storytelling, critiques, etc.). the scholarly contributions found here, however, should not be read as “research articles” aimed as providing probable “solutions” to the injustices of privilege and oppression in mathematics education. but rather, read as academic essays of invitation to join in scholarly discourse across differences in critically examining privilege and oppression as mathematics educators (i.e., mathematics education researchers, teacher educators, classroom 2 privilege and oppression in the mathematics preparation of teacher educators (prompte) conference (funded by create for stem institute through the lappan-phillips-fitzgerald cmp 2 innovation grant program), michigan state university, battle creek, mi, october 2012. any opinions, findings, and conclusions or recommendations expressed herein are those of the authors and do not necessarily reflect the views of the funding agency. stinson & spencer guest editorial stinson, d. w., & spencer, j. a. (eds.). (2013). privilege and oppression in the mathematics preparation of teacher educators [special issue]. journal of urban mathematics education, 6(1). 3 teachers, and mathematicians). in other words, they are to be read as inclusive continuations or extensions of the critical conversations that were begun at the prompte conference. each of the eight essays included, authored or co-authored by 14 conference participants, challenges all those who are concerned with providing a humanizing and empowering (mathematics) education for all children (bartolomé, 1994; freire, 1970/2000) to break the deafening sound of silence of privilege and oppression in mathematics education specifically and in schools and society generally. in the opening essay, “strong is the silence: challenging interlocking systems of privilege and oppression in mathematics teacher education,” beth herbeleisenmann, tonya gau bartell, m. lynn breyfogle, kristen bieda, sandra crespo, higinio dominguez, and corey drake, explore the interlocking systems of privilege and oppression in relationship to themselves as mathematics teacher educators and in the preparation of new mathematics teacher educators. beth and colleagues provide justifications in calling for multiple conversations about privilege and oppression at all levels of mathematics education. they present their justifications from both personal perspectives, sharing their own struggles in engaging in such conversations, and scholarly perspectives, drawing upon the research literature that supports such conversations. joel amidon, in his essay, “teaching mathematics as agape: responding to oppression with unconditional love,” theorizes an ideal relationship between students and mathematics that is functional, communal, critical, and inspirational through teaching mathematics as an act of unconditional love. he begins his theorizing by asking: what do i do from my position of power and privilege as a mathematics teacher, researcher, and teacher educator to interrupt oppression and enable everyone the opportunity and expectation of success in mathematics and in life? this question is a solid beginning—to dismantle privilege and oppression, one must first come to understand it and see how it operates. in their essay, “‘all for one and one for all’: negotiating solidarity around power and oppression in mathematics education,” victoria (vicki) hand and imani masters goffney reflect on the tensions inherent in standing with and speaking on behalf of communities. they provide “cautionary tales” in regards to the intensions and impact of actions as they speak back to the burgeoning group of mathematics education researchers who focus on equity and justice. vicki and imani discuss how “equity” researchers might build the research base of equity work in mathematics education. but caution that in an effort to present a unified face to the larger mathematics education community, equity work runs the risk of marginalizing the multiplicity of possible voices within the field. tonya gau bartell and kate johnson provide lists of both privilege and oppression as they begin to unpack the invisible knapsack of privilege in mathematics education research in their essay, “making unseen privilege visible in mathematics education research.” tonya and kate take a macro look at the field of mathematics ed stinson & spencer guest editorial stinson, d. w., & spencer, j. a. (eds.). (2013). privilege and oppression in the mathematics preparation of teacher educators [special issue]. journal of urban mathematics education, 6(1). 4 ucation research, asking mathematics education researchers to assess their own privilege. they warn that a lack of self-examination often leads to paternalistic stances and solutions to the problems of mathematics teaching and learning. tonya and kate speak most directly to those whom the field of mathematics education is centered— researchers—calling for researchers to examine their positionality and to consider how the questions they ask might stymie victories for students most in need. judit moschkovich describes principles for equitable mathematics teaching practices for english language learners and outlines guidelines for materials to support such practices in her essay, “principles and guidelines for equitable mathematics teaching practices and materials for english language learners.” judit contends that the “language of mathematics” should not be understood merely as a list of vocabulary or technical words with precise meanings, but rather as the multiple communicative competencies necessary for equitable participation in high cognitive demand mathematical tasks. the principles and guidelines she provides stress the importance of creating learning environments that support all students in engaging in rich mathematical activities and discourse. in their essay, “advocating for equitable mathematics education: supporting novice teachers in navigating the sociopolitical context of schools,” craig willey and corey drake present questions that novice mathematics teachers might ask at the personal, interpersonal, institutional, and cultural levels that hold the potential to disrupt the too often oppressive dominant discourses of mathematics teaching and learning. given the current out-of-control era of accountability, they argue it is increasingly necessary that teacher educators assist preand in-service teachers to develop a critical consciousness about the sociopolitical context of schooling and to assume an activist stance. craig and corey propose ways that mathematics educators might assist those with overwhelmingly little power (new teachers) to do the hard work—the work of disrupting privilege and oppression in the mathematics classroom. laura mcleman and joyce piert share some of their journey, through a backand-forth professional dialogue, as they seek to make sense of what it might mean to prepare secondary mathematics preservice teachers to teach mathematics for social justice in their essay, “considering the social justice mathematical journey of secondary mathematics preservice teachers.” laura and joyce’s dialogue becomes centered on the question: should all college-level mathematics courses be taught through a lens of social justice? their dialogue exposes the kinds of stalemates that too often occur at the collegiate mathematics level. in the midst of their dialogue, laura and joyce have an epiphany: in order to help preservice teachers rethink mathematics— and in order for them to rethink mathematics themselves—they realize that they must take a leap (of faith of sorts). and, in similar style, anita wager and kristin whyte, in their essay, “young children’s mathematics: whose home practices are privileged?,” share a professional dialogue about the ways in which issues of power and privilege emerge in pre stinson & spencer guest editorial stinson, d. w., & spencer, j. a. (eds.). (2013). privilege and oppression in the mathematics preparation of teacher educators [special issue]. journal of urban mathematics education, 6(1). 5 school classrooms when teachers endeavor to build on children’s home and school mathematical experiences. throughout the dialogue, anita and kristin focus on three questions: what home mathematics practices do pre-k teachers privilege? how does privileging particular practices reinforce or interrupt historical power structures in teacher/parent and teacher/child relationships? which families and children do teachers privilege and how might that privileging oppress others? they claim that teachers need to have explicit conversations about what, how, and who they privilege and what the consequences of that power to privilege might have in their work with families and children. we believe that the essays within the pages of this jume special issue (and the extensive scholarship cited throughout) provides ample entry points for critically challenging and productively discomforting conversations about privilege and oppression among colleagues and students (and others) as well as within oneself—a sort of pedagogical tool, if you will. we hope and trust that the conversations begun at the prompte conference and continued or extended here will be a catalyst for critical soul-searching, asking oneself (and one another): “am i wringing as much bias out of myself as i can” (obama, 2013, ¶23)? in what we hope to be collective efforts of “becoming a more perfect union—not a perfect union, but a more perfect union” (obama, ¶25), we challenge mathematics educators to take up president obama’s charge. in the absence of justice for far too many, mathematics educators, as people of authority, must put ourselves, our positions, our power—our privilege—on the line. unafraid and unashamed, we must ask questions from the margins, evaluate and re-evaluate our stances, and critique our work for the purpose of justice in our field specifically, and in schools and society generally. the conversations at the prompte conference and continued or extended here, take the stand that change begins at home. privilege and oppression is not a figment of “other peoples’” imagination, but holds a great deal of explanatory power related to achievement and success differentials in mathematics in the united states (and throughout the globe). for the tide to change in regard to mathematics opportunities, we, as mathematics educators, must be vigilant in examining and re-examining our work, our commitments, and ourselves. then, we must do the hard work of making things right—“doing everything we can to encourage the better angels of our nature” (obama, ¶25). references bartolomé, l. i. (1994). beyond the methods fetish: toward a humanizing pedagogy. harvard educational review, 64, 173–194. freire, p. (2000). pedagogy of the oppressed (m. b. ramos, trans.; 30th anniv. ed.). new york, ny: continuum. (original work published 1970) obama, b. h. (2013). remarks by the president on trayvon martin. retrieved from http://www.whitehouse.gov/the-press-office/2013/07/19/remarks-president-trayvon-martin. http://www.whitehouse.gov/the-press-office/2013/07/19/remarks-president-trayvon-martin journal of urban mathematics education december 2012, vol. 5, no. 2, pp. 182–187 ©jume. http://education.gsu.edu/jume morgin jones williams is an instructor of mathematics at spelman college and a secondyear doctoral student at georgia state university, p.o. box 3978, atlanta, ga 30303, email: mjones137@student.gsu.edu. her research interests include examining the lived experiences of undergraduate african american women in mathematics. elijah porter is a high school mathematics teacher at columbia high school in decatur, ga and a second-year doctoral student at georgia state university, p.o. box 3978, atlanta, ga 30303, email: eporter9@student.gsu.edu. his research interests include mathematics, culture, and linguistics. book review reclaiming the right to the city: a book review of the new political economy of urban education: neoliberalism, race, and the right to the city 1 morgin jones williams spelman college elijah porter columbia high school n the past 30 years or so, the neoliberal agenda or, more generally, neoliberalism has forcefully pervaded the changing landscape of “urban” education and negatively influenced education policies; too often resulting in alarming negative consequences on the teaching and learning experiences of historically marginalized students and their communities. in her new book the new political economy of urban education: neoliberalism, race, and the right to the city, lipman (2011) describes neoliberalism as an ensemble of economic and social polices, forms of governance, and discourses and ideologies that promote individual self-interest, unrestricted flows of capital, deep reductions in the cost of labor, and sharp retrenchment of the public sphere. neoliberals champion privatization of social goods and withdrawal of government from provision for social welfare on the premise that competitive markets are more effective and efficient. neoliberalism is not just “out there” as a set of polices and explicit ideologies. it has developed as a new social imaginary, a common sense about how we think about society and our place in it. (p. 6) throughout the book, lipman demonstrates the negative consequences of neoliberalism by placing the reader in the middle of the current events in chicago public schools as well as by providing a historical timeline that has lead to its current troubling state of affairs. in this review, we provide a brief overview of lipman’s 1 lipman, p. (2011). the new political economy of urban education: neoliberalism, race, and the right to the city. new york, ny: routledge. pp. 208, $36.95 (paper), isbn 978-0-415-80224-6 http://www.routledgementalhealth.com/catalogs/education_policy_and_politics/1/9/ i http://www.routledgementalhealth.com/catalogs/education_policy_and_politics/1/9/ jones williams & porter book review journal of urban mathematics education vol. 5, no. 2 183 critique on urban education and the neoliberal agenda. we then make connections between lipman’s analysis and our own lived urban schooling experiences. we hope that each of our personal accounts offers insight into the power of neoliberalism and its detrimental effects on the nation’s public education system—and its students. the new political economy at a glance throughout her latest book, lipman (2011) takes great effort to make clear her points of concern. she begins with a comprehensive analysis of neoliberalism as the force that launched and continues to fuel society’s actions to empower corporations. she continues by describing, in great detail, the destruction of what arguably is the last great public service that society has to offer—public education. lipman presents an analysis detailing how public education is being obliterated across the country. she does so by documenting the historical footprint of the penetration of corporations into public education, highlighting efforts to thrust students into stem (science, technology, engineering, and mathematics) fields. lipman (2011) chronologically frames her analysis and guides the reader through a revealing tour of the chicago public school system, which she describes as a “‘laboratory’ for neoliberal policy experiments” (p. 23). she demonstrates how this misfortune of experimentation is nothing more than a carbon copy of what has occurred and continues to occur in cities across the nation. lipman channels her personal knowledge of chicago’s past, critically assessing the educational reform efforts instituted by the city’s policy-making team, and passionately argues for expanding critical research in this area of inquiry. throughout the book, she illustrates the importance of her refusal to accept ideologies that support a neoliberal society. lipman concludes the book by stressing that through diagnosis a prognosis is possible, and that every person has a role to play in the pursuit of social justice and the fight against corporate take-over. she advocates for every individual to become a change agent in her or his community so that we may collectively create equitable possibilities for students (and teachers) and improve the learning possibilities for all children in the nation’s public schools. lipman (2011) clearly details her research efforts and analytical techniques so that the reader may appreciate her constructionist epistemology and critical agenda. the book contains seven chapters: (1) introduction (2) neoliberal urbanism and education policy, (3) dismantling public schools, displacing african americans and latino/as, (4) racial politics of mixed-income schools and housing, (5) venture philanthropy, (6) choice and empowerment, and (7) education and the right to the city. each chapter contributes to the overarching theme of the book: the need for a critical reexamination of the nation’s public education system, particularly the policies that affect marginalized groups. moreover, each jones williams & porter book review journal of urban mathematics education vol. 5, no. 2 184 chapter provides a template for measuring the extraordinary quality of her critical analysis. lipman (2011) assists the reader in developing a relevant timeline and examining the relationship amongst the city, the citizens, and corporate america. using lipman’s analysis as a compass, the reader is able to see a shift in the priorities of public policy over time: from improving the living conditions of all citizens in the 1970s, to growing u.s. corporate interests in the 1980s, to globalizing corporate america in the 1990s. the 2000s and beyond are characterized by the ongoing dismantling of the public sphere, including the public education system. lipman’s detailed fieldwork coupled with the prevailing nature of the current assault on public education makes this book a must read for those who wish to understand more deeply the complexities of urban education, including urban mathematics education. introspections on a neoliberal society in the following section, we provide introspections on lipman’s (2011) book by presenting brief narratives of our own lived urban schooling experiences within a neoliberal society. both narratives, we believe, exemplify the reality of the dangerously uncritical “new social imaginary” (p. 6) of neoliberalism. morgin’s introspection as an african american woman living in a neoliberal society, lipman’s (2011) book compelled me to reconsider the daily challenges marginalized students face compared to my own public schooling experiences in virginia beach, virginia. to some extent, i was aware of the racial complexities that impact our society, particularly in school settings. throughout my schooling experiences, i quickly learned to negotiate the politics that constantly burden public education through the support of a family of educators. in high school, my accelerated courses were predominately occupied by white students; there were only a few african american students granted access to advanced courses. together, my african american peers and i coped with the competitive nature of the schooling process, supporting each other’s academic efforts and exceptional achievements along the way. the isolation that we felt in school was certainly indicative of a larger, more serious problem. reflecting upon my high school experiences brings back memories of students’ individualistic attitudes. this individualism still pervades the culture as a trace of neoliberal values. lipman’s (2011) critical observations made me realize that public school education has been largely a training ground for neoliberal thought. my peers knew that in order to be considered successful they should at jones williams & porter book review journal of urban mathematics education vol. 5, no. 2 185 tend the nation’s top colleges and universities—and a majority of white students did just that. for african american students, the expectations were slightly different. it was evident that we did not operate on a level playing field, but all students were expected to achieve academically and support, in one form or another, the agenda of capitalism. due to the rise of corporate america, many students were encouraged to learn more about stem initiatives and to pursue careers in those fields. because of my early “grooming,” i decided to earn an undergraduate and master’s degree in mathematics and applied mathematics, respectively. i forecasted my future career as a mathematician and banked on lucrative opportunities provided to me by the profession. the schooling experiences of today’s students are not much different from my own: stem fields are presented everywhere and by everyone as the key to success. as a graduate of and undergraduate mathematics instructor at spelman college, i empathize with the african american women that i teach each semester. in my role as an instructor at a historically black college for women, i understand the impact of the neoliberal agenda on students’ thinking inside my classroom. lipman (2011) highlighted that “[social] policies are, in part, discourses—values, practices, ways of talking and acting—that shape consciousness and produce social identities” (p. 11). at the beginning of each semester, i ask my undergraduate mathematics students what they want to be when they “grow up” and why. more often than not, my students have declared or are thinking about declaring a major in the stem fields, and i enthusiastically support their academic goals and professional aspirations. nonetheless, i believe that some of these students may be simply perpetuating the ideals sustaining neoliberalism in this country as they acquire dreams of climbing the “corporate ladder.” it is clear that my undergraduate students have developed their own perceptions of what it means to live and survive in a neoliberal society, and consider spelman college an ideal starting point on their journey to success. importantly, based on the selectivity of the institution, spelman college attracts women from around the world. it is evident that my undergraduate students feel the pressure to align themselves with those undergraduate students who attend ivy league colleges and universities. they believe that acquiring an undergraduate degree from spelman will inevitably increase the brightness of their futures. while that might be true for some graduates, others may confront the sobering reality that they have to work much harder than white students because of the persistent limiting effects of racism (and patriarchy) within our society. it is important for undergraduate students to recognize how the neoliberal agenda operates in their school community, and in society at large. it is equally important that educators engage in teaching and learning practices that motivate students to enact change in the public sphere. only then will there be possibilities for equitable education and opportunity for all. it is my hope that lipman’s (2011) voice will jones williams & porter book review journal of urban mathematics education vol. 5, no. 2 186 inspire individuals to critically assess and bravely confront the ongoing issues within the u.s. educational system, and society at large. elijah’s introspection as an african american male, growing up during the time period in which lipman (2011) suggests that cities prioritized the citizenry (the 1970s), i remember my parents’ accounts of receiving public assistance in terms of food, housing, and employment. the civil rights acts had not long before been signed into law, and the nation was recovering from several riots across the country. i would say that my educational opportunities were top-notch. it is immaterial to debate if the opportunities made available to me by the city were obligatory or moral. during the 1980s, i was directly impacted by the harmonious partnership amongst the detroit public school system, the michigan post-secondary educational system, and corporate america. lipman (2011) reports that during this decade the cities focused on the growth of corporate america. specifically, there was a major push to grow the mathematics, science, and technology fields in detroit and throughout michigan. as “the motor city,” money, opportunity, and momentum all intersected in detroit during that time period. in large part, many of my friends and i are in stem careers today due to the money, opportunity, and momentum created and invested in us during our school years. it can be argued that the number of minorities currently in stem fields today is due directly to the efforts of cities like detroit and what lipman describes as the push to grow corporate america. the globalization of corporate america, as lipman (2011) calls it, was quite evident during the 1990s. i noticed this globalization effort as a member of a corporate team that trained workers in another country during the mid-1990s. i was employed for 15 years as an engineer for a major american computer company; for three of those years i trained workers in europe to do the work that we were doing here in the united states. when i finished training the european workers, my entire department was moved to europe. i was not privy to the consequences of my actions, but lipman has provided a detailed analysis of the long-term effects of actions similar to mine and the efforts of thousands of others to globalize corporate america. a major economic downturn occurred in the united states in the early 2000s: the burst of the dot-com bubble. the impact of my earlier global actions finally became clear to me and, as a result, i changed careers and entered the education field during the early 2000s. as lipman (2011) points out, the beginning of the dismantling of public education can be specifically tracked to this time period, and i have experienced some part of its destruction every day since i became a teacher. the closing of schools, the lack of resources, the constant fluctuation of “standards,” the infusion of charter schools, and the influence of funds, whether it jones williams & porter book review journal of urban mathematics education vol. 5, no. 2 187 is from the federal government, corporate america, or philanthropic groups, all have had devastating effects on public education. as a high school mathematics teacher, i have witnessed firsthand the low morale that has developed among some of the teachers and students. i also have seen the high levels of frustration amongst teachers, which has led to scores of teachers retiring or exiting the profession. all of these claims are expounded upon in the cause and effect analysis of lipman’s research. if i were not on the inside experiencing everything that she has documented in her book, i might be tempted to believe that she has exaggerated her point. however, i am on the inside, and i see how these ideas are made manifest in urban settings. i applaud her activist scholarship. reclaiming the right to the city together, we admire lipman’s (2011) examination of the relationship between the public education system and corporate america. she pushes the reader into deep self-reflection. her discourse is penetrating and demands the reader to select a side: the empowerment of children, youth, and communities or the empowerment of corporations. lipman gives the reader the necessary information to make an informed choice, providing her own evaluation, and making clear that simply sitting idly and observing the shifts in society makes known a person’s choice. her work details a gloomy tale, but she opens the reader’s mind to the possibilities of change. going forward, we ought to educate student learners about the power of neoliberalism and help them to better understand that “education is integral to a movement to reclaim the city” (lipman, 2011, p. 161). as educators, we strongly believe that individuals must activate their own agency so that a critical mass aimed at “reclaiming the city” can mobilize. this action, we believe, will result in a new political economy. references lipman, p. (2011). the new political economy of urban education: neoliberalism, race, and the right to the city. new york, ny: routledge. journal of urban mathematics education december 2013, vol. 6, no. 2, pp. 1–6 ©jume. http://education.gsu.edu/jume david w. stinson is an associate professor of mathematics education in the department of middle and secondary education in the college of education, at georgia state university, p.o. box 3978, atlanta, ga, 30303; e-mail: dstinson@gsu.edu. his research interests include exploring socio-cultural, -historical, and -political aspects of mathematics and mathematics teaching and learning from a critical postmodern theoretical (and methodological) perspective. he is a cofounder and current editor-in-chief of the journal of urban mathematics education. editorial on being a hardliner on issues of race and culture in mathematics education research1 david w. stinson georgia state university n the past, when i have been a discussant or respondent at conferences, after i have provided my remarks, i am often accused of being somewhat of a “hardliner” when it comes to the inclusion of issues of “race”/ethnicity and culture, or, more generally, the challenges and promises of exploring “diversity” (broadly defined) in mathematics education research. so this afternoon, it’s with great pleasure that i provide some briefs remarks in response to professor na’ilah nasir’s (2013) plenary address “why should mathematics educators care about race and culture?” i believe that my hardliner image has evolved over the years because more often than not i offend folks (unintentionally) by strongly arguing for an explicit and clear focus on the issues of race and culture in their projects (see, e.g., stinson, 2011). this call for an explicit and clear focus is especially evident when projects have been positioned under the larger—and i might add, increasingly popular—umbrella of “equity work” in mathematics education research. i believe that such projects should keep both culture and mathematics education in the foreground (and here, when i say mathematics education, i am including not only the teaching and learning of mathematics but also the discipline). in actuality, i believe that all mathematics education research should pay serious attention to issues of race, culture, and diversity, broadly defined—but that’s just me. the increasing popularity of positioning projects under the “equity” umbrella is clearly evident in grant proposals and submitted manuscripts; given that, the words equity and its derivative, “diversity,” have become increasingly important within the discourses of funding agencies and editorial boards. that is to say, it appears that more and more folks are “positioning” their research as equity or diversity projects. but more often than not, i can see the complexities of mathematics education in the 1 this editorial is a revised version of remarks delivered at the 35th annual meeting of the north american chapter of the international group for the psychology of mathematics education, chicago, il, november 15, 2013; the remarks were in response to professor na’ilah suad nasir’s (2013) plenary address “why should mathematics educators care about race and culture?” (see stinson’s reaction to nasir for accompanying powerpoint presentation.) i http://education.gsu.edu/jume mailto:dstinson@gsu.edu http://www.pmena.org/pastconferences/2013/index.html stinson editorial journal of urban mathematics education vol. 6, no. 2 2 foreground—after all, we’re mathematics educators. but the complexities of issues around race, culture, diversity, or equity, in general, somehow slip in the background or are left to the reader to make some kind of implicit connections. or, worst yet, these issues are stripped of their complexities and reduced to “labels” or “categories” in which children, youth, and communities belong; unfortunately, the latter is too often the case (see, e.g., lubienski & bowen, 2000; parks & schmeichel, 2012). as i make this critique of some of the work positioned under the equity umbrella and mathematics education research in general, i clearly understand the difficulty of keeping issues of race, culture, or diversity, broadly defined, equally in the foreground along with issues of mathematics teaching and learning. it was not too long ago that the jume editorial team provided what we believed to be a much needed space for intellectual discourse around the very issue of the importance of maintaining a “both–and” approach in mathematics education research by publishing a collection of critical commentaries (see battista, 2010; confrey, 2010; martin, gholson, & leonard, 2010). these commentaries were in response to kathleen heid’s journal for research in mathematics education editorial “where’s the math (in mathematics education research)?” (heid, 2010) and to a national council of teachers of mathematics research presession symposium “keeping the mathematics in mathematics education research” (ball, battista, guershon, thompson, & confrey, 2010). we can hypothesize about the many reasons that folks have the tendency to let the issues of race, culture, or again, more broadly, issues of diversity slip to the background, or to avoid them altogether:  restrictions on the length of manuscripts (that is, can one do justice to both culture and mathematics in a single grant proposal or manuscript?);  concerns or fears of “political correctness” in talking about issues such as race, racism, and white supremacy (that is, white folks of a certain age in the united states have been reared in a discourse of political correctness where it is not “proper” to talk about race and other such “uncomfortable” things.); or  lacking the knowledge of how to engage in the sheer volume of literature that addresses issues of race, culture, gender, language, socioeconomic class, and so forth (that is, in our “formal” schooling in becoming mathematics educators and researchers, how much time was devoted in our doctoral programs to exploring, in meaningful ways, larger sociocultural and -political issues of human existence, and how they relate to mathematics and mathematics teaching and learning?). in short, doing race work, culture work, diversity work, or equity work in mathematics education research is just hard to do.2 as i struggle in doing both–and in my own work, i often return to a diagram that has become quite familiar: the instructional triangle. this diagram, originat 2 for a collective discussion of the challenges and promises of doing race and culture work in mathematics education research and teacher education, see jume special issue: volume 6, number 2 (stinson & spencer, 2013). http://ed-osprey.gsu.edu/ojs/index.php/jume/issue/view/12 http://ed-osprey.gsu.edu/ojs/index.php/jume/issue/view/12 stinson editorial journal of urban mathematics education vol. 6, no. 2 3 ing from a consortium for policy research in education paper (cohen & ball, 1999), has become somewhat of a standard model when considering the teaching and learning context. the model was further refined specifically for mathematics education in the book adding it up: helping children learn mathematics (national research council, 2001). i return to this figure in my own research to keep me grounded in thinking about what my work specifically has to do with the dynamics of mathematics and mathematics teaching and learning as i bring issues of race, culture, and diversity, broadly defined, to the foreground. other researchers have provided extensions, elaborations, or rethinkings of the instructional triangle. for instance, nipper and sztajn (2008) extend the instructional triangle into the challenges of mathematics teachers’ professional development. herbst and chazan (2012) elaborate on the instructional triangle to illustrate how the nature of mathematics instructional activity might help in justifying teachers’ actions in mathematics teaching. and bullock and i (stinson & bullock, 2012) rethink the instructional triangle as we apply critical postmodern theory and place each of the vertices under erasure (cf. derrida, 1974/1997). but the extension, elaboration, or rethinking that i turn to most often—and has become my standard—is the one provided by weissglass (2002) in figure 1. figure 1. the many factors that affect student learning (weissglass, 2002, p. 35). from “inequity in mathematics education: questions for educators,” by j. weissglass, 2002, the mathematics educator, 12(2), p. 35. copyright 2002 by the mathematics education student association. reprinted by permission. stinson editorial journal of urban mathematics education vol. 6, no. 2 4 as i have argued elsewhere (stinson, 2006), i believe that weissglass appropriately positions the triangle in its proper perspective. in that, when doing research in mathematics education—or dare i say, ethical research in mathematics education—explorations of mathematics teaching and learning must become much broader than what is possible within the confines of the initial instructional triangle (cohen & ball, 1999). it is important to note, however, that throughout the construction of the original model, cohen and ball (1999) consistently made reference to the “environmental” contexts in which the instructional triangle is embedded. but in specifically naming some of these socio-cultural, -historical, and -political contexts— contexts that too often marginalized particular students, families, and communities—weissglass (2002), i believe, is asking us to adopt a degree of social consciousness and responsibility in seeing the wider socio-cultural and -political picture of mathematics education (gates & vistro-yu, 2003). adopting such a stance requires us to delve deeper into how the social, political, cultural, and economic discourses of society in general affect the construction of students, teachers, and mathematics—and the possibilities and impossibilities of equitable and just mathematics teaching and learning. in short, it requires taking the “socio-political turn” in mathematics education research (gutiérrez, 2013, p. 40). in her talk this afternoon, i believe that professor nasir (2013) has asked us to engage in the ethical act of adopting a degree of social consciousness and responsibility in seeing the wider social and political picture of mathematics teaching and learning. and here my remarks are specific to some of the work that she and her colleagues from the national science foundation learning in informal and formal environments (life) center are engaged in currently (see http://lifeslc.org). in particular, i pull from a paper titled “learning pathways: a conceptual tool for understanding culture and learning” (nasir et al., 2013). in this paper, professor nasir and colleagues describe a developing framework for “conceptualizing learning as occurring along culturally organized learning pathways—the sequences of consequential participations and transitions in learning activities that move one toward greater social recognition as competent in particular learning domains and situations” (p. 2). what struck me about professor nasir and colleagues’ (2013) developing culturally organized framework for learning is that it is, concurrently, simple and complex. and one really has to possess poststructural sensibilities for this seemly contradictory remark to not be contradictory. nevertheless, the learning pathways draw attention to—  the resources students have access to (or not);  the ways that students are positioned as learners (or not); and  the role that identity—that is, the process of becoming—plays in learning. http://life-slc.org/ http://life-slc.org/ stinson editorial journal of urban mathematics education vol. 6, no. 2 5 according to the culturally organized framework, there are four key characteristics to learning pathways. characteristic 1 – learning pathways are taken up in relation to identities, and have a relational, affective, and motivational component: key here is the acknowledgement that a student’s identity (or her or his becoming) can be supported (or not) by the normalizing discourses and discursive practices, and that identity has a critical influence on a students’ motivation to continue on particular learning pathways (or not). characteristic 2 – learning pathways are socially constructed by self and others, and they build up over multiple instances: key here is the acknowledgement that learning pathways are iterative, building up over multiple instances with significant social others being important in supporting (or not) the construction and maintenance of particular learning pathways. characteristic 3 – learning pathways are made up of related sets of practices and routines, which over time support repertoires of practices, often organized with one or more goals in mind: key here is the acknowledgement that learning pathways are constructed and constituted through socially and historically accepted discourses and discursive practices that are made available (or not), and are shaped and reshaped over multiple times, in both informal and formal spaces. and characteristic 4 – learning pathways include enactments of privilege and marginalization that occur in relation to structural constraints and supports from families and institutions: key here is the acknowledgement that structures and the normalizing processes and practices of institutions serve to marginalize some students as they privilege others, and that absent of support from families members (extended or otherwise) some learning pathways are effectively closed off for certain students. so going back to the instructional triangle—after all, it is mathematics and mathematics teaching and learning that we’re researching. what if we overlay the instructional triangle with professor nasir and colleagues’ (2013) learning framework that conceptualizes learning as occurring along culturally organized learning pathways? but then again, for me, that just brings us back to figure 1. so, i guess, in the end, similar to professor nasir, i am a hardliner when calling for an explicit and clear focus on issues of race, ethnicity, culture, language, socio-economic class, and so on when doing ethical work in mathematics teaching and learning. —but then again, that’s just me. references ball, d. l., battista, m., harel, g., thompson, p. w., & confrey, j. e. (2010, april). keeping the mathematics in mathematics education research. research symposium at the national council of teachers of mathematics research presession, san diego, ca. battista, m. t. (2010). engaging students in meaningful mathematics learning: different perspectives, complementary goals. journal of urban mathematics education, 3(2), 34–46. retrieved from http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/115/58. http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/115/58 stinson editorial journal of urban mathematics education vol. 6, no. 2 6 cohen, d. k., & ball, d. l. (1999). instruction, capacity, and improvement. the consortium for policy research in education. retrieved from http://www.cpre.org/instruction-capacity-and-improvement. confrey, j. (2010). “both and”—equity and mathematics: a response to martin, gholson, and leonard. journal of urban mathematics education, 3(2), 25–33. retrieved from http://edosprey.gsu.edu/ojs/index.php/jume/article/view/108/53. derrida, j. (1997). of grammatology (g. c. spivak, trans., corrected ed.). baltimore, md: johns hopkins university press. (original work published 1974) gates, p., & vistro-yu, c. p. (2003). is mathematics for all? in a. j. bishop, m. a. clements, c. keitel, j. kilpatrick, & f. k. s. leung (eds.), second international handbook of mathematics education (vol. 1, pp. 31–73). dordrecht, the netherlands: kluwer. gutiérrez, r. (2013). the sociopolitical turn in mathematics education. journal for research in mathematics education, 44, 37–68. herbst, p., & chazan, d. (2012). on the instructional triangle and sources of justification for action in mathematics teaching. retrieved from http://deepblue.lib.umich.edu/handle/2027.42/91281. heid, m. k. (2010). where’s the math (in mathematics education research)? journal for research in mathematics education, 41, 102–103. lubienski, s. t., & bowen, a. (2000). who’s counting? a survey of mathematics education research 1982–1998. journal for research in mathematics education, 31, 626–633. martin, d. b., gholson, m. l., & leonard, j. (2010). mathematics as gatekeeper: power and privilege in the production of knowledge. journal of urban mathematics education, 3(2), 12–24. retrieved from http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/95/57. nasir, n. s. (2013, november). why should mathematics educators care about race and culture? plenary address delivered at the 35th annual meeting of the north american chapter of the international group for the psychology of mathematics education, chicago, il. nasir, n. s., barron, b., pea, r., goldman, s., stevens, r., bell, p., & mckinney de royston, m. (2013). learning pathways: a conceptual tool for understanding culture and learning. manuscript submitted for publication. national research council. (2001). adding it up: helping children learn mathematics. in j. kilpatrick, j. swafford, & b. findell (eds.), mathematics learning study committee, center for education, division of behavioral and social sciences and education. washington, dc: national academy press. nipper, k., & sztajn, p. (2008). expanding the instructional triangle: conceptualizing mathematics teacher development. journal of mathematics teacher education, 11, 333–341. parks, a. n., & schmeichel, m. (2012). obstacles to addressing race and ethnicity in the mathematics education literature. journal for research in mathematics education, 43, 238–252. stinson, d. w. (2006). african american male adolescents, schooling (and mathematics): deficiency, rejection, and achievement. review of educational research, 76, 477–506. stinson, d. w. (2011). “race” in mathematics education research: are we a community of cowards? [editorial]. journal of urban mathematics education, 4(1), 1–6. retrieved from http://edosprey.gsu.edu/ojs/index.php/jume/article/view/139/83. stinson, d. w., & bullock, e. c. (2012). transitioning into contemporary theory: critical postmodern theory in mathematics education research. in l. r. van zoest, j. j. lo, & j. l. kratky (eds.), proceedings of the 34th annual meeting of the north american chapter of the international group for the psychology of mathematics education (pp. 1163–1169). kalamazoo, mi: western michigan university. stinson, d. w., & spencer, j. a. (eds.). (2013). privilege and oppression in the mathematics preparation of teacher educators [special issue]. journal of urban mathematics education, 6(1). weissglass, j. (2002). inequity in mathematics education: questions for educators. the mathematics educator, 12(2), 34–39. http://www.cpre.org/instruction-capacity-and-improvement http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/108/53 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/108/53 http://deepblue.lib.umich.edu/handle/2027.42/91281 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/95/57 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/139/83 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/139/83 journal of urban mathematics education december 2017, vol. 10, no. 2, pp. 1–7 ©jume. http://education.gsu.edu/jume david w. stinson is professor of mathematics education in the department of middle and secondary education in the college of education and human development, at georgia state university, p.o. box 3978, atlanta, ga, 30303; e-mail: dstinson@gsu.edu. his research interests include exploring socio-cultural, -historical, and -political aspects of mathematics and mathematics teaching and learning from a critical postmodern theoretical (and methodological) perspective. he is a co-founder and current editor in chief of the journal of urban mathematics education. editorial beyond white privilege: toward white supremacy and settler colonialism in mathematics education david w. stinson georgia state university as i write, i try to remember when the word “racism” ceased to be the term which best expressed for me exploitation of black people and other people of color in this society and when i began to understand that the most useful term was “white supremacy.” – bell hooks here are numerous incidents over the past several months that bring into stark relief that society at large, here in the united states and around the globe, just possibly has been thinking, reading, talking, researching, writing, presenting, and so forth about the wrong thing; an inference i make from the bell hooks (1995, p. 184) statement above. perhaps in this twenty-fifth anniversary year of cornel west’s (1993/1994) powerful book race matters it is time to “flip the coin,” so to speak, so that we might begin to think, read, talk, research, write, present, and so forth about the other side of the coin: white supremacy. perhaps the statement has moved beyond race matters and its derivative white privilege matters toward white supremacy matters; a point that w. e. b. du bois (1920/1999), i believe, explicitly made nearly 100 years ago in his essay “the souls of white folk.”1 of course, white supremacy is what most black folk—laypersons and scholars alike, of the 1 “the souls of white folk” is the second essay in du bois’s (1920/1999) collection of essays darkwater: voices from within the veil. in celebration of the hundredth anniversary of du bois’s (1903/1989) collection of essays the souls of black folk, the editors (2003) of the monthly review reprinted “the souls of white folk,” they wrote: on the hundredth anniversary of the souls of black folk we are once again face to face with the ongoing absence of “racial democracy” at home and with an imperialism that walks naked abroad. “the souls of white folk,” like the souls of black folk before it, remains required reading. (p. 44) the souls of black folk and darkwater: voices from within the veil are available freely online: https://www.gutenberg.org/files/408/408-h/408-h.htm https://www.gutenberg.org/files/15210/15210-h/15210-h.htm t http://education.gsu.edu/jume mailto:dstinson@gsu.edu https://www.gutenberg.org/files/408/408-h/408-h.htm https://www.gutenberg.org/files/15210/15210-h/15210-h.htm stinson editorial journal of urban mathematics education vol. 10, no. 2 2 past and present—have always been thinking, reading, talking, … and so forth about; it is just that many? most? white folk—laypersons and scholars alike, of the past and present—choose not to listen. (the choice to not listen is just one of the numerous privileges that white supremacy continuously and consistently affords all white folk.) nonetheless, over the past several months, it has been interesting to listen for how the terms race, racism, whiteness, white privilege, and especially, white supremacy have been taken up and used by both white and non-white folk since the campaign, election, and inauguration of “america’s first white president” (coates, 2017, para. 6). these terms are clearly being taken up and used differently as well as provoking different reactions today than they did say just 18 or 24 months ago. i certainly have felt the shift in folk’s reactions as i continue to push into the idea that white supremacy is the most useful term to express the current and historical exploration of black folk and other folk of color in the united states and around the globe—in mathematics education and in society at large. for instance, although exploring the discourses and discursive practices of racism has been at the center of my research and scholarship, in my earlier work (e.g., stinson, 2006, 2008), i mention the term white supremacy in a cursory manner, almost as a footnote. it was not until a symposium at the 2014 national council of teachers of mathematics research conference in new orleans, louisiana (united states) that i began to equate white privilege/supremacy as the flipside to racism (stinson, 2014). in 2016 at the mathematics education and contemporary theory conference 3 in manchester, england (united kingdom), i made the absence of researching white supremacy in mathematics education—both in my work and in the larger mathematics education community—the concluding talking point of my presentation (stinson, 2016). and then in 2017 at the 9th international mathematics education and society conference in volos, greece (european union), i argued that the virtual absence, numerically evidenced, of researching white supremacy in mathematics education has been a strategic discursive practice (stinson, 2017). as i reflect on these last three experiences, i can easily say that the seen and heard reactions to the very term white supremacy in mathematics education have become increasingly more emotional and more resistant.2 that is to say, as i continue to push myself, and others, into moving beyond racism and white privilege toward a new space of critically examining and deconstructing white supremacy, i 2 it is important to note that these experiences were with largely white audiences in the united states, united kingdom, and european union; as valoyes-chávez and martin (2016) argue: the meanings of race and racial categories are created, politically contested, and re-created in any given sociohistorical and geopolitical context as a way to maintain boundaries of difference related to domination and oppression. … no matter what country (e.g., the usa, south africa, brazil, and throughout the european union), these meanings emerge to shape all social structures and institutions in a given society…, including mathematics education. (p. 1) stinson editorial journal of urban mathematics education vol. 10, no. 2 3 have encountered some impassioned reactions. these reactions have come from academic colleagues as well as from family members and long-time friends. maybe it has been my most recent approach—all white folk are inherently white supremacists—that has evoked such visceral reactions. my reactions to their reactions: i have stopped speaking to certain family members, i have continued to seek entry points with some long-time friends, and i have searched the literature in hopes of engaging academic colleagues. it was while searching the literature some months ago that i discovered the anne bonds and joshua inwood’s (2016) essay “beyond white privilege: geographies of white supremacy and settler colonialism,” which inspired both the title of this editorial and the possible beginnings of a new approach of getting others to push into the exploration and deconstruction of white supremacy—at least in mathematics education. limited space does not provide for a complete theoretical argument (a forthcoming project); therefore, here i just present some of the larger ideas from bonds and inwood in hopes of provoking some productive thinking and questioning around the analytic frames white supremacy and settler colonialism in mathematic education. some descriptions or definitions of white supremacy and settler colonialism, pulling from bonds and inwood’s synthesis, are in order first however. with these definitions in mind, i then ask mathematics educators (i.e., mathematics classroom teachers and teacher educators, mathematics education researchers and scholars, and mathematicians) to participate in a thought experiment; but first, the definitions. bonds and inwood (2016), in their definitions of white supremacy and settler colonialism, make explicit the differences between the analytic frames white privilege and white supremacy and between (post) colonialism and settler colonialism. they do so by pulling from a wide range of research and scholarship found in a variety of intellectual fields. one key aspect of bonds and inwood’s definitions is the positioning of both white supremacy and settler colonialism in historicized rather than historical contexts—historicized contexts locate the frames in the here and now rather than the past. in making their argument for engaging with the analytic frames white supremacy and settler colonialism, bonds and inwood do not entirely dismiss the more commonly used frames of white privilege and colonialism. but rather, they show how these commonly used frames are incomplete in identifying and documenting the ongoing violence perpetrated by the hegemony of the white racial frame (cf. feagin, 2013). to move beyond white privilege, bonds and inwood (2016) identify racism and white privilege as mere symptoms and white supremacy as the disease—their interest is in the disease. white supremacy simply defined, according to bonds and inwood, “is the presumed superiority of white racial identities … in support of the cultural, political, and economic domination of non-white groups” (pp. 719–720). white supremacy, therefore, “is the defining logic of both racism and privilege as stinson editorial journal of urban mathematics education vol. 10, no. 2 4 they are culturally and materially produced” (p. 720, emphasis in original). white supremacy as an analytic frame highlights— both the social condition of whiteness, including the unearned assets afforded to white people, as well [as] the processes, structures, and historical foundations upon which these privileges rest. european and, later, north american colonists created and developed a logic of race that placed white, european men at the pinnacle of the social hierarchy and all others in various positions of subordination… . these imaginations valorized whiteness and sanctioned the violence of white domination, enslavement, and genocide while bolstering eurocentric understandings of land use, private property, and wealth accumulation… . white supremacy is not only a rationalization for race; it is the foundational logic of the modern capitalist system and must be at the center of efforts to understand the significance of whiteness… . (p. 720) through a historicized understanding, white supremacy then is no longer located only in historical pasts or extremist groups but rather “reveals its stubborn endurance and the ways its every-day logics are reproduced through spectacular and mundane violences that reaffirm empire and the economic, social, cultural, and political power of white racial identities” (p. 721). the acknowledgment of the enduring violences (e.g., macro and micro racial aggressions) of the empire and the economic, social, cultural, and political power of white supremacy is what distinguishes settler colonialism from colonialism (bonds & inwood, 2016). settler colonialism is positioned in the here and now, a permanent and “unfolding project [that] involves the interplay between the removal of first peoples from the land and the creation of labor systems and infrastructures that make the land productive” (p. 721). that is to say, settler colonialism— licenses the disappearance of indigenous peoples, the expropriation of indigenous spaces, and makes others infinitely exploitable and/or expendable (e.g., slaves, immigrant labor, prisoners). it is thus foundational in establishing processes that separate humanity into distinct groups and in placing those groups into a larger hierarchy. the political, economic, and social processes necessary to contain, exterminate, and permanently occupy territory are premised on a continuously reworked white supremacist dialectic that underwrites racial capitalism. (p. 721) settler colonialism, then, as a historicized process, similar to white supremacy, is no longer located only in historical pasts or conquering empires. but rather, settler colonialism is a dialectic that “drives the socio-spatial logics of contemporary settler colonial nationalism and identity and is not only central to the production of white supremacist discourses, but the very materiality of whiteness itself” (pp. 721– 722). so, with these definitions of white supremacy and settler colonialism in mind, i now ask mathematics educators to participate in a simple thought experiment by reaching back to 1984—the publication year of the first “equity” special stinson editorial journal of urban mathematics education vol. 10, no. 2 5 issue of the journal for research in mathematics education (jrme), the leading mathematics education journal in the united states. below, in its entirety, is the editorial of the first special issue, written by the then editor and associate editor of jrme, jeremy kilpatrick and laurie hart reyes (1984), respectively; the special issue was guested edited by westina matthews (1984): the national council of teachers of mathematics has been instrumental in making mathematics educators more aware of the special problems faced by members of minority groups in learning mathematics. the council has a long history of involving members of minority groups in its activities, but its sponsorship of the core conference on equity in mathematics, held at reston, virginia, in february 1981, began a new phase of concern and positive action. the jrme editorial board has for some time been interested in bringing to the attention of our readers various aspects of research into the learning of mathematics by minorities. when we learned that westina matthews had been assembling a collection of manuscripts on the topic, we invited her to serve as the guest editor for a special issue of the journal. matthews identified a set of potential manuscripts. working with us and with the authors, she reduced the set somewhat, obtained revisions, and emerged with a balanced and polished collection of articles that together portray the status of research on minorities and mathematics in the united states today. the manuscripts were given a final editorial review at the meeting of the editorial board in october 1983. research on the learning of mathematics by minorities, as noted in several articles in this issue, has followed in the footsteps of research on the learning of mathematics by women. unfortunately, the climate of funding for research has become less favorable just as researchers working with minority students have begun to explore some deeper questions. there are, however, indications that private foundations—such as carnegie, ford, and rockefeller—will continue to support research on minorities and mathematics. we applaud their efforts, and we hope to be able to publish manuscripts representing the fruits of that research in the near future. efforts to improve the learning of mathematics by minorities are often hampered by a lack of information about successful work done elsewhere. one effort to improve communication among people interested in such efforts is the minorities and mathematics network, organized in 1981 and coordinated by westina matthews. the network now contains over 200 members who share resources, ideas, and research findings. international communication is also important. in editing the manuscripts for this issue, we were struck by the limited number of references to research conducted beyond the borders of the united states. surely there must be a body of work that has been done in other countries as they attempt to provide a sound education in mathematics to the members of minority groups among their citizens. we hope to provide a continuing forum in the jrme so that reliable knowledge on the learning of mathematics by minorities is shared as widely as possible with people who can put that knowledge into practice. (p. 82) now for a foucauldian thought experiment (see foucault, 1966/1994, 1969/ 1972): think about how the first sentence back in 1984 might have read if we had the tools to think with back then that we do today: stinson editorial journal of urban mathematics education vol. 10, no. 2 6 the national council of teachers of mathematics has been instrumental in making mathematics educators more aware of the special problems faced by [the] white supremacist dialectic that underwrites racial capitalism in [the teaching and] learning [of] mathematics. read through the 1984 editorial again. what other different “statements” 3 might have been possible? what else might have changed? what might the mathematics education research community have begun to think, read, talk, research, write, present, and so forth about back then if only we had chosen to listen to what most black folk—laypersons and scholars alike—were always thinking, reading, talking, … and so forth about? where might we be today if white folk had just chosen to listen to black and other folk of color? – what will we (you) choose to listen to today? references bonds, a., & inwood, j. (2016). beyond white privilege: geographies of white supremacy and settler colonialism. progress in human geography, 40(6), 715–733. coates, t-n. (2017). the first white president. the atlantic. retrieved from https://www.theatlantic.com/magazine/archive/2017/10/the-first-white-president-ta-nehisicoates/537909/ du bois, w. e. b. (1989). the souls of black folk (bantam classic ed.). new york, ny: bantam books. (original work published 1903) du bois, w. e. b. (1999). darkwater: voices from within the veil. new york, ny: dover. (original work published 1920) editors. (2003). the souls of white folk: w. e. b. du bois. monthly review, 55(6), 44. feagin, j. r. (2013). the white racial frame: centuries of racial framing and counter-framing. new york, ny: routledge. foucault, m. (1972). the archaeology of knowledge (a. m. sheridan smith, trans.). new york, ny: pantheon books. (original work published 1969) foucault, m. (1994). the order of things: an archaeology of the human sciences. new york, ny: vintage books. (original work published 1966) hooks, b. (1995). killing rage: ending racism. new york, ny: henry holt and company. kilpatrick, j., & reyes, l. h. (1984). editorial. journal for research in mathematics education, 15(2), 82. matthews, w. (ed.) (1984). minorities and mathematics [special issue]. journal for research in mathematics education, 15(2). stinson, d. w. (2006). african american male adolescents, schooling (and mathematics): deficiency, rejection, and achievement. review of educational research, 76(4), 477–506. stinson, d. w. (2008). negotiating sociocultural discourses: the counter-storytelling of academically (and mathematically) successful african american male students. american educational research journal, 45(4), 975–1010. 3 for a brief discussion on statements and foucault, see stinson (2010). https://www.theatlantic.com/magazine/archive/2017/10/the-first-white-president-ta-nehisi-coates/537909/ https://www.theatlantic.com/magazine/archive/2017/10/the-first-white-president-ta-nehisi-coates/537909/ stinson editorial journal of urban mathematics education vol. 10, no. 2 7 stinson, d. w. (2010). how is it that one particular statement appeared rather than another?: opening a different space for different statements about urban mathematics education [editorial]. journal of urban mathematics education, 3(2), 1–11. retrieved from http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/116/69 stinson, d. w. (2014, april). how (and why) researching “race” without researching racism/white privilege became political normalized in mathematics education research. in n. nasir (chair), theorizing racism: unpacking supremacy, privilege, and justice in mathematics education. symposium conducted at the national council of teachers of mathematics research conference, new orleans, la. stinson, d. w. (2016, july). normalizing race in mathematics education research as a strategic discursive practice. talk delivered at the mathematics education and contemporary theory conference 3, manchester, united kingdom. stinson, d. w. (2017). researching race without researching white supremacy in mathematics education research: a strategic discursive practice. in a. chronaki (ed.), proceedings of the 9th international mathematics education and society conference (mes9, vol. 2, 901–912). volos, greece: mes9. valoyes-chávez, l., & martin, d. b. (2016). exploring racism inside and outside the mathematics classroom in two different contexts: colombia and usa. intercultural education, [online], 1–14. west, c. (1994). race matters. new york, ny: vintage books. (original work published 1993) http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/116/69 journal of urban mathematics education july 2017, vol. 10, no. 1, pp. 1–6 ©jume. http://education.gsu.edu/jume david w. stinson is professor of mathematics education in the department of middle and secondary education in the college of education and human development, at georgia state university, p.o. box 3978, atlanta, ga, 30303; e-mail: dstinson@gsu.edu. his research interests include exploring socio-cultural, -historical, and -political aspects of mathematics and mathematics teaching and learning from a critical postmodern theoretical (and methodological) perspective. he is a co-founder and current editor in chief of the journal of urban mathematics education. editorial in search of defining ethics in (mathematics) education research? david w. stinson georgia state university his editorial is inspired by an essay by paul ernest (2012), “what is our first philosophy in mathematics education?,” which i recently uncovered as i was preparing for my summer courses. in the introduction, ernest asks— can mathematics education have a first philosophy? is there a branch of philosophy that is a sine qua non for mathematics education research and possible its practice as well? are there philosophical assumptions that cannot be avoided in pursuing any inquiries whatsoever in our field? can these assumptions be located in one branch of philosophy? (p. 8) ernest offers five alternative “candidates” as a response to this singularly focused set of questions. two candidates are general branches of philosophy—the philosophy of mathematics and the philosophy, if you will, of critical theory—and the other three candidates are general areas of philosophical inquiry—ontology, epistemology, and ethics.1 throughout the essay, ernest provides justifications of why both the philosophy of mathematics and the philosophy of critical theory as well as ontology and epistemology fall short in providing a first philosophy for mathematics education research and practice. after eliminating the first four candidates, ernest (2012) makes a fourpronged argument for ethics as a first philosophy. ethics, he claims, “enters into mathematics education research in several ways” (p. 13). first, ethics is at the center of the research process with respect to seeking informed consent, causing no harm or detriment, and ensuring confidentiality for all those involved. ernest believes that any research that does not conform to these most basic standards “is ethically flawed and its knowledge claims are suspect” (p. 13). second, mathematics education researchers are participating “in the great, age-old human conversation that sustains and extends our common knowledge and cultural heritage,” as such 1 these are the branches of study that are concerned with the nature of being, the theory of knowledge, and the principals of moral behavior, respectively. t http://education.gsu.edu/jume mailto:dstinson@gsu.edu stinson editorial journal of urban mathematics education vol. 10, no. 1 2 “we and others benefit and grow” (p. 13.) third, the species of human beings depends on its survival by sharing in ethical social and life behaviors with fellow humans. and fourth, drawing on emmanuel levinas (1906–1995) and his ethics as first philosophy,2 ernest states, according to levinas— we owe a debt to the other that precedes and goes beyond reasons, decisions, and our thought processes, and precedes and exceeds any attempt to understand the other. our infinite responsibility to the other person is, of course, ethical: “ethics precedes ontology […] ethics primarily signifies obligation toward the other, that it leads to the law and to gratuitous service, which is not a principle of technique” (levinas, 1987, p. 183). (p. 13) in the end, ernest (2012) contends that positioning ethics as a first philosophy for mathematics education research enables us (i.e., the larger mathematics education research community) “to rethink and re-evaluate some of the taken-for-granted commonplaces of our practices” (p. 14), which opens up new possibilities for theorizing and researching mathematics teaching and learning. ernest’s 2012 essay is neither his first discussion on ethical considerations in the field (see, e.g., ernest, 19913), nor is he the only mathematics education scholar to explore directly the ethical implications of our work as researchers. nearly two decades ago, judith sowder (1998) explicitly placed ethics at the center of the mathematics education research process in her contribution to the international congress on mathematical instruction study mathematics education as a research domain: a search for identity (sierpinska & kilpatrick, 1998). given the shift from largely quantitative methodology to qualitative methodology that occurred in mathematics education research during the 1980s (lester & lambdin, 2003), sowder believed that new questions with respect to ethics should be explored (see also adler & lerman, 20034). much of her discussion can be characterized as aligning with ernest’s (2012) first argument for ethics as a first philosophy: ethical considerations in the most basic standards of conducting research, for example, purpose, informed consent, minimizing harm, maximizing benefits, confidentiality, data use, data interpretation, and so forth. she concluded, stating: “ethical decision making is difficult. there are times when we face conflicting ethical demands, and a decision must be made—a decision with which we want to be satisfied in the long run” (p. 440). 2 see atweh and brady (2009) and neyland (2004) for discussions of levinas’s ethics as first philosophy applied to mathematics teaching and learning. 3 traces of concern for ethical considerations in the field of mathematics education are found throughout ernest’s prolific body of scholarship. 4 see also andersson and le roux (2017) for a discussion of ethical considerations in writing in mathematics education research. http://www.scholar.google.com/citations?user=u29ilr8aaaaj&hl=en stinson editorial journal of urban mathematics education vol. 10, no. 1 3 uncovering ernest’s 2012 essay this summer reignited my own struggles, concerns, and doubts in becoming an ethical critical postmodern5 mathematics education scholar and researcher. these struggles, concerns, and doubts came into my being as i entered the field as a researcher and have grown exponentially (it seems) when i consider the entirety of the roles that i have occupied in the field: participant, researcher, presenter, writer, reviewer, editor, and so forth. my struggles (and so forth) originated when my journey into postmodern theory began to expose the fault line of research done within the humanist tradition (st. pierre, 2000). within the ruptures of this fault line, researcher ethics emerged, for me at least, as the primary concern of education research—becoming an issue that was not completely addressed by the inquiries of an institutional review board (guillemin & gillam, 2004). uncovering ernest’s essay this summer, therefore, gave me solace, if you will, that i was somewhat on the “right” track, back then and now. being “well schooled” as an education researcher in the early 2000s, i was carefully taught that there was a crisis in representation (marcus & fischer, 1986) and an end to innocence (van maanen, 1995) in the work that we do as (mathematics) education researchers. nevertheless, as i acknowledge, both then and now, a crisis in representation and an end to innocence, i continue to ask: is everything dangerous? i argue, yes, while using foucault’s (1983/1997) reconfiguration of the word dangerous: “my point is not that everything is bad, but that everything is dangerous, which is not exactly the same as bad. if everything is dangerous, then we always have something to do” (p. 256). it is the coupling of marcus and fischer’s (1986) and van maanen’s (1995) arguments and foucault’s (1983/1997) statement, and postmodern theory in general, that continues to present me with some troubling questions: given that there is still work to do, how do i (we) go about doing that work, ethically? does showcasing and monitoring my (our) subjectivities address ethical concerns (glesne, 1999; peshkin, 1988)? or does presenting thick descriptions (geertz, 1973)? or is it reconceptualizing validity (lather, 1986)? although helpful, i believe, engaging in a combination of these methodological procedures (and others) in our research is only a starting point in confronting the crisis in representation, as we acknowledge that our work is not innocent and is always dangerous. as a critical postmodern researcher, the aspect of the research process that i focus on most, in order that i might sleep at night, is my ethics. in making the foregoing statement, i understand that i might be seeking a metanarrative around ethics, 5 often the words postmodernism and poststructuralism are used interchangeable in the literature; however, there are acknowledged differences in the terms (for a brief discussion see st. pierre, 2000, pp. 506–507). following walshaw (2004), i use the term postmodern as a general term that attempts to capture the nuances of both words (also see ernest, 1998). see stinson and bullock (2012, 2015) for discussions on the phrase critical postmodern in mathematics education research. stinson editorial journal of urban mathematics education vol. 10, no. 1 4 which i have incredulity toward (lyotard, 1979/19846). but in considering ethics as a first philosophy (ernest, 2012), i somehow get out of the metanarrative quandary. … i think. nonetheless, i do not get complete peace of mind by showcasing and monitoring my subjectivity or by presenting thick descriptions, neither from searching for disconfirming evidence nor conducting triangulation of data (silverman, 2000). although these methodological procedures are important components of the research process, i get most of my peace of mind by engaging, continuously and chaotically, my ethics. but how do i talk about ethics? how do i “represent” or “define” ethics? how do i talk about and represent engaging in continuous and chaotic ethics without establishing a metanarrative around ethics, which begins to surveil and discipline ethics (foucault, 1975/1995)? shouldn’t ethics always be unsurveilled, undisciplined, continuous, and chaotic engagement? oops, is the aforementioned statement the start of a metanarrative? then again, would a metanarrative around ethics be such a bad thing? i state explicitly here that i do not wish to engage in a critical discussion about ethics (see, e.g., dewey, 1932/1985), a discussion that has been ongoing since the question what is ethics? was even possible to ask. i do, however, discuss, although briefly, how i have come to think about framing my researcher ethics and provide a definition of ethics that i attempt to continuously and chaotically engage throughout the multiple roles i occupy in the field of mathematics education research. guillemin and gillam (2004) identify two different dimensions of ethics in research: “procedural ethics and ‘ethics in practice’” (p. 262). they define procedural ethics as those ethical issues most often addressed by research ethics committees (e.g., institutional review boards). and they define “ethics in practice” as the ongoing day-to-day ethical issues that arise throughout the research process (e.g., the disclosure of sensitive information from a research participant). although guillemin and gillam perceive continuity between the two dimensions, they frame “ethics in practice” within reflexivity. reflexivity, in research, requires thinking about the researcher’s positionality and how the process of conducting research affects the study and the human relationships developed throughout the study (glesne, 1999). guillemin and gillam claim that in being reflexive the researcher becomes alert not only to issues related to knowledge creation (i.e., epistemology) but also to the ethical issues of research. in so doing, they believe, the researcher adopts “a continuous process of critical scrutiny and interpretation, not just in relation to the research methods and the data but also to the researcher, participants, and the research context” (p. 275). 6 lyotard (1979/1984) believed that metanarratives are foundational in supporting “universal truths” that serve to legitimate modern culture; metaor grand-narratives are evident in “the dialectics of spirit, the hermeneutics of meaning, the emancipation of the rational or working subject, or the creation of wealth” (p. xxiii). stinson editorial journal of urban mathematics education vol. 10, no. 1 5 it is guillemin and gillam’s (2004) concept of reflexivity and ethics in practice that best frames how i bring into play a definition of ethics, which offers a means for thinking and rethinking ethical issues throughout the research process, no matter what role i am playing. the dalai lama (1999), in his book ethics for the new millennium, suggests that ethics is an understanding that we all desire happiness, and we all seek to avoid suffering. thus, he believes what is entailed…is not an admission of guilt but…a reorientation of our heart and mind away from self and toward others. to develop…an attitude of mind whereby, when we see an opportunity to benefit others, we will take it in preference to merely looking after our own narrow interests. but though, of course, we care about what is beyond our scope, we accept it as part of nature and concern ourselves with doing what we can. (p. 162–163) the dalai lama’s definition of ethics does not require one to solve the “world’s problems”; it simply requires one to seek ways of assisting in others’ happiness and security concurrently with her or his own, decentering oneself to attempt to become the other. becoming the other is never possible; it is, however, i believe, the honest attempt that is an ethical act. moreover, such a definition of ethics, i believe, requires an ethics of care of the self and, in turn, care of others (foucault, 1984/1988). in the end, as i live, think, and interact within a critical postmodern existence, there appears to be a bottomless abyss of ethical issues that trouble the research process (and the field of mathematics education in general). yet, by continuously and chaotically engaging my ethics throughout the research decision-making process, i believe, i successfully negotiated some of these issues—doing dangerous work, ethically. and yes, i will continue to take up ernest’s (2012) suggestion: ethics as a first philosophy for mathematics education research (and practice). – what is your first philosophy? references adler, j., & lerman, s. (2003). getting the description right and making it count: ethical practice in mathematics education research. in a. bishop, m. a. clements, c. keitel-kreidt, j. kilpatrick, & f. koon-shing leung (eds.), second international handbook of mathematics education (vol. 2, pp. 441–470). dordrecht, the netherlands: springer. andersson, a., & le roux, k. (2017). toward an ethical attitude in mathematics education writing. journal of urban mathematics education, 10(1), 74–94. retrieved from http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/303/215 atweh, b., & brady, k. (2009). socially response-able mathematics education: implications of an ethical approach. eurasia journal of mathematics, science & technology education, 5(3), 267– 276. bstan-`dzin-rgya-mtsho-dalai lama xiv. (1999). ethics for the new millennium. new york, ny: riverhead books. dewey, j. (1985). ethics. in j. a. boydston (ed.), john dewey: the later works, 1925–1953 (vol. 7). carbondale, il: southern university of illinois press. (original work published 1932) http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/303/215 stinson editorial journal of urban mathematics education vol. 10, no. 1 6 ernest, p. (1991). the philosophy of mathematics education. london, united kingdom: falmer press. ernest, p. (1998). a postmodern perspective on research in mathematics education. in a. sierpinska & j. kilpatrick (eds.), mathematics education as a research domain: a search for identity (vol. 1, pp. 71–85). dordrecht, the netherlands: springer. ernest, p. (2012). what is our first philosophy in mathematics education?. for the learning of mathematics, 32(3), 8–14. foucault, m. (1988). the history of sexuality. volume 3: the care of the self (r. hurley, trans.). new york, ny: vintage books. (original work published 1984) foucault, m. 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(2004). mathematics education within the postmodern. greenwich, ct: information age. journal of urban mathematics education july 2017, vol. 10, no. 1, pp. 113–121 ©jume. http://education.gsu.edu/jume viveka o. brown is an assistant professor in the department of mathematics, spelman college, 350 spelman lane, box 306, atlanta, ga 30314; vborum@spelman.edu. her research interest examines equity issues in mathematics. in particular, her focus is on black women and their experiences in mathematics. joycelyn wilson is a senior instructor in the department of mathematics, spelman college, 350 spelman lane, box 306, atlanta, ga 30314; jwilso20@spelman.edu. her interest includes operations research, active learning techniques for the classroom, and solving mathematics and logic puzzles. book review hidden figures no more: a book review of hidden figures: the american dream and the untold story of the black women mathematicians who helped win the space race1 viveka o. brown spelman college joycelyn wilson spelman college the best thing about breaking a barrier was that it would never have to be broken again. – margot lee shetterly hen you think of the “figures” 3, 1.3, and 0.6, what comes to mind? one probably would not realize that these figures represent the average percentages of black women in the united states who earned their bachelor’s, master’s, and doctoral degrees, respectively, in mathematics between the years of 2003–2012 (national science foundation, 2015). with such low figures, it is no wonder why the notion of black women in mathematics is practically nonexistent. thus, discovering and reading margot lee shetterly’s (2016) non-fiction book hidden figures: the american dream and the untold story of the black women mathematicians who helped win the space race (i.e., hidden figures) was a welcoming revelation. finally, black women mathematicians, often ignored and invisible, are now the protagonists of a true story. throughout the pages of hidden figures, shetterly takes readers into the personal and professional lives of four black women mathematicians: dorothy vaughn, mary jackson, katherine johnson, and christine darden. these women, individually and collectively, continuously broke a variety of gender, racial, and social barriers during their time working for the national aeronautics and space administration (nasa) at the langley research center in hampton, virginia. 1 shetterly, m. l. (2016). hidden figures: the american dream and the untold story of the black women mathematicians who helped win the space race. new york, ny: marrow. 368 pp. isbn 978-0-06-236359-6 (hb), $27.99 https://www.harpercollins.com/9780062363596/hidden-figures w http://education.gsu.edu/jume mailto:vborum@spelman.edu mailto:jwilso20@spelman.edu https://www.harpercollins.com/9780062363596/hidden-figures brown & wilson book review journal of urban mathematics education vol. 10, no. 1 114 shetterly’s (2016) book takes readers on an exhilarating journey of breaking barriers and opening doors for women—particularly black women—at nasa. while readers become intrigued by the women’s success and progression inside of nasa, the realities of segregation and racism outside of the langley facility is also explored. it is quickly realized that progress in one area does not imply advancement everywhere. shetterly provides an historical account, highlighting events that played a major role in american (education) history such as sputnik and the brown v. board of education decision. her active writing style allows readers to place themselves in the same positions and situations faced by these black women mathematicians and to relive the experiences they encountered. in many ways, shetterly’s prose allows black women to see themselves as mathematicians. it creates a surge and excitement not just for black women in mathematics but all women in science, technology, engineering, and mathematics (stem) fields. in this review, we not only explore the nuances and specific themes of hidden figures but also provide personal reflections on how the book impacted our lives as black women mathematicians. in addition, we briefly discuss the hidden figures book circle and semester-long hidden figures colloquium both created at spelman college to further highlight how the book influenced faculty and students. background information motivation for hidden figures margot lee shetterly (2016), the author of hidden figures, is a former investment banker, media consultant, and entrepreneur. she grew up in hampton, virginia, and was familiar with nasa’s langley research center given that her father worked there as an atmospheric scientist. often when visiting her father at langley, shetterly and her siblings recalled seeing several other african americans. it was therefore common for her to see blacks who were in science or working for nasa at church, in her neighborhood, and throughout the hampton community. later, through listening to stories from her father about some of the mathematicians and engineers at nasa, shetterly realized that there was so much she did not know about the women who worked at langley. while being privileged to hear about some of them through her father’s stories, shetterly understood that there existed an audience of people who would never know of these hidden women unless someone shared their stories. intrigued by their stories and driven by her curiosity to know more, shetterly set out to research the black women who worked for nasa at the langley research center, known as the west area computers. brown & wilson book review journal of urban mathematics education vol. 10, no. 1 115 overview of chapters, figures, and their influence at nasa as one begins to read hidden figures, one recognizes the various forms of symbolism in the book. throughout the 23 chapters, the titles alone will have one feeling nostalgic and desiring to recall and review historical events that took place during that time. chapter titles such as mobilization, manifest destiny, war birds, and with all deliberate speed will have many historians and history enthusiasts reliving important events that shaped american history. additionally, stem readers will also be enthused with chapter titles such as the area rule, turbulence, angle of attack, and degrees of freedom. although there are explanations given within the text to why the chapter titles were selected, the book focuses on the lives of four african american women mathematicians: dorothy vaughn, mary jackson, katherine johnson, and christine darden. in the early 1940s, the national advisory committee for aeronautics (naca), later known as nasa, recruited the first african american women “computers” (nasa, n.d.). prior to this decade, human computers were positions held only by white women. during this time, a young civil rights and labor activist, asa phillip randolph, a close friend of eleanor roosevelt, demanded that president roosevelt open lucrative war jobs for negro applicants. after the persistence of randolph’s labor initiatives, including threatening to bring thousands of negroes to the capital to protest the president, president roosevelt issued an executive order in 1941 banning employment discrimination by federal agencies (shetterly, 2016). thus, the recruitment of african american women for the human computing positions began. due to segregation at the time, however, the black women of the west area were kept separate from the white women computers (shetterly, 2016). when reading hidden figures, readers learn about the endless glass ceilings that were broken by these black women (shetterly, 2016). dorothy vaughn, who began working for nasa in 1943, paved the way for black women computers in the west computing area. vaughn was resilient and through her perseverance and determination at nasa, she became the first black supervisor of the west computing area in 1949. to avoid losing her job and eliminating the future positions of black women in the west computing area, she taught herself the programming language fortran, making herself indispensable at nasa. under vaughn’s leadership, mary jackson began working as a human computer in 1951. in 1958, after overcoming many hurdles, jackson became the first black female engineer at nasa. during her 20-year stint of working as an engineer, authoring and coauthoring several papers, jackson noticed the inequities of langley’s female professionals. as a result, she decided to leave the engineering field, for the greater good, and take a demotion to become langley’s federal women’s program manager. in the early 1950s, the segregation of the women computers began to change (shetterly, 2016). after integrating west virginia university’s graduate mathematics program, katherine johnson joined the west area computers in 1953. within a brown & wilson book review journal of urban mathematics education vol. 10, no. 1 116 few weeks of joining, johnson moved to the space task force. she was the first woman to attend meetings that detailed the specifics regarding the logistics and mathematics for the space flights. in 1962, while preparing for the orbital flight of john glenn, astronauts were not comfortable with the accuracy of mechanical computers. johnson was requested to verify the calculated trajectories for what became the first successful orbital flight by humans for the united states. the west area computers officially ended in 1958. during the west area years, there were up to 80 african american women computers. even after 1958, african american women continued to play a vital role, now being integrated within different departments at nasa (shetterly, 2016). in fact, in 1967, katherine johnson mentored a young african american computer named christine darden who had just started her career at nasa. darden challenged the norms of nasa’s culture by approaching supervisors about why women with similar or more education were sent to computing pools, while men were placed in engineering groups. the response she received was that no one had ever complained. within weeks of her meeting with the supervisor, darden was assigned to an engineering group where she worked on sonic boom research. although the west area computers ended in 1958, these black women’s legacy, perseverance, and tenacity of breaking barriers paved the way for future african american women and women in general at nasa. professional evaluation shetterly’s (2016) thoughtful wording of the book’s title and chapters make it an exciting read. when reading across the chapter titles, readers may become introspective wishing to connect the titles with the stem or historical content in that section. throughout the book, shetterly maintains a nice balance between juxtaposing the women’s professional lives and careers at nasa with their personal lives and families at home. hidden figures would be a great read for k–12 and higher education students across the globe, especially those in the united states. specifically, students in urban mathematics classrooms will benefit by exposure to historical figures they may relate to in gender and/or race. additionally, hidden figures can be an excellent motivator for teachers to engage students in the discussion of stem and historical events. young women may find the book to be an inspiration for them to pursue a career in a field that has been historically dominated by white men. for black females, hidden figures allows young women to see others who look like them in a field where they seem almost invisible. hidden figures is an inspiration for men as well. it allows men to see the benefits of having equity across gender and race, evident when john glenn decided not to orbit the earth until katherine johnson confirmed the computer’s calculations. brown & wilson book review journal of urban mathematics education vol. 10, no. 1 117 as black female mathematicians, we desire to hear about all the stories of the women who worked at nasa during this time. hidden figures is an inspiration for future women in stem, especially black women. readers will gain knowledge about the role of women in the aeronautical field and beyond. as shetterly (2016) states, “there was virtually no aspect of twentieth-century defense technology that had not been touched by the hands and minds of female mathematicians” (p. 189). the significance of this alone is paramount, especially in fields that have often been regarded as white male dominated arenas. non-mathematics readers, however, might find it difficult at times to follow the mathematical terminology used throughout the book. nevertheless, shetterly contextualizes the mathematical ideas to assist readers with comprehending the ideas being communicated. personal reflections movies are often filmed in and around the city of atlanta, which is quickly becoming known as the hollywood of the south. sometimes movie production equipment and trailers hinder our access to and from campus at spelman college (a women’s historically black college in atlanta, georgia). a few years ago, students, staff, and faculty were excited that hollywood movie production was literally nextdoor on the campus of morehouse college (a men’s historical black college), making a movie about black women mathematicians. after investigating and discovering who these women were, we learned of the book hidden figures. we were both ecstatic to read the book; someone who shared our gender and race wrote the mathematical language and jargon in the text. although we are from a different generation than the main figures of hidden figures, reading the book motivated us to reflect on our own struggles and trials and tribulations that we encountered (and continue to encounter) during our academic journeys toward becoming mathematicians. viveka’s reflection growing up in an urban area and being a first-generation college student, my family valued education but knew little about deciding on careers and majors in college. while developing my love of mathematics, i was not aware that it was a predominantly male field. although there were no professors who looked like me until my ph.d. program, i was oblivious to the fact that my presence in mathematics was an exception and not the rule. nonetheless, i soon realized that prejudices due to gender and race held true in mathematics when i encountered a professor who always called on the male students. furthermore, when female students did ask questions he would speak down to us or ignore our presence altogether. i do, however, remember breaking the silence of women in his class one day by challenging his brown & wilson book review journal of urban mathematics education vol. 10, no. 1 118 reasoning. after one of my female classmates asked a valid question and was dismissed, i quickly interjected and explained why her reasoning was correct. he agreed. sadly, such incidents were all too common during my undergraduate years. other micro-aggressions occurred throughout my undergraduate and graduate programs while attending predominantly white institutions. these micro-aggressions included, for example, non-black peers excluding me from study sessions, non-black peers speaking to me only when my white friends were present, and non-black peers erasing work they had completed on the board when i walked in the room. too often my classmates held preconceived notions about my mathematical prowess simply due to my gender and/or race. hearing the stories of the women in hidden figures allowed me to connect with their experiences on a deep personal level. other black women mathematicians shared similar experiences—i was not alone. their (our) stories inspire an intrinsic motivation to dismantle the norms of mathematics being a gender and race specific domain. joycelyn’s reflection as a child, i always had a love of mathematics. in high school, i was encouraged by teachers to pursue my passion by majoring in mathematics. taking their advice, i enrolled in a 5-year bachelor’s of science/master’s of science program in mathematics at clark atlanta university, a historical black university. up to that point, i, like shetterly, saw mathematics role models who looked like me. going to primary and secondary schools as well as college in an urban area, i often saw women of color in stem areas who encouraged and supported me in my endeavors. it was not until i attended graduate school for my ph.d. that very few people looked like me anymore. i was no longer encouraged by faculty to persevere, and i was even told that graduate school was not for me. however, with a strong family and community outside of school supporting me, i was able to push forward. while reading hidden figures, i could directly relate to the plight of the women in the book. at home, they had the support of their families and friends and the support of other black women “computers” within the walls of langley. however, they consistently had to prove themselves and work harder to get into positions that were generally held by white men. i, like these women, had to prove myself not only to my professors but also to my classmates who assumed that i was not as smart or capable as they were in mathematics. while i had learned of the contributions of katherine johnson about ten years prior to the book’s release, i only knew a few things about her and did not know of the other women in the book. while reading the book, i was so intrigued and excited to not only learn about the mathematical contributions of these women but also to learn more background information about their lives. one of the most exciting things for me is that all communities, even outside of stem fields, will know of these women and their trailblazing accomplishments. brown & wilson book review journal of urban mathematics education vol. 10, no. 1 119 with respect to our personal reflections, shetterly’s (2016) book motivates us to do more for black women and their visibility in mathematics. moreover, this book incites an internal rage that stems from this history being knowingly kept or “hidden” from us. to combat the disparities in the number of women mathematicians, we must acknowledge the astounding roles that these women mathematicians played at nasa. one can only imagine the untapped talent and motivation that may have sparked young girls’ dreams hearing the stories of these women mathematicians. hidden figures book circle and colloquium faculty members in the department of mathematics at spelman college have always found it important to share the accomplishments of women mathematicians with their students. as a part of a mathematics seminar course, students are asked to research a black woman mathematician and present their findings to the class. while katherine johnson is usually among the women researched, little was known about the other women who worked at nasa. upon learning about the book hidden figures, we decided that the it provided an opportunity to engage students in a setting outside of the classroom to learn more about black women mathematicians. a hidden figures book circle was formed, where nine mathematics majors, one physics major, three mathematics faculty members, and spelman’s president gathered to read and discuss the book. all those who joined the book circle appeared to enjoy the three reading circle sessions, where the book was discussed by chapters and then as a whole at the culminating meeting. students pointed out that they enjoyed reading about the mathematics that they were currently learning in their courses and the subtle mathematics terminology that was sprinkled throughout the book in chapter titles and within the reading. one student stated: “shetterly had a strong command of words, vocabulary was not too cumbersome, and her style was very engaging. it reminded me of a diary and a textbook at the same time.” another student expressed that she “enjoyed that the book was written in a manner that not only shared the details of the characters’ lives but also did it in a story format chronicling current events happening at the same time.” the circle participants discussed the work-life balance and the sacrifices that these women endured while working for nasa. the women in the book had support from their spouses and families and sometimes had to live away from their families for periods of time for work. one student found the book eye opening given how it unveiled hidden history and illuminated the still relevant challenges faced by the women in the book. shetterly’s book provoked discussion on what could be done now to prevent future women in stem from being hidden. in addition to the hidden figures book circle at spelman college, first-year brown & wilson book review journal of urban mathematics education vol. 10, no. 1 120 students from different majors at spelman college took a semester-long colloquium entitled hidden figures: unheralded black women who expand our imagination. in this course, students explored the women of hidden figures and similar hidden figures across all disciplines. they also examined the work of black women who have broken barriers in fields within and outside of stem. the following three questions guided the discussions and assignments in the course: 1. what does it mean to be a hidden figure? 2. who are other hidden figures and what roles have they played in society? 3. why and, perhaps more importantly, how have these important figures been hidden and their stories untold? students enrolled in this course were required to read hidden figures as well as excerpts from other texts that discussed female hidden figures regardless of race or discipline. at the conclusion of the course, students had to develop ways to prevent future hidden figures in society. overall, the course was engaging for students because it allowed them to see women, in particular those who look like them, in roles where their acknowledgment has been limited and/or hidden. concluding thoughts hidden figures takes readers on a historical mathematical space adventure. reading about the achievements of the four protagonists and other black women computers mentioned in the book will enlighten and encourage readers. transformative was a word used by one of our students during our discussion group to describe the book, and we agree that this text was transformative indeed. hidden figures can motivate one to begin or continue to pursue a career in mathematics. additionally, it spans across generations and allows students to reflect and realize that there are no limits to their future aspirations. as shetterly (2016) states, “to keep moving forward, they [the black women computers] needed to take advantage of every opportunity to make themselves as valuable as possible to the laboratory” (p. 139). becoming trailblazers, breaking barriers, and setting standards allowing these women to go from hidden to visible accomplished this feat. therefore, the charge to all who read hidden figures is to support, expose, and encourage future women in mathematics and other stem fields to create their own trajectories of success and work to ensure that they are hidden figures no more. references national aeronautics and space administration. (n.d.). from hidden to modern figures. retrieved from: https://www.nasa.gov/modernfigures https://www.nasa.gov/modernfigures brown & wilson book review journal of urban mathematics education vol. 10, no. 1 121 national science foundation. (2015). women, minorities, and persons with disabilities in science and engineering: 2015. national center for science and engineering statistics – special report nsf 15-311. arlington, va: national science foundation. retrieved from https://www.nsf.gov/statistics/women/ shetterly, m. l. (2016). hidden figures: the american dream and the untold story of the black women mathematicians who helped win the space race. new york, ny: marrow. https://www.nsf.gov/statistics/women/ microsoft word final bullock vol 7 no 1.doc journal of urban mathematics education july 2014, vol. 7, no. 1, pp. 1–6 ©jume. http://education.gsu.edu/jume     erika c. bullock is an assistant professor of mathematics education in the department of instruction and curriculum leadership in the college of education, health and human sciences, at the university of memphis, 419a ball hall, memphis, tn, 38152; e-mail: erika.bullock@memphis.edu. her research interests include exploring urban mathematics education curriculum and policy from a critical postmodern and historical perspective. she is associate to the editor-in-chief of the journal of urban mathematics education. editorial danger: ghetto ahead? erika c. bullock university of memphis love attending conferences! i enjoy research, writing, and teaching, but there is something electric about stepping away from the obligations of home, being in the conference environment, meeting with old and new colleagues, sharing my work, and learning from others. as an assistant professor, i am in the midst of the joy and stress of building a scholarly identity; balancing my responsibilities for research, teaching, and service; and navigating both the politics of my institution and of mathematics education as a discipline. sometimes conference connections reveal that the challenges that i face are not unique to my experience or my institution; sometimes the revelation is that someone across the country or around the world has already found a way to address those challenges. academic work can be isolating; conferences help to build and sustain the vital connections that contribute to our knowledge bases and to our support systems. conference organizers often use special interest groups to facilitate collaboration among attendees. the annual meetings for both the association of mathematics teacher educators (amte) and the north american chapter of the international group for the psychology of mathematics education (pme-na), for example, have begun to incorporate divers strands and working groups including some targeted toward equity in mathematics education. these equity sessions and strands serve a two-fold purpose. primarily, they place issues of equity and access at the center of empirical and conceptual conversations. secondly, they become a reunion of sorts for equity-minded mathematics educators and allow us to come together in the name of forming and sustaining an equity agenda in mathematics education. i attended the amte annual meeting for the first time this year. amte has much to offer mathematics educators; i recommend attending if you have not done so. this year, amte’s equity task force debuted the learn & reflect equity strand, which included a day of focused sessions, a set of reflection questions, and a debriefing period. the learn & reflect equity strand appears to operate differently from the pme-na equity working group with which i was more familiar. the former is an organizational strand for submitted sessions and papers with a culminating open debriefing session. the pme-na equity working group i       bullock editorial journal of urban mathematics education vol. 7, no. 1 2 meets three times during the conference to address on an issue, set of issues, or project of interest. it is not attached to submitted conference papers. my intention was to participate in the learn & reflect equity strand for the entire day as i do with the pme-na equity working group. however, as is customary with academic conferences, i had to make choices about the sessions i would attend, which meant that i was not able to attend all sessions of interest due to scheduling conflicts. before i continue, please allow me to state without question that, although amte is the site of this vignette, my comments here are not about amte as an organization or a conference. i am also not addressing any particular presenter or participant. all of the sessions i attended were rich and generated different questions. this experience was a catalyst that caused me to “rethink my rethinking” (stinson, 2004, p. xx), a process that continues even as i record my thoughts in this editorial. i enjoyed and gleaned much from the sessions that i attended within and outside of the learn & reflect equity strand. while each presenter skillfully offered her or his expertise, there were several moments when i questioned what was said or unsaid: the deficit-oriented language used to discuss black and brown children and pre-service teachers; the term “urban” used as proxy for poor and black or brown; and the absence of cultural relevance in conversations about structuring teacher preparation programs, courses, and field experiences. i also observed that i did not recognize any equity-minded colleagues in the session audiences who might also notice these infractions. i dismissed their absence, remembering that the learn & reflect equity strand’s sessions were occurring at the same time. later that evening, however, i began to think about the number of sessions that ran that day and the number of opportunities to address the types of issues that i saw in the ones i attended. my reflection brought me to a question: as a community of mathematics educators who ascribe to equity agendas, are we missing opportunities to advance these agendas when we choose to engage within the community while the prevailing discourses continue around us? this question is not easily answered. of course it is important for mathematics educators interested in equity to have time together. these conferences offer an excellent opportunity for us to share and strategize. but what happens while we are talking? is there a way for us to take advantage of our time together while also engaging with the larger community to participate in a cross-pollination of ideas across the landscape of mathematics education? although i tend to find labels constraining, i will identify myself as an equity-minded urban mathematics educator, and i will address a collective “we” with the assumption that the majority of my readers will be people who are also equityminded. as such, i have assumed certain responsibilities within the mathematics education community. those responsibilities include advocating for quality math       bullock editorial journal of urban mathematics education vol. 7, no. 1 3 ematics experiences for all children and holding the mathematics education community accountable for its rhetorical and actual treatment of historically marginalized people. my experience at amte prompted me to consider what it means to take on these responsibilities in the larger mathematics education arena. it also caused me to turn attention toward jume and special issues of journals such as the journal for research in mathematics education (jrme; gutiérrez, 2013) and the journal of mathematics teacher education (jmte; strutchens, 2012) as spaces set aside to address issues related to equity in mathematics education. i appreciate the community of scholars that i meet during equity-focused conference sessions and in the pages of jume and other journals. these spaces refresh and inspire me and provide hope in an era of mathematics education scholarship during which challenges to the status quo are deemed unwelcome or irrelevant and are often dismissed under the banner “where’s the math?”1 one can understand how, facing such a response, mathematics educators who make such challenges find comfort and camaraderie in spaces designed to embrace such approaches. there are obvious benefits to these spaces, but we also sacrifice a measure of opportunity to plant seeds of consideration for equity in other spaces. in the following section, i consider further the questions generated from my experience at the conference and how we can use these spaces both to build strength together and to use our collectivity to address larger discourses in mathematics education in strategic ways. the good, the bad, and the dangerous equity strands, working groups, and specialized journals are important for building community and for sharing such work with others. however, while we sit in targeted sessions with like-minded colleagues, the conference continues around us. while we enjoy a well-executed special journal issue on equity, the journal continues with little consideration for equity or affiliated issues in its pages (martin, 2003; parks & schmeichel, 2012). in addition to our collaborative work, we also have a responsibility to address the larger discourse in mathematics education that largely ignores or mishandles issues of equity.2 without our voices, the status quo continues unchecked. it is not our purpose to be necessarily adver                                                                                                                 1 for discussions of acceptability and inclusivity in mathematics education, see battista (2010); confrey (2010); heid (2010); martin (2003); martin, gholson, and leonard (2010); and parks and schmeichel (2012). 2 references to my perceptions of our responsibilities to engage with the mainstream of mathematics education do not absolve those in the mainstream from responsibilities to engage with us. it seems, however, that such engagement is most likely to occur only if we bring the questions to them.       bullock editorial journal of urban mathematics education vol. 7, no. 1 4 sarial, but rather to introduce questions, assert objections, and respond to the sometimes inadequate ways in which the mathematics education community talks about, represents, and engages with populations that it has historically ignored or maligned. so what is the issue? foucault (1983) beckons us to soberly consider our actions: “my point is not that everything is bad, but that everything is dangerous, which is not exactly the same as bad. if everything is dangerous, then we always have something to do” (pp. 231−232). here, foucault’s use of danger should not induce fear, but rather inspire action. he is calling us to be vigilant and to guard against complacency. considering these words in the context of equity-oriented spaces in mathematics education gives me pause. i have begun to wonder if the proliferation of spaces amenable to mathematics educators whose work lies on the margins of the field is helpful (good), harmful (bad), or potentially both (dangerous). in other words, what does it mean to consider these spaces as dangerous rather than simply good (opportunities to share in a targeted and welcoming space) or bad (opportunities that isolate equity-oriented scholars from mainstream conversations)? taking foucault’s (1983) lead, i problematize equity-specific spaces by considering the political ramifications of their existence. consider the possible perspectives of those who grant such requests. could it be that providing special journal issues or conference strands is a means to continue marginalizing equity work? perhaps granting our requests could be considered a win-win scenario in which we get what we want and the establishment gets to continue without material change and with the opportunity to say that the forum is open and inclusive. is it possible that the segregation of space allows those who are unwilling or unable to engage equity discussions to be as comfortable as we are because there are few voices of question or dissent? does the segregation in the name of creating progressive space reify the very marginalization that these spaces purport to address? as i have discussed my concerns with others, the response that i receive (and a sentiment that i often share) is that we enjoy being able to talk “equity talk” without having to justify or explain our positions; we enjoy having forums where we can submit our written work without worrying about having to address the often-dismissive “where’s the math?” question (heid, 2010). i, too, am grateful for these opportunities, but i have to ask of this community the same question that i ask myself: have i (we) become complacent in my (our) segregation? these questions pull me toward the process of ghettoization as a means to describe what could be happening to/in the equity community.3 skovsmose and penteado (2011) characterize ghettoes in the mathematics classroom as spaces                                                                                                                 3 spatial limitations preclude me from exploring these ideas fully in this editorial. here, i intend to introduce my questions and current thinking. i will expound in future work.       bullock editorial journal of urban mathematics education vol. 7, no. 1 5 that “emerge through complex processes of differentiation, where lack of prestige, poverty and stigmatization turns into general discourses, which in turn coagulate as ghetto-walls…which obstruct exchanges of meaning” (pp. 87−88). they argue that ghettoes form “when differentiation turns into an us–them formulation, and labeling turns into a stigmatization” (p. 87). thus, the purpose of equity-oriented mathematics education becomes clear: “an education for equity and quality must try to act against all processes, social and educational, which make part of the formation of ghettoes in the classroom” (p. 87). extending skovsmose and penteado’s argument beyond the classroom to mathematics education in a broader sense, there seems to exist the same us–them formulation that the authors observed in classrooms (i am guilty of using it here). our lack of recognition within the larger mathematics education conversation has caused us to seek formal spaces that, if we are not careful, could be enclosed by semi-permeable “ghettowalls.” i wonder if, as a community, we are in danger of creating—or allowing others to create for us—equity ghettoes in mathematics education where opportunity for camaraderie and mutual engagement can lead to self-marginalization. a charge to keep more seasoned scholars have fought to create forums like these focal conference strands, journal special issues, and jume to provide an outlet for scholars whose work is neither well-received by nor well-represented in mainstream venues. as a junior scholar, it is a privilege to take advantage of these hard-won opportunities, but i also recognize the need to continue the work that our forebears have begun to move issues of equity and access into the larger mathematics education conversation. i urge us not to be satisfied with the mere existence of such spaces. we must keep before us our goals to pursue quality and equitable mathematics education experiences for all children and avoid the construction of ghettowalls around us. one generation of mathematics educators has fought to create spaces where our work can be represented; what is the next generation’s fight? one of the many positive outcomes from the spaces i have identified is the revelation that there is a growing mass of equity-oriented mathematics educators. i charge all of us to think about how we can use this critical mass and these spaces strategically. one strategy could be to attend sessions in pairs or triads and to raise at least one question in each session related to issues of equity. we could also leverage social media to create opportunities for connection outside of conferences. these opportunities could help to combat the isolation that we may feel in our home institutions and to encourage collaborative and collective action. conference meetings, then, can be opportunities to touch base regarding sustained efforts and to debrief about what we hear in the larger community and how the group and use its expertise to affect dominant discourses.       bullock editorial journal of urban mathematics education vol. 7, no. 1 6 we are in a moment in mathematics education in which there are more opportunities for equity work than ever before. i am proud of the advances that have been made, but there is still work to be done and i love the mathematics education community and the children, teachers, and communities that i serve too much to be satisfied with what we have. my charge to you, scholar-friends, is to consider with me how we might take the opportunities that we have been given and use them strategically to achieve our primary goals. i invite you to contact me so that we can think, plan, and act together. acknowledgements special thanks to my dear friends and colleagues nathan alexander, maisie gholson, christopher jett, and gregory larnell for their critical feedback on this editorial. your support is invaluable and your thoughtfulness continually inspires me. references battista, m. t. (2010). engaging students in meaningful mathematics learning: different perspectives, complementary goals. journal of urban mathematics education, 3(2), 34–46. retrieved from http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/115/58 confrey, j. (2010). “both and”—equity and mathematics: a response to martin, gholson, and leonard. journal of urban mathematics education, 3(2), 25–33. retrieved from http://edosprey.gsu.edu/ojs/index.php/jume/article/viewfile/108/53 foucault, m. (1983). on the genealogy of ethics: an overview of work in progress. in h. l. dreyfus, & p. rabinow, michel foucault: beyond structuralism and hermeneutics (pp. 229– 252). chicago, il: the university of chicago press. gutiérrez, r. (ed.) (2013). special equity issue. journal for research in mathematics education, 44(1). heid, m. k. (2010). where’s the math (in mathematics education research)? journal for research in mathematics education, 41(2), 102–103. martin, d. b. (2003). hidden assumptions and unaddressed questions in mathematics for all rhetoric. the mathematics educator, 13(2), 7–21. martin, d. b., gholson, m. l., & leonard, j. (2010). mathematics as gatekeeper: power and privilege in the production of knowledge, journal of urban mathematics education, 3(2), 12– 24. retrieved from http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/95/57 parks, a. n., & schmeichel, m. (2012). obstacles to addressing race and ethnicity in the mathematics education literature. journal for research in mathematics education, 43(3), 238– 252. skovsmose, o., & penteado, m. g. (2011). ghettoes in the classroom and the construction of possibilities. in b. atweh, m. graven, w. g. secada, & p. valero (eds.), mapping equity and quality in mathematics education (pp. 77–90). dordrecht, the netherlands: springer. stinson, d. w. (2004). african american male students and achievement in school mathematics: a critical postmodern analysis of agency. dissertations abstracts international, 66 (12). (umi no. 3194548) strutchens, m. (ed.) (2012). foregrounding equity in mathematics teacher education. journal of mathematics teacher education, 15(1). journal of urban mathematics education july 2013, vol. 6, no. 1, pp. 35–44 ©jume. http://education.gsu.edu/jume tonya gau bartell is an assistant professor in the department of teacher education at michigan state university, 620 farm lane, erickson 116n, east lansing, mi, 48824; email: tbartell@msu.edu. her research focuses on the tools and experiences that can support teachers’ development of equitable pedagogical practices with explicit attention to social justice, culture, race, and power in mathematics education. kate r. johnson is a doctoral candidate in the department of teacher education at michigan state university and will be an assistant professor in the department of mathematics education at brigham young university fall 2013. email: johnson@mathed.byu.edu. her research focuses on the identities of mathematics teachers in learning about and teaching mathematics for social justice. making unseen privilege visible in mathematics education research tonya gau bartell michigan state university kate r. johnson michigan state university in this essay, the authors begin to “unpack the invisible knapsack” of mathematics education research privilege. they present short statements representing the multiplicity of their respective identities; acknowledging that efforts to understand privilege and oppression are often supported and constrained by identities. the authors then present three lists generated as they identify experiences as mathematics education researchers that they may have taken for granted; two lists are from the perspective of privilege and the other is from a position of marginalization. the multiple lists reflect that a person can be simultaneously oppressed in some ways and privileged in others. the authors conclude by inviting others to join the discussion about the invisible knapsack of mathematics education research privilege. keywords: equity, mathematics education research, privilege the true focus of revolutionary change is never merely the oppressive situation which we seek to escape, but that piece of the oppressor which is planted deep within each of us. – audre lorde (1984, p.123) athematics education research in the united states, similar to all social systems, is embedded in interlocking systems of privilege and oppression. generally in such systems, particular groups are granted privilege based on birthright or other unearned means (as opposed to “effort” and/or “perceived” intelligence or ability) and privileged people begin to believe their personal qualities warrant inclusion in the group. in turn, lack of membership is viewed as lack of effort or personal flaw (e.g., the myth of meritocracy, young, 1958). in other words, the unseen dimensions of privilege and oppression become normalized, dehumanizing both subordinate and dominant groups (freire, 1970/2000) in ways that perpetuate the unjust systems of oppression and domination (bell, 1997). m bartell & johnson making unseen privilege visible stinson, d. w., & spencer, j. a. (eds.). (2013). privilege and oppression in the mathematics preparation of teacher educators [special issue]. journal of urban mathematics education, 6(1). 36 we define privilege as unearned benefits and advantages granted to people in dominant groups at the expense of people in oppressed groups, “whether they want those privileges or not, and regardless of their stated intent” (allies for change, n.d., p. 1; as presented at propmte 1 ). privilege plays out in particular situations in daily life as a result of a variety of systems. the way privilege operates within such systems often goes unseen and requires explicit unpacking from positions of privilege and marginalization (mcintosh, 2011). scholars, such as mcintosh (2011), scalzi (2005), and schlosser (2010), have begun to list some of the associated privileges or marginalizations for the systems of race, class, and religion, respectively. similar to the way mcintosh (2011) “came to see white privilege as an invisible package of unearned assets which [she] can count on cashing in each day, but about which [she] was ‘meant’ to remain oblivious” (p. 121), we have come to see mathematics education research as a “package of unearned assets” that needs to be unpacked and exposed. it is our intention that exposing mathematics education research privileges will engender a sense of responsibility and accountability for mathematics education researchers to begin the process of dismantling and changing such systems by first acknowledging their many and vast unseen dimensions. in other words, mathematics education researchers need to understand how mathematics education research privilege plays out in order to begin to dismantle and disrupt oppressive systems. here, we begin to “unpack the invisible knapsack” (mcintoch, 2011) of mathematics education research privilege. first, we present short statements representing the multiplicity of our fragmented and continuously shifting and evolving identities, as our work to understand privilege and oppression is supported and constrained by whom we are. next, we present three lists generated as we attempted to identify conditions of our experiences as mathematics education researchers that we may take for granted. two are from the perspective of privilege and the other is from a position of marginalization. these multiple lists reflect that a person can be simultaneously oppressed in some ways and privileged in others (freire, 1970/2000). our positionalities we cannot separate ourselves from whom we are, and we recognize that our worldviews influence the privileges that we see as well as those that we do not 1 privilege and oppression in the mathematics preparation of teacher educators (prompte) conference (funded by create for stem institute through the lappan-phillips-fitzgerald cmp 2 innovation grant program), michigan state university, battle creek, mi, october 2012. any opinions, findings, and conclusions or recommendations expressed herein are those of the authors and do not necessarily reflect the views of the funding agency. bartell & johnson making unseen privilege visible stinson, d. w., & spencer, j. a. (eds.). (2013). privilege and oppression in the mathematics preparation of teacher educators [special issue]. journal of urban mathematics education, 6(1). 37 see. these positionalities are shaped by the multiplicity of our fragmented identities (e.g., along lines of race, class, sexuality, able-ness, etc.), including positions of power into which society has placed us as well as our personal life experiences within and around these identities (maher & tetreault, 1994). specific to the work we present here, our life experiences impact the positionality we bring to this analysis, informing the questions we ask and the interpretations we draw (foote & bartell, 2011); thus, we share some information about our positionality. 2 tonya – i am a white, able-bodied, heterosexual woman who grew up in a lower middle-class catholic family in rural minnesota. i taught high school mathematics for 6 years in both traditional and alternative public school settings. i have a ph.d. from the university of wisconsin-madison; the national science foundation (nsf) funded my degree. i am now a tenure-track faculty member at michigan state university; i am a co-pi on an nsf-funded project. i am a mother, and my daughter is in kindergarten in a public school. my research focuses on the tools and experiences that can support teachers’ development of equitable pedagogical practices with explicit attention to social justice, culture, race, and power in mathematics education. kate – i am a white, hearing, 3 heterosexual, upper middle-class, adult-convert mormon woman. i am an only child. my father served in the united states navy until i was 25 years of age. i have a master’s degree from the university of pittsburgh in the education of students who are deaf and hard of hearing and taught high school mathematics for 4 years at a school for the deaf. i have attended michigan state university for my doctoral studies. some of my degree has been funded through working on an nsf-funded project. i have accepted a tenure-track assistant professor position to begin fall 2013. my research focuses on mathematics teachers’ identities in the context of learning about and teaching mathematics for social justice. 2 for additional details about the positionality we bring to this work, please contact us by email. note that one danger of sharing our positionality briefly and somewhat categorically is that we are not able to fully unpack the ways in which our positionality plays out in the analyses presented. we acknowledge that these statements are necessary but not complete elaborations of our positionality. 3 the capital letter “h” on hearing is one way to acknowledge how deafness can be viewed as a culture (as opposed to as a medical condition) (cf. padden & humphries, 1988; lane, 1992). bartell & johnson making unseen privilege visible stinson, d. w., & spencer, j. a. (eds.). (2013). privilege and oppression in the mathematics preparation of teacher educators [special issue]. journal of urban mathematics education, 6(1). 38 generating lists of privileges drawing on the work of mcintosh (2011), we generated three lists as we attempted to identify conditions of our experiences as mathematics education researchers that we may take for granted. the first is from the perspective of the field of mathematics education and considers the privileges we believe all mathematics education research shares (as opposed to social studies or english education research, for instance). the second is from the perspective of institutional privilege and considers unearned privilege granted through institutional association, both from the perspective of being a faculty member employed at a privileged institution and from the perspective of being a doctoral student at that institution. the third list illuminates privileges in the field of mathematics education research from our position as researchers who study a traditionally marginalized body of work. in creating these lists, we aimed to consider privilege and oppression in relation to the identity of a “mathematics education researcher.” though compelled to include additional items on our list explicitly related to race, class, gender, religion, age, ability, sexual orientation, and language—and these factors are still present, as systems of oppression are interlocking (cf. hardiman & jackson, 1997)—we felt that a focus on mathematics education researcher privilege in particular was important. certain statements such as “i can say mathematics education research historically publishes research about people like me” did not make the lists, as this privilege is not reflective of privilege earned from being a mathematics education researcher, but rather from being a white mathematics education researcher. systems of privilege and oppression are the subject of some research in mathematics education; however, the privileges associated with the community of mathematics education research itself are rarely considered. additionally, the first two lists begin with “i can” statements, while the third list begins with “i cannot” statements. that is, here we write wearing two hats (at least): (a) as mathematics education researchers who have benefitted from institutional and field privilege within mathematics education, and (b) as mathematics education researchers whose work has been too often marginalized within the field. our method of compiling the lists is not the only way one could or should examine privilege in mathematics education research. on the contrary, much work is needed in mathematics education research that explicitly names and unpacks systems of oppression (cf. gutstein, 2006) and also interrogates systems of privilege, such as using whiteness theory to explore researcher identity (cf. gregson, 2011) and questioning the privileging of english in mathematics education research (cf. meaney, 2013) (bartell, bieda, breyfogle, crespo, dominguez, drake, & herbel-eisenmann, 2013). bartell & johnson making unseen privilege visible stinson, d. w., & spencer, j. a. (eds.). (2013). privilege and oppression in the mathematics preparation of teacher educators [special issue]. journal of urban mathematics education, 6(1). 39 the field of mathematics education research  i can be sure funding is available for my field.  i can be sure there is a job in my field.  i can tell people what my field is and they think i’m really smart.  i can go into a classroom and evaluate a teacher’s practice.  i can work the educational system to privilege my child.  i can be sure travel funding is available to travel to conferences around the world.  i can attend multiple conferences in any given year because there are a variety of conferences in my field.  when i’m told about the institution of schooling, i can be sure the people in my field built it.  i can go into a pre-k–12 school and see the content of my field being taught.  i can get a mortgage because there’s job security in my field. 4  i can be sure the content of my field is represented in a major portion of various standardized tests.  i can see my field represented in general public discussions of education issues. institutional affiliations 5  i can be pretty sure that if i ask to talk to the “person in charge” i will be facing a person from an institution similar to mine.*  i can turn down jobs because there are so many available to me.  i can be asked to chair any dissertation committee based on my pedigree.  i can be sure of my preparation to be a faculty member in my field because of the graduate institution i attended.  i can publish in the top-tiered journals in my field because of the institution in which i work.  i can attend multiple conferences because there are institutional funds to support me.  i can access grant-writing support in various forms. 4 we recognize that property values in many geographic areas make mortgages unavailable to many professionals, but the point here is that in comparison to other fields, mathematics education researchers are better positioned to obtain mortgages. we acknowledge and understand that getting a mortgage is also mediated by race, language, and other factors, as is each statement on these lists, but this statement is meant to describe the ways in which job stability and security (again in mathematics education research in contrast to other fields such as social studies or english education) can facilitate one’s ability to obtain a mortgage. 5 asterisks reflect statements created directly from mcintosh’s (2011) list. bartell & johnson making unseen privilege visible stinson, d. w., & spencer, j. a. (eds.). (2013). privilege and oppression in the mathematics preparation of teacher educators [special issue]. journal of urban mathematics education, 6(1). 40  i can be pretty sure i’ll be asked to give a colloquium at another institution.  i can be assured that my institutional infrastructure can support external funding.  i can get a job regardless of my area of research.  if i wish to switch institutions, i can be sure that all other institutions are still options for me.*  i can be pretty sure that my colleagues at this new institution would respect me.*  i can, if i wish, arrange to be in the company of people from an institution similar to mine at mathematics education research meetings most of the time.* mathematics education research  i cannot publish without first debunking the presumed cultural neutrality of my field.  i cannot open the table of contents of top-tiered journals in my field and see research that legitimates my own.  i cannot be sure funding is available for my work.  i cannot publish without justifying “where’s the math” within a particular framing of what mathematics is and what counts as mathematics.  i cannot remain unschooled in the language of the dominating discourses about mathematics education research.  i cannot generally look at the work of those with authority in my field and see work similar to mine. discussion the above lists were created in an attempt to make unseen privilege in mathematics education research visible. though these statements are written as “i” statements, it should not be taken to mean that they are only personal. rather, they reflect broader systems of privilege and oppression in which all people operate. these systems play out in personal ways, such as in conscious or unconscious actions or attitudes that maintain oppression. these personal interactions, however, are shaped by institutional policies and procedures as well as by society’s cultural norms, which can perpetuate implicit and explicit values (e.g., definition of “good” or “normal”) that bind institutions and individuals. it is the interactions of these multiple levels simultaneously that create interlocking systems of oppression which serve to normalize unseen dimensions of privilege. it is these systems that we aim to illuminate and, in turn, begin to disrupt and change. bartell & johnson making unseen privilege visible stinson, d. w., & spencer, j. a. (eds.). (2013). privilege and oppression in the mathematics preparation of teacher educators [special issue]. journal of urban mathematics education, 6(1). 41 as mentioned previously, these lists are a sample of possible lists. for the sake of our argument—an attempt to make visible privilege within mathematics education research—we chose not to include additional items specifically related to factors such as race, class, gender, sexual orientation, and language. this is not to say that these factors do not exist or are not at play. rather, they are always present and intersect with mathematics education research privilege in various ways. for instance, a latino faculty member with a spanish accent who conducts research in an area traditionally marginalized in the field may not only have to prove “where the mathematics is” (see martin, gholson, & leonard, 2010) in their research but also that they are competent in mathematics. thus, it is impossible to separate these roles. it is important to highlight that these lists are specifically focused on mathematics education research and the privilege therein. these lists do not consider the ways in which mathematics education research might not have privilege (e.g., in relation to mathematics as a discipline), which would be an interesting and important analysis to conduct. caution is necessary, though, as it might be easy to dismiss mathematics education research privilege when one also notes how mathematics education research is oppressed, thus never unpacking the privileges. for example, sometimes white people dismiss examinations of their privilege related to whiteness because they instead associate with their marginalization as poor people (or other areas of oppression), perhaps feeling that the focus on race “invalidates their oppressions” or that these “oppressions make them ‘less’ racially privileged” (diangelo, 2006, p. 52). thus, we cautiously encourage continued exploration of the ways mathematics education is not privileged as well as the ways in which it is. as we worked to create these lists, we experienced some discomfort. for instance, we found it easier at times to talk about marginalization than to talk about privilege. perhaps this ease of naming marginalization is because when speaking from a place of marginalization it is made apparent that we are part of a group that has some solidarity or connectedness through such marginalization. in contrast, when speaking from a place of privilege there is a sense of “othering” marginalized groups, or engagement in a process, either unwillingly or willingly, that steals others’ humanity (freire, 1970/2000). we both noted that we felt ill several times as we created these lists. we recognize that we, too, live within and are constrained by systems of social oppression that “trap” us and “confine us to roles and prescribed behavior” (hardiman & jackson, 1997, p. 20). but that acknowledgement doesn’t make it any easier to stomach. at the same time, speaking from a place of marginalization was uncomfortable, in that it is scary to speak from a place where you do not hold power. yet, part of the reason we were able to speak from that position was because of the institutional privilege from which we benefit. bartell & johnson making unseen privilege visible stinson, d. w., & spencer, j. a. (eds.). (2013). privilege and oppression in the mathematics preparation of teacher educators [special issue]. journal of urban mathematics education, 6(1). 42 we also found that we could at times abate our discomfort with some items that we could choose to admit to or not based on our position of privilege as mathematics education researchers in an institution with privilege. we reflected on the fact that in some contexts, we do not acknowledge what we study (be that a focus on mathematics education or a focus on social justice). for example, the privilege grants us the choice to say that we study teachers instead of saying that we study mathematics teaching and learning, so that we don’t have to hear the negative discourses back (e.g., “i hate math,” “i was never good at math”). we are denying our privilege to align ourselves with the marginalized field of teaching in order to not have to deal with the negative reactions. alternatively, we might frame our work as focused on mathematics education, and omit that race and class are central ideas in our research. perhaps this omission is part of what made it possible for us to write this piece—we come to these lists both from positions of privilege and of marginalization. moving forward understanding and acknowledging privilege is not enough. when mathematics education researchers have not critically examined their own place in the systems of privilege and oppression, they frequently bring a deficit model and exhibit behaviors that are patronizing because they view this work through a lens of charity rather than justice. (bartell et al., 2013, p. 227) although tacitly identifying the ways in which systemic privilege exists is a necessary condition, it is not sufficient. mathematics education researchers must illuminate how privilege plays out on a more personal level by examining one’s position of privilege, both by reading lists such as those presented here or looking closely at one’s own work. without this type of critical examination, individual researchers in particular, and the field in general, risk bringing a paternalistic view to teachers and students. this view can often be seen in research that does not respect the work and practices of teaching or focuses on illuminating what teachers and students do not know. instead, we posit that the field needs to both acknowledge (or make visible) the privilege from which it benefits and then consider how to disrupt the perpetuation of privilege. at the same time, for those traditionally marginalized in various ways, the privilege bestowed upon mathematics education researchers is an important and often strategic tool used to address inequity and injustice and serves as a means to push from the margins. it is therefore also important to consider how one might use one’s privilege for positive change. we liken the duality of raising awareness and taking action to the notion of reading and writing the world with mathematics that gutstein (2006) described. to that end, we see two sets of actionable items. first, we wish to specifically invite others to dialogue with us about the invisible knapsack of mathematics educa bartell & johnson making unseen privilege visible stinson, d. w., & spencer, j. a. (eds.). (2013). privilege and oppression in the mathematics preparation of teacher educators [special issue]. journal of urban mathematics education, 6(1). 43 tion research privilege described here. we acknowledge that our positionality has affordances and constraints in illuminating these unearned assets. therefore, we wonder:  what are we missing as white people? as women? as “insiders”? as people who view our work as marginalized?  what might other marginalized mathematics education researchers notice?  what would discussions with colleagues in other content areas illuminate?  what additional privileges would people in other disciplines see? second, we wish to encourage ourselves and other researchers to consider possible ways to disrupt and dismantle privilege. mcintosh (2011) asked: “what will we do with such knowledge? … [will we] choose to use unearned advantage to weaken hidden systems of advantage? … [will we] use any of our arbitrarily awarded power to try to reconstruct power systems on a broader base?” (p.125). in the context of mathematics education research, these questions provoke a related set of ideas. here, we raise a few questions and speculate some possible answers:  how might mathematics education researchers come together to think about understanding and confronting privilege afforded the field of mathematics education research? we might encourage interdisciplinary work or drawing on and citing research outside of mathematics education. we might develop ways to spread the access to funding we have across other disciplines.  how might we begin to dismantle and disrupt institutional privilege? we might consider making curriculum vitae and cover letters institutionally blind during the hiring process. we might consider not including institutional affiliation on nametags at conferences.  how might we begin to dismantle and disrupt privileging certain mathematics education research? we might consider abolishing “special issues” so as to reframe all research as central. we might consider amending the peer review process in an effort to broaden “what counts” as mathematics education research (cf. martin, gholson & leonard, 2010). the questions posed here are intended to move our discussions forward and invite others into this conversation. every mathematics education researcher, we be bartell & johnson making unseen privilege visible stinson, d. w., & spencer, j. a. (eds.). (2013). privilege and oppression in the mathematics preparation of teacher educators [special issue]. journal of urban mathematics education, 6(1). 44 lieve, is actively benefitting from the privileges we describe here and, therefore, has a responsibility to engage in disrupting these simultaneously oppressive systems. references allies for change. (n.d.) glossary of terms. retrieved from http://www.alliesforchange.org/documents/glossary.pdf. bartell, t., bieda, k., breyfogle, m. l., crespo, s., dominguez, h., drake, c. & herbel-eisenmann, b. (2013). strong is the silence: challenging systems of privilege and oppression in mathematics teacher education. in m. berger, k. brodie, v. frith, & k. le roux (eds.), proceedings of the 7th international mathematics education and society conference (mes 7; vol. 2, pp. 223–231). cape town, south africa: mes 7. retrieved from http://www.mes7.uct.ac.za/bartell%20et%20al._paper.pdf. bell, l. a. (1997). theoretical foundations of social justice education. in m. adams, l. a. bell, & p. griffin (eds.), teaching for diversity and social justice: a sourcebook (1st ed., pp. 1–15). new york, ny: routledge. diangelo, r. j. (2006). my class didn’t trump my race: using oppression to face privilege. multicultural perspectives, 8(1), 52–56. foote, m. q., & bartell, t. g. (2011). pathways to equity in mathematics education: how life experiences impact researcher positionality. educational studies in mathematics, 78, 45–68. freire, p. (2000). pedagogy of the oppressed (m. b. ramos, trans.; 30th anniv. ed.). new york, ny: continuum. (original work published 1970) gregson, s. a. (2011). negotiating social justice teaching: one full-time teacher’s practice viewed from the trenches. journal of research in mathematics education, 44, 164–198. gutstein, e. (2006). reading and writing the world with mathematics: toward a pedagogy for social justice. new york, ny: routledge. hardiman, r., & jackson, b. w. (1997). conceptual foundations for social justice courses. in m. adams, l. a. bell, & p. griffin (eds.), teaching for diversity and social justice: a sourcebook (1st ed., pp. 16– 29). new york, ny: routledge. lane, h. (1992). the mask of benevolence: disabling the deaf community. san diego, ca: dawn sign press. lorde, a. (1984). sister outsider. new york: crossing press. maher, f., & tetreault, m. k. (1994). the feminist classroom. new york, ny: basic books. martin, d. b., gholson, m. l., & leonard, j. (2010). mathematics as gatekeeper: power and privilege in the production of knowledge. journal of urban mathematics education, 3(2), 12– 24. retrieved from http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/95/57. mcintosh, p. (2011). white privilege: unpacking the invisible knapsack. in p. s. rothenberg (ed.), white privilege: essential readings on the other side of racism (4th ed., pp. 121–125). new york, ny: worth. meaney, t. (2013). the privileging of english in mathematics education research, just a necessary evil? in m. berger, k. brodie, v. frith, & k. le roux (eds.), proceedings of the 7th international mathematics education and society conference (mes 7; vol. 1, pp. 65–84). cape town, south africa: mes 7. retrieved from http://www.mes7.uct.ac.za/meaney_plenary.pdf. padden, c. & humphries, t. (1988). deaf in america: voices from a culture. cambridge, ma: harvard university press. scalzi, j. (2005, september 3). being poor – whatever. [web log post]. retrieved from http://whatever.scalzi.com/2005/09/03/being-poor/. schlosser, l. w. (2010). christian privilege. in m. adams, w. j. blumenfeld, r. castañeda, h. w. hackman, m. l. peters, & x. zúñiga (eds.), readings for diversity and social justice (2nd ed., pp. 246– 247). new york, ny: routledge. young, m. (1958). rise of the meritocracy. new brunswick, nj: transaction. http://www.alliesforchange.org/documents/glossary.pdf http://www.mes7.uct.ac.za/bartell%20et%20al._paper.pdf http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/95/57 http://www.mes7.uct.ac.za/meaney_plenary.pdf http://whatever.scalzi.com/2005/09/03/being-poor/ journal of urban mathematics education july 2012, vol. 5, no. 1, pp. 44–54 ©jume. http://education.gsu.edu/jume denise natasha brewley is an assistant professor at georgia gwinnett college in the school of science and technology, 1000 university center lane, lawrenceville, ga 30043, email: dbrewley@ggc.edu. her research interests include identity, understanding how communities of practice developed in mathematics spaces, and creating significant learning experiences for students taking undergraduate mathematics courses. college mathematics literacy workers of the young people’s project chicago: a community of practice denise natasha brewley georgia gwinnett college n college, i decided to pursue mathematics as a discipline of study despite my apprehension concerning my own “average” mathematical abilities. i was counseled repeatedly that only mathematically talented students should pursue a major in mathematics, implying that there was some “natural” mathematical talent that one either had or didn’t have. to disprove that assumption, i studied mathematics anyway and committed myself to working diligently even when the content was difficult to grasp. one of the most important decisions i made, which transformed my understanding and the way i thought about mathematics, was to work as a peer tutor in the college’s mathematics tutoring laboratory. as a tutor, my ability to conceptualize mathematics in broader ways began to take shape, contributing to my own mathematics literacy. more importantly, i had the opportunity to help struggling students. through the practice of tutoring my peers, to improve their performance in mathematics, my confidence and understanding also grew. engaging in mathematics through conversations with my peers strengthened my knowledge of the subject and my ability to communicate mathematical concepts to others. in retrospect, i recognize that this experience helped to shape how i saw myself as a mathematics doer and knower (i.e., this experience helped to shape my mathematics identity). mathematics identity, as defined by martin (2007), “refers to the dispositions and deeply held beliefs that individuals develop about their ability to participate and perform effectively in mathematical contexts and to use mathematics to change the conditions of their lives” (p. 150). beyond building confidence in my mathematics ability, the peer tutoring experience ultimately helped to shape my mathematics literacy—my conceptual understanding of mathematical topics—and my identity—my broader sense of self outside of mathematics. my experience as a peer tutor motivated me to become a mathematics educator so that i could continue to help others understand and engage in mathematics. my engagement with mathematics continues to deepen my mathematical content knowledge, influence how i see myself, and creates possibilities for thinking about mathematics and doing mathematics in new and different ways. i brewley community of practice bullock, e. c., alexander, n. n, & gholson, m. l. (eds.). (2012). proceedings of the 2010 philadelphia and 2011 atlanta benjamin banneker association conferences – beyond the numbers [special issue]. journal of urban mathematics education, 5(2). 45 given the importance of the tutoring experience in my own life, i wanted to study whether other students’ experiences as individuals helping others to learn mathematics in an out-of-school context positively influenced their identities and roles within their local communities. the chicago chapter of the young people’s project (ypp) provided a venue to pursue this question. the research question that guided the project was what identities and roles do african american college mathematics literacy workers see themselves having in their local communities as a result of their participation in the community of practice, the young people’s project chicago? here, i begin by briefly explaining the necessity of mathematics literacy for citizenship. i then give a background of young people’s project, community of practice and modes of belonging, and explain how ypp constitutes a community of practice (wenger, 1998)—a useful construct for considering questions of identity. mathematics literacy for citizenship the need for complex and sophisticated mathematical knowledge and problem solving has grown over the past 100 years and is strongly attributed to technological advancement. many african american students are unable to take full advantage of careers in the sciences and technologies due to limited access to meaningful mathematics. in the 1960s, poor mississippi sharecroppers had to demonstrate literacy and a modest interpretation of the constitution in order to vote (moses, 1994; moses & cobb, 2001). today a similar demonstration is necessary for black students in mathematics who must make the case for themselves that acquiring mathematics literacy is an issue of civil rights in that every citizen should be able to access quality mathematics education to become mathematically literate (moses, 1994; moses & cobb, 2001). when we prepare students for citizenship, we prepare them to take a position on issues, utilize their voices effectively, and to deal with situations critically as they arise (rudduck, 2007). some have suggested that there is a need to establish communities where students can become engaged in and excited about doing mathematics, and where they can take an active role in the teaching and learning of mathematics with their peers outside of school. the work of the ypp is a response to this call. ypp and ypp chicago the ypp is a youth-driven organization affiliated with the algebra project 1 1 the algebra project is a national initiative, rooted in the u. s. civil rights movement, that is carried out in schools and afterschool programs. the main purpose of the ap is to improve the mathematics literacy of young people of color underserved by existing education reform efforts in order for them to gain political and economic power and access to opportunities. brewley community of practice bullock, e. c., alexander, n. n, & gholson, m. l. (eds.). (2012). proceedings of the 2010 philadelphia and 2011 atlanta benjamin banneker association conferences – beyond the numbers [special issue]. journal of urban mathematics education, 5(2). 46 (moses & cobb, 2001) founded in jackson, mississippi in 1996. the ypp was created as an after-school mathematics initiative by an alliance of african american algebra project graduates who wanted to take an active role in their community by preparing youths to become more mathematically proficient. ypp has three main objectives: (a) to use mathematics literacy to develop youth leaders and organizers who would radically change the quality of education and life in their communities; (b) to develop young people as facilitators, mentors, and advocates for mathematics literacy; and (c) to assist a target population of algebra project students and non-algebra project students to successfully complete algebra by eighth or ninth grade in order to enter a college preparatory mathematics sequence in high school (moses, 2006). ypp works to achieve these objectives through the development and implementation of mathematics games and activities in schools. ypp chicago was started in 2002 through local city partnerships, with the objective of operating training hubs throughout the city of chicago for the development of college mathematics literacy workers. ypp chicago was “founded on the belief that there is work that young people can and must do to change the conditions of their lives and that math literacy work was a good place to start” (ypp, n.d., ¶1). they also believe that african american students in economically depressed areas, like chicago, are disenfranchised the most when it comes to mathematics education and that mathematics literacy work is a necessity in those communities. as it relates to mathematics, it is rare to find grassroots organizations that execute mathematics programs in urban settings that are led by black students. the work of ypp offers a way for african american students to exercise their voice in schools while reinforcing what matters to them most. in the larger project, i wanted to investigate how the role of mathematics teaching among peers mediates the mathematics identities of the african american students who serve as peer facilitators. considering the ypp as a social context for engagement in mathematics, i also wanted to know what other aspects of their identities are shaped by the mathematics literacy work in which they participate. through this active involvement, african american students can effect change in their communities for the purpose of liberation for themselves and others. in this sense, liberation refers to the connection that is made by students in what is being learned and how it informs what they know, think, and engage in their world. communities of practice and modes of belonging a community of practice (cop) is a collective group unified by common interests where members interact regularly in order to create and improve what they learn and share over time (wenger, 1998). cops are an important part of our eve brewley community of practice bullock, e. c., alexander, n. n, & gholson, m. l. (eds.). (2012). proceedings of the 2010 philadelphia and 2011 atlanta benjamin banneker association conferences – beyond the numbers [special issue]. journal of urban mathematics education, 5(2). 47 ryday lives as we are each a part of a number of cops both implicitly and explicitly. cops range in their level of formalness and involvement; some individuals are core members, and others hold peripheral membership. cops contain a combination of three fundamental elements: (a) a domain of interest or mutual engagement; (b) a community or joint enterprise; and (c) a focus on practice or shared repertoire (wenger, 1998; wenger, mcdermott, & snyder, 2002). from an identity point of view, there are three modes of belonging that are part of learning and constructing identity: engagement, imagination, and alignment. engagement describes interactions as an ongoing collaboration with members in communities that can change. imagination is the ability of members to create new images of themselves beyond time and space. alignment is the mode of belonging in which all efforts, such as energies, actions, and practices, come together to produce coordinated activities. i consider ypp chicago to be a cop because of its community-based mathematics literacy work. there is a common mission statement that unifies mathematics literacy workers and affirms mathematics literacy workers’ purpose, inspires members to participate, gives members meaning and a context for their outreach work, and guides their learning and the knowledge they produce. ypp chicago members also share a commitment to improve young people’s understanding of numbers through mathematics literacy activities in which they participate. in this community setting, mathematics literacy workers also form social bonds through prolonged interactions with each other. methods in this article, i report on two participants, who are college mathematics literacy workers (cmlws), naomi and demarcus. 2 naomi was a 21-year-old african american female student at a large urban university majoring in african american studies, gender studies, and performance studies. she was born and raised in the south, but relocated to chicago to attend college. she had worked with nonprofit organizations since she was 14 years of age and learned of ypp through her involvement in other youth-oriented work. at the time of this study, she had been involved in ypp work for nearly one year. demarcus was also a native of the south who relocated to chicago to attend college. he was a 21-yearold african american student studying at a large urban university. his major was african american studies with a concentration in history. demarcus realized that teaching was the career he wanted to pursue and his work with ypp enabled him to improve his teaching skills and his interactions with young people. at the time of the study, demarcus had worked with ypp for about six months. 2 proper names throughout are pseudonyms. brewley community of practice bullock, e. c., alexander, n. n, & gholson, m. l. (eds.). (2012). proceedings of the 2010 philadelphia and 2011 atlanta benjamin banneker association conferences – beyond the numbers [special issue]. journal of urban mathematics education, 5(2). 48 the study took place at abelin preparatory high school, where the cmlws along with high school mathematics literacy workers (mlws) participated in daily workshop training in preparation to teach mathematics games to elementary school children. abelin is part of the chicago public school system and was founded in 1998 as a neighborhood charter school located on the west side of chicago. approximately 670 students attended abelin from 9th through 12th grade at the time of data collection. ninety-seven percent of the students enrolled were african american, and 3% latina/o. ninety-two percent of the student population were eligible for the free-and-reduced lunch program, and 97% of the students who attended the school resided in neighborhoods on the west side of chicago. the work of the cmlws in workshop training sessions consisted of critical reflections about what activities worked and did not work and interactions among attendees as they developed and learned about the mathematics activities that would be used in the elementary schools. the duration of the workshop training sessions was over a 4-week period. they were conducted after school for about three hours a day, for 4 days, for a total of 12 hours each week. the overall focus of workshop training sessions was for cmlws to help mlws to develop a deep understanding of a mathematically rich game, called flagway. 3 the workshop training consisted of five components: (a) building mlws’ basic mathematics competency with numbers 2 through 100, (b) building mlws’ facilitation skills, (c) building mlws’ awareness of community social issues, (d) teaching mlws how to effectively play flagway, and (e) planning for implementation of flagway at elementary schools. data were comprised of 92 hours of observations, two semi-structured interviews of each participant ranging from one to two hours, student work from mathematical tasks of the mathematics concepts taught in the workshop training, and participant reflections on their daily workshop trainings. to answer the overarching research question, i drew from each cmlw’s view of mathematics literacy, their interpretation of the ypp’s mission statement, and how they sought to embody the expectations of the mission in their mathematics literacy work. the mission statement was developed by its members in 2006, and provides a framework for cmlws’ practice as they engaged in mathematics literacy work. thematic analysis was used to shed light on how each cmlw viewed mathematics literacy and embraced the mission of ypp, and how their view of the mission statement shaped their work and identity as mathematics literacy workers within the community of practice. 3 the flagway game was developed to help young people expand their understanding of natural numbers, through exploration of mathematically rich numeracy activities. brewley community of practice bullock, e. c., alexander, n. n, & gholson, m. l. (eds.). (2012). proceedings of the 2010 philadelphia and 2011 atlanta benjamin banneker association conferences – beyond the numbers [special issue]. journal of urban mathematics education, 5(2). 49 identities and roles in mathematics literacy work looking beyond mathematics identity, holland, lachicotte, skinner, and cain (2003) define identity as “a concept that figuratively combines the intimate or personal world with the collective space of cultural forms and social relations” (p. 5). they start with the premise that “identities are lived in and through activity and so must be conceptualized as they develop in social practice….they are important bases from which people create new activities, new worlds, and new ways of being” (p. 5). through social interaction with individuals that are around us, there is growth and development in how we see ourselves and who we hope to become (gonzalez, 2009; holland et al., 2003). as gonzalez (2009) affirms, “identities affect agency and action” (p. 27). in this sense, identities are the broad ways in which we see ourselves and the roles that we play are how we take up our identities and put them to work. an “agent of change” and “doer of mathematics” were two identities that were common themes between naomi and demarcus. agent of change. an agent of change, in this context, is defined as someone who purposely worked toward creating social, cultural, or behavioral change in society or in others through his or her work or actions. a doer of mathematics is someone willing to engage in thinking deeply about mathematical ideas (brewley, in press). naomi and demarcus discussed in length how their work with young people in ypp helped to inform how they view their broader purpose in society. they had several revelations about themselves as a result of their work in ypp. naomi explained that her purpose in doing mathematics literacy work was “to make sure that abelin [could] survive....i can leave, and abelin will still be a functioning site with students who are impacted in a positive way.” she further explained that she believed her purpose was to be able “to affect those students in a way that they were inspired to do something in their community, or they feel like they have done something to their community.” naomi also stated, “whether it be i start teaching again through other…nonprofit organizations or working for human rights campaigns, or working for social change, social policy…i just have to make changes.” in regards to how she viewed herself in doing mathematics literacy work, it can be described as activism. naomi contended: i contextualize the entire mission statement with bob moses’s bigger picture of being a civil right activist and mathematics being a civil right….i should be able to access education so that i can exercise my full human potential. [in] a lot of minority communities and low-income communities, that’s not what’s happening….we started off with trainers, then the instructors, and then the cmlws, and then we spread ourselves all around all areas of chicago. and i think i just really think that it [was] sort of supposed to be a movement….like the word radical [italics added] to me means we are not taking this anymore. we are not going to sit here and let our students just continue to be below math literacy rates. like we are not going to do it. brewley community of practice bullock, e. c., alexander, n. n, & gholson, m. l. (eds.). (2012). proceedings of the 2010 philadelphia and 2011 atlanta benjamin banneker association conferences – beyond the numbers [special issue]. journal of urban mathematics education, 5(2). 50 demarcus felt “willing to undergo or figure out any new way of instituting certain forms of education that will make these things more exciting, to build confidence, and to [help the students be] more willing to take anything on that comes in their path.” demarcus often referred to the need of making mathematics more enjoyable for the population of students that he had worked with daily. he argued: “i felt almost as though my education was somewhat of an injustice…so the combination for striving for social equality and improving the lives and the achievement potential for kids like me…led me…on trying to be an educator.” demarcus reiterated that his mission working with youth is in alignment with the young people’s project: i essentially feel as though the mission with the young people’s project is essentially my mission in working with youth….my goal in education is to empower….i want to help build self-sufficient students. and i mean self-sufficient in that i want them to be of the character where they can go into a situation with as much confidence in their heart and success or failure to have the strength to take it on. naomi and demarcus collectively believed that their role was to help uplift young people in the community to achieve a higher level of mathematical literacy. they spoke about this empowerment in a variety of ways. naomi saw her role as being proactive in helping others understand mathematics while shifting how mathematics is perceived in the local community. naomi also spoke to the importance of minority and low-income communities having access to education in order for them to exercise their full human potential. demarcus believed that his role in ypp was to build competent and self-sufficient students and to help them to be prepared for situations that may arise by having the confidence to meet any challenge, no matter if they succeeded or failed. in his own conception, demarcus reiterated how education was used as a tool to disenfranchise black people. for him, it was the combined effort of “striving for social equality and improving the lives and the achievement potential for kids” that fueled his efforts in the community and in becoming an educator. doer of mathematics. naomi demonstrated a high comfort level in doing mathematics, and could be considered a confident doer of mathematics (martin, 2000, 2007). she believed that she could do mathematics easily and was generally “good” at the subject. working in the ypp workshop training provided her further opportunity to engage in mathematics. through opportunities that were provided to her earlier in her schooling, she understood that possessing strong mathematics skills was an invaluable tool that afforded opportunity and access. she explained: it’s like so [important] for minorities to be skilled in that area because it allows them to excel….it opens a lot of doors. like if i hadn’t won that competition. i got a two thousand dollar scholarship. i got a laptop….just so many doors, so many opportunities, just for, you know, my math skills. and so, it’s really important. brewley community of practice bullock, e. c., alexander, n. n, & gholson, m. l. (eds.). (2012). proceedings of the 2010 philadelphia and 2011 atlanta benjamin banneker association conferences – beyond the numbers [special issue]. journal of urban mathematics education, 5(2). 51 naomi also pointed out that she did not hesitate to do mathematics even if it required more thinking time of her than others. she stated, “i really think…i enjoy math, and i like doing math, and i love discovering it, and i love figuring stuff out….and i think coming from that standpoint…the fact that it takes me longer to do math, and i’m still not intimidated by it.” here, naomi echoes the necessity of embracing struggle when doing mathematics. she reifies the importance of struggle with new mathematical ideas as part of the learning process. demarcus possessed a comfort level in doing mathematics that developed over time. in many respects, his confidence grew as a result of his work in the ypp workshop training and ypp in general. an increasingly confident doer of mathematics is defined as an individual who demonstrates a sense of confidence or willingness to do mathematics that developed through prolonged engagement. demarcus attributed some of the challenges he had with mathematics to how he saw himself, his own perceived ability, and his enthusiasm. he also indicated that he had some apprehension about engaging in mathematics from prior schooling experiences because he perceived his peers to be better at the subject than he was. his desire to do mathematics evolved in a positive way the more he engaged in mathematics through the workshop training and with the students. he realized that as a result of his work in ypp, he had a renewed interest to do mathematics problems that he found challenging. demarcus provided the following insight about doing mathematics: i consider myself a person [that is] not too good at math. and i consider myself a person unwilling to do math for a large portion of my life. i feel as though in that respect, i can relate a lot to the kids that i work with….like [the workshop training has] improved my willingness to do math in everyday situations. i can definitely say before this program, i’d see [certain] number[s] and refuse to touch it. and i felt as though, [if the workshop training] was able to have this effect on me, then it would be possible for it, the kids that i work with, to have this effect [on them]. so, i guess one of the bigger influences on me is knowing this is possible. demarcus recognized that although his disposition toward mathematics had evolved, some of the young people with whom he worked still struggled with developing a desire to do mathematics. while both of the participants were considered to be doers of mathematics, naomi could be considered a confident doer of mathematics who was intrinsically motivated. naomi believed that mathematics was a challenging discipline to study but chose to engage in it anyway. naomi also expressed a general sense of enjoyment when she did mathematics. demarcus’ increased confidence in doing mathematics was developed over time and grew from his initial uneasiness, which came from earlier experiences in his schooling. demarcus made a decision that he brewley community of practice bullock, e. c., alexander, n. n, & gholson, m. l. (eds.). (2012). proceedings of the 2010 philadelphia and 2011 atlanta benjamin banneker association conferences – beyond the numbers [special issue]. journal of urban mathematics education, 5(2). 52 would be more confident in his approach to mathematics and he attributed this change with his work in ypp workshop trainings. discussion and conclusion the identities found in cmlws are supported by research that have documented other outcomes that were associated with students involvement in afterschool program initiatives that were not limited to academic achievement. these findings included motivation to succeed in school and an increased commitment in learning (mahoney, cairns, & farmer, 2003), a higher self-esteem and improved emotional adjustment and interpersonal skills (barber, eccles, & stone, 2001; gerstenblith et al., 2005; mahoney, 2000; mclaughlin, 2000) as a result of student participation. several of the reasons that led to membership in ypp, as described by naomi and demarcus, are echoed in findings from other studies that have examined the factors that contribute participation in community or school-based activities. these included friend endorsements of the afterschool activities (huebner & mancini, 2003), activities found to be fun as motivation for participation (gambone & arbreton, 1997), opportunities to learn (strobel, kirshner, & mclaughlin, 2008), and the acquisition of new skills and involvement in the community (perkins et al., 2007). also in accord with mclaughlin’s (2000) findings, youth that participated in afterschool programs in urban settings wanted to participate in something greater than themselves. these students selected programs where they could make an impact in their community, posses some autonomy in decision making, participate in a learning environment with committed adults, and reflect consistently on how well the program was going for them. at the symposium in the breakout session following the symposium at the 2011 benjamin banneker association conference, there was one reoccurring question that conference participants seemed to ask: what other organizations are doing work similar to the young people’ project? although there are several initiatives aimed to helping students in urban settings improve their mathematical skills, there is a dearth of grassroots organizations such as ypp that are run by young people interested in mathematics education related initiatives. participants of the breakout session suggested a further exploration of additional outlets and resources where mathematics literacy work could take place and is taking place. certainly, there are other community spaces where students have opportunities to build multiple mathematics skills, social awareness, which ultimately impacts their identities. community-based organizations like the algebra project has brewley community of practice bullock, e. c., alexander, n. n, & gholson, m. l. (eds.). (2012). proceedings of the 2010 philadelphia and 2011 atlanta benjamin banneker association conferences – beyond the numbers [special issue]. journal of urban mathematics education, 5(2). 53 continued this work since they first began developing mathematics curriculum in the early 80s. the algebera project approach to teaching and learning mathematics steps away from traditional methods learned in schools (moses & cobb, 2001). one other local community based organization discussed in the breakout session was the nsoromma school. 4 located in atlanta, georgia, the nsoromma school is an afterschool and weekend enrichment program that uses an african-centered approach in developing the character and the academic acumen of students in the area of mathematics, science, and engineering. in discussions with breakout session attendees, the benjamin banneker association (bba) was also considered as a site where those interested improving the mathematics education of youth could come together. as a national organization, bba’s scope extends beyond teachers to include teacher educators, researchers, and administrators. the bba with its mission devoted to mathematics education advocacy, leadership, and professional development in supporting teachers in providing african american students high quality education could be an additional stakeholder at the forefront of this effort. one suggestion was made to host workshops through organizations like bba around the country where school district administrators, educators, researchers, parents, students, and other community members could come together to share their individual expertise, discuss community issues, and generate solutions to aid in improving what students take away from their schools and classrooms as it relates to mathematics. if the goal is to see the brilliance of our black children actualized, community stakeholders should come together to figure out how to solve the problems of black student achievement in mathematics. references barber, b., eccles, j., & stone, m. (2001). whatever happened to the jock, the brain, and the princess? young adult pathways linked to adolescent activity involvement and social identity. journal of adolescent research, 16, 429–455. brewley, d. n. (in press). mathematics literacy for liberation and liberation in mathematics literacy: the chicago young people's project as a community of practice. in j. leonard & d. martin (eds.), the brilliance of black children in mathematics: beyond the numbers and toward new discourse. charlotte, nc: information age. gambone, m., & arbreton, a. (1997). safe havens: the contributions of youth organizations to healthy adolescent development. philadelphia, pa: public/private ventures. gerstenblith, s. a., soulé, d. a., gottfredson, d. c., lu, s., kellstrom, m. a., womer, s. c., & bryner, s. l. (2005). after-school programs, antisocial behavior, and positive youth development: an exploration of the relationship between program implementation and changes in youth behavior. in j. mahoney, r. larson, & j. eccles (eds.), organized activities as contexts of development: extracurricular activities, after-school and community programs (pp. 457–478). mahwah, nj: erlbaum. 4 for more information on the nsoromma school, visit www. http://www.nsoromma.org/. http://www.nsoromma.org/ brewley community of practice bullock, e. c., alexander, n. n, & gholson, m. l. (eds.). (2012). proceedings of the 2010 philadelphia and 2011 atlanta benjamin banneker association conferences – beyond the numbers [special issue]. journal of urban mathematics education, 5(2). 54 gonzalez, l. (2009). teaching math for social justice: reflections on a community of practice for high school math teachers. journal of urban mathematics education. 2(1), 22–51. retrieved from http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/32/13. holland, d., lachicotte, w., jr., skinner, d., & cain, c. (2003). identity and agency in cultural worlds. cambridge, ma: harvard university press. huebner, a. j., & mancini, j. a. (2003). shaping structured out-of-school time use among youth: the effects of self, family, and friend systems. journal of youth and adolescence, 32, 453– 463. mahoney, j. l. (2000). participation in school extracurricular activities as a moderator in the development of antisocial patterns. child development, 71, 502–516. mahoney, j. l., cairns, b. d., & farmer, t. (2003). promoting interpersonal competence and educational success through extracurricular activity participation. journal of educational psychology, 95, 409–418. martin, d. b. (2000). mathematics success and failure among african-american youth: the roles of sociohistorical context, community forces, school influence, and individual agency. mahwah, nj: erlbaum. martin, d. b. (2007). mathematics learning and participation in the african american context: the co-construction of identity in two intersecting realms of experience. in n. s. nasir & p. cobb (eds.), improving access to mathematics: diversity and equity in the classroom (pp. 146–158). new york: teachers college press. mclaughlin, m. w. (2000). community counts: how youth organizations matter for youth development. washington, dc: public education network. moses, o. (2006). the young people’s project: strategy for organizational expansion. chicago, il: young people’s project. moses, r. p. (1994). remarks on the struggle for citizenship and math/science literacy. journal of mathematical behavior, 13, 107–111. moses, r. p., & cobb jr., c. e. (2001). radical equations: civil rights from mississippi to the algebra project. boston, ma: beacon press. perkins, d. f., borden, l. m., villarruel, f. a., carlton-hug, a., stone, m. r., & keith, j. g. (2007). participation in structured youth programs: why ethnic minority urban youth choose to participate—or not to participate. youth society, 38, 420–442. rudduck, j. (2007). student voice, student engagement, and school reform. in d. thiessen, & a. cook-sather (eds.), international handbook of student experience in elementary and secondary school (pp. 587–610), dordrecht, the netherlands: springer. strobel, k., kirshner, b., o’donoghue, j., & mclaughlin, m. (2008). qualities that attract urban youth to after-school settings and promote continued participation. teachers college record, 110, 1677–1705. wenger, e. (1998). communities of practice. cambridge, united kingdom: cambridge university press. wenger, e., mcdermott, r., & snyder, w. m. (2002). cultivating communities of practice: a guide to managing knowledge. boston, ma: harvard business school. the young people’s project. (n.d.). young people’s project math literacy + social change: history. retrieved from http://www.typp.org/history. http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/32/13 http://www.typp.org/history microsoft word final rousseau anderson vol 7 no 1.doc journal of urban mathematics education june 2014, vol. 7, no. 1, pp. 7–18 ©jume. http://education.gsu.edu/jume celia rousseau anderson is an associate professor in the department of instruction and curriculum leadership at the university of memphis, 401d ball hall, memphis, tn 38152; email: croussea@memphis.edu. her research interests include equity and opportunity to learn in mathematics education.   commentary place matters: mathematics education reform in urban schools celia rousseau anderson university of memphis n the inaugural issue of this journal, william tate (2008a) outlined a few of the challenges related to research on urban mathematics education, including the need for the development of theories that take seriously the geography of opportunity. he urged scholars to “build theories and models that realistically reflect how geography and opportunity in mathematics education interact” (p. 7). this type of geospatial perspective “calls for the addition of a geographic lens that focuses on place and space as important contextual variables. a geospatial view increases our understanding of education…by framing research in the context of neighborhoods, communities, and regions” (tate, jones, thorne-wallington, & hogrebe, 2012, p. 426). in this commentary, i plan to revisit the call put forward by tate (2008a) and to explore some of the issues that models of mathematics education in the urban setting must accommodate, paying particular attention to mathematics education reform at the school level. while mathematics education research has often focused at the level of the classroom (rousseau anderson & tate, 2008), there are emerging calls for attention to shift from individual classrooms to consider the process of reform at the school or district level. as gamoran and colleagues (2003) acknowledge, we have multiple “existence proofs” of individual classrooms in which teaching for understanding occurs. what we lack is a clear model of the factors that contribute to the success of those classrooms from an organizational perspective. yet, the significance of the institutional context with regard to reform cannot be ignored. according to cobb and smith (2008), there is a substantial body of research that raises doubts regarding “an implicit assumption that underpins many reform efforts, that teachers are autonomous agents in their classrooms who are unaffected by what takes place outside the classroom door” (p. 234). in other words, the institutional setting matters. investigating the role of the institution and conditions of the organization becomes crucial in order to fully understand classroom practice and to move beyond existence proofs to take reform to scale (cobb & jackson, 2011; cobb & smith, 2008). i rousseau anderson commentary journal of urban mathematics education vol. 7, no. 1 8 taking mathematics reform to scale could involve a whole school or an entire district. however, whether schoolor district-wide, the process of instructional improvement across multiple classrooms involves design (cobb & jackson, 2011; cobb, mcclain, de silva lamberg, & dean, 2003; cobb & smith, 2008). tate (2004) has asserted the need to take an engineering approach to opportunity to learn in mathematics education. such an approach would involve a “learn to build” orientation that uses existing research to design more effective and equitable systems. this engineering orientation is reflected in the work of cobb and his colleagues involving efforts to design change at scale (cobb & jackson, 2011; cobb et al., 2003; cobb & smith, 2008). they acknowledge, however, that there are relatively few examples of successful design at the organizational level. moreover, according to cobb and jackson (2011), the history of large-scale improvement efforts that involved significant changes in teachers’ instructional practices is primarily one of failure. we contend that this unfortunate record is due in large part to the inability of research to inform the design and implementation of comprehensive systems of supports aimed at building and sustaining district and school capacity for instructional improvement. (p. 26) while the need to design instructional improvement is not limited to a particular school or district type, it is arguably most critical for urban settings. there is an urgent need to develop and test models that can be used for the design of more effective urban schools and districts (cobb & smith, 2008; tate, 2008a). while any model of school or district reform should involve the development of “testable conjectures about the constraints and affordances of the institutional setting” (cobb et al., 2003, p. 21), the specific conditions of urban schools are crucial to consider. as such, my purpose in this commentary is not to report on a particular design effort. rather, i seek to offer design considerations that are specific to urban schools. my intent is to contribute to theoryand model-building regarding urban mathematics education by outlining how the design process for mathematics reform might be influenced by the geospatial context. introducing the case: three rivers to help frame the discussion, i offer the following case of a school that i am calling three rivers.1 three rivers middle and high school is located in a large urban school district in the southern united states. the school is similar, in many respects, to other schools in the neighboring community and larger district. the                                                                                                                           1 some of the information about the school, district, and community is drawn from local news sources. however, to include these sources in the reference list would potentially compromise the anonymity of the school and teachers. for this reason, i indicate when information comes from local media outlets but do not list the articles in the reference list. rousseau anderson commentary journal of urban mathematics education vol. 7, no. 1 9 school population is 98% african american and 2% latina/o. eighty-nine percent of students in the school are categorized as “economically disadvantaged” by the state. and similar to several schools in the surrounding community, three rivers has not fared well under the current accountability system. student achievement, particularly in mathematics, has been significantly below district and state averages. the school has struggled to consistently keep pace with annual growth benchmarks established by the state (sometimes meeting the growth targets one year and missing them the next). comparable to other urban schools described in the mathematics education literature, instruction at three rivers does not, on the whole, reflect the reform goals of teaching for understanding that mathematics educators have been promoting for several years (cobb & jackson, 2013; gamoran et al., 2003; spencer, 2012). observations of mathematics classrooms reveal a general pattern of “traditional” instructional practices (e.g., teacher modeling of procedures followed by student independent practice). although the school has adopted curricular materials that support teaching for understanding, these materials are often supplemented (or supplanted) by worksheets or previously adopted textbooks that align more closely with the standardized test. in all fairness, however, there are glimpses of teaching for understanding at three rivers, but it has not yet taken hold in a systemic way. moreover, the practices associated with teaching for understanding are largely viewed by teachers as in conflict with preparing students for the high-stakes standardized tests⎯tests that tend to shape curricular, instructional, and assessment decisions through the school year. thus, i would argue that three rivers is an example of a school in need of a design. improvement of the quality of instruction will require that the localized and intermittent examples of teaching for understanding spread across the school and become part of the institutional fabric such that teaching for understanding becomes one of the taken-for-granted characteristics of the organization. mathematics reform in urban schools there are existing theories of how this systemic reform might be accomplished (gamoran et al., 2003; tate & rousseau, 2007). for example, cobb and smith (2008) have outlined a theory of action for designing schools and larger organizations for instructional improvement. their theory encompasses five components intended to provide the supports necessary to establish reform at an institutional level: (a) a coherent system of supports for instruction; (b) teacher networks; (c) mathematics coaches’ roles in supporting teacher learning; (d) school leaders’ practices; and (e) district leaders’ practices. as cobb and jackson (2011) note, their theory of action includes multiple components and interactions. they hypothesize that all of the components must be established for maximum instructional improvement. moreover, the improvement efforts aligned with these ac rousseau anderson commentary journal of urban mathematics education vol. 7, no. 1 10 tions should be based on “useful knowledge about the relations between the institutional settings in which teachers work, the institutional practices they develop in those settings, and their students’ mathematical learning” (cobb & smith, 2008, p. 249). understanding the impact of the institutional setting is arguably of particular importance when considering the design of instructional improvement in urban schools. for example, gamoran and his colleagues (2003) reported on the process of mathematics and science reform at six different sites located in various parts of the united states. the authors divided the six cases into three groups: sites that met the challenges of reform; sites that initially met the challenges of reform but did not demonstrate long-term growth or stability; and sites that failed to meet the challenges of reform. notably, three of the six sites were located in urban school districts. two of the urban sites made up the least-successful category⎯those cases where little progress was made toward reform. the third urban site fell into the middle category⎯reform was initially successful but did not sustain or grow over time. the bifurcation of the sites into successful or relatively successful suburban or rural-suburban sites and less successful urban sites is noteworthy and points to the need to take seriously the specific factors shaping reform in urban schools. according to gamoran and colleagues (2003), the cases of the two unsuccessful urban sites “point to the difficulties in meeting the challenges so prevalent in urban environments” (p. 104). in particular, gamoran and his colleagues highlight the obstacles faced by teachers in these schools in light of various district policies, particularly those related to assessment. the role of testing constrained both time and curriculum in the urban sites as they “contain a built-in tension between accountability systems and building resources to support teaching for understanding” (gamoran et al., p. 169). addressing this tension is one of the challenges of design for mathematics education in urban schools (cobb et al., 2003). another issue of particular importance for the reform of urban schools involves the characteristics of the teacher population and the capacity that teachers in urban schools bring to the reform process. teacher preparation, credentialing, and experience are important factors with regard to instructional effectiveness (darling-hammond, 2010; rousseau-anderson & tate, 2008; tate, 2008b). yet, these characteristics of teacher capacity, particularly in urban schools, are influenced by a variety of factors, including teacher shortages, teacher turnover, and the presence of alternative pathways to teaching. shortages of qualified teachers are clear impediments to instructional improvement in mathematics and are of particular concern in urban schools. according to ingersoll and perda (2009), “contemporary educational thought holds that one of the pivotal causes of inadequate school performance is the inability of schools to adequately staff classrooms with qualified teachers, especially in fields rousseau anderson commentary journal of urban mathematics education vol. 7, no. 1 11 such as mathematics and science” (p. 1). in fact, ingersoll and perda report that, of the school subjects included in their national study, mathematics experienced the most serious hiring and recruitment problems. the results indicated that 54% of secondary schools had job openings for mathematics teachers and about 41% of these indicated serious difficulties filling these openings. in other words, 22% of all secondary schools in the national sample had difficulties filling mathematics positions with qualified teachers. yet, these shortages were not driven as much by lack of production of new teachers as by turnover of existing teachers: “turnover is a major factor behind the problems that many schools have staffing their classrooms with qualified mathematics, science, and other teachers” (ingersoll & merrill, 2013, p. 23). moreover, we know that turnover is not equitably distributed across states, districts, or schools within districts. according to ingersoll and merrill (2013), 45% of teacher turnover of all public schools in 2004−05 took place in just 25% of schools. high-poverty, high-minority, urban, and rural schools had the highest rates of turnover. yet, as noted in the cases highlighted by gamoran and colleagues (2003), teacher turnover can disrupt professional communities in the process of reform. thus, as we consider the factors shaping mathematics reform in urban schools, the role of teacher turnover must be part of the model. one strategy for addressing turnover and the subsequent teacher shortfall in hard-to-staff schools and districts has been to provide alternative pathways to teaching. these pathways reduce the requirements for initial entry to teaching, allowing teachers to begin teaching before completing all of the requirements for licensure (clark et al., 2013; darling-hammond, holtzman, gatlin, & heilig, 2005; heilig & jez, 2010). one of the primary examples of these alternative pathways is teach for america (tfa). according to its website (www.teachforamerica.org), the organization serves 48 sites, including several of the nation’s urban school districts. while the overall number of tfa corps members is small, relative to the larger population of u.s. teachers, the expedited pathway to teaching represented by tfa and the concentration of the program in high-minority, low-income school districts makes tfa a noteworthy entity in any discussion of reform and teacher capacity in urban schools. it is beyond the scope of this commentary to review the recent research on the effectiveness of teach for america and other alternative pathway programs. however, it is worth noting that reports of teacher effectiveness have been mixed. some researchers have concluded that tfa corps members are making a positive impact on mathematics teaching and learning in the high-needs schools that they serve and are more effective than teachers from other alternative pathway programs or traditional teacher licensure programs (clark et al., 2013; kane, rockoff, & staiger, 2008). other researchers have not found such positive outcomes when comparing tfa corps members to other teacher populations, particu rousseau anderson commentary journal of urban mathematics education vol. 7, no. 1 12 larly when tfa teachers are compared to completers of traditional teacher preparation programs. according to darling-hammond and her colleagues (2005), large, well-controlled, longitudinal studies have shown that “teachers who entered teaching without full preparation…were significantly less effective when they started than fully prepared beginning teachers working with similar students” (p. 47). while the results have not been entirely straightforward, two points are worth highlighting when considering how programs such as tfa might fit into models of school-level mathematics reform. first, even in studies in which the effectiveness of tfa corps members is described as positive, relative to other populations, their longevity is not comparable to traditionally prepared teachers. for example, in a study by kane and colleagues (2008) of new york city schools, the 5-year retention rate for traditionally certified teachers in the population was approximately 50%, compared to only 18% for tfa corps members. thus, one consideration with regard to the role of tfa in urban mathematics education involves the issue of turnover. yet, a second issue raised by the research on tfa and other alternative pathway programs is the significance of the local context and larger teacher supply in that setting. for example, darling-hammond and colleagues (2005) note that tfa operates in districts that hire many uncertified teachers. “our analyses suggest that in contexts where many teachers have little preparation and where there is high turnover, tfa may make a positive contribution” (p. 21). darlinghammond and her colleagues observe that, because tfa corps members make a two-year commitment, they can actually provide more stability than the existing teacher pool in some urban districts. similarly, while kane and his colleagues (2008) acknowledged the much higher turnover rate of tfa corps members over 5 years, they argued that the relative effectiveness of tfa teachers potentially offsets this turnover. in both cases, these conclusions point to the importance of understanding teacher capacity in context. the interplay of teacher shortages, teacher turnover, and the strategies that are employed to address those shortages, particularly in the schools serving low-income students and students of color, contribute to a complex picture with regard to teacher capacity and mathematics reform in urban schools. models of urban mathematics education must account for this complexity. moreover, design strategies for organizations must also be sensitive to these forces shaping teacher capacity. a return to three rivers similar to other urban schools, the teacher supply in the district in which three rivers is located reflects national patterns of shortages in secondary mathematics. for example, according to a local media source, school district officials rousseau anderson commentary journal of urban mathematics education vol. 7, no. 1 13 reported that over 80 vacancies in mathematics were posted in the 2013−14 school year with only 20 certified applicants. moreover, this number of vacancies does not include local charter schools and the schools that are operated by the state, as these schools are responsible for their own hiring. in fact, at one point during the 2013–14 school year, a local organization that serves as an umbrella for teacher recruitment for the area listed 162 open positions in mathematics within its partner schools. the district in which three rivers is located, similar to other urban districts, has turned to alternative providers to fill teacher positions. specifically, national organizations such as teach for america and the new teacher project operate as providers of local teachers, with tfa as the largest of the alternative route programs. while a state evaluation of teacher preparation highlights the effectiveness of tfa relative to other teacher preparation programs (including most of the state’s traditional university-based programs), the state report also indicates that only 37% of the tfa corps members included in the evaluation continued teaching past the 2-year teaching obligation. these larger trends within the district with regard to teacher supply and credentialing in mathematics are reflected in the teacher population at three rivers middle and high school. at the start of the 2013−14 school year, only three of the nine mathematics teachers held a regular teaching license (one teacher held a license in secondary mathematics and the other two were licensed for the elementary and middle grades). four of the nine teachers who started the 2013−14 school year teaching mathematics were tfa corps members.2 in addition to the tfa corps members, three teachers were employed on a “transitional” license. this emergency credential allows individuals with undergraduate degrees to teach for up to 3 years while completing the requirements for licensure. with regard to experience, seven of the nine teachers who started the 2013–14 school year at three rivers were within their first 3 years of full-time teaching. thus, the majority of the school’s mathematics teachers had not completed traditional teacher preparation programs before beginning teaching, and, as a whole, they were relatively inexperienced. in addition, the school reflected patterns of teacher turnover evident in urban districts. two of the teachers who began the 2013−14 school year left mid-year. a licensed teacher who had previously taught at three rivers replaced one of the teachers. a tfa corps member, who had been teaching at the elementary school level, replaced the other teacher (bringing the number of tfa corps members to five). in this way, several of the larger trends with regard to                                                                                                                           2 one of the tfa corps members had completed an undergraduate teacher training program and held a regular teaching license. as such, this individual is represented in the count of tfa corps members and in the number of licensed teachers. rousseau anderson commentary journal of urban mathematics education vol. 7, no. 1 14 teacher supply, teacher turnover, and alternative pathways to teaching are evident in the case of three rivers. before considering what this means for the task of design, i should also raise additional geospatial considerations relevant to this case. three rivers middle and high school is located in a larger community, which i refer to, by the same name. the community of three rivers is an incorporated area of approximately 40,000 residents within a larger u.s. city. over a period of several years, the three rivers area has suffered from the loss of large plants that provided bluecollar jobs for many in the community. according to a local newspaper, the unemployment rate in the community is 17%; and 6 out of 10 children in the community of over 40,000 live within the federal definition of poverty. the community is served by three high schools. the graduation rates of these schools for the 2012−13 school year were 58%, 41.6%, and 84.1%. only one of the three schools approached the statewide graduation rate of 86.3%. these relatively low graduation rates are also reflected in post-secondary attainment in the community. according to the u.s. census bureau, in the zip code that encompasses most of three rivers, only 45% of the population of residents 25 years or older hold a high school diploma, and only 7.1% of residents hold a bachelor’s degree or graduate/professional degree. moreover, these post-secondary outcomes are consistent with achievement patterns in the three community high schools. average act composite scores for eleventh graders at the three schools during the 2012–13 school year were 14.6, 13.9, and 16.2, compared to the state average of 19.1. yet, the struggles of the three rivers schools are not only reflected in the relatively low graduation rates and levels of post-secondary attainment. under the current accountability system within the state, schools whose achievement levels put them in the bottom 5% statewide are subject to takeover. during the 2012−13 school year, 83 schools in the state were identified in the bottom 5%. of these 83 schools, 12 were located in the three rivers community. thus, although three rivers represents only about 0.6% of the state population, their schools made up over 14% of those in the lowest-performing bracket. in fact, 12 out of the 18 elementary, middle, or high schools located in three rivers were in the bottom 5% of schools statewide. building models of urban school reform in mathematics so, what does this mean for research on urban mathematics education? i submit that the case of three rivers, while far more complex than i have been able to describe in this space, points to some of the issues that models of urban mathematics education must be able to accommodate. design strategies and theo rousseau anderson commentary journal of urban mathematics education vol. 7, no. 1 15 ries of change for urban schools must account for these influences on the institution. first, efforts to understand mathematics reform in urban schools must take into account the role of policy at various levels. as cobb and smith (2008) note, school and district policies are part of the institutional setting that shapes teacher instructional practice. moreover, the case of three rivers is not unlike the urban sites described by gamoran and colleagues (2003) in the manner in which state policies regarding school and teacher accountability appeared to influence reform (or the lack thereof). however, it is important to note that the institutional settings of urban schools, such as three rivers, also intersect with broader policy initiatives that, while sometimes instantiated through state or district policies, can take on a life of their own. as lipman (2012) has previously outlined in this journal, for example, urban mathematics education operates within a larger (neoliberal) reform agenda. the increased presence of teach for america as well as the state takeover of low-performing schools can be considered as examples of this larger reform agenda within the local three rivers context. it would be impossible to ignore the role of these policy conditions in any theory or model of urban mathematics education that takes seriously the dynamics of reform in urban schools. moreover, the design of instructional improvement must be informed by knowledge of this policy context. in addition, i submit that the case of three rivers lends further support to tate’s (2008a) call for geospatial models of urban mathematics education. for example, while the issues of assessment and accountability likely affect schools statewide, the specific pressures experienced by teachers at three rivers middle and high school must be understood in the context of the surrounding community. twelve of the 18 schools in three rivers are already subject to state takeover, with several schools in the area being turned over to charter management organizations in the upcoming school year. this situation raises key questions related to the local school context and the institutional setting, and these questions would potentially impact the design of interventions at this site. in particular, what impact does the proximity of multiple takeovers have on teachers’ and administrators’ levels of comfort with respect to mathematics reform? given the school restaffing that accompanies takeover, how willing are teachers and leaders to shift from a test-driven, teacher-centered pedagogy in the midst of these stakes? my point in raising these questions is not to provide a justification for lack of reform. rather, my intent is to highlight the contextual factors that we must take into account as we build models of school change. only by considering the specific conditions of this school’s geographic setting can we fully understand the factors shaping mathematics reform and instructional improvement in this location. moreover, knowledge of the geospatial context also helps us to recognize the intergenerational factors shaping reform in this setting. for example, just as rousseau anderson commentary journal of urban mathematics education vol. 7, no. 1 16 the teacher capacity conditions cannot be understood without knowledge of teacher supply, the shortage of mathematics teachers must be situated in an intergenerational context. the larger school district in which three rivers is located serves over 100,000 students. a recent newspaper article reported that, of those, fewer than 6% of eleventh graders are considered “college ready” on the basis of act benchmarks. and, as previously noted, fewer than 8% of adults in the three rivers community have undergraduate degrees. in this setting, the mathematics teacher shortage is a predictable consequence of long-term under-education. it is perhaps not surprising, then, that less than half of the mathematics teachers at three rivers middle and high school are from the local area, and none are from the community itself. what impact does this lack of connection to the community have on mathematics teaching and learning at this school? and, more generally, how can we situate the intergenerational impact of chronically underperforming schools within models of school reform and the design of interventions? conclusion in summary, i argue that theories or models of urban mathematics reform must be sensitive to policy, situated geospatially, and attentive to the intergenerational influence of the local context. our efforts to design interventions for schools such as three rivers will potentially be limited in their long-term impact if considerations of these factors are absent. for example, models of mathematics reform at scale include professional development as a key component (cobb & smith, 2008; gamoran et al., 2003; tate & rousseau, 2007). yet, the design of professional development in settings such as three rivers must be informed by knowledge of the geospatial context in order to fully account for the factors shaping reform. each unique geospatial context raises particular design issues. how would we leverage professional development to promote mathematics reform in a high-pressure accountability context? what are the design considerations for professional development involving the various actors (teachers, school leaders, coaches) in a setting in which the accountability stakes are so high? similarly, a model that includes consideration of the contextual factors shaping teacher capacity raises particular design questions. given the interplay of teacher shortages, alternative pathways to teaching, and teacher turnover, how do we design effective professional development for urban schools? how do we design professional development opportunities that can leverage reform in a context in which teachers possess limited teaching experience or previous preparation? additionally, how can professional communities sustain themselves in the midst of substantial teacher turnover (gamoran et al., 2003), particularly given that some of the turnover is a common outcome of the alternative teaching pathways? finally, what are the intergenerational considerations of teacher capacity as they rousseau anderson commentary journal of urban mathematics education vol. 7, no. 1 17 shape urban mathematics reform? more to the point, what are the intergenerational solutions for issues of limited teacher capacity in urban schools? “place matters” in the study of urban mathematics education (hogrebe & tate, 2012). schools such as three rivers are situated at the nexus of a myriad of factors that influence the capacity to move from “existence proofs” of classrooms to whole-school implemention of mathematics reform. yet, as cobb and jackson (2011) assert, “the issue of how to support instructional improvement on a large scale continues to be under-researched. as a consequence, research can currently provide only limited guidance to district and school leaders who aim to improve the quality of mathematics teaching” (p. 6). thus, there is a need for research to support model development and testing to help us “learn to build” in urban spaces (tate, 2004, 2008a). until we construct more nuanced models that are sensitive to all of these factors operating in urban settings and their multiple interactions, we will continue to find that urban schools, such as three rivers, populate the “failed” category of urban mathematics reform (gamoran et al., 2003). what i have learned from the example of three rivers is the need for models and designs that take into account the fact that place matters. references clark, m., chiang, h., silva, t., mcconnell, s., sonnenfeld, k., erbe, a., & puma, m. (2013). the effectiveness of secondary math teachers from teach for america and the teaching fellows programs (ncee 2013-4015). washington, dc: national center for education evaluation and regional assistance, institute of education sciences, u.s. department of education. cobb, p., & jackson, k. (2011). towards an empirically grounded theory of action for improving the quality of mathematics teaching at scale. mathematics teacher education and development, 13(1), 6−33. cobb, p., & jackson, k. (2013). lessons for mathematics education from the practices of african american mathematics teachers. teachers college record, 115(2), 1−14 cobb, p., mcclain, k., de silva lamberg, t., & dean, c. (2003). situating teachers’ instructional practices in the institutional setting of the school and district. educational researcher, 32(6), 13−24. cobb, p., & smith, t. (2008). the challenge of scale: designing schools and districts as learning organizations for instructional improvement in mathematics. in t. wood, b. jaworski, k. krainer, p. sullivan, & d. tirosh (eds.), international handbook of mathematics teacher education (vol. 3, pp. 231−254). rotterdam, the netherlands: sense. darling-hammond, l. (2010). the flat world and education: how our commitment to equity will determine our future. new york, ny: teachers college press. darling-hammond, l., holtzman, d., gatlin, s. j., & heilig, j. v. (2005). does teacher preparation matter? evidence about teacher certification, teach for america, and teacher effectiveness. education policy analysis archives, 13(42), 1−51. gamoran, a., anderson, c., quiroz, p., secada, w., williams, t., & ashmann, s. (2003). transforming teaching in math and science: how schools and districts can support change. new york, ny: teachers college press. rousseau anderson commentary journal of urban mathematics education vol. 7, no. 1 18 heilig, j. v., & jez, s. j. (2010). teach for america: a review of the evidence. boulder, ca and tempe, az: education and the public interest center & education policy research unit. hogrebe, m., & tate, w. (2012). place, poverty, and algebra: a statewide comparative spatial analysis of variable relationships. journal of mathematics education at teachers college, 3, 12−24. ingersoll, r., & merrill, l. (2013). seven trends: the tranformation of the teaching force. philadelphia, pa: consortium for policy research in education ingersoll, r., & perda, d. (2009). the mathematics and science teacher shortage: fact and myth. philadehphia, pa: consortium for policy research in education kane, t., rockoff, j., & staiger, d. (2008). what does certification tell us about teacher effectiveness? evidence from new york city. economics of education review, 27(6), 615−631. lipman, p. (2012). neoliberal urbanism, race, and equity in mathematics education. journal of urban mathematics education, 5(2), 6−17. retrieved from http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/191/116 rousseau anderson, c., & tate, w. (2008). still separate, still unequal: democratic access to mathematics in u.s. schools. in l. english (ed.), handbook of international research in mathematics education (pp. 299−318). new york, ny: routledge. spencer, j. (2012). views from the black of the math classroom. dissent, 59(1), 76−80. tate, w. (2004). access and opportunities to learn are not accidents: engineering mathematical progress in your school. tallahassee, fl: southeast eisenhower regional consortium for mathematics and science education at serve. tate, w. (2008a). putting the “urban” in mathematics education scholarship. journal of urban mathematics education, 1(1), 5−9. retrieved from http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/20/9 tate, w. (2008b). the political economy of teacher quality in school mathematics: african american males, opportunity structures, politics and method. american behavioral scientist, 51(7), 953−971. tate, w., jones, b., thorne-wallington, e., & hogrebe, m. (2012). science and the city: thinking geospatially about opportunity to learn. urban education, 47(2), 399−433. tate, w., & rousseau, c. (2007). engineering change in mathematics education: research, policy, and practice. in f. lester (ed.), second handbook of research on mathematics teaching and learning (vol. 2, pp. 1209−1246). greenwich, ct: information age. developing preservice teachers’ multicultural mathematics dispositions journal of urban mathematics education july 2012, vol. 5, no. 1, pp. 31–43 ©jume. http://education.gsu.edu/jume dorothy y. white is an associate professor of mathematics education in the college of education at the university of georgia, 105 aderhold hall, athens, ga 30605, e-mail: dywhite@uga.edu. her research focuses on equity and culture in matheamtics education to prepare and support mathematics teachers of diverse student populations, and the development of models for collaboratie planning and teacher learning communities. eileen c. murray is a professor of secondary education in the school of education at the state university of new york at new paltz, 1 hawk drive, new paltz, ny 12651, e-mail: drmurraye@gmail.com. her research interests include the influence of reflective teaching cycles on teachers’ capactity to reflect on their practice, their mathematical knowledge, and their ability to teach all students. victor brunaud-vega is a doctoral candidate in the department of mathematics and science education at the university of georgia, 105 aderhold hall, athens, ga 30605, e-mail lbrunaud@uga.edu. his research interests include multicultural education, teacher collaboration, and teacher professional development using existing structures in schools. discovering multicultural mathematics dispositions dorothy y. white eileen c. murray victor brunaud-vega the university of georgia state university of new york at new paltz the university of georgia culture refers to the consistent ways in which people experience, interpret, and respond to the world around them; it represents the “ways of being” of a collective population…. culture is a feature of all human groups and is shaped by historical, social, political, economic, and even geographical factors…. additionally, culture can be reinforced through contacts with social institutions such as places of worship and schools. (marshall, 2002, p. 8) here is a well-documented relationship between culture and learning with studies highlighting how mathematics classroom cultures act as a context that supports or constrains different forms of knowledge (boaler, 2006; gutierrez & rogoff, 2003; nasir, hand, & taylor, 2008). nasir and colleagues argue, “mathematics classrooms are inherently cultural spaces where different forms of knowing and being are being validated” (p. 206). as u. s. public school populations become increasingly diverse (aud et al., 2011), we must prepare teachers to work effectively with all students especially in mathematics where too many black students are underperforming. according to kitchen (2005), mathematics teachers can be prepared to meet the needs of an increasingly diverse student population by learning how to understand and recognize students’ cultural backgrounds while engaging and challenging students in mathematics. there is an emerging literature that suggests establishing preservice teachers’ (psts) dispositions toward culture in mathematics helps them to understand that “no culture is monolithic; every culture consists of multiple subcultures” (leonard, brooks, barnes-johnson, & berry, 2010, p. 267) and that culturally responsive teaching (gay, 2000, 2002; ladson-billings, 2000; leonard, 2008) can t white et al. multicultural dispositions bullock, e. c., alexander, n. n, & gholson, m. l. (eds.). (2012). proceedings of the 2010 philadelphia and 2011 atlanta benjamin banneker association conferences – beyond the numbers [special issue]. journal of urban mathematics education, 5(2). 32 be used with all students. as schools and colleges of education are responsible for challenging prejudices and cultural biases of psts, and for finding strategies to shape these beliefs (sleeter, 2001), mathematics methods courses can provide an opportunity to explicitly address issues of culture and diversity. furthermore, because teachers’ dispositions toward students’ cultural background play an important role in their teaching practices and effectiveness (thornton, 2006), it is important to find ways to develop psts’ dispositions toward multiculturalism. in particular, as we prepare psts to educate black children, we need to look beyond the often documented “achievement gap” that has encouraged the idea that black students are deficient in mathematics. focusing solely on achievement fails to take into account how and why black students experience mathematics as they do and how mathematics learning and participation are racialized forms of experience (martin, 2007). when we focus on experience rather than achievement, we are better able to consider how we can prepare teachers to work effectively with not only all students but also specifically with black students. studies that document the relationship between teacher knowledge and increased student achievement fail to question “teachers’ dispositions and beliefs about who can or cannot learn mathematics, who is math literate and who is not, and why they believe what they do” (martin, 2007, p. 14). to understand how to best prepare teachers to teach black children, we have to think about dispositions and beliefs as well as how teachers act on their perceptions given that these actions will help shape the cultural, racial, and mathematical identities of black students. the study reported in brief here, suggests that a cultural awareness unit provides a reasonable starting point to examine psts’ dispositions to culture in mathematics. in this paper, we use written responses from the cultural awareness unit to describe psts’ multicultural mathematics dispositions. after a brief overview of the relevant literature, we define multicultural mathematics dispositions and its use as the conceptual framework undergirding this study. we then present the results of the study and propose implications for mathematics methods courses. relevant literature the literature on teaching for diversity puts forward that many forms of pedagogy are effective. teachers can incorporate multiculturalism (kitchen, 2005), social justice (gutstein, 2003; martin, 2003), or culturally responsive pedagogy into their classrooms. cultural responsive pedagogies use “the cultural knowledge, prior experiences, frames of reference, and performance styles of ethnically diverse students to make learning encounters more relevant to and effective for them” (gay, 2000, p. 29). however, many white teachers assume that culture is something other (non-white) people have (sleeter, 2000, 2001) and that white et al. multicultural dispositions bullock, e. c., alexander, n. n, & gholson, m. l. (eds.). (2012). proceedings of the 2010 philadelphia and 2011 atlanta benjamin banneker association conferences – beyond the numbers [special issue]. journal of urban mathematics education, 5(2). 33 mathematics is culture free. given that white teachers comprise the majority of public school teachers (aud, et al., 2011), we need to prepare all psts to understand the role of culture in the teaching and learning of mathematics. that is, psts need to become critically conscious of their own view of the world and cultural socialization in mathematics to understand how it affects their attitudes and behaviors toward people and cultures of other ethnic groups (gay, 2002; villegas & lucas, 2002). teacher disposition can provide a more comprehensive perspective towards the construction of a teacher’s identity in the context of a multicultural classroom. dispositions are “habits of mind including both cognitive and affective attributes that filter one’s knowledge, skills, and beliefs and impact the action one takes in classroom or professional setting” (thornton, 2006, p. 62). teacher educators must consider psts’ dispositions in order to help psts develop awareness and sensitivity to diverse learners. de freitas (2008) explored psts’ resistance to include issues of social justice in school mathematics. twelve psts analyzed their classroom practices and wrote self-study narratives as a form of critical reflection. through the narratives, they became aware of how the mathematics classroom represents a place of enculturation and a place where particular teacher and students’ identities are developed. some psts realized that their teaching practices repeated the same patterns of exclusion for underrepresented students, but were unable to recognize how their “teacher identities” (p. 53) were related to the sociopolitical framing and power relations in the mathematical classroom. de freitas found that self reflection had a profound influence on psts’ awareness of mathematics classroom cultures and recommended implementing some form of reflection in mathematics method courses. kidd, sanchez, and thorp (2008), in their study of 19 elementary psts’ dispositions and teaching practices, reported five experiences that impacted psts’ dispositions. these included readings concentrating on issues of race, culture, poverty, and social justice, critical reflection, and discussion. the researchers concluded that opportunities for critical reflection and discussions were essential to changes in dispositions. similarly, dunn (2005) studied elementary psts’ reflections on diversity activities in a mathematics method course. she found that in order to engage in critical reflection, psts need to experience a paradigm shift and a phase of disequilibrium while their beliefs about mathematics content, teaching, learning and students are challenged. she recommended challenging psts preconceived notions of diverse learners and encouraging critical reflection to broaden psts’ vision of teaching and learning mathematics. garmon’s (2004) case study of a white female pst identified important factors that facilitated multicultural awareness and attitudes toward diversity. more specifically, he identified three dispositional factors: (a) openness to diversity, (b) white et al. multicultural dispositions bullock, e. c., alexander, n. n, & gholson, m. l. (eds.). (2012). proceedings of the 2010 philadelphia and 2011 atlanta benjamin banneker association conferences – beyond the numbers [special issue]. journal of urban mathematics education, 5(2). 34 self-awareness/self-reflectiveness, and (c) commitment to social justice. garmon concluded that personal experiences with diversity coupled with the opportunity for processing them “may be critical to developing greater multicultural awareness and sensitivity” (p. 212). thus, dispositional factors may determine how ready a pst is to learn from his or her multicultural educational experiences and may predict the success of programs designed to develop psts’ awareness and sensitivity. the aforementioned literature suggests that teacher education courses that engage psts in critical reflection through discussions about culture and diversity promote awareness of the role of culture in the teaching and learning process. by challenging psts’ beliefs and awareness, teacher educators can support the development of psts’ critical reflection and responsive dispositions. responsive dispositions are necessary to prepare teachers to work in culturally diverse classrooms in order to “reverse the cycle of underachievement and educational disadvantage for diverse learners” (dunn, 2005, p. 144). however, we need to further identify the essential factors that comprise critical reflection and responsive dispositions in mathematics education. the multicultural mathematics dispositions framework extending the scholarship of garmon (2004) to mathematics education, we developed the construct multicultural mathematics dispositions (mcmd), which are characterized by three dispositional factors: openness, self-awareness/selfreflectiveness, and commitment to culturally responsive mathematics teaching. each construct of mcmd and its relation to garmon’s work is described in table 1. table 1 comparing garmon’s dispositional factors and mcmd garmon mcmd openness: “receptiveness (i.e., open-mindedness) to others’ ideas or arguments, as well as receptiveness to diversity” (p. 202). openness is receptiveness to the role of culture in teaching and learning mathematics, including being open to: (a) others’ cultures and arguments about teaching and/or learning mathematics, (b) the idea that different cultures may think about and do mathematics differently than oneself, (c) the inclusion of culture in mathematics classrooms, and (d) the value of using culturally responsive strategies to teach mathematics. white et al. multicultural dispositions bullock, e. c., alexander, n. n, & gholson, m. l. (eds.). (2012). proceedings of the 2010 philadelphia and 2011 atlanta benjamin banneker association conferences – beyond the numbers [special issue]. journal of urban mathematics education, 5(2). 35 self-awareness/selfreflectiveness: “having an awareness of one’s own beliefs and attitudes, as well as being willing and/or able to think critically about them” (p. 202). self-awareness/self-reflectiveness is perceiving the differences between one’s own culture and other cultures. this entails: (a) awareness of personal culture beyond recognizing differences from others; (b) awareness of personal beliefs about the influence of culture on teaching and learning mathematics or mathematics classroom culture; and (c) the ability to think critically about those issues. commitment to social justice: “a sense of social justice as a commitment to equity and equality for all people in society” (p. 202). commitment to culturally responsive pedagogy includes: (a) understanding students’ cultures and different ways of incorporate culture in teaching, (b) holding high expectations for all children, and (c) exposing children to rigorous mathematics. mcmd should encourage mathematics teachers to see mathematics as a cultural activity and their role as a mediator between students’ culture and mathematical learning. methods context the study took place in the college of education at a large southeastern university. three cohorts of psts (n = 76) enrolled in mathematics methods courses, taught by the first author, participated in the cultural-awareness unit. two of the cohorts were elementary education majors and the third consisted of middle school education majors. most psts were white females, except for one african american female, two african american males, and four white males. cultural-awareness unit the cultural awareness unit in this mathematics methods course allowed psts to express their mcmd as they learned how to teach mathematics. we focused on psts’ awareness of the role of culture in the teaching and learning of mathematics, stereotypes about who can do mathematics, and strategies to teach mathematics. the unit consisted of: (a) article search and reflection, (c) class discussions, and (c) post-discussion reflection. each part is discussed below with attention to how it enabled us to characterize the three constructs of mcmd among the participants. article search and reflection. this first assignment required psts to search for, read, and write a reflection on an article that addressed teaching and/or learning mathematics to students who are culturally different from them. prior to this assignment we did not discuss culture to avoid influencing psts views about cul white et al. multicultural dispositions bullock, e. c., alexander, n. n, & gholson, m. l. (eds.). (2012). proceedings of the 2010 philadelphia and 2011 atlanta benjamin banneker association conferences – beyond the numbers [special issue]. journal of urban mathematics education, 5(2). 36 ture as they wrote their reflections. having the freedom to select a culture required them to express self-awareness, given that they had to recognize their own culture to find differences between their culture and the culture in the article. reflecting on the strategies for infusing black culture in the mathematics classroom helped psts reveal their openness to the idea that different cultures may think about and do mathematics differently and their valuation of using culturally responsive strategies in mathematics. this assignment was expected to be enlightening and informative about psts’ points of view (taylor & sobel, 2001). class discussions. the second part of the unit included class discussions of the article reflections, the nature of mathematics classrooms, and how culture influences students’ access to mathematics. the discussions lasted two days and began with the kola nut welcoming activity (ukpokodu, 2002). ukpokodu suggests psts are more likely to engage in discussions when they are welcomed into a safe space. instead of kola nuts, we passed a bowl of hershey kisses to mirror a southern custom of welcome and to set the tone for the class discussion. next, each pst shared his or her article with the class. this allowed them to learn from each other and to further their openness to other ways of thinking about doing, learning, and teaching mathematics. the activities of the second day helped psts think critically about the role of culture in mathematics education. we began with marshall’s (2002) definition of culture and had the psts discuss their personal cultures by completing a cultural toolkit. in this activity they listed up to 10 characteristics to describe their culture (e.g., southern: likes sweet tea and football; values education and religion). they then reflected on whenever they had been part of the “other” or nondominate culture, how they felt and what they did. rather than present psts with a particular scenario to critique, we wanted them to talk about their personal experiences to further develop their self-awareness/self-reflectiveness. finally, psts thought about and discussed how teachers’ cultures, students’ cultures, and mathematics content interact in classrooms. we considered various stereotypes related to who is and is not perceived as being good at mathematics, the implications of stereotypes on students’ mathematics learning, and how teachers can combat some of these stereotypes. post-discussion reflection. in the last part of the unit, psts reflected on the previous activities and discuss how those activities influenced their views about the role of culture, teaching, and learning mathematics. these final reflections allowed us to note the aspects of the cultural awareness unit that seemed to influence psts’ views and commitment to teaching culturally responsive mathematics. participants the cultural awareness unit required preservice teachers to learn about teaching mathematics to a culture other than their own. students chose to learn white et al. multicultural dispositions bullock, e. c., alexander, n. n, & gholson, m. l. (eds.). (2012). proceedings of the 2010 philadelphia and 2011 atlanta benjamin banneker association conferences – beyond the numbers [special issue]. journal of urban mathematics education, 5(2). 37 about a variety of cultures, including black (african-american, caribbean, or african), hispanic and latina/o, native american, urban, low ses, and ancient egyptian. forty-two percent (n = 32) of the preservice teachers chose to explore black culture and were included in this study. thirty were white females, one white male, and one african-american female. in this paper, we describe the mcmd of the 32 psts as they provide insight into the preparation of teachers of black children. data analysis thematic analysis (braun & clarke, 2006) was used to analyze the data. thematic analysis is “a method for identifying, analysing and reporting patterns (themes) within data” (p. 79). first, we read all the article reflection papers to categorize the culture psts explored. then we selected only those papers related to black culture (n = 32) for further analysis. according to patton (2002), “purposeful sampling focuses on selecting information rich cases whose study will illuminate the questions under study” (p. 230). next, we read the article-reflection and post-discussion papers to identify the presence of the mcmd constructs. for each paper, we highlighted and analyzed passages that showed evidence of the constructs as understood from the psts’ perspectives. any coding questions or disagreements were discussed until we reached consensus. results we discovered passages relating to at least one of the mcmd constructs in 28 out of 32 psts’ written work. we were unable to identify any passages relating to mcmd in the remaining four psts work. for these psts, their comments were generic in nature. for example, when students talked about teachers in general, rather than personal experiences or understandings, we did not code the passage as mcmd: “presenting multiple views of mathematics allows the teacher to touch all types of learners.” these comments were vague, and seemed to be paraphrasing the article or another’s ideas, or did not display an understanding of culture or culturally relevant teaching. openness nineteen psts demonstrated some form of openness. they were open to learning about black culture and ways to include culture in the classroom. through this unit they learned more ways to think about and do mathematics: learning about other culture’s differences and appreciating these differences opens our minds up to the way children in our class perform math. it is so important to respect children’s cultural differences because as teachers we need to realize that they white et al. multicultural dispositions bullock, e. c., alexander, n. n, & gholson, m. l. (eds.). (2012). proceedings of the 2010 philadelphia and 2011 atlanta benjamin banneker association conferences – beyond the numbers [special issue]. journal of urban mathematics education, 5(2). 38 may have different views on things and may have been taught different mathematical methods. we were influenced by this article to not only embrace cultural differences, but to learn something new from someone’s culture whether it be the way they perform math, the clothing they wear, customs, etc. [hazel] most psts were open to classroom strategies that make connections between mathematics and real world contexts. they saw the value of creating mathematics problems from black students’ lived experiences: another strategy that was discussed in class was relating mathematics to everyday life and to the culture of your students. i found this extremely interesting because i have never thought of using hair braiding to learn mathematics. i think that as a teacher i will relate math to anything i can that will help my students learn…i think it does make a difference if you use real life ideas that students are interested in for the premises in math problems. [karen] in addition, some psts saw learning more about students’ cultures and using culturally relevant strategies would help their students learn mathematics and make them better teachers: “it is this type of connection with culturally relevant ideas and practices that will excite our students and help them become invested in their mathematical learning.” self-awareness/self-reflectiveness most psts (n = 24) were aware of and could identify aspects of their own personal culture, such as gender, race, socioeconomic status, and language. some mentioned family and community characteristics, including religion and country of origin as being a part of their culture. they identified their culture and how it was different than black culture. however, oftentimes their self-awareness was at the surface level and included assumptions about black students: one of the biggest differences between me and the students discussed in the article is that i am caucasian and they are african american. i am originally from a small town, and i grew up very far from the city. however, these children live and attend school in the middle of a bustling metropolis. i am used to slow paced, uneventful, quiet days where these students are used to the total opposite. in my community, church is a big part of our culture, but i feel it is an even bigger part in this african american community. through it is not stated in the article, i am assuming that the students come from a low socioeconomic status, which also differs from me. [karmen] self-awareness/self-reflectiveness includes the ability to reflect on personal beliefs about culture in teaching and learning mathematics. several psts reflected back on the mathematics classroom cultures they experienced as students of mathematics and became aware of different practices for teaching mathematics. white et al. multicultural dispositions bullock, e. c., alexander, n. n, & gholson, m. l. (eds.). (2012). proceedings of the 2010 philadelphia and 2011 atlanta benjamin banneker association conferences – beyond the numbers [special issue]. journal of urban mathematics education, 5(2). 39 one pst wrote about traditional mathematics teaching and how her parents stressed this culture at home: i attended public schooling where the teacher taught me particular algorithms as a way and the only way to compute. i was never allowed to be creative and invent my own way to solve a problem. it was always the teacher’s way or it was wrong, even if i arrived at the correct answer. my teachers were all about memorizing the formula instead of understanding the concept. i was rarely ever taught two different ways to solve one specific problem. my parents were also taught this way so they reinforced this idea of mathematics. however, many of the people mentioned in this article are from diverse backgrounds where they were taught a different way to think about mathematics. [alma] the few psts who thought critically about the role of culture in mathematics education, reflected on their personal culture and how minorities are viewed in society: the african american urban population also differs from myself in the way that they are viewed by others. i am a caucasian coming from a suburban, middle class area. i believe that society at large would consider someone like myself as having a good chance of having a successful and accomplished career and personal life. [kim] finally, psts reflected on how they were treated differently based on personal characteristics and the role of stereotyping. the following excerpts exemplify how stereotyping in mathematics classrooms privileged or disadvantaged psts as mathematics students: i was usually in gifted or accelerated mathematics classes, so my teachers were continuously challenging me to go beyond the basic answer and understand my thought processes behind it. i feel that because classmates and i were caucasian, many of our teachers assumed we could handle the workload that we were given, even if some of us were not academically adept in that regard. [rebecca] i really came to realize how i have been stereotyped and even how that happens to many of my friends…it seemed that some teachers came to expect less from me because i played sports. sometimes i would play into this and not live up to my potential. my favorite teachers were the ones that often pushed me the hardest and had high expectations for me…i want to try and have an environment that doesn’t really rely on stereotypes. i know that it will be up to me to monitor myself so that i do not stereotype my students. i also know that the less i stereotype the better the environment will be for learning math. [bob] commitment to culturally responsive mathematics teaching ten psts expressed commitment to infusing issues of black culture into their future mathematics classrooms. they wrote that they would teach mathemat white et al. multicultural dispositions bullock, e. c., alexander, n. n, & gholson, m. l. (eds.). (2012). proceedings of the 2010 philadelphia and 2011 atlanta benjamin banneker association conferences – beyond the numbers [special issue]. journal of urban mathematics education, 5(2). 40 ics by including: students’ own strategies, students’ culture, or culturally responsive strategies to teach mathematics. as judy noted, “i feel that these activities would have really benefited me as a learner. i plan to use one or some of these activities in my future classroom.” others committed to abandoning the traditional way they were taught mathematics. leslie’s quote is an example, “rather than following ineffective methods that my mathematics teachers used, i plan on using strategies from this article to benefit each of my unique students.” conclusions and implications the literature on preparing psts to teach culturally diverse students highlights the need for critical reflection and discussion of issues of diversity (kidd et al., 2008; leonard & evans, 2008), and the importance of developing multicultural dispositions (dunn, 2005; garmon, 2004). our unit was designed for psts to make explicit their multicultural mathematics dispositions. the purpose of this study was to examine preservice teachers’ mcmd during a cultural awareness unit in a mathematics methods course. we exposed them to culturally different ways of doing and teaching mathematics, asked them to reflect on their own mathematical experiences (de freitas, 2008; dunn, 2005), and the ways race, class, and culture influence learning (kidd et al., 2008). the class discussions allowed psts to share what they learned and invited others to ask questions or talk about their own article and experiences, which supported mcmd. similar to garmon’s (2004) finding, our psts illustrated openness to new understandings about black culture, strategies to connect black culture with mathematics, and the importance of culturally responsive strategies. de freitas (2008) notes that reflecting on mathematics experiences help psts “understand the intersections between their experiences in school mathematics and the cultural framing of those experiences” (p. 50). in this study, most psts were willing and capable of thinking about how their cultures differ from black culture, and how their experiences in mathematics classrooms influenced their views about what it means to do and learn mathematics. further, most psts saw the value of using culturally responsive strategies in the classroom, but only ten were ready to commit. therefore, while several psts’ were self-aware about their own culture, their self-awareness/self-reflectiveness remained superficial and included assumptions about black students. the unit made us aware of such preconceived notions of black students, but additional research is needed to understand how psts’ comments could be situated in mcmd and what additional coursework could help develop mcmd. mathematics methods courses provide an opportunity to develop mcmd by tackling issues of culture, diversity, and mathematics. psts need direction to understand the multiple layers of culture, the cultures surrounding them and those white et al. multicultural dispositions bullock, e. c., alexander, n. n, & gholson, m. l. (eds.). (2012). proceedings of the 2010 philadelphia and 2011 atlanta benjamin banneker association conferences – beyond the numbers [special issue]. journal of urban mathematics education, 5(2). 41 they will create when they teach. otherwise they will continue to create classroom cultures and engage in classroom practices that perpetuate limited opportunities and barriers for students to learn and do mathematics. this study highlights the potential of mcmd in the preparation of teachers in general, and mathematics teachers of black students in particular. our cultural awareness unit provides a reasonable starting point for the development of productive practices and habits of mind in cohorts of psts. in the future, we would like to build on this initial study to better understand what the presence, or absence, of certain constructs might tell us about psts. in this study, we only focused on the presence of mcmd. our analysis suggests that there are levels of openness, self-awareness/self reflectiveness, and commitment. in particular, we need a better understanding of how we might think about different levels of each construct and what these levels can tell us about psts’ experiences, needs, and potential as future mathematics teachers of black children. at the symposium the conversations at the symposium allowed us to learn about current research. the mcmd framework was well received with several attendees stating that it would allow them to think more critically about the comments they hear from psts and classroom teachers. additionally, many others expressed an interest in using the unit in their methods classes. our presentation, and the presentations of others, highlights the need for researchers and educators to define what we want black children to know and do in mathematics and the cultural norms that need to exist in mathematics classrooms to achieve these goals. we need to support the education of black children and mcmd provides a tool to achieve this. references aud, s., hussar, w., kena, g., bianco, k., frohlich, l., kemp, j., & tahan, k. (2011). the condition of education 2011 (nces 2011-033). united states department of education, national center for education statistics. washington, dc: u.s. government printing office. boaler, j. (2006). students in an urban high school learn to appreciate diversity and respect multiple viewpoints as they help one another succeed in mathematics. educational leadership, 63(5), 74–78. braun, v., & clarke, v. (2006). using thematic analysis in psychology. qualitative research in psychology, 3, 77–101. de freitas, e. (2008). troubling teacher identity: preparing mathematics teachers to teach for diversity. teaching education, 19(1), 43–55. white et al. multicultural dispositions bullock, e. c., alexander, n. n, & gholson, m. l. (eds.). (2012). proceedings of the 2010 philadelphia and 2011 atlanta benjamin banneker association conferences – beyond the numbers [special issue]. journal of urban mathematics education, 5(2). 42 dunn, t. k. (2005). engaging prospective teachers in critical reflection: facilitating a disposition to teach mathematics for diversity. in a. j. rodriguez & r. s. kitchen (eds.), preparing mathematics and science teachers for diverse classrooms: promising strategies for transformative pedagogy (pp. 143–158). mahwah, nj: erlbuam. garmon, m. a. (2004). changing preservice teachers’ attitudes/beliefs about diversity: what are the critical factors? journal of teacher education, 55, 201–213. gay, g. (2000). culturally responsive teaching: theory, research, and practice. new york: teachers college press. gay, g. (2002). preparing for culturally responsive teaching. journal of teacher education, 53, 106–116. gutiérrez, k. d., & rogoff, b. (2003). cultural ways of learning: individual traits or repertoires of practice. educational researcher, 32(5), 19–25. gutstein, e. (2003). teaching and learning mathematics for social justice in an urban, latino school. journal for research in mathematics education, 34, 37–73. kidd, j. k., sanchez, s. y., & thorp, e. k. (2008). defining moments: developing culturally responsive dispositions and teaching practices in early childhood preservice teachers. teaching and teacher education, 24, 316–329. kitchen, r. s. (2005). making equity and multiculturalism explicit to transform the culture of mathematics education. in a. j. rodriguez & r. s. kitchen (eds.), preparing mathematics and science teachers for diverse classrooms: promising strategies for transformative pedagogy (pp. 33–60). mahwah, nj: erlbaum. ladson-billings, g. (2000). fighting for our lives: preparing teachers to teach african american students. journal of teacher education, 51, 206–214. leonard, j. (2008). culturally specific pedagogy in the mathematics classroom: strategies for teachers and students. new york, ny: routledge. leonard, j., brooks, w., barnes-johnson, j, & berry, r. q., iii (2010). the nuances and complexities of teaching mathematics for cultural relevance and social justice. journal of teacher education, 61, 261–270. leonard, j., & evans, b. r. (2008). math links: building learning communities in urban settings. journal of urban mathematics education, 1(1), 60–83. retrieved from http://edosprey.gsu.edu/ojs/index.php/jume/article/view/5/5. marshall, p. l. (2002). cultural diversity in our schools. belmont, ca: wadswoth. martin, d. b. (2003). hidden assumptions and unaddressed questions in mathematics for all rhetoric. the mathematics educator, 13(2), 7–21. martin, d. b. (2007). beyond missionaries or cannibals: who should teach mathematics to african american children. the high school journal, 91(1), 6–28. nasir, n. s., hand, v., & taylor, e. v. (2008). culture and mathematics in school: boundaries between “cultural” and “domain” knowledge in the mathematics classroom and beyond. review of research in education, 32, 187–240. patton, m. (2002). qualitative research & evaluation methods (3rd ed.). thousand oaks, ca: sage. sleeter, c. e. (2000). epistemological diversity in research on preservice teacher preparation for historically underserved children. review of research in education, 25, 209–250. sleeter, c. e. (2001). preparing teachers for culturally diverse schools: research and the overwhelming presence of whiteness. journal of teacher education, 52, 94–106. taylor, s. v., & sobel, d. m. (2001). addressing the discontinuity of students’ and teachers’ diversity: a preliminary study of preservice teachers’ beliefs and perceived skills. teaching and teacher education, 17, 487–503. thornton, h. (2006). dispositions in action: do dispositions make a difference in practice? teacher education quarterly, 33(2), 53–68. http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/5/5 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/5/5 white et al. multicultural dispositions bullock, e. c., alexander, n. n, & gholson, m. l. (eds.). (2012). proceedings of the 2010 philadelphia and 2011 atlanta benjamin banneker association conferences – beyond the numbers [special issue]. journal of urban mathematics education, 5(2). 43 ukpokudu, n. (2002). breaking through preservice teachers’ defensive dispositions in a multicultural education course: a reflective practice. multicultural education, 9(3), 25–33. villegas, a. m., & lucas, t. (2002). preparing culturally responsive teachers: rethinking the curriculum. journal of teacher education, 53, 20–32. goals , aimed at forging links between the educational and commercial sectors journal of urban mathematics education july 2012, vol. 5, no. 1, pp. 1–7 ©jume. http://education.gsu.edu/jume maisie l. gholson is a recipient of nsf graduate research fellows program in stem education and doctoral student in mathematics education in the curriculum and instruction department in the college of education, at the university of illinois at chicago, 1040 w. harrison street, m/c 147, chicago, il, 60607; e-mail: mghols2@uic.edu. her research interests include the intersection of racial and mathematics identities, identity development through talk, and black children’s experiences in first-year algebra courses. erika c. bullock is a doctoral candidate in mathematics education in the department of middle-secondary education and instructional technology in the college of education, at georgia state university, p.o. box 3978, atlanta, ga, 30303; e-mail: ebullock1@student.gsu.edu. her research interests include exploring teacher professionalism and mathematics education policy from a critical postmodern theoretical (and methodological) perspective. she is a southern regional education board doctoral fellow and the assistant to the editor-in-chief of the journal of urban mathematics education. nathan n. alexander is a doctoral candidate in mathematics education in the department of mathematics, science, and technology at teachers college, columbia university and a research analyst at the research alliance for nyc schools, 285 mercer street, 3rd floor, new york, ny, 10003; email: nna2106@tc.columbia.edu. his research interests include academic peer networks, supplementary education, and self-efficacy in mathematics and science, specifically, among low income students and black and latina/o youth. guest editorial on the brilliance of black children: a response to a clarion call maisie l. gholson erika c. bullock nathan n. alexander university of illinois– chicago georgia state university teachers college, columbia university n this special issue of the journal of urban mathematics education (jume), we believe a new precedent was set in the mentoring and development of mathematics education scholars in which we were fortunate enough to play a role. as three doctoral students from three different institutions, we were given the extraordinary opportunity and unique responsibility to serve as co-editors for the proceedings of the 2010 philadelphia and 2011 atlanta benjamin banneker association (bba) conferences under the supervision of the editor-in-chief, dr. david w. stinson. relying on the previous experience of the assistant to editor-in-chief, erika c. bullock, we managed the process from the issuing of invitations to the conference speakers to the final round of revisions. during the editing process we found ourselves continually referencing the clarion call by dr. danny b. martin1 (martin, 2011), who urged attendees of the 1 unfortunately, dr. martin was not able to contribute to this special issue. the powerpoint slides from his presentation, and those of the other 2011 conference presenters, are available on the benjamin banneker association website (http://www.bannekermath.org). i about:/blank gholson et al. guest editorial bullock, e. c., alexander, n. n, & gholson, m. l. (eds.). (2012). proceedings of the 2010 philadelphia and 2011 atlanta benjamin banneker association conferences – beyond the numbers [special issue]. journal of urban mathematics education, 5(2). 2 2011 bba conference to engage in research and argumentation with the brilliance of black children as axiomatic. it was evident in our conversations and interactions with each manuscript that we did not see this call as an exercise in sloganeering; rather, it moved us individually and collectively. in other words, taking the brilliance of black children as an axiom seriously disrupted our sense of doing the work of mathematics education research related to black children. we understood this call as a challenge to our work as emerging scholars and, in this immediate case, as editors. although we acknowledged that editing (i.e., really conceptualizing) with black brilliance in mind would require modest shifts in our own thinking, we were unprepared for the intense and insidious gravity of our own deficit thinking, given we believed ourselves to be progressive thinkers, promising scholars, and black nonetheless. in this editorial we assume the responsibility to address some of the major issues that we have wrestled—and continue to wrestle—with as it relates to black mathematics education research. thinking axiomatically about black children in the talk, “proofs and refutations: the making of black children in mathematics education,” martin (2011) took a note from imre lakatos to describe the production of mathematical knowledge relating to black children as a series of axioms, conjectures, and counterexamples.2 according to martin, there are two lines of argumentation within the process of proof and refutation, about black children, both of which are equally problematic: (1) “black children are mathematically illiterate and intellectually inferior to white and asian children;” and, alternatively, (2) “black children are brilliant.” both are put forth as conjectures and lead mathematics education scholars to produce knowledge that maintains the racial hierarchy of mathematics ability, wherein black children are positioned at the bottom. these conjectures also place those who hold opposing views in a position to produce counterexamples that disprove the conjectures, resulting in a stalemate. 2 in his presentation, dr. martin defined these terms as follows: an axiom is a logical statement that is assumed to be true. axioms are not proven or demonstrated, but rather considered to be self-evident. axioms serve as starting points for deducing and inferring other truths. a conjecture is a proposition that is unproven but is thought to be true and has not been disproven. a counterexample is an exception to a proposed general rule. counterexamples are used to show that certain conjectures are false. gholson et al. guest editorial bullock, e. c., alexander, n. n, & gholson, m. l. (eds.). (2012). proceedings of the 2010 philadelphia and 2011 atlanta benjamin banneker association conferences – beyond the numbers [special issue]. journal of urban mathematics education, 5(2). 3 for example, in the first conjecture, knowledge is produced via examples that focus on the inferiority of black children’s mathematical ability, namely, the throngs of literature describing “achievement gaps” in standardized tests (see, e.g., u.s. department of education, 2003). of course, this conjecture places critical scholars who wish to refute such examples in the untenable position of proving that black children are not mathematically illiterate and inferior. in this case, as martin notes, knowledge production continues to reify notions of deficiency. contrary to how it may appear, the second conjecture, “black children are brilliant,” does not escape this pitfall. in this case, researchers concentrate their efforts on proving that black children are brilliant, yet such evidence and examples are simply refuted by a return to the status quo, where de facto constructions of black children as less than brilliant persist. the only escape from this quagmire is to treat the brilliance of black children not as a conjecture, but as axiomatic—a self-evident, starting point for deducing and inferring other truths. martin (2011) gives three key points for moving forward:  we must accept, and insist on, the brilliance of black children as axiomatic.  we must avoid the trap of having to prove that black children are brilliant.  we must avoid generating arguments, logic models, and counternarratives requiring proof that black children are not brilliant. we suggest that taking martin’s (2011) axiomatic stance calls into question many aspects of current scholarship. we first felt this need for questioning when discussing the lifeblood of black mathematics education scholarship—critical race theory (crt). one of the key tenets of crt (see jett, this issue) is the counternarrative (as opposed to the master or dominant narrative), which works to refute hegemonic claims of black inferiority. consider that much of the scholarship on black students in primary, secondary, and college in the last ten years relied on counternarratives generated by “successful” (read: brilliant) black children, adolescents, and young adults. despite honorable intentions, these counternarratives—and particularly the focus on successful black children—reinforced the notion that successful black children are the exception, not the rule. we certainly have the privilege of retrospection in reviewing this work and we recognize the imminent need at the time to push against the wave of deficit research on black children in mathematics education of the 1980s and 1990s (which continues today). we also recognize that the bba conference marked a moment with potential to pivot towards new forms of argumentation, that is, new axioms on which to build new truths about black children. our intent here is not to criticize our predecessors who have advanced the conversation regarding black children in mathematics by researching success and shining the light of research and teacher education efforts on black children. to the contrary, we extend our gratitude toward those who have come before us and look toward the future to consider how our gholson et al. guest editorial bullock, e. c., alexander, n. n, & gholson, m. l. (eds.). (2012). proceedings of the 2010 philadelphia and 2011 atlanta benjamin banneker association conferences – beyond the numbers [special issue]. journal of urban mathematics education, 5(2). 4 landscape of responsibility is changing as we take this sociopolitical turn in mathematics education and mathematics education research (gutiérrez, 2010). we do not purport to have answers; we are armed only with questions and an invitation to enter into a constant state of dis-ease by continually reflecting upon the questions: what do i really believe about black children and their abilities? how does my work reflect those beliefs? and given that black children are brilliant, how does this affect my research agenda? questioning prevailing axioms some may suggest that taking an axiomatic stance on black children’s brilliance is more intellectual posturing than substance. yet, we only need to be reminded of the current axioms on which knowledge about black children are built to see the need for change. the recent tragedy of trayvon martin serves as an unfortunate reminder of the axiom: “black children are criminals.” this position dictates that the very presence of a black male adolescent is cause for suspicion and presumed criminality—a criminal so heinous, in fact, that his presence poses an imminent threat against which the “victim” must “stand his ground.” this axiom of criminality mediates black children’s lived experiences in and out of the school system, as evidenced by the spring release of a u.s. department of education study that found black and latina/o students are three and a half times more likely to be suspended or expelled than white students (adams & robelen, 2012). the axioms that we choose have a material reality and, in trayvon’s case, a deadly reality. an equally pernicious axiom that remains unspoken, unquestioned, and often undetected is “white children are the standard.” under this axiom, black children’s test scores, behavioral and socioemotional patterns, as well as their dress and speech are subject to comparison of a fictitious, normalized white child. we (you may count yourself among us) have occasionally felt that pang of pride, however momentary, when a class or school of black children are highlighted on the nightly news with their khaki pants, collared shirts, and blazers and singing in unison. these black children create the optics of normalization; they appear to “do school” in an idyllic manner that is non-threatening to white middle-class sensibilities and is subconsciously part of our desire to see black children reflected as valuable and precious in larger society. this moment of pride comes with a cost—a clear subtext—namely, that black children are only valuable to the extent that they reflect whiteness. the axiom of white standardization harkens frantz fanon’s (1967) classic indictment in black skins, white masks: “for the black man [and woman] there is only one destiny. and it is [to become] white” (p. 10). thus, in the realm of mathematics education, the destiny of black children has been confined to closing a so-called “racial achievement gap.” by assuming gholson et al. guest editorial bullock, e. c., alexander, n. n, & gholson, m. l. (eds.). (2012). proceedings of the 2010 philadelphia and 2011 atlanta benjamin banneker association conferences – beyond the numbers [special issue]. journal of urban mathematics education, 5(2). 5 white children as our standard, we limit our imagination for generations of black children to nothing more than what most of us have been taught since childhood: to work twice as hard and to be considered just as good as the average white child. moreover, maintaining the axiom of white standardization requires that we cede to a set of beliefs that disallow black children from being standard-bearers, because they are inevitably lacking in some “objective” capacity. despite obvious problems, the axiom of standardization persists and is repeatedly invoked with every insinuation of an achievement gap. with an axiomatic stance of black brilliance, we consider achievement gap rhetoric to be a relic of deficit-research with diminishing returns and, thus, should be avoided, particularly, when used in the defense of black children in mathematics. a challenge to the community so, what does it mean for the next generation of mathematics education scholars to take an axiomatic stance of black brilliance? what possibilities does this stance create? what possibilities does it eliminate? our work is most often a reflection of who we are and what we value. even methodological approaches that claim little or no influence from researcher subjectivity are unable to avoid the residue of researcher bias. this evidence of our positionality as researchers often not easily seen but becomes evident through careful reflection. as we consider the prevailing axioms about black children, some of which we discussed above, we must assess our own complicity in the perpetuation and reproduction of these discourses through even those elements of our research that seem insignificant. the way that we select participants, frame interview or research questions, write up our research, solicit grant funding, or even focus on particular students during classroom observations are all influenced by and evidence of the axioms that we choose about teaching, learning, research, and black children. of course, this call to reflective caution is not limited to researchers. to the mathematics educator: how do you talk about black children in the presence of the young, white, middle-class, and female preservice teachers who overwhelmingly fill your classrooms (martin, 2009; walker, 2007)? do you dismiss subtle statements of deficiency as the comments of an exhausted and over-extended teacher? this call is one for accountability throughout the community of mathematics educators and researchers who are concerned with black children. we must contemplate our own missteps, gently critique those of our colleagues, and remain open to that critique. we believe that exercising this reflexivity in the development of axiomatic stances is different than merely shifting discourses or reframing the phenomenon of black children’s mathematics education experiences. an axiomatic stance of brilliance transcends the offensive position (e.g., proving black children are brilliant) and defensive position (e.g., refuting black children’s gholson et al. guest editorial bullock, e. c., alexander, n. n, & gholson, m. l. (eds.). (2012). proceedings of the 2010 philadelphia and 2011 atlanta benjamin banneker association conferences – beyond the numbers [special issue]. journal of urban mathematics education, 5(2). 6 illiteracy or inferiority) involved in framing and forecloses on the endless cycle of proving black children’s brilliance. a new axiom of black brilliance signals a new set of research questions and a new approach to mathematics teacher education that have nothing to do with black children’s achievement, as their ability and potential is no longer in question. although treating brilliance as axiomatic may forestall the surge of interview-based research in black mathematics education (given counternarratives are no longer necessary as proof), new possibilities abound. research questions can move beyond offensive or defensive positioning to locate and highlight unique characteristics of black students, teachers, and classrooms. in response to different questions, the axiom of brilliance may encourage the influx of re-discovered methodologies such as microethnography, sociolinguistics, and (critical) discourse analysis that make familiar (and strange) the circuitous rhythms of black mathematics classrooms and communities. finally, new questions and rediscovered methodologies will facilitate new modes of data representation and connections to the broader sociopolitical structures in the schools, communities, states, and nation. equity in the face of brilliance to conclude, we note that the axiomatic stance of brilliance indexes a new conceptualization of equity research, wherein inequity, disparity, and marginalization are perhaps backdrops, but not foci for our questions and arguments. equity based on the conjecture of black brilliance begs for measurement, comparison, and legitimation, whereas an approach to equity based on axioms demands attention and remuneration on principle, not evidence. in other words, we can no longer afford to make equity arguments on evidentiary grounds, we have learned from those before us that no amount of evidence or proof will be sufficient to mandate systemic change—there is no silver study. we believe that forceful moral argumentation is the way forward for systemic change under the axiom of brilliance. we arrive with these arguments, questions, and conclusions humbly and largely by virtue of the incredible opportunity to serve as editors. for that, we are particularly thankful. as we considered this idea of axiomatic brilliance through editing the manuscripts in this issue and discussing those manuscripts and this editorial, we encountered a crisis of our own assumptions. for three young scholars who embarked upon the doctoral process with hopes of changing the lived experiences of black children, their teachers, their communities, and their schools, this process of “rethinking [our] rethinking” (stinson, 2004, p. xx) left us feeling rather uneasy. despite our discomfort, we persist, knowing (or hoping) that our continual questioning of our own motivations will allow us to remain true to the people and communities that we intend to serve. gholson et al. guest editorial bullock, e. c., alexander, n. n, & gholson, m. l. (eds.). (2012). proceedings of the 2010 philadelphia and 2011 atlanta benjamin banneker association conferences – beyond the numbers [special issue]. journal of urban mathematics education, 5(2). 7 authors’ note we, the guest editors of the special issue of the journal of urban mathematics education, would like to thank dr. david stinson and dr. jacqueline leonard for extending this incredible opportunity. we appreciate your confidence in us and your ongoing support. references adams, c. j., & robelen, e. w. (2012, march 6). civil rights data shows detention disparities. edweek. retrieved from http://www.edweek.org/ew/articles/2012/03/07/23data_ed.h31/html. fanon, f. (1967). black skins, white masks. new york, ny: grove press. gutiérrez, r. (2010). the sociopolitical turn in mathematics education research. journal for research in mathematics education, 41. retrieved from http://www.nctm.org/publications/article.aspx?id=31242. martin, d. b. (2009). researching race in mathematics education. teachers college record, 111, 295–338. martin, d. b. (2011). proofs and refutations: the making of black children in mathematics education. lecture presented at the 2011 benjamin banneker association conference, atlanta, ga. stinson, d. w. (2004). african american male students and achievement in school mathematics: a critical postmodern analysis of agency. dissertation abstracts international, 66(12). (umi no. 3194548) united states department of education. (2003). status and trends in the education of blacks. (nces 2003034), by k. hoffman & c. llagas. project officer: t. d. snyder. washington, dc: national center for education statistics. retrieved from http://nces.ed.gov/pubs2003/2003034.pdf walker, e. n. (2007). preservice teachers’ perceptions of mathematics education in urban schools. the urban review, 39, 519–540. http://www.edweek.org/ew/articles/2012/03/07/23data_ed.h31/html http://nces.ed.gov/pubs2003/2003034.pdf microsoft word final zavala vol 7 no 1.doc journal of urban mathematics education july 2014, vol. 7, no. 1, 55–87 ©jume. http://education.gsu.edu/jume   maria del rosario zavala is an assistant professor in the department of elementary education at san francisco state university’s graduate college of education, 1600 holloway ave, burk hall 191, san francisco, ca 94312; email mza@sfsu.edu. her research interests include culturally responsive mathematics teaching, bilingual teacher education, and equity in mathematics education for latina/o and other marginalized youth.   latina/o youth’s perspectives on race, language, and learning mathematics maria del rosario zavala san francisco state university in this article, the author employs critical race theory (crt) and latino critical theory (latcrit) to examine latina/o students’ narratives of learning mathematics in a multi-lingual, urban high school. intersectionality as a tenet of latcrit is introduced as an important way to understand how students talk about the roles of race, language, and other central identities in their mathematics identity development as well as how they believe race may or may not matter in other people’s mathematics achievement. the author’s analysis illustrates how mathematics identities are co-constructed in relation to racial, linguistic, and gendered narratives of latina/o youth. in general, the study adds empirical evidence to previous research on the difficulties that high school students encounter when articulating how race matters to their own identities in academic subjects and highlights the nuanced ways latina/o students make connections between race, mathematical achievement, and schooling experiences in and through narratives of school success and failure. keywords: critical race theory, latina/o critical race theory, mathematics education, mathematics identity, urban education he underachievement of latina/o students in mathematics is repeatedly framed as “race-based” testing outcomes and seldom explored from the perspective of latina/o students’ lived experiences in mathematics classrooms (gutiérrez, 2008; martin, 2009). research that does account for the experiences of latina/o youth in mathematics classrooms rarely analyzes their perspectives as learners of mathematics and rarely positions them as possessing agency in their own mathematics learning (gutiérrez, 2008). although research and contemporary theories of learner identities suggest that both language and race matter in the development of mathematics identities (e.g., martin, 2007; spencer, 2009; turner, dominguez, maldonado, & empson, 2013), recent research on latina/o youth has focused primarily on linguistic identity (e.g., moschkovich, 2002; turner, gutiérrez, simic-muller, & díez-palomar, 2009; zahner & moschkovich, 2011). this focus often leaves the nuances of the intersections of race and language underresearched. in many ways, the picture of how latina/o youth might navigate their t zavala race, language, and mathematics   journal of urban mathematics education vol. 7, no. 1 56 mathematics identities remains incomplete, especially when articulating the roles of their racial and linguistic identities in their own learning. in this article, i draw on culturally responsive literature on latinas/os in mathematics education and research on students’ schooling experiences from critical race theory (crt) and latina/o critical race theory (latcrit) perspectives to explore being a learner of mathematics with seven latina/o students in two algebra i classrooms. using martin’s (2000) construct of mathematics identity to analyze interviews and focus group data, three research questions guided the analysis: 1. what perspectives do latina/o students have about their own mathematics identities? 2. how do they describe the roles of language and race in learning mathematics? 3. how do these perspectives inform how students display agency in their mathematics education? relevant literature the analysis presented here draws from two growing bodies of literature on latina/o students and mathematics teaching and learning: intersections of race and mathematics identity and intersections of language and mathematics identity. as noted, the issues of race and language tend to be compartmentalized in the literature: an either–or (race or language) approach rather than a both–and (both race and language) approach. there are a few exceptions (see, e.g., gutiérrez, 2002; gutstein, 2003), but more studies are needed to illustrate the nuanced ways in which latina/o students navigate the two identities simultaneously. racial identities in learning mathematics within the field of mathematics education, researchers and scholars have documented students’ perspectives on mathematics and larger discourses of who can be successful in mathematics (see, e.g., barajas-lopez, 2009; boaler & greeno, 2000; jansen, 2008; martin, 2000; mcgee & martin, 2011). esmonde (2009), in a recent literature review, noted that although identity (broadly defined) has been increasingly noted in mathematics education research, researchers most often have not dealt adequately with the role of socially constructed identities such as gender, race, sexual orientation, or ability labels. in general, researchers tend to either ignore these identities or to position them as pre-determining factors in student learning. though scholars have researched the role of racial and ethnic identities of latina/o youth in a broad school context (see, e.g., barajas & ronnkvist, 2007; valenzuela, 1999), few studies in mathematics education have examined the intersections of racial identities and learning mathematics for latina/o youth (notable exceptions include gutiérrez, 2002; gutstein, 2003). fewer of these stud zavala race, language, and mathematics   journal of urban mathematics education vol. 7, no. 1 57 ies employ methodologies that privilege student voices (few exceptions inlude, jilk, 2011; gutiérrez, willey, & khisty, 2011).1 jilk’s (2011) study provides an exception as she investigated the mathematics identities of latinas at railside high school, where mathematics was taught through complex instruction (cohen, 1994; cohen & lotan, 1997). focusing on the case of amelia, a student who emigrated from mexico, jilk situates salient identity as negotiated across multiple communities of practice. jilk used a narrative approach to discuss how amelia’s self-description as a “liberal”—someone who willingly expressed her opinions and authored her own destiny—allowed her to choose to participate in the way mathematics was taught at her school because the practices aligned with this salient piece of her identity. in her analysis, jilk demonstrated how amelia’s constructed herself as a liberal in response to traditional cultural values associated with being a woman in mexico. jilk argued that amelia displayed agency by participating in her mathematics class and acting on the perceived intersections of her identity as a liberal and the practices found in her classroom at railside, which included, for example, actively engaging with others in collective problem solving. jilk’s (2011) study is a reminder that what students find most salient about their identities may not be the neatly packaged macro-labels that are often of interest to researchers: race, class, gender, and so on. however, there is still a need to understand how students perceive these dimensions as they negotiate their mathematics educations (martin, 2009), given that they have been documented to be real barriers to school success, motivational factors for academic achievement, and catalysts for social action such as school walk outs (mcgee & martin, 2011; rivas & chavous, 2007; solórzano & bernal, 2001; steele, 1997). scholars have documented the stereotypes that latina/o students face in schools that serve to position them as violent, illegal, and alien (see, e.g., solórzano, 1997). at the same time, stereotypes around asian students’ high achievement in mathematics simultaneously essentialize asian students (lee, 2009), as well as position latina/o students as less capable (martin, 2009). some of the research by scholars who focus on african american students has addressed important intersections between discourses of african american student achievement and the success of african american students in mathemat                                                                                                                 1 scholars in the united states have assumed various positions concerning whether the label “latino” constitutes a racial, cultural, ethnic, or other identity. some scholars have also considered different ways to write about the intersectionality of multiple identities (i.e., gendered, political, historical, situated) that can describe the experiences of latina/o people (anzaldúa, 2007; valle & torres, 2000). those scholars concerned with this plurality note that the field has yet to capture the “fluid and transformable” essence of latina/o identities (estela zarate, bhimji, & reese, 2005, p. 97). in this study, i use race and ethnicity where adopted by scholars in discussing their literature, but adopt the language of race in my own analysis (see lopez, 1997). zavala race, language, and mathematics   journal of urban mathematics education vol. 7, no. 1 58 ics (see, e.g., russell, 2013). martin’s (e.g., 2000, 2006, 2007) body of research on race and mathematics identities is influential in the field of mathematics education. prior to his work, mathematics education, in general, lacked a nuanced discussion of racial identity and mathematics achievement. martin’s multilayered framework examined the multiple contexts that affect african american students’ socialization into mathematics; it pushed scholarship in mathematics identity to examine the historical positioning of students as people with racial identities who navigate racial stereotypes and racialized hierarchies of success (including racialized mathematics experiences). in extending this body of research, mcgee and martin (2011) examined how undergraduate black mathematics and engineering majors navigated stereotypes of achievement. they noted that despite a preponderance of evidence that stereotype threat can negatively affect intellectual performance, “little is known about how black students in particular manage racial stereotypes” (p. 1349), including using them as motivating factors. mcgee and martin argued that students constructed blackness on their own terms; therefore, the process of navigating stereotypes was not predetermined. rather, students used the negative stereotype as the impetus to display positive agency and defy it, perhaps needing to “prove something to these people” (cory, interview in mcgee & martin, 2011, p. 1367). their research provides a significant knowledge base for examining black student success in mathematics. nonetheless, there is a lack of similar research for latina/o youth. gaps in the research addressing how race affects latina/o students’ mathematics learning may be due, in part, to difficulties students face in describing race in relation to one’s own learning when the dominant discourse suggests race is not a barrier to one’s ability to learn. in her ethnography of a multi-lingual and multiethnic california high school, pollock (2004) elaborated on what she calls “the reality of race’s fluctuating relevance to [students’] own relationships and lives” (p. 47). pollock’s analysis showed that neither adults nor students consistently spoke to a singular way that race mattered in navigating school. the participants’ comments indicated that race’s relevance to their lives was complex. on the one hand, when asked how race impacted their lives, students might have offered statements such as “we are all the same” or “race is important” without elaboration. on the other hand, their everyday talk with peers, about peers, and with teachers revealed deviations from the racial scripts of how race was or was not supposed to matter. an important take away from pollock’s detailed ethnography is that race talk was complicated and situational for students and teachers, and conspicuously absent from teacher-to-student talk even though teacher-student conflict “felt” racialized to students (p. 61). in my prior study in which latina/o youths self-reported on learning mathematics (zavala, 2009), participants portrayed factors influencing mathematics zavala race, language, and mathematics   journal of urban mathematics education vol. 7, no. 1 59 achievement as colorblind (bonilla-silva, 2006), or not influenced by race. students argued that race should not be seen as a barrier to one’s ability to learn mathematics. student participants shared the view that race does not determine how well you do in school. as freshman yenni put it during the focus group session, “we are all just human beings (zavala, 2009, p. 38). one student described his racial identity in a more complex way, and also provided an example of how u.s.-born status may intersect with racial identity as students navigate their mathematics identities. andrew, an english-dominant, u.s.-born, latino student with mexican parents described how he negotiated feeling like a “white guy” when compared to the dominant forms of “being mexican” he perceived in his school: well, i don’t show a lot of mexican stuff, at all. being mexican, i am mexican, but i just don't wear the los angeles t-shirts and baggy jeans and stuff like that, like all the mexicans do here. i’m just another type of mexican, that looks like, i guess, a white guy. yeah, like being a mexican-american, but myself and mostly leaning towards the american part, i guess. (zavala, 2009, p. 15) andrew addressed intersections of “being himself” with racial identities and his identity as an american. he noticed that he did not look like what he perceived to be the “typical” mexican at his school. however, he refused to choose between his two identities, claiming both his mexican and american identities, for which he evoked the dominant discourse that americans are white. andrew’s case highlights the value of intersectionality as an approach to analyzing how latina/o students make sense of their experiences as learners. extant literature in mathematics education does not address cases like andrew enough. thus, more research is necessary to further understand intersections of racial identity and mathematics identity for latina/o youth. linguistic identities and learning mathematics according to research literature on latina/o students who are emergent bilinguals (eb),2 students can be more engaged in mathematics and have greater access to mathematics when they work in their first language (gutiérrez, 2002; gutstein, lipman, hernandez, & de los reyes, 1997; khisty, 1995; moschkovich, 2002, 2007; turner et al., 2013; zahner & moschkovich, 2011). however, few of                                                                                                                 2 emergent bilingual refers to students who are dominant in a language other than english and whom schools may classify as english learners. i use emergent bilingual instead of english learner to acknowledge the language resources the student does possess, rather than positioning students as lacking in language. as garcía (2009) wrote, “categorizing children as leps (limited english proficient) or eps (english proficient) is a dubious construction that misleads educators and that robs emergent bilinguals of languaging and educational possibilities” (p. 323). zavala race, language, and mathematics   journal of urban mathematics education vol. 7, no. 1 60 these studies focus on how eb latina/o youth learn mathematics in a high school setting. in gutiérrez’s (2002) study of a high school mathematics department with a track record of success for latina/o students, she found the department’s success could be attributed, in part, to policies that promoted learning mathematics in spanish for students who preferred it. she also found that strategies typically used in elementary school classrooms (e.g., having students work in groups) were equally successful in this high school. according to moschkovich (2002), latina/o eb students do not use two languages simply as a tool to translate vocabulary from one language to another. rather, she argued that the field should assume a sociocultural stance toward learners, viewing them as participants in multiple discourse communities. such a focus counters deficit notions of latina/o eb students as lacking english, and opens up space for analyzing their mathematics identities as people who bring resources to learning mathematics, such as ways of communicating in other discourse communities. a sociocultural perspective of eb students also allows for consideration of how their linguistic identities can function across settings, and the intersection of linguistic identities and mathematics identities. mathematics education researchers who focus on latina/o youth have yet to examine the intersections of linguistic and mathematics identities for latina/o youth who are english dominant. too often they focus on barriers to learning rather than differential access to discourse. in related research, scholars have shown that bilingual students who are highly proficient or dominant in english may still use spanish as they engage in mathematics (see, e.g., zahner & moschkovich, 2011). zahner and moschkovich also suggest that students may draw on hybrid linguistic identities, as a spanish and english speaker, to negotiate their mathematics identities. however, questions remain regarding how latina/o students who do not speak spanish perceive the role of language in learning mathematics (i.e., if an english-only identity matters for them in multi-lingual classrooms). moreover, prior research on linguistic identity most often does not include student voice. with few exceptions, the literature does not include voices of latina/o youth narrating their own experiences as mathematics learners. in addition, the research on racial identity does not sufficiently examine its intersections with language. language is a consistent theme in research on latina/o youth, suggesting a need to examine further the intersections of language and race in the mathematics identity development of latinas/os. the study detailed here contributes to scholarship at the intersection of racial and linguistic identities of latina/o youth by analyzing mathematics identity through the theoretical lenses of crt and latcrit. in the following section, i provide the definition for mathematics identity on which i base my analysis. i also describe how i draw upon crt and latcrit as theoretical perspectives to analyze the mathematics identities of latina/o youth. zavala race, language, and mathematics   journal of urban mathematics education vol. 7, no. 1 61 a critical lens on latina/o mathematics identity and agency defining mathematics identity this study builds on theories that allow examination of the intersections of self, place, and discourses of race and language. martin (2006) defined mathematics identity as: dispositions and deeply held beliefs that individuals develop, within their overall self-concept, about their ability to participate and perform effectively in mathematical contexts and to use mathematics to change the conditions of their lives. a mathematics identity encompasses a person’s self-understanding of himself or herself in the context of doing mathematics (i.e., usually a choice between a competent performer who is able to do mathematics or an incompetent performer unable to do mathematics, but often flowing back and forth). (pp. 206–207) though martin does not draw explicitly on crt or latcrit in his work, his definition is an important starting point for analyzing the mathematics identities of students from non-dominant backgrounds. he argues that mathematics identities are co-constructed with academic and social identities. mathematics identities are negotiated over time, contributing to the socialization that allows students to see themselves as mathematical people. mathematics educators require theoretical tools to understand how students navigate racial, linguistic, gendered, and other social identities in relation to learning mathematics. crt and latcrit provide such theoretical tools. scholars who privilege latina/o voice use crt and latcrit to examine latina/o students’ experiences as part of a collective history of marginalization within the system of schooling in the united states that requires them to navigate broader discourses of oppression and resistance in their education (cammarota, 2004; fernandez, 2002; perez huber, 2010; solórzano & bernal, 2001; solórzano & yosso, 2001, 2002; wassell, fernandez hawrylak, & lavan, 2010). crt is particularly useful to examine and challenge prevalent colorblind (bonilla-silva, 2006) and “culturally neutral” (ladson-billings & tate, 1995) assumptions in mathematics. a fundamental tenet of crt is to re-contextualize seemingly a-historical or abstract accounts of persons of color within historical and political contexts that illuminate whose interests are being served, and for what purpose (solórzano & yosso, 2001). the stories students tell about who they are as mathematical people have roots in multiple contexts, are situated in broader discourses of achievement and access, and contain notions of deeply seeded attitudes students have towards themselves and how learning mathematics works in general. although crt and latcrit focus on the same goal—understanding and dismantling multiple forms of oppression—they are distinct (valdes, 1996). crt mainly focuses on race, and while race is important, scholars of latcrit argue that zavala race, language, and mathematics   journal of urban mathematics education vol. 7, no. 1 62 researchers must consider multiple constructs specific to the experiences of latinas/os in the united states, such as language, culture, ethnicity, immigration status, phenotype, and sexuality (espinoza, 1990; garcia, 1995) to understand the experiences of latinas/os in the united states. as bernal (2002) described, critical race-gendered perspectives avoid binary power relationships and, instead, look at the way that multiple identities intersect to inform the experiences and epistemologies of people of color. building on this idea, she wrote, “latcrit is a theory that elucidates latinas/latinos’ multidimensional identities and can address the intersectionality of racism, sexism, classism, and other forms of oppression” (p. 108). critical race and latcrit theorists argue that educational systems and discourses have both the power to oppress and empower. therefore, educators are most deeply informed about latina/o students when they analyze their experiences through a lens of intersectionality (bernal, 2002; solórzano & yosso, 2001). student agency and testimonio in this study, i follow powell’s (2004) concept of mathematical agency for students of color, which he defines as “the mathematical ideas and reasoning evidenced from learners’ individual initiative to define or redefine as well as build on or go beyond the specificities of mathematical situations on which they have been invited to work” (p. 43). powell argued that it is important to understand the mathematical agency of students of color because both failure and success can be located within the same set of social, economic, and school conditions, although dominant discourses generally portray these conditions as mechanisms that produce failure. the research literature presents examples of how latina/o students exhibit mathematical agency including how they see themselves as people who construct mathematical knowledge and use it in meaningful ways (pitvorec, willey, & khisty, 2011; gutstein, 2007), and how they use personal agency to resist unfair schooling practices (fernandez, 2002). in spite of these examples, it is important for the mathematics education research community to continue to document the mathematical agency of latina/o students as scholars seek to understand the connections between how these students make sense of their own experiences and how they feel empowered to act to learn mathematics. latcrit also offers some important tools for understanding latina/o student agency. solórzano and yosso (2002) outline what they call a “critical race methodology” for education, which focuses on the stories and experiences of students of color. they propose that scholars can use the counterstories offered by students of color as they share their testimonios as a tool for exposing, analyzing, and challenging the majoritarian stories of racial privilege. testimonio privileges the experiences of people marginalized by institutions such as schooling within a u.s. context, and highlights the way they show agency as they navigate these settings (solórzano & yosso, 2002). zavala race, language, and mathematics   journal of urban mathematics education vol. 7, no. 1 63 a particular feature of many latina/o students’ testimonios is productive resistance (fernandez, 2002). the notion of productive resistance helps scholars to challenge deficit views of latina/o youth, and instead highlight latina/o student resistance to schools as institutions that were never designed to support their identities and cultures. thus latina/o students show agency in both how they may succeed within the constraints of schooling, and how they may resist schooling practices and redefine success on their own terms (fernandez, 2002; perez huber, 2010; yosso, villalpando, bernal, & solórzano, 2001). methods scholars who use crt and latcrit argue that narratives function as a means to understand the lived experiences of students who have been traditionally marginalized, and whose experiences are largely absent from scholarship (cammarota, 2004; fernandez, 2002; solórzano & yosso, 2001). as solórzano and yosso (2001) wrote: crt in education recognizes that the experiential knowledge of students of color is legitimate, appropriate, and critical to understanding, analyzing, and teaching about racial subordination in the field of education. in fact, critical race educational studies view this knowledge as a strength and draw explicitly on the student of color’s lived experience by including such methods as storytelling, family histories, biographies, scenarios, parables, cuentos, chronicles, and narratives. (p. 473) therefore, i employed a qualitative design based on latcrit methodology of testimonio to respond to research questions anchored in latina/o students’ experiences. though this analysis is part of a broader qualitative case study of two algebra i classrooms, in this iteration, i privilege the student interview and focus group data in which students’ interpret their own experiences. i use other data sources to triangulate or add depth to students’ narratives, not to contradict their self-narrated accounts. i conducted this study in the context of a multi-racial and multi-lingual school in an urban setting in the pacific northwestern united states (office of superintendent of public instruction, 2011). i found two classes whose demographic composition of the class roughly approximated the racial data of the school (approximately 20% latina/o) and attempted to recruit all the latina/o students in these classes. ultimately, three latina/o students in ms. williams’ class and four latina/o students in mr. anderson’s class elected to participate (see appendix a).3 only one or two additional latina/o students in each classroom de                                                                                                                 3 all proper names throughout are pseudonyms. zavala race, language, and mathematics   journal of urban mathematics education vol. 7, no. 1 64 clined to participate. diversity in the cases appears somewhat limited given that all students are of mexican heritage and are either first generation or immigrants to the united states. in other ways, the cases represent diversity among latina/o youth in characteristics such as number of years in the united states, el (english language) or sl (spanish language) proficiency, age, and continuity of education. there were some potentially interesting data that i did not collect as they departed from the study’s focus, such as previous schooling contexts, socioeconomic differences, family migration patterns, or students’ roles in their home communities. however, the data provided capture multiple dimensions of these latina/o students’ lived experiences as people who participate in multiple communities and who bring their histories with them into the classroom. in a u.s. national context of increasingly diverse urban schools, this group of latina/o students that participate in multi-lingual and multi-racial mathematics classroom settings have important insights to share about their mathematics learning experiences and how those experiences impact their mathematics identities and, ultimately, mathematics achievement. data collection individual interview and focus group data were the primary sources analyzed for this article, with observational data used in some cases to situate students’ testimonios. in this study, i attempted to reposition myself as researcher, build a cultural bridge (gay, 2010), and gain affinity with the group by sharing my background, my commitments to equity in mathematics education, and my language skills in spanish. my goal was to align myself with the group in some ways to establish rapport (glesne, 2006). in spite of these efforts, i did not eliminate the power differential between the students as participants and myself as researcher. additionally, there was some discomfort among the young people related to discussing race, identity, and mathematics. they were not accustomed to addressing the types of questions i asked. as a participant-observer in their mathematics classrooms, i found other ways to build rapport through informal conversation and opportunities to engage in mathematics with the students. these interactions also gave me insight into students’ mathematical competencies. in the case of the students who preferred to speak spanish in their interviews, the fact that i could engage in spanish with them allowed access into their selfexpressions as spanish speakers. this spanish language engagement assisted both the students and me to be more comfortable in the interview setting. the first interview focused on educational and personal history, attitudes toward mathematics, perceptions of the utility of mathematics, and descriptions of who can be good at mathematics. in the second interview, i used stimulated recall (gass & mackey, 2000), allowing students to watch a brief classroom video episode (selected by me) featuring the student participating in whole-class discus zavala race, language, and mathematics   journal of urban mathematics education vol. 7, no. 1 65 sion. the video prompted discussion of what it meant to participate in class and students’ impressions of the teacher and other students. we also discussed peer relationships and experiences within that classroom. the second interview is where students who expressed a preference for spanish explicitly discussed language preferences because discussing their classroom experiences brought language issues to the surface. english-preferring and english-dominant students explored attitudes about language in the focus group setting, as they did not raise language issues in the individual interviews. i used my knowledge of language preference, length of time in the united states, and participants’ age to organize three focus groups: samuel, anita, and ignacio (students who grew up primarily in the united states); rubén and luis (younger recent immigrant students); and marco and julieta (older recent immigrant students). julieta did not participate in the focus group due to leaving school. consequently, marco, her focus group partner, completed the focus group protocol as a third interview. in the focus group, we looked at a bar graph from the school district that indicated a relationship between ethnicity and academic achievement (see figure 1). i used the graph as a vehicle to discuss beliefs about race and racial stereotypes. we also watched a short selection of classroom video in which multiple distinct languages (english, spanish, at least one distinct african language, and tagalog in mr. anderson’s class; english and spanish in ms. williams’ class) were being used at once. the video prompted a discussion of the role language plays in mathematics learning. sample protocols from focus groups and interviews are available in appendix b. analysis these data sources were used to write in-depth case reports of each focal student (miles & huberman, 1984), covering the following dimensions: educational history; experiences learning and attitudes towards mathematics; perceptions of the utility of mathematics; descriptions of who each student seemed to be in class, including their perception of particular roles each of them played; and perceptions of how race, language, and culture matter (or not) in learning mathematics. in this article, i focus on three themes that emerged from analysis of all seven focal students narratives: (a) the utility of mathematics knowledge in relation to self, (b) the role of race in learning mathematics, and (c) the role of language (english and other) in learning mathematics. for analytic purposes, it is useful to explore these themes as unique focal points for consideration, but that does not mean that they do not subsume, overlap, or influence each other. consistent with a latcrit perspective, i looked for how these and other layers of identity intersected in latina/o youth’s mathematics learning experiences and how the students exhibited agency in learning mathe zavala race, language, and mathematics   journal of urban mathematics education vol. 7, no. 1 66 matics. in particularly, i looked across these areas for how students exhibited mathematical agency (powell, 2004) and analyzed what dimensions of their identities seemed to support their abilities to exhibit agency. in addition to findings across students, i present one student’s case of exhibiting mathematics agency in detail, to demonstrate how mathematical agency was supported by a unique combination of academic and social identity resources. in the next section, i present findings from the collective voices of the latina/o youth in this study and their beliefs about the proposed key aspects of mathematics identities. appendix b contains a summary of identifying characteristics and notes on key findings discussed in the next section. findings the first set of findings focuses on general beliefs students held about themselves as mathematics learners. these findings add to extant literature on perspectives latina/o students hold about their mathematics abilities, and connections between mathematics and self. following these findings is a more detailed analysis of the focal students’ perspectives on racial and linguistic identities and their intersections with mathematics identities. findings on attitudes toward the role of race in their own mathematics learning show that students most-often adopted a colorblind (bonilla-silva, 2006) stance, though intersectionality of immigrant status, years in the united states, understanding of family history, and other personal factors may have helped some students to articulate how race matters in their own mathematics experiences. finally, i use the case of one latina, julieta, to demonstrate how linguistic identities played a key role in the formation of strategic partnerships between eb and fully bilingual students. i also consider how julieta used the strategic partnership to exhibit mathematical agency and how this agency was both supported by and functioned to support various facets of her mathematics identity. the utility of mathematics: connections to self and future goals all of the students in these cases believed themselves to be capable of learning mathematics and that mathematics was important for their futures. these are important ideas to establish from the beginning, as latina/o students’ positive views of themselves as mathematicians and importance of mathematics remains under-documented in the literature (exceptions include gutiérrez, willey, & khisty, 2011; gutstein, 2003, 2006;). however, students’ attitudes toward mathematics varied, indicating that having faith in one’s own capabilities is not the same as liking mathematics. one student said that, although he liked mathematics, he did not like his current mathematics class (ignacio, interview 1). in addition, zavala race, language, and mathematics   journal of urban mathematics education vol. 7, no. 1 67 students articulated two main uses of mathematics: to get ahead academically and vocationally or to use in daily life. three students elaborated on particularly deep ties between their overall descriptions of themselves and mathematics. anita, who wanted to be an elementary school teacher, described mathematics as important for her to learn as a future teacher (anita, interview 1). marco was an older student in this study who had already graduated from a preparatoria4 in mexico. when asked why he likes mathematics, he described how liking mathematics, persisting in it, and feeling good about himself for persisting in problem solving where interrelated: “me gusta sentirme si tengo un problema difícil, hacerlo y sentir o, puedo, yo puedo, yo se que yo puedo, y me hace sentir bien [i like to feel that if i have a difficult problem after i do it i can feel like oh, i can, i can, and i know that i can which makes me feel good about myself]” (marco, interview 1). julieta described how being good at mathematics was important for multiple activities in everyday life, and criticized her classmates for maybe thinking that “los números solo representan pandilla [numbers are just good for gang use],” such as tagging territory. julieta went on to say that she thought this perspective was silly because numbers are good for everything. she shared that, in addition to liking mathematics, learning mathematics was very important for her overall sense of independence, including not having to rely on anyone else to control her finances (julieta, interview 1). these findings illustrate that, to varying degrees, mathematics was important to long-term goals latina/o students established for themselves and, in some cases, was deeply connected to students overall sense of self. these points of view provide a foundation for understanding students’ perspectives about the roles of racial and linguistic identities in learning mathematics. the role of race: colorblindness, complexities, and cultural attribution in latina/o youth perspectives race mattered in different ways for students in this study. most of the group touted a colorbind (bonilla-silva, 2006) perspective, while two students who grew up in the united states expressed more complex ideas of how race does and does not matter in learning mathematics. these two participants linked racial identity to mathematics identity through motivation. furthermore, students expressed awareness of but did not necessarily challenge stereotypes that position asian students as smart in mathematics, describing asian american success as a function of observable cultural practices. in particular, two recent immigrant students located and named racial stereotypes about asian american school success,                                                                                                                 4 participants report that a preparatoria is essentially high school in mexico, covering grades 10 through 12. zavala race, language, and mathematics   journal of urban mathematics education vol. 7, no. 1 68 insisting that asian americans deserved all the a’s they earned because they studied a lot, whereas, when it came to mexicans, it was more about who has “ganas [motivation].” a colorblind perspective versus race “does and doesn’t” matter. most students expressed a view that mathematics achievement is a matter of individual motivation and race is not and should not be a factor. this perspective cut across the lines of who was a recent immigrant and who was born in the united states, with all recent immigrants taking this stance and samuel, born and raised in the northwestern united states, also aligning with this thinking. marco’s explanation is representative of the students who held this belief. in his interview, he explained that what matters most is that an individual student focuses and wants to learn: interviewer: ¿eso es decir que si todos se enfocan podrán sacar la calificación que deseen? [you are saying that if everyone focused they could get whatever they wanted (grade)?] marco: pues, sí exactamente, todos somos iguales, todos tenemos la misma capacidad. como dice el dicho ‘querer es poder’, y si uno no quiere, no puede; sería imposible poder. aunque sea más difícil. [well, yes exactly, well we’re all equal, we all have the same capabilities, it’s like the saying that ‘caring is power’, and if you don’t care you can’t do it, it’s impossible to do it. even the person with the most difficulties.] (marco, interview 3) marco’s position that everyone is equal and possesses the same capacity privileged individual effort over contextual constraints such as navigating a racial hierarchy of success in mathematics (martin, 2009). there are elements of meritocracy at work here in that his explanation does not take into account a critical perspective on schooling as an institution that has systematically marginalized latinas/os. marco’s position as a nearly twenty-year-old student in an algebra i classroom—even though he already has a high school diploma from mexico—made his statement even more curious. he enrolled in school in the united states because he wanted to learn english. he enrolled in traditional high school classes such as algebra, in which he excelled and found enjoyment. marco described his former high school in mexico as very small, and of the more than 30 students who started the first year, “no mas alcanzamos a graduar ocho [only eight of us graduated]” (marco, interview 1). he attributed his ability to graduate from high school in mexico to his individual motivation, charging those who did not finish with not wanting it enough. examining marco’s comments in the context of his prior school success, his affinity for mathematics, and his success in school in the united states reveals the multiple layers of his mathematics identity and helps to contextualize his color zavala race, language, and mathematics   journal of urban mathematics education vol. 7, no. 1 69 blind stance. furthermore, he spent little time in the u.s. schools, but as successful and happily engaged in the process. marco represents a success story: a student who cares a lot about and is engaged in mathematics. these characteristics may situate his ideas of why racial hierarchies did not apply to how he perceived learning mathematics. anita and ignacio expressed a different perspective, indicating that their racial identities sometimes matter in relation to their mathematics identities. they were two of the ninth graders in the study and also two students who grew up in the u.s. school system, with ignacio living in the united states all his life and anita immigrating from mexico at age six. in our second focus group meeting, i asked the group about how language could be connected to a larger sense of self. anita shared that speaking spanish is an important way she lets people know she is mexican, and making others aware of her heritage is important to her: interviewer: so some people say that language is also how you express a part of your identity. like choosing to speak in different languages at different times. what do you think about that? anita: well sometimes when i meet new people, and most of the time they think i’m white, if i’m with a friend who speaks spanish i try to speak spanish so they recognize that i’m not white and they don’t judge me by my cover. (anita, in focus group with samuel & ignacio) it is important to note that anita feels she “passes” for white, so she uses language (spanish) to communicate her racial identity. understanding how anita positions herself as a mexican and how it is important for her that other people do not mistake her for white also helps to situate the way she described how her racial identity and mathematics identity intersect in the following excerpt. later in the same focus group meeting, we discussed racial identities in relation to school achievement. anita said: well, for me i feel bad about myself sometimes for not trying the best i can, and not really doing as well as i could be doing on quizzes and stuff, and that brings me back to thinking that i could end up like my mom or my dad or like my cousin like washing dishes or whatnot. and then it makes me try harder, but then i forget about it. and i’ll go back down again. (anita, in focus group with samuel & ignacio) using a latcrit lens, anita’s comments sit at the intersection of her mexican latina identity, her family, their immigration status, and how they are racialized in the united states. anita perceived herself as having opportunity and advantage and considered herself fortunate for having familial support. when asked about experiences with racism in their mathematics classrooms, ignacio and anita did not describe any personal experiences. however, anita zavala race, language, and mathematics   journal of urban mathematics education vol. 7, no. 1 70 connected possible outcomes for herself to the discrimination her family members had experienced as adult immigrants, suggesting that she thought about what her racial identity meant in relation to her academic achievement. she recounted the story of an aunt whose degree from a mexican university was devalued in the united states. she spoke with an awareness that, in the united states, the way she identified as mexican meant she, too, could experience the devaluation and, ultimately, dehumanization that her relatives experienced as spanish-speaking mexican immigrants. this awareness affirmed her need to excel in mathematics. in this way, she addressed the pressure to be a good student so that she could be better positioned than her parents and her aunt had been. stereotypes and racial hierarchies: asian americans and latinas/os. when questioned explicitly about racial stereotypes, none of the students who completed the focus group protocol could think of a stereotype that applied to latina/o or mexican students related to academics. rather, when asked to name any stereotype about latina/o youth, students named “we are illegal” and “we’re violent” as the two dominant stereotypes. in their second focus group session, samuel and anita described the illegal stereotype: interviewer: so, have we heard any stereotypes about latinos in school, or math learning? samuel: (smiling) we’re illegal. interviewer: we’re illegal? anita: yeah. (samuel laughs and looks at the ceiling) interviewer: that’s a big one, right? where have you heard that come up? anita: well, we kind of like, well the people we hang around with, we kind of joke around about it. samuel: yeah. anita: we’re like, oh we’re alien! because you know how like when you fill out for passports and anything it says list your alien number here and something? i find it kind of like insulting for us to be called aliens, but… interviewer: yeah, say more about that. anita: well we kind of joke around about it so we don’t feel that bad. interviewer: so it’s more like a joke among people who share that experience? anita: like when black people call themselves the n-word and stuff. interviewer: what do you think about that kind of thing though? anita: well, i don’t know. because you know how people say only black people can say the n-word because they’re black. interviewer: mhmm. anita: i guess it’s kind of the same thing, what i just said about aliens and stuff. interviewer: so as long as you are on the inside of it you mean? anita: i guess. because you can’t just go up to somebody and say the nword and like they’ll get like offended, because just like they can’t zavala race, language, and mathematics   journal of urban mathematics education vol. 7, no. 1 71 come up to us and call us aliens, we’ll get offended too. so i guess you have to be one to say it, i guess. in making sense of the way latinas/os joke about being illegal, anita positioned “the n-word” as holding a similar power among african americans that the word illegal holds among latinas/os. anita uses this parallel to illustrate the way that stereotypes have an insider–outsider quality. she and her friends use humor to negotiate the “illegal alien” stereotype, but they find the same stereotype offensive when employed by an outsider. the illegal alien stereotype was not the only topic for which latina/o students drew on their understanding of patterns in other racial groups to make sense of their own. when looking at the district grade data disaggregated by ethnicity (see figure 1), all students sought to explain why asian american students would have the highest grades by describing their perceptions of asian students’ cultural practices. rubén and luis, who had a focus group together and who coconstructed their ideas about how asian and asian american students excelled, juxtaposed that group’s achievement with their perception of mexican and latina/o student underachievement. marco speculated that asian students got the most a’s because of “how they are brought up.” in their focus group, ignacio, anita, and samuel speculated that asian student achievement might be a function of how they study. ignacio said, “they’re known for that,” suggesting that the stereotype for asian and asian american students was based on a history of observable patterns of achievement and an expectation for high performance. during his focus group with rubén, luis noticed that the bar representing the percentage of asian american students who received a’s was significantly higher than the other grades. when i asked for his explanation, rubén proposed that all the asians were smart, at which point i engaged them in an examination of this belief: luis: no, yeah, this is true. interviewer: is it true or a stereotype? luis: well, it’s true. interviewer: but how did you learn that? luis: well you can tell. rubén: porque se van y andan con salen del bus leyendo. andan comiendo y con sus papel escribiendo. [because they go they walk with they get of the bus reading. they go along eating and writing papers (at the same time)] interviewer: so all the asians you know are very studious… y nosotros? cuál tenemos? [and us? what (stereotypes) do we have?)] luis: that we’re violent. [luis and rubén laugh] zavala race, language, and mathematics   journal of urban mathematics education vol. 7, no. 1 72 interviewer: that we’re violent? luis: you can see it. we’re violent. you’ll never see us like that, up like the asians [points to the bar graph for as] (rubén & luis, focus group) as the discussion continued, luis and rubén laughed more about how the violent stereotype was not part of their experience despite being positioned as a universal characteristic ascribed to latinas/os. interestingly, in the end, luis suggests that being violent is related to why you will not see latina/o grade achievement as high as asian and asian american students, suggesting that the stereotype can simultaneously seem ridiculous to him, but also hold some explanatory power for latino underachievement.     figure 1: district high school grade data by ethnicity.5 in spite of rubén and luis’ resistance to generalizations about latinas/os, they seemed to make similar sweeping statements about asian american students. the exchange in the focus group calls attention to how what may be perceived as a positive stereotype (asian students are smart) is applied as capital-t truth to the monolithic category of asian americans, which feeds the model minority myth (museus & kiang, 2009). it could also be that because their own perspectives are situated in their experiences as young mexican men who primarily socialize with other mexicans, they may not be oriented towards thinking about collective op                                                                                                                 5 reproduced directly from a publicly available district report. zavala race, language, and mathematics   journal of urban mathematics education vol. 7, no. 1 73 pression or struggle when their experiences suggest every mexican is a different individual who makes their own choices. this position could support luis’s and rubén’s beliefs that race is not some kind of a determining factor in their experience. but on the other hand, luis responded to my prompt to address “us” (“that we’re violent?”) by asserting a group identity and making a claim about mexicans or latinos in general, if you take my bid to address “us” into account. in asserting a group identity, he positioned a collective mexican as less academically talented than asian americans: “you’ll never see us like that, up like the asians.” this statement also captures how, in some way, luis was acquainted with the stereotypical racial hierarchy of success in mathematics, and raises questions about the extent to which he had internalized the racial hierarchy. the findings around racial identities in learning mathematics suggest that students navigated racialized hierarchies of success. they attributed success among other racial groups to collective cultural practices, while perceiving their own failure or success as a function of caring about education or having enough motivation. for anita and ignacio, racial identities had a more complex role in this motivation process because of their families’ histories of struggle in the united states. students also grappled with stereotypes or observed cultural patterns among other racial groups as they made sense of their own experiences and within-group stereotypes. anita also addressed intersections of her racial and linguistic identities in how she lets people know she is not white, but mexican. in the next section, i explore further the role of language and linguistic identities. the role of language: linguistic identities, agency, and academic partnerships the english dominant students recognized the importance of specific mathematics vocabulary for learning mathematics. although they did not find language of instruction to be as important to their mathematics learning, they did express that language was a likely issue for eb students. additionally, they justified the use of multiple languages in their mathematics classes by considering that it was probably easier for students to learn mathematics in their first language. the significance of learning mathematics in spanish becomes more salient when situated in the linguistic worlds of the recent immigrant eb students, who reported speaking english only when in school. for these participants, language of instruction was a critical element of their mathematics learning. marco, julieta, and rubén indicated a preference for learning mathematics in spanish. luis, the fourth recent immigrant student, preferred to learn mathematics in english, although he could be observed in class using both english and spanish. these students’ accounts of their experiences revealed two important themes. first, in order to understand what is happening in the mathematics classroom and the textbook, the student must have access to the appropriate mathematics vocabulary. second, students who do not speak english “properly” may find the mathematics class zavala race, language, and mathematics   journal of urban mathematics education vol. 7, no. 1 74 room to be a threatening place. julieta saw teaching mathematics in spanish as a tool to help students for whom english is not their first language: julieta: obviamente estamos en un país inglés, y hay que hablar inglés y practicarlo más que nada. pero para tener un poquito más de ayuda con los estudiantes creo que deberían tener gente que pueda, um, translate explicar mejor en su primera lengua y así entender mejor y tener mejores calificaciones. [obviously we are in an englishspeaking country, and so we have to speak english and practice it as much as possible. but i think that to provide the students a little more help i think there needs to be people who can, um, translate, and explain better in our first language, and in that way understand better and get better grades.] (julieta, interview 1) in their focus group, luis and rubén expressed difficulty with understanding mathematics vocabulary: luis: well (what’s hard) for me it’s the words. i misunderstand at first…i mean the actual words—las palabras [the words]. rubén: sí, las matematicas tiene palabras mas difíciles, que a nosotros no entienden…el vocabulario de las matemáticas en mas complejo que inglés. [yes, mathematics has harder words that we don’t understand…the vocabulary of mathematics is more complex than english.] (luis and rubén, focus group) luis and rubén’s comments support the notion that there are three languages at work in their classrooms: spanish, english, and the language of mathematics. similar to other students in this study who had been in the united states three years or less, marco did not speak english if he did not have to. in his third interview, he summarized the way spanish dominates the worlds of the recently immigrated students: “i never speak in english. en la casa, puro español, en el trabajo, puro español, en la clase puro español. hasta mi jefe, que habla español porque ha estado en españa dos anos. solamente cuando hablo con los maestros. [i never speak in english. in the house, pure spanish, at work pure spanish, in class pure spanish. even with my boss, who spent two years in spain. it’s only when i talk with my teachers.]” in spite of his choice to use spanish in his daily life, marco recognized the importance of learning english. in fact, marco’s purpose for enrolling in high school in the united states after graduating from high school in mexico was to learn english. marco described the breakdown of when he spoke spanish and when he spoke english in his mathematics class: zavala race, language, and mathematics   journal of urban mathematics education vol. 7, no. 1 75 marco: hablo español cuando pido ayuda a los estudiantes hispanos, o a un maestro que habla español, porque lo explican mejor en mi idioma, así entiendo mejor y entiendo perfectamente lo que están diciendo. y los momentos en que tengo que hablar en inglés son cuando pido ayuda a la maestra. necesito hablar en inglés para entender lo que está diciendo. o, a veces, a un estudiante que tampoco es de aquí y no habla español. me explico en inglés, tengo que hablar inglés. [i speak spanish when i ask for help from the other hispanic students, or help from a teacher who speaks spanish, because they explain it better in my language because i understand better and perfectly what they are saying. and the moments i speak english are when i have to ask the teacher for help. i have to speak english to understand what she is saying. or sometimes a student who is from here and doesn’t speak spanish. if they explain to me in english, i have to speak in english.] (marco, interview 3) the teacher who speaks spanish that he is referring to is the spanish-language instructional aid who was present two or three times per week. marco’s description aligns with the experiences of other eb students in this study who preferred learning mathematics content in spanish while also knowing that it was important to engage in english sometimes, especially when talking with the mathematics teacher. the case of julieta: navigating linguistic identities and exhibiting agency some students used language in the classroom to display agency in forming strategic partnerships for their own benefit. julieta was passionate about her own learning and critical of the students in her mathematics class that she perceived as not taking advantage of their opportunities to learn mathematics. as we watched a video clip of her classroom in her second interview, she indicated who she thought was an intelligent person in the class, and then went on to criticize other classmates and to classify her own feelings as jealousy: julieta: ah sí, [pointing to screen] kayla ayuda. es muy inteligente y ayuda, pero la mayoría no lo es, no sé. casi nunca preguntan porque tienen pena, o simplemente no quieren trabajar. algunos de los estudiantes son [pause] tontos, sí tontos. porque si yo supiera hablar inglés como ellos, siempre les haría preguntas, y no sé – [oh yes, (pointing to screen) kayla is helpful. she is very intelligent and helpful, but most of the class is not, i don’t know. they almost never ask questions because they’re embarassed or they simply don’t want to work. some of the students are (pause) stupid, yes stupid. because if i knew english like they do, i would always be asking questions, and i don’t know–] interviewer: ¿y por qué ? [and why? -] zavala race, language, and mathematics   journal of urban mathematics education vol. 7, no. 1 76 julieta: me siento, creo, creo que más bien siento jealous? [i feel, i believe, i believe that more than anything i feel jealous?] interviewer: jealous, ¿te sientes celosa? [jealous you feel jealous?] julieta: sí, porque ellos saben perfectamente bien el inglés y no preguntan, y ellos pueden entender todos los problemas, todas la palabras del libro de matemáticas, y yo no. si yo fuera ellos yo me la pasaría leyendo matemáticas, pero no sé bien que significan algunas palabras, así es que no puedo. [yes, because they know english perfectly well and don’t ask questions, and they can understand all the problems all the words in the math book, and i can’t. if i was them, i would spend my time reading about mathematics, but i don’t know what all the words mean very well, so i can’t.] (julieta, interview 1) julieta passionately identified accessing the mathematics content (pasando leyendo matemáticas) and asking questions as an advantage english speakers have over her, and criticized them for not embracing that privilege. her perspective was that these young english-speaking students in the algebra i class did not understand how well positioned they were to do something she wished she could, just because they grew up speaking english. she perceived these students to be wasting their advantage. because instruction and all materials were in english, julieta’s jealousy was partly a critique of how she could not access mathematics in ways that they could. julieta addressed her disadvantage by doing mathematics in spanish. this strategy took some creativity on her part because, over time, the teacher asked the language instructional aid to work with julieta exclusively in english. julieta sought samuel, a partner who could do mathematics with her in spanish. ms. williams arranged her class in groups of four; julieta and samuel were seated together in a group.6 in her interviews, julieta named samuel as important to her mathematics learning. their partnership supported julieta’s linguistic and mathematics identities. in the quote below, julieta explained the importance of her relationship with samuel, and her preference to work with samuel rather than the adult assigned to work with her (the language instructional aid): cada vez le pregunto (a la ia) y me contesta en inglés, y a mí me gustaría más que me contestara en español para entender mejor [pause⎯continues in english] that’s why, maybe i – le estoy preguntando más a samuel que a ella, porque samuel me está respondiendo en español. y por eso, veo que no estoy practicando tanto, y sé que ella es maestra, pero ya no me responde en español, pero samuel sí. [every time i ask the instructional aid she responds to me in english and i would prefer if she                                                                                                                 6 ms. williams had not seated them together intentionally and did not know that samuel spoke spanish. zavala race, language, and mathematics   journal of urban mathematics education vol. 7, no. 1 77 would respond in spanish to understand better (pause⎯continues in english) that’s why, maybe i—i’m asking samuel more than her, because samuel is responding to me in spanish. and for that i notice that i’m not practicing as much and i know she’s a teacher, but she’s stopped responding to me in spanish, but samuel does.] (julieta, interview 2) later in the interview, when asked about whether she thought she played a particular role in the class, julieta elaborated on how her partnership with samuel was important not only for accessing mathematics content but also for allowing her to feel like part of the classroom community. she described how she felt “no fuera del círcuclo, sino dentro” [not outside the circle, but instead inside]: por ejemplo, preguntándole a las personas de mi grupo en español, y creo que así siento que estoy participando, trato de poner la mayor atención posible para entender el problema, y no sentirme fuera del círculo, sino dentro. [for example, asking people (in my group) in spanish, and i believe that is how i feel i’m participating in the group, and trying to put all the attention that i can into understanding the problem, and feeling that i’m not out of the circle, but instead inside.] (julieta, interview 2) she described “preguntándole a las personas en español,” indicating that she spoke in spanish and samuel passed her ideas on to the group. this process allowed her to contribute to the group’s ideas. julieta participated in the class on her own terms, using her linguistic identity as a resource. julieta used her agency to initiate a strategic partnership with samuel. through this partnership, she simultaneously maintained her linguistic and mathematics identities, which enabled her to feel like a participant in her small group. as a part of the community, she saw herself as a mathematics learner in positive ways. julieta initiated a friendship that would benefit her mathematics education. it was the intersectionality of multiple layers of julieta’s identity in this case, how she liked mathematics, how she connected mathematics to a broader sense of self, her linguistic identity, as well as her initiative, that together capture how she demonstrated mathematical agency in her classroom. discussion the findings presented here assist mathematics education researchers and scholars to understand the layers of experience (identities, agency, and participation) that impact the mathematics education of latina/o youth. overall, this study adds evidence to how crt and latcrit are useful to understand the multiple identities that intersect in the lives of latina/o youth and inform the agency they exhibit in learning mathematics and negotiating their identities within mathematics contexts. focusing on students’ perspectives was also crucial to understand their lived experiences in the tradition of crt and latcrit scholars (fernandez, 2002; zavala race, language, and mathematics   journal of urban mathematics education vol. 7, no. 1 78 perez huber, 2010; sólorzano & yosso, 2001). in particular, julieta’s case adds documentation to how students resist schooling practices and exhibit mathematical agency in productive ways within their classrooms. what we learn across the cases is that race and language are complex factors that influence how these latina/o youth become mathematical people. likewise, the variety of experience among latina/o youth shows how the particular mathematics identities these youth negotiate are impacted by racial stereotypes and their linguistic identities in relation to the official language of instruction. findings around student perspectives on how racial identities matter in learning mathematics show latina/o students perceive racial identity to be more salient for other racial groups than for their own. rather than tell a counterstory about race and achievement, the students who subscribed to the colorblind stance of mathematics achievement were drawing on a dominant narrative of achievement-motivation in which race does not matter. only two students (anita and ignacio) who grew up in the united states described the role of race in their own educations as more complex, and tied to motivation. anita’s description of using her heritage as a resource is a counterstory to the dominant colorblind perspective, although even her own testimonio shows an intricate link between her racial identity and her personal motivation. however, as she grapples with how her racial identity is a resource for her learning, she resists negative stereotypes in ways that increase options for what it means to be mexican and to learn mathematics, making new discourses (gee, 2001) for latinas/os to navigate in mathematics (zavala, 2009). similarly, other scholars have addressed the complex effects of race for high school students (esmonde, brodie, dookie, & takeuchi, 2009; fernandez, 2002; pollock, 2004). as pollock (2004) noted, “taking cues from youth, we can keep creating moments to talk about racial categorization as a human and contestable process, even while keeping race labels strategically available for analyzing social inequality” (p. 43). race is implicated, but in ways that may be contestable and difficult to articulate at this point in their learning trajectories. there is also a need to dig deeper into the implications for academic success that are suggested by navigating stereotypes of latina/o youth as illegal or violent. these stereotypes may not speak directly to latina/o achievement in mathematics, but they perpetuate an image of latinas/os as violent and alien, rather than belonging and intellectually or mathematically resourceful (solórzano, 1997). luis’ accounts of how latinas/os are violent and not as high achieving as asian and asian americans also suggests that he may be grappling with internalized racism (padilla, 2001). though luis does not speak directly to anglo-superiority, which padilla (2001) argues is part of internalized racism, he does participate in self-defeating behavior and, thus, is an active participant in his own oppression. zavala race, language, and mathematics   journal of urban mathematics education vol. 7, no. 1 79 the consequences then for his academic achievement in mathematics are dire⎯luis has his own internal work to do before he can realize his full potential. another issue that emerges from the students’ accounts is how to reconcile students’ expressed beliefs about race with what we observe as scholars in the field who conceptualize mathematics classrooms as racialized spaces (martin, 2009) and what we have come to understand from research that suggests attitudes toward race matter in learning mathematics (spencer, 2009). are we wrong? should we take the perspectives of youth as a sign that race really does not matter, if that is the way they narrate their experiences? in this study, it is important to note that, from the student perspective, ideas of who can be mathematically successful are related to racial stereotypes of success in mathematics. the students support these racialized dimensions of learning mathematics when they positioned themselves as less capable than asian and asian americans. as problematic as their perceptions of asian and asian american students may be, they also insist that their own success is not predetermined by race. again, the complexities in talking about race that pollock (2004) uncovered are important to making sense of the ways students can simultaneously notice racialized patterns in others while maintaining a colorblind stance for themselves. the students who hold views that race does not matter may use this approach as a coping mechanism because they do not have a means to engage deeper power issues that they have experienced elsewhere, but have not come to recognize in their mathematics classrooms. the findings around linguistic identities and mathematics identities suggest that latina/o students may use strategic peer relationships to resist schooling practices in ways that seem productive for learning mathematics. for julieta, engaging in mathematics in spanish with samuel was a way that she also exhibited mathematical agency (powell, 2004) by taking initiative to position herself as a contributor to the group’s mathematical reasoning. at the same time, the way she exhibited agency to learn mathematics in spanish can be seen as resisting the dominant story of bilingual latina/o youth in u.s. schools that privileges an end goal of learning english over learning academic subjects in their language of choice. while scholars have written about competing hypotheses for why bilingual students would use spanish and english to learn mathematics, very little has been documented about students actively resisting learning mathematics in english by circumventing the teacher in favor of a competent peer. this form of resistance was productive because it facilitated julieta’s learning. though she expressed jealousy towards her classmates who spoke english but did not use it to participate in class, she may have been speaking from her positions as a marginalized participant in a classroom where the language of instruction was english. she wanted to exhibit more mathematical agency, and she found a way to do it. even though julieta had an instructional aid assigned to work with her, when that adult stopped working with her in spanish or was not present, julieta was creative. in zavala race, language, and mathematics   journal of urban mathematics education vol. 7, no. 1 80 this way, julieta’s story interweaves her linguistic and mathematics identities as she engaged in productive resistance. her experiences represent a counternarrative to the dominant notion of latinos/as as hardworking but passive. as the research on bilingual latina/o students in secondary mathematics settings suggests, students use multiple resources to learn mathematics (gutiérrez, 2002; zahner & moschkovich, 2011). one mode of access that mathematics education researchers should consider is how their racial and linguistic identities intersect to inform how they display agency in learning mathematics. julieta and samuel’s relationship had clear linguistic affinity, but they also had shared heritage. that is to say, those relationships were not formed with white or other spanish-speaking students. questions remain regarding how racial and linguistic identities are co-implicated in mathematics identity negotiation, and close attention to peer groups may be a way to understand how they both manifest in mathematics identity negotiation. further questions also remain that are specific to how gendered identities are also implicated in mathematics identity negotiation. mathematics teachers in the united states are under such pressure to cover a large volume of content in the midst of high-stakes accountability measures that attending to how big ideas like racial and linguistic identities manifest in the classroom can seem overwhelming. some may find such ideas irrelevant to the mathematics classroom. however, an important implication from this research and others is that racial and linguistic identities do matter to learners in the classroom and impact their engagement. a key implication from this research for mathematics teachers is to learn how the aspects of identity analyzed here, specifically those related to race and language, may be important in the lives of their own students. observing how students are navigating these aspects of their identities can help teachers to make informed pedagogical decisions including, but not limited to, supporting multiple forms of talk, making students’ first language more central for participation, or challenging racial stereotypes of who has the authority to be a mathematical resource in the classroom. teachers will need support to engage their students in frank discussions of stereotypes and achievement in their classrooms. at the same time, teachers need time to reflect on their own identities as mathematics learners and teachers, and the privilege and power they bring to the classroom (aguirre, mayfield-ingram, & martin, 2013). we should not expect mathematics teachers to do this crucial work alone. mathematics coaches and school administrators need to help re-think how instructional time in mathematics should be spent and commit to supporting teachers to grapple with these issues. zavala race, language, and mathematics   journal of urban mathematics education vol. 7, no. 1 81 conclusion the perspectives of these latina/o youth suggest there is more to explore about how students use racial stereotypes to make sense of other racial groups’ academic achievement and how these stereotypes manifest as reality. the findings presented add depth and complexity to the experiences of students who we might see as all being affiliated by the label latina/o. their variety of experiences and differences in how agency is enacted, give us insight into what matters to them about learning mathematics. their perspectives are key to understanding what they believe are important aspects of becoming different kinds of mathematical people⎯people with goals and people with promise. given the unique history of racism in schooling experiences of latinas/os in the united states (solórzano & yosso, 2002) , and the historical tension between bilingual education and englishonly education (ovando, 2003), there is an academic imperative to continue examining the multiple influences on how latina/o youth come to see themselves as successful mathematical people, and to address racial and linguistic discourses in our research. within the monolithic category of latina/o we find a range of experiences with languages other than english, and a range of ways people identify racially, ethnically, and culturally. crt and latcrit can provide an important perspective on the experiences of young people learning 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(2009, april). “i’m smart, but i just don’t want to put myself out there”: latino/a high school students’ perspectives on mathematics and identity. paper presented at the annual meeting of the american educational research association, san diego, ca. zavala race, language, and mathematics   journal of urban mathematics education vol. 7, no. 1 86 appendix a characteristics, attitudes, and beliefs of focal students name mathematics is useful for? how would you describe yourself as a mathematics learner? does race matter? how? does language matter? how? marco it is important in everyday life and for mental development; mathematics is hard, but useful capable, hard worker, and likes to solve problems no – individual attitude matters most yes – english language proficiency is the advantage other students have in class over non-english speakers interview language: español teacher: ms. w grade: 12 age: 19 years in us: 2 previous schooling: méxico (graduated from preparatoria) julieta it is useful for multiple things in life (provided many examples); linked it to independence enjoys doing mathematics; recalls crucial positive experiences learning mathematics julieta left school before completing her final interview about the role of race in learning mathematics yes – english is the only thing between her and the mathematics; jealous of english speaking students who don’t understand what privilege they have interview language: español teacher: ms. w grade: 11 age: 19 years in us: 2 previous schooling: méxico (primaria/secundaria on and off) samuel knows it is important for some things (non-specific) capable, “normal” student, not excited about mathematics; it is easy no – individual attitude matters most; everyone is equal yes – if you are proficient in both english and spanish, then your job might be to help others who are not able to access the mathematics in english interview language: english teacher: ms. w grade: 9 age: 15 years in us: 15 previous schooling: us pacific northwest ignacio it is necessary for college, not a lot else has changed from feeling good at mathematics in elementary school, to not good anymore after sixth grade no – individual attitude is what matters yes – can knock you down because people may note expect much from you yes – for people who don’t speak english interview language: english teacher: mr. a grade: 11 age: 19 years in us: 2 previous schooling: us los angeles rubén it is useful for everything you do and especially for securing your future career capable, “normal” student, but is not excited about mathematics no – individual attitude is what matters most yes – could matter for why asian students are successful yes – explaining the vocabulary of mathematics clearly is what matters most; if it can be explained in spanish is betters; non-english speakers are at a disadvantage interview language: español teacher: mr. a grade: 9 age: 15 years in us: 3 previous schooling: méxico (primaria) anita it is a part of everyday life. it is important for her plans to be a teacher that she learn math well capable and interested in learning; puts pressure on herself to work hard when not happy with her mathematics grades no – individual attitude is what matters most yes – can remind you that you need to do better to take advantage of your opportunities yes – both to help people learn mathematics who do not speak english, and because it is how she expresses her mexican identity interview language: english teacher: mr. a grade: 9 age: 15 years in us: 9 previous schooling: us pacific northwest luis it is necessary for everything (non-specific) does not think he is very good at it, and points out he likes to work with people “smarter” than him so he can learn no – individual attitude is what matters most. yes – could matter for why asian students are successful yes – learning vocabulary matters most; people may tease you if you do not speak english well enough interview language: english/español teacher: mr. a grade: 11 age: 17 years in us: 3 previous schooling: méxico (on and off) zavala race, language, and mathematics   journal of urban mathematics education vol. 7, no. 1 87 appendix b sample interview questions and focus group protocols sample student interview questions the questions below are taken from interview protocols 1 and 2 and are meant to share kinds of questions asked of the participants. these are neither entire data collection protocols nor represent the order questions were asked necessarily. 1. self-descriptions and meanings a. if someone didn’t know you, what are some words you would use to describe yourself? b. how would you describe yourself as a student? what do you do well as a student? what do you feel you need to improve? i. tell me about your study habits, and how you are doing in school. ii. probe around cultural, familial connections c. what do you think of (this school)? i. what is important to you at this school, so far? ii. do you think you can be yourself at school? iii. what are your first impressions of your teachers? iv. what do you think this year is going to be like for you as a student? 2. mathematics class/id a. how would you describe yourself as a math student? i. are you “good” at math? what does it mean to be good at mathematics? b. take a moment and think about how you learn different things (how you learned to cook, or play soccer, or whatever). how do you learn math? c. what will it take to be good at mathematics this year? (i know you haven’t been in school very long, but what you think right now?) d. is learning mathematics the same as being good at math? so, if you are good at math does that mean you are going to learn a lot of math? is there a difference? 3. impressions of language, race, gender, class: a. what languages do you hear students speaking in your class? what about yourself? b. do you think learning math is related to race, class, language, gender, or any other bit of information about a person? how? c. how do you think being latina/o plays a role in how you learn math? d. for spanish speakers: how do you think speaking spanish plays a role in how you learn math? sample focus group questions 1. let’s look at some data from your school district. this is demographically unpacked data about who gets what kind of grades. a. what do you see here? where are you in this data? b. what do you notice about the achievement of hispanic students? why do you think this is so? c. what does this data have to do with learning mathematics? 2. let’s watch a video from your class and get our brains going—think about the languages people speak and might be speaking in this class. a. what languages do you hear students speaking in your class? what about yourself? how do students seem to be using multiple languages to learn mathematics? how else is language used? (some people would say that language is also how you express your identity, what do you think about that?) b. last time we talked about how there are stereotypes around different races of people. it makes me wonder, in your class, have you ever heard or experienced racism? do people joke about race (racist language, racist actions, anything like that)? what about at school in general? i. do you think race plays a role in learning math? how? ii. do you think race plays a role in how people participate or not in math class? how? journal of urban mathematics education december 2012, vol. 5, no. 2, pp. 1–5 ©jume. http://education.gsu.edu/jume david w. stinson is an associate professor of mathematics education in the department of middle-secondary education and instructional technology in the college of education, at georgia state university, p.o. box 3978, atlanta, ga, 30303; e-mail: dstinson@gsu.edu. his research interests include exploring socio-cultural, -historical, and -political aspects of mathematics and mathematics teaching and learning from a critical postmodern theoretical (and methodological) perspective. he is a co-founder and current editor-in-chief of the journal of urban mathematics education. editorial mathematics educators and the “math wars”: who controls the discourse? david w. stinson georgia state university he lead story in the daily online publication inside higher ed on october 15, 2012 was titled “casualty of the math wars” (jaschik, 2012). it provided a context for professor jo boaler’s 1 actions in posting the essay “jo boaler reveals attacks by milgram and bishop: when academic disagreement becomes harassment and persecution” (boaler, 2012a) on her stanford university website. professor boaler’s essay chronicles, in detail, the professional and personal attacks she has experienced since joining the stanford faculty in 1998 by two mathematicians—james milgram (stanford university, emeritus) and wayne bishop (california state university los angeles). as evident by the title of the inside higher ed article, jaschik (2012) places professor boaler’s actions within the larger context of the so called “math wars.” schoenfeld (2004), in his participant-observer historical review of the wars, states that the underlying issues or questions being contested by two opposing camps are more than a century old: “is mathematics for the elite or for the masses? are there tensions between ‘excellence’ and ‘equity’? should mathematics be seen as a democratizing force or as a vehicle for maintaining the status quo?” (p. 253). probable responses to these questions are significantly different depending on which camp controls the discourse: the traditionalists or the reformers. the traditionalists’ camp claims that standards-based, reform curricula are superficial and undermine “classical” mathematical values—milgram and bishop clearly reside 1 editor’s note: we, the editorial team at jume, support our colleague, professor jo boaler, in her actions of going public with the harassment and persecution (i.e., academic bullying) that she has experienced through the unrestrained professional and personal attacks by james milgram and wayne bishop. others can show support of professor boaler by signing the change.org petition the community of mathematics educators: join in defending fundamental values, initiated by university of georgia regents professor jeremy kilpatrick. t http://www.insidehighered.com/news/2012/10/15/stanford-professor-goes-public-attacks-over-her-math-education-research http://www.stanford.edu/~joboaler/ http://www.stanford.edu/~joboaler/ http://www.stanford.edu/~joboaler/ http://www.change.org/petitions/the-community-of-mathematics-educators-join-in-defending-fundamental-values?utm_campaign=action_box&utm_medium=twitter&utm_source=share_petition http://www.change.org/petitions/the-community-of-mathematics-educators-join-in-defending-fundamental-values?utm_campaign=action_box&utm_medium=twitter&utm_source=share_petition stinson editorial journal of urban mathematics education vol. 5, no. 2 2 here. whereas the reformers’ camp claims that reform-oriented curricula such as imp or cpmp 2 reflect a deeper, richer view of mathematics—this of course is where professor boaler resides. my intent here is not to provide a detailed discussion of the math wars; it has been done elsewhere (see, e.g., becker & jacob, 2000a, 2000b 3 ; davison & mitchell, 2008; herrera & owens, 2001; kilpatrick, 2001; o’brien, 2007; reys, 2001; schoen, fey, hirsh, & coxford, 1999; schoenfeld, 2004). but rather i provide some background (albeit brief) to explain why i believe jaschik placed professor boaler’s recent actions within the larger context of the math wars. it is interesting to note that jaschik’s (2012) article is not the first time that james milgram and wayne bishop have been named as traditionalist culprits in the math wars who have hindered reform in mathematics teaching and learning. in a march 2000 phi delta kappan article, becker and jacob (2000b, also see 2000a) name both milgram and bishop among members of “a powerful group of parents and mathematicians who manipulated information and played off of the public’s perception of our ‘failing schools’ to acquire political clout” (p. 530), which, in turn, was used to substantially revise california’s school mathematics policy in the late 1990s. specifically, becker and jacob outline how this undeserved “political clout,” in many ways, silenced mathematics educators and k-12 mathematics teachers during the process of revising california’s school mathematics policy. they write: a unique feature of california’s new school mathematics policy is the influential role of university mathematicians. four stanford university mathematics professors substantially revised the standards in 1997, and three mathematics professors wrote the sample problems for the framework in 1998. two math professors wrote key sections of the framework’s discussion for teachers and then, on 22 september 1999, led the department of education presentation for publishers, explaining what was expected of them when they submit materials for adoption in august 2000. two other mathematics professors judged (and extensively rewrote) the curriculum for the professional development provided, for which $43 million will be available during 2000–01. to our knowledge, none of these mathematicians ever taught in k-12 schools, and throughout their work on policy, there was never a publicly scheduled session for them to interact with k-12 teachers. mathematics professors also ran the math content review panels for the billion-dollar material adoption that was competed by the state board during the summer 1999. through these actions, the state board made it clear whose voice would count and whose would be ignored. (p. 531) in concluding their discussion, becker and jacob (2000b) claim that school 2 for information about the interactive mathematics program (imp), see http://mathimp.org; for information about the core-plus mathematics project (cpmp), see http://www.wmich.edu/cpmp/. 3 see haimo and milgram (2000) for a response to becker and jacob (2000b). http://mathimp.org/ http://www.wmich.edu/cpmp/ stinson editorial journal of urban mathematics education vol. 5, no. 2 3 mathematics policy which once held “‘teaching for understanding’ as its centerpiece has vanished from the california mathematics education landscape, and mastery of procedure skills is now the order of the day in the state’s standards, framework, standardized assessments, and professional development” (pp. 535– 536). overall, becker and jacob’s purpose in outlining the events that unfolded in the late 1990s is to bring to light their bewilderment of how mathematicians managed to replace mathematics educators and classroom teachers in leading the development and implementation of california’s school mathematics policy. in other words, how did those with expertise in mere mathematics content knowledge replace those with expertise not only in mathematics content knowledge but also in how students come to learn mathematics and how teachers might best teach mathematics? this “replacement” was (is) most problematic. battista (1999) argues: to perform a reasonable analysis of the quality of mathematics teaching requires an understanding not only of the essence of mathematics but also of current research about how students learn mathematical ideas. without extensive knowledge of both, judgments made about what mathematics should be taught to schoolchildren and how it should be taught are necessarily naïve and almost always wrong. (p. 433) i believe that most, if not all, mathematics educators and classroom teachers as well as most “mathematically sane” 4 mathematicians would agree with battista’s (1999) argument. but how did such a replacement happen in california in the late 1990s? battista claims that traditionalists exploited the “‘talk show/tabloid’ mentality of americans” and provided them “with hearsay, misinformation, sensationalism, polarization, and conflict as they attempt[ed] to seize control of school mathematics programs and return them to traditional teaching” (p. 425). a walk through any school today would confirm that traditionalists did indeed win the battle of the 2000s—both the bush administration’s no child left behind act and the obama administration’s race to the top fund have secured a return to traditional practices. nonetheless, wining the battle is not winning the war. new battles always provide for different possibilities. it is within the context of a new battle—the battle of 2010s—with its different possibilities that i like to place professor boaler’s recent actions. from a poststructural perspective, i like to think of professor boaler’s actions as a coun 4 mathematically sane “has been created to provide insights into the reform of mathematics teaching in the schools by making a compelling case that changes in our nation’s mathematics programs are imperative for our students’ future success and for the economic health of our nation”; see http://mathematicallysane.com for more information. of particular interest might be web links to two ted talks: teaching kids real math with computers by conrad wolfram and math class needs a makeover by dan meyer. http://mathematicallysane.com/ stinson editorial journal of urban mathematics education vol. 5, no. 2 4 termove in the math wars, seizing control of the discourse and thus, the power. 5 professor boaler (2012b), in her plenary talk at the 34th annual conference of the north america chapter of the international group for the psychology of mathematics education (pme-na) held recently in kalamazoo, michigan, spoke directly about such efforts, providing her own insights of communicating mathematics education research to broader audiences and the importance of leading the discourse in mathematics education reform. but controlling or leading the discourse does not mean that there is no room for scholarly disagreement. scholarly disagreement is beneficial (and needed) for intellectual growth. 6 nonetheless, what professor boaler has demonstrated by going public and taking control of the discourse is that she will no longer be bullied. we might all take a cue from professor boaler in this regard. when it comes to issues of mathematics teaching and learning, mathematics educators and classroom teachers should not stand for being bullied in our own sandbox. references battista, m. t. (1999). the mathematical miseducation of america’s youth: ignoring research and scientific study in education. phi delta kappan, 80, 425–433. battista, m. t. (2010). engaging students in meaningful mathematics learning: different perspectives, complementary goals. journal of urban mathematics education, 3(2), 34–46. retrieved from http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/115/58. becker, j. p., & jacob, b. (2000a). look at the details: a reply to deborah haimo and james milgram. phi delta kappan, 82, 147–148. becker, j. p., & jacob, b. (2000b). the politics of california school mathematics: the anti-reform of 1997-99. phi delta kappan, 81, 529–537. boaler, j. (2012a). jo boaler reveals attacks by milgram and bishop: when academic disagreement becomes harassment and persecution. retrieved from http://www.stanford.edu/~joboaler/. boaler, j. (2012b). scaling up innovation: using research to make a difference. plenary talk delivered at the 34th annual conference of the north america chapter of the international group for the psychology of mathematics education, kalamazoo, mi. confrey, j. (2010). “both and”—equity and mathematics: a response to martin, gholson, and leonard. journal of urban mathematics education, 3(2), 25–33. retrieved from http://edosprey.gsu.edu/ojs/index.php/jume/article/view/108/53. davison, d. m., & mitchell, j. e. (2008). how is mathematics education philosophy reflected in the math wars? the montana mathematics enthusiast, 5(1), 143–154. haimo, d. t., & milgram, r. j. (2000). professional mathematicians comment on school mathematics in california. phi delta kappan, 82, 145–146. 5 see stinson (2009) for a brief discussion of how discourse and power are re-inscribed within poststructural theory. 6 jume recently played a roll in demonstrating such benefits through the scholarly exchange within its pages regarding the question where’s the math in mathematics education research? (see battista, 2010; confrey, 2010; martin, gholson, & leonard, 2010). http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/115/58 http://www.stanford.edu/~joboaler/ http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/108/53 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/108/53 stinson editorial journal of urban mathematics education vol. 5, no. 2 5 herrera, t. a., & owens, d. t. (2001). the “new new math”?: two reform movements in mathematics education. theory into practice, 40(2), 84–92. jaschik, s. (2012, october 15). casualty of the math wars, inside higher ed. retrieved from http://www.insidehighered.com/news/2012/10/15/stanford-professor-goes-public-attacksover-her-math-education-research. kilpatrick, j. (2001). understanding mathematical literacy: the contribution of research. educational studies in mathematics, 47, 101–116. martin, d. b., gholson, m. l., & leonard, j. (2010). mathematics as gatekeeper: power and privilege in the production of knowledge. journal of urban mathematics education, 3(2), 12– 24. retrieved from http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/95/57. o’brien, t. c. (2007). the old and the new. phi delta kappan, 88, 664–668. reys, r. e. (2001). curricular controversy in the math wars: a battle without winners. the phi delta kappan, 83, 255–258. schoen, h. l., fey, j. t., hirsch, c. r., & coxford, a. f. (1999). issues and options in the math wars. phi delta kappan, 80, 444–453. schoenfeld, a. h. (2004). the math wars. educational policy, 18, 253–286. stinson, d. w. (2009). the proliferation of theoretical paradigms quandary: how one novice researcher used eclecticism as a solution. the qualitative report, 14(3), 498–523. retrieved from http://www.nova.edu/ssss/qr/qr14-3/stinson.pdf. http://www.insidehighered.com/news/2012/10/15/stanford-professor-goes-public-attacks-over-her-math-education-research http://www.insidehighered.com/news/2012/10/15/stanford-professor-goes-public-attacks-over-her-math-education-research http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/95/57 http://www.nova.edu/ssss/qr/qr14-3/stinson.pdf journal of urban mathematics education july 2013, vol. 6, no. 1, pp. 45–57 ©jume. http://education.gsu.edu/jume judit moschkovich is a professor of mathematics education in the education department at the university of california, santa cruz, 1156 high street, santa cruz, ca 95064; email: jmoschko@ucsc.edu. her research uses sociocultural approaches to examine mathematical thinking and learning, mathematical discourse, and mathematics learners who are bilingual, learning english, and/or latino/a. principles and guidelines for equitable mathematics teaching practices and materials for english language learners1 judit moschkovich university of california santa cruz in this essay, the author describes principles for equitable mathematics teaching practices for english language learners (ells) and outlines guidelines for materials to support such practices. although research cannot provide a recipe for equitable teaching practices for ells, teachers, educators, and administrators can use this set of research-based principles and guidelines to design equitable mathematics instruction, developing their own approaches to supporting equitable practices in mathematics classrooms. the recommendations presented use a complex view of mathematical language as not only specialized vocabulary but also as extended discourse that includes syntax, organization, the mathematics register, and discourse practices. the principles and guidelines stress the importance of creating learning environments that support all students (but specifically those learning english) in engaging in rich mathematical activity and discussions. keywords: english language learners, mathematics education he purpose of this essay is to describe principles for equitable mathematics teaching practices for english language learners (ells) and outline guidelines for materials to support such practices. the approach to equity used here is based on gutiérrez’s (2009, 2012) discussion of four dimensions of equity: access, achievement, identity, and power. using these dimensions, i contend that ells need access to curricula, classroom practices, and teachers shown to be effective in supporting the mathematical academic achievement, identities, and practices of these students. i define equitable teaching practices for students who are learning english in mathematics classrooms as those that (a) support mathematical reasoning, conceptual understanding, and discourse—because we know such practices lead to learning important mathematics, and (b) broaden participa 1 the principles and guidelines described and outlined here are informed by a sociocultural and situated perspective on mathematical thinking, on language, and on bilingual mathematics learners; for details of this framework see moschkovich, 2002, 2007b, 2010. t moschkovich principles and guidelines for ells stinson, d. w., & spencer, j. a. (eds.). (2013). privilege and oppression in the mathematics preparation of teacher educators [special issue]. journal of urban mathematics education, 6(1). 46 tion for students who are learning english—because we know that participation is connected to opportunities to learn. to support mathematical reasoning, conceptual understanding, and discourse, classroom practices need to provide all students with opportunities to participate in mathematical activities that use multiple resources to do and learn mathematics. to broaden participation, classroom practices need to provide all students with opportunities to use multiple ways of engaging in classroom discourse. equitable classroom practices, then, are fundamentally focused on honoring student resources, in particular, the “repertoires of practices” (gutiérrez & rogoff, 2003) that students bring to the classroom. equitable mathematics classroom practices for ells should be informed by knowledge of students’ experiences with mathematics instruction, language history, and educational background (moschkovich, 2010). teachers need to know details of a student’s history with formal schooling, for example, which grades they attended, where, and in what language (or languages). they should have some information about their language history, for example, are they literate in their home language, what is their reading and writing competence in the home language. some students may not have had any formal instruction in the language spoken at home. another important piece of information is the students’ history with school mathematics instruction: when they had mathematics classes, in what language, and for which topics. 2 we often hear that “academic language” is important for english language learners, but this phrase can have multiple meanings. interpretations of this phrase often reduce the meaning of “academic language in mathematics” to single words or technical vocabulary. in contrast, the recommendations for teaching practices and materials described here are based on research and a view of language that run counter to commonsense notions of language. these principles and guidelines use a more complex view of mathematical language as not only specialized vocabulary but also as extended discourse that includes syntax, organization, the mathematics register (halliday, 1978), and discourse practices (moschkovich, 2007c). the phrase “the language of mathematics” is used here not to mean a list of vocabulary or technical words with precise meanings, but rather the communicative competence necessary and sufficient for competent participation in mathematical discourse practices (moschkovich, 2012). while learning vocabulary is necessary, it is not sufficient. in other words, learning to communicate mathematically and participate in mathematical discussions is not a matter of merely learning vocabulary. during discussions in mathematics classrooms, students are learning to describe patterns, make generalizations, and use representations to support their claims. the question is not whether students who are ells should learn vocabulary, but rather how instruction can 2 for more details on equitable practices see moschkovich, in press. moschkovich principles and guidelines for ells stinson, d. w., & spencer, j. a. (eds.). (2013). privilege and oppression in the mathematics preparation of teacher educators [special issue]. journal of urban mathematics education, 6(1). 47 best support students to learn vocabulary as they actively engage in mathematical reasoning about important mathematical topics. therefore, the principles and guidelines presented here stress the importance of creating learning environments that support all students (but specifically those learning english) in engaging in rich mathematical activity and discussions. enacting the recommended principles and guidelines requires that teachers develop skills and strategies for leading, supporting, and orchestrating mathematical discussions, whether these occur in small groups or with the whole class. a review of the research suggests that professional development that has an impact on student achievement provides “adequate time for professional development and ensures that the extended opportunities to learn emphasize observing and analyzing students’ understanding of the subject matter” (american educational research association [aera], 2005). two other characteristics of effective professional development include linking professional learning to teachers’ real work and using actual curriculum materials. therefore, professional development can support teachers in learning these skills and strategies through long-term work in the context of particular mathematics topics, for example, focusing on teacher questions to support student algebraic (driscoll, 1999) or geometric thinking (driscoll, dimatteo, nikula, & egan, 2007). these skills also can be supported through long-term professional development that exposes teachers to examples of best practices for supporting mathematical discussions and engages teachers in reading about discourse in mathematics classrooms (e.g., moschkovich, 1999, 2007c; o’connor & michaels, 1993; sherin, 2002; stein, engle, smith, & hughes, 2008), watching classroom video (e.g., chapin, o’connor, & anderson, 2003; sherin & van es, 2005), lesson study (e.g., fernandez, 2005), and so on. these skills and strategies for teaching mathematics are fundamental to supporting students in the common core state standards (ccss), the standards for mathematical practice, and teaching mathematics for understanding, and are essential for supporting ells. 3 principles for equitable mathematics instruction for ells the following sections summarize (briefly) research relevant to principles for equitable mathematics instruction for ells. the summary includes: (a) research-based recommendations for effective instruction for ells (in general, not 3 there are materials available that specifically address teaching mathematics to ells. there are also materials that, although they do not target ells in particular, can be used to support teachers in learning to orchestrate mathematical discussions (e.g., five practices for orchestrating productive mathematics discussions [stein & smith, 2011] and classroom discussions: using math talk to help students learn, grades 1-6 [chapin, o’connor, & anderson, 2003]). see http://www.corestandards.org/math for the ccss for mathematical practice. http://www.corestandards.org/math moschkovich principles and guidelines for ells stinson, d. w., & spencer, j. a. (eds.). (2013). privilege and oppression in the mathematics preparation of teacher educators [special issue]. journal of urban mathematics education, 6(1). 48 specific to mathematics); (b) research-based recommendations for effective instruction in mathematics (for all students, not ells in particular); and (c) research-based recommendations for effective mathematics instruction specific to ells that is aligned with the ccss. a principled approach to teaching mathematics to ells would include characteristics from each section. what is effective instruction for ells? although it is difficult to make generalizations about the instructional needs of all students who are learning english, instruction should be informed by knowledge of students’ experiences with mathematics instruction, language history, and educational background (moschkovich, 2010). in addition, research suggests that high-quality instruction for ells that supports student achievement has two general characteristics: a view of language as a resource rather than a deficiency, and an emphasis on academic achievement, not only on learning english (gándara & contreras, 2009). research provides general guidelines for instruction for ells. overall, students who are labeled as such are from non-dominant communities and need access to curricula, teachers, and instructional techniques proven to be effective in supporting the academic success of ells. the general characteristics of such environments are that curricula provide “abundant and diverse opportunities for speaking, listening, reading, and writing” and that instruction should “encourage students to take risks, construct meaning, and seek reinterpretations of knowledge within compatible social contexts” (garcia & gonzalez, 1995, p. 424). teachers with documented success with students from non-dominant communities share some characteristics (garcia & gonzalez, 1995): (a) a high commitment to students’ academic success and to student-home communication, (b) high expectations for all students, (c) the autonomy to change curriculum and instruction to meet the specific needs of students, and (d) a rejection of models of their students as intellectually disadvantaged. curriculum policies for ells in mathematics should follow the guidelines for traditionally underserved students (aera, 2006), such as instituting systems that broaden course-taking options and avoiding systems of tracking students that limit their opportunities to learn and delay their exposure to college-preparatory mathematics coursework. what is effective mathematics instruction? according to a review of the research (see hiebert & grouws, 2007), mathematics teaching that makes a difference in student achievement and promotes conceptual development in mathematics has two central features. first, teachers and students attend explicitly to concepts; second, teachers should give students the time to wrestle with important mathematics. mathematics instruction for ells moschkovich principles and guidelines for ells stinson, d. w., & spencer, j. a. (eds.). (2013). privilege and oppression in the mathematics preparation of teacher educators [special issue]. journal of urban mathematics education, 6(1). 49 should follow these general recommendations for high-quality mathematics instruction, for example, by encouraging students to explain their problem-solving and reasoning (aera, 2006; stein, grover, & henningsen 1996). what is effective mathematics instruction for ells aligned with the ccss? first and foremost, mathematics instruction that is aligned with the ccss means teaching mathematics for understanding (hiebert, 1997). all students should use and connect multiple representations, share and refine their reasoning, and develop meaning for symbols. mathematics instruction for ells should align with the ccss, particularly in these four ways:  balance conceptual understanding and procedural fluency. instruction should balance student activities that address important conceptual and procedural knowledge and connect the two types of knowledge (hiebert, 1997; hiebert & grouws, 2007).  maintain high cognitive demand. instruction should use high cognitive demand mathematical tasks and maintain the rigor of tasks throughout lessons and units (stein, grover, & henningsen, 1996; stein, smith, henningsen, & silver, 2000).  develop beliefs. instruction should support students in developing beliefs that mathematics is sensible, worthwhile, and doable (schoenfeld, 1992).  engage students in mathematical practices. instruction should provide opportunities for students to engage in mathematical practices such as solving problems, making connections, understanding multiple representations of mathematical concepts, communicating their thinking, justifying their reasoning, and critiquing arguments (for the ccss for mathematical practice see http://www.corestandards.org/math). recommendations for mathematics instruction for ells effective instruction for ells should have the principles previously noted; these principles are important for mathematics instruction generally and mathematical instruction that is aligned with the ccss specifically. in addition, there are several recommendations that are specific to mathematics instruction for ells. instruction for ells should not emphasize low-level language skills over opportunities to actively communicate about mathematical ideas. research on language and mathematics education provides general guidelines for instructional http://www.corestandards.org/math moschkovich principles and guidelines for ells stinson, d. w., & spencer, j. a. (eds.). (2013). privilege and oppression in the mathematics preparation of teacher educators [special issue]. journal of urban mathematics education, 6(1). 50 practices for teaching ells (moschkovich, 2010). mathematics instruction for ells should address more than vocabulary and support ells’ participation in mathematical discussions as they learn english. instruction should draw on multiple resources available in classrooms (objects, drawings, graphs, and gestures) as well as home languages and experiences outside of school. below, i expand on these general guidelines by providing four recommendations to guide teaching practices.  recommendation #1: focus on students’ mathematical reasoning, not accuracy in using language. instruction should focus on uncovering, hearing, and supporting students’ mathematical reasoning, not on accuracy in using language (moschkovich, 2010). instruction should focus on recognizing students’ emerging mathematical reasoning and focus on the mathematical meanings learners construct, not the mistakes they make or the obstacles they face. instruction needs to first focus on assessing content knowledge as distinct from fluency of expression in english so that teachers can then build on, extend, and refine students’ mathematical reasoning. if we focus only on language accuracy, we miss the mathematical reasoning.  recommendation #2: focus on mathematical practices, not language as single words or vocabulary. instruction should move away from simplified views of language and interpreting language as vocabulary, single words, grammar, or a list of definitions (moschkovich, 2010). an overemphasis on correct vocabulary and formal language limits the linguistic resources teachers and students can use to learn mathematics with understanding. if we only focus on accurate vocabulary, we can miss how students are participating in mathematical practices. instruction should provide opportunities for students to actively use mathematical language to communicate about and negotiate meaning for mathematical situations. instruction should provide opportunities for students to actively engage in mathematical practices such as reasoning, constructing arguments, expressing structure and regularity, and so on.  recommendation #3: recognize the complexity of language in mathematics classrooms and support students in engaging in this complexity. language in mathematics classrooms is complex and includes multiple: representations (objects, pictures, words, symbols, tables, graphs); modes (oral, written, receptive, expressive); kinds of written texts (textbooks, word problems, student explanations, teacher explanations); kinds of talk moschkovich principles and guidelines for ells stinson, d. w., & spencer, j. a. (eds.). (2013). privilege and oppression in the mathematics preparation of teacher educators [special issue]. journal of urban mathematics education, 6(1). 51 (exploratory, expository); and audiences (presentations to teacher, peers, by teacher, by peers).  recommendation #4: treat everyday and home languages as resources, not obstacles. treating home or everyday language as obstacles limits the linguistic resources for communicating mathematical reasoning (moschkovich, 2007d, 2009). everyday language and academic language are interdependent and related—not mutually exclusive. everyday language and experiences are not necessarily obstacles to developing academic ways of communicating in mathematics (moschkovich, 2002, 2007a, 2007b, 2007c). all students, including ells, bring linguistic resources to the mathematics classroom that can be employed to engage with activities designed to meet the ccss. as students continue to expand their linguistic repertoires in english, students can use a wide variety of linguistic resources—including home languages, everyday language, developing proficiency in english, and nonstandard varieties of english—to engage deeply with the kinds of instruction called for in the ccss (bunch, kibler, & pimentel, 2012). guidelines for mathematics practices and materials for ells 4 the guidelines described here are adapted from and based, in part, on work by the understanding language mathematics workgroup. that work, currently under development, aims to provide general guidelines and instructional principles that hold promise for maximizing alignment between mathematics instruction for ells and the ccss for mathematical practice. the work by this discipline specific workgroup (which i am a member) has informed, and been informed by, efforts on the part of the more general understanding language (ul) workgroup that is developing key principles for instruction intended to guide educators and administrators as they work to help ells meet standards in various content areas. as the mathematics workgroup conducted our work, i developed the following guidelines for mathematics instructional materials. the purpose of these guidelines was to develop a shared understanding of how instructional materials and approaches for teaching ells in mathematics might be framed in ways that are aligned with the ccss. these guidelines draw in part on papers prepared for the january 2012 understanding language conference at stanford university (http://ell.stanford.edu/papers/practice) and were modeled after the guidelines for english language arts (ela) materials (bunch, 2012). the guidelines described, 4 these guidelines were developed using the understanding language project’s english language arts unit guidelines as a model (see bunch, 2012). http://ell.stanford.edu/papers/practice moschkovich principles and guidelines for ells stinson, d. w., & spencer, j. a. (eds.). (2013). privilege and oppression in the mathematics preparation of teacher educators [special issue]. journal of urban mathematics education, 6(1). 52 while developed to correspond with the ul project-wide principles and parallel the ela guidelines, are distinct in that they specifically address the ccss for mathematics and are intended to inform the adaptation of mathematics instructional materials to address the needs of ells. 5 1. engage students in the eight ccss for mathematical practice. when designing instruction, consider how students will participate in the eight standards for mathematical practice across the various modes of communication (reading, writing, listening, speaking) that students might use during instruction. it is not necessary to include every practice in every lesson; the goal is to provide students opportunities to actively participate in these mathematical practices when possible and appropriate. ccss for mathematical practice 1. make sense of problems and persevere in solving them 2. reason abstractly and quantitatively 3. construct viable arguments and critique the reasoning of others 4. model with mathematics 5. use appropriate tools strategically 6. attend to precision 7. look for and make use of structure 8. look for and express regularity in repeated reasoning when considering #6 during instruction for ells, it is important to remember that emerging language may sometimes be imperfect and that mathematically precise statements need not to be expressed in full sentences. it is also crucial to recognize that mathematical precision lies not only in using the precise word but also in making precise mathematical claims. 2. keep tasks focused on high cognitive demand, conceptual understanding, and connecting multiple representations. mathematics instruction for ells should follow the general recommendations for high-quality mathematics instruction: (a) focus on mathematical concepts and the connections among those concepts; and (b) use and maintain high cognitive demand mathematical tasks, for example, by encouraging students to explain their problem solving and reasoning (aera, 2006; stein et al., 5 neither these guidelines nor the “understanding language principles” should be confused with the publisher’s criteria for the common core state standards in mathematics, a more extensive document intended for commercial textbook companies and curriculum developers that was prepared by the council of chief state schools officers and others independent from the work of understanding language and which does not focus explicitly on ells. moschkovich principles and guidelines for ells stinson, d. w., & spencer, j. a. (eds.). (2013). privilege and oppression in the mathematics preparation of teacher educators [special issue]. journal of urban mathematics education, 6(1). 53 1996). explanations and justifications need not always include words. instruction should support students in learning to develop oral and written explanations, but students also can show conceptual understanding by using pictures (e.g., a rectangle as an area model to show that two fractions are equivalent or how multiplication by a positive fraction smaller than one makes the result smaller). 3. facilitate students’ production of different kinds of reasoning. instruction and materials should provide opportunities for students to produce different types of mathematical reasoning (i.e., algebraic thinking, geometric thinking, statistical thinking, etc.) and to share and compare reasoning. instruction needs to include different language functions (purposes) such as describing, comparing, explaining, and arguing. although sentence frames can be useful scaffolds, these should be used flexibly and fluidly, more as sentence starters than rigid formulas for producing perfect sentences. 4. facilitate students’ participation in different kinds of participation structures. students should have opportunities to participate in a spectrum of participation structures—from informal collaborative group interactions to formal presentations—in ways that allow them to use their linguistic resources (e.g., first language, everyday language) and cultural resources (e.g., alternative algorithms). materials should provide structures that allow students to collaborate with others, articulate ideas, interpret information, share explanations, present their solutions, and defend claims. teacher led discussions are only one setting for mathematical discussions and instruction should support student participation in classroom mathematical discussions in other settings such as in pairs or in small groups. when creating these different structures, consider student proficiencies not only in english but also in mathematics as well as literacy in their first language. 5. focus on language as a resource for reasoning, sense making, and communicating with different audiences for different purposes. activities calling students’ attention to features of language (e.g., grammatical structures, vocabulary, and conventions of written and oral language) should only occur in conjunction with, and in the service of, engagement with the mathematical ideas, mathematical practices, and multiple representations at the heart of high cognitive demand mathematical tasks. there are many ways to address vocabulary, including introducing, using, and reviewing. the pre-teaching of vocabulary should be carefully considered. moschkovich principles and guidelines for ells stinson, d. w., & spencer, j. a. (eds.). (2013). privilege and oppression in the mathematics preparation of teacher educators [special issue]. journal of urban mathematics education, 6(1). 54 vocabulary should not be introduced in isolation, but instead be included in activities that involve high cognitive demand mathematical work: reasoning, sense making, explaining, comparing solutions, and so on. when introducing new vocabulary, it is useful for students to first have a successful and engaging experience discussing their mathematical reasoning and developing their conceptual understanding, then later label, discuss, and review the vocabulary, having first grounded meanings in actually doing mathematics. 6. prepare students to deal with typical texts in mathematics. typical written texts in mathematics include not only word problems and mathematics textbooks but also other students’ written explanations that are shared in small groups and a teacher’s or a student’s solution written on the board. typical written texts also include assessment problems and scenarios for modeling. oral texts include explanations, descriptions of solutions, conjectures, and justifications. the goal of instruction should not necessarily be to “reduce the language demands” of a written text, but instead to provide support and scaffolding for ells to learn how to manage complex text in mathematics. there are several reasons to not adapt the language of a task: (a) changing the language of a task can change the mathematical sense of the task; (b) it is not yet clear which adaptations are best to make for which students, for which purposes, or at which times; (c) instruction should support students in understanding complex mathematical texts as they are likely to appear in curriculum and assessment materials; and (d) experiences that allow ells to engage with authentic language used in mathematics (with support) can provide opportunities for their continued language development. closing thoughts equity and social justice considerations require that ells have access to high-quality and effective mathematics instruction. currently, we do not have a set of empirical studies showing that a specific curriculum, teaching approach, or instructional practice is the cause for an effect on the learning, achievement, or motivation for ells. however, we have decades of research on effective teaching for students from non-dominant communities, even if not specifically in mathematics. we also have reviews of research pointing to the general characteristics of effective mathematics teaching, not specific to ells but still relevant. the recommendations summarized here are an attempt to collect what we already know while we continue to conduct more research relevant to mathematics teaching for ells. moschkovich principles and guidelines for ells stinson, d. w., & spencer, j. a. (eds.). (2013). privilege and oppression in the mathematics preparation of teacher educators [special issue]. journal of urban mathematics education, 6(1). 55 when i attended the privilege and oppression in the mathematics preparation of teacher educators (prompte 6 ) conference, i was involved in work with the understanding language mathematics workgroup. at that time, i had just completed the first phase of a project developing resources for teachers to address the needs of ells in their mathematics instruction. the goal of that project was to develop materials to illustrate how mathematical tasks aligned with the ccss can be used to support mathematics instruction for ells. 7 during the prompte conference, i decided to use that work to also develop a set of general principles for designing instruction and reviewing materials because i hoped these principles could provide resources for mathematics educators. i left prompte deeply committed to doing something that could inform practice. the set of principles outlined here is thus a result, not only of my work with the understanding language project but also of the discussions and conversations at prompte. my intention in this essay was not to provide a perfect definition of equitable teaching practices for ells, but rather to establish some common ground using reviews of relevant empirical research. it is my sincere hope that the principles, recommendations, and guidelines provided prove useful for designing equitable mathematics instruction, reviewing curriculum materials, and supporting mathematics educators in preparing new teachers. acknowledgments the following people were writers for the annotated mathematics tasks that served to generate these guidelines for instructional materials: grace davila coates, vinci daro, lucy michal, katherine morris, cody patterson, nora ramirez, and jeanne f. ramos. the writing of these guidelines also benefitted from the advice of several “critical friends.” as part of the understanding language initiative, george bunch, phil daro, maria santos, judith scott, guadalupe valdes, and aida walqui provided advice on these guidelines. the following were reviewers for the mathematics resources: harold asturias, sylvia celedón-patichis, alma ramirez, susie hakansson, erin turner, and steven weiss. 6 privilege and oppression in the mathematics preparation of teacher educators (prompte) conference (funded by create for stem institute through the lappan-phillips-fitzgerald cmp 2 innovation grant program), michigan state university, battle creek, mi, october 2012. any opinions, findings, and conclusions or recommendations expressed herein are those of the authors and do not necessarily reflect the views of the funding agency. 7 the workgroup members used tasks from two publicly accessible curriculum projects: inside mathematics (see http://www.insidemathematics.org) and mathematics assessment project (see http://map.mathshell.org/materials/index.php ). members of the workgroup developed the materials and a team of experts reviewed the materials; all materials developed will be available online at the understanding language website (see http://ell.stanford.edu/). http://www.insidemathematics.org/ http://map.mathshell.org/materials/index.php http://ell.stanford.edu/ moschkovich principles and guidelines for ells stinson, d. w., & spencer, j. a. 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(2002). a balancing act: developing a discourse community in a mathematics classroom. journal of mathematics teacher education, 5, 205–233. sherin, m., & van es, e. (2005). using video to support teachers’ ability to notice classroom i nteractions. journal of technology and teacher education, 13, 475–491. stein, m. k., engle, r. a., smith, m. s., & hughes, e. k. (2008). orchestrating productive mathematical discussions: five practices for helping teachers move beyond show and tell. mathematical thinking and learning, 10, 313–340. stein, m. k., grover, b., & henningsen, m. (1996). building student capacity for mathematical thinking and reasoning: an analysis of mathematical tasks used in reform classrooms. american educational research journal, 33, 455–488. stein, m. k., & smith, m. (2011). five practices for orchestrating productive mathematics discussions. reston, va: national council of teachers of mathematics. stein, m. k., smith, m. s., henningsen, m. a., & silver, e. a. (2000). implementing standardsbased mathematics instruction: a casebook for professional development. new york, ny: teachers college press. http://www.nctm.org/news/content.aspx?id=22838 http://ell.stanford.edu/publication/mathematics-common-core-and-language journal of urban mathematics education july 2013, vol. 6, no. 1, pp. 58–70 ©jume. http://education.gsu.edu/jume craig willey is an assistant professor of teacher education and mathematics education at indiana university school of education at iupui, 902 w. new york street – es 3156, indianapolis, in, 46236; e-mail: cjwilley@iupui.edu. his research focuses on the preparation and development of mathematics teachers of latinas/os and other bilingual student populations, as well as the improvement of mathematics curriculum to increase access for and enhance engagement of bilingual students. corey drake is an associate professor and director of teacher preparation in the department of teacher education at michigan state university, 620 farm lane, room 118a, east lansing, mi, 48824; e-mail: cdrake@msu.edu. her research interests include the design of teacher education experiences and contexts to support new teachers in learning to connect to students’ resources and knowledge bases, as well as the use of curriculum materials as learning tools for teachers. advocating for equitable mathematics education: supporting novice teachers in navigating the sociopolitical context of schools craig willey indiana university purdue university indianapolis corey drake michigan state university in this essay, the authors situate elementary mathematics teacher preparation in a broader, sociopolitical context, one that includes historical patterns of educational privilege and oppression. the authors attend to the effects of “reform” movements that encompass a vast array of stakeholders and interests as well as the growing significance of federal education policy on mathematics teacher education. in particular, they highlight the tensions involved in prospective teachers’ first experiences in attempting to make sense of how research-based theories of learning and practice intersect with local schooling realities. the authors present questions that novice mathematics teachers might ask at the personal, interpersonal, institutional, and cultural levels; questions which hold the potential to disrupt dominant discourses and initiatives in favor of discourses that reframe mathematics education opportunities for oppressed youth in the united states. keywords: mathematics education, novice teachers, oppression, sociopolitical context, urban education mathematics education class for pre-service teachers (psts) reads “exploring area and perimeter – the case of isabelle olson,” a chapter in smith, silver, and stein’s (2005) book improving instruction in geometry and measurement: using cases to transform mathematics teaching and learning. the particular case of isabelle olson portrays the story of a teacher who is committed to providing her seventh-grade students with rich mathematical tasks (i.e., tasks that do not clearly indicate immediate solutions to students). thus, the story also is about the students’ experiences of mathematical disequilibrium, which characa willey & drake navigating the sociopolitical context of schools stinson, d. w., & spencer, j. a. (eds.). (2013). privilege and oppression in the mathematics preparation of teacher educators [special issue]. journal of urban mathematics education, 6(1). 59 terizes the state of confusion felt by students as they engage in “productive struggle” (schoenfeld, 2013; warshauer, 2011) and persist with novel, multi-faceted problems. below is the mathematical task given to students in the case: ms. olson’s 7 th -grade class at roosevelt middle school will raise rabbits for their spring science fair. the class will use some portion of the school building as one of the sides of its rectangular rabbit pen and will use the fencing that was left over from the school play to enclose the other three sides of the pen. if ms. olson’s class wants its rabbits to have as much room as possible, what would the dimensions of the pen be? try to organize your work so that someone else who reads it will understand it. (pp. 24–25) details provided in the case clearly describe the tensions experienced by the teacher and her students: the students crave more assistance and guidance, the teacher struggles to construct questions that provoke thought but do not steer students in a pre-determined direction. in essence, ms. olson struggles to support students’ development of critical, mathematical habits of mind while working with urban students who have been strongly socialized in a particular set of sociomathematical norms and values. 1 for example, it has been our experience working with urban fourth-graders that children have often internalized the following protocol when presented with a word problem: (a) isolate and pull out the numbers (if those numbers are not already provided to them in the form of an equation), and (b) identify key words that index the operation that is to be performed on these numbers. this approach may or may not have been explicitly taught by the teacher; nonetheless, it is the net effect of thousands of mathematical experiences in the students’ first years of formal schooling (kamii & dominick, 1998), especially in classes with high percentages of students of color (brenner, 1998; ladson-billings, 1997; lipman, 2004). as mathematics teacher educators, we have seen many psts—particularly those preparing to teach in urban schools—share ms. olson’s will to implement a mathematics program that represents a radical departure from the mathematics instruction with students of color that predominates nationwide in their attempts to re-define what it means to do, learn, and be good at mathematics. they aspire to ground their curriculum in realistic problem contexts, ones that necessitate that students work together, thus pushing against the math-as-individual activity paradigm. the problems, psts often proclaim, should be intentionally designed so that kids are not just maneuvering mathematically through the problem’s various 1 mathematical habits of mind: to understand the nature of a problem, develop conjectures, test conjectures, disprove conjectures with counterexamples, collect and organize data, generalize, and so on. mathematical habits of mind are consistent with the standards for mathematical practice advocated for in the common core state standards initiative (see http://www.corestandards.org/math). http://www.corestandards.org/math willey & drake navigating the sociopolitical context of schools stinson, d. w., & spencer, j. a. (eds.). (2013). privilege and oppression in the mathematics preparation of teacher educators [special issue]. journal of urban mathematics education, 6(1). 60 steps, neatly laid out in parts a–d, but instead are living the problem, determining the parameters of the problem, and collaboratively determining the most sensible approaches. the nature of the mathematical activities demands that students dialogue, organize their work, and make impromptu presentations to either the teacher or their classmates. simply put, this is mathematical activity re-defined, and this re-definition of mathematics and mathematical processes has important implications, particularly for historically marginalized students. this commitment, however, requires flexible time—perhaps not conforming neatly to existing curriculum pacing guides. despite an awareness of the kinds of mathematics learning experiences that ought to be afforded to students, the current context of schooling—steeped in traditional structures and practices and simultaneously experimenting with impactful (and restrictive) curricular and pedagogical interventions and mandates—provides teachers with few opportunities and resources to alter the way mathematics might be experienced in their classrooms. there is a significant tension between the will of the psts (supported by ms. olson’s example and the common core state standards for mathematical practice) and the pragmatic realities of schooling today. as psts experience their first field placements, this tension is often revealed in comments such as:  i’m not sure we could spend two or three days on this activity given all that needs to be covered [for the standardized test].  i would be surprised if students would persist, or have the stamina, to work on a problem like this for two days. i would have given up right away.  my students have never shared their strategies before.  i think the teacher should have offered more guidance to the students, maybe have gone over area and perimeter before the problem.  i do not see problems like this in my classroom’s mandated curriculum materials. these kinds of comments, which invariably surface each semester, represent particular mathematics ideologies and a struggle to envision mathematics teaching and learning in ways that are significantly different than the ways in which psts experienced mathematics as students. this gap between the ways in which psts learned mathematics and the ways in which they are being asked to teach mathematics is not new (ball, 1990). however, what is new—or at least exacerbated— in the current sociopolitical context is that the kind of teaching exemplified by ms. olson is simultaneously promoted by the adoption of the common core state standards for mathematical practice and, at the same time, constrained by a number of other federal, state, and local education mandates related to curriculum, instruction, and assessment. this tension created by simultaneous and conflicting willey & drake navigating the sociopolitical context of schools stinson, d. w., & spencer, j. a. (eds.). (2013). privilege and oppression in the mathematics preparation of teacher educators [special issue]. journal of urban mathematics education, 6(1). 61 policies is particularly acute in urban schools and classrooms; that is, while the mathematics education field is getting more articulate as to what constitutes strong mathematics pedagogy and corresponding student activity, the increasing pressures of punitive accountability structures are felt disproportionately on urban schools, resulting in a compromised curriculum and pedagogy and an often unstable school culture that revolves around test performance (lipman, 2004). moreover, when we consider the distinct ideologies that prevail around what is needed to remediate students who register achievement levels below grade level— disproportionately latina/o and african american students—we complicate matters even further. with a focus on the tensions experienced by psts and novice teachers, this essay aims to highlight the struggles currently endured by urban schools as part of a larger “reform” movement that encompasses a vast array of stakeholders and interests as well as the growing significance of state and federal education policy on mathematics teacher education. we suggest that the current political climate exacerbates schooling practices that already have the effect of sorting students into mathematical proficiency groups, ultimately contributing to lasting mathematical identity formations, many of which are negative. we claim that this context is oppressive not only for pre-k–12 students but also for psts and practicing teachers. given this reality, we aim to equip psts and novice teachers with ways they can disrupt oppressive mathematics teaching and learning arrangements by asking pointed questions of themselves, their peers, administrators, and lay people—all in an effort to shift the discourse and highlight the tensions that exist at various levels of the educational system around quality mathematics learning and teaching. background (briefly) on inequitable mathematics experiences critical, or anti-oppressive, mathematics education has been an emergent topic for both researchers and practitioners for decades (see, e.g., frankenstein, 1989, 1992; gutstein, lipman, hernandez, & de los reyes, 1997; khisty, 1995; secada, 1992). yet, achieving the vision of equitable mathematics teaching and learning practices remains elusive, especially for black and latina/o youth. the last thirty years or so have proven to be an intense struggle to push against prevalent beliefs around who can learn mathematics, elevate the realities of black and brown students in mathematics classrooms, and advocate solutions to address their mis-education (gutiérrez & irving, 2012; martin, 2009). gutiérrez (2012) argues this work is still “in its infancy” (p. 26). in general, a large portion of the field’s attention has been committed to understanding and improving preand in-service teachers’ mathematics content knowledge (ball, lubienski, & mewborn, 2001). while teachers’ mathematics willey & drake navigating the sociopolitical context of schools stinson, d. w., & spencer, j. a. (eds.). (2013). privilege and oppression in the mathematics preparation of teacher educators [special issue]. journal of urban mathematics education, 6(1). 62 content knowledge and mathematics knowledge for teaching (hill et al., 2008) are certainly important in the context of urban students’ mathematical achievement, some researchers have proposed that we might be overlooking other significant factors contributing to the mathematical success of youth of marginalized populations (e.g., milner, 2013). martin (2007, 2009), for example, raises serious questions about what makes a “qualified” or “effective” mathematics teacher of black students. assuming that mathematics learning is a racialized experience (martin, 2006), it is not appropriate to treat mathematics as a neutral body of knowledge or set of skills to be acquired indiscriminately. surely, issues of relevance, meaning making, and interaction need to be considered and scrutinized. with respect to latina/o learners, many of whom are learning mathematics in a second language, issues of language and discourse are just now beginning to capture the attention of wider audiences (moschkovich, 2012). it has been a difficult journey to illuminate the mathematics learning process as language intensive and to dispel myths that mathematics is a “universal language.” certainly, with the rise of the mathematical practice of the common core state standards, attention to the role of mathematical communication has increased. as we also consider the fact that schools historically do not recognize nor capitalize on latinas/os’ lived experiences, this attention to the dynamics and complexities of language holds great promise to reverse the dismal state of mathematics education amongst latinas/os (gándara & contreras, 2009). though this picture of inequitable mathematics and schooling opportunities is becoming clearer, issues of privilege and oppression (i.e., neglecting students’ cultural and intuitive mathematics knowledge; granting mathematical authority to only the teacher, the textbook, or a few outstanding students; leaving unchallenged current constructions of what it means to do and learn mathematics) are still too infrequently included in mathematics teacher preparation programs to help novice teachers understand—and develop agency within—the sociopolitical complexities of mathematics learning environments. it is this agency we hope to ignite through this initial attempt at developing a meaningful line of questions for psts and novice teachers to use to disrupt current discourses around mathematics, youth, and educational reforms (i.e., interventions) and policies. overview (briefly) of state and federal initiatives u.s. education policy is firmly in the accountability era. though this movement started long before no child left behind, this reauthorization effort of the elementary and secondary education act (u.s. department of education, 2001) catapulted schooling into corporate-style accountability at all levels (anagnostopoulos, rutledge, & jacobsen, 2013). a distinct byproduct of this movement is a spotlight on “failing” schools—narrowly defined and grossly under willey & drake navigating the sociopolitical context of schools stinson, d. w., & spencer, j. a. (eds.). (2013). privilege and oppression in the mathematics preparation of teacher educators [special issue]. journal of urban mathematics education, 6(1). 63 nuanced—and a corresponding public discourse portraying teachers and schools as inadequate to help students reach pre-determined standards. the accountability era was ushered in long before standards were clearly defined or metrics to assess learning were adequately designed and validated. by all accounts, we are engaged in these tasks of definition and design decades after rigid accountability systems were put in place. nonetheless, the u.s. department of education is expanding its influence and elevating the role of high-stakes assessments, data, and corresponding punitive measures through the use of competitions for federal resources (e.g., race to the top) that, to a significant degree, depend on the adoption of the common core state standards and participation in one of the two federally-funded test consortia (mcneil, 2013). the effects of these accountability systems, based on the increased use of high-stakes tests, have been felt in schools in all corners of the united states. scholars argue, however, that the effects are amplified in schools serving students of color and the poor (lipman, 2004; valenzuela, 2005). with great detail, lipman (2004) shows what the culture of urban schools—schools that districts and private education management organizations (emos) are eager to take over— feels like for students, teachers, and administrators when the pressures to perform on tests wholly consume all aspects of school life. in a recent blog titled “are we decimating the teaching profession?” (sept. 11, 2012), diane ravitch implicates our obsession with tests and test scores as a major reason why veteran and novice teachers alike are leaving the profession, resulting in first-year teachers being the most populous sub-group of teachers. as she points out, this cannot bode well for students. to compound matters further, there has been a dramatic increase in the educational intervention industry. with each new fad, a flurry of development and moneymaking opportunities arise, consuming a significant portion of district and school budgets (lipman, 2004). take, for example, response to intervention (rti), a widely adopted approach “to the early identification and support of students with learning and behavior needs”; upon screening of all students, “struggling learners are provided with interventions at increasing levels of intensity to accelerate their rate of learning” (rti action network, 2011). the ideas behind rti might be reasonable, but the interventions rarely amount to more than “reteaching” (martinez & young, 2011). moreover, it often leads to the practice of segmenting and labeling both students and time (i.e., rti time, tier 2 intervention time, etc.), with little thought given to what we are “doing” to, for, or with children (artiles, bal, & king-thorius, 2010). the point here is to highlight our complicity in implementing programs that don’t necessarily have the uniform or positive impact that is intended. as another example—and one that is making huge ripples in the education landscape—the development of the common core state standards and the two willey & drake navigating the sociopolitical context of schools stinson, d. w., & spencer, j. a. (eds.). (2013). privilege and oppression in the mathematics preparation of teacher educators [special issue]. journal of urban mathematics education, 6(1). 64 new assessment consortia are being held up as the “silver bullet” that will achieve educational equity across and within schools. while the standards for mathematical practice hold the potential to make a significant contribution to mathematics pedagogy at scale in the united states, as foreshadowed through the teaching of ms. olson, the common core state standards have been adopted based on a premise that the current education “problems” are a result of: (a) a lack of curricular clarity and uniformity across the nation, (b) a lack of alignment between the current education system and the needs of local and national economies, and (c) overwhelmingly poor instructional practices. while the standards were designed to influence not only what is taught but also how it is taught, the balance between changes in content and changes in pedagogy may well be determined by the nature of the assessments that are developed. again, this reinforces a particular type of top-down educational policy, one that uses accountability measures (i.e., standardized tests representing so-called “assessment advances”) as levers for educational change. given the wide array of problems that the common core state standards portends to solve, it is worth our time to consider and assess its underlying assumptions. more importantly, we need to engage in a conversation around what it means for the standards to have been developed within the context of current educational and economic institutions. why is it that the standards do not mention the root causes of the achievement gap, namely, racism, oppression, privilege, power, and poverty? the absence of these considerations suggests that the standards and the corresponding assessments are not necessarily positioned well to create more equitable mathematics learning arrangements for urban and other marginalized youth. finally, these forces (i.e., policies controlling time and space, the growth of the education industry, the standardization of curriculum and assessment) and tensions have resulted in the implementation of new accountability systems for prek–12 teachers and schools and, by extension, for teacher preparation programs. as schools continue to work to comply with rti, mandated and scripted curriculum materials, the adoption of the common core state standards, and numerous other policies governing curriculum, instruction and assessment, they are, in some states, also struggling to design and implement new teacher evaluation systems. given the constraints and tensions involved in trying to negotiate multiple mandates, teachers, schools, and districts are increasingly less willing or able to work with prospective teachers at all, much less support them in innovating in the ways exemplified by ms. olson. these laws and policies steering practicing teachers’ work, accompanied by external definitions of “quality” teaching, compromise efforts to prepare new, innovative teachers like ms. olson, the same teachers that schools and districts will be interviewing in a few short months or years. with willey & drake navigating the sociopolitical context of schools stinson, d. w., & spencer, j. a. (eds.). (2013). privilege and oppression in the mathematics preparation of teacher educators [special issue]. journal of urban mathematics education, 6(1). 65 limited access to and participation with the daily operations of classrooms and schools, psts’ development is short-changed. at a time when navigating the sociopolitical landscape of mathematics teaching and learning grows increasingly complex, teacher preparation programs and school districts are looking to each other for help; new roles and responsibilities are emerging. it is becoming clearer that psts need a different kind of preparation, one largely anchored in the clinical experience (grossman, 2010; zeichner, 2010). this inevitably demands that university-school partnerships are strengthened and pre-k–12 schools assume an amplified role in teacher preparation. without a deliberate and sustained investment in the preparation of psts, schools will be faced with an increased workload in terms of the induction and support of novice teachers, a challenging task given that schools and districts often “lack an understanding of the learning and needs of beginning teachers and of the resources required to create effective [induction] programs” (feiman-nemser, 2003, p. 25). at the same time, these partnerships are increasingly constrained by policies and mandates that reduce incentives for pre-k–12 schools to work with psts or for teacher education programs to work in struggling schools at all, much less in support of the development of the kinds of pedagogies exemplified by ms. olson. teachers as disruptors: four levels to influence oppressive arrangements given this reality, it is increasingly important that teacher preparation programs help teachers develop a critical consciousness about the sociopolitical context of schooling and assume an activist stance to both “play the game and change the game” (gutiérrez, 2008). even within a time where it feels like there is little flexibility and support to innovate meaningful mathematics teaching practices, teachers can still own a curriculum and pedagogy that is relevant to young learners and reflects different mathematical objectives and outcomes, as illustrated in “the case of isabelle olson.” in this spirit, we propose that both preand inservice teachers consider and maximize their sphere of influence at four different levels: personal, interpersonal, institutional, and cultural (batts, 1998; 2002). 2 2 these levels were highlighted for us at the convening of the privilege and oppression in the mathematics preparation of teacher educators (prompte) conference. consequently, it led us to consider the ways in which they could be useful in our work with pre-service and novice teachers. privilege and oppression in the mathematics preparation of teacher educators (prompte) conference (funded by create for stem institute through the lappan-phillips-fitzgerald cmp 2 innovation grant program), michigan state university, battle creek, mi, october 2012. any willey & drake navigating the sociopolitical context of schools stinson, d. w., & spencer, j. a. (eds.). (2013). privilege and oppression in the mathematics preparation of teacher educators [special issue]. journal of urban mathematics education, 6(1). 66 the personal level refers to the ways in which we recognize our privilege and the oppression of others; it also represents how we might challenge our own deeply held belief systems (and their origins) around what it means to live in a stratified society. the interpersonal level signifies the ways in which our interactions with other individuals might disrupt oppressive arrangements. particular attention might be given, for example, to the language we use with others, the authorities we invoke, and the perspectives we privilege. the institutional level refers to the arrangements within organizations and institutions that either perpetuate or challenge oppressive relationships. similar to other levels, this level is likely to be met with strong resistance as we challenge seemingly “efficient” or “effective” modes of operating (e.g., transitioning to test preparation activities as standardized tests near). seemingly benign practices need to be scrutinized for the sometimes-tacit role they play in subjugating individuals or groups of people. finally, the cultural level refers not to particular groups that share customs and values as “culture” is conventionally used, but rather to the broader audiences that both directly and tangentially influence the ways in which people are either oppressed or liberated. at this level, we must ask: in what ways do i contribute to or challenge prevalent public discourses that affect privileged and oppressive arrangements? given the pressures described above that are particularly acute for novice teachers and in urban settings, we feel a tension in asking psts and novice teachers to do more by posing these questions and advocating for themselves and their students at each of these levels of the system. at the same time, part of our responsibility as teacher educators is to support psts in learning to disrupt privilege and oppression in mathematics education—in part through these questions—in order to be able to teach in the ways exemplified by ms. olson. in a job where students’ development is directly connected to teachers’ own willingness to grow; where pedagogical practices and interactional style can serve to empower or alienate students; where teachers have an important role to play in building-level, decision-making processes with serious implications; and, where teachers’ livelihood is strongly affected by the sentiments and support of the general public, it is crucial that we confront oppression and privilege at all levels. how we view and promote ourselves as teachers and teacher educators; how we resist being cast as self-serving and incompetent; how we frame and discuss urban youth, and elevate their capacity and accomplishments; these discourses matter! as luke (1995) points out, shifts in discourses and the corresponding social movements (or vice versa) are effective vehicles to influence our own and others’ educational experiences and realities (luke, 1995). in this vein, freire (1985) reminds us, “as a referent for change, education represents a form of action that emerges from a join opinions, findings, and conclusions or recommendations expressed herein are those of the authors and do not necessarily reflect the views of the funding agency. willey & drake navigating the sociopolitical context of schools stinson, d. w., & spencer, j. a. (eds.). (2013). privilege and oppression in the mathematics preparation of teacher educators [special issue]. journal of urban mathematics education, 6(1). 67 ing of the languages of critique and possibility” (p. 16). as such, below are some initial and representative questions that psts and novice teachers might ask themselves—or others—to disrupt oppression as it manifests in student-teacher interactions, classroom pedagogy, school-level policy or initiatives, and the broader public discourse. personal. how am i privileged? in what ways are my views of teaching mathematics to students of color influenced by my own upbringing? to what degree do i want for them what i had/have? what kinds of power am i granted as i make mathematics curricular and pedagogical decisions? how well do i understand how this privilege and power affect others? why do i teach mathematics to students of color? interpersonal. how might i interact with students in a way that reflects their histories and helps promote positive mathematics identities? how might my conversations with my peers and mentor teachers acknowledge students’ social and mathematical positioning and re-frame the learner and her or his needs? how might i come to understand my mentor teacher’s (or colleague’s) history, identities, and perspective? which critiques do i have of my mentor teacher? how might i transform my critique of my mentor teacher into thoughtful, provocative, and action-driven questions? institutional. which mathematics initiatives, structures, and norms are present? who do these initiatives and norms serve? in what ways do they reinforce or dismantle patterns of underachievement and/or student success? how might i raise awareness around mis-guided educational initiatives—or those initiatives that are well-intended, but improperly implemented or poorly resourced—that too often lead to the narrowing of the curriculum? what kinds of evidence would convincingly illuminate the mathematical practices and initiatives that do not afford students of color with the kind of mathematical experiences we would want for our own children? on what grounds do i endorse or resist proposed mathematics programs or policies at my school or within my district? cultural/larger public sphere. how might i, through my dialogue and actions, challenge the current public discourse around the ineptitude of teachers and learners—particularly around what it means to be a mathematics teacher of urban students? how might i incorporate mathematical activity that involves and supports our community? how might i publicize my students’ mathematics work and growth (beyond test scores)? how might i articulate and defend other components of students’ mathematical achievement (e.g., identity)? concluding thoughts we have to refocus on teaching as intellectual and ethical work, something beyond the instrumental and the linear. we need to understand that teaching requires willey & drake navigating the sociopolitical context of schools stinson, d. w., & spencer, j. a. (eds.). (2013). privilege and oppression in the mathematics preparation of teacher educators [special issue]. journal of urban mathematics education, 6(1). 68 thoughtful, caring people to carry it forward successfully, and we need, then, to commit to becoming more caring and more thoughtful as we grow into our work. this refocusing requires a leaning outward, a willingness to look at the world of children—the sufferings, the accomplishments, the perspectives, and the concerns— and an awareness, sometimes joyous but just as often painful, of all that we find. and it requires, as well, a leaning inward—inbreathing, in-dwelling—traveling toward self-knowledge, a sense of being alive and conscious in a going world (ayers, 2006, p. 271). as we (and others) have pointed out, there are many mathematics reform efforts occurring simultaneously in our cities, counties, states, and at the federal level. there are also many drivers: philanthropists, commerce leaders, politicians, and parents among them. teachers carry the burden of implementing policies and initiatives, whether they have advocated for them or not, and they are held increasingly accountable for an array of outcomes, many beyond their control and representative of someone else’s agenda. there are a myriad of consequential curricular and work-related decisions being made that limit the times and spaces that teachers have available to make their own decisions about what and how to teach children mathematics. for example, teachers seem to have less autonomy in how they spend their planning time, as professional learning communities and their corresponding agendas tend to be determined with little input from teachers, or their instructional time, as rti, test preparation, and mandated curricula each claim their own portion of classroom time. while we continue to work within teacher education programs to help psts develop and utilize innovative pedagogies and critical perspectives, we, as teacher educators and researchers, also need to be more active in supporting teachers as they move into these roles and understand the nature of privilege and oppression in mathematics teaching and learning, particularly in urban settings. while many may argue the u.s. education landscape looks grim, we continue to recognize the importance of engaging in the struggle to envision a different paradigm of school operations and relay the crucial role teachers play in enacting the shifts we want to see. mathematics, in particular, is entrenched in historical traditions and granted high status by society. in some ways, it makes sense that those who possess the status of being proficient mathematically, and who have voice in determining what criteria it takes to be successful at mathematics, would be vested in the status quo and reluctant to radically re-define the way we teach, learn, and use mathematics. we need to be cognizant of the ways in which privilege and oppression rear their head in our classrooms, schools, and district offices. teachers, conscious of the limitations of their mathematical experiences, recognize the need for change, and also the challenges that come with it. now, it is up to us all to sharpen our sociopolitical lenses in order to notice and disrupt manifestations of privilege and oppression in mathematics education. willey & drake navigating the sociopolitical context of schools stinson, d. w., & spencer, j. a. 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(unpublished doctoral dissertation). university of texas at austin, austin, tx. zeichner, k. (2010). rethinking the connections between campus courses and field experiences in college-and university-based teacher education. journal of teacher education, 61(1-2), 89–99. http://www.edweek.org/ew/index.html http://dianeravitch.net/2012/09/18/are-we-decimating-the-teaching-profession/ http://www.rtinetwork.org/ http://www2.ed.gov/policy/elsec/leg/esea02/index.html journal of urban mathematics education july 2015, vol. 8, no. 1, pp. 24–30 ©jume. http://education.gsu.edu/jume gareth bond is an undergraduate student in the college of education at the university of saskatchewan, canada; email: gjb492@mail.usask.ca. he is looking forward to beginning his teaching career and furthering his studies. egan j. chernoff is an associate professor in the college of education at the university of saskatchewan – 28 campus drive saskatoon sk s7n0x1 canada; email: egan.chernoff@usask.ca. he is an ardent user of social media for mathematics education; follow him @matthewmaddux. commentary mathematics and social justice: a symbiotic pedagogy gareth bond university of saskatchewan egan j. chernoff university of saskatchewan athematics can be defined as “the science of pattern and order” (van de walle, folk, karp, & bay-williams, 2009, p. 10). but because there is often a perceived spectrum of approachability to mathematics (based on common misconceptions that envision the subject as a sort of elitist wizardry) it is important to bear in mind different definitions of mathematics when exploring applications of mathematics in the classroom. this is especially true when considering the instruction of mathematics for social justice. traditional stigmas have led many to view mathematics and social justice as being positioned on opposing ends of a spectrum describing quantitative and qualitative reasoning and, thus, unsuitable for integration. garii and appova (2013) noted that many new teachers struggle with the idea of integrating mathematics and science with social justice issues because their own limited understandings of mathematics (and science) cannot accommodate the notion. it is this fundamental comprehension of mathematics as an approachable and understandable science that can shatter the illusion of academic segregation and begin the integration of mathematical understanding into a realistic and holistic field of academic study. the study of social justice is increasingly in need of empirical methods to describe, defend, and advise the critical analysis of the systems of domination and subjugation that permeate human power structures. the study of mathematics, which often needs a meaningful context in which abstraction and anxiety can be nullified, is the ideal symbiotic partner for the study of social justice in the greater pursuit of equipping students with the effective tools needed to thrive in the 21st century. without a literacy of mathematics and social justice, students will be at the mercy of sociopolitical and economic systems of oppression. to make the science of mathematics available to every student, teachers should first create a classroom environment in which the learning of mathematics is accessible to all students. van de walle and colleagues (2009) explored the issue of unequal accessibility (in the context of gender) and posit that such problems are “largely a function of the educational environment” (p. 101). indeed, the “traditionm http://education.gsu.edu/jume mailto:gjb492@mail.usask.ca mailto:egan.chernoff@usask.ca bond & chernoff commentary journal of urban mathematics education vol. 8, no. 1 25 al” classroom environment in which north american school children learn mathematics has produced a dominance of white men in mathematical arenas (steele, james, & barnett, 2002). as curricula mandate an approach of equal opportunity (e.g., saskatchewan ministry of education, 2010), it is vital that all students be given inclusive and engaging mathematical instruction. van de walle and colleagues (2009) affirmed this mandate by their assertion that the inclusivity of mathematics education must be addressed by changes in the classroom environment. teachers must shift the traditional environment of the classroom significantly to facilitate an equitable accessibility for all students. the need for an alternative approach is clear. the national council of teachers of mathematics’ (nctm) recommendations for mathematics education outlined in the principles and standards for school mathematics (2000) suggest that teachers focus on five process standards in their classrooms: problem solving, reasoning and proof, communication, connections, and representations (bossé, lee, swinson, & faulconer, 2010, p. 263). this approach decentralizes the teacher’s authoritative role and places much of the learning in the hands of the students as they communicate and critique their reasons and representations among each other. this decentralizing increases the chances for students to make meaningful connections with the material and reduces the risk of instructional bias, on a personal or institutional scale, from excluding historical marginalized groups in classrooms. it is not difficult to understand how these process standards can be a step toward teaching mathematics in an equitable manner. however, mathematical skills learned in this way are still at risk of being abstracted, devoid of authentic context, and isolated from applications to social justice. taking into consideration particular nctm goals, set, arguably, as a “first step” toward weaving the studies of mathematics and social justice, teachers should ensure that the mathematical problems and concepts presented can provide insights into authentic social justice issues within the context of the learners’ communities. there are surely skeptics who doubt that this can be done effectively. fortunately, the radical math website (see http://www.radicalmath.org) provides a wealth of free lesson plans, which effectively integrates the interests of mathematics and social justice within the fabric of community (see also gutstein & peterson, 2013). one such lesson, “community voices heard: statistics – survey project,” examines age, gender, sex, and economics in the community with the statistically informed critical lens of social justice (osler, 2007). the study begins with an exploration of the statistical processes through some engaging problem solving and discussion. ultimately, the students are responsible for surveying people in their own community or school on critically relevant topics. a result of this approach, according to one of the project’s rubrics, is to provide students with the ability to “compare relevant sets of data” with a variety of tables and graphs created with the very data they collected; in order to, as stated by the authors, “determine key findings http://www.radicalmath.org/ bond & chernoff commentary journal of urban mathematics education vol. 8, no. 1 26 from [the] data” (p. 32). the mathematical demands of the rubric are ambitious and the focus of the activities gives the students significant and empirical insights into the authentic and pertinent social justice issues within their communities. when mathematics is taught in this way, it may move beyond a passively equitable implementation toward an active one, that is, an instructional method in which the dynamic, authentic, and critical approaches potentially reflect the subjects of study. embracing an actively progressive approach toward teaching mathematics for social justice also has the potential to arm students with the authentic tasks and tools needed to develop their mathematical and critical abilities. this type of learning may also perpetuate itself, as both students and teachers become active in creating increasingly interrelated “real-world” connections with mathematics and social justice in their communities. having witnessed a successful demonstration of mathematics and social justice working in conjunction, some skeptics may still question the need to integrate these two fields of study. in addition to the strong case that can be made regarding the powerful intrinsic motivations that can arise within learners through such a partnering of subjects, a critical understanding of economics must also be considered a primary reason for integration. a survey of financial literacy across 28 nations by jump$tart coalition for personal financial literacy (2013) showed that the vast majority of teenaged students are worried about the impact of the economic recession and would like to learn about finances in school before entering into the possibility of losing money in the real world. the study also showed that in the united states the average credit card debt is $15,266.00, the average mortgage debt is $149,667.00, and the average student loan debt is $32,559.00. clearly, an understanding of the mathematics involved in debt and other financial operations is required. a thoughtfully crafted lesson could utilize a variety of local financial actions and entities in a comprehensive and critical study. such a lesson could provide a window of understanding, in advance, of the pitfalls and responsibilities of adult expectations within the sphere of modern economics. it could also address the need for the basic skill set required to function within the current financial parameters of a given community. however, the critical aspect of such a lesson might not be sufficient. social justice is, ultimately, about understanding and correcting the macrosystemic power dynamics that perpetuate the conditions in which we live. by conducting a critical study of one’s community, students can reach a partial understanding of such global power systems. to fully embrace an integrated approach of mathematics and social justice, teachers could contextualize the functions and limitations of their students’ communities with these broad global power systems—and they could do so with the empirical science of mathematics. this integration presents a unique challenge, as the size and scope of such massive power structures exist at a near abstract level of complexity and size. cox (2003) described how this bond & chernoff commentary journal of urban mathematics education vol. 8, no. 1 27 void of understanding exists in american tax systems: “big statements like, ‘the benefits … go mainly to households in the top 1 percent tax bracket,’ tend not to tell the whole story” (¶ 4). the numbers are so large, and the distribution so skewed, that simple sentences fail to deliver accurate meaning. cox suggested that a reconciliation of understanding could be achieved by constructing “a scale model of household income in america” (¶ 5). this could be an excellent project for a classroom struggling to conceptualize debt, wages, unemployment, and industry in their own community. gutstein (2006), in his observations of mathematics and social justice in a classroom environment, noted that mathematics evolved into a “necessary and powerful analytical tool that students used to study their sociopolitical existence” (p. 70). this critical approach, contextualizing authentic social justice issues with mathematical representation, enhances student understandings of local authentic entities and systems. but, more importantly, it solidifies or creates knowledge of hitherto abstract global entities and systems. by integrating, through inquiry-driven projects, an understanding of power systems and community within a mathematical context, students continually expand their mathematical abilities; they gain increasingly more powerful insights into the power systems that permeate the world and define the contexts in which their local communities exist (garii & appova, 2013). this deep understanding of the contexts that governs an individual within a community, within a nation, within global organizations of power, can provide students with distinct advantages. in addition to mathematical prowess, such students can be equipped with an awareness and understanding that might help them utilize the forces that shape events within their local sociopolitical and economic communities. it is important to keep the goals of social justice education firmly in mind. although social justice may seem like a secondary goal to a teacher who wishes their students to know their “basic math facts,” it is of primary importance and should not be isolated, in any way, from other goals in the mathematics classroom. bartell (2013) described education as being “intricately linked to economic, political, and social power structures in society that serve to perpetuate inequality in both schools and society” (p. 129). because the classroom itself is one of the primary sources of the socialization that shapes social inequality, an uncritical pedagogy serves only to enforce existing systems of dominance and inequity. if, as the saskatchewan curriculum (2010) broad areas of learning suggest, teachers should aim to inspire “engaged citizens” with a “passion for lifelong learning” who will contribute to the “environmental, social, and economic sustainability of local and global communities” (p. 22), students, then, should understand, at the very least, and not be marginalized by the systemic dominance associated with race, age, religion, gender, sex, and all other aspects of society. ideally, mathematics and social justice should be used to arm all students with the tools to not only succeed in so bond & chernoff commentary journal of urban mathematics education vol. 8, no. 1 28 ciety but also to critique social systems, disrupt inequalities, and be engaged in social transformation. bartell (2013) asserted that for effective integration of social justice in classrooms to occur, oppression must be fought “with, not for” (p. 131) the oppressed. furthermore, adair (2008) insisted that when integrating social justice into the curriculum, pedagogy “should depend on the community context in which we are teaching and with the individual experiences of our students” (pp. 413–414). in other words, mathematics and social justice must be studied in authentic contexts (e.g., community-based projects). social justice must also be taught with authentic empathy in the classroom. this means diversifying instructional methods to provide multiple entry points for personal connections with the mathematical content being presented. these entry points should be student centered; therefore, a considerable effort is required to facilitate an understanding of each student’s sociocultural and sociohistorical lived experiences. the monitoring of student progress with consistent and dynamic feedback is also necessary. this approach aims to selfempower marginalized “voices” in the classroom and provides a starting point for teaching social justice with (not for) the oppressed. honoring and valuing student voice through input, feedback, and authenticity is a pedagogical practice that embodies social justice and decentralizes teacher-centric authority. it is democratic. too often the great fault of social justice is that as a stand-alone qualitative social science it often widens the divide between practitioners and deniers; practitioners are often aggressive in their pursuit of justice to the point where deniers feel vilified and become aggressively defensive. because socialized dominance is a controversial and dissonant subject, much of the discussion surrounding it degrades to an intersection of opposing opinions. skovsmose (1994) claimed that mathematics provides a system for analyzing and understanding injustices in society. because mathematics, as a science, has a practical foundation in observation and representation, these characteristics can establish a dialogue that circumvents, or at least deemphasizes, emotional connections to socially critical arguments. for example, by employing the empirical facilities of mathematics in data collection, statistical interpretation, and graphical representation, one might make an argument that can only be rebutted with an equally well-researched presentation. the lengthy time involved in the formulation of such presentations also helps to reduce the chances of emotionally charged and reactionary responses. nolan (2009) described the same mathematical niche from a different perspective. she claimed that students are at a disadvantage when striving to understand, communicate, and argue social justice issues without literacy in the scientific methods of mathematics. nolan’s argument described the need for a unity in social justice and mathematics with greater strength because she recognizes that mathematics is beyond simply useful—it is necessary for constructive participation in social justice dialogues. bond & chernoff commentary journal of urban mathematics education vol. 8, no. 1 29 mathematics can be the light that illuminates the writing on the wall. increasingly, academics, government and civic organizations, and the media are presenting mathematically derived findings that describe dangerous power inequities in north american society. fischer, colton, kleiman, and schimke (2004) produced a report on the economic hardships of the middle and lower classes in new york that called for regulations and accountability in the economic sector well before the crash of 2008. killewald (2013) published a study which, contrary to a widespread misconception that racism has ended or is declining, documented the greatest disparity in 25 years between median household wealth values among white and black families in 2009: a ratio of 20:1. while compelling arguments exist for the integration of mathematics and social justice in pedagogy and content, it is ultimately each individual teacher’s decision. this ultimate decision presents a problem because many new teachers struggle to see mathematics as anything beyond “a tool to find a correct answer to a problem, rather than a way to characterize community decision making or understanding” (garii & appova, p. 206). a further complication that darling-hammond, french, and garcia-lopez (2002) explained is the “lifelong” commitment of “effort, perseverance, and reflection” that is required of social justice teachers (p. 4). learning how to teach for social justice is neither quick nor easy; learning to reinvent the traditional conceptualizations and applications of mathematics is difficult, bordering on anathematic to many. this difficulty presents a formidable barrier to the establishment of socially critical mathematics. in spite of traditional misconceptions to the contrary, mathematics and social justice are two fields of study that can exist in a truly symbiotic pedagogy. mathematics is a uniquely well-suited partner with social justice because it can model, through replicable and empirical demonstrations, the nature and intersections of global and local power systems in a way that students can comprehend. social justice provides engaging, empowering, and authentic contexts for projects in which mathematics skill sets can come alive and transcend the traditional limited and abstract operations that have isolated and discouraged too many students for too long. mathematics can be employed to argue social justice issues without the succumbing to the pitfalls of emotional backlash. social justice can elicit intrinsic motivation in students, which inspires growth far beyond the basic traditionally requisite mathematical skill set. the two strands are best contextualized authentically within the local community. social justice presents unique challenges for teachers because it is a lifelong and dynamic process. it requires the maintenance of extensive and constantly evolving understandings of every student in a given classroom. teachers of social justice must also be dedicated to applying socially critical pedagogies within the classroom environment. mathematics has been so heavily conditioned into many teachers’ minds as an abstract calculation tool that many will blindly defend it as such while others, alienated by the traditional approach, will openly declare their hatred of the subject. the problems that suppress the integration of mathemat bond & chernoff commentary journal of urban mathematics education vol. 8, no. 1 30 ics and social justice pedagogy and content are eclipsed only by the ever-increasing need for such pedagogy and content to be unilaterally implemented in schools. students need socially critical mathematics and local, national, and global communities need students (i.e., citizens) who can interpret, articulate, and act upon social justice issues with the science of mathematics at their command. references adair, j. k. (2008). everywhere in life there are numbers: questions for social justice educators in mathematics and everywhere else. journal of teacher education, 59(5), 408–415. bartell, t. g. (2013). learning to teach mathematics for social justice: negotiating social justice and mathematical goals. journal for research in mathematics education, 44(1), 129–163. bossé, m. j., lee, t. d., swinson, m., & faulconer, j. (2010). the nctm process standards and the five es of science: connecting math and science. school science and mathematics, 110(5), 262–276. cox, s. (2003). astronomical incomes. independent media institute. retrieved from: http://www.radicalmath.org/docs/astronomicalincomes.doc darling-hammond, l., french, j., & garcia-lopez, s. (2002). learning to teach for social justice. new york, ny: teachers college press. fischer, d. j., colton, t., kleiman, n. s., & schimke, k. (2004). between hope and hard times: new york’s working families in economic distress. new york, ny: center for an urban future. garii, b., & appova, a. (2013). crossing the great divide: teacher candidates, mathematics, and social justice. teaching and teacher education, 34, 198–213. gutstein, e. (2006). reading and writing the world with mathematics: toward a pedagogy for social justice. new york, ny: routledge. gutstein, e., & peterson, b. (2013). rethinking mathematics: teaching social justice by the numbers (2nd ed.). milwaukee, wi: rethinking schools. jump$tart coalition for personal financial literacy. (2013). making the case for financial literacy. retrieved from: http://jumpstart.org/assets/state-sites/la/files/downloads/making-the-case2013.pdf killewald, a. (2013). return to being black, living in the red: a race gap in wealth that goes beyond social origins. demography, 50(4), 1177–1195. national council of teachers of mathematics. (2000). principles and standards for school mathematics. reston, va: national council of teachers of mathematics. nolan, k. (2009). mathematics in and through social justice: another misunderstood marriage? journal of mathematics teacher education, 12(3), 205–216. osler, j. (july, 2007). lesson plan: community voices heard. retrieved from: http://www.radicalmath.org/docs/communityvoicesheard_teacher.doc saskatchewan ministry of education. (2010). saskatchewan curriculum. regina, sk: saskatchewan ministry of education. skovsmose, o. (1994). towards a philosophy of critical mathematics education. dordrecht, the netherlands: kluwer. steele j., james, j. b., & barnett, r. c. (2002). learning in a man’s world: examining the perceptions of undergraduate women in male-dominated academic areas. psychology of women quarterly, 26(1), 46–50. van de walle, j. a., folk, s., karp, k. s., & bay-williams, j. m. (2009). elementary and middle school mathematics: teaching developmentally (3rd canadian ed.). toronto, canada: pearson allyn & bacon. http://www.radicalmath.org/docs/astronomicalincomes.doc http://jumpstart.org/assets/state-sites/la/files/downloads/making-the-case-2013.pdf http://jumpstart.org/assets/state-sites/la/files/downloads/making-the-case-2013.pdf http://www.radicalmath.org/docs/communityvoicesheard_teacher.doc journal of urban mathematics education july 2012, vol. 5, no. 1, pp. 8–20 ©jume. http://education.gsu.edu/jume jacqueline leonard is the director of the science and mathematics teaching center at the university of wyoming and professor of mathematics education, 1000 e. university avenue., dept. 3992, laramie, wy 82071, e-mail: jleona12@uwyo.edu. her research interests include access and opportunity in mathematics education and critical pedagogy, such as teaching for cultural relevance and social justice in mathematics classrooms. erica r. davila is an associate professor of education and coordinator of urban education at arcadia university, 450 s. easton rd. glenside, pa 19150, e-mail: davilae@arcadia.edu. her research aims to highlight lived experiences as they unfold within the school context, and how focusing on students’ lives might impact pedagogy and curriculum development. david w. stinson is an associate professor of mathematics education in the department of middle-secondary education and instructional technology in the college of education, at georgia state university, p.o. box 3978, atlanta, ga, 30303; e-mail: dstinson@gsu.edu. his research interests include exploring socio-cultural, -historical, and -political aspects of mathematics and mathematics teaching and learning from a critical postmodern theoretical (and methodological) perspective. he is a co-founder and current editor-in-chief of the journal of urban mathematics education. beyond the numbers: a benjamin banneker association conference series jacqueline leonard university of wyoming erica r. davila arcadia university david w. stinson georgia state university hen asked to think of a mathematician, who comes to mind? rené déscartes (1596–1650, french)? isaac newton (1642–1727, english)? or is it carl gauss (1777–1885, german)? when thinking about a mathematician, westerners rarely think of anyone other than a white man of european origin—and even more rare, a woman of any cultural heritage. in fact, most westerners are unfamiliar of benjamin banneker’s name, much less his legacy as a mathematician. societal discourses in too many ways continue to position mathematics as a discipline primarily reserved for elite white men. children and youth, and people of all ages, internalize these discourses and, in turn, continue to imagine the mathematician as a white, middle-aged, balding or wild-haired man (picker & berry, 2000). the einstein-ish silhouette readily comes to mind. even when popular media attempts to diversify images of the mathematician and make mathematics “cool,” the image of the white, wild-haired man is more times than not reified: recall the cbs network crime series numb3rs. in short, the “white male math myth” (stinson, 2010, p. 3) and its apparent permanence continues to frame people’s perceptions of mathematics participation and achievement and, in turn, assists in constituting the mathematics (education) enterprise all together as a “white institutional space” (martin, 2010, p. 65). the negative consequences of this whiteness of mathematics continue to be played out inside apartheid (re)segregated and “integrated” schools and classw leonard et al. a conference series bullock, e. c., alexander, n. n, & gholson, m. l. (eds.). (2012). proceedings of the 2010 philadelphia and 2011 atlanta benjamin banneker association conferences – beyond the numbers [special issue]. journal of urban mathematics education, 5(2). 9 rooms across the nation (kozol, 2005). for example, beginning as early as first grade, students are frequently tracked into either highor low-level mathematics courses based on standardized test scores and teachers’ perceptions of the mathematical abilities of “other people’s children” (delpit, 1995). in too many classrooms, black and brown (and female) children pick up their belongings and march off to another teacher’s classroom for “low-level” mathematics instruction (oakes, ormseth, bell, & camp, 1990; stinson, 2004). the rationale for this shuffling and sorting of children between highand low-level courses is that it is easier for teachers to address students’ needs. but in racially integrated schools the hue of students in the low-level courses is usually black and/or brown. in other words, white students are rarely found in the low-level courses; likewise, black and brown students are rarely found in the high-level courses. even nearly 60 years after the brown v. board of education decision, too many black students have had neither equitable access to nor learning opportunities for the study of advanced mathematics (leonard, napp, & adeleke, 2009; moses & cobb, 2001; spencer, 2009; tate, 1995). therefore, the driving force behind the beyond the numbers conference series was to push against the negative consequences of the whiteness of mathematics, moving beyond the numbers of aggregated “achievement gap” data and toward new discourses about black children and mathematics. the conference series the benjamin banneker association (bba) beyond the numbers conference series—june 2010 philadelphia and november 2011 atlanta—was hosted by temple university and arcadia university (philadelphia) and georgia state university (atlanta), and sponsored and generously funded by the national science foundation. 1 the year 2011 marked the 25th anniversary of bba as an activist organization that advocates for high-quality instruction and “levels of excellence” (hilliard, 2003, p. 138) in mathematics for black children in pre-k–16 school settings. 2 as part of its advocacy work, yearly, bba recognizes exemplary work among teachers of black children and honors the outstanding work of black students at its annual meeting held during the national council of teachers of mathematics (nctm) annual meeting and exposition. benjamin banneker association 1 national science foundation dr-k12 grant – mathematics attainment and african american students: discourse from multiple perspectives (award # 0907896 and 0910672). the conference series also included a leadership summit held at the university of colorado denver in october 2010; the purpose of the summit was to debrief the june 2010 philadelphia conference and to plan the november 2011 atlanta conference. texas instruments and eta cuisenaire provided additional nominal financial support. 2 the founding members of bba were benjamin dudley, edgar edwards jr., william greer, harriett haynes, marie jernigan, genevieve knight, and dorothy strong. leonard et al. a conference series bullock, e. c., alexander, n. n, & gholson, m. l. (eds.). (2012). proceedings of the 2010 philadelphia and 2011 atlanta benjamin banneker association conferences – beyond the numbers [special issue]. journal of urban mathematics education, 5(2). 10 also has held stand-alone conferences that focus on educational issues specifically related to the mathematics education of black children and youth. prior to the beyond the numbers conference series, bba had sponsored four other stand-alone conferences. previous conferences dr. carol malloy, the bba president from 1997–1999, organized the first stand-alone conference in easton, maryland in august 1996; this conference resulted in the nctm publication challenges in the mathematics education of african american children: proceedings of the benjamin banneker association leadership conference (malloy & brader-araje, 1998). dr. anthony scott, the bba president from 2003–2005, organized the second stand-alone bba conference in philadelphia, pennsylvania in april 2004; this conference was held in conjunction with the nctm annual meeting and exposition and co-sponsored by temple university. and dr. lou matthews, the bba president from 2007–2009, organized two stand-alone conferences: the national leadership summit on the mathematics education of black children: an agenda for impact 07, held at and co-sponsored by georgia state university in atlanta, georgia, november 2007; and teaching, learning, and research of african students: unlocking the doors of excellence, held in little rock, arkansas, november 2008. the beyond the numbers conference series built upon the successes of previous conferences and established a precedent for bba in sponsoring stand-alone conferences dedicated to the teaching and learning of black students every two years or so. such specifically focused conferences are needed to draw attention to the ever-changing critical issues surrounding mathematics access and opportunity for black children and youth. rationale, goal, and objectives the rationale for the conference series in general was based on research regarding the racial identity and educational attainment of black children and youth. data suggest that beliefs about self and race relate to black youths’ educational and social development through their attitudes and self-evaluations around education (e.g., the stronger racial pride, the stronger attachment to academics) (chavous et al., 2003). additionally, research has shown that providing all children and youth with opportunities to learn rigorous mathematics is the crucial element in diversifying the pool of human talent in stem (science, technology, engineering, and mathematics) undergraduate and graduate degree programs and in stemrelated professions (tyson, lee, borman, & hanson, 2007). despite these findings, a recent study of high school dropouts participating in the federal job corps program (comprised mostly of black and latina/o youth) identified difficulties leonard et al. a conference series bullock, e. c., alexander, n. n, & gholson, m. l. (eds.). (2012). proceedings of the 2010 philadelphia and 2011 atlanta benjamin banneker association conferences – beyond the numbers [special issue]. journal of urban mathematics education, 5(2). 11 and disaffection with mathematics as the primary reason for dropping out (viadero, 2005). moreover, although black (and hispanic) youth enter college with the same level of interest in stem fields as their white and asian peers, they often fail to persist in stem majors at the same rate (anderson & kim, 2006). therefore, the major goal of the conference series was to bring districtand school-level administrators and classroom teachers as well as teacher educators and researchers in educational policy, psychology, sociology, and mathematics, science, and urban education together to discuss both the challenges and opportunities for black students in mathematics. coupled with this goal was an overarching activist agenda: to move discussions about black children and mathematics away from discourses of deficiency or rejection and toward discourses of achievement 3 (stinson, 2006). for over a decade now, research documenting the success stories of black students’ mathematics achievement has been available in the mathematics education literature (e.g., berry, 2008; jett, 2010; martin, 2000; mcgee & martin, 2011; moody, 2000; nzuki, 2010; stinson, 2010; and walker, 2006). this research serves as an impetus for districtand school-level administrators and classroom teachers to depart from familiar deficit discourses about black students and mathematics and to embrace different discourses for participation and achievement and teaching and learning. for instance, culturally relevant and critical pedagogies, which effectively use the cultural, social, and intellectual capital that black students bring to the classroom, show promise in increasing both mathematics participation and achievement for black (and brown) students (gutstein, 2006; leonard, 2008). but improving outcomes in mathematics for black children and youth requires districtand school-level administrators and classroom teachers (and education stakeholders in general) to critically examine current instructional practices (and polices), to build black students’ mathematics competency from the early grades to middle school, and to offer supports and incentives for rigorous mathematics in high school and college. with an aim of improving black students’ participation and achievement in mind, the specific objectives for the conference series were to bridge research and practice, to identify best practices for teaching mathematics to black children and youth, to strengthen the preparation of “highly qualified” urban mathematics teachers, and to expand the research agendas of seasoned and developing researchers in mathematics education. a secondary component for the series (but of equal value), and similar to the previous bba conferences, was to continue to develop a critical mass of districtand school-level administrators, classroom teachers, teacher educators, and education researchers who focus on issues of equity 3 here, the concept of “achievement” or “brilliance” is not restricted to high test scores or exceptional grades; many students achieve neither but are, nevertheless, highly intelligent and capable (leonard & martin, in press). leonard et al. a conference series bullock, e. c., alexander, n. n, & gholson, m. l. (eds.). (2012). proceedings of the 2010 philadelphia and 2011 atlanta benjamin banneker association conferences – beyond the numbers [special issue]. journal of urban mathematics education, 5(2). 12 and race in mathematics, while refusing to participate in the all too common “‘gap-gazing’ fetish” (gutiérrez, 2008, p. 357). this secondary component is clearly evident in the titles of and questions explored at each conference. the title of the june 2010 philadelphia conference was beyond the numbers: celebrating the best of how teachers teach and african american students learn mathematics, and the title of the november 2011 atlanta conference was beyond the numbers: the brilliance of black children in mathematics. both conferences addressed four interrelated questions: 1. in what ways do school structures and institutional policies (i.e., lack of certified mathematics teachers, low student expectations, disproportionate discipline policies, tracking policies, etc.) impact black students’ “success” in school and in mathematics? 2. (philadelphia) in what ways can technological tools and other forms of multimedia be used to motivate and encourage black students to learn rigorous mathematics and persist in their mathematics education? (atlanta) in what ways do highly qualified mathematics teachers understand issues of race and equity, and how might teacher education programs develop highly qualified mathematics teachers for urban schools? 3. what is the nexus of race and identity for black students in the deep south and other spaces where black students are the majority (e.g., washington, dc; baltimore, maryland; milwaukee, wisconsin; and chicago, illinois), and how does racial identity and individual agency impact their mathematics attainment? 4. what are the best practices that facilitate learning and mathematical empowerment among black students, and how might culturally relevant and social justice pedagogies assist in developing academic success, cultural competence, and critical consciousness among black students? structure and evaluation 4 both conferences had 15 to 20 invited speakers who addressed topics relevant to the four questions noted above (for details of speakers and topics see conference programs: philadelphia and atlanta). 5 professor gloria ladson-billings (past president of the american educational research association) was the keynote speaker for the 2010 philadelphia conference, and professor joyce king was the keynote speaker for the 2011 atlanta conference. in addition, presidents and past presidents of both nctm (henry kepner [philadelphia] and linda gojak 4 data from this section were pulled from the two evaluation reports prepared by dr. sukey blanc of creative research and evaluation services; copies of both reports are available upon request. 5 editors’ note: all invited speakers, including the keynote speakers, were invited to contribute to this jume special issue. http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/182/109 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/181/108 leonard et al. a conference series bullock, e. c., alexander, n. n, & gholson, m. l. (eds.). (2012). proceedings of the 2010 philadelphia and 2011 atlanta benjamin banneker association conferences – beyond the numbers [special issue]. journal of urban mathematics education, 5(2). 13 [atlanta]) and bba (jacqueline leonard [philadelphia] and cheryl adeyemi [atlanta]) were featured speakers. both conferences had similar structures: during each day of the 2-day conferences, attendees participated in two back-to-back, 1hour research symposia, followed by four to six, 1-hour concurrent breakout sessions. each symposium included three to four speakers presenting research papers organized around a specific theme (e.g., “cognitive approaches to learning mathematics” and “black children and mathematics success”). the concurrent breakout sessions, facilitated by symposium speakers, provided conference attendees with an opportunity to discuss in small groups the research presented during the symposia with other attendees and the speaker in-depth. in addition to the small-group breakout sessions, both conferences presented attendees with several opportunities for networking; breakfast, lunch, dinner, and a networking reception were provided for conference attendees. these activities provided attendees with additional opportunities to build interdisciplinary relationships between and within school districts and universities and to plan future projects that might lead to the formulation of new questions for both research and practice. data for the evaluation of both conferences included evaluator observations made during each 2-day conference, informal conversations with attendees and conference planners, attendee surveys, and a small number of follow-up interviews with attendees from the atlanta conference. there were approximately 80 people in attendance at each conference, including conference planners, volunteers, and speakers. and at each conference, approximately 50 surveys were completed. data analyses from both conferences indicate that the conference series was overwhelmingly successful in providing new theoretical perspectives and practical applications for research and practice, helpful information about untapped resources and technology, and interdisciplinary learning and networking opportunities for future projects. in addition to being a stimulating opportunity for learning and networking, an overwhelming majority of attendees reported that they plan to utilize ideas from the conference series in their research and practice. below is a small sampling of open-ended survey responses from the atlanta conference: incorporating crp [culturally relevant pedagogy] as a model for my teacher candidates to help them understand how to teach for social justice…[the] theoretical frameworks, personal narratives regarding the relevance of these dialogues, conceptualizing dispositions [were all useful]. this is all relevant because i work with pre-service teachers. i learned about activities to anchor math in students’ lives in open-ended ways where the students determine what is relevant. leonard et al. a conference series bullock, e. c., alexander, n. n, & gholson, m. l. (eds.). (2012). proceedings of the 2010 philadelphia and 2011 atlanta benjamin banneker association conferences – beyond the numbers [special issue]. journal of urban mathematics education, 5(2). 14 [the speakers] really helped us (through discussions) to rethink the language that we use when speaking of the excellence of young children mathematically. the speakers reaffirmed the importance/role culture plays in a student's identity and thus academic achievement. highlights of keynote speakers as previously noted, the keynote speaker for the philadelphia conference was dr. gloria ladson-billings, professor of curriculum and instruction and educational policy studies and keller family chair in urban education at the university of wisconsin – madison. the title of dr. ladson-billings’ (2010) address was the meaning of culture in mathematics education. she began with a statement by cornel pewewardy, the indigenous american (comanche and kiowa) educator, which went something like: “we don’t need to put culture into education. what we need is to put education in our culture.” before educational structures and systems were put into place, culture was there and remains a salient part of our everyday lives. in essence, “there is no activity that is not situated” in a cultural context (lave & wenger, 1991, p. 33). hollins (1996) defined culture in three parts. first, culture includes artifacts and behavior: artifacts refer to visual and performing arts and culinary practices while behavior refers to social interaction patterns, ceremonies, rituals, and dress. second, culture is the social and political relations and points of view that are shared by people bound together by history. third, culture is affective behavior and intellect. thus, culture guides the reasoning, emotions, and actions of a particular group of people. each part of this definition implies that culture is part of the socialization process where cultural knowledge is passed down by elders or significant others. ladson-billings used the word culture as a mnemonic during her address to focus on specific components that are necessary to improve mathematics participation and achievement for all children:  c = contexts  u = uses  l = language  t = teaching  u = understanding  r = relationships  e = expectations the contexts in which students live imbue culture. culture is their home and community, and school culture (banks, 1993). when it comes to learning mathematics, it is important that teachers link these two cultures together to help students cross borders from their home and community culture to school culture leonard et al. a conference series bullock, e. c., alexander, n. n, & gholson, m. l. (eds.). (2012). proceedings of the 2010 philadelphia and 2011 atlanta benjamin banneker association conferences – beyond the numbers [special issue]. journal of urban mathematics education, 5(2). 15 (leonard, 2008). mathematics is used in many different types of contexts. from bartering and trading with different monetary systems to using mathematics for social justice and empowerment, mathematics is a tool that can be used to understand and relate to the world and interact with people of all cultures. language, which ladson-billings distinguished between lower-case “d” and upper-case “d” discourses (gee, 1989), is evident in mathematics as students learn and explain big ideas and connect those ideas to community issues. teaching should be focused on student learning as well as helping students to develop cultural competence and sociopolitical consciousness. understanding is also a vital part of communicating cultural knowledge. ladson-billings (2010) asked: how do we know what we know? what kind of evidence do we consider “good” enough? it is important that all viewpoints are valued in the mathematics classroom. relationships are also critically important. before teachers can teach the children in their classrooms, they must know and value who their students are and where they come from (nieto, 2002). finally, teachers must hold high expectations of all children. research has shown that instruction is more effective, particularly in inner-city schools, when teachers have “high expectations” for all of their students (brophy, 1983; edmonds, 1979). culture is the center of our lives—it defines us and provides us with a foundation upon which to learn. culture was the focus of the keynote address at the atlanta conference as well. dr. joyce king (2011), professor of social foundations and benjamin e. mays chair of urban teaching, learning, and leadership at georgia state university, delivered the address academic & cultural excellence in mathematics: transformative education for human freedom. she began with the ma’atian theory of knowledge for the practice of human freedom: we want to be found worthy. ma’ta, the ancient egyptian concept for truth, order, law, and justice, was personified as the goddess ma’ta who regulated the stars and the seasons, setting order to the universe. in keeping with the conference title, dr. king provided contemporary exemplars of the brilliance of black children: stephen stafford, the decatur, georgia 13-year-old morehouse college freshman; james black jr., the brooklyn, new york middle school chess champion; and khadijah williams, the los angles, california homeless girl and harvard university student. she used these exemplars of brilliance as the backdrop to her discussion of the national alliance of black school educators (nabse). specially, she highlighted nabse’s 53-page, 1984 report: saving the african american child: a report of the task force on black academic and cultural excellence (the task force was lead by asa hilliard and barbara sizemore). she drew attention to and discussed the following statements pulled from the report:  academic excellence cannot be reached without cultural excellence. leonard et al. a conference series bullock, e. c., alexander, n. n, & gholson, m. l. (eds.). (2012). proceedings of the 2010 philadelphia and 2011 atlanta benjamin banneker association conferences – beyond the numbers [special issue]. journal of urban mathematics education, 5(2). 16  african american children must be given the opportunity to experience an appropriate cultural education that gives them an intimate knowledge of, and which honors and respects, the history and culture of our people.  “excellence” in education is much more than a matter of high test scores on standardized minimum or advanced competency examinations.  excellence must prepare a student for self-knowledge and to become a contributing problem-solving member of his or her own community and in the wider world as well.  no child can be ignorant of or lack respect for his or her own unique cultural group and meet others in the world on an equal footing. in many ways, the 1984 report critically questioned the fundamental ideology of school desegregation in the 1954 brown v. board of education decision, and its subsequent results: the racial composition of a school, when considered alone, does not necessarily have a substantial positive effect on academic performance of african american children. significant evidence does not exist to support any claim that racial mixing alone has contributed to the excellence in the academic growth of the masses of african american students. it is not simply the addition of african americans to a previously all-white school that makes a positive difference; it is the elimination of many of the negative factors within the school and the teaching and learning process, african american or european american, that enhances growth and development. (as cited in lemons-smith, 2008, p. 91) dr. king concluded her address by outlining the transformative curriculum and pedagogy of the songhay club, a group of teachers and graduate students, lead by her, that are teaching (and learning) in local atlanta schools and communities:  african-centered scholarship is used to create values-based, standards-aligned lessons that link learning to heritage knowledge;  criterion standards for contextualized teaching are used to “re-member” african heritage;  students learn in order to serve the school and the community;  students experience a community-building classroom environment;  teachers (and graduate students) use culturally authentic assessment for visionary parent education; and  collectively, teachers, students, parents and community members are producing knowledge for and about the community. next steps and concluding remarks the benjamin banneker association conferences in philadelphia and atlanta provided yet another point of departure in moving toward new discourses about leonard et al. a conference series bullock, e. c., alexander, n. n, & gholson, m. l. (eds.). (2012). proceedings of the 2010 philadelphia and 2011 atlanta benjamin banneker association conferences – beyond the numbers [special issue]. journal of urban mathematics education, 5(2). 17 black children and mathematics. both conferences offered a balance among prek–12 educators, teacher educators, and scholars and researchers as well as young emerging scholars and researchers who are committed to and passionate about strengthening the mathematics participation and achievement of black children and youth. nonetheless, to consider next steps is crucial. the theoretical and practical knowledge presented at the conferences must be disseminated beyond those who attended if substantive change in the teaching practices and learning opportunities in mathematics for black children and youth is to occur. to that end, two writing projects aim to advance the work that was presented in philadelphia and atlanta. evidently, one is this special issue of the journal of urban mathematics education (jume); it includes proceedings papers from some of the symposium speakers and a coauthored editorial from the special issue guest coeditors (each attended at least one of the conferences). the contributing authors to this special issue (and the coedited volume, described below) include a range of researchers and scholars in mathematics (and science) education from graduate students who are ready to carry the torch as the next generation of advocates to some of the most recognized names in the field whose expertise and science has been crucial in changing the discourses and questioning educational policies and practices that work counter to the brilliance of black children. the second writing project is a coedited volume by jacqueline leonard and danny bernard martin to be published by information age press (expected winter 2012). the title of the volume the brilliance of black children in mathematics: beyond the numbers and toward new discourse is pulled directly from the conference series; most of its contributing authors were symposium speakers from the conference series as well. the edited volume in many ways is an extension of the ongoing advocacy work of bba, with a specific focus on bringing to light the brilliance of black children and youth. leonard and martin (in press) in the preface to the book, explain: this volume is unique in its focus. the authors explicate the experiences of black learners across contexts, using diverse theoretical and conceptual perspectives, and critically analyze extant research with respect to those experiences. rather than reify failure, we give attention to black students’ success and resiliency. the conception of brilliance adopted for this volume is not restricted to high test scores or exceptional grades. many students do not achieve either but are, nevertheless, highly intelligent and capable. they are able to demonstrate their brilliance in non-school contexts and in their ordinary everyday mathematical lives. they are also able to demonstrate their brilliance in schools, but it may be often overlooked. in this volume, we bring it to light. the volume is organized into five sections. section one takes a sociocultural and -historical perspective on mathematics education as it relates to black children. section two focuses on policy implications brought about by no child leonard et al. a conference series bullock, e. c., alexander, n. n, & gholson, m. l. (eds.). (2012). proceedings of the 2010 philadelphia and 2011 atlanta benjamin banneker association conferences – beyond the numbers [special issue]. journal of urban mathematics education, 5(2). 18 left behind as it relates to charter schools and assessment. the third section centers on learning and learning environments and explores mathematics learning among black children in particular content areas. the fourth section addresses black student racial identity and school success (broadly defined). and the final fifth section focuses on preparing teachers to embrace the brilliance of black children. all in all, the beyond the numbers conference series was yet another component of a movement that aims to bring to the fore the need for research and practice which brings to light the brilliance of black children in mathematics. the ongoing dialogue from the philadelphia and atlanta conferences, this jume special issue, and the leonard and martin coedited volume all offer reverberating voices calling out for new discourses about black children and mathematics. these discourses, however, must include an honest and critical look at the inequities and injustices that continue to exist in the schooling and life experiences of too many black children while exhibiting the numerous examples of education for liberation that empowers both black students and their teachers. this struggle to change the discourse is not anything new—various local, state, and national community, educational, and political organizations, including advocacy organizations such as the benjamin banneker association, have fought for years in efforts to make the brilliance of black children (in mathematics) visible. all of those who love (black) children must continue to engage in that effort: what the best and wisest parent wants for his own child, that must the community want for all of its children. any other ideal for our schools is narrow and unlovely; acted upon, it destroys our democracy. – john dewey (1915/1990) references anderson, e., & kim, d. 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(2010). negotiating the “white male math myth”: african american male students and success in school mathematics. journal for research in mathematics education, 41. retrieved from http://www.nctm.org/publications/article.aspx?id=26303. tate, w. f. (1995). school mathematics and african american students: thinking seriously about opportunity-to-learn standards. educational administration quarterly, 31, 424–448. tyson, w., lee, r., borman, k. m., & hanson, m. a. (2007). science, technology, engineering, and mathematics (stem) pathways: high school science and math coursework and postsecondary degree attainment. journal of education for students placed at risk, 12, 243–270. viadero, d. (2005, march 16). math emerges as big hurdle for teenagers: h. s. improvement hinges on ‘critical subject,’ education week, p. 1. walker, e. n. (2006). urban high school students’ academic communities and their effects on mathematics success. american educational research journal, 43, 43–73. http://www.nctm.org/publications/article.aspx?id=26303 microsoft word final wamsted vol 3 no 2.doc journal of urban mathematics education december 2010, vol. 3, no. 2, pp. 155–159 ©jume. http://education.gsu.edu/jume john o. wamsted is a third-year, mathematics education doctoral student in the department of middle-secondary education and instructional technology, in the college of education at georgia state university, p.o. 3978 box atlanta, ga, 30303-3978, email: jwamsted1@student.gsu.edu. he is also a mathematics teacher at benjamin elijah mays high school, atlanta, ga, where he has taught 9th grade mathematics for the past 5 years. his research interests center around autoethnography and its investigation into the intersections of white privilege and critical race theory. book review a mathematics teacher looks at mathematics educators looking at mathematics education: a review of culturally responsive mathematics education1 john o. wamsted benjamin elijah mays high school although it is common among many scholars and policy makers who attempt to discuss what is best for african-american learners, we do not ignore or fail to discuss our own positionality and subjectivity. – martin and mcghee, 2009, p. 215 n an attempt to eschew this failure to note my own positionality and subjectivity—the avoidance of author that lather (2007) calls part and parcel with the performance of scientism—i will, from the outset, let the authorial cat out of the bag. i am both a full-time secondary mathematics educator as well as an aspiring doctor of philosophy—at once both present teacher and future researcher. i am white, while my students are 99% african american and 1% hispanic; this makes me not only a teacher at what ladson-billings (2006) calls an apartheid school but also a rather conspicuous “other” (delpit, 1995). i am currently in my fifth year of teaching—all at this same school—and, having struggled mightily at times to connect curriculum to culture, would seem to be an ideal audience for greer, mukhopadhyay, powell, and nelson-barber’s edited volume culturally responsive mathematics education. the nascent researcher in me believes that i, and teachers like me all around the country, could not help but benefit from a view through a lens more focused on the advantages that culture brings to our efforts to teach mathematics and, on the one hand, i was hopeful that this volume would sharpen that focus. on the other hand, however, as a teacher in the trenches, i am usually less than sanguine about the ability of any sort of writing to proffer more than pyrite-promises. loaded as i am with both the eternal optimism of the researcher and the perpetual pessimism of the teacher—though the 1 greer, b., mukhopadhyay, s., powell, a. b., & nelson-barber, s. (eds.). (2009). culturally responsive mathematics education. new york: routledge. 400 pp., $67.95 (paper), isbn 978-08058-6264-5 http://www.routledge.com/books/details/9780805862645/ i wamsted book review journal of urban mathematics education vol. 3, no. 2 156 lighter for having confessed them up front—i proceed from here, janus-faced, with my review. organization of the book as educators, little else could be more important than for us to deeply understand how our narrow vision of culture, bias, prejudice, and discrimination all creep consciously or unconsciously into the subject matter we present, the pedagogy we implement, and what we fail to do to minimize or eliminate them from our instruction. – barta and brenner, 2009, p. 87 the book is rather tightly organized into two large sections: (a) foundations and backgrounds, and (b) teaching and learning; loosely, of course, we could rename these two sections theory and practice. d’ambrosio contributed the foreword—fittingly, as paulo freire is cited in seemingly every chapter, while ethnomathematics is alluded to nearly as often (and explicitly detailed in two separate chapters). the theory section begins with swetz’s wonderful survey of culture, history, and the concurrent development of mathematics; it then continues on from ernest’s attempt at a re-articulation of the philosophy of mathematics, through gutstein’s exploration of economic policy and the attendant mathematical response, before closing with miller-jones and greer’s timely look at testing practices and how these contribute to the notable inter-cultural achievement gaps. overall, the section provides a wide view of culture, education, and their intertwined relationship—solid reading for any mathematics teacher whose head is not buried in the sand of platonism, who might believe there is more to education than just the transmission of “the truth.” the practice section is both more narrowly focused and more predictable than its predecessor. batting leadoff is gay with a broad look at culturally responsive mathematics teachers, followed by three heavy hitters: martin on african american children, moses on the algebra project, and lipka on the cultural knowledge of the yup’ik alaskan natives. civil is positioned in the seventh spot of the lineup with her look at latina mothers, followed by davis, hauk, and latiolais’ examination of culturally responsive college level mathematics—this only peek into postsecondary education tucked into the nether regions of the book, batting last. i call this section “narrowly focused” because it is replete with specificity; each set of authors brings a different cultural perspective coupled with actual and abundant mathematical applications. i call it “predictable” because, except for the surprising piece on the college level, i am fairly certain that i had previously read everything present. to continue with my baseball analogy, it was rather like watching the late 1990s yankees win three world series in a row: impressive and amazing, but each game seeming much as if i had seen it wamsted book review journal of urban mathematics education vol. 3, no. 2 157 play out in exactly the same manner before. tight, beautiful, skilled, to be sure; however, nothing unusual. as a researcher, as a teacher… how can middle-class, monolingual european-american math teachers work better with students who are predominately of color, attend schools in poor urban communities, and are often multilingual? – gay, 2009, p. 189 as an aspiring doctor of philosophy who one day hopes to teach future teachers inside a college of education, i found the book to be a resource both pedagogically rich and potentially problematic. rich on account of both the breadth of topics covered in the theory section as well as the testimonial power of the heavy hitters from the practice section; in particular, i found swetz’s survey of history and culture to be both interesting and illuminating, and i am becoming of the increasing opinion that one could never reach an intellectual limit on reading gutstein or martin. in this day and age of an impending “minority-majority,” the power of these authors to inspire the potentially overwhelmed white teacher to action and efficacy cannot be minimized. here, however, is where i worry that the book has gone mildly astray. specifically, i wonder who, in fact, is their audience? i am a third-year doctoral student in a mathematics education program; i have read almost all of these authors before, some several times, and felt—as stated above—that i did not see much that i would consider new. thus, it would seem, i am not the audience for this book. i would recommend this book highly to a new cohort of doctoral students, fresh into their first year, most still surveying a broad swath of literature as they seek to find their own voice, their individual passions. to be more specific, if ever granted the ability to teach first year doctoral students i will almost certainly use this work in whole or in part. i must report, however, that i would only use it for advanced graduate students, as much of the theory section assumes a familiarity with specific corners of academia that might not be a part of the typical beginning graduate student’s argot. we have here an academically advanced tome—scaring the pants off of novice graduate students in order to expose them to culturally responsive mathematics hardly seems efficacious. the problem of arcane academic language will only exacerbate as the book is disseminated downward, from graduate schools to undergraduate, from postsecondary to secondary, from high school to middle school. as a teacher, however, i desire desperately for these ideas to make it out of our nation’s college campuses and into our neighborhood schools. thus, in an effort to prevent the baby being thrown out with the bathwater, i recommend the book in piecemeal, wamsted book review journal of urban mathematics education vol. 3, no. 2 158 thinking of my many colleagues who might benefit from an introduction to topics such as these. swetz, though lengthy, would be the piece i would most advise reading from the theory section. cruising the globe both geographically and historically—aggregating mathematical tidbits while making a case for the cultural development of mathematics—seems, on paper, an impossible task; swetz, however, performs this task with seeming ease. glancing back over my copy, i find notes scribbled all over the margins: “cool,” “wow,” “bold,” “relevant.” i find it hard to picture any lover of mathematics—from the college professor to the high school student—not feeling similarly. any mathematics teacher unfamiliar with robert moses and the algebra project must read moses, maxwell, and davis’s chapter. moses’s story should serve as an inspiration for any teacher to think, “how might i do this week’s lesson just a bit differently than i have done it before?” lipka, yanez, and andrewihrke’s chapter on the yup’ik provided me with at least one geometrical idea for my class—this when i thought i had all but “mastered” the concept of the quadrilateral. i think that theirs is the broadest look at what can be done in the classroom when culture is foregrounded. civil and quintos’s chapter on latina mothers would also be beneficial to most teachers, i believe, as it would provide a sort of view through the looking-glass: what exactly do parents think about all of this, anyways? all three of these chapters are accessible, practical, and inspirational; any current or pre-service teacher could not help but glean something from them. if i were to teach a class of undergraduates someday, or have free reign to assign reading to my colleagues, these four chapters—swetz, moses et al., lipka et al., and civil and quintos’s—would make my short list. concluding thoughts the idea of culturally responsive education...is widely understood but, so the familiar argument goes, isn’t mathematics, and more particularly the teaching of mathematics, culture-free? – greer, mukhopadhyay, nelson-barber, and powell, 2009, p. 1 spring (2008) defines culture as “socially transmitted behavior patterns, ways of thinking and perceiving the world, arts, beliefs, institutions, and all other products of human work and thought” (p. 3). unless we are willing to adopt a sort of platonic ideal pertaining to mathematics, we are stuck—under the auspice of this definition—regarding mathematics as a product of human work, and, thus, a part of our culture. it is, of course, those two words—our culture—that cause so much trouble for so many people; whose culture, exactly, is meant here? the power of this book lies in part in its ability to debunk the myth of mathematics as a western invention, passed down from the ancient greeks to the enlightenment europeans to the modern day “first world” in some unbroken chain of unbridled wamsted book review journal of urban mathematics education vol. 3, no. 2 159 genius. while it is true that the mathematics we learn today in school is a largely westernized mathematics, what the authors of this volume want us as educators to understand is that this does not necessarily need be so. the yu’pik were doing mathematics long before the white man “discovered” what would soon be renamed america; could not this parallel product of human work be equally as rich as the westernized mathematics that has been so normalized? i conclude with a quote from delpit (1995) who writes, “i would like to suggest that if one does not see color, then one does not really see children” (p. 177). i use delpit’s statement to address those who think that mathematics both is and should remain culture-free. if this is so, then i, as a white mathematics teacher, am free to ignore the racial and cultural differences of my black and hispanic students as i teach them the only academic discipline that is fully pure, gentle, and clean. given—as i believe—that this illusion of cultural independence is not true (skovsmose, 2005), then delpit’s words apply: choosing not to see culture in mathematics would be tantamount to choosing not to see children. for the mathematics teacher who wishes to see children, who believes that to do so they need to improve their ability to see culture, culturally responsive mathematics education is an excellent step in a most excellent direction. references barta, j., & brenner, m. e. (2009). seeing with many eyes: connections between anthropology and mathematics. in b. greer, s. mukhopadhyay, a. b. powell, & s. nelson-barber (eds.), culturally responsive mathematics education (pp. 85–109). new york: routledge. delpit, l. (1995). other people’s children: cultural conflict in the classroom. new york: the new press. gay, g. (2009). preparing culturally responsive mathematics teachers. in b. greer, s. mukhopadhyay, a. b. powell, & s. nelson-barber (eds.), culturally responsive mathematics education (pp. 189–205). new york: routledge. greer, b., mukhopadhyay, s., nelson-barber, s., & powell, a. b. (2009). introduction. in b. greer, s. mukhopadhyay, a. b. powell, & s. nelson-barber (eds.), culturally responsive mathematics education (pp. 1–7). new york: routledge. ladson-billings, g. (2006). from the achievement gap to the education debt: understanding achievement in u. s. schools. educational researcher, 35, 3–12. lather, p. a. (2007). getting lost: feminist efforts toward a double(d) science. new york: state university of new york press. martin, d. b., & mcgee, e. o. (2009). mathematics literacy and liberation: reframing mathematics for african-american children. in b. greer, s. mukhopadhyay, a. b. powell, & s. nelson-barber (eds.), culturally responsive mathematics education (pp. 207–238). new york: routledge. skovsmose, o. (2005). travelling through education: uncertainty, mathematics, responsibility. rotterdam, the netherlands: sense. spring, j. (2008). the intersection of cultures: multicultural education in the united states and the global economy (4th ed.). new york: erlbaum. parecer do conselho editorial journal of urban mathematics education december 2013, vol. 6, no. 2, pp. 62–80 ©jume. http://education.gsu.edu/jume milton rosa is professor of mathematics education at centro de educação aberta e a distância (cead) at the universidade federal de ouro preto (ufop), morro do cruzeiro, bauxita, 35.400-000, ouro preto, minas gerais, brasil; email: milton@cead.ufop.br. his research interests are mathematics education, educational leadership, mathematics history, ethnomathematics, curriculum and instruction, sociocritical mathematical modeling, ethnomodeling, linguistics and mathematics, and distance education. daniel clark orey is professor emeritus of mathematics and multicultural education at california state university, sacramento where he served from 1987 to 2011. he is currently professor of mathematics education in the centro de educação aberta e a distância at the universidade federal de ouro preto, morro do cruzeiro, bauxita, 35.400-000, ouro preto, minas gerais, brasil; email: oreydc@cead.ufop.br. his research interests are mathematics education, multicultural education, curriculum and instruction, mathematics history, ethnomathematics, sociocritical mathematical modeling, ethnomodeling, linguistics and mathematics, and distance education. ethnomodeling as a research theoretical framework on ethnomathematics and mathematical modeling milton rosa universidade federal de ouro preto brazil daniel clark orey universidade federal de ouro preto brazil in this article, the authors discuss a pedagogical approach that connects the cultural aspects of mathematics with its academic aspects in which they refer to as ethnomodeling. ethnomodeling is the process of translation and elaboration of problems and questions taken from systems that are part of the daily life of the members of any given cultural group. here, the authors offer an alternative goal for research, which is the acquisition of both emic and etic forms of knowledge for the implementation of ethnomodeling. they also offer a third perspective on ethnomodeling research, which is the dialectical approach, which makes use of both emic and etic knowledge. finally, the authors define ethnomodeling as the study of mathematical phenomena within a culture because it is a social construct and culturally bound. keywords: dialectical approach, ethnomathematics, ethnomodeling, emic and etic approaches, mathematical modeling hen researchers investigate knowledge possessed by members of distinct cultural groups, they may be able to find unique mathematical ideas, characteristics, procedures, and practices that we consider ethnomathematics, which is used to express the relationship between culture and mathematics. in this regard, the prefix ethnodescribes characteristics related to the cultural identity of a group such as language, codes, values, jargon, beliefs, food and dress, habits, and physical traits; while the term mathematics expresses a broad view of mathematics, which includes ciphering, arithmetic, classifying, ordering, inferring, and modeling (d’ambrosio, 2001). w http://education.gsu.edu/jume mailto:milton@cead.ufop.br mailto:oreydc@cead.ufop.br rosa & orey ethnomodeling journal of urban mathematics education vol. 6, no. 2 63 however, an outsider’s understanding of cultural traits is always an interpretation that may emphasize only their inessential features, which can be considered as the misinterpretation of this distinctly unique and culturally specific mathematical knowledge. the challenge that arises from this understanding is how culturally bound mathematical ideas can be extracted or understood without letting the culture of the researcher and investigator interfere with the culture of the members of the cultural group under study. this lack of interference happens when the members of distinct cultural groups have the same interpretation of their culture, which is named an emic approach as opposed to an outsider’s interpretation, which is named the etic approach. the concepts of emic and etic were first introduced by the linguist pike (1954) who drew upon an analogy with two linguistic terms: phonemic— which is considered as the sounds used in a particular language—and phonetic—which is considered as the general aspects of vocal sounds and sound production in that language. in other words, all the possible sounds human beings can make constitute the phonetics of the language. however, when people actually speak a particular language, they do not hear all its possible sounds. in this regard, as modeled by linguists, not all sounds make a difference because they are locally significant. this means that they are the phonemics of that language. researchers, investigators, educators, and teachers who take on an emic perspective believe that many factors such as cultural and linguistic backgrounds, social norms, moral values, and lifestyles come into play when mathematical ideas, procedures, and practices are developed by the people of their own culture. different cultural groups have developed different ways of doing mathematics in order to understand and comprehend their own cultural, social, political, economic, and natural environments (rosa, 2010). furthermore, every cultural group has developed unique and often distinct ways to mathematize their own realities (d’ambrosio, 1990). mathematization is the process in which members from distinct cultural groups come up with different mathematical tools that can help them to organize, analyze, solve, and model specific problems located in the context of their own real-life situation (rosa & orey, 2006). these tools allow these members to identify and describe specific mathematical ideas, procedures, or practices in a general context by schematizing, formulating, and visualizing a problem in different ways and discovering relations and regularities. frequently, local mathematical practices are simply analyzed from a western view by translating daily problems to academic mathematics through mathematization (eglash, bennett, o’donnell, jennings, & cintorino, 2006) without considering the cultural aspects of these practices. it is important that researchers mathematize local mathematical practices because modeling techniques may be used to translate these practices into academic mathematics. on the other hand, an ethnomathematics perspective attempts to apply modeling to establish relations between local conceptual framework and the mathematical ideas, procedures, and practices developed rosa & orey ethnomodeling journal of urban mathematics education vol. 6, no. 2 64 by the members of distinct cultural groups. in this context, mathematics arises from emic rather than etic origins (eglash et al., 2006). we must, therefore, search for alternative methodological approaches in order to record mathematical ideas, procedures, and practices that occur in different cultural contexts. one alternative approach to this methodology is called ethnomodeling, which is considered an application of ethnomathematics that adds a cultural perspective to the modeling process by studying mathematical phenomena within a culture, which are social constructions and culturally bound (rosa & orey, 2010a). because it is a dynamic research program that is in permanent change and evolution, when justifying the need for a culturally bound view on mathematical modeling, our sources are rooted on the theoretical base of ethnomathematics (d'ambrosio, 1990). the rationale of this theoretical perspective is to discuss how ethnomodeling is the study of the mathematical ideas and procedures of local communities that uses dialectical relationships between emic and etic approaches on those mathematical practices. the contributions of this article to the mathematic education research literature is to bring to light a distinction about the types of processes that ethnomathematical and modeling research may include such as emic, etic, and dialectical approaches into the ethnomodeling research field. this context allows us to state that the purpose of this article is to offer an alternative goal for research, which is the acquisition of both emic and etic knowledge for the implementation of ethnomodeling in classrooms. in order to achieve this goal, this article is structured in a way that guides the readers in sections that discuss the application of ethnomathematics along with the application of modeling techniques. this approach prepares readers to perceive the connection of cultural aspects of mathematics with its academic aspects. ethnomathematics ethnomathematics as a research paradigm is wider than traditional concepts of mathematics, ethnicity, or any current sense of multiculturalism. ethnomathematics is described as the arts and techniques (tics) developed by members from diverse cultural and linguistic backgrounds (ethno) to explain, to understand, and to cope with their own social, cultural, environmental, political, and economic environments (mathema) (d’ambrosio, 1990). ethno refers to distinct groups identified by cultural traditions, codes, symbols, myths, and specific ways of reasoning and inferring. detailed studies of mathematical procedures and practices of distinct cultural groups most certainly allow us to further our understanding of the internal logic and mathematical ideas of diverse groups of people. as depicted in figure 1, we consider ethnomathematics as the intersection of cultural anthropology, mathematics, and mathematical modeling, which is used to help us understand and connect diverse mathematical ideas rosa & orey ethnomodeling journal of urban mathematics education vol. 6, no. 2 65 and practices found in our communities to traditional and academic mathematics (rosa, 2000). figure 1. ethnomathematics as an intersection of three research fields. source: rosa (2000) ethnomathematics, as well, is a program that seeks to study how students have come to understand, comprehend, articulate, process, and ultimately use mathematical ideas, procedures, and practices that enable them to solve problems related to their daily activities (rosa, 2000). this holistic context helps students to reflect, understand, and comprehend relations among all components of systems under study. in this regard, educators should be empowered to analyze the role of students’ ethnoknowledge in the mathematics classroom (borba, 1990), which is acquired by students in the process of learning mathematics in culturally relevant educational systems. ethnomodeling ethnomodeling is the study of mathematical ideas and procedures elaborated by members of distinct cultural groups. it involves the mathematical practices developed, used, and presented in diverse situations in the daily life of the members of these groups (rosa & orey, 2010a). this context is holistic and allows those engaged in this process to study mathematics as a system taken from their own reality in which there is an equal effort to create an understanding of all components of these systems as well as the interrelationship among them (d’ambrosio, 1993; bassanezi, 2002; rosa & orey, 2003). researchers and investigators such as ascher (2002), eglash (1999), gerdes (1991), orey (2000), urton (1997), and rosa and orey (2009) “have revealed [in their research] sophisticated mathematical ideas and practices that include geometric principles in craft work, architectural concepts, and practices in the activities and artifacts of many indigenous, local, and vernacular cul rosa & orey ethnomodeling journal of urban mathematics education vol. 6, no. 2 66 tures” (eglash et al., 2006, p. 347). these concepts are related to numeric relations found in measuring, calculation, games, divination, navigation, astronomy, modeling, and a wide variety of other mathematical procedures and cultural artifacts (eglash et al., 2006). in this context, ethnomodeling may be considered as the intersection area of cultural anthropology, ethnomathematics, and mathematical modeling, which can be used “as a tool towards pedagogical action of an ethnomathematics program, students have been shown to learn how to find and work with authentic situations and real-life problems” (rosa & orey, 2010a, p. 60). figure 2 shows ethnomodeling as an intersection of three research fields. figure 2. ethnomodeling as an intersection of three research fields. source: rosa & orey (2010a) researchers such as eglash and colleagues (2006) and rosa and orey (2006) use the term translation to describe the process of modeling local cultural systems (emic), which may have a western academic mathematical representation (etic). in other words, ethnomathematics makes use of modeling by attempting to use it to establish a relationship between the local conceptual framework (emic) and the mathematics embedded in relation to local designs. on the other hand, often indigenous designs are merely analyzed from a western view (etic) such as the applications of the symmetry classifications from crystallography to indigenous textile patterns (eglash et al., 2006). in some cases, “the translation to western mathematics is direct and simple such as counting systems and calendars” (eglash et al., 2006, p. 347). however, there are cases in which mathematical ideas and concepts are “embedded in a pro rosa & orey ethnomodeling journal of urban mathematics education vol. 6, no. 2 67 cess such as iteration in bead work, and in eulerian paths in sand drawings” (eglash et al., 2006, p. 348). in our point of view, the act of translation applied in this process is best referred to as ethnomodeling in which “mathematics knowledge can be seen as arising from emic rather than etic origins” (eglash et al., 2006, p. 349). in this context, the emphasis of ethnomodeling takes into consideration the essential processes found in the construction and development of mathematical knowledge, which includes often curious and unique aspects of collection, creativity, and invention. it is impossible to imprison mathematical concepts in registers of univocal designation of reality because there are distinct systems that provide an unambiguous representation of reality as well as universal explanations (craig, 1998). this means that mathematics cannot be conceived as a universal language because its principles, concepts, and foundations are not the same everywhere (rosa, 2010). in this regard, “the choice among equivalent systems of representation can only be founded on considerations of simplicity, for no other consideration can adjudicate between equivalent systems that univocally designate reality” (craig, 1998, p. 540). the processes of production of mathematical ideas, procedures, and practices operate in the register of the interpretative singularities regarding the possibilities for a symbolic construction of knowledge in different cultural groups (rosa & orey, 2006). mathematical phenomena, modeling, and ethnomodeling throughout history, researchers and investigators have made extensive use of mathematical modeling procedures ranging from statistical methods for the elucidation of patterns in behavior to the mathematical representations of the logic processes of indigenous and local conceptual systems. mathematical modeling has been considered by some to be a pedagogical tool and by others as a way to understand anthropological and archaeological research (read, 2002). yet others have decried the use of mathematical, and in particular statistical and quantitative, modeling as fundamentally in opposition to a humanistic approach to understanding human behavior and knowledge that takes into account contingency and historical embeddedness and, in turn, decries universality. however, we argue that traditional mathematical modeling does not fully take into account the implications of cultural aspects of human social systems. the cultural component in this process is critical; it “emphasize[s] the unity of culture, viewing culture as a coherent whole, a bundle of [mathematical] practices and values” (pollak & watkins, 1993, p. 490) that are incompatible with the rationality of the elaboration of traditional mathematical modeling process. in the context of mathematical forms of knowledge, however, what is meant by the cultural component varies widely. it ranges from viewing mathematical practices as procedures learned, acquired, and transmitted to members of cultural groups across generations to mathematical practices rosa & orey ethnomodeling journal of urban mathematics education vol. 6, no. 2 68 viewed as made up of abstract symbols with an internal logic (rosa, 2000). if the former is considered, then it is the process by which knowledge transmission takes place from one person to another, which is central to elucidating the role of culture in the development of mathematical knowledge (d’ambrosio, 1993). if the latter is considered, then culture plays a far reaching and constructive role with respect to mathematical practices that cannot be induced simply through observation of these practices (eglash et al., 2006). in this regard, mathematical knowledge developed by members of a specific cultural group consists of abstract symbol systems whose form is the consequence of a unique internal logic. these symbols are then learned through instances of usage within this cultural context. these members also learn about what is derived from those instances, which helps them to form a cognitively based understanding of the internal logic of their mathematical symbolic systems. the cognitive aspects needed in this procedure are primarily decision -making processes by which the members of cultural groups either accept or reject an ethnomodel as part of their own repertoire of mathematical knowledge. we believe that the conjunction of these two scenarios appears to be adequate enough to encompass the full range of cultural mathematical phenomena. in effect, there are two ways in which we recognize, represent, and make sense of a mathematical phenomenon that impinges upon our sensory system. first, there is a level of cognition that we share, to varying degrees, with the members of our own and other cultural groups. this level would include cognitive models that we may elaborate on at a non-conscious level, which serves to provide an internal organization of external mathematical phenomena in order to provide the basis upon which a mathematical practice takes place. second, there is a culturally constructed representation of external mathematical phenomena that also provides its internal organization. however, this representation arises through the formulation of abstract and conceptual structures that provides forms and organizations for external mathematical phenomena. in other words, cultural constructs provide representations for systems taken from reality. the implications for mathematical modeling are that models of cultural constructs are considered as symbolic systems organized by the internal logic of members of cultural groups. we agree with eglash and colleagues (2006) and rosa and orey (2010b) who argued that models built without a first-hand sense for the world being modeled should be viewed with suspicion. researchers and investigators, if not blinded by their prior theory and ideology, should come out with an informed sense of distinctions that make a difference from the point of view of the mathematical knowledge of the work being modeled. in doing so, they should, in the end, be able to tell outsiders (etic) what matters to insiders (emic). ethnomodeling privileges the organization and presentation of mathematical ideas and procedures developed by the members of distinct cultural groups in order to facilitate its communication and transmission across genera rosa & orey ethnomodeling journal of urban mathematics education vol. 6, no. 2 69 tions. these members construct ethnomodels of mathematical practices found in sociocultural systems (rosa & orey, 2010b), which link cultural heritage with the development of mathematical practice. it is our understanding that this approach may help the organization of the pedagogical action that takes place in classrooms through the use of the emic and etic aspects of this mathematical knowledge. the emic and etic constructs of ethnomodeling in the ethnomodeling approach, the emic constructs are the accounts, descriptions, and analyses expressed in terms of the conceptual schemes and categories that are regarded as meaningful and appropriate by the members of the cultural group under study (lett, 1996). this means that an emic construct is in accordance with the perceptions and understandings deemed appropriate by the insider’s culture. the validation of the emic knowledge comes with a matter of consensus, which is the consensus of local people who must agree that these constructs match the shared perceptions that portray the characteristic of their culture (lett, 1996). in other words, the emic approach tries to investigate mathematical phenomena and their interrelationships and structures through the eyes of the people native to a particular cultural group. it is paramount to note that particular research techniques used in acquiring emic mathematical knowledge has nothing to do with the nature of that knowledge. in this regard, the “emic mathematical knowledge may be obtained either through elicitation or observation because it is possible that objective observers may infer local perceptions” (lett, 1996, p. 382) about mathematical ideas, procedures, and practices developed through history. it is necessary to state that etic constructs are considered accounts, descriptions, and analyses of mathematical ideas, concepts, procedures, and practices expressed in terms of the conceptual schemes and categories that are regarded as meaningful and appropriate by the community of scientific observers, researchers, and investigators (lett, 1996). an etic construct is precise, logical, comprehensive, replicable, and observer-researcher independent. in so doing, the validation of the etic knowledge thus becomes a matter of logical and empirical analysis, in particular, the logical analysis of whether the construct meets the standards of comprehensiveness and logical consistency, and then the empirical analysis of whether or not the mathematical concept has been replicated (lett, 1996). it is important to emphasize that particular research techniques used in the acquisition of etic mathematical knowledge has no bearing on the nature of that knowledge. the etic knowledge may be obtained at times through elicitation as well as observation, because it is entirely possible that native informants possess scientifically valid knowledge (lett, 1996). in this sense, we agree with d’ambrosio (1990) who states that researchers and investigators rosa & orey ethnomodeling journal of urban mathematics education vol. 6, no. 2 70 have to acknowledge and recognize that members of distinct cultural groups possess scientifically mathematically valid knowledge. the dialectical approach in ethnomodeling research the emic perspective is concerned about differences that make mathematical practices unique from an insider’s point of view. we argue that emic ethnomodels are grounded in the mathematical ideas, procedures, and practices that matter to the insiders’ view of the mathematical world being modeled. on the other hand, many ethnomodels are etic in the sense that they are built on an outsider’s view about the mathematical world being modeled. in this context, etic ethnomodels represent how the modeler thinks the world works through systems taken from reality while emic ethnomodels represent how people who live in such worlds think these systems work in their own reality. we also would like to point out that the emic perspective always plays an important role in ethnomodeling research, yet the etic perspective should also be taken into consideration in this process. in this perspective, the emic ethnomodels sharpen the question of what an agent-based model should include to serve practical goals in modeling research. thus, mathematical ideas and procedures are etic if they can be compared across cultures using common definitions and metrics. on the other hand, the focus of the analysis of these aspects are emic if mathematical ideas, concepts, procedures, and practices are unique to a subset of cultures that are rooted on the diverse ways in which etic activities are carried out in a specific cultural setting. currently, the debate between emic and etic is one of the most intriguing questions in ethnomathematics and mathematical modeling research since researchers continue to deal with questions such as: 1. are there mathematical patterns that are identifiable and/or similar across cultures? 2. is it better to focus on these patterns particularly arising from the culture under investigation? while emic and etic are often thought of as creating a conflicting dichotomy, they were originally conceptualized as complementary viewpoints (pike, 1967). according to this context, rather than posing a dilemma, the use of both approaches deepens our understanding of important issues in scientific research and investigations (berry, 1999). a suggestion for dealing with this dilemma is to use a combined emic-etic approach, rather than simply applying emic or etic dimensions to study and examine mathematical procedures and practices employed by the members of distinct cultural groups. a combined emic-etic approach requires researchers to first attain emic knowledge developed by the members of the cultural groups under study. this may allow researchers to put aside their cultural biases so that they may be able to become rosa & orey ethnomodeling journal of urban mathematics education vol. 6, no. 2 71 familiar with the cultural differences that are relevant to the members of these groups (berry, 1990). usually, in ethnomodeling research, an emic analysis focuses on a single culture and employs descriptive and qualitative methods to study a mathematical idea, concept, procedure, or practice of interest. its focus is on the study within a cultural context in which the researcher tries to examine relative internal characteristics or logic of the cultural system. in this perspective, meaning is gained relative to the context and therefore not easily, or if at all, transferable to other contextual settings. for example, it is not intended to compare the observed mathematical patterns in one setting with mathematical patterns in other settings. this means that the primary goal of an emic approach is a descriptive idiographic orientation of mathematical phenomena because it puts emphasis on the uniqueness of each mathematical idea, procedure, or practice developed by the members of cultural groups. thus, if researchers and educators wish to highlight meanings of these generalizations in local or emic ways, then they need to refer to more precisely specified mathematical events. in contrast, an etic analysis would be comparative, examining many distinct mathematical cultural practices by using standardized methods (lett, 1996). this means that the etic approach tries to identify lawful relationships and causal explanations valid across different cultures. thus, if researchers and educators wish to make statements about universal or etic aspects of mathematical knowledge, these statements need to be phrased in abstract ways. on the other hand, an etics approach may be a way of examining the emics of members of cultural groups because it may be useful for looking deeply at, discovering, and elucidating emic systems that were developed by members of different cultural groups (pike, 1954). in so doing, while traditional concepts of emic and etic are important points of view for understanding and comprehending cultural influences on mathematical modeling, we would like to propose a distinctive view on ethnomathematics and modeling research, which is referred as a dialectical approach (martin & nakayama, 2007). in this approach, the etic perspective claims that the mathematical knowledge of any given cultural group will have no necessary priority over its competing emic meaning. according to this point of view, it is necessary to depend “on acts of ‘translation’ between emic and etic perspectives” (eglash et al., 2006, p. 347). in other words, the cultural specificity may be better understood with the background of communality and the universality of theories and methods and vice versa. in this context, these claims must be verified with methods independent of the subjectivity of the observer and researcher in order to achieve a scientific character. therefore, it becomes important to analyze the insights that have been acquired through subjective and culturally contextualized methods. the rationale behind the emic-etic dilemma is the argument that mathematical phenomena in their full complexity can only be understood within the context of the culture in which they occur. rosa & orey ethnomodeling journal of urban mathematics education vol. 6, no. 2 72 the wine barrel: the dialectical ethnomodel the ethnomodel that offers an ethnomodeling methodology example was elaborated by a group of brazilian students who studied wine production (bassanezi, 2002). the motivation of their study was to find the volume of wine barrels by applying the techniques learned by ancestors of the wine producers who came to southern brazil as italian immigrants in the early twentieth century. since then the production of wine has become an essential agricultural activity for the economy of that brazilian region. the ethnomodeling process. in order to conduct their research, initially, students visited wineries to conduct interviews with the wine producers. subsequently, they collected data that were supplemented by the literature review on the chosen theme. the ethnological and historical research of the construction of wine barrels theme was the first stage of the ethnomodeling process. in the ethnological study, students identified characteristics of this particular group so that they were able to understand some of the cultural elements that shape their mathematical thinking (bassanezi, 2002). in this context, students found out that, in addition to producing wine, wine producers construct their own wooden wine barrels by using geometric schemes inherited from their ancestors in italy. during their research, students also found out that to construct these barrels with pre-established volume, it is necessary for the wine producers to cut wooden staves to fit perfectly. this process drew the attention of the students who were interested in exploring the mathematical ideas the wine producers were using in their geometric schemes. for example, figure 3 shows a geometric scheme made by the wine producers in the construction of wine barrels. figure 3. geometric scheme made by wine producers in the construction of wine barrels. source: bassanezi (2002, p. 47) rosa & orey ethnomodeling journal of urban mathematics education vol. 6, no. 2 73 in the scheme in figure 3, l is the maximum width of the stave,  is the width to be determined and  is the fitting angle between the staves, which depend on the initial width of the stave l and the volume required for the wine barrel. in figure 4, the r is the radius of the base of the larger circle that represents the base of the barrel, r is the radius of the smaller circle that represents the barrel cover and h is the height of the barrel. the wine producers construct barrels shaped like a truncated cone by interlocking wooden staves whose dimensions are 2.5 cm in length and width ranging from 5cm to 10cm (bassanezi, 2002). figure 4. wine barrel shaped like a truncated cone source: bassanezi (2002, p. 48) in order to determine the volume of the wine barrel, wine producers approximated its volume by applying a procedure called averaging cone (bassanezi, 2002), which is used in the construction of wooden wine barrels. it is important to state here that the volume determined by the averaging cone formula is an approximation of the volume obtained by applying the academic formula that provides the volume of the truncated cone. the averaging cone formula is given by: hrv m  2  they also apply the averaging radius procedure, which is given by formula ii: by replacing formula ii into formula i, formula iii is given by: rosa & orey ethnomodeling journal of urban mathematics education vol. 6, no. 2 74 figure 4 also shows that the fitting angle  between the two wooden staves is obtained by considering that:  r is the radius of the base of the wine barrel.  l is the width of the wooden stave of the wine barrel in its base.  all juxtaposed wooden staves form a circumference at the base of the wine barrel. in this process, it is possible to observe that the scheme used in figure 3 is an orthogonal projection of one of the wine barrel wooden staves as shown in figure 5. figure 5. orthogonal projection of a wine barrel wooden slate source: bassanezi (2002, p. 49) according to the etic approach by developing the mathematical model used in academic mathematics, the volume of the truncated cone is given by the formula:  22 3 1 rrrrhv   in the emic approach by developing the ethnomodel used by the wine producers the volume of the wine barrel is given by the formula: h rr v         2 2  the construction of the wine barrel process is an excellent example that typifies the connection between ethnomathematics and mathematical modeling rosa & orey ethnomodeling journal of urban mathematics education vol. 6, no. 2 75 (d’ambrosio, 2002) through ethnomodeling. thus, this method presents an approximated calculation of the area of the volume of the wine barrel as employed by the members of this specific cultural group. some considerations about the ethnomodeling wine barrel construction an emic observation of this mathematical practice sought to understand it for constructing wine barrels from the perspective of local dynamics and relationships as influenced within the culture of the wine producers themselves. on the other hand, an etic perspective uses some aspects of academic mathematics to translate this mathematical practice in order to amplify the understanding of those from a different cultural background by explaining this practice from the point of view of the outsider. in this context, the emic viewpoint clarifies intrinsic cultural distinctions while the etic perspective seeks objectivity as an outside observer across cultures. this is the dialectical approach, which concerns the stability of relationships between these two different cultural approaches. in our point of view, both perspectives are essential to understand human behaviors (pike, 1996), especially, social and cultural behaviors that help to shape mathematical ideas, procedures, and practices developed by the members of distinct cultural groups. finally, in order for this process to be successful as well as mathematics to be valued as a discipline whose contents can be considered as human creations, it is necessary to understand and modify the environment we live. in this regard, we can use ethnomodeling in order to link theory into practice by the inclusion of the dialectical approach into the mathematics curriculum. the dialectical approach into a mathematical curriculum classrooms cannot be isolated from their communities because they form part of a well-defined cultural practice. classrooms form learning environments that facilitate pedagogical practices, which are developed by using an ethnomodeling approach (rosa, 2010). when students come to school, they bring with them values, norms, and concepts that they have acquired in their home-community cultural environments. some of these concepts are mathematical in nature (bishop, 1994). however, mathematical concepts of the school curriculum are presented in a way that may not be directly related to the cultural backgrounds of the students. it has been hypothesized that low attainment in mathematics could be due to a lack of cultural consonance in mathematics curriculum. in this regard, the pedagogical elements necessary to develop a mathematics curriculum are found in the school community itself (bakalevu, 1998). thus, the field of ethnomodeling presents tangible possibilities for educational initiatives that introduce mathematical ideas via rich problems that engage students in doing mathematics and will aid them in developing the mathematical reasoning and problem-solving abilities used by expert problem solvers (rosa, 2010). rosa & orey ethnomodeling journal of urban mathematics education vol. 6, no. 2 76 according to this context, mathematical knowledge of the members of distinct cultural groups combined with western-mathematical knowledge systems may result in a dialectical approach to mathematics education. in this regard, academic mathematics (etic) is efficient and appropriate to solve many problems and there is no reason to replace it. however, local mathematical practices (emic) are good in solving other kinds of problems. the combination of these two domains of ethnomathematics offers greater possibilities to understand and explain different problem-solving situations (rosa & orey, 2010a). in this context, an emic analysis of a mathematical phenomenon is based on local structural or functional elements of a particular cultural group while an etic analysis is based on predetermined general concepts external to that cultural group (lovelace, 1984). the emic perspective provides internal conceptions and perceptions of mathematical ideas and concepts while the etic perspective provides the framework for determining the effects of those beliefs on the development of the mathematical knowledge. in this perspective, the acquisition of mathematical knowledge is based on the applications of current mathematics curriculum, which may be assessed based on multiple instructional methodologies across various cultures. in this regard, it could indeed be that one of the reasons for failure of students’ achievement in many educational systems is that curriculum developers by using a one size fits all program have ignored unique emic perspectives regarding the recognition of distinct cultural backgrounds within the schools. a dialectal approach supports the recognition of the existence of other mathematical knowledge systems, which are found in many schools and urban centers. in other words, an ethnomodeling curriculum provides an ideological basis for learning with and from the diverse cultural and linguistic elements presented by members of distinct cultural groups (rosa & orey, 2010a). in this kind of curriculum it would be crucial to understand that etic constructs are mathematical ideas, procedures, and practices that are assumed to apply in all cultural groups while emic constructs only apply to members of specific cultural groups. this means that there is concern for cultural bias occurring if educators and researchers assume that an emic construct is actually etic (eglash et al., 2006), which results in an imposed etic perspective in which a culture-specific idea is wrongly imposed on the members of another cultural group. this kind of curriculum is intended to make school mathematics more relevant and meaningful to students because it is based on students’ knowledge, which allows teachers to have more freedom and creativity to choose academic mathematical topics to be covered in the lessons. it is through dialogue that teachers can apply mathematical themes that help students to elaborate the mathematics curriculum. in this regard, teachers engage students in the critical analysis of the dominant culture as well as the analysis of their own culture through an ethnomodeling perspective. it also means that it is necessary to investigate the conceptions, traditions, and mathematical rosa & orey ethnomodeling journal of urban mathematics education vol. 6, no. 2 77 practices of a particular cultural group with the intention of incorporating these concepts into the mathematics curriculum (rosa, 2010). a classroom environment in which teachers are applying an ethnomodeling curriculum would be full of examples that draw on the students’ own experiences and makes use of experiences common in their cultural environments. these examples would be vehicles for communicating mathematical ideas, which themselves would remain relatively unchanged. in other words, ethnomathematics aims to draw from the students’ cultural experiences and practices of the individual learners, the communities, and the society at large (rosa, 2010). this means that ethnomathematics uses these cultural experiences as vehicles to not only make mathematics learning more meaningful but also, and more importantly, to provide students with the insights of mathematical knowledge as embedded in their social and cultural environments (rosa & orey, 2008). while it seems that only the members of a specific cultural group living within the culture can provide an emic perspective of the mathematical knowledge that is generated by their own culture, when teachers and local members of a community come together in research/study groups they can find creative ways in which to use elements of their own culture, knowledge, and language in the elaboration of curricular activities and pedagogical practices. they also can create zones of safety in which resistance to conventional practices can be expressed and innovative approaches to schooling can be investigated and practiced. the work of these teacher groups may have theoretical implications for community-based teacher preparation. factors influencing development of these groups and their ability to affect change need to be discussed along with the challenges of transferring their cultural creations to the wider institutions of schooling. according to this context, teachers and members of the community decide what mathematics content needs to be taught to students. in our opinion, this is how teachers may reconcile between the many emics that students from distinct cultural backgrounds bring to the classrooms. an ethnomodeling curriculum that combines key elements of local and academic knowledge in a dialectical approach helps students to manage knowledge and information systems taken from their own cultural background and creatively apply this knowledge to other situations. this means that ethnomodeling can be considered part of a critical mathematics education because it provides a learning process in which teachers encourage the use of multiple sources of knowledge from different cultural contexts. in this approach, acquired knowledge is centered, located, oriented, and grounded on the cultural backgrounds of the students, which could be applied and translated appropriately by them and thus equip them to be fully productive locally and globally. according to rosa and orey (2010b), ethnomodeling is a pedagogical approach valuable for reaching this goal. the nature of the previous mathematical knowledge of the students lends themselves to the principle of sequencing in curriculum development. by giving educators the freedom to start with previous mathematical knowledge and rosa & orey ethnomodeling journal of urban mathematics education vol. 6, no. 2 78 experience of their students, we can move from the familiar to the unfamiliar and from the concrete to the abstract in the process of promoting the acquisition of mathematical knowledge (rosa & orey, 2006) and should include moves from emic to etic perspectives and vice versa. in this dialectical context, an ethnomodeling curriculum provides the underlying philosophy for knowledge generation and exchange within and between distinct systems of mathematical knowledge. this concept of an ethnomodeling curriculum approach ensures a more balanced integration of the affective domain of educational objectives that are essential to the recognition and utilization of the students’ previous knowledge. final considerations today, numerous diverse indigenous and local mathematical knowledge systems and traditions are at risk of becoming extinct because of the rapidly changing natural and cultural environments and a fast pacing economic, social, environmental, political, and cultural changes occurring on a global scale. many local mathematical practices disappear because of intrusion and imposition of foreign etic knowledge and technologies or from the development concepts that promise short-term gains or solutions to problems faced by the members of a specific cultural group without considering their emic knowledge to solve these problems. not unlike the loss of global tropical rainforests, the tragedy of the impending disappearance of indigenous and local knowledge is most obvious when a diversity of skills, technologies, and cultural artifacts, problem solving strategies and techniques, and expertise are lost to all of us before being understood and/or achieved and archived. defined in that manner, the usefulness of the emic and etic distinction is evident. like all human beings, researchers, educators, and teachers have been enculturated to some particular cultural worldview. they, therefore, need a means of distinguishing between the answers they derive as enculturated members and the answers they derive as observers. defining emics and etics in epistemological terms provides a reliable means of making that distinction. in this perspective, culture is a lens, shaping reality; it can be considered a blueprint, specifying a plan of action. at the same time, a culture is unique to a specific group of people. by utilizing the research provided by both emic and etic approaches through ethnomodeling, we gain a more complete understanding of the cultural groups of interest. in this article, the ethnomodeling process was investigated from an ethnomathematical perspective as the cultivation of grape vines and the production of wine barrels are strongly linked to the history and culture of people in the southern region of brazil. we have offered an alternative goal for research, which is the acquisition of both emic and etic knowledge for the implementation of an ethnomodeling research. emic knowledge is essential for an intuitive and empathic understanding of mathematical ideas of a culture and it is essential for conducting effective ethnographic fieldwork. furthermore, emic knowledge is a valuable rosa & orey ethnomodeling journal of urban mathematics education vol. 6, no. 2 79 source of inspiration for etic hypotheses. etic knowledge, on the other hand, is essential for cross-cultural comparison, the essential components of ethnology, because such comparison necessarily demands standard units and categories. we also offered here a third approach on ethnomodeling research, which is the dialectical approach that makes use of both emic and etic knowledge and understandings through the processes of dialogue. references ascher, m. 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(1997). the social life of numbers: a quechua ontology of numbers and philosophy of arithmetic. austin, tx: university of texas press. http://www.rc.unesp.br/igce/matematica/bolema/site%2026/resumo%20rosa%20e%20orey.doc http://www.rc.unesp.br/igce/matematica/bolema/site%2026/resumo%20rosa%20e%20orey.doc journal of urban mathematics education july 2013, vol. 6, no. 1, pp. 81–95 ©jume. http://education.gsu.edu/jume anita a. wager is an assistant professor in the department of curriculum & instruction at the university of wisconsin-madison, 225 n. mills st., madison, wi 53706; email: awager@wisc.edu. her research interests include equity in mathematics education, teacher education for social justice, and culturally responsive early childhood mathematics. kristin whyte is a graduate student in the department of curriculum & instruction at the university of wisconsin-madison, 1025 w. johnson st., madison, wi 53706; email: kwhyte@wisc.edu. her research interests include early childhood education, home-school relationships, multicultural perspectives on education, and qualitative research methods. young children’s mathematics: whose home practices are privileged? anita a. wager university of wisconsin madison kristin whyte university of wisconsin madison in this essay, the authors share a professional dialogue about the ways in which issues of power emerge in preschool classrooms when teachers endeavor to build on children’s home and school mathematical experiences and understanding. from different perspectives, as early childhood and mathematics education researchers, the authors discuss ways in which data from teacher interviews and discussions collected during a professional development program provide evidence of whose knowledge is privileged. the authors use the dialogue to explore what, how, and who pre-k teachers most often privileged in their work with children and families in mathematics. and what effect that privileging had on power relationships. keywords: early childhood education, mathematics education, power relations, privilege, professional development or the past three years, we (anita and kristin) have been working with pre-k teachers in a professional development (pd) designed to support culturally and developmentally responsive mathematics practices. the focus of the pd was to weave together understandings of early childhood education, funds of knowledge, and early mathematics so that teachers would expand their understanding of ways to draw on children’s multiple resources. last fall, as we were driving home from one of the pd classes, anita shared some of the topics that were discussed at a conference she had attended on privilege and oppression in mathematics education (prompte 1 ). this impromptu discussion led to a conversation about how power structures in preschool classrooms (i.e. relationships be 1 privilege and oppression in the mathematics preparation of teacher educators (prompte) conference (funded by create for stem institute through the lappan-phillips-fitzgerald endowment), michigan state university, battle creek, mi, october 2012. any opinions, findings, and conclusions or recommendations expressed herein are those of the authors and do not necessarily reflect the views of the funding agency. f wager & whyte whose home practices are privileged stinson, d. w., & spencer, j. a. (eds.). (2013). privilege and oppression in the mathematics preparation of teacher educators [special issue]. journal of urban mathematics education, 6(1). 82 tween teachers and parents and teachers and children) were strengthened, or interrupted, when teachers endeavored to build on children’s home and school mathematical experiences and understanding. we found it interesting to explore this idea from our different areas of expertise (kristin’s expertise in early childhood education and anita’s in mathematics education). thinking back to interviews, observations, and conversations with teachers in our pd, we had noticed two perspectives. on one side, there were teachers who privileged the diverse mathematical practices and experiences that families and children shared at home. and on the other side, and more common, were those teachers who privileged the home mathematics activities that aligned most with their own experiences and perspectives regarding mathematics. given the focus of the pd on culturally responsive mathematics teaching, and the fact that all of the teachers in our pd identified as white, middle-class women whereas the children in their classrooms were from diverse ethnic, linguistic, and economic backgrounds, we wanted to understand what, how, and who pre-k teachers privileged in their work with children and families in mathematics. and what effect that privileging had on power relationships in the classroom. in other words, how did teachers use the power inherent in the privilege they possess as adults and teachers from a dominant group to interrupt or reify the power relationships between children and teachers, and teachers and parents? these questions, central to our work in understanding teachers’ perspectives and actions, motivated this essay. in deciding how to present our conversation about privilege in pre-k mathematics, we found the dialogic methodology used by mccarthy and moje (2002) in their work on identity particularly helpful. these scholars addressed their research question about why identity matters through a 4month electronic mail exchange that they presented as a conversation. through their conversation the authors explored theories of identity, research studies on identity and literacy, and implications for pre-k–12 literacy practices. we too initiated an electronic exchange, but modified mccarthy and moje’s approach by responding to data from our study of the pd through a conversation on a google doc in which we discussed power and privilege in pre-k classrooms. we consider this essay an initial exploration into how power and power structures, and privilege and privileging played out in our pd. we draw on foucault (1982) in considering the way power (the ability to do something or act in a particular way) may be used to identify what knowledge counts. thus in the traditional power relationship between adult and child (canella, 2008), the adult/teacher identifies what counts as knowledge, which may occur by “privileging” or giving preference to particular types of mathematical practices. the power adults/teachers possess is something they can take advantage of as a result of adult privilege. similar to white privilege, adult privilege brings with it an “invisible package of unearned assets” (mcintosh, 1988, p. 1), “which is often abused to wager & whyte whose home practices are privileged stinson, d. w., & spencer, j. a. (eds.). (2013). privilege and oppression in the mathematics preparation of teacher educators [special issue]. journal of urban mathematics education, 6(1). 83 the disadvantage of children” (mcintosh, 2012, p. 199). the other power structure we explore is that between teacher and parent, where teachers are positioned as the professional and expert and parents are the clients (lareau, 2000). teachers’ position as experts is particularly damaging when they hold deficit views of parents (pushor & murphy, 2004). our initial question was about how teachers privileged certain home practices in mathematics and what that implied about power relationships between children, families, and teachers. to address the first part of the question, we identified data from interviews with teachers in which they responded to questions about building on home mathematics practices. next, we each read through the data, making comments and asking questions to each other about what teachers’ perspectives implied about power. these comments began a dialogue about findings from the data and triggered recollections of examples from other data sources. in the remainder of the essay, we present “dialogues” in which we provide examples of teachers’ comments or conversations followed by our discussion of those comments. we then summarize our findings to respond to the questions that emerged through this process:  what home mathematics practices did pre-k teachers privilege?  how did privileging particular practices reinforce or interrupt historical power structures in parent/teacher and child/teacher relationships?  which families and children did teachers privilege and how might that privileging have oppressed others? dialogue 1: what counts and establishing power differentials the following quote is betty’s (one of the teachers in the pd) response to the questions: what have you learned from families about their home practices that has allowed you to be more successful in your practice? what about mathematics practices in particular? we found betty’s response to be typical of those teachers who assumed they would have to help families see the mathematics in their everyday experiences: when i meet with families i ask questions like: “do you count with your child when you go out in the stores?” “do you look at different patterns and designs?” just trying to get them to help me to understand. “what can you share with me that you do with your child that i can see is a math-related activity. and is it important to you?” ... i don’t know if they [families] just don’t sense that mathematics is really as vitally important. because it’s all around us. when i start pointing out to them, when you go to wal-mart or when you go to mcdonald’s you’ll see patterns, you can count things; you can look at designs of things and shapes. oh, my gosh. so, sometimes it’s bringing an awareness. …to the parents that, yeah this is important. this is valu wager & whyte whose home practices are privileged stinson, d. w., & spencer, j. a. (eds.). (2013). privilege and oppression in the mathematics preparation of teacher educators [special issue]. journal of urban mathematics education, 6(1). 84 able and you want to encourage your child to learn these things just as much as what are the letters in wal-mart when we pull up on the store or mcdonald’s. kristin: betty expresses that “mathematics is everywhere” and then quickly slips into what she wants families to do to promote mathematics learning. the kind of mathematics that counts to betty seems to be driven by the standards she is expected to teach toward, and it makes sense that this is the kind of mathematics that she sees. anita: if this is how she engages with families, it is pretty clear that she holds the power in terms of identifying what math-related activities families should be doing—what “counts” as mathematics. in particular, her comment about what she “can see” as mathematics rather than what the families consider to be mathematical. what do you think she means when she asks: “is this important to you?” how do you think families might interpret that? kristin: i keep thinking about the possible responses to the question. a parent is probably not going to say, “well, you know, mathematics is not important to our family or my child’s education.” so what is the alternative, “yes, mathematics is important to our family.” these responses both feel close to meaningless. if she asks that question during the same time she’s talking about “counting at stores,” it seems obvious that what betty really wants to know is if the family is doing particular kinds of mathematics activities. imagine how a parent whose child is just starting school might hear this question. some families could enter prekindergarten already thinking about what they are doing wrong, while some might have what they are doing at home with their children reinforced. i think this example shows how what a teacher privileges can easily become reinforced from the very beginning of a child’s schooling. anita: you are so right—and what concerns me most is that because this is happening before children ever enter the pre-k–12 system families whose home practices do not align with schooling are shut down (or out) from the start. this shutting down may be even more evident with mathematics as it is an area that many families (particularly those who have been historically marginalized) are not comfortable (remillard, 2005). i wonder if, in some ways, we as teacher educators encourage this act of telling families what to do. in reviewing articles in teacher journals, i found this idea of telling was reinforced. and, in articles such as hansen’s (2005) on the abcs of connecting with families about mathematics, the “what to do” often came from a white, middle-class perspective. even research studies make suggestions such as “teachers need to help parents and students label home activities as wager & whyte whose home practices are privileged stinson, d. w., & spencer, j. a. (eds.). (2013). privilege and oppression in the mathematics preparation of teacher educators [special issue]. journal of urban mathematics education, 6(1). 85 mathematical activities” (gautreau, kirtman, & guillaume, 2011), which positions teachers as the ones with the expertise rather than parents. dialogue 2: learning from families here, chela (another teacher) is responding to a question asked at the beginning of the school year: what have you learned from families that you have successfully incorporated into your practice? chela was representative of those teachers who took an asset based approach in considering home practices: i’ve been speaking with the mother much more and in a system of we want to work together as a partnership here. i tell her, you have a lot of experience with this child and you know way more as i’m only two weeks into it and so what can you do, what is it that you do at home that i can do here? what kind of the same language can be used? sometimes we can do some of the same activities to just make it familiar and so that the student has a little bit more success. kristin: chela is one of the teachers who really seemed to talk concretely about trying to learn from families. often, i think the teachers talked about the importance of having strong partnerships, but the actual relationships, when they talked about their practices with families, reminds me of how lareau (2000) saw home-school relationships reflective of a professional/client relationship, but not chela. in this passage, chela shows that she truly wants input from parents because she thinks home practices should have a place in the classroom. anita: this quote is a great example of ways to rethink parent/teacher relationships so that by viewing parents as resources, the teacher makes them the authority (civil, guevara, & allexsaht-snider, 2002). she also goes beyond asking what the family does by asking about the language they use as they engage—by language, i think she is referring to discourse practices, which research has shown may be very different in home than in school (heath, 1982). it reminds me of some of her learning stories and her family outreach project. chela learned from home visits and observation that her child enjoyed playing games, counting food, and loved donuts. so she designed a game similar to hi-ho cheerio but using donuts. some might argue that she was only drawing on the child’s interests, much like hedges (2011) interpretation of funds of knowledge, rather than incorporating family practices. but i would argue that we can’t say what meaning game playing had in the family—it may have contributed to the families “household well-being” and thus an example of incorporating the child and family’s funds of knowledge (moll, amanti, neff, & gonzález, 1990, p. 133). wager & whyte whose home practices are privileged stinson, d. w., & spencer, j. a. (eds.). (2013). privilege and oppression in the mathematics preparation of teacher educators [special issue]. journal of urban mathematics education, 6(1). 86 dialogue 3: moving beyond cooking projects we found some situations in which there seemed to be evidence of movement toward an asset view of family practices. however, the kinds of family practices that were built upon made us wonder whether or not the teacher was only supporting what she was comfortable with. in this segment of her interview, emma shared how some of the mathematics activities in the classroom built on children’s experiences in the home: you know, a lot of it comes from the cooking projects. we have some shared cooking projects, and a lot of times, that’ll come with the special weeks or even the holiday things where you have a special treat and it’s something that can be made with kids. we often ask, “hey, can we do that?” and that—you know, the measuring, the pouring, all of the counting the scoops, and that kind of stuff. that’s usually where it comes in. you know, other than if they bring stuff in from home that they want to count, or like bringing—sometimes families will bring in pinecones from their yard, not a lot—i mean, i don’t know. kristin: this provides an example of how she brings home mathematics into the classroom. anita: yes, but it is the home mathematics she is expecting to be there— pinecones in a yard (what if there is no yard) and cooking and measuring. cooking and measuring is such a “go to” context for bridging home and school mathematics but families have multiple practices for measuring that may not include scoops (wager, 2012). she seems to be making assumptions that the cooking she is doing in school reflects the cooking in the children’s homes. kristin: well, i think she is giving actual examples of home practices families and children elected to bring into the classroom. however, i hear your point. often, i think early childhood teachers bring in food-related projects to connect with families who are different than them and that’s where a family’s culture being reflected in the classroom ends. anita: it is hard to know—were those projects brought in because those families felt they had agency to do so? were there other families that didn’t bring projects in because they didn’t know they could, didn’t know if their experiences would be validated? but you have more experience with this teacher and so i am interpreting a small portion of a comment without any context. it is interesting to think about the methodology behind what we are doing, discussing snippets of data having varying degrees of knowledge about the teacher or the context. i also think it is important to think about the lens i bring in. am i looking for a deficit perspective? wager & whyte whose home practices are privileged stinson, d. w., & spencer, j. a. (eds.). (2013). privilege and oppression in the mathematics preparation of teacher educators [special issue]. journal of urban mathematics education, 6(1). 87 kristin: i don’t think you’re looking for a deficit perspective, i think you have a healthy amount of skepticism when it comes to cooking projects. so, what if a family brings in a cooking activity to do with the class and emma maps school mathematics language on to it? is that bridging home and school? is it not acknowledging how the family enacts mathematics practices in their home? anita: i’d say yes, in this case it is bridging home and school and acknowledging family practice. this situation is a common issue we face when thinking about how to (and how teachers) use family practices to engage with mathematics. to what extent are the things teachers are doing in the classroom reflecting the embedded mathematical practices that families engage with in various activities (gonzález, andrade, civil, & moll, 2001). on the other hand—it is a start. she is thinking about what families do that can provide a context for mathematics in school. dialogue 4: young children and grown-up conversations the following exchange is based on an observation in birdie’s classroom. she is chatting with the children during lunchtime about what they did over the weekend: naeem: i got into a fight at my house. birdie: naeem, don’t forget to eat your corn. did anyone do anything fun over the weekend? tim: i went to the zoo. birdie: in town? and you got new boots? and you did too? wow—lots of new clothes. amaya: i went to chuck e. cheese’s with my baby sister and then we was about to go somewhere else…. birdie: what did you do at chuck e. cheese’s? birdie: did you do anything fun at all this weekend, leilani? leilani: nothing. birdie: nothing! did you play outside? no! did you stay in bed all weekend? and what about you naeem, did you do something with daddy this weekend? naeem: he played some games, then went to bed and work. i didn’t go to chuck e. cheese’s. kristin: this exchange doesn’t necessarily have to do with math, but while looking in nvivo for a different paper, i came across this powerful example of what teachers privilege from an observation in birdie’s classroom. it has to do with not wanting “unfun” things to be brought into the classroom. anita: it isn’t about mathematics in this instance, but it is an example of a teacher finding out about what children do outside of school and the teachers in the pd wager & whyte whose home practices are privileged stinson, d. w., & spencer, j. a. (eds.). (2013). privilege and oppression in the mathematics preparation of teacher educators [special issue]. journal of urban mathematics education, 6(1). 88 are learning how to connect mathematics in school to what children do out of school. in birdie’s case, it seems she is looking to learn about what children do outside of school that meets her definition of “fun.” leilani had to compare what she did to going to chuck e. cheese’s and it didn’t seem to be as fun so she did “nothing” fun. is birdie shutting children out of the conversation if their weekends don’t match up to the fun of chuck e. cheese’s? doesn’t this privileging of particular activities suggest that the out of school experiences birdie has to draw on for mathematics activities will be limited to particular kinds of activities, from certain children? thus other children may not get the opportunity to see their experiences reflected in the classroom. kristin: right, and thinking of naeem, not only did naeem not get to see his experiences reflected in the classroom, he learned that parts of someone else’s life counts more than his. anita: it reminds me of gutiérrez’s (2007) windows and mirrors—in birdie’s class those children like naeem who don’t share birdie’s experiences will always be looking through a window at the experiences of more privileged children whereas amaya gets to see her home experiences reflected in school. kristin: and then what do the children who are looking in do? do they find other ways to get attention? there were a couple other moments in this observation that seemed like naeem was seeking negative attention from birdie: birdie: [naeem] we don’t belch at the table. that’s not polite. you may do that at home, but we don’t do that at school. naeem: i can do it at home. birdie: ok, but not at school. because he couldn’t get attention for talking about the fight he finds memorable from his weekend (his mirror), i wonder, is he doing things on purpose to get in trouble because that’s how he learned he could get attention in this classroom? i feel like it’s easy for situations like this to arise in classrooms. this situation seems true in early childhood classrooms because our conceptions of childhood are wrapped up in ideas of fun and innocence (johnny, 2006), which doesn’t make talking about things like fighting feel easy or appropriate (zipin, 2009). that being said, i do think there are ways to talk about “uncomfortable” topics with young children. anita: speaking of uncomfortable topics, i can’t let this conversation go without bringing up birdie’s comments about “lots of new clothes.” this comment brings to mind a remark clover, another teacher, made two years ago. she and her group wager & whyte whose home practices are privileged stinson, d. w., & spencer, j. a. (eds.). (2013). privilege and oppression in the mathematics preparation of teacher educators [special issue]. journal of urban mathematics education, 6(1). 89 were planning a lesson connecting mathematics to social justice using the children’s book those shoes (boelts & jones, 2009) when it suddenly occurred to her how often we comment on new clothes or shoes that children wear without regard to how those children who don’t have new things must feel. kristin: i think talking about clothes feels like an easy, “nice” thing to talk about, giving and receiving compliments feels good and little kids beam when you talk about their new shoes. i think they’ve learned some of these lessons before they enter school so children who are used to receiving compliments about their appearance seek that kind of attention. when i taught pre-k if a little person would come in and proudly show me their new shoes, of course, i would share in their excitement. however, as a classroom practice, we would talk about the importance of giving compliments about how we treated each other. consequently, the children did give and seek out more compliments about their interactions with each other and their learning than they did their appearance. dialogue 5: what children bring to the classroom the following section of an interview with amanda provides an interesting contrast to “what should be fun” and who identifies what is fun. amanda began by talking about what happened at lunch with one of her children and shifted to a discussion about objects she has children bring in from home to share: amanda: one little girl last year was eating a hamburger. and we had peas that day. and she took her finger, and i just watched her. and she poked holes all the way around her hamburger bun. and then she counted her peas as she placed them in the holes. interviewer: now someone would say that’s playing with your food, but somebody amanda: i know. i—i just looked at her and i said, “that’s really neat.” and then when her dad came in that night i said, “you know, spontaneous learning. and she’s counting.” amanda: i try to encourage them to bring stuff in. and we usually have a morning time where i read stories and it’s just as easy to drop off a story if somebody brings something in. and they’re all learning from it. and it seems like the more children that do that, the more they want to bring in. and—and typically like on show and tell days it’s usually a toy. and i always leave a note for the parents, “please think of something other than a toy.” interviewer: ah. now why is that? amanda: why is that? because there’s so many things in everyday life. and i think that children think they have to bring something that’s fun and exciting for the class. but the girl that brought in the apple, which is very simple, she told me, “amanda, if you cut it this way, there’s a star inside.” so you know what that child has done at home because they already came in with that. so we cut it that way, and then we talk about how many compartments there are, how many seeds there are. wager & whyte whose home practices are privileged stinson, d. w., & spencer, j. a. (eds.). (2013). privilege and oppression in the mathematics preparation of teacher educators [special issue]. journal of urban mathematics education, 6(1). 90 anita: amanda presents an interesting example of privileging the child in the child-teacher relationship. rather than taking control as some might by worrying about “playing with her food,” amanda allows the peas and bun example to continue and sees it as a site for engaging with mathematics. kristin: i agree; this is a great example child-centered pedagogy. amanda is both creating and allowing learning to flow from the child by starting with the child’s interests and actions. anita: the other thing i find interesting here is how she positions fun. for birdie “fun” was what she expected young children to consider fun. to amanda, any object can be interesting and revealing about children’s home practices. kristin: yes, but what would amanda do if this topic was more sensitive or violent? i bet birdie would be thrilled if naeem wanted to bring in an apple or count peas on his hamburger bun. i do think this is a great example of how a teacher can support both child initiated learning and get clues into what’s important to a child’s family. however, i still wonder how we, as early childhood educators, feel about what place “heavier” conversations have with young children. if early childhood classrooms are supposed to be protective places for young children, i sometimes think teachers consider conversations about things like who can afford new clothes to be unsafe teaching topics. if that’s the case, i have to wonder, unsafe for whom? anita: i can’t say how amanda would address a more critical issue. from my perspective, many of these topics can be connected to or explored through mathematics, much the way clover and her group were planning to do. dialogue 6: mathematizing home practices versus worksheets another theme that teachers often raised was the appropriateness of workbooks. sophie is responding to the interview question: have families ever shared with you strategies that they use in terms of mathematics at home?: sophie: not really. i mean a lot of families, in all honesty, they just buy a workbook from walgreens and have their kids practice writing their numbers at home—which isn’t bad, but i mean, that’s just how they practice math, and they don’t really see the real-life connection that could be—that is possible with math. especially, for younger children. interviewer: have you been working with families and learned some things about how they use mathematics at home? sophie: i guess mostly in kindergarten, i’ve just tried to get families to incorporate mathematics into their everyday lives. counting settings at the table, counting differ wager & whyte whose home practices are privileged stinson, d. w., & spencer, j. a. (eds.). (2013). privilege and oppression in the mathematics preparation of teacher educators [special issue]. journal of urban mathematics education, 6(1). 91 ent like objects maybe when they’re driving along, counting signs, counting cars; when they’re out on a walk, maybe counting their steps or something, i mean, just trying to incorporate that into their everyday life as far as, you know, that skills has been going. anita: it is interesting that sophie states that families haven’t really shared how they use mathematics at home but then says they do workbooks. how does she know this if they haven’t shared? is it an assumption she has made based on observations at the drug store? kristin: maybe she’s responding this way because some early childhood pedagogy does not encourage using worksheets with young children (copple & bredekamp, 2009). if this is something sophie believes then maybe workbooks don’t count? maybe she’s trying to think of mathematics practices that are more reflective of a play-based pedagogy—i know that’s what the district she’s working in is pushing for four-year-old kindergartners. anita: then she goes on to talk about what she encourages families to do. this remark is similar to betty’s comments from dialogue 1. kristin: definitely. sophie makes me think about when i was teaching and how i didn’t feel comfortable using worksheets as teaching tools with four-year-olds, but that parents often asked for them. i was being guided by what i learned in my teacher preparation program about developmentally appropriate practices and i think families are guided by their own sets of experiences from when they were in school (lightfoot, 2003). also, to be honest about it, even though i wouldn’t use worksheets in my classroom and i would talk to families about why i felt more contextual ways of teaching mathematics were more effective; i definitely loved those mathematics workbooks when i was little. so it’s tricky, but i think we need to examine why we are teaching in particular ways and then ask ourselves: how do i teach mathematics this way and validate what the families are doing; are there things i can change about my teaching, why or why not; how do i show how i see how much they’re caring about their children’s education when they’re buying that workbook? i also think we need to consider both whether or not we are doing the child an injustice if we don’t talk to their family about what we see as best practices. however, at the same time, we need to think about if by telling a family to “count signs while they’re driving” are we changing a family’s ways of being with their children and how do we feel about that? when pondering these questions, i always came around to thinking about the importance of balance. i took solace in the idea that by continuously thinking complexly about these kinds of questions and pushing my comfort levels hopefully i ended up creating more space for families in our classroom. wager & whyte whose home practices are privileged stinson, d. w., & spencer, j. a. (eds.). (2013). privilege and oppression in the mathematics preparation of teacher educators [special issue]. journal of urban mathematics education, 6(1). 92 what did we learn about privilege in pre-k mathematics? we embarked on our dialogue with a goal of answering three questions: what home mathematics practices did pre-k teachers privilege? how did privileging particular practices reinforce or interrupt historical power structures in teacher/parent (wallard, 1932; laureau, 2000) and teacher/child (canella, 2008) relationships? which families and children did teachers privilege and how might that privileging have oppressed others? through our discussion of the data presented, it is not surprising that we found our three questions were connected. in situations when teachers privileged their view of what home mathematics should look like, it reinforced existing teacher-child and teacher-parent power structures and oppressed those children and families who did not share the same experiences. on the other hand, in those instances in which teachers privileged children’s and family’s experiences and voices interrupted historical practices and repositioned power and agency to families and children. in the following brief discussion, we respond to our questions based on what and how teachers privileged home mathematics practices when teaching young children. we then conclude with a few considerations for other mathematics teacher educators involved in this work. what practices were privileged there were two approaches to the ways teachers thought about mathematics in the home: (a) an appreciation for families’ diverse mathematical practices, or (b) an expectation for home mathematical practices that teachers considered “appropriate.” emma (dialogue 3) and amanda (dialogue 5) privileged the practices of families and children. they both sought out what families did and thought about how to incorporate those practices in the classroom. in emma’s case, she “added” the mathematics to what she saw families doing, whereas amanda recognized the mathematics that children brought in. in so doing, both teachers provided a mirror (gutiérrez, 2007) through which the children could see their home practices reflected. furthermore, the teachers shifted agency to the children who, as a population, have historically been oppressed (cannella, 2008). birdie (dialogue 4), on the other hand, privileged those out-of-school practices that aligned with what she considered as appropriate and interesting school topics (e.g., going to chuck e. cheese’s, having new clothes). by focusing on the practices of children whose experiences were considered appropriate, birdie maintained the privileged status of those children, thereby oppressing others. sophie offered an interesting example of privileging home practices over what she saw as school-like practices in the home such as workbooks. one might expect that in the current standards-driven climate, workbooks would be seen as a positive practice, yet early childhood “standards” call for play-based learning, thus wager & whyte whose home practices are privileged stinson, d. w., & spencer, j. a. (eds.). (2013). privilege and oppression in the mathematics preparation of teacher educators [special issue]. journal of urban mathematics education, 6(1). 93 workbooks are the antithesis of standards. as such sophie (as we found with many of our teachers) disdained worksheets. the home mathematics practices that teachers did or did not find appropriate may have been connected to the early childhood practice of mathematizing activities children engage with in the classroom (ginsburg, 2006). as a result, early childhood teachers may expect families to do the same (e.g., counting oranges as they are placed in the cart at the market). how practices were privileged from our perspectives, teachers viewed their role in one of two ways: (a) learning from families and bringing those ideas in the classroom, or (b) “telling’” families what they should be doing (graue, kroeger, & prager, 2001; pushor & murphy, 2004). chela (dialogue 2) saw families as a resource and endeavored to learn from them. by asking families for specific examples of how they interacted with their children at home and explaining how she would use the family’s practices to support classroom experiences, she shifted power to the family positioning them as experts. others, such as betty (dialogue 1) and sophie (dialogue 6), reified their role as the expert by making suggestions of mathematics practices families should do, rather than asking families about their own practices. in posing yes-or-no questions about those practices she expected or approved of, betty provided recommendations for ways families could support their children’s mathematics learning, which may or may not have been disconnected from family practices. those teachers who assumed it was their responsibility to teach the families, rather than learn from them, both created and maintained the power reminiscent of a professional/client relationship rather than a collaborative relationship with families. parting thoughts our dialogue revealed to us how simple it seems to reinforce power structures in the relationships between teacher/family and teacher/child, and there are direct consequences for things that teachers hold dear: collaborative relationship, child-centered pedagogy, early math learning. we suggest that teachers need to have explicit conversations about what they privilege and what the consequences of that power to privilege might have in their work with families and children. in some cases, the funds of knowledge framework used in the pd provided teachers with a “feel good” approach to pedagogy yet “still reflected a pervasive power relationship that positions the educator as one who can pick and choose those aspects of students’ lives that ‘belong’ in the realm of the classroom” (rodriguez, 2013, p. 94). we recognize that this move toward learning about and incorporat wager & whyte whose home practices are privileged stinson, d. w., & spencer, j. a. (eds.). (2013). privilege and oppression in the mathematics preparation of teacher educators [special issue]. journal of urban mathematics education, 6(1). 94 ing children’s mathematical funds of knowledge in the classroom is difficult, particularly when it challenges existing structures and practices. it also decenters teachers, forcing them to disconnect from their established practices—particularly if their experiences are different from those of the child they teach. as we move forward, we will consider how funds of knowledge can provide opportunities for disrupting current inequitable power structures/practices hopefully shifting “whose” knowledge counts. finally, during our dialogue, issues arose about our role in supporting and studying teachers as they learn to work with families and children in culturally responsive ways. we remind ourselves as we do this work to assure that we set teachers up for success in the questions we ask and assignments we give them and to watch out for our own biases so as not to reify the power relationship between researcher/facilitator and teacher. acknowledgments the work presented here was supported in part by a grant from the national science foundation, grant number 1019431. any opinions, findings, and conclusions or recommendations expressed herein are those of the authors and do not necessarily reflect the views of the national science foundation. references boelts, m., & jones, n. z. 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(1992). funds of knowledge for teaching: using a qualitative approach to connect homes and classrooms. theory into practice, 31(4), 132– 141. pushor, d. & murphy, b. (2004). parent marginalization, marginalized parents: creating a place for parents on the school landscape. alberta journal of educational research, 50, 221– 235. remillard, j. t. (2005). rethinking parent involvement: african american mothers construct their roles in the mathematics education of their children. school community journal, 15(1), 51– 73. rodriguez, g. m. (2013). power and agency in education: exploring the pedagogical dimensions of funds of knowledge. review of research in education, 37, 87–120. wager, a. a. (2012). incorporating out-of-school mathematics: from cultural context to embedded practices. journal of mathematics teacher education, 15, 9–23. waller, w. (1932). the sociology of teaching. new york, ny: john wiley. zipin, l. (2009). dark funds of knowledge, deep funds of pedagogy: exploring boundaries between lifeworlds and schools. discourse: studies in the cultural politics of education, 30, 317–331. journal of urban mathematics education july 2015, vol. 8, no. 1, pp. 17–23 ©jume. http://education.gsu.edu/jume danny bernard martin is chair of curriculum and instruction and professor of mathematics at the university of illinois at chicago, 1040 w. harrison street, chicago, il, 60607; email: dbmartin@uic.edu. his research has focused primarily on understanding the salience of race and identity in black learners’ mathematical experiences, taking into account sociohistorical and structural forces, community forces, school forces, and individual agency. commentary the collective black and principles to actions1 danny bernard martin university of illinois at chicago for years i labored with the idea of reforming the existing institutions of society, a little change here, a little change there. now i feel quite differently. i think you’ve got to have a reconstruction of the entire society, a revolution of values.2 – dr. martin luther king, jr. ood morning. i want to begin by thanking the organizers of this plenary session for extending me an invitation to participate. this time of year is especially busy and being able to break away to discuss important issues with colleagues from across the country is certainly appreciated. in that spirit, let me say that it is my pleasure to share the podium with deborah, dan, and steve. by show of hands, how many people have had a chance to read principles to actions: ensuring mathematics success for all (national council of teachers of mathematics [nctm], 2014)? if you have not read it, i encourage you to do so. i believe there are many take-aways worthy of further discussion and analysis. in terms of my own take-aways, there are five that i would like to focus on. first, it is clear to me that principles to actions is a political document. it advances particular views and visions of mathematics teaching and learning. these views are so strongly worded that other possibilities and visions are pretty much ruled out. in fact, the word “non-negotiable” is used in relation to the recommendations. the political nature of the document is confirmed on the copyright page where it reads: “principles to actions: ensuring mathematics success for all is an official position of the national council of mathematics teachers as approved by the nctm board of directors, february 2014” (p. ii). so, it is not just a book, it is an official position. my second take-away is that despite the strong tone of the document, the actual content of principles to actions will be familiar to most of you if you have read 1 this commentary is a revised version of remarks made at the national council of teachers of mathematics research conference plenary session “turning the common core into reality in every math classroom,” delivered on april 15, 2015 in boston, ma. (other invited panelists included deborah loewenberg ball, dan meyer, and steven leinwand.) 2 see vincent harding’s 2008 book martin luther king: the inconvenient hero, page 98. g http://education.gsu.edu/jume mailto:dbmartin@uic.edu martin commentary journal of urban mathematics education vol. 8, no. 1 18 previous documents published by nctm, including the 1989 curriculum and evaluation standards for school mathematics, the 2000 principles and standard for school mathematics, the 2006 curriculum focal points for prekindergarten through grade 8 mathematics: a quest for coherence, and focus in high school mathematics: reasoning and sense making, published in 2009. beyond these documents, one can also recognize the influence of the national research council’s adding it up: helping children learn mathematics published in 2001 and two publications from the conference board of mathematical sciences, the mathematical education of teachers i and ii, published in 2001 and 2012, respectively. as a result, much of what is contained in principles to actions is not new. what might appear to be new is the merging of high-leverage practices and essential teaching skills into a set of eight “mathematics teaching practices” in the same vein as the eight standards for mathematical practice found in the common core state standards initiative. 3 these mathematics teaching practices include things like: (a) establishing mathematics goals to focus learning; (b) implementing tasks that promote reasoning and problem solving; (c) posing purposeful questions; (d) supporting productive struggle in learning mathematics; and (e) eliciting and using evidence of student thinking. again, nothing new here. some would say that these practices just represent aspects of good teaching. a third, slightly more problematic, take-away for me is that principles to actions reflects a deep and unequivocal commitment to the common core by nctm even as it seems that elements of the common core movement are starting to unravel (see, e.g., kirp, 2014; ravitch, n.d.). this unequivocal support can be found early in the document where there is a partial restatement of nctm’s position statement on the common core.4 i think it is worth repeating here: the widespread adoption of the common core state standards for mathematics presents an unprecedented opportunity for systemic improvement in mathematics education in the united states. the common core state standards offer a foundation for the development of more rigorous, focused, and coherent mathematics curricula, instruction, and assessments that promote conceptual understanding and reasoning as well as skill fluency. this foundation will help to ensure that all students are ready for college and the workplace when they graduate from high school and that they are prepared to take their place as productive, full participants in society. (p. 4) the fourth take-away, reflected in the common core position statement and the essential elements of principles to actions is the continued focus on equity and the rhetoric of “mathematics for all” (martin, 2003, 2011) that was expressed in the 1989 and 2000 standards documents. the concerns for equity expressed in 3 see http://www.corestandards.org/math/practice/. 4 see http://www.nctm.org/ccssmposition/. http://www.corestandards.org/math/practice/ http://www.nctm.org/ccssmposition/ martin commentary journal of urban mathematics education vol. 8, no. 1 19 principles to actions, like earlier documents, make note of the need to ensure mathematics success for all students with particular expressions of concern for african american, latin@, indigenous,5 and poor students; that is, those who have been the least well served by school-based mathematics education. this is a 26-year-old message, couched in a 400-year-old quest for equity in the united states. in fact, as i read principles to actions, it seemed that the emphasis on equity and mathematics success for all was repeated on every other page. the repetitive nature of this commitment certainly got me thinking. on one hand, it might be reasonable to applaud nctm’s persistent message on issues of equity and mathematics for all. on the other hand, the inequitable outcomes that are the focus of nctm’s 26-year lament have also happened on their institutional watch and in the context of all previous recommendations. so, taking the latter response as my cue, i am going to say that perhaps it is time to take a more critical look at nctm’s equity-oriented message and politics. i understand this could be a risky move. i may be in violation of the adage: don’t bite the hand that is feeding you. my final take-away from principles to actions focuses on nctm’s framing of the obstacles that could hinder their vision for mathematics teaching and learning. these obstacles are framed in terms of unproductive beliefs on the part of stakeholders. i want to push on this framing and raise some specific points of concern about the equity and mathematics for all messages relative to principles to actions and to nctm as an organization. as i noted, principles to actions is a political document. it is also true that nctm is a political organization. it speaks for and to particular audiences for political purposes. it advances social and political agendas and attempts to shape the prevailing social order, particularly in the realm of education. on the copyright page of principles to actions, there is a statement that reads in part: “the national council of teachers of mathematics is the public voice of mathematics education” (p. ii, emphasis added). in reading principles to actions as a political document and thinking more deeply about nctm’s equity advocacy, several questions emerged for me. the first set of questions that i considered is: who is this document written for? who are the primary audiences? beyond any surface level considerations and possibilities, who is this document really written for? the second set of questions is: what are the underlying appeals that are being made to these primary audiences? what are the politics associated with these appeals? 5 the original text of the plenary used the term native american; i change that term here to reflect the pre-invasion and pre-colonial identities of people from indigenous nations. in fact, the remaining terms in this list are social constructions and reflect their use in the racialized social system of the united states. my use of these terms also reflects their use in principles to actions. martin commentary journal of urban mathematics education vol. 8, no. 1 20 based on the answers to the first two sets of questions, my final question asks: moving forward, what stance will i take toward nctm and its professed commitment of ensuring equitable mathematics learning of the highest quality for all students? let me take up the questions in the order that i raised them. who is this document written for? who are the primary audiences? the obvious response is that nctm is targeting mathematics teachers, mathematics education researchers, and mathematics education policy makers. in terms of demographics, we know that each of these spaces—practice, research, and policy—is predominantly white. for example, about 85% of the u.s. teaching force is white (feistritzer, griffin, & linnajarvi, 2011). we also know that the research and policy domains are also characterized by a largely white demographic. in my research, i refer to mathematics education and research contexts as white institutional spaces (see, e.g., martin, 2008, 2011, 2013). moreover, i am going to estimate that the membership of nctm— encompassing teachers, researchers, and others—is about 90% white. based on sheer numbers alone, nctm is a white organization and the primary audiences for principles to actions are largely white audiences. my second set of questions asks: what are the underlying appeals that are being made in principles to actions? what are the politics associated with these appeals? i claim that the underlying appeals are to white rationality, white sensibilities, and white benevolence. these appeals are not specific to principles to actions. the history of the united states is littered with examples of equity-oriented policies that have had to appeal to white rationality and benevolence. the appeals that are implicit in principles to actions also include a form of interest convergence. interest convergence refers to the fact that gains for minority groups coincide with white self-interests and materialize at times when some type of breakthrough for minority groups is needed, usually for the sake of world appearances or the imperatives of international competition (see, e.g., bell, 1980). converging interests in principles to actions take this form: engage in mathematics education via the teaching practices and essential elements outlined here and all students will benefit, not just those identified as white, but also those identified as african american, latin@, indigenous, and poor (i.e., the collective black).6 6 i borrow the term the collective black from eduardo bonilla-silva (2002, 2004), who proposes: that the bi-racial order typical of the united states, which was the exception in the world-racial system, is evolving into a complex and loosely organized tri-racial stratification system. … specifically, i argue the emerging tri-racial system will be comprised of ‘whites’ at the top, an intermediary group of ‘honorary whites’ similar to the coloureds in south africa during the formal apartheid (fredrickson 1981), and a nonwhite group or the ‘collective black’ at the bottom. … i hypothesize that the white group will include ‘traditional’ whites, new ‘white’ immigrants and, in the near future, totally assimilated white latinos…lighter-skinned multiracials (rockquemore and brunsma 2002), and other sub-groups; the intermediate racial group or honorary whites will comprise most light-skinned latinos (e.g., most cubans and segments of the mexican and puerto martin commentary journal of urban mathematics education vol. 8, no. 1 21 the unspoken, hidden reality in principles to actions is that potential benefits to the collective black are metered by whites and white design and are contingent on parallel benefits to whites. principles to actions could never have been written to focus solely on gains for the collective black. this statement is true because most systems and institutions in our society, including mathematics education, are not set up to serve the collective black. the hard truth is that the outcomes and inequities lamented over in principles to actions and previous documents are precisely the outcomes that our educational system is designed to produce. equity-oriented slogans, statements about idealized outcomes, and tweaks to teaching or curricular practices within this system do not change this fact. a more honest framing of mathematics reform and policy would speak to the fact that school-based mathematics education for the collective black is placed largely in the hands of whites or in the hands of non-whites who are often positioned to preserve white interests. this recipe is a familiar one. we have seen it, for example, in missionary-oriented efforts from decades ago. today, we see it in efforts like teach for america. my late colleague william watkins (2001) wrote a book titled the white architects of black education: ideology and power in america, 1865–1954. this book reminds us that negative outcomes for the collective black relative to white interests are not really problems but actually support larger social and political agendas. my own view is that this form of education, one that is rooted in appeals to white rationality and white benevolence, is a colonizing form of education, not a liberating education or an education characterized by equitable access to opportunity. for example, framing mathematics education solely in service to college and career readiness, for example, glosses over the commodification of students as future workers in favor of their participation in a system that has long oppressed many of them. principles to actions says little about critical mathematical literacy to understand and change that system. these calls also bypass the limited capacity of higher education to serve the students it currently tries to serve. increased access, which i strongly support, is still likely to result in greater selectivity, bias, and backlash against the collective black, in many cases to maintain white interests. as some of you know, early resistance to algebra for all, for example, was rooted in such a backlash, stemming from the interest-preserving concerns of white middle and upper-class parents. more recently, u.s. secretary of education arne duncan noted how resistance to the common core overlaps with white interests. he stated that some of the opposition to the common core state standards has come from “white suburban moms who—all of a sudden—their child isn’t as brilliant as they rican communities), japanese americans, korean americans, asian indians, chinese americans, filipinos, and most middle eastern americans; and, finally, that the collective black group will include blacks, dark-skinned latinos, vietnamese, cambodians and laotians. (bonillasilva, 2004, pp. 932–933) martin commentary journal of urban mathematics education vol. 8, no. 1 22 thought they were, and their school isn’t quite as good as they thought they were” (strauss, 2013). wrapping up, i want to say that despite the fact that nctm is a political organization and produces political documents like principles to actions, i would argue that the organization and document do not go far enough in arguing for a decolonizing form of mathematics education. instead, it is rooted in an implicit benevolent appeal and the provision of accommodations that will allow african american, latin@, indigenous, and poor students to enjoy contingent benefits of the system that is not set up for them or by them. according to revolutionary and philosopher frantz fanon (see, e.g., 1965), a minimum outcome of decolonization, including a decolonized education, is that the last shall become first. according to fanon, the process of decolonization should have such a violent character that it completely dismantles existing systems of oppression. requests or negotiations for white benevolence would have no place in a decolonizing (re)form of mathematics education. so, reframing my final question posed earlier, i am left to ask: does this document represent, symbolically and in spirit, the kind of disruptive violence to the status quo that can move the last to first? can it truly help in improving the collective conditions—not isolated examples of success—of african american, latin@, indigenous, and poor students? by success, i do not mean slow growth and incremental gains. i predict when the dust settles on common core, we will move on to some other reform and there will, once again, be statements about the need for equity and mathematics for all. with respect to nctm, i invite you to consider the question: is nctm the kind of organization that is capable of facilitating the kind of violent reform necessary to change the conditions of african american, latin@, indigenous, and poor students in mathematics education? references bell, d. a., jr. (1980). brown v. board of education and the interest-convergence dilemma. harvard law review, 93, 518–533. bonilla-silva, e. (2004). from bi-racial to tri-racial: towards a new system of racial stratification in the usa. ethnic and racial studies, 27(6), 931–950. bonilla-silva, e. (2002). we are all americans!: the latin americanization of racial stratification in the usa. race and society, 5(1), 3–16. conference board of mathematical sciences. (2001). the mathematical education of teachers. providence, ri and washington, dc: american mathematical society and mathematical association of america. conference board of mathematical sciences. (2012). the mathematical education of teachers ii. providence, ri and washington, dc: american mathematical society and mathematical association of america. fanon, f. (1965). the wretched of the earth. new york, ny: grove press. martin commentary journal of urban mathematics education vol. 8, no. 1 23 feistritzer, c. e., griffin, s., & linnajarvi, a. (2011). profile of teachers in the us, 2011. washington, dc: national center for education information. fredrickson, g. m. (1981). white supremacy: a comparative study of american and south african history. new york, ny: oxford university press. kirp, d. l. (2014, december 27). rage against the common core. the new york times. retrieved from http://www.nytimes.com/2014/12/28/opinion/sunday/rage-against-the-common-core.html?_r=0 martin, d. b. (2003). hidden assumptions and unaddressed questions in mathematics for all rhetoric. the mathematics educator, 13(2), 7–21. martin, d. b. (2008). e(race)ing race from a national conversation on mathematics teaching and learning: the national mathematics panel as white institutional space. the montana math enthusiast, 5(2-3), 387–398. martin, d. b. (2011). what does quality mean in the context of white institutional space? in b. atweh, m. graven, w. secada, & p. valero (eds.), quality and equity agendas in mathematics education (pp. 437–450). new york, ny: springer. martin, d. b. (2013). race, racial projects, and mathematics education. journal for research in mathematics education, 44(1), 316–333. national council of teachers of mathematics. (1989). curriculum and evaluation standards for school mathematics. reston, va: national council of teachers of mathematics. national council of teachers of mathematics. (2000). principles and standards for school mathematics. reston. va: national council of teachers of mathematics. national council of teachers of mathematics. (2006). curriculum focal points for prekindergarten through grade 8 mathematics: a quest for coherence. reston, va: national council of teachers of mathematics. national council of teacher of mathematics. (2009). focus in high school mathematics: reasoning and sense making. reston, va: national council of teachers of mathematics. national council of teachers of mathematics. (2014). principles to actions: ensuring mathematics success for all. reston, va: national council of teachers of mathematics. national research council. (2001). adding it up: helping children learn mathematics. in j. kilpatrick, j. swafford, & b. findell (eds.), mathematics learning study committee, center for education, division of behavioral and social sciences and education. washington, dc: national academy press. ravitch, d. (n.d.) diane ravitch’s blog: a site to discuss better education for all – archives for category: common core. retrieved from http://dianeravitch.net/category/common-core/ rockquemore, k. a., & brunsma, d. l. (2002) beyond black: biracial identity in america. thousand oaks, ca: sage. strauss, v. (2013, november 16). arne duncan: ‘white suburban moms’ upset that common core shows their kids aren’t ‘brilliant’. the washington post. retrieved from http://www.washingtonpost.com/blogs/answer-sheet/wp/2013/11/16/arneduncan-white-surburban-moms-upset-that-common-core-shows-their-kids-arent-brilliant/ watkins, w. h. (2001). the white architects of black education: ideology and power in america, 1865–1954. new york, ny: teachers college press. http://www.nytimes.com/2014/12/28/opinion/sunday/rage-against-the-common-core.html?_r=0 http://dianeravitch.net/category/common-core/ http://www.washingtonpost.com/blogs/answer-sheet/wp/2013/11/16/arne-duncan-white-surburban-moms-upset-that-common-core-shows-their-kids-arent-brilliant/ http://www.washingtonpost.com/blogs/answer-sheet/wp/2013/11/16/arne-duncan-white-surburban-moms-upset-that-common-core-shows-their-kids-arent-brilliant/ microsoft word final davis et al vol 7 no1.doc journal of urban mathematics education july 2014, vol. 7, no. 1, pp. 96–106 ©jume. http://education.gsu.edu/jume   julius davis is an assistant professor of mathematics education in department of teaching, learning and professional development in the college of education at bowie state university, 14000 jericho park road, center for learning and technology building 233n, bowie, md 20715; email: jldavis@bowiestate.edu. his research focuses on african american students’ k–12 mathematical experiences and african american mathematics teachers’ mathematical experiences and praxis in urban areas. vanessa r. pitts bannister is an assistant professor of mathematics education in the college of education at the university of south florida, 4202 e. fowler ave., edu105, tampa, fl 33620; email: pittsbannister@usf.edu. her research interests include teacher and student knowledge in the areas of algebra and rational numbers, teachers’ pedagogical and content knowledge with respect to curriculum materials, and equity and diversity issues in mathematics education. jomo w. mutegi is an associate professor of science education and director of the center for the advancement of stem education (ucase) in the school of education at indiana university-purdue university indianapolis, 902 west new york street, es 3132, indianapolis, in 46202; email: jmutegi@iupui.edu. his research focuses on the underrepresentation of african americans in science and science-related careers. book review hip-hop and mathematics: a critical review of schooling hip-hop: expanding hip-hop based education across the curriculum1 julius davis bowie state university vanessa r. pitts bannister university of south florida jomo w. mutegi indiana universitypurdue university indianapolis early hhbe [hip-hop based education] practices have taken place most often in language arts and english education classrooms because of rap music’s clear and intuitive connections to the written, spoken, and poetic word. however, researchers and practitioners must forge meaningful connections to other disciplines, including those (like math and science) that are alleged to be culturally neutral. (hill & petchauer, 2013, p. 3) here has been a call for researchers and practitioners to use hip-hop based education (hhbe) practices in mathematics education (hill & petchauer, 2013). in the edited book schooling hip-hop: expanding hip-hop based education across the curriculum, hill and petchauer contend that extant hhbe literature has produced a clear and persuasive reason to use hhbe practices in educational settings. hill and petchauer assemble eight chapters from new and veteran hhbe scholars in the united states and abroad to expand the use of hip-hop beyond english into other disciplines, specifically, and the use of hip-hop based ed                                                                                                                           1 hill, m.l., & petchauer, e. (eds.). (2013). schooling hip-hop: expanding hip-hop based education across the curriculum. new york, ny: teachers college record. pp. 208, $29.95 (paper) isbn 080-7-75431-5 http://store.tcpress.com/0807754315.shtml t davis et al. book review journal of urban mathematics education vol. 7, no. 1 97 ucational practices, generally. while rap2 is the main hhbe strategy used by researchers and practitioners, hill and petchauer call for the use of other dimensions of hip-hop “such as djing or turntablism, b-boying/b-girling, graffiti writing and visual art, fashion, language, or spoken-word poetry” (p. 2). nevertheless, the editors do not offer a definition of hhbe that serves as a unifying thread throughout the book. hill and petchauer (2013) note, “the overwhelming majority of hhbe scholarship has failed to broaden the bounds of possibility for theorizing, researching, or implementing hip-hop based educational practices” (p. 2). the contributors to this edited volume seek to provide guidance in helping to broaden the possibilities of hhbe into other disciplines. in this review, we discuss the possibilities of this work in urban mathematics education for african american students. organization of the book the volume is divided into two sections of four chapters that are intended to expand hhbe into new intellectual directions. part i: aesthetics, worldviews, and pedagogies of hip-hop addresses the intersection between hip-hop and educational practice in a range of disciplines and settings. in chapter 1, emdin (2013) asserts to go beyond rap text to focus on incorporating the rap cypher3 battle rapping,4 and what he calls reality pedagogy in science education. in chapter 2, petchauer (2013) focuses on justice-oriented teaching and democratic curriculum through the hip-hop aesthetics of kinetic consumption and autonomy/distance with african american pre-service teachers. in chapter 3, peterson (2013) continues the focus on college students in an undergraduate hip-hop based composition course. he focuses on the hip-hop aesthetics of sampling, freestyling, and remixing to design a pedagogical and theoretical foundation for the composition course. in chapter 4, wilson (2013) focuses on southern hip-hop, higher education, and historically black institutions using the hiphop2020 curriculum project                                                                                                                           2 rap is just one of the elements of hip-hop. hip-hop refers to art forms and street influence that consist of 11 elements: (a) rapping, (b) singing, (b) b-boying/b-girling, (d) djaying, (e) graffiti, (f) beatboxin, (g) street fashion, (h) street language, (i) street knowledge, and (j) street entrepreneurialism (krs-one as cited in bridges, 2011; also see chang, 2005; jeffries, 2011). hip-hop also refers to how a person acts, walks, dresses, looks, and talks; in this context; it is both an art form and lifestyle. we do not refer to hip-hop as a culture. 3 a rap cypher involves people getting together in a circle taking turns freestyling, rapping prewritten or pre-thought-out rhymes. 4 a rap battle or battle rapping involves two people rapping against one another in front of an audience who determines the rapper with the best lyrical content and flow. davis et al. book review journal of urban mathematics education vol. 7, no. 1 98 (see http://fourfourbeatproject.org/hiphop2020/) to promote leadership development of college students. part ii: curricula, courses, and pedagogies with hip-hop explore hhbe programs and their complexities. in chapter 5, irby and hall (2013) focus on hhbe professional development provided to practicing teachers interested in using hip-hop in diverse schooling contexts. chapter 6 focuses on two hip-hop based afterschool programs for youth in montreal, canada where low, tan, and celemencki (2013) advocate for teachers to use rap as text and creative practice to conceptualize it as an aesthetic, cultural, and imaginative production. international perspectives of hhbe continue in chapter 7 with pardue (2013) who examines the use of hip-hip as a political project in são paulo, brazil with youth from poor, working class backgrounds to learn about what it means to be brazilian. the book concludes by returning hhbe to the united states in chapter 8 where stovall (2013) describes a social studies college bridge course that examines the current wave of gentrification and urban renewal in chicago. hip-hop based education in science education in order to forge meaningful connections between mathematics education and hhbe, mathematics educators will have to cross disciplinary boundaries to evaluate hhbe as an appropriate tool for the field. we would like to take it a step further by questioning whether hhbe is appropriate for african american students learning mathematics in urban schools. the use of hip-hop in classrooms was largely intended to reach african american students in urban schools (irby & hall, 2013). as no chapters in this book address mathematics education directly, we choose to focus a larger part of our review on emdin’s (2013) chapter on using hip-hop based educational practices in science education, the affiliate to mathematics education. we pay close attention to low and colleagues (2013) and stovall’s (2013) chapters where the focus is on youth, using them to draw parallels to work being done in mathematics education and to glean insight into hhbe for african american youth in mathematics education. we also focus on petchauer (2013) and irby and hall’s (2013) chapters where the focus is on pre-service and in-service teachers’ use of hhbe to think critically about how preand inservice mathematics teachers might use hip-hop based educational practices. emdin (2013) argues that culturally relevant approaches that do not consider hip-hop culture are most often ineffective approaches to teaching urban youth. he asserts that culturally relevant pedagogy (see, e.g., ladson-billings, 1994) is ineffective when theorists and teachers do not consider that urban youth are deeply immersed in hip-hop culture. emdin argues further that misidentifying “hip-hop as just a musical genre and not a culture, limits research and practice in urban schools from moving beyond the dead, mechanical, and formal approach to in davis et al. book review journal of urban mathematics education vol. 7, no. 1 99 struction that is prevalent in urban schools” (p. 12). in response to these approaches, emdin advocates for five main concepts/steps (the 5 c’s) that teachers should use to engage in hhbe. the 5 c’s comprise what he calls reality pedagogy. the 5 c’s of reality pedagogy include: (a) cogenerative dialogues, (b) coteaching, (c) cosmopolitanism, (d) context incorporation, and (e) content development (emdin, 2013). cogenerative dialogues are “structured dialogues in which the teacher and four to six students discuss the science classroom” (p. 20). he argues that these discussions should be structured like a rap cypher where participants form a circle, have equal turns to speak, and support one another in their roles. emdin views co-teaching as a hip-hop performance where the artists prepare for a performance. in his view, the student should take on traditional teacher roles such as planning and implementing a lesson. cosmopolitanism is a philosophical principle that, emdin contends, is a part of hip-hop, and is based on the idea that all students are responsible for each other. he purports that teachers need to see how hip-hop youth exhibit cosmopolitanism in their lives as a means to bring function into the classroom. context incorporation involves teaching practices that use analogy and simile as a strategy similar to what rap artists use in their lyrics. additionally, context integration supports bringing items from students’ communities into the classroom and connecting the items to hip-hop and science. the last c, content development, involves teachers being willing to admit they do not always have all the information and to share with students how they acquire new knowledge. emdin’s reality pedagogy offers useful suggestions for involving youth in science classrooms and getting them to take responsibility for one another. from emdin’s (2013) perspective, cogenerative dialogues, rap cyphers, and rap battles are designed for urban youth to engage in science talk that results in a better understanding of science content. his explanation of how to use rap cyphers and battles appears to fall short of demonstrating how urban youth gain indepth knowledge of science content. it is also unclear whether cogenerative dialogues are about science content or the science classroom. in one part of the chapter, emdin writes about science content with cogenerative dialogues while in other places he writes about students offering suggestions for improving the class and being able to showcase their perspective on any classroom situation. he views hip-hop as a key component to actively engaging students in science classrooms, helping students in urban areas learn science, and making science culturally relevant to them. emdin’s use of culturally relevant pedagogy appears to focus on using hip-hop to help students achieve academic success and cultural competence of hip-hop culture. he does not discuss or describe how to develop urban youth’s critical consciousness. davis et al. book review journal of urban mathematics education vol. 7, no. 1 100 developing the critical consciousness of african american students in urban areas critical consciousness is an essential component of culturally relevant pedagogy (ladson-billings, 1994) that emdin’s (2013) reality pedagogy fails to address. culturally relevant pedagogy advocates for african american youth to develop critical consciousness of racism, classism, and other forms of oppression as a means to improve their lived realities (ladson-billings, 1994). emdin’s reality pedagogy falls short of raising students’ critical consciousness and changing the conditions of their communities or lived realities in urban areas. however, stovall (2013) engages youth in raising their consciousness in chicago at lawndale/little village school for social justice (sojo) in a social studies unit entitled “hiphop, urban renewal, and gentrification.” urban renewal and gentrification are important issues in urban communities throughout the nation. stovall makes connections between gentrification and urban renewal in chicago and new york city using reports, hip-hop, social studies texts, and rap lyrics. stovall’s (2013) urban renewal unit was relevant to his students because their neighborhood was experiencing the effects of massive gentrification. stovall describes how he collaborated with eric (rico) gutstein (a colleague at the university of illinois at chicago who also works with sojo) and sojo faculty and students to develop curricula and lessons in mathematics and social studies. gutstein (see, e.g., 2013) teaches mathematics for social justice and he has also taken up the issue of gentrification in mathematics as a means of helping students to develop sociopolitical awareness and to see themselves as change agents in their community and society. his work also draws heavily on ladson-billings’ (1994) notion of culturally relevant pedagogy, more specifically, helping students to develop critical consciousness to take action to change their lived realities. without addressing critical consciousness, emdin’s notion of using rap cyphers or rap battles as part of his reality pedagogy is no different from teachers who simply rap or use rap text or rap videos in the classroom without any critical examination of rap music, the artist lifestyle, or the communities they rap about. low and colleagues (2013) draw attention to critical rap/hip-hop consciousness and pedagogies to prepare youth to critically think about the world and to deal with their investment in some of the most oppressive representations of hiphop. the authors address the tensions with hip-hop culture’s depiction of violence, misogyny, race, and materialism, and how these issues impact the schooling of youth. these scholars assert that hhbe scholars and practitioners have been so focused on working to legitimize hip-hop based educational practices in schools that they either have ignored or disparaged the ways youth engage in oppressive elements of hip-hop culture. low and colleagues focus specifically on how racialized minority youth construct identities connected to hip-hop and how davis et al. book review journal of urban mathematics education vol. 7, no. 1 101 these identities are constantly being constructed and reconstructed in relation to other cultures, communities, and affiliations. on the one hand, low and colleagues emphasize that they are “wary of reifying notions of ‘hip-hop identities’” (p. 119), a concern that we share, especially as it relates to african american youth in urban communities and schools learning mathematics. on the other hand, however, emdin’s (2013) focus on hip-hop appears to advocate for urban youth to adopt hip-hop identities without question. emdin essentializes all urban youth as being immersed in hip-hop culture, identifying with hip-hop culture, and wanting to be taught using hip-hop. conversely, shockley (in press) argues that having african american students develop hip-hop identities and refer to hip-hop as a culture creates identity confusion and interferes with them developing healthy cultural identities connected to african culture. in mathematics education, martin (2007) argues that educators must assume responsibility for helping african american students to develop healthy racial, academic, and mathematics identities. these identities have played a major role in helping african american students achieve at high levels in mathematics (berry, 2003; berry & mcclain, 2009; stinson, 2004; thompson & davis, 2013). teachers use of hip-hop based education hill and petchauer (2013) suggest that hhbe is intended for preand inservice teachers, but little is known about the lives of these teachers. in the united states, most preand in-service teachers are white and many of them possess very little knowledge of hip-hop music or hip-hop based educational practices (irby & hall, 2013). petchauer’s (2013) chapter focuses on african american preservice teachers; however, it does not provide insight into how these teachers used hhbe practices because they did not implement the practices with actual students in a classroom. irby and hall (2013) provide insight into how practicing teachers use hhbe. they contend that most research on hhbe focuses on teacherresearchers in urban areas who elect to document their educational practice and little research has been conducted on non-researching k–12 practitioners interested in using hhbe without publishing their work in scholarly journals. the authors report findings of 63 non-researching veteran and novice teachers in philadelphia who attended one or more of four professional development workshops that focused on hhbe practices. the findings suggest that it is important to understand: (a) teacher identity, (b) the perspectives of teachers who do not identify with hip-hop, and (c) the diversity of k–12 practicing teachers interested in using hhbe practices because their perspectives are not reflected in the extant research literature. in irby and hall’s (2013) study, the majority of the teachers (51%) taught in philadelphia, but only nine of these teachers taught in philadelphia public davis et al. book review journal of urban mathematics education vol. 7, no. 1 102 schools; most of the participants (74%) taught in private or public charter schools in urban areas. the majority of the teachers were elementary and middle school teachers (80%) with fewer being secondary school teachers (9%), and a population of teachers (11%) that did not work with a specific grade level or with k−12 students. most of the teachers (60%) lived in the surrounding suburban or rural areas of philadelphia and commuted to teach in the city’s schools. only 40% of the teachers lived in philadelphia. irby and hall found that non-researching teachers interested in using hip-hop lacked knowledge of hip-hop and their motives for using hhbe were not “situated in the theoretical and practical objectives of critical and culturally relevant pedagogies” (p. 112). in our review of the mathematics education literature, it appears that, as irby and hall (2013) report, hhbe practices that have been used by practicing teachers are not being published in scholarly journals. similar to the book editors, we were unable to find hhbe journal articles published by mathematics educators. our search for insight into how mathematics educators use hhbe practices led us to several web-based sources where mathematics teachers and students were using rap and rap videos to memorize formulas, to learn mathematical facts, to improve vocabulary, and to increase test scores. hill and petchauer (2013) caution against such “rappin teachers” who rap or use recorded raps to promote memorization of facts as they advocate for a deeper understanding of hip-hop aesthetics and epistemology and how they are connected to students’ lives and specific content area practices (e.g., science, mathematics). however, most practicing teachers (including mathematics teachers) who use hip-hop based educational practices in classrooms do not situate their work in critical and/or culturally relevant pedagogies (irby & hall, 2013). hhbe and the mathematics education of african american students: important considerations the use of hhbe for african american students in mathematics education should be approached from five areas of caution. first, there is an agenda being advanced in mathematics education to conduct liberatory research and to provide african american students with a liberatory mathematics education (martin, 2009a; martin & mcgee, 2009). in martin’s (2009a) edited volume mathematics teaching, learning and liberation in the lives of black children, he assembled mathematics educators of african descent and others who are committed to providing african american children with a meaningful mathematics education to “change the direction of research on black children and mathematics” (p. vi) with a focus on the theme of liberation. simply rapping, using rap text, or sharing rap videos in mathematics classrooms without any critical examination of rap music, the artist lifestyle, or the communities they rap about is not liberatory. martin and davis et al. book review journal of urban mathematics education vol. 7, no. 1 103 mcgee argue, “any relevant framing of mathematics education for african americans must address both the historical oppression that they have faced and the social realities that they continue to face in contemporary times” (p. 210). we support and advocate for scholarship and pedagogy that produce liberatory outcomes for african american youth. jett’s (2009) review of martin’s (2009a) book echoes our sentiments about advancing liberatory mathematics scholarship and pedagogy. we ask: where does providing african american students with a liberatory mathematics education fit into the current research and pedagogical approaches being advanced in hhbe? second, the theoretical underpinnings of hip-hop pedagogy remain undertheorized (hill, 2009). the same is true for hhbe in k–12 settings. the underdevelopment of hhbe theoretically contributes to practitioners’ lack of understanding of what does and does not constitute hhbe practices. for most mathematics practitioners, it appears that the use of rap music or videos constitutes hip-hop pedagogy. most iterations of hhbe claim to draw on critical pedagogy and culturally relevant pedagogy (emdin, 2013; hill, 2009; irby & hall, 2013). however, most practicing teachers who use hhbe in their classrooms do not appear to develop african american students’ critical consciousness or prepare them to take action or to change their lived realities, which are all key components of critical pedagogy and culturally relevant pedagogy. we ask: what are the key tenets of hhbe that should guide teachers in general and mathematics teachers in particular to achieve liberatory outcomes? third, there is a push to develop african american students’ racial, cultural, and mathematics identities (berry & mcclain, 2009; martin, 2007; thompson & davis, 2013). scholarly literature in mathematics education indicates that racial and mathematics identity development is important for black students to determine what it means to be a black mathematics learner (berry & mcclain, 2009). while the development of black students’ racial identity is important, we think that it is important to distinguish between racial and cultural identities. thompson and davis (2013) argue that there is a difference between racial and cultural identity development among african american students in mathematics settings. to these scholars, racial identity development pertains to the ways social constructions of race shape black students’ racial identity development whereas cultural identity pertains to african american students’ developing ethnic identities that connect them to their cultural heritage in africa. here, we are concerned about how the cultivation of hip-hop identities will impact the current line of scholarship devoted to positively developing african american students’ racial, cultural, and mathematics identities. hhbe studies have shown that african american students are developing hip-hop identities that often run counter (low et al., 2013) to them developing healthy cultural identities of what it means to be a person of african descent (davis, in press; murrell, davis et al. book review journal of urban mathematics education vol. 7, no. 1 104 2002). both davis (in-press) and murrell (2002) argue that african american youth must develop healthy cultural identities connected to the traditions, history, and heritage of people of african descent. as a mathematics education community, we have to decide if consciously promoting the development of hip-hop identities will interfere with african american students developing healthy racial, cultural, and mathematics identities. in our view, advocating for the development of hip-hop identities is not aligned with current efforts to promote the development of healthy positive identities in mathematics education. however, we think that mathematics educators will have to determine how to address african american students’ development of hip-hop identities because many of them are consciously and/or unconsciously developing these identities. fourth, there has been a push to understand how social constructions of race, racism, and other forms of oppression impact african american students’ mathematical experiences and lived realities (see, e.g., martin, 2009a, 2009b, 2009c). martin’s (2009b) teachers college record article sparked much conversation in mathematics education about race and racism that had been silenced. he examined the ways that race and racism are conceptualized in society and how these social constructions of race and racism inform mathematics education researchers, policymakers, and practitioners. martin called for researchers, policymakers, and practitioners to examine how social constructions of race and racism shape the mathematics education landscape. we ask: where does hhbe stand on addressing social constructions of race, racism, and other forms of oppression that are prevalent in the lives, schooling, and mathematics education of african american students? many scholars suggest that hip-hop addresses issues of race, racism, and other forms of oppression in urban communities and society (hill & petchauer, 2013). if this is the case, addressing issues of race, racism, and other forms of oppression should be a salient feature of hhbe. those concerned with african american children’s well-being must act with a sense of urgency to address issues of race, racism, and other forms of oppression because african american students’ lives are at stake (martin, 2009c). lastly, there has been a paradigm shift in mathematics education to focus on successful or high-achieving african american students. the following factors have been found to contribute to african american students’ high achievement and persistence in mathematics: (a) early opportunities to learn mathematics; (b) parental, guardian, and extended family support and advocacy; (c) advanced mathematics courses and programs; (d) teacher and peer support and encouragement; (e) involvement in extracurricular activities; and (f) spiritual beliefs (berry, 2003; ellington, 2006; stinson, 2004). there is a clear line of scholarly research focused on african american students’ success and high-achievement as opposed to a narrow focus on their failures. thompson and davis (2013) argue that research on high-achieving african american students in mathematics must shift davis et al. book review journal of urban mathematics education vol. 7, no. 1 105 from a focus on individual mathematics achievement to a focus on collective mathematics achievement. they also argue that there has to be collective responsibility for ensuring both “high-” and “low-performing” african american students have an opportunity to achieve in mathematics. we ask: how does hhbe promote high achievement in african american students that can complement or advance efforts in mathematics education? the authors of this edited book suggest that hhbe scholars and the current hhbe literature have made a clear and persuasive argument to use hhbe practices and to expand the practices into in mathematics education. as it stands, we are yet convinced that hhbe practices in mathematics education should be used to teach african american students mathematics in urban schools as the authors of this edited book suggest. references berry, r. q., iii (2003). voices of successful african american male middle school mathematics students (unpublished doctoral dissertation). university of north carolina, chapel hill, nc. berry, r. q., iii, & mcclain, o. l. 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(2013). the rap cypher, the battle, and reality pedagogy: developing communication and argumentation in urban science education. in m. l. hill, & e. petchauer (eds.), schooling hiphop: expanding hip-hop based education across the curriculum (pp. 11−27). new york, ny: teachers college press. gutstein, e. (2013). whose community is this? mathematics of neighborhood displacement. rethinking schools, 27(3), 11−17. hill, m. l. (2009). beats, rhymes, and classroom life: hip-hop, pedagogy and the politics of identity. new york, ny: teachers college press. hill, m. l. & petchauer, e. (eds.) (2013). schooling hip-hop: expanding hip-hop based education across the curriculum. new york, ny: teachers college press. irby, d. j., & hall, b. h. (2013). fresh faces, new places: moving beyond teacher-researcher perspectives in hip-hop based education research. in m. l. hill, & e. petchauer (eds.), schooling hiphop: expanding hip-hop based education across the curriculum (pp. 95−117). new york, ny: teachers college press. jeffries, m. p. (2011). thug life: race, gender, and the meaning of hip hop. chicago, il: university of chicago press. jett, c. c. (2009). mathematics, an empowering tool of liberation?: a review of mathematics teaching, learning and liberation in the lives of black children. journal of urban mathematics education, 2(2), 66−71. retrieved from http://ed-osprey.gsu.edu/ojs/index.php/jume/article/viewfile/48/21 davis et al. book review journal of urban mathematics education vol. 7, no. 1 106 ladson-billings, g. (1994). the dreamkeepers: successful teachers of african american children. san francisco, ca: jossey-bass. low, b., tan, e., & celemencki, j. (2013). the limits of “keepin’ it real”: the challenges for critical hip-hop pedagogies of discourses of authenticity. in m. l. hill, & e. petchauer (eds.), schooling hip-hop: expanding hip-hop based education across the curriculum (pp. 118−136). new york, ny: teachers college press. martin, d. b. (2007). beyond missionaries or cannibals: who should teach mathematics to african american children? the high school journal, 91(1), 6−28. martin, d. b. (ed.). (2009a). mathematics teaching, learning and liberation in the lives of black children. new york, ny: routledge. martin, d. b. (2009b). researching race in mathematics education. teachers college record, 111(2), 295−338. martin, d. b. (2009c). little black boys and little black girls: how do mathematics education and research treat them? in s. l. swars, d. w. stinson, & s. lemons-smith (eds.), proceedings of the 31st annual meeting of the north american chapter of the international group for the psychology of mathematics education (pp. 22−41). atlanta, ga: georgia state university. martin, d. b., & mcgee, e. o. (2009). mathematics literacy for liberation: reframing mathematics education for african-american children. in b. greer, s. mukhophadhay, s. nelson-barber, & a. powell (eds.), culturally responsive mathematics education (pp. 207−238). new york, ny: routledge. murrell, p. c. (2002). african-centered pedagogy: developing schools of achievement for african american children. new york, ny: state university of new york press. pardue, d. (2013). who are we? hip-hoppers’ influence in the brazilian understanding of citizenship and education. in m. l. hill, & e. petchauer (eds.), schooling hip-hop: expanding hip-hop based education across the curriculum (pp. 28−46). new york, ny: teachers college press. petchauer, e. (2013). “i feel what he was doin’”: urban teacher development, hip-hop aesthetics, and justice-orientated teaching. in m. l. hill, & e. petchauer (eds.), schooling hip-hop: expanding hip-hop based education across the curriculum (pp. 28−46). new york, ny: teachers college press. peterson, j. b. (2013). rewriting the remix: college composition and the educational elements of hiphop. in m. l. hill & e. petchauer (eds.), schooling hip-hop: expanding hip-hop based education across the curriculum (pp. 28−46). new york, ny: teachers college press. shockley, k. (in press). theoretical musing about hip-hop scholarship from an african-centered perspective. african american learners. stinson, d. w. (2004). african american male students and achievement in school mathematics: a critical postmodern analysis of agency. dissertations abstracts international, 66 (12). (umi no. 3194548) stovall, d. (2013). hip-hop and the new response to urban renewal: youth, social studies, and the bridge to college. in m. l. hill, & e. petchauer (eds.), schooling hip-hop: expanding hip-hop based education across the curriculum (pp.115−166). new york, ny: teachers college press. thompson, l., & davis, j. (2013). the meaning high-achieving african american males in a urban school ascribe to mathematics. the urban review, 42(4), 490−517. wilson, j. a. (2013). the mc in y-o-u: leadership pedagogy and southern hip-hop in the hbcu classroom. in m. l. hill, & e. petchauer (eds.), schooling hip-hop: expanding hip-hop based education across the curriculum (pp. 28−46). new york, ny: teachers college press.   microsoft word 6 final guerra et al vol 7 no 2.doc journal of urban mathematics education december 2014, vol. 7, no. 2, pp. 55–75 ©jume. http://education.gsu.edu/jume paula guerra is an assistant professor of mathematics education in the department of elementary and early childhood education at kennesaw state university, 1000 chastain road, kennesaw, ga 30144, usa; email: pguerral@kennesaw.edu. her research interests include social justice and mathematics education, ell mathematics schooling, and the mathematics schooling of girls, in particular, girls of color. woong lim is an assistant professor in the department of secondary and middle grades education at kennesaw state university, 1000 chastain road, kennesaw, ga, 30144, usa; email: wlim2@kennesaw.edu. his research interests include the interrelation between language and mathematics, gender studies in mathematics education, and inclusive classroom practice. latinas and problem solving: what they say and what they do paula guerra kennesaw state university woong lim kennesaw state university in this article, the authors present three adolescent latinas’ perceptions of ideal mathematical competencies, their perception of their individual “abilities” in mathematics, and their work on a mathematics problem-solving task. results indicate that these latinas recognize flexible mathematics as the ideal mathematical competency in problem solving but demonstrate rigid mathematics in the problem-solving task. reasons for the discrepancy between the three latinas’ perceptions of ideal mathematical competencies and their own work on mathematical tasks are discussed. implications related to opportunities for girls of color to pursue careers in stem fields are discussed as it relates to flexible problem-solving skills in mathematics. keywords: latinas, mathematics education, problem solving, stem hildren have different schooling experiences that can be shaped by race and gender and that influence their decisions regarding career choices. in particular, classroom mathematical experiences can help explain divergent aspects of mathematics among boys and girls as well as how children perceive their own mathematical “abilities.” despite a growing interest in science, technology, engineering, and mathematics (stem), women of color remain underrepresented in stem fields. although girls might enjoy studying mathematics and science at school, they are less likely to pursue careers in stem than boys; this leads, in part, to the “leaky pipeline” in the fields, a disparity in representation of women in stem today (de welde, laursen, & thiry, 2007). for example, although women represent nearly half of those awarded bachelor’s degrees in mathematics, they earn only 27% of doctoral degrees in mathematics (national science foundation [nsf], 2013). similarly, women are underrepresented in physics (30%), computer science (23%), and engineering (13%; nsf, 2013). c guerra & lim latinas’ problem solving journal of urban mathematics education vol. 7, no. 2 56 two decades ago, catsambis (1994) discussed the lack of research focusing on women’s experiences in mathematics; she noted: “women of color are the most underrepresented group in mathematics and science, but few researchers have specifically studied their educational experiences” (p. 201). more recently, varley-gutiérrez (2009) made similar claims about gaps in the literature, explaining that “little (if any) mathematics education research speaks specifically to girls of color or to a feminist of color perspective in relation to mathematics,” and that “there is an urgency to include the voices of women of color in re-envisioning mathematics education [so that it can be] used as a tool to transform society to be more just” (p. 49). with the growing numbers of latinas/os in u.s. schools, the case of latinas and their underrepresentation in stem merits careful attention. in particular, few studies exist on the complex intersection of latinas and their mathematics learning demonstrated by problem solving. using qualitative case study methodology, we present three adolescent latinas’ perceptions of their mathematics schooling, of the mathematics that they produce and value, and of success in mathematics. in doing so, we examine the adolescent latinas’ perceptions of success in mathematics and mathematical competency compared to their demonstrated competencies on a specific mathematical task. two research questions guided the study: 1. what are the latinas’ perceptions of different ways in which they do mathematics? 2. what kind of mathematics did, in fact, these latinas demonstrate in problem solving, and how does it compare to their identification with mathematical ability? theoretical framework latina/o critical theory (latcrit) was used to elicit, explain, and understand the participants’ comments and their interactions with each other and with the researchers. latcrit holds that multiple forms of oppression can affect latinas/os and that such racial experiences permeate our education with regard to the ways latinas/os experience race, class, gender, and issues of language, immigration, ethnicity, culture, and identity (huber, 2010). solórzano and yosso (2002) defined critical race methodologies in such a way that they could help shed light on issues that affect minoritized groups when used as a lens for research. latcrit is a theoretical approach that foregrounds race and racism in all aspects of the research process. however, it also challenges the separate discourses on race, gender, and class by showing how these three elements intersect to affect the experiences of students of color. racism, for example, is a category of analysis, but its intersection with other forms of subordination, such as sexism or class guerra & lim latinas’ problem solving journal of urban mathematics education vol. 7, no. 2 57 discrimination, serves as a lens for analysis (solórzano & delgado, 2001; solórzano & yosso, 2002). likewise, latcrit positions latinas at the multiple intersections of gender, class, and race. latcrit offers a liberatory or transformative solution to racial, gender, and class subordination and focuses on the racialized, gendered, and classed experience of students of color. furthermore, it positions these experiences as sources of strength. finally, it uses the interdisciplinary knowledge base of ethnic studies, women’s studies, sociology, history, humanities, and legal studies to better understand the experiences of students of color (solórzano & yosso, 2002). using a latcrit lens to study the experiences of latinas in school mathematics specifically enabled the researchers to critically examine the notion that the functions and ideological purpose of schooling are colorblind, objective, merit-based, neutral, and offer equal opportunities (solórzano & delgado, 2001; solórzano & yosso, 2002). when a study includes girls of color as participants to examine racialized experiences of individuals in the learning of traditionally whiteand maledominated subjects at school (e.g., mathematics), there needs to be a clear effort to examine how their gender, race, and/or social class influenced their perceptions about various aspects of school mathematics. literature review stereotyping in mathematics stereotyping in mathematics—related to gender and ethnicity specifically— is common inside and outside of schools. gender stereotyping leads some girls to hide their interest in mathematics, fearing that this interest could challenge their femininity (mendick, 2005a, 2005b; walkerdine, 1989). kiefer and sekaquaptewa (2007) examined gender identification and stereotyping among women enrolled in college calculus courses. they reported that women with low gender identification and low gender stereotyping performed best on examinations, and women with high gender identification and stereotyping were not inclined to pursue careers in mathematics. they also suggested that those who were not expressive but were suspected as accepting that women have lower mathematical competence than men were not likely to choose a career in mathematics. teachers’ beliefs about girls’ success in mathematics may also influence female students’ relationships with mathematics. for example, walkerdine (1989) pointed out how teachers talked about girls’ mathematics success being a result of their hard work or “effort.” in contrast, in the same classrooms, teachers believed that boys who were not achieving at the same rate as the girls were still had potential and were not trying hard enough. in fact, researchers have pointed to the inadequacy in how girls are taught and socialized into mathematics rather than to in guerra & lim latinas’ problem solving journal of urban mathematics education vol. 7, no. 2 58 nate differences between men and women (battey, kafai, nixon, & kao, 2007; boaler, 2002; carr, jessup, & fuller, 1999; clewell & campbell, 2002; fennema, carpenter, jacobs, franke, & levi, 1998). however, gender stereotyping is a complex issue and includes more than whether women participate in mathematical activities or careers (hyde, fennema, ryan, frost, & hopp, 1990). for instance, mendick (2005b) equated doing mathematics with “doing masculinity” (p. 235). ethnic stereotyping in mathematics is also prevalent in schools. regarding the participation of latinas/os in mathematics courses, their achievement in mathematics, and participation in stem careers, researchers have found that latinas/os are overrepresented in low-ability classes (catsambis, 1994) and underrepresented in high-ability mathematics courses and stem careers (zarate & gallimore, 2005). according to gutiérrez (1999, 2002), even though the scores for latinas/os in mathematics have improved, the improvement has been mostly in basic skills. this underachievement “has serious life consequences for earning potential and for participation in an increasingly technological society” (gutiérrez, 2002, p. 1048). gutiérrez (1999) also stated that not only are latinas/os’ and african americans’ scores below those of white students, but they also tend to score significantly lower in advanced placement (ap) courses and college entrance examinations. using data from the national educational longitudinal study of 1988, catsambis (1994) concluded that by eighth grade, fewer adolescent girls than boys decide to pursue a career in mathematics or science, with female african americans and latinas being the least likely to do so. latinas and mathematics women of color are underrepresented across all science and engineering fields. statistics for latinas are particularly troubling: the percentage of latinas in undergraduate engineering programs was 2%, while the percentage of latinas in graduate engineering programs was less than 1% (nsf, 2013). however, these statistics are not surprising when you look at k–12 experiences. mcwhirter, valdez, and caban (2013) reported in their study that 41 high school latinas experienced barriers such as lack of financial and language resources, negative peer influences, and discrimination from teachers and peers. related to mathematics specifically, catsambis (1994) found that latinas tend to have less confidence in their abilities to do mathematics, report higher levels of mathematics anxiety, and appear to have less interest in mathematics. for instance, many latinas believed that mathematics courses would not be useful for them. catsambis also found that latinas were afraid to ask questions in mathematics classrooms and were less likely to express enjoyment in studying mathematics when compared to male students. zarate and gallimore (2005) studied factors that impact college enrollment guerra & lim latinas’ problem solving journal of urban mathematics education vol. 7, no. 2 59 for latinas/os and reported that teachers and school counselors play important roles in latinas’ educational shaping. additionally, jilk (2006) found that firstgeneration immigrant latina students’ mathematical successes were related to the identities they constructed in schools and at home. furthermore, she criticized the views that english learners cannot learn difficult mathematics and that parents and the cultures of the immigrants might cause the students’ academic struggles. with first-generation latinas who were “low performing” students, pyne and means (2013) conducted a case study and found that hidden social and institutional discourses created stress, struggles, and doubts with regard to success in society. as a response to those societal influences, weisgram and bigler (2007) claimed that girls might benefit from learning about feminism and gender equality in order to disavow the notion of women’s low achievement in mathematics and science. problem solving and invented algorithms problem solving is one of the essential functions of stem. there is a growing body of literature indicating that problem solving in mathematics offers students opportunities to experience how their mathematical knowledge creates solutions to problems and also helps students to develop a deeper understanding of mathematical thinking and reasoning (lubienski, 2000; millard, oaks, & sanders, 2002). adeleke (2007) claimed that problem solving dwells on the use of conceptual and procedural knowledge, and other researchers found that it takes balanced strategies of using both types of knowledge to achieve success in problem solving (hiebert, 1986; national research council [nrc], 2001; sfard, 1991). conceptual approaches in problem solving, for instance, enable students to employ an integrated understanding of mathematical ideas. procedural approaches to problem solving allow students to execute related procedures to solve problems (nrc, 2001; rittle-johnson, siegler, & alibali, 2001). however, hiebert and lefevre (1986) argued that the two approaches are hard to separate, as they actually support each other. rittle-johnson and siegler (1998) found that articulating how procedures and concepts interact is critical to understanding various methods in problem solving. in describing ways in which one conducts problem solving, adeleke (2007) explained that those with high conceptual understanding apply a host of related, but possibly unknown, procedures to problem solving, and others may apply skills in a routine and rigid manner with fluency; overall, learners tend to use one primarily over another. fennema and colleagues (1998) found that girls solved mathematics items primarily using taught algorithms, while boys used invented algorithms. this difference is troubling when given that success in mathematics, particularly in the stem-related fields, has been connected to a student’s ability to produce invent guerra & lim latinas’ problem solving journal of urban mathematics education vol. 7, no. 2 60 ed mathematical solutions (adams & hamm, 2011; carr et al., 1999; fennema et al., 1998). adding to this, carr and colleagues (1999) found that girls too often do not reflect on their solutions nor ask the hows and whys when solving mathematical problems. failing to consistently ask these higher-order questions can translate into limiting the opportunity for girls to remain engaged in what the national council of teachers of mathematics (nctm) considers to be “doing” mathematics. for stein, smith, henningsen, and silver (2000), doing mathematics consists of engaging in complex and non-algorithmic thinking, as well as exploring and understanding mathematical concepts, processes, and relationships. thus, it seems critically important to look carefully at female students’ experiences in mathematics, particularly those women from underrepresented groups. methods setting this study was part of a larger research project investigating latinas’ schooling experiences related to mathematics. the research took place in an elementary school in the southwest united states. the student population of the district is 53.6% latina/o, 23.6% white, 11.5% african american, 7.7% native american, and 3.4% asian/pacific islander. the class to which the participants belonged did not reflect the demographics of the school, with only five latinas/os in a class of 25 students. according to the state, the school was not meeting adequate yearly progress. participants the participants were three eighth-grade latinas⎯viviana, rocío, and teresa (all pseudonyms)⎯who were enrolled in the school’s honors track program. honor student latinas were chosen for this project as representative of traditional notions of success in mathematics, with recognition that these notions must be critically examined. for example, good grades in mathematics classes may not mean that latinas are learning the kind of mathematics that society and participants value. alternatively, these grades could mean that latinas perform well in the current assessment system of their mathematics classrooms. viviana showed a strong personality during interviews. she frequently stated that she did not like mathematics. rocío was a quiet girl who took time when she spoke. rocío stated that she wanted to be an architect in the future. teresa demonstrated a modest disposition and stated that being an honors student was challenging. during the time of the study, the three girls were taking geometry and had taken algebra the previous year. viviana, rocío, and teresa were all firstgeneration latinas, and their family members emigrated from mexico. they guerra & lim latinas’ problem solving journal of urban mathematics education vol. 7, no. 2 61 spoke fluent english like native speakers and declined to speak spanish with the interviewer, who spoke spanish as the first language. the participants chose to speak spanish only when the interview involved meeting with their parents. design the participants met with the interviewer nine times from september 2010 to march 2011. there were four focus group interviews, four follow-up individual interviews, and one individual working session with a mathematics task. group interview questions prompted participants to share their experiences learning mathematics in classroom and home settings and also to share their unique experiences they had at school because they were latinas. during the individual follow-up interviews, participants had an opportunity to talk more about the topics discussed in the group interviews. during the mathematical task working sessions, participants engaged in a problem solving activity and to explain their reasoning. additionally, the interviewer asked the participants to solve one equation that could have been a key component to finding solutions to the problem solving activity. in addition to interviews and working sessions with viviana, rocío, and teresa, the researchers conducted one interview session with the girls’ families. for the family interview, viviana, rocío, and teresa provided questions of interest they wanted to ask their parents regarding their mathematics schooling and the influence of parents and family. finally, the researchers had informal conversations with the participants’ teacher. data collection during data collection, the researchers built a collaborative relationship with the participants (erickson, 1986) in order to gain access. for example, they made efforts to establish trust and ultimately gain access to the experiences that participants shared and their views of mathematics, maintaining it throughout the process of data collection. table 1 shows the timetable for the data collection and the different central topics for the meetings. the focus groups were designed to encourage the participants to open up to the interviewer in an environment where the girls could support each other and help recall their experiences with mathematics, which included school and home experiences. through these meetings, the researchers identified the community that the girls belonged to, how they saw this community, and what kind of participation they had had in it. by being in a group with others who shared similar lived experiences, the participants were able to discuss topics of common knowledge understood by each of them but not necessarily by others outside their community. guerra & lim latinas’ problem solving journal of urban mathematics education vol. 7, no. 2 62 the researchers used individual interviews to further investigate the participants’ statements from the focus group interviews by giving them opportunities to explain what they meant and why they made specific statements. the individual conversations allowed the participants to share details that they may have wanted to keep private. these interviews were semi-structured but with enough freedom to have an open conversation in a safe environment, pursuing emerging themes. the mathematics working session aimed to compare what the participants talked about while referring to mathematics, including the ways they do mathematics. table 1 data collection timetable month topic of focus group meeting – date individual interview date september mathematics autobiography – 9/16 9/30 october what is necessary to succeed in mathematics – 10/7 10/1 november why do others not succeed in mathematics – 11/19 11/4 december non-applicable (n/a) 12/2 february our parents and mathematics – 2/4 (participants prepared an interview with their parents) n/a march no focus group meeting; individual mathematics working session during final interview 3/23 all of the group meetings were videoand audio-recorded, and all individual interviews were audio-recorded. each videoand audio-tape was transcribed for analysis. the researchers also took notes during the working session and collected the work that the students produced. another source of data was the informal conversations the researchers had with the teacher at the school in the mathematics classroom after classes were over for the day. the researchers took notes on the teacher’s comments. additionally, when the teacher heard something the girls said in the group interviews—the teacher sometimes came to the room to get materials or make plans for the next day’s class—she was allowed to make a comment. the teacher was a white woman in her 20s. she portrayed a calm, positive demeanor and demonstrated passion for her students. the three latinas expressed respect toward the teacher and stated that they liked her very much. data analysis following erickson’s (1986) methodology, data was reviewed multiple times in search of critical events (considering their frequency of appearance in the data, as well as the effect and emphasis the girls placed upon them) that defined (a) what each girl thought about her school mathematics experiences, (b) the different “kinds” of mathematics, and (c) the mathematics she valued and actually guerra & lim latinas’ problem solving journal of urban mathematics education vol. 7, no. 2 63 demonstrated. the data were examined critically to determine counter-arguments that could serve to disprove the relevance of assertions made from the analysis of the data. when evidence was found that some assertions could be explained through verbalizations the girls made, the assertions were accepted with the supporting evidence. some assertions were discarded when contradictory evidence was found. according to erickson (1986), this process provides evidence for the assertions that were generated in an inductive way. assertions supported by multiple sources of data were accepted as the strongest claims. to determine the strongest claims, the researchers compared all the pieces of evidence, weighing and comparing proand against-evidence. this process of finding the strongest claims was made in two instances: one for all data and another for each girl’s individual data. the researchers often referenced the recordings to ensure that conclusions based on the transcriptions matched the interviews—an advantage that erickson noted for the use of video. limitations one limitation of this study is the impossibility of generalization. this study aims not to create a general case for all successful adolescent latinas in mathematics classrooms in the united states but rather to bring light to the issue. this study can inform researchers’ design studies with larger populations using this study as baseline data. another limitation regarding the design and the tools of data collection is the lack of classroom observations. classroom observations could have been useful to tie the participants’ accounts to the classroom environment and the behaviors demonstrated in the teaching and learning of mathematics in the classroom. classroom observations would have provided the researchers with firsthand information about the mathematics favored in the classroom, instead of relying on descriptions based on the participants’ and teacher’s perspectives. findings rigid mathematics vs. flexible mathematics the latina participants stated that smart people do mathematics quickly. all three girls, however, presented varying degrees of agreement with regard to competency in problem solving. while talking about those who demonstrate a high level of mathematics, teresa said, “[smart students] don’t want…like…work out guerra & lim latinas’ problem solving journal of urban mathematics education vol. 7, no. 2 64 the problems and the steps” (ii1, oct. 1).1 she linked the idea of problem solving with fluency in executing the host of steps she needed to work out: i have a lot of friends, like, guys that are in my class, they just open…. they don’t even need the teacher to explain. they just open the textbook, and they just start looking at examples really quick, and they are like boom. they really quickly got the idea. (ii1, oct. 1) teresa, who described rocío and viviana as smart, needed more steps to “see” the mathematics: i know [rocío] will get the answer really quick, or [viviana] will do like short steps, and i need to see like all the problems to understand it…. [people] should not be ashamed at trying to do memory things, because most things are in your mind. (fg2, oct. 7) rocío said she was successful in mathematics and used problem-solving skills to describe mathematical competence. rocío explained that she, “like boys,” preferred hard mathematics problems that “take more [than basic] information” to result in successful solutions. rocío said she did not do her mathematics “step by step” like in textbooks, as if “one follows a prescribed recipe” (ii2, oct. 1). she added that, after solving problems in her head, the actual timeconsuming task was writing the steps down as the teacher required: i can do mental math fast… i do it in my head. i use less steps than [teresa and viviana]…. but then i need to figure out the steps for the teacher…. we need to show our work and write it out…it takes time. (fg2, oct. 7) rocío also explained why she did not choose easy problems to solve by drawing a parallel between her and boys’ mathematics: i think [boys] think like higher… level or something sometimes. they think about the problem, for instance, like as it gets harder, they think more about it. like the easiest problems they don’t really care…. i don’t really care either. (fg2, oct. 7) she also said: “i do [mathematics] like boys…. they just leave the easier problems behind; they don’t really care. they think the harder problems, the more information they are gonna get out of it” (ii2, oct. 1). according to rocío, the mathematics she could do was the kind that would prepare her for a career in architecture: “well, my dad, he’s always been there, telling me specially to do my math homework because he thinks it’s really im 1 individual interviews are noted as ii, followed by the number and the date. focus groups are noted as fg. example: the first focus group, which took place on september 16: fg1, sept. 16. guerra & lim latinas’ problem solving journal of urban mathematics education vol. 7, no. 2 65 portant, specially for architecture and engineering, and that’s what i wanna be” (fg1, sept. 16). later she added: “this year we are doing geometry. so i think i still have a lot to go through to become an architect” (ii1, oct. 1). rocío spoke about her career goal in the context of her ethnicity: “there are a lot of latino poets and writers famous, but not architects or engineers. i want to show [people] latinos can do it” (ii2, oct. 1). all the girls agreed that mathematics “just comes naturally” and explained that some people are simply better at it “naturally”: interviewer: why do they [boys] do it [mathematics] so quickly? (short pause) any ideas? rocío: i don’t know. teresa: just comes. interviewer: what? all of three: it just comes. rocío: comes naturally, i guess. interviewer: what about for girls? does it just come naturally as quickly too? rocío: well, to me it does but… i learn differently so…. (fg2, oct. 7) unlike rocío, teresa stated that word problems were sometimes difficult for her: teresa: i have some problems with those. i read them, but then i try to go back again, try to understand what they are asking, but i don’t know how to like…. i don’t know how to put the equations together like…. interviewer: why would that be? teresa: i don’t know. maybe ’cause i don’t try to think higher, but i should. and… i don’t know. that’s hard. (fg2, oct. 7) teresa identified the cause of her struggle as a lack of effort and cited “setting up the problem” and “writing the equations” as the best approach to solve problems in classrooms. often the girls relied on memorized routines to solve mathematical problems: viviana: when we are young, we do stuff the long way, but now there are like…formulas and sure-ways, shortcuts… teresa: and if we memorized something from the past, it’s easy to…when we are using formulas; it’s just easier to…. i don’t know… put all together. (fg2, oct. 7) teresa was not satisfied with her performance. she discussed not being sure about whether she should change her style of doing mathematics to emulate the ways boys do mathematics: guerra & lim latinas’ problem solving journal of urban mathematics education vol. 7, no. 2 66 i have a lot of friends that go to another school…guys…. they don’t need a lot of questions. they just read it and they know…. they know what to do. i ask a lot of questions. i was embarrassed at first, but if i don’t ask, then i don’t know what to do. i don’t know…. maybe i should change…. i don’t know.” (ii2, oct. 1) rocío stated⎯and teresa agreed⎯that viviana did not need as many steps to solve a mathematics problem like other girls (e.g., teresa) but still needed more than rocío. the teacher agreed by saying, “yes, they are smart and very hard working. viviana needs a little help, and teresa needs a little more. rocío is ahead of the bunch…. it is pure effort. pure effort” (conversation with teacher on nov. 19). even though teresa struggled with application problems, viviana and rocío claimed these were fun and made them connect mathematics to the real world: viviana: i don’t like math. interviewer: really? why is that? you are so successful at it. viviana: i don’t like it. rocío: sometimes [it] is boring. interviewer: really? rocío: sometimes [it] is good, like when we connect with science or something, so we see how it is. viviana: yeah, like with problems. rocío: yeah, problems. viviana: everything is connected with math. it is not boring when it is connected. (fg1, sept. 16) viviana described mathematics in the textbook as being one type of mathematics where memorizing rules and following steps were synonymous with success in schools. while talking about those steps, viviana was proud that she took fewer steps than teresa did. in the last meeting with a mathematical task, the participants completed a mathematics problem and solved an equation. in the next section, the problemsolving experience is described. the latinas’ performance in problem solving viviana, rocío, and teresa were given the following problem: i have 1 2/3 cups of milk. my recipe calls for 2 1/2 cups. by how much do i need to reduce the other ingredients in the recipe so that i can use 1 2/3 cups instead of 2 1/2? the participants solved the problem while the interviewer observed what they were writing. application problems were not the focus of their mathematics class, guerra & lim latinas’ problem solving journal of urban mathematics education vol. 7, no. 2 67 according to the teacher. the teacher said: “sadly, we don’t have time for that. problem solving takes time. it is great, but we need to cover so many things, we stick to the workbook.” (conversation with teacher on mar. 23) one step the three latinas commonly took after finishing reading the problem was converting the fractions to decimals, demonstrating that they were fluent with calculation. although the participants stated that they understood the problem, they struggled with identifying key mathematical ideas of the problem as well as expressing their thinking algebraically. they read the problem repeatedly. viviana said: “we need this milk, but we have less…. we can’t make the cake. reduced…smaller cake?” (ii5, march 23). they went over the numbers, writing them down, saying them out loud. it took them 5 to 6 minutes to start writing their work on the paper. the participants stated that the percentage by which the other ingredients were reduced corresponded to the percentage by which the milk was reduced, but they could not translate the key concept to algebraic expressions or equations. when the interviewer changed the problem during the interview so the percentage was half of the original, the participants were able to solve the problem. however, they still could not solve the original problem. rocío struggled to find the percentage and said that the wording in the problem was confusing. she started by finding the difference between the amount of milk in the original recipe and the amount that she had available. she said she wanted to find the percentage of reduction. rocío was persistent in using decimals instead of fractions. it took her longer to solve the problem because she was working with a repeating decimal (which made the computations difficult). also, it was difficult for rocío to guess a number multiplied by 2.5 to result in 1.667 (i.e., 2.5x = 1.667). the interviewer suggested writing an equation for rocío to solve: (2 1/2)x = 1 2/3. when the equation was proposed, rocío solved this equation quickly but still struggled to understand how the equation may represent an algebraic path in the problem. similar to rocío, teresa quickly converted the fractions to decimals. she said that the applications to real-life situations in these types of problems were confusing. as it became clear that she had grown frustrated with the problem, the interviewer suggested a slightly different situation, where the original recipe used 2 cups and the cook had 1 cup only in the kitchen. then, the interviewer asked about the relationship of these numbers (one being double the other) and how much of the whole “cake” we would bake if we had 1 cup when we needed 2, in an attempt to tie the mathematics with the real-world situation of the problem. teresa decided to find the difference between the numbers, similar to what rocío did. she said that the new recipe should be reduced by 50%. the interviewer also asked teresa if the original recipe called for 3 cups of sugar (moving into another ingredient where she would have to keep the same guerra & lim latinas’ problem solving journal of urban mathematics education vol. 7, no. 2 68 relationship), how much she would need for the reduced (to half) recipe. with this suggestion, teresa became frustrated. after 13 minutes passed with failed attempts at the problem, the interviewer provided the same equation she had provided to rocío. teresa solved this equation without difficulty. she told the interviewer: “word problems are harder because you have to set up the equation” (fg5, mar. 24). viviana, similar to rocío, found the difference between the amount she needed and the amount she had with decimals. then she tried to convert this difference back to fractions but was not successful. the interviewer presented viviana with the easier version of the problem provided to teresa. viviana said that she wanted to work with percentages because she thought that was what the problem aimed for: “like…if it was half, like you [the interviewer] said, then it will be 50%. i need to know the percentage” (fg3, nov. 19). applying her thinking in the simpler version of the original problem, viviana was able to figure out the solution to the original problem. summary the participants described mathematical behaviors in problem solving in two categories. first, they talked about one type of mathematical behavior as rigid or “textbook-like,” consisting of steps and rules to memorize and follow. they described the other type of mathematical behavior in problem solving as fast mathematics or flexible mathematics that is not bound by rules, as it just “happens naturally” immediately after reading a problem. the three latinas indicated that they considered the fast and “naturally” occurring mathematics superior to the methods that are typically presented in the textbooks as a rigid set of procedures. in problem-solving sessions, however, the latinas demonstrated performance in rigid mathematical behavior closer to the level of mathematics that they considered inferior. the girls struggled to translate their mathematical thinking into algebraic structures and demonstrated limited mathematical reasoning. instead, the participants were fluent in solving explicit equations that could be solved through procedural routines. the girls solved the equation by adhering to a set of rules and procedures they had been taught. these ideas are explored further in the next section. discussion different notions of doing mathematics the three latinas recognized varying degrees of school mathematics. their descriptions of mathematical competencies of problem solving were primarily two-fold: flexible and rigid. in flexible mathematics, the student will quickly guerra & lim latinas’ problem solving journal of urban mathematics education vol. 7, no. 2 69 grasp the key elements of a problem-solving task, devise creative strategies, and produce solutions. we believe flexible mathematics aligns with the kind of mathematical behaviors of the learner who effectively uses her or his conceptual knowledge base and is willing to take risks to produce a solution to a problemsolving situation (adeleke, 2007; hiebert, 1986). the three girls used adjectives such as smart, fast, “just happening in one’s head,” or high-level mathematics to describe flexible mathematics. in rigid or algorithmic mathematics, the student would recall memorized rules, use prescribed procedures, and apply taught strategies. the participants also described rigid mathematics as low-level or mathematics “like in textbooks.” flexible mathematics could be in opposition to algorithmic mathematics and useful to devise strategies in unknown problem-solving situations. rigid mathematics could be useful to execute efficient calculations in a clear application of formulas or common strategies of problem solving. this comparison parallels the differences in using an invented algorithm versus taught strategies (fennema et al., 1998). rocío, viviana, and teresa knew that some students, mostly boys, could perform and relate to flexible mathematics more positively than toward rigid mathematics. for the most part, mathematics educators agree that neither boys nor girls have some special “innate” ability toward mathematics (campbell, 1995, 1997; fennema, 1996). the participants in this study, however, showed less confidence in their mathematical abilities, which support conclusions drawn in prior research (e.g., catsambis, 1994). the findings of this study add that some latina students associate flexible mathematics with boys’ mathematical ability and rigid mathematics with girls’ mathematical ability. for example, the girls showed excitement describing how fast one can work out a solution in problem solving. rocío associated herself with flexible mathematics by saying that she learned differently than other girls who rely on procedures and taught strategies. rocío described her way of doing mathematics as “different” from that of most girls and more similar to what boys do. the mathematics these latinas considered “higher” did not reflect the nature of mathematical learning at school. rocío explained that the longest part of mathematics work was translating what was in her head into procedures so that the teacher could see her work. teresa talked about higher mathematics as something students at a different school do. these statements reveal a glimpse of the school mathematics the three latinas experienced in which problem solving was not emphasized and mathematics curricula and pedagogy were different from school to school. the teacher confirmed this difference when she said there was no time for problem solving (or, at least not in this teacher’s classroom). the occurrence of bias in mathematics classrooms was evident in the study. rocío’s teacher, with whom the interviewer talked informally before and after all the interviews and focus groups, agreed with the idea of rocío being the one do guerra & lim latinas’ problem solving journal of urban mathematics education vol. 7, no. 2 70 ing mathematics faster and in “fewer steps.” she added that it was “pure effort working for her.” according to the teacher, putting forth effort was a characteristic common among the three girls: “all of them, they work really hard for the class, and they are very responsible. they are doing very well in this class.” (conversation with the teacher, nov. 19). this comment confirms what much of the existing literature says: teachers more times than not believe girls’ success in mathematics is due to their efforts alone, not their intelligence or skill (damarin, 1995; forbes, 2002; walkerdine, 1989). the teacher also agreed that teresa had to put in more work toward her success than the others. however, teresa’s conception about mathematical competency was centered on rote learning and memorization. when asked about the skills to be good at mathematics, she said: “[people] should not be ashamed at trying to do memory things because most things are in your mind” (fg2, oct. 7). this path of memorizing as the way to learn mathematics neither aligns with what the nctm (2000) defines as doing mathematics nor matches the statements the latinas provided during the study when they explained their success and what they were able to do. nctm recommends problem solving, yet the girls’ schooling experiences did not provide learning opportunities to experience the nctm recommendations. nevertheless, while navigating the rigid mathematics available to them at school, the girls were awarded the possibility of being in the honors track, which could be part of the conflicting schooling to which the latinas adhere, but it may fail to build the necessary skills (i.e., problem solving) to be successful in the stem fields. we note the teacher stated teresa was the least successful of the three girls, but she could, ironically, be the one who methodically adheres to the rigid mathematics that are common to school mathematics, as she explained: because a lot of people i know, they just know the answer, or they just do it in their head and can just write an answer real quick. but i actually go through all the steps and make like a long list so i can actually understand it. [motioned to rocío and added] ’cause i know she [rocío] will get the answer really quick, or she [viviana] will do like short steps, and i need to see like all the problems to understand it. (fg2, oct. 7) teresa’s schooling also shaped her perception about higher mathematics. while teresa was cautious about her mathematics, she clearly still valued the same mathematics rocío did: fast, in your head, and not prescribed by books or even teachers. these girls valued highly being fast and doing work in their heads. to explain why girls sometimes do not achieve at the same level boys do, viviana said: “i can do mental math fast” (ii2, oct. 1). what viviana shared about mathematics and how she achieved success fit with the other two narratives. her evaluation of mathematics was similar to rocío’s, and, like rocío, she related to this guerra & lim latinas’ problem solving journal of urban mathematics education vol. 7, no. 2 71 better way of solving mathematical problems when she showed some disdain for the textbook’s way of doing mathematics by following steps—sometimes many steps. in sum, these girls indicated that they learned rigid mathematics at school but see the boys as capable of doing flexible mathematics and believe the current school curriculum offers few opportunities to experience the balance between the two kinds of mathematics. the mathematics produced the mathematics the three girls demonstrated here suggests that they could not engage higher-level flexible mathematics. the mathematics the participants demonstrated appeared to be the rigid mathematics they described as lower-level mathematics. this outcome is not unexpected, considering problem solving was not a regular part of the mathematical learning opportunities available at their school (or, at least not available to these three latinas). both rocío and viviana chose to use additive thinking instead of multiplicative thinking in the working session. even though finding the difference could have helped them solve the problem, the fact that the participants did not use the difference to calculate the percentage of reduction from the original recipe suggest their solution path considered little about the relationships of the quantities involved in the problems. the participants appeared to struggle with justifying procedures or strategies, but they were proficient with lower levels of cognitive demand tasks (stein et al., 2000) such as calculations. as carr, jessup, and fuller (1997) reported, the girls were not comfortable asking the hows and whys or explaining their thinking and reasoning. we noted that the participants talked about their success in mathematics, especially with problem solving and how good they were at it. problem solving contributes to cognitive development, and the benefits of using problem solving motivate students’ learning of mathematics (turner, celedon-pattichis, & marshall, 2008). however, in this case, the participants could not perform well in the problem-solving task presented. discrepancies existed between the latinas’ expectations and their demonstrated skill sets. we posit some reasons to make sense of the discrepancy. these girls could simply be narrating the image of successful and higher mathematics that they perceive from society. the partcipants may echo what they hear from society about the ideal skills necessary to be successful in careers in stem fields. alternatively, and more implicitly, the girls could be expressing their desire to grow from the biased notion of women’s low achievement in mathematics in which flexible mathematics can be a valuable tool for success (weisgram & bigler, 2007). it is important to reiterate here that neither the mathematics curriculum nor the teacher’s pedagogy appeared to nurture the specific skills and knowledge necessary for problem solving. the teacher stated that problem solving was not the guerra & lim latinas’ problem solving journal of urban mathematics education vol. 7, no. 2 72 main focus of their mathematics courses, claiming that too much material from the textbook had to be “covered,” and they had to stick to it by drilling. this focus may echo the status quo of mathematics instructions for girls in many parts of the country; as boaler (2002) argued, the learning environment created by teachers and schools does not always support girls’ needs. it is essential to ask whether latinas, and girls in general, have access to learning opportunities to develop such skills and knowledge that support their perception of good mathematics and continued interest in stem fields. researchers have found that teachers, parents, and peers influence latinas’ perceptions of academic achievement and motivation to purse stem professions (frome & eccles, 1998; mcwhirter, et al., 2013; robnett & leaper, 2013). teachers who interact more with boys, question boys’ thinking and strategies for problem solving, and who ask girls only lower thinking skill questions (e.g., a numeric answer or a rule that they were meant to memorize), are not helping girls learn mathematics that they can use later in their careers. these actions may lead girls to learn more structured and less inventive mathematics. thus, it is not surprising that girls choose to give memorized answers and use taught algorithms and traditional strategies at a higher rate than boys (fennema et al., 1998). it could also explain why girls, at times, appear to struggle with problem solving and vocalizing flexible mathematics strategies in classrooms. concluding remarks one finding this study depicted is that these three latinas wanted to be able to perform the higher-level flexible mathematics but fail to do so in the specific problem-solving task presented. it also revealed that the mathematics instruction in which these girls were provided (i.e., lower-level rigid mathematics) did little to enhance flexible problem-solving skills. why the three girls thought, felt, and performed that way, and why their opportunities to learn flexible mathematics were limited, are questions that should guide the work that needs to be done if the under-representation in stem is to be changed. latinas, as illustrated here, may do well in school mathematics in which problem solving is avoided and rigid mathematics is the only way to sustain their success. we ask, then: how do latinas respond when flexible mathematics becomes the key skill not only in the mathematics curriculum but also in the pedagogical practices? when they struggle with flexible mathematics, does this lead them to stop pursuing careers in stem fields? could this explain part of the reason why latinas are underrepresented in the stem fields? and from a latcrit perspective, how does the intersectionality of race, gender, language, immigration status, and so on, confound the mathematics learning opportunities provided to latinas? research needs to continue to investigate the connections between lati guerra & lim latinas’ problem solving journal of urban mathematics education vol. 7, no. 2 73 nas’ access to opportunities to learn problem solving through the use of flexible mathematics and the underrepresentation of latinas in stem fields. future studies could examine a larger group of latinas and consider providing latinas access to multiple mathematical tasks including in-depth problem solving in order to examine latinas’ mathematics achievement and career choices. in the end, it is important to preempt girls, in general, and girls of color, in particular, from gendered and racialized learning experiences and opportunities that hinder them from choosing a career in stem fields (weisgram & bigler, 2007). equally important would be providing learning opportunities for girls of color with an aim toward increasing the knowledge and skills that are relevant in stem fields, such as flexible problem-solving skills, which could be significant in addressing the underrepresentation of women of color in stem fields. references adams, d., & hamm, m. 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(1989). feminity as performance. oxford review of education, 15(3), 267–279. weisgram, e. s., & bigler, r. s. (2007). effects of learning about gender discrimination on adolescent girls’ attitudes toward and interest in science. psychology of women quarterly, 31(3), 262–269. zarate, m. e., & gallimore, r. (2005). gender differences in factors leading to college enrollment: a longitudinal analysis of latina and latino students. harvard educational review, 75(4), 383–408. journal of urban mathematics education july 2015, vol. 8, no. 1, pp. 119–127 ©jume. http://education.gsu.edu/jume alexia mintos is a ph.d. candidate in the department of curriculum and instruction at purdue university – 100 n. university street, west lafayette, in, 47907; email: amintos@purdue.edu. her research interests include equity in secondary mathematics teacher education and student transitions to science and mathematics disciplines at the university level. book review a notice to novices: what can we learn from “how should i know?”: a book review of how should i know? preservice teachers’ images of knowing by heart in mathematics and science1 alexia mintos purdue university athleen nolan’s (2007) book, how should i know? preservice teachers’ images of knowing by heart in mathematics and science, is a “critical qualitative study of mathematics and science epistemologies” (p. 33), particularly preservice elementary teachers’ (psets) views of what it means to know mathematics and science. in this book, nolan uses data from individual and focus group interviews and observations to describe the experiences of psets as both learners and future teachers. as nolan suggests, the main point of this book is, “that this research is not about the content of the subjects so much as it is about the preservice teachers’ perceptions of, and experiences in knowing in these subject areas” (p. 32). she seeks to understand how and why these perceptions and experiences come about. nolan uses these experiences to critique the conditions and epistemologies that exclude some groups of learners from fully engaging in mathematics and science learning and proposes alternative ways of teaching and learning mathematics and science. nolan uses multiple voices and perspectives in this work, including those of the participants, colleagues, and other scholars from a variety of disciplines (e.g., education, psychology, social sciences, and visual arts) to discuss methodological choices and broader epistemological issues. she leverages these voices to interrogate what it means to learn and teach mathematics and science and proposes alternatives to commonly accepted norms and practices in mathematics and science education. the participants’ perspectives are used to frame a vision for helping all preservice teachers (psts) to experience success in teaching and learning mathe 1nolan, k. t. (2007). how should i know? preservice teachers’ images of knowing (by heart) in mathematics and science. rotterdam, the netherlands: sense. pp. 256, $54.00 (paper), isbn 9789087902124 https://www.sensepublishers.com/catalogs/bookseries/new-directions-inmathematics-and-science-education/how-should-i-knowr/ k http://education.gsu.edu/jume mailto:amintos@purdue.edu https://www.sensepublishers.com/catalogs/bookseries/new-directions-in-mathematics-and-science-education/how-should-i-knowr/ https://www.sensepublishers.com/catalogs/bookseries/new-directions-in-mathematics-and-science-education/how-should-i-knowr/ mintos book review journal of urban mathematics education vol. 8, no. 1 120 matics and science. some particular strengths of this book, besides nolan’s multivocal approach to inquiry, include her reflexivity and transparency, her use of a variety of narrative tools (e.g., metaphor, poetry, narrative, pictures, format, and author reflections), and the interweaving of theoretical perspectives throughout the text. my motivation for this review is aligned with my goal as a novice researcher to find exemplars of qualitative research informed by various theoretical perspectives and research methodologies. to facilitate the development of qualitative research in a climate of continued focus on empirical research in education, it is important to highlight qualitative studies and publications that are unique, innovative, and rigorous, but that also have practical implications for mathematics, science, and teacher education (st. pierre, 2002). i am also interested in seeing applications of the many ontological and epistemological perspectives and learning theories that i have read. i find it valuable to see these theories exhibited in qualitative research to develop a clearer conception of these perspectives and an awareness of what they look like experientially. i also think that it is important to imagine how these theories can be linked to urban mathematics education and mathematics teacher education. summary of the content in the beginning chapters of this book, nolan introduces us immediately to the participants, eight white female psets. first, they are introduced with a transcript of their discussion of “reasons and/or influences behind [their] beliefs about math and science knowing” (nolan, 2007, p. 1) and then more formally with contextual information. nolan presents this transcript simultaneously with a lesson about constructing a kaleidoscope. the use of metaphor is a prominent tool that nolan uses to highlight specific themes and to critique dominant discourses and ways of knowing in mathematics and science education. nolan introduces the participants and describes how they became part of the study and particular experiences, characteristics, or perceptions they share with her. in the preface following the introduction, titled “postmodern conscienceness: reflections on light,” nolan provides an overview of the book and brings attention to specific experiences and discourses in mathematics which shaped her participants’ conceptions of science and mathematics. she highlights the focus on gender and gendered experiences in mathematics and science. she also reveals the significance of the language, formatting, artistic tools, and metaphors she uses in the book. nolan explains the epistemologies that inform her theoretical framework and methodological choices in data collection and re-presentation. for example, postmodern epistemology frames her choice of re-presenting a variety of participants’ stories that counteract the dominant discourse about mathematics and mintos book review journal of urban mathematics education vol. 8, no. 1 121 science knowledge and practice. borrowing a term from peters and lankshear (1996), nolan suggests that her participants’ narratives could best be viewed as a “kaleidoscope of counternarratives…since the voiced experiences of the preservice teachers run counter to and challenge the official grand narratives” (p. 30). this assertion can be seen as amplifying voices and featuring stories told in psets’ words that are not prominently featured in discussions on what counts as knowledge, what it means to learn mathematics, and personal experiences of students who struggle with mathematics. nolan (2007) chooses to focus on light as a metaphor for the “explorations of preservice teachers’ thinking on what it means to know in mathematics and science” (p. 31). this chapter gives insight into the researcher’s choices and provides some significant background that prepares the reader for the unique journey that follows in the remaining chapters. nolan uses feminist epistemologies to critique light and vision as the dominant representations of knowledge and ways of knowing in mathematics and science. overview of chapters in chapter one, nolan discusses the sources of knowledge, unpacks the conceptualization of light as a metaphor for knowledge, and highlights the resulting implications for teachers and learners. she draws from feminist and postmodern thought to challenge accepted conceptions of knowledge and enlightenment. she defines luminous light sources as those individuals who are considered teachers and experts in mathematics and science, and non-luminous light sources as students and others who absorb the knowledge transmitted by luminous light sources. nolan argues that this conceptualization is problematic because it assumes a fixed conception of knowledge. it could also perpetuate the impossible expectations we have for new teachers to become instant luminous sources of knowledge. nolan draws on the light metaphor because it has many properties and characteristics that are figuratively linked to knowledge and knowing, but she also demonstrates its limitations. while nolan critiques the conception of knowledge as light, she also seeks to reconfigure this metaphor to address its problematic aspects as it relates to knowledge and how it is shared. chapter two is divided into three parts: (a) “part i: the rectilinear propagation of light,” (b) “part ii: particle and wave theories of light,” and (c) “part iii: formation of shadows.” in part i, nolan draws from seminal literature (e.g., bruner, 1996; dewey, 1938; von glasersfeld, 1996) to discuss and critique the traditional model of education and the need to revisit the configuration of an instructional model of “information transfer from teacher and textbook to student” (p. 84). she also presents aspects of a progressive model of education that involves engaging students’ wills and building on what they already know. in this section, she concludes with a discussion of the impact of perceptions and attitudes on views of learning. in part ii, nolan compares the particle and wave theories of light with mintos book review journal of urban mathematics education vol. 8, no. 1 122 theories of knowledge. this chapter draws on the ambiguity around theories related to the physical nature of light. in physics, light is considered both a particle and a wave as it has properties of both. nolan argues that an analogous conceptualization is appropriate for understanding. she notes that it is valuable to have pieces of information related to particular concepts in mathematics and science, but it is also important to see how these pieces fit into a larger framework of continuously connected ideas. part iii is partly devoted to participants’ experiences as female learners in mathematics and science and how these experiences are mitigated by teachers, the curriculum, and gender-focused ideologies. nolan presents quotes from participants that highlight the gender biases that seemed to be ingrained in the discourse they experienced in mathematics and science. these messages, whether intentional, careless, or playful, seemed to discourage the psets in the study from actively pursuing mathematics and science or at least communicated the possibility that they were not the right fit for advanced studies in mathematics and science. overall, this chapter delves into the messages expressed and internalized by psets as they experienced mathematics and science education. chapter three focuses on answering the question of how the learner comes to know and how learners, particularly the participants in this study, communicate about their knowledge of and interactions in mathematics and science. nolan also discusses the refraction metaphor for knowing. during the refraction process, light is transformed within refractive materials and the transformations vary with the nature of those materials; analogously, the process of coming to know takes place within the learner and knowledge transformation takes place when the learner actively constructs his or her knowledge. this process is also unique to the learner; thus, it is unrealistic for educators to have identical learning and achievement goals for all students. in chapter four, nolan restates, summarizes, and further unpacks some of the notable statements made by participants. for example, an exchange is quoted where some participants express their enjoyment of inquiry-based lessons in science, but are unsure of how these lessons might help students learn the “underlying concepts” that would be building blocks for their later science classes. she then harnesses participant experiences and expressed thoughts to highlight the importance of experiential learning, which is supported by scholars and teacher educators with multiple worldviews. she proposes “reimag(in)ing” mathematics and science education as open, engaging, and creative activities and reconceptualizing knowing and knowledge by recognizing past experiences and building on them to cultivate deeper understanding and meaningful learning experiences for all students. her proposal to find ways to broaden access and engagement in mathematics and science, especially to students who have typically been underserved, aligns with the goals of urban education and those who enact emancipatory paradigms and pedagogies. in the next section, i discuss the strengths of nolan’s work. mintos book review journal of urban mathematics education vol. 8, no. 1 123 distinctive features of this study that indicate credibility nolan uses multiple pieces of data from her participants’ and her own reflections to provide rich description in conjunction with literature that strengthens credibility in the text. tracy (2010) defines credibility as “the trustworthiness, verisimilitude and plausibility of the research findings” (p. 840). some tenets of credibility are thick description, crystallization, triangulation, multivocality, and member reflections. credibility is also closely tied to the ethical qualities of the work (howe & eisenhart, 1990; tracy, 2010). data from the participants were obtained through multiple individual and focus group interviews, member-checking interviews, and some informal written communication. multiple voices are re-presented throughout the book. every chapter from introduction to conclusion includes a mix of participant, scholar, and the author voices. particularly noteworthy is that while she interprets the meaning of participant responses, nolan presents their own words and unpacks her interpretations in conversations shared throughout the book. support from a variety of scholarship moss and colleagues (2009) support the conception that rigorous research should demonstrate an awareness of the history, ethics, and philosophy of the chosen phenomenon, problem, or methodology and should build on the work of past researchers. nolan highlights the results of other studies to support her claims and to provide a foundation for her arguments. for example, she draws on other studies to claim that many psets enter their teacher education programs with a palpable dislike or disinterest in mathematics and science (both teaching and learning). she notes one of the conclusions of hill’s (1997) study that “in mathematics methods courses, most psts view mathematics as a set of rules and procedures to be memorized” (nolan, 2007, p. 81). this quote is consistent with her opinion that psts’ conceptions of mathematics will not only influence their attitudes toward mathematics, but also the pedagogical strategies they enact and their willingness to teach in ways that are different from what they experienced as learners. nolan situates all of the ideas she discusses in prior work and includes summaries of research-related concepts in excerpts called “inside research” found throughout the book. she also incorporates colleagues’ perspectives by including their quotes in the “responsibilities” snippets distributed throughout the chapters. reflexivity another distinctive feature of this work is the reflexivity of the researcher. this reflexivity is a particular strength exhibited by nolan’s transparency, sincerity, and openness, which are hallmarks of ethical studies (tracy, 2010). nolan is open about herself and her experiences throughout the text by revealing her qualifica mintos book review journal of urban mathematics education vol. 8, no. 1 124 tions, experiences in teaching, and feelings about mathematics and science. even though she is well qualified in mathematics and physics, nolan expresses the following sentiment: “the hairs on the back of my neck stand on end, and i break out in a cold sweat when i think someone is about to ask me a question that tests my ‘knowledge’ in math and science” (nolan, 2007, p. 21). these insights into nolan’s background, research interests, methodological choices, tensions, and thought processes also make her motivations and biases visible to the reader. this transparency demonstrates a high level of trustworthiness and provides additional perspective and context for interpreting the study and its findings. reflexions are passages inserted into the text to explore topics in more depth, expand on or clarify statements in the text, and provide glimpses into nolan’s thought processes. inside (my) research sections give added insight into her decisions about the project, details about her reasoning, and dilemmas she experienced in data collection and analysis. use of learning theories another distinctive feature of the book is how nolan was able to implicitly and explicitly integrate a variety of learning theories throughout her work. because her work is motivated by postmodern assumptions, crystallization and multivocality are essential parts of her work. nolan also uses a feminist lens to deconstruct, challenge, and critique the common assumptions about knowledge and knowing, especially in mathematics and science. because nolan uses feminist theory prominently in this work, a focus on gender is evident at multiple levels. she highlights the achievements of women in science and mathematics, and seeks to counter the common discourses that women are not suited for mathematics and science by giving participants pseudonyms inspired by women who made significant contributions in mathematics and science (e.g., physicist ursula franklin and mathematics ecologist evelyn pielou). she shares different stories of women in mathematics and science through the “her story” features. through her use of multiple epistemological and pedagogical lenses, she also seeks to strengthen the case for multiple and alternative perspectives on teaching and learning to accommodate more women, rather than trying to change women to fit mathematics and science. she uses feminist epistemology to unpack the notion of the expert and posits that the view of “knowledge as light” and seeing as a metaphor for knowing can be interpreted as gendered and hegemonic. moreover, nolan often points to the literature for solutions for more engaging pedagogy, especially ways to engage students as active participants in science and mathematics learning. enactivist theory posits that the individual is not just an observer, but also an active participant in the surrounding world, both physically and cognitively (ernest, 2010). freire (1970/2000) argues against the “banking” model of learning or the knowledge as commodity view. he urges active engagement of students in the learning process to ensure that it is meaningful to them (lave, 1996). mintos book review journal of urban mathematics education vol. 8, no. 1 125 nolan (2007) proposes that science should be viewed as “participation with natural phenomena” (p. 195). she also envisions mathematics and science as corporal endeavors involving the whole individual. she proposes that these disciplines mix inquiry with drama, storytelling, reflection, or writing and that there should be a focus on the journey or the process of problem solving, rather than right or wrong answers. overall, nolan recommends that we reconceptualize mathematics teaching and learning as active, rather than passive pursuits. discussion this book provides a valuable contribution to the body of work related to qualitative studies that use unique approaches to inquiry. the experiences and perspectives shared from the participants and unpacked with the help of other scholarly work prove to be an effective combination. nolan uses her original framework of ideas related to inquiry and knowledge for the organization of the text. there are ideas expressed in this book that teacher educators, teachers, psts, and other stakeholders in education can benefit from hearing. the postmodern and feminist perspectives she describes inform her data collection, analysis, interpretation, and representation consistently. her use and interpretation of epistemologies and learning theories is implicitly and explicitly represented throughout the narrative as she discusses perspectives about knowing mathematics and science. these theories not only influence what she chooses to say, but how she chooses to present data, existing literature, and her own ideas. particularly powerful was the prominent role that the participants’ perspectives and voices play in the text; this approach is consistent with the feminist paradigm, which informs the research design. the participants’ perspectives are integrated into every topic allowing the reader to envision some aspects of mathematics and science education through their eyes. nolan also makes a case for the deconstruction of mathematics and science discourse, pedagogy, and learning as we know it. her presentation of concerns related to norms that could exclude some learners challenge others to consider the social and political implications of their actions as educators, mathematicians, and scientists. while this work does not emphasize racial or cultural diversity nor is it situated in an urban setting, the critical lens that she uses and the emphasis on empowering participants can inform the work of scholars in urban mathematics education. this critique could also be extended to ways of knowing which exclude or marginalize the experiences and funds of knowledge (moll, amanti, neff, & gonzalez, 1992) of students from african american, urban, or culturally diverse backgrounds because these ways of knowing are often undervalued or overlooked in conventional schooling. critiquing discourses and opening dialogue about alternative ways of knowing mathematics can help to inform educators’ positioning of students in urban settings as more empowered and active mathematics learners. mintos book review journal of urban mathematics education vol. 8, no. 1 126 conceptions of mathematics as an active endeavor enriched by multiple perspectives, cultures, and experiences align with the work of educators who seek to empower marginalized students in mathematics and the teachers who work with them (e.g., moll et al., 1992; tate, 1995; turner et al., 2012). in addition, nolan also questions whether high quality research should only include studies designed to limit researchers’ interactions with participants, require rigid objectivity, and exclude researchers from the narrative. nolan’s methodological example could also lend credence to the investigative approaches of researchers who work with marginalized populations and employ participatory or liberatory epistemologies and approaches, especially within urban contexts. the distinctive features of qualitative methodology she describes could inform research studies in urban mathematics and science education, particularly those studies that seek to broaden participation in urban communities at all levels and to amplify diverse voices that are not typically included in conversations about knowledge, teaching, and learning in mathematics and science. nolan’s work also highlights the importance of learning from the perspectives and experiences of future teachers. in particular, she shows that it is essential for all psts to reflect on their lived experiences as mathematics and science learners in the contexts of their teacher education programs, and for mathematics and science teachers, teacher educators, and researchers to listen to and reflect on the meanings of these experiences. this work would be helpful in understanding and addressing the lack of diverse teacher candidates who perhaps may have responded to negative learning experiences in mathematics and science by choosing alternative paths of study at the university level or not completing coursework in preparation for teaching careers. this point is particularly important given the need for diverse teacher candidates in urban settings because of the cultural and social capital they may bring to the table. having future teachers reflect on their experiences as mathematics and science learners could also be very enlightening for teacher educators, especially if examined through racial or cultural lenses. it could also be of value to those interested in seeing how psets view mathematics and science to inform the creation of learning opportunities to support positive conceptions of teaching and learning among diverse students. more generally, this book might help teachers think about ways that they create mathematics and science classrooms where all learners feel encouraged to participate and thrive, especially our most vulnerable students. references bruner, j. s. (1996). the culture of education. cambridge, ma: harvard university press. dewey, j. (1938). experience and education. new york, ny: collier books. ernest, p. (2010). reflections on theories of learning. in b. sriraman & l. english (eds.), theories of mathematics education: seeking new frontiers (pp. 39–47). berlin heidelberg, germany: springerverlag. mintos book review journal of urban mathematics education vol. 8, no. 1 127 freire, p. (2000). pedagogy of the oppressed (m. b. ramos, trans; 30th anniversary ed.). new york, ny: continuum. (original work published 1970) hill, l. (1997). just tell us the rule: learning to teach elementary mathematics. journal of teacher education, 48(3), 211–221. howe, k., & eisenhart, m. (1990). standards for qualitative (and quantitative) research: a prolegomenon. educational researcher, 19(4), 2–9. lave, j. (1996). teaching, as learning, in practice. mind, culture, and activity, 3(3), 149–164. moll, l. c., amanti, c., neff, d., & gonzalez, n. (1992). funds of knowledge for teaching: using a qualitative approach to connect homes and classrooms. theory into practice, 31(2), 132–141. moss, p. a., phillips, d. c., erickson, f. d., floden, r. e., lather, p. a., & schneider, b. l. (2009). learning from our differences: a dialogue across perspectives on quality in education research. educational researcher, 38(7), 501–517. nolan, k. t. (2007). how should i know? preservice teachers’ images of knowing (by heart) in mathematics and science. rotterdam, the netherlands: sense. peters, m., & lankshear, c. (1996). postmodern counternarratives. in h. a. giroux, c. lankshear, p. mclaren, & m. peters (eds.), counternarratives: cultural studies and critical pedagogies in modern spaces (pp. 1–40). new york, ny: routledge. st. pierre, e. a. (2002). comment: “science” rejects postmodernism. educational researcher, 31(8), 25–27. tate, w. f. (1995). returning to the root: a culturally relevant approach to mathematics pedagogy. theory into practice, 34(3), 166–173. tracy, s. j. (2010). qualitative quality: eight “big-tent” criteria for excellent qualitative research. qualitative inquiry, 16(10), 837–851. turner, e. e., drake, c., mcduffie, a. r., aguirre, j., bartell, t. g., & foote, m. q. (2012). promoting equity in mathematics teacher preparation: a framework for advancing teacher learning of children’s multiple mathematics knowledge bases. journal of mathematics teacher education, 15(1), 67–82. von glasersfeld, e. (1996). aspects of radical constructivism and its educational recommendations. in l. p. steffe & p. nesher (eds.), theories of mathematical learning (pp. 307–314). mahwah, nj: erlbaum. microsoft word 3 final civil vol 7 no 2.doc journal of urban mathematics education december 2014, vol. 7, no. 2, pp. 9–20 ©jume. http://education.gsu.edu/jume marta civil is a professor of mathematics education and the roy f. graesser endowed chair in the department of mathematics at the university of arizona, 617 n. santa rita ave., tucson, az 85721; email: civil@math.arizona.edu. her interests include cultural, social, and language aspects in the teaching and learning of mathematics; linking in-school and out-of-school mathematics; and parental engagement in mathematics. commentary why should mathematics educators learn from and about latina/o students’ in-school and out-of-school experiences? marta civil the university of arizona have been studying the complexities of bridging in-school and out-of-school mathematics for quite a long time. my motivation for doing this work is grounded in my initial experiences with the funds of knowledge for teaching (fkt) project (gonzález, moll, & amanti, 2005a) and has continued throughout the years as i have been working with largely low-income, mexican american1 communities in tucson, arizona. all throughout this time, i have raised questions related to connecting in-school and out-of-school mathematics (civil, 2002, 2007, 2014). these questions have to do with what is mathematics? where is the mathematics? how are different mathematical practices valued, by whom, and where? in civil (2014), i raise further questions about what is competence? what is the interplay between task, setting, and engagement? and how do languages (as in english and spanish, for example) and affective elements interact when doing mathematics? in particular, i discuss these questions in relation to in-school and out-of-school settings. for example, the case of alberto, a young immigrant student from mexico, portrays a child who is seen as quite competent in his home/community environment, but not succeeding in the school environment (civil, 2014; civil & andrade, 2002). what should we learn from this case? why is this case relevant to our research and our teacher preparation efforts? i believe that mathematics educators (teachers, school administrators, university faculty, etc.) need to take a more holistic approach toward the mathematics education of students, and in particular of marginalized students. in this commentary, i expand upon this belief through the “voices” of several students and their parents with whom i have interacted over the years. but first i provide some general context for my research, in terms of who the participants are and how i approach my work.                                                                                                                 1 here, i use the term mexican american rather than latina/o to more closely reflect the communities in which this work is located. i civil commentary journal of urban mathematics education vol. 7, no. 2 10 context as i mentioned earlier, most of my work is located in schools and communities that are largely mexican american. thus, while i have worked with other marginalized students, my focus here will be on students of mexican origin and their parents. even though i use the terms mexican american or of mexican origin i want to underscore the variability among the children and families with whom i have worked over the years. most of them have some connection to mexico. in some cases, the connection is quite current, in that they may be recent immigrants or have family members living in mexico, in the bordering area. while in other cases, it is a historical connection, in that they trace back their histories to when this southwest region of the united states was part of mexico (sheridan, 1995). some are “documented” and some are not. some were born in the united states and some were born in mexico. some speak english only or spanish only; some speak both languages, and some have one of the many indigenous languages from mexico as first language. furthermore, it is important to understand that these situations occur within families. that is, in a given family, one or both parents and some of the children may be “undocumented” but some may have been born in the united states or are documented; the parents may speak mostly spanish, while the children may be bilingual. my approach to research is heavily influenced by an ethnographic tradition. in that sense, i spend considerable time in the community building rapport with its members, just being there. also, whenever possible, i like to interact with the students, their parents,2 and their teachers. that is, i argue that to gain an understanding of students’ experiences with in-school and out-of-school mathematics, we need to talk to the various people that are likely to play a role in these students’ experiences. starting with the fkt project, i developed a particular interest in working with parents and mathematics education. thus, much of my work focuses on parents’ perceptions of their children’s mathematics education, as well as on their own interests, uses of, and questions about mathematics. for me, an ideal situation is the case where i have developed a solid rapport with a parent (usually a mother), often through activities (e.g., workshops, classroom and home visits), and i also have developed a rapport with her children through my work in the classroom and conversations with them. i am describing my approach not in the “traditional” sense of research method—though of course, i have collected data through interviews, classroom observations, and so on—but to me the relevant point is the importance of what many mothers describe as confianza (trust). confianza is a key component of the fkt project. as gonzález, moll, and amanti (2005b) write, “when there is sincere interest in both learning about and learning                                                                                                                 2 in most of my work parent actually refers to mother (most often) or father, but i also use the term to refer to other primary care takers such as grandparents. civil commentary journal of urban mathematics education vol. 7, no. 2 11 from a household, relationships and confianza can flourish” (p. 6). in civil (2001), i elaborate on the concept of confianza and illustrate it through the voices of some of the mothers, as in the following excerpt: when i integrated into the group de las señoras, for me the most important foundation was the confianza that each one offered me…. i can say that all that i now know and have learned has been accomplished by means of the confianza. (p. 175) developing confianza takes time and its results are not necessarily immediately measureable. i am concerned with the pressures that funding agencies, policy makers, and others are putting on interventions and studies that “produce outcomes” and measure things, usually in a limited amount of time, as if the complexity of children’s lives could be put on hold, manipulated a bit, and “desirable” outcomes will follow. what i hope to convey in this commentary is that pretending that we can “improve” marginalized students’ mathematical learning opportunities without taking into account their lived experiences, is educationally naïve at best. some of these lived experiences involve navigating different worlds (e.g., literally geographically, mexico and the united states, as well as home and school), different languages, negative perceptions (e.g., views of immigration), fears (e.g., their “status” in the united states), and areas of expertise that grow out of these lived experiences and that may be different from our own experiences and expertise. in what follows, i discuss the relevance for mathematics education of students’ in-school and out-of-school experiences by focusing on two themes: language and culture. considerations around language my focus here is on the students for whom spanish plays a role in their everyday life, either because they speak it, or a close member in their family speaks it. one of the schools where colleagues and i carried out several activities over five years is in a primarily mexican american neighborhood, where spanish is very present (in the local businesses and among the people in the community, particularly the adults). the school, like all schools in arizona at the time of this work (and currently), is under a repressive law against bilingual education.3 still, this was a school where one could hear spanish easily in the hallways, in the                                                                                                                 3  in 2000 arizona voters passes proposition 203 that severely restricted bilingual education requiring that ells be placed in structured english immersion (sei) classrooms with instruction only in english. furthermore, in 2008–09 the 4-hour english language development model was implemented. this model calls for 4 hours per day of english language instruction. “it is the only state in the country with such an arrangement. whereas proposition   203 indicated that all teachers could teach content, albeit in english only, the 4-h eld block established  stringent instructional  procedures for its teachers” (rios-aguilar, gonzález canché, & sabetghadam, 2012, p. 49).   civil commentary journal of urban mathematics education vol. 7, no. 2 12 classrooms when working in small groups, and in the office (the staff was bilingual). so, this was not a school where spanish was not welcomed (although the instruction did follow the mandate of the law). yet, i was not aware that several students i met spoke spanish until i either asked them if they did or heard them speak with a family member (usually a parent or grandparent). i wonder, does language become associated with certain settings? what kinds of messages are students getting that dictate what language to use when? in the following exchange between penny, a fourth grader in that school, and me, we see her thinking on when to use which language: penny: [in mexico] mostly spanish; it’s only spanish that i hear all over. my dad usually talks english, but in mexico he says, “no hables inglés, hablas español.” (don’t speak english, speak spanish.) my tío (uncle) also says that. marta: ok. why do you think they say that? penny: um because you aren’t at a specific place to talk one language like in school. if you have a friend that talks spanish you should talk spanish to them, but in school you talk english and, and…and at your house some people talk english or spanish. marta: ok, is that what your parents tell you; your dad tells you that? penny: um mostly my tío. he tells my cousins and me. he says, “don’t talk english at this house.” penny was one of the students that i did not know spoke spanish until she told me she did. as she indicates in the excerpt above, english was the language of school and when not in school, the language used depended on the place and people involved. while penny may have felt comfortable navigating both languages and knowing when to use one or the other (we do not know that), this is not necessarily the case for other students who need to make choices and who do not feel free to use whatever language they want. for example, the excerpt below is what a mother shared when her son entered kindergarten: julia: when my son entered school, in kinder, he wanted to go back [to mexico] because one at home well one speaks spanish and the neighborhood kids speak spanish and everything is spanish. and when all of a sudden he entered kinder he told me: “you know what mommy, i don’t want to go to school because everyone speaks english and i don’t understand them at all. it’s a world of english, this is not my world; my world is spanish.” he would say, “i want to go back to [town in mexico] because over there it’s my world”; he would say, “my world is spanish.” over the years, i have collected several examples from parents reporting how hard it was for their children when they did not know english well yet. while this may not be surprising, what i want to stress is that parents mostly shared the emotional toll that this situation created for their children and for themselves as civil commentary journal of urban mathematics education vol. 7, no. 2 13 parents. parents shared stories about their children coming back from school crying and wanting to drop out. they also shared their difficulties when trying to help their children with homework. even in subjects like mathematics where they had the knowledge to help them, they could not always do it because of the language difference, creating a barrier between children and parents (acosta-iriqui, civil, díez-palomar, marshall, & quintos-alonso, 2011). in this commentary, i am looking at language from a political point of view, given that “as educators, to ignore the political underpinnings of school language policies would be irresponsible” (civil & planas, 2012, p. 72). setati (2005) argues: decisions about which language to use, how to use it, and for what purpose are both pedagogic and political…. if we are to explain language practices in a coherent and comprehensive way, we must go beyond the cognitive and pedagogic aspects and consider the political aspects of language use in multilingual mathematics classrooms. (p. 451) in a survey of mathematics education and immigrant students (in different parts of the world), not knowing the language of instruction was seen as one of the main problems (civil, 2012a). this notion of seeing language as a problem exemplifies the pervasive deficit view toward marginalized students—in this case, those whose first language is not the language of instruction. elsewhere (civil & planas, 2012; planas & civil, 2013), planas and i have argued for the need to move away from this problem focus toward a resource orientation in which languages are seen as resources toward the teaching and learning of mathematics. furthermore, planas and i argue for the need to understand the socio-political context of our work, and the role that the different languages involved play in positioning students as learners of mathematics: “ultimately, we cannot separate the ‘language issue’ from the socio-political context in which students are embedded” (planas & civil, 2013, p. 376). so, why are language issues relevant to mathematics education? and here, i mean beyond the perhaps more obvious answer of teaching mathematics to english language learners (ells). in fact, i argue that while we need to continue to seek ways to support ells in their learning of mathematics, this is not enough, particularly in settings where the use of home language(s) is not supported. we need to address the complexity of language ideology in the classroom (civil, 2011b). in prior writings (e.g., civil, 2011b; civil & planas, 2012), i have articulated my dilemma when working with a group of middle school students, most of whom were recent immigrants and spanish dominant. by encouraging the use of spanish in their small group work and even in the whole class presentations, i was able to document rich mathematical discussions (see, e.g., civil, 2011b; 2012b). while from a mathematics education point of view i thought it was powerful, i civil commentary journal of urban mathematics education vol. 7, no. 2 14 was unaware of the conflicting messages that the students were getting. these students were in a segregated environment (apart from the non-ell students) for most of their school day (quite a few articles have been written on the current language policy in arizona; see, e.g., combs, dasilva iddings, & moll, 2014; gándara & orfield, 2012; rios-aguilar, gonzález canché, & sabetghadam, 2012). students were aware of the segregation and, in fact, were concerned about their learning. in an interview with the mothers of two of the students in this class, they shared their children’s (ernesto and larissa) concern with being in section a of the school (this is where the classrooms for ells were located): roxana: ernesto says that he wants to go higher. he is going for, he says: “i want to get to my final goal …i haven’t reached it yet. … i am striving to get there.” he says that he’s not very convinced of being there [in section a]. he wants more. mila: larissa feels embarrassed. she says, “mom i am embarrassed to go to section a.” interviewer: and what does it mean to be in section a? what is the difference? mila: well that they speak a lot of spanish, that they hardly know any english. roxana: my son says that it’s more spanish there. he says: “mom, just imagine that we are back in mexico, with the teachers from mexico because now i even get mixed up because they explain more in spanish than english. and i am with the expectation that they are going to talk to me in english and i am thinking in english…. i get mixed up, because i want them to talk to me in english and the teacher can’t because there are quite a few children who don’t understand english well. and the teacher opts to speak spanish first and when [she] starts talking in english, i am already all tangled up in knots. i am already confused, and i can’t get untangled.” and that is why he wants to go where “the class in general, from start to finish, [is] in english.” (see civil & menéndez, 2011) below is an excerpt from an interview with another ell from section a, an eighth grader, cecilia, whom i knew since 6th grade. as we can see, cecilia is conflicted about section a. she does not like being there because it feels that she is not moving forward, not learning enough, but at the same time, she likes the people there, because “everybody is mexican like me and we talk”: marta: how do you like being in section a? cecilia: i don’t like it. marta: how come? cecilia: because when i was in 6th grade, i had all the classes in here; and when i was in 7th grade, and now. marta: and what is it that you don’t like? cecilia: que no salgo de la misma (that i don’t get out of the same place)… marta: so, what else don’t you like about section a? civil commentary journal of urban mathematics education vol. 7, no. 2 15 cecilia: i like section a because everybody is mexican like me and we talk, and yeah, i like it. marta: you like being in section a? cecilia: no, i, i like the people in section a, the persons in… marta: got you! the students? cecilia: yeah. marta: the students in section a. got you. but if you could choose, where would you be? cecilia: in section b [a different set of classrooms for non-ells]. marta: in section b. if now you were to start eighth grade, if this was august instead of april… cecilia: section b. marta: in section b. ok. and why? why do you think that, that… cecilia: i would learn more. marta: and why do you think you’d learn more in section b? cecilia: like i said, …all the people speak english and…i have to speak english too. it is rather ironic that while the intent behind the segregation was presumably that the students would learn english quicker, the experience for these students was quite the opposite. while the instruction may have been in english (with clarification in spanish when needed), in general spanish was very present in section a. most of the students i interviewed who were in section a expressed that they did not think they were learning as much as they wished. they had an awareness that “something was wrong” with being in section a and the goal was to move out of it. these were young people navigating a different culture and language, a different approach to schooling (many of them had been in the united states for three or fewer years), trying to fit in with the “regular kids” while at the same time being classified as “section a.” i encouraged them to use spanish in their mathematics explanations, which opened up the patterns of participation in mathematical discussions (civil, 2011b). however, i have been wondering since about what other messages i was unintentionally sending them, such as “your english is not quite there, use spanish?” or simply ignoring their struggles with power issues associated to different languages. as i look back, i believe that besides wanting the students to be able to do and talk about mathematics in any way they felt comfortable, my intent was also to value and affirm their home language (which it is important to note, is my and the classroom teacher’s home language, too). but that is a political positioning that i do not think came through for the students. that is, we did not engage in discussions with the students about language ideology. and this is coming from a mathematics educator who learned mathematics in a language (french) other than my home language (spanish), grew up in catalonia at a time where the catalan language was repressed, and then moved to the united states, where obviously english is not my home language. i share this personal story to put things in per civil commentary journal of urban mathematics education vol. 7, no. 2 16 spective. i have had ample opportunity to experience linguistic diversity and power issues around different languages. i know how important language is as part of one’s identity (ruiz, 2010), and i also know how unsettling it can be to feel different because of one’s language and how that language is perceived with respect to other language(s). yet, i still feel that in my work i am not addressing this “language issue” head on. i should make it clear that these issues around language are not only about settings where one’s home language is repressed and devalued, as in the situation i have illustrated with the middle school students in arizona. even in dual-language settings where the school is trying to make sure that the non-dominant students’ language is valued, the power issues are still present (cervantes-soon, 2014; see civil, 2012b for a discussion on the participation of latina/o students in the mathematics classroom in a dual-language setting). as valdés (1997) cautions, “bilingualism can be both an advantage and a disadvantage, depending on the student’s position in the hierarchy of power” (p. 420). and so, as i reflect on the comments and experiences of students like penny, larissa, ernesto, and cecilia, and mothers like julia, mila, and roxanna, i wonder: what should we do as mathematics educators? should we bring up issues around valorization of language and language policy when we teach prospective/practicing teachers? how do we raise awareness among mathematics educators that these considerations about language are important in the mathematics classroom and include more than just approaches to teach mathematics to ells? similar to martin, gholson, and leonard (2010), i have a focus on the mathematics when i work with students like penny, larissa, ernesto, and cecilia, as well as when i work with parents, teachers, and prospective teachers. i want to know how they think about mathematics, but also as martin and colleagues point out: yet, for many scholars, including ourselves, subsequent efforts to add needed complexity to the understanding of learners, their social realities, and the forces affecting these realities have led them (and us) to take social, sociopolitical, and critical turns in their (our) work, away from overly narrow concerns with mathematics content. these turns have made salient many issues not typically pursued in mathematics education research, including issues of identity, language, power, racialization, and socialization. (p. 15) one of these “salient many issues not typically pursued in mathematics education research” is learning about the backgrounds and experiences of marginalized students. in what follows, i offer some brief remarks on what that learning may look like and why it is relevant to mathematics education. considerations around culture first of all, my definition of culture is based on gonzález’s (2005) work in the fkt project, where culture is defined as lived experiences: “we have interrogated civil commentary journal of urban mathematics education vol. 7, no. 2 17 many of the assumptions of a shared culture, and have chosen instead to focus on ‘practice,’ that is, what it is that people do, and what they say about what they do” (p. 40). learning about and from students’ (and their families’) lived experiences is a key premise in the fkt project. to illustrate, let me go back to the case of penny. in her interview, penny shared that she went to mexico quite often, almost every weekend, to a ranch that was about three hours away from tucson. at the ranch she rode horses, and in her description she conveyed both her expertise and confidence with this practice. from a mathematics funds of knowledge approach, there are rich opportunities for learning experiences building on penny’s knowledge of international travel (mexico – united states) (e.g., my first experience in the fkt project was with a module around money that originated in part from students’ experiences with u.s. and mexican currency) and on her knowledge of horses and ranch life (for an example of a module based on her students’ experiences with ranches and horses, see amanti, 2005). while these connections between out-of-school experiences and in-school mathematics learning opportunities are important and are in fact what first attracted me to a project such as fkt, i think that finding the actual connection to the mathematics may be less significant than making the connection with the student and his or her family. it is these connections that help develop confianza, which has proven to be so important in my work with parents and children. i argue that this confianza that grows out of a real interest in understanding students’ lived experiences allows us to also make connections toward their mathematical learning. by developing confianza with the middle school students mentioned earlier, their interactions often combined what one could describe as social chat (others might characterize it as “off task” and miss the learning going on). this social chat drew upon cultural elements such as humor and metaphors, which at times became tools for the actual solution of a problem (see, e.g., civil, 2011a). through this confianza, i was able to get to know the students better. for example, octavio did not participate much in the mathematics classroom when i first met him. as we worked on encouraging mathematical discussions, i found out that octavio liked to argue. his peers referred to him as “alegador” (argumentative). this interest in engaging in arguments became an asset to what i would describe as his passionate participation in mathematical discussions (civil, 2011a; civil, 2012b). most of my work within this notion of confianza has been with parents. through listening to parents, i have learned about their experiences with school mathematics (in many cases, as children growing up in mexico), as well as their children’s experiences, often as they navigate a system that is quite different from their parents’ (civil & planas, 2010). these conversations with parents also point to their perceptions of power issues as they realize that their knowledge of and experiences with mathematics are not even acknowledged in their children’s civil commentary journal of urban mathematics education vol. 7, no. 2 18 schools. for instance, one of the mothers, in a discussion around barriers and opportunities for a stronger communication between school and community, commented: the first problem is that when the teacher send papers, the teachers want to do a better job with the kids that come from mexico, but they don’t start thinking that it is not just the kids, it is the parents and they go together [my emphasis]. this mother wanted her knowledge and experiences acknowledged. similar to others parents in the group, there was a concern that the school did not ask them about their approaches to doing mathematics. while some still went ahead and shared their knowledge with their children, they described a tension as their children were caught between the school approaches and those from home. for me, this is an area that needs to be addressed. while i have learned much from, with, and about children and their families, particularly through my work with the parents, what i see lacking is real dialogue between teachers and parents. here, i argue for the need to take a holistic approach to the mathematics education of marginalized students, an approach that takes into account their lived experiences and that brings together all the parties involved in their education. as the mother in quote above reminds us, parents and children go together. what do we need to do to get parents, children, teachers, and mathematics educators to go together? acknowledgment the data presented here comes primarily from the center for the mathematics education of latinos/as (cemela), funded by the national science foundation, grant no. esi-0424983. references acosta-iriqui, j., civil, m., díez-palomar, j., marshall, m., & quintos-alonso, b. (2011). conversations around mathematics education with latino parents in two borderland communities: the influence of two contrasting language policies. in k. téllez, j. moschkovich, & m. civil (eds.), latinos/as and mathematics education: research on learning and teaching in classrooms and communities (pp. 125–147). charlotte, nc: information age. amanti, c. (2005). beyond a beads and feathers approach. in n. gonzález, l. c. moll, & c. amanti, (eds.), funds of knowledge: theorizing practice in households, communities, and classrooms (pp. 131–141). mahwah, nj: erlbaum. cervantes-soon, c. g. (2014). a critical look at dual language immersion in the new latin@ diaspora. bilingual research journal, 37(1), 64–82. civil, m. (2001) parents as learners and teachers of mathematics: towards a two-way dialogue. presented at the seventh international conference of adults learning mathematics – a research conference, tufts university, medford, ma. july, 2000. in m. j. schmitt & k. safford-ramus (eds.), (2001). adults learning mathematics – 7: a conversation between researchers and practitioners (pp. 173–177). cambridge, ma: alm & ncsall. civil commentary journal of urban mathematics education vol. 7, no. 2 19 civil, m. (2002). everyday mathematics, mathematicians’ mathematics, and school mathematics: can we bring them together? in m. brenner & j. moschkovich (eds.), everyday and academic mathematics in the classroom. journal of research in mathematics education mongraph, no. 11 (pp. 40–62). reston, va: national council of teachers of mathematics. civil, m. (2007). building on community knowledge: an avenue to equity in mathematics education. in n. nasir & p. cobb (eds.), improving access to mathematics: diversity and equity in the classroom (pp. 105–117). new york, ny: teachers college press. civil, m. (2011a). lessons learned from the center for the mathematics education of latinos/as: implications for research with non-dominant, marginalized communities. in j. clark, b. kissane, j. mousley, t. spencer, & s. thornton (eds.), mathematics: traditions and [new] practices—proceedings of the 34th annual conference of the mathematics education research group (merga) and of the 23rd biennial conference of the australian association of mathematics teachers (aamt) (pp. 11–24). alice springs, australia. civil, m. (2011b). mathematics education, language policy, and english language learners. in w. f. tate, k. d. king, & c. rousseau anderson (eds.), disrupting tradition: research and practice pathways in mathematics education (pp. 77–91). reston, va: national council of teachers of mathematics. civil, m. (2012a). mathematics teaching and learning of immigrant students: an overview of the research field across multiple settings. in b. greer & o. skovsmose (eds.), opening the cage: critique and politics of mathematics education (pp. 127–142). rotterdam, the netherlands: sense. civil, m. (2012b). opportunities to learn in mathematics education: insights from research with “non-dominant” communities. in t. y. tso (ed.), proceedings of the 36th conference of the international group for the psychology of mathematics education (vol. 1, pp. 43–59). taipei, taiwan. civil, m. (2014). stem learning research through a funds of knowledge lens. cultural studies of science education. doi 10.1007/s11422-014-9648-2 civil, m., & andrade, r. (2002). transitions between home and school mathematics: rays of hope amidst the passing clouds. in g. de abreu, a. j. bishop, & n. c. presmeg (eds.), transitions between contexts of mathematical practices (pp. 149–169). boston, ma: kluwer. civil, m., & menéndez, j. m. (2011). impressions of mexican immigrant families on their early experiences with school mathematics in arizona. in r. kitchen & m. civil (eds.), transnational and borderland studies in mathematics education (pp. 47–68). new york, ny: routledge. civil, m., & planas, n. (2010). latino/a immigrant parents’ voices in mathematics education. in e. grigorenko & r. takanishi (eds.), immigration, diversity, and education (pp. 130–150). new york, ny: routledge. civil, m., & planas, n. (2012). whose language is it? reflections on mathematics education and language diversity from two contexts. in s. mukhopadhyay & w-m. roth (eds.), alternative forms of knowing (in) mathematics (pp. 71–89). rotterdam, the netherlands: sense. combs, m. c., dasilva iddings, a. c., & moll, l, c. (2014). 21st century linguistic apartheid: english language learners in arizona public schools. in p. w. orelus (ed.), affirming language diversity in schools and society: beyond linguistic apartheid (pp. 23–34). new york, ny: routledge gándara, p., & orfield, g. (2012). segregating arizona’s english learners: a return to the “mexican room”? teachers college record, 114(9), 1–27. gonzález, n. (2005). beyond culture: the hybridity of funds of knowledge. in n. gonzález, l. c. moll, & c. amanti (eds.), funds of knowledge: theorizing practice in households, communities, and classrooms (pp. 29–46). mahwah, nj: erlbaum. civil commentary journal of urban mathematics education vol. 7, no. 2 20 gonzález, n., moll, l. c., & amanti, c. (eds.) (2005a). funds of knowledge: theorizing practice in households, communities, and classrooms. mahwah, nj: erlbaum. gonzález, n., moll, l. c., & amanti, c. (2005b). introduction: theorizing practices. in n. gonzález, l. c. moll, & c. amanti, (eds.), funds of knowledge: theorizing practice in households, communities, and classrooms (pp. 1–24). mahwah, nj: erlbaum. martin, d. b., gholson, m. l., & leonard, j. (2010). mathematics as gatekeeper: power and privilege in the production of knowledge. journal of urban mathematics education, 3(2), 12– 24. retrieved from http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/95/57 planas, n., & civil, m. (2013). language-as-resource and language-as-political: tensions in the bilingual mathematics classroom. mathematics education research journal, 25(3), 361– 378. rios-aguilar, c., gonzález canché, m. s., & sabetghadam, s. (2012). evaluating the impact of restrictive language policies: the arizona 4-hour english language development block. language policy, 11(1), 47–80. ruiz, r. (2010). reorienting language-as-resource: anticipations and adumbrations of languageas-resource. in j. petrovic (ed.), international perspectives on bilingual education: policy, practice, and controversy (pp. 155–172). charlotte, nc: information age. setati, m. (2005). teaching mathematics in a primary multilingual classroom. journal for research in mathematics education, 36(5), 447–466. sheridan, t. e. (1995). arizona: a history. tucson, az: the university of arizona press. valdés, g. (1997). dual-language immersion program: a cautionary note concerning the education of language-minority students. harvard educational review, 67(3), 391–429. microsoft word final china vol 7 no 1.doc journal of urban mathematics education july 2014, vol. 7, no. 1, pp. 88–95 ©jume. http://education.gsu.edu/jume   erivn j. china is a second-year mathematics education doctoral student in the department of middle and secondary education, in the college of education at georgia state university, 30 pryor street, atlanta, ga 30303-3978; e-mail: echina1@student.gsu.edu. his research interests include post-secondary mathematics teacher preparation and preparing african american students for post-secondary stem disciplines. currently, mr. china is a mathematics instructor and program coordinator for learning support mathematics at southern crescent technical college. book review and then there was light: a book review of the brilliance of black children in mathematics: beyond the numbers and toward new discourse1 ervin j. china georgia state university had the privilege of studying science and mathematics with three mathematically brilliant black men. we had diverse life experiences—different socioeconomic statuses, different family dynamics, and different schooling experiences. nonetheless, we had the necessary standardized test scores to be admitted to college, where we chose to study mathematics. we began our journey with a promising start in calculus ii, a class of few freshmen. as first-year college students, our advanced placement provided us the rare opportunity to tutor seniors taking college algebra, pre-calculus, and calculus courses. we continued to excel throughout our college matriculation. when we were not tutoring our classmates or leading calculus recitations, we were mentoring young children at title i schools in the neighborhood. the four of us graduated at the top of our department and went on to pursue graduate studies in science, technology, engineering, and mathematics (stem) disciplines. indeed, one of us holds a terminal degree in aerospace engineering from a prestigious catholic university in the midwest. the other two also hold advanced degrees in aerospace engineering, with one working for “big oil” in the southwest and the other in the northeast for the united states government. i hold an advanced degree in mathematics and am currently pursuing a ph.d. in mathematics education at georgia state university. although our paths diverged after our undergraduate experience, brilliance continues to unite us. there are several definitions of brilliance, one being the                                                                                                                           1 leonard, j., & martin, d. b. (eds.). (2013). the brilliance of black children in mathematics: beyond the numbers and toward new discourse. charlotte, nc: information age. pp. 398, $45.99 (paper), isbn 978-1623960797 http://www.infoagepub.com/products/the-brilliance-of-blackchildren-in-mathematics i china book review journal of urban mathematics education vol. 7, no. 1 89 ability to exude great brightness or light. i am partial to this definition because i believe it most fitting that the universe brought the four of us together to study mathematics at the nation’s premiere school for educating black men—a school whose motto is et facta est lux (roughly translated as “and there was light”). i chose to include this narrative, because, like leonard and martin (2013), i wanted to begin the discussion of the mathematics achievement of black students with brilliance. the brilliance of black children in mathematics: beyond the numbers and toward new discourse (or brilliance) is a much-needed compilation of manuscripts that does just what the book’s subtitle states—it moves beyond the numbers and toward new discourse. when the subject of mathematical achievement of african american children arises, it is often accompanied by phrases such as “the statistics show that black students perform significantly lower than white and asian students on our nation’s standardized tests.” this book challenges the deficit perspective of the so-called black-white achievement gap, examines why such a gap exists, and suggests new and innovative ways that educators can help (black) children manifest their often obstructed brilliance in mathematics. brilliance is presented in five sections—cultural-historical perspectives, policy and black children’s mathematics education, learning and learning environments, student identity and student success, and preparing teachers to embrace the brilliance of black children. rather than summarizing each of these sections, i provide an analysis of a few themes that resonated with me in my reading. exemplars of brilliance brilliance begins with a thorough and informative history of the mathematically brilliant african american, benjamin banneker. the story of banneker, whom history credits with helping to build our nation’s capital and publishing a series of almanacs (among many other contributions to science, mathematics, and society more broadly), immediately serves to debunk the myth that mathematics is a white and male domain (stinson, 2013). the reader learns the rich history of brilliant african american mathematicians (in addition to banneker) like david blackwell and euphemia haynes, as these often unrecognized scholars are placed at the forefront and provided their rightful place alongside the likes of dedekind, erdös, and other mathematicians we read about in western mathematics texts. their contributions to the field of mathematics permanently dispose of thomas jefferson’s (1781) assertion that blacks are “in reason, inferior as they could scarcely be found capable of tracing and comprehending the investigations of euclid…and that in imagination, they are dull, tasteless, and anomalous” (p. 232). china book review journal of urban mathematics education vol. 7, no. 1 90 brilliance also delivers powerful counter-narratives of mathematically successful african americans. it provides the reader insight as to how mathematically successful african americans might negotiate du bois’s (1903/1996) doubleconsciousness or “sense of always looking at one’s self through the eyes of others” (p. 5) while learning mathematics and developing healthy mathematics identities. the volume explores racial identity and the role race and racism plays in african american students’ mathematics success. many of these students feel the burden to prove themselves academically in classrooms where they are often the only african american student in a sea of white, asian, and indian students. additionally, these mathematically successful african american students recall influential teachers that developed caring relationships with them and worked aggressively to challenge the “whites-only” face of mathematics. finally, the reader learns about the schooling and racialization experiences of the mathematically brilliant mrs. gant—an 83 year-old wife, mother, grandmother, great-grandmother, and “self-proclaimed math person” (gholson, 2013, p. 53). mrs. gant comes from a large family of high school-educated siblings and parents who valued education. mrs. gant recalls how mathematics, particularly algebra, was her favorite subject in school and how her teachers recognized her brilliance. indeed, she is so brilliant that her classmates often copied her work, thereby forcing her teachers to provide her with an exemption from assessments in an effort to prevent her classmates from cheating. but perhaps the most captivating part of mrs. gant’s story is that she unequivocally rejects the notion of the racial achievement gap between black students and their white counterparts. in her words, “i don’t believe it. …now it’s true it [sic] might be a lot of black students who don’t figure good, but then i would say there are a lot [of] black students who do” (gholson, 2013, p. 70). this grandmother, these african american mathematicians, and these mathematically successful african students defy the numbers and serve as a testament to the mathematical brilliance of the black child. a historical practice of policy and curricula that exclude early in the book, the contributing authors frame the discussion of brilliance by beginning with an intensive history and critical review of k−12 mathematics education in the united states (berry, pinter, & mcclain, 2013). with the ussr’s launch of sputnik in the 1950s, the united states began to reform mathematics curricula based on fears that u.s. children lacked the necessary mathematical skills to compete with the russians, thus threatening the nation’s future security. with the help of university mathematicians, the united states went from a mathematics curriculum that focused on minimum elementary-level competencies meant to prepare students for everyday work requirements to more rigorous china book review journal of urban mathematics education vol. 7, no. 1 91 mathematics competencies such as abstract algebra, topology, and set theory—a curriculum that the writers call “new math.” the problem was that black students neither had access to this new mathematics curriculum nor the new pedagogical techniques that came along with it. the millions of the dollars the government spent to identify the best and brightest young minds were, in effect, reserved for the best and brightest young white minds. in the face of such exclusion, mathematics teachers in segregated southern schools taught with demoded, hand-me-down textbooks from the white schools. in spite of these obstacles, they exhibited high-quality teaching and demanded greatness from their students. these segregated black schools offered advanced mathematics classes and required students to complete algebra i before they could graduate, further supporting the claim that these teachers and schools had high expectations of their students. some of these teachers, seeing the mathematical brilliance of their students, taught them mathematics that went beyond geometry and algebra, such as calculus. however, when school officials discovered that black schools were teaching mathematics content that the white schools were not, they quickly stopped the practice and eliminated such course offerings. in many southern schools, integration resulted in black students often being placed into “low-level” classes that did not prepare them for post-secondary education. in the segregated schools, these same students would have been enrolled in rigorous, more advanced mathematics courses. there, they would have had teachers who set high academic standards and realized their brilliance. berry and colleagues (2013) argue that this policy of exclusivity has had long-lasting negative effects on black students as evidenced by the racial composition of mathematics courses we see in u.s. classrooms today. in fact, i can recall being one of only a handful of black students in my advanced trig-analytics course and the only black student in my high school advanced placement (ap) calculus course. when educators find themselves consumed with achievement-gap rhetoric, i urge them to reject this deficit perspective and consider the history of mathematics education in this country that has long excluded the black child. to embrace her/his culture is to embrace her/his brilliance as i read brilliance, it became apparent that there are many ways we, as mathematics teacher educators, can begin preparing teachers to recognize the brilliance of black children, but the most commonly discussed approaches in this book involve a reform of mathematics policy, curriculum, and assessment (see e.g., matthews, jones, & parker, 2013; tawfeeq & yu, 2013; chahine, 2013). matthews and colleagues (2013) suggest beginning with culturally relevant and specific pedagogy and cognitively demanding mathematical tasks that have meaning in the lives of our (black) students. admittedly, i had mixed feelings about china book review journal of urban mathematics education vol. 7, no. 1 92 these ideas. i thought to myself, “what is culturally relevant pedagogy? are the authors suggesting that i teach the quadratic formula behind the backdrop of the latest 2 chainz song? if so, this approach deeply offends my sensibilities.” moreover, i thought “are the authors reifying the idea that plagues so many of my students that just because i don’t find something useful right now, it’s not worth knowing?” in an effort to better understand the authors’ point, i was compelled to read a book my doctoral advisor gave me entitled the dreamkeepers: successful teachers of african american children written by gloria ladson-billings (2009). in her words, “the primary aim of culturally relevant teaching is to assist in the development of a ‘relevant black personality’ that allows [black] students to choose academic excellence yet still identify with african and african american culture” (p. 20). she goes on to give an example of how a teacher might incorporate culturally relevant pedagogy: for example, let us examine how a fifth-grade teacher might use a culturally relevant style in a lesson about the u.s. constitution. she might begin with a discussion of the bylaws and articles of incorporation that were used to organize a local church or african american civic association. thus the students learn the significance of such documents in forming institutions and shaping ideals while they also learn that their own people are institution-builders. (p. 20) after reading this excerpt, culturally relevant teaching became clearer to me. when reading brilliance the reader will learn about the preparations of practicing teachers and how they, in an effort to become culturally relevant pedagogues, use a framework for culturally relevant, cognitively demanding mathematics tasks to “re-engineer” (matthews, jones, & parker, 2013) their mathematics classroom content and make mathematics meaningful for their students. the text also highlights the innovative algebra project curriculum that was implemented with a cohort of high school students in miami as they learned how to add and subtract integers (eraso, 2013). the contributing authors of brilliance suggest mathematics simply cannot be taught in a meaningful and effective way if it is not related to the student’s culture. when educators and policy makers modify their teaching practices, policies, and curricula to include culture, the brilliance of the black child, which we know is there, will begin to manifest itself. suggestions for future work brilliance focuses heavily on k−12 students, educators, curriculum, and policy. we read about the elementary schooling experiences of a grandmother (gholson, 2013), receive a thorough history of k−12 mathematics education in the united states (berry et al., 2013), and learn about school curricula that have china book review journal of urban mathematics education vol. 7, no. 1 93 historically excluded the life experiences and culture of the black child (berry et al., 2013). as an african american college mathematics instructor and program coordinator for developmental education, i would have liked for the discussions to include the mathematics experiences of college students and the history of the post-secondary culture that often neglects the brilliance of black college freshmen. many first-year college students place in remedial studies courses because of poor performance on standardized placement tests, and these students are overwhelmingly black and hispanic (bonham, 2012). how do these students navigate knowing that their respective colleges label them as mathematically deficient? furthermore, how do these students navigate a classroom space where they are simply a number—just one out of a room of 100 students, for example, in a lecture hall where there are few culturally responsive pedagogues on deck (jett, 2012)? it is my hope that critical mathematics education researchers who are interested in race and equity issues address some of these questions in their future work. final thoughts and conclusion in his collection of essays entitled the souls of black folk, william edgar burghardt du bois (1903/1996) opens with the statement, “between me and the other world, there is ever an unasked question…how does it feel to be a problem? i answer seldom a word” (p. 3). i now wonder how many of my teachers perceived me⎯a poor kid from one of the many “broken homes” on the south side of sumter, south carolina attending the newly built elementary school⎯as a problem; after all, i always felt like a problem. i had trouble focusing when it came time to take standardized tests. i could sense their frustration and disgust when i did not grasp the concepts taught in class well enough to complete my homework or when i scored poorly on end-of-unit assessments. i recall when my fifth grade teacher denied me the opportunity to play during recess after i, in his words, “blew off” his test. he never inquired about how i prepared for tests at home or what other factors may have contributed to my low performance. he did not even spend the forgone recess time reviewing with me what i had missed on the test. to him, i was just a problem. it was not until i entered middle school and experienced my first african american mathematics teacher that i realized my mathematical brilliance. this teacher, unlike many of my elementary school teachers, developed a caring relationship with me that reached beyond the classroom. she “set high expectations for academic success and disrupted school mathematics as a white institutional space” (stinson, jett, & williams, 2013, p. 227). to her, i was hardly a problem. i was a diamond in the rough but a diamond sure enough. china book review journal of urban mathematics education vol. 7, no. 1 94 the brilliance of black children in mathematics: beyond the numbers and toward new discourse is a powerful, eye-opening read that resonates with me because it offers a new and refreshing discourse about black children and their mathematics achievement. instead of focusing on the perceived achievement gap between black children and their white counterparts or viewing black children as problems in need of “fixing” (stinson et. al, 2013), this compilation begins with the brilliance of the black child. i found the section on student identity and student success particularly inspiring as i read the narratives of mathematically successful black students, many of whom earned degrees in stem disciplines. in reading about how these students developed healthy mathematics identities while learning to negotiate “what it means to ‘be african american’ in the context of doing mathematics” (mcgee, 2013, p. 250), i could not help but think about my classmate, peter,2 with whom i recently reconnected at my high school reunion. peter was placed in the low-achieving, technical preparatory track in high school. an administrator told him that he was not four-year college material and that he should consider going directly into the workforce upon graduation. but our guidance counselor,a black woman who happened to complete her undergraduate education at a historically black college,realized that peter was capable of more than settling for a minimum-wage job at the local factory. she realized his brilliance and encouraged him to apply to college. today, peter holds the bachelor of science degree in mathematics—a feat our discouraging administrator surely never dreamed peter brilliant enough to accomplish. i invite parents, especially those of the black children enrolled in special education or lower tracks, to read this book and use it as a tool as they question their children’s teachers and administrators and inquire why their children are not being equally counseled into gifted and talented programs. additionally, i invite all educators and administrators to read brilliance to see that african americans have always been achievers in mathematics and to share that legacy of achievement with their students. they will see that mathematics is in our blood, as it was in the blood of our ancestors (e.g. banneker, blackwell, fuller, henson, etc.) (leonard & beverly, 2013) and that african americans “have the inborn capacity to accomplish just as much as any nation of twelve million anywhere in the world ever accomplished” (du bois, 1935, p. 333). references berry, r. q., iii, pinter, h. h., & mcclain, o. l. (2013). a critical review of american k−12 mathematics education, 1900−present: implications for the experiences and achievement of black children. in j. leonard, & d. b. martin (eds.), the brilliance of black children in mathematics: beyond the numbers and toward new discourse (pp. 247–272). charlotte, nc: information age.                                                                                                                           2 the name peter is a pseudonym used to protect my classmate’s anonymity. china book review journal of urban mathematics education vol. 7, no. 1 95 bonham, b. r. (2012). developmental mathematics: challenges, promising practices, and recent initiatives. journal of developmental education, 36(2), 14–21. chahine, i. (2013). ethnomathematics in the classroom: unearthing the mathematical practices of african cultures. in j. leonard, & d. martin (eds.), the brilliance of black children in mathematics: beyond the numbers and toward new discourse (pp. 195–218). charlotte, nc: information age. du bois, w. e. b. (1935). does the negro need separate schools? journal of negro education, 4(3), 328–335. du bois, w. e. b. (1996). the souls of black folk (penguin classics ed.). new york, ny: penguin. (original work published 1903) eraso, m. (2013). adolescents learn addition and subtraction of integers using the algebra project’s curriculum process. in j. leonard, & d. martin (eds.), the brilliance of black children in mathematics: beyond the numbers and toward new discourse (pp. 151–169). charlotte, nc: information age. gholson, m. (2013). the mathematical lives of black children: a sociocultural-historical rendering of black brilliance. in j. leonard, & d. b. martin (eds.), the brilliance of black children in mathematics: beyond the numbers and toward new discourse (pp. 55–76). charlotte, nc: information age. jefferson, t. (1781). notes on the state of virginia, query xvi. retrieved from http://www.pbs.org/jefferson/archives/documents/frame_ih198154.htm jett, c. c. (2012). let’s produce culturally responsive pedagogues on deck. democracy & education, 20(2), 1–5. ladson-billings, g. (2009). the dreamkeepers: successful teachers of african american children. san francisco, ca: jossey-bass. leonard, j., & beverly, c. l. (2013). the history, brilliance, and legacy of benjamin banneker revisited. in j. leonard, & d. b. martin (eds.), the brilliance of black children in mathematics: beyond the numbers and toward new discourse (pp. 3–21). charlotte, nc: information age. leonard, j., & martin, d. b. (eds.). (2013). the brilliance of black children in mathematics: beyond the numbers and toward new discourse. charlotte, nc: information age. matthews, l. e., jones, s. m., & parker, y. a. (2013). advancing a framework for culturally relevant, cognitively demanding mathematics tasks. in j. leonard, & d. b. martin (eds.), the brilliance of black children in mathematics: beyond the numbers and toward new discourse (pp. 123– 150). charlotte, nc: information age. mcgee, e. o. (2013). growing up black and brilliant: narratives of two mathematically high-achieving college students. in j. leonard, & d. b. martin (eds.), the brilliance of black children in mathematics: beyond the numbers and toward new discourse (pp. 247–272). charlotte, nc: information age. stinson, d. w. (2013). negotiating the “white male math myth”: african american male students and success in school mathematics. journal for research in mathematics education 44(1), 69–99. stinson, d. w., jett, c. c., & williams, b. a. (2013). counterstories from mathematically successful african american male students: implications for mathematics teachers and teacher educators. in j. leonard, & d. b. martin (eds.), the brilliance of black children in mathematics: beyond the numbers and toward new discourse (pp. 221–245). charlotte, nc: information age. tawfeeq, d. a., & yu, p. w. (2013). methods of studying black students’ mathematical achievement. in j. leonard, & d. b. martin (eds.), the brilliance of black children in mathematics: beyond the numbers and toward new discourse (pp. 79-94). charlotte, nc: information age. journal of urban mathematics education july 2013, vol. 6, no. 1, pp. 28–34 ©jume. http://education.gsu.edu/jume victoria hand is an assistant professor of mathematics education in the school of education at the university of colorado, 249 ucb, boulder, co 80309; email: victoria.hand@colorado.edu. her research interests include culture, learning and identity in mathematics education, situative and critical perspectives on mathematics learning, and teacher noticing for equitable mathematics instruction. imani masters goffney is an assistant professor in the department of curriculum and instruction in the college of education, at the university of houston, 436 farish hall, houston, tx 77204; email: idoggney@uh.edu. her research focuses on equity and mathematics education and on interventions designed to improve its quality and effectiveness, especially for socially, linguistically, and ethnically diverse students. “all for one and one for all”: negotiating solidarity around power and oppression in mathematics education victoria hand university of colorado, boulder imani masters goffney university of houston in this essay, the authors, as participants of the privilege and oppression in the preparation of mathematics teachers educators conference, reflect on tensions inherent in standing with and speaking on behalf of communities in an attempt to build and signal solidarity with them. they describe this tension in relation to their membership in the community of researchers who study equity in mathematics education. a particular exchange that arose during whole group discussion at the conference seeded a conversation around other situations they have encountered in this community, and led to the development of a set of “cautionary tales” for the field. keywords: mathematics education research, power and oppression oming together as researchers, teacher educators, and at a basic level, as human beings, participants at the privilege and oppression in the preparation of mathematics teachers educators conference (prompte 1 ) were asked to interrogate and grapple with the multiple intersectionalities of individual privilege and oppression at different levels of social activity (e.g., personal, social, cultural, and institutional). in the process of doing this work, the facilitators from allies for change (http://www.alliesforchange.org/allytrainers.html) taught conference participants about the possibility and potential of being allies for one another, for mathematics teachers, and for children in schools (duncan-andrade & morrell, 2007; katsarou, picower, & stovall, 2010), as participants attempted to name and inter 1 privilege and oppression in the mathematics preparation of teacher educators (prompte) conference (funded by create for stem institute through the lappan-phillips-fitzgerald cmp 2 innovation grant program), michigan state university, battle creek, mi, october 2012. any opinions, findings, and conclusions or recommendations expressed herein are those of the authors and do not necessarily reflect the views of the funding agency. c http://www.alliesforchange.org/allytrainers.html hand & masters goffney all for one and one for all stinson, d. w., & spencer, j. a. (eds.). (2013). privilege and oppression in the mathematics preparation of teacher educators [special issue]. journal of urban mathematics education, 6(1). 29 rupt structures and processes that marginalize groups of people. while we (vicki and imani) see this ally work as critically important, there is also a tension inherent in standing with and speaking on behalf of communities in an attempt to build and signal solidarity with them (freire, 1970). the impact of such a stance is in part a reflection of the privilege the ally has garnered through predominant power structures. it also gives the impression that allies understand all of the experiences of the people and communities they stand for. in this essay, we reflect on this tension by describing a particular exchange that arose during whole group discussion at the conference, and relate this to other situations, or “cautionary tales,” we have encountered in our experience as researchers who study equity in mathematics education. the exchange several times during the workshop, a white, male, english-fluent participant attempted to speak on behalf of the group of participants. he was promptly reprimanded by the allies for change facilitators, who instructed us to speak only for ourselves and not on behalf of others. their point was to disrupt the privileged position that could make invisible realities and experiences present in the room, and to recognize that no one can truly put themselves in the shoes of another person. this participant expressed frustration with the way his comments were being received by the facilitators. he pointed out that his purpose was to build solidarity among us, as a group of scholars whose research is often positioned at the margins of mathematics education research. the public chastisement left the two of us (vicki and imani) feeling torn about the feeling of empowerment from the interruption of privilege, and the discomfort around the interpretation of our friend’s remarks by people who did not know him or his leadership in social justice mathematics education. we note that a key element of this tension is what the facilitators referred to as “intent versus impact,” in which one’s intentions may be honorable, but the impact of one’s actions can have (unintended) destructive consequences. throughout the conference, the facilitators encouraged us to deeply consider the impact of our actions, versus the intentions behind them. we view this type of reflection as inextricably bound up in ally work. in the following sections, we unpack three “cautionary tales” that emerged from our (vicki and imani’s) conversations around the situation previously noted. for each one, we identify a key tension and offer suggestions for our work as allies and mathematics educators. researchers who have come before us have articulated these tensions and suggestions; our point here is to link them to our work as allies and our growth as a research community. hand & masters goffney all for one and one for all stinson, d. w., & spencer, j. a. (eds.). (2013). privilege and oppression in the mathematics preparation of teacher educators [special issue]. journal of urban mathematics education, 6(1). 30 “our” view of equity as the community of mathematics education researchers who study issues of equity grows, we have increased opportunities to define concepts like “equity,” “diversity,” and “social justice” for the broader field (for a critical discussion see lubienski, 2008; lubienski & gutiérrez, 2008; gutiérrez, 2008). we note that this opportunity also has its drawbacks. our community, like any other community, operates within and through power structures. these structures are yoked to dominant hierarchies, even as we actively seek to disentangle ourselves from them (bourdieu, 1991). this situation has inevitably positioned some of us (albeit not necessarily white, middle-income, english-speaking men) with more clout in the field. as these individuals offer their perspectives on equity in research journals and at conferences, the circulation of these perspectives makes it difficult for members with less status to assert alternative ideas. this statement is not to say that some individuals possess more power than others, but that the proliferation of certain ideas itself organizes a power structure (foucault, 1980). thus, while our intention may be to enable our equity agenda to play a more prominent role in discussions within mathematics education, the impact of the discursive structures within our profession (such as tenure and academic publishing) limits the range of perspectives that come to be recognized as central to the equity or social justice agendas. suggestions for allies:  instead of providing a definition of equity, focus on the processes and outcomes you want to enable (cochran-smith, 2004; dime, 2007);  include the ideas of colleagues—in particular, junior scholars—in your work; and  play around with the concepts of equity, social justice, and diversity within your work and discuss the affordances and constraints of different conceptualizations of them (wager & stinson, 2012). the methods “we” use to study it “we” can imply consensus and agreement on both the substance and the methods of our research in equity in mathematics education. many of the participants at the conference shared a deep commitment to social justice; yet employ very different methods and strategies for working on these issues. variety across these approaches include studying the relation of learning mathematics to social action (bartell, 2013; frankenstein, 1983; gutstein, 2005; skovsmose, 1994a), the effects of positioning in the classroom (esmonde & langer-osuna, 2010; hand, 2010; turner, gutierrez, & sutton, 2011), the structure of classroom discourse practices (herbel-eisenmann, choppin, wagner, & pimm, 2012; moschkovich, hand & masters goffney all for one and one for all stinson, d. w., & spencer, j. a. (eds.). (2013). privilege and oppression in the mathematics preparation of teacher educators [special issue]. journal of urban mathematics education, 6(1). 31 2010), the affordances of teaching practices (aguierre & zavala, 2013; battey, 2013; masters-goffney, 2010; meaney, trinick, & fairhall, 2013; rubel & chu, 2012; turner et al., 2012), and others. accepting this range as natural and even welcome is a core tenet of social justice (north, 2006). however, we often train graduate students in a narrow range of approaches, without either experience in or at the very least discussion of other methods. we also limit our own exposure to other methods when we work within the same networks of individuals and draw primarily on their work in our research. while it is important to grow our cadre of researchers who analyze issues of power and marginalization in mathematics education (gutiérrez, 2007; martin, 2009; skovsmose, 1994b; stinson & bullock, 2012; tate, 1995; valero & zevenbergen, 2004), this focus on growth should not come at the expense of the natural hybridity that we seek to flourish. another way that we have experienced the marginalization of methodological approaches within our community is through questions about the evidentiary basis of our claims that are often derived from a lack of understanding about the philosophical and theoretical foundations of these approaches. for example, parks and schmeichel (2012) argue that researchers tend to shy away from analyses that link practices in mathematics classrooms and education to broader sociopolitical structures, in part, due to the lack of a venue in the field for discussions around the complexity of identity and power in learning. their argument was based, in part, on a twostaged review of mathematics education literature: broadly surveying research articles over a 10-year period that included descriptors related to “mathematics” and “race” and carefully reading research articles in the journal for research in mathematics education (jrme) between 2008 and 2011. in particular, parks and schmeichel found that more times than not research articles in jrme treated race and ethnicity as independent variables, and that most of the articles that did address race and ethnicity in substantive ways were found in the recent jrme special equity issue (three out of the five; january 2013). they attribute these tendencies to predominant discourse structures within the field of mathematics education that “[place] an additional burden on researchers who want to write about identities in detailed and political ways, because they must use valuable space to justify why this matters” (p. 247). suggestions for allies:  seek out methodological approaches across disciplines and frameworks and discuss these with students; and  be as detailed as possible about the relation between your research methods, framework, and specific research questions. hand & masters goffney all for one and one for all stinson, d. w., & spencer, j. a. (eds.). (2013). privilege and oppression in the mathematics preparation of teacher educators [special issue]. journal of urban mathematics education, 6(1). 32 why “you” should pay attention to it many of the participants share a common struggle with finding opportunities to discuss their work, given that it is often relegated to special themed issues in journals (e.g., jrme, special equity issue, january 2013) or strands at national conferences labeled as equity or social justice. often, the work also is removed from mainstream conversation in mathematics education where discussions continue to be focused on being “neutral” (martin, 2003). because these ideas are not often taken up centrally, as part of the field, it can be tempting for us to take advantage of any and all opportunities to advocate for attention being paid to social justice, equity, and diversity. examples of opportunities of this kind include serving as a discussant at national conferences, or as a reviewer for education research journals. when in this situation, it can be second nature for us to frame or assess the conversation from the perspective of an equity agenda, and to forego addressing the topic intended by the author. this situation again runs into issues of intent versus impact. first, while it is critically important to interrogate privilege and oppression in any research in mathematics education, doing so without care may inadvertently marginalize the presenters or authors’ work. second, it can weaken how other people view work on social justice if we are unable to build connections between this topic and other themes in mathematics education. finally, for authors who are members of underrepresented groups and early career academics, having their ideas and work ignored or significantly critiqued in the very public space of a national conference, or even in the confidential journal review process can reinforce their marginalized status. when any of these situations occur, we argue that they work against the interest of social justice because the author inadvertently pays the price for increasing the visibility of these issues. suggestions for allies:  carefully consider the identity and position of the authors or presenters whose work is being addressed, and  make explicit connections between the work being discussed and core issues of equity and social justice to help others learn new ways of seeing this work. final thoughts we encourage our colleagues as they enact positions as allies in the disruption of oppression in mathematics to take up the notions of “intent” and “impact” and carefully consider the implications these lenses can have on their activities. although many efforts around social justice, equity, and diversity in mathematics education research and teaching are well intended, the impact of these has been hand & masters goffney all for one and one for all stinson, d. w., & spencer, j. a. (eds.). (2013). privilege and oppression in the mathematics preparation of teacher educators [special issue]. journal of urban mathematics education, 6(1). 33 negative for particular groups of people, within particular contexts. in the three cautionary tales, we argued that advocating social justice and equity within mathematics education requires the field to allow equity to be broadly defined and to create opportunities for junior colleagues to join and have status in the discussions around these issues. we also argued for senior scholars in the field to carefully attend to the ideas and experiences of junior colleagues who are building expertise and find ways to value their work in public settings. additionally, we advocate for thoughtfully weaving equity, social justice, and diversity into the fabric of the field of mathematics education—in each teacher preparation course, throughout different research agendas, and in our service. using a dual lens of intent versus impact can provide a more inclusive set of ideas for deliberately attending to these issues and strengthen and build expertise in the field of mathematics education. references aguierre, j., & zavala, m. (2013). making culturally responsive teaching explicit: a lesson analysis tool. pedagogies: an international journal, 8(2), 163–190. bartell, t. g. (2013). learning to teach mathematics for social justice: negotiating social justice and mathematical goals. journal for research in mathematics education, 44, 129–163. battey, d. (2013). “good” mathematics teaching for students of color and those in poverty: the importance of relational interactions with instruction. educational studies in mathematics, 82, 125–144. bourdieu, p. (1991). language and symbolic power. boston, ma: polity. cochran-smith, m. (2004). walking the road: race, diversity, and social justice in teacher education. new york, ny: teachers college press. dime (2007). culture, race, power, and mathematics education. in f. lester (ed.), handbook of research on mathematics teaching and learning (2nd ed., pp. 405–434). charlotte, nc: information age. duncan-andrade, j., & morrell, e. (2007). the art of critical pedagogy: possibilities for moving from theory to practice in urban schools. new york, ny: peter lang. esmonde, i., & langer-osuna, j. m. (2010). power in numbers: student participation in mathematical discussions in heterogeneous spaces. journal for research in mathematics education, 44, 288–315. foucault, m. (1980). power/knowledge: selected interviews and other writings, 1972–1977. new york: the harvester press. frankenstein, m. (1983). critical mathematics education: an application of paulo freire's epistemology. journal of education, 165, 315–339. freire, p. (1970). pedagogy of the oppressed (m. b. ramos, trans.). new york, ny: herder and herder. gutiérrez, r. (2007). (re)defining equity: the importance of a critical perspective. in n. nasir & p. cobb (eds.), improving access to mathematics: diversity and equity in the classroom (pp. 37–50). new york, ny: teachers college press. gutiérrez, r. 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(1994b). towards a philosophy of critical mathematics education. boston, ma: kluwer. stinson, d. w., & bullock, e. c. (2012). critical postmodern theory in mathematics education research: a praxis of uncertainty. educational studies in mathematics, 80, 41–55. tate, w. f. (1995). returning to the root: a culturally relevant approach to mathematics pedagogy. theory into practice, 34, 166–173. turner, e., drake, c., mcduffie, a. r., aguierre, j., bartell, t. g., & foote, m. q. (2012). promoting equity in mathematics teacher preparation: a framework for advancing teacher learning of children's multiple mathematics knowledge bases. journal of mathematics teacher education, 15, 67–82. turner, e., gutierrez, r. j., & sutton, t. (2011). student participation in collective problem solving in an after-school mathematics club: connections to learning and identity. canadian journal of science, mathematics and technology education, 11, 226–246. valero, p., & zevenbergen, r. (eds.). 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(2012). teaching mathematics for social justice: conversations with educators. reston, va: national council of teachers of mathematics. journal of urban mathematics education december 2013, vol. 6, no. 2, pp. 81–85 ©jume. http://education.gsu.edu/jume james s. ewing is a fourth-year doctoral student in teaching and curriculum at the graduate school of education at syracuse university, 150 huntington hall, syracuse, ny 13244; email: jsewing@syr.edu. he taught elementary school in spain for 24 years. his research interests include elementary mathematics education, english language learners, and pre-service teachers. book review the standards for mathematical practice and hybrid spaces: a review of empowering science and mathematics education in urban schools 1 james ewing syracuse university mpowering science and mathematics education in urban schools by edna tan and angela calabrese barton with erin turner and maura varley gutiérrez (2012) is a useful and timely book for a broad range of educators. tan and calabrese barton have backgrounds in science education, while mathematics teaching and learning is included with contributions from turner and varley gutierrez, who both have backgrounds in mathematics education. throughout the book, the authors discuss how teachers might provide equitable access for “urban” students in mathematics and science by creating hybrid spaces—spaces where schools connect with students’ personal and home lives. the authors’ main argument woven throughout the book: to achieve social justice in mathematics and science it is necessary to focus on equity (not equality) and empowerment. tan and calabrese barton intentionally use the phrase “empowering learning environments” as a means to envision an education that engages youth in learning and using mathematics and science not only as a tool but also a means for change (p. 14). in each of the chapters, there are different iterations of the argument that literacy in mathematics and science should aim for more than mere functional literacy but also, and perhaps more importantly, critical literacy. they support their argument with four practical studies described as “hybrid spaces in action” (p. 17). collectively, the studies are grounded in critical ethnography—a methodological approach intended to expose injustices and break down or blur the researcher/researched binary. as an elementary school teacher in spain for 24 years, and now as a current doctoral student and future teacher educator and researcher who has research interests in elementary mathematics education and english language learners, the 1 tan, e., calabrese barton, a., turner, e. e., & varley gutiérrez, m. (2012). empowering science and mathematics education in urban schools. chicago, il: the university of chicago press. pp. 224, $29.00 (paper), isbn 978-0226037981 http://press.uchicago.edu/ucp/books/book/chicago/e/bo13181197.html e http://education.gsu.edu/jume mailto:jsewing@syr.edu http://press.uchicago.edu/ucp/books/book/chicago/e/bo13181197.html ewing book review journal of urban mathematics education vol. 6, no. 2 82 title of the book alone caught my interest. many of the pre-service teachers (psts) whom i have taught in my elementary mathematics methods courses are from the suburbs. during their field and student teaching placements in urban schools, they struggle to provide environments that nurture their urban students’ empowerment. furthermore, many of the psts have shared that their own elementary mathematics learning experiences were disconnected from their personal and home lives. therefore, i seek out literature that offers suggestions on how teachers might connect mathematics teaching and learning to students’ lives as well as how teachers might create learning environments that fill students with a sense of (self) empowerment. during my methods course, psts are assigned literature to read which they are to connect to the eight standards for mathematical practice as outlined by the common core state standards initiative (ccss). 2 these practices are used as an overall guiding framework for the methods course. here, i use the standards throughout the review to illustrate both the usefulness and timeliness of the book as well as how mathematics classroom that are hybrid spaces might assist in achieving the objectives of the standards for mathematical practice. ccss for mathematical practice mp1 – make sense of problems and persevere in solving them mp2 – reason abstractly and quantitatively mp3 – construct viable arguments and critique the reasoning of others mp4 – model with mathematics mp5 – use appropriate tools strategically mp6 – attend to precision mp7 – look for and make use of structure mp8 – look for and express regularity in repeated reasoning chapters 1 and 2 – setting the stage to set the stage, tan and calabrese barton (2012) begin by providing vignettes about two science teachers who changed the suggested district curriculum in order to develop and teach lessons that connected to students’ lives. they show that students were empowered by the lessons because they were allowed to discover science for themselves rather than “practicing the routines of knowledgeable others” (p. 12). tan and calabrese barton contend that the discourse of mathematics (and science) for all “needs to be recast to be emergent of the interest, 2 for complete details of the eight standards for mathematical practice, see http://www.corestandards.org/math/practice. http://www.corestandards.org/math/practice ewing book review journal of urban mathematics education vol. 6, no. 2 83 needs, concerns, locations, and conditions of those who participate” (p. 11). rather than stripping students’ lived experiences from these “technical” discourses, tan and calabrese barton suggest making the discourses accessible to all by infusing students’ lives into the discourse as a “process of cultural production” (p. 10). in these hybrid spaces of learning the lines that too often separate schooling from the lived experiences of students (and teachers) are blurred if not altogether erased—and yes, even within the technical discourses of mathematics and science. curriculum changes made by the teachers described in the vignettes were in alignment to mp1: make sense of problems and persevere in solving them. many of the psts that i have observed misinterpret this practice to mean that it is their role to explain the mathematical tasks to their students over and over until they “make sense” of them. during these well-intended efforts, teachers tend to offer too much support and decrease the rigor of the mathematics. if students become too dependent on the teacher, they often do not persevere in solving the problems. in contrast, tan and barton’s (2012) equitable approach of empowering students by connecting the discourses of mathematics and science to students’ lives encourages them to make sense of problems and persevere. when students’ lives become part of the mathematics and science discourse, students are more willing to persist with problems until they are solved. chapters 3, 4, 5, and 6 – reporting four studies in chapter three, turner (2012) discusses findings from her study of predominately african american sixth graders from an overcrowded school. in a class discussion, students complained that their school’s facilities were inferior to a magnet school located in the same building with a high population of white students. the teacher and students designed and completed mathematical projects to support their hypothesis that the resources in their part of the building were inferior. these projects were examples of “critical mathematical agency” (p. 53): students not only learning a deep understanding of mathematics but also applying mathematics to right an injustice. one student’s project posited that the girls’ bathroom shared among students and adults was too small to meet their needs. she used mathematical concepts such as area, ratios, and fractions to support her hypothesis. turner also provides an example of when critical mathematical agency was not possible because it did not fit the guidelines of being mathematically rigorous (i.e., critical mathematics is not less rigorous mathematics). two boys wanted their project to focus on the idea that there were too many poles where they played basketball. although this problem tied into their personal lives and was an act of social injustice—other schools have more space to play basketball— counting poles was not a rigorous mathematical approach for sixth graders. ewing book review journal of urban mathematics education vol. 6, no. 2 84 turner’s explanation of critical mathematical agency aligns with mp4: model with mathematics. in chapter four, tan and calabrese barton (2012) report on how a seventhgrade science teacher told stories, referred to as narrative pedagogy, to capture her students’ interests. they make the point that in “traditional” science classes the textbook (and/or teacher) is typically the authority, whereas storytelling introduces students to multiple points of view. thus narrative pedagogy encourages students to be more critical as they determine which of the stories might be the most accurate. tan and calabrese barton find that storytelling builds stronger ties between and among the teacher, the students, and the content. here, the teacher initiates a story and students are able to critique her (or his) story and construct their own stories using the discourses of mathematics and science. tan and calabrese barton contend that students are empowered “when individual narratives are woven into the educational content of the curriculum” (p. 81). the focus of this chapter, narrative pedagogy, aligns with mp3: construct viable arguments and critique the reasoning of others. in chapter five, tan and calabrese barton (2012) provide details of another example in which students are empowered through hybrid spaces. a community club allowed youth to use “slang” and music in their science video projects. as opposed to traditional classes where the teacher is the expert and students listen passively these students acquired an expertise in their projects as a result of the teacher linking science to the students’ lives. as previously noted, being empowered in hybrid spaces has implications for mp1: make sense of problems and persevere in solving them. it stands to reason that students cannot make sense of problems and persevere if their culture is not considered. thus the argument by tan and calabrese barton that teachers should consider urban students’ culture is both appropriate and consistent with mp1. in chapter six, varley gutiérrez (2012) discusses her findings from a study about fifth-grade girls who protested the closing of their school in which they calculated the time and money it would take for students to travel to the new school. the students constructed a persuasive argument to convince the school board to change their decision. in order to do so, the girls had to understand the board’s argument at a deep level to develop counter arguments. the mathematical activities described in this chapter align with several of the mps. for example, being able to understand the school board’s point of view and developing a viable argument to convince them not to close the school aligns with mp3: construct viable arguments and critique the reasoning of others as well as mp4: model with mathematics. ewing book review journal of urban mathematics education vol. 6, no. 2 85 chapter 7 – summarizing the book tan and calabrese barton (2012) conclude the book by summarizing the importance of empowering youth by creating hybrid spaces in mathematics and science classrooms. according to them, there has been recent attention in the literature about such spaces (see, e.g., gutiérrez, baquedano-lopez, & tejeda, 1999), especially as a proposed solution for closing the so called “achievement gap” between white students and students of color. nonetheless, tan and calabrese barton claim that their book is one of the few that goes beyond explanations and offers concrete examples of students actually engaged in hybrid spaces in mathematics and science. concluding thoughts i have shown how the mathematics (and science) activities and learning environments (in school and out of school) discussed throughout empowering science and mathematics education in urban schools might be aligned with some of the standards for mathematical practice. here, i have highlighted only a few such cases; in truth, a close reading of the book reveals that hybrid mathematics (and science) classrooms align with all eight of the standards. but these hybrid spaces go beyond the standards in an important way: hybrid spaces can/do facilitate students (and teachers) empowerment. for example, the use of tools in mathematics is discussed in mp5: use appropriate tools strategically. the students (and teachers) described throughout the book did indeed use tools appropriately and strategically. but the authors illustrate how students (and teachers) might go deeper; students (and teachers) not only use tools, mathematics and science becomes a tool. in the end, empowering mathematics and science hybrid learning spaces are those that engage youth in learning and using mathematics and science as both a tool and a context for social change (p. 14). i strongly recommend this book to teachers, teacher educators, parents, and anyone interested in looking for ways to (self) empower youth in and through the discourses of mathematics and science. references gutiérrez, k., baquedano-lopez, p., & tejeda, c. (1999). rethinking diversity: hybridity and hybrid language practices in the hybrid space. mind, culture, & activity: an international journal, 6, 286–303. tan, e., calabrese barton, a., turner, e. e., & varley gutierrez, m. (2012). empowering science and mathematics education in urban schools. chicago, il: the university of chicago press. journal of urban mathematics education december 2017, vol. 10, no. 2, pp. 66–105 ©jume. http://education.gsu.edu/jume laurie h. rubel is a professor in the department of secondary education at brooklyn college of the city university of new york, 2900 bedford avenue, brooklyn, ny 11210; email: laurie.rubel@gmail.com. her research interests include equity in mathematics education, probabilistic thinking, and teaching mathematics for spatial justice. equity-directed instructional practices: beyond the dominant perspective laurie h. rubel brooklyn college city university of new york in this article, the author synthesizes four equity-directed instructional practices: standards-based mathematics instruction, complex instruction, culturally relevant pedagogy (crp), and teaching mathematics for social justice (tmfsj). the author organizes these practices according to the dominant and critical axes in gutiérrez’s (2007a) equity framework. among 12 teachers from 11 schools in a large urban school district, the author presents case studies of 3 teachers who excelled with the aforementioned dominant equity-directed practices but struggled with the critical practices of connecting to students’ experiences called for in crp and critical mathematics called for in tmfsj. the analysis explicitly explores the role of whiteness in these struggles. the author presents implications and recommendations for mathematics teacher education on how to better support teachers for equitable teaching that includes these critical equity-directed practices. keywords: complex instruction, culturally relevant pedagogy, standards-based instruction, teaching mathematics for social justice, urban mathematics education ost black and latinx1 students in u.s. cities attend schools hyper-segregated2 by race and socioeconomic class (milner, 2013; orfield, kuscera, & siegelhawley, 2012). school segregation—currently, accelerated by neoliberal processes of gentrification—is confluent with inequalities in teacher qualifications, experience, and turnover rates; advanced course offerings; money spent per student and condition of facilities; as well as deficit orientations to students and their families and communities (anderson, 2014; kitchen & berk, 2016; lipman, 2016; martin & larnell, 2013). additionally, there is a mismatch between the students in urban schools and a teaching force that is largely white and middle-class (chazan, brantlinger, clark, & edwards, 2013; martin, 2007). despite these structural and systemic inequalities, if school achievement falls short compared to “better” resourced schools (often white, suburban schools), differences are then typically at 1 in some places in this article, i refer inclusively to these and other marginalized groups as “students of color.” 2 frankenberg, siegel-hawley, and wang (2010) define hyper-segregated schools as those with at least 90% of its students from “racial/ethnic minority groups,” or at least 90% of its students white. m http://education.gsu.edu/jume mailto:laurie.rubel@gmail.com rubel equity-directed practices journal of urban mathematics education vol. 10, no. 2 67 tributed to the students and their families and communities for what is perceived as a lack of ability, effort, “grit,” values, or parenting (battey & franke, 2015; martin, 2009b; milner, 2012; nasir & de royston, 2013; pollak, 2013). hyper-segregated schools serving black and/or latinx students in u.s. cities are vulnerable to a cycle in which teachers’ deficit views of students fuel low expectations (martin, 2009b; milner, 2012; rousseau & powell, 2005; stinson, 2006). low expectations are known to manifest in didactic pedagogy organized around remediation, rote drill and practice, and preparation for standardized tests (anyon, 1997; grant, crompton, & ford, 2015; rist, 1970/2000; thadani, cook, griffis, wise, & blakey, 2010), even though such pedagogy is less effective for students from marginalized, underserved groups (e.g., diversity in mathematics education center for learning and teaching, 2007; franke, kazemi, & battey, 2007; ladsonbillings, 1997; silver, smith, & nelson, 1995; spencer, 2009). weak results on external assessments reinforce deficit views about students to reboot a cycle of low expectations for students and about teaching (nasir, cabana, shreve, woodbury, & louie, 2014; rubel & chu, 2012). supporting teachers in “urban” schools to acknowledge and modify their deficit views of students, increase expectations for students and for their own teaching, and develop or improve a robust set of equitydirected instructional practices disrupts this cycle (aguirre et al., 2013; rubel & chu, 2012; silver & stein, 1996; turner et al., 2012). my goal is to present a research-based argument focused on teaching mathematics in hyper-segregated urban schools that moves away from a “failure-focused” master narrative (martin & larnell, 2013, p. 376). i begin with an overview of research about a set of four equity-directed instructional practices advocated for urban schools and synthesize these practices using gutiérrez’s (2007a) equity framework consisting of dominant and critical dimensions. next, i present three cases of white teachers—those who demonstrated excellence with equity-directed practices that correspond to dominant dimensions of equity, on the one hand, but struggled with practices that correspond to critical dimensions, on the other. the goal is not to pin responsibility for systemic inequities on any individual teachers but rather i heed dutro, kazemi, balf, and lin’s (2008) claim that “teachers’ attempts—with all of their flaws and complexity—can provide rich texts for teachers to study collectively” (p. 295) and present this analysis accordingly. [facing] race in (urban) mathematics education i foreground “race” in this literature review because of the significance of whiteness in the united states in reproducing subordination and widening society’s opportunity gaps in and through mathematics education (battey & leyva, 2016; martin, 2009a, 2009b, 2012; nasir, snyder, shah, & ross, 2012; spencer, 2009; stinson, 2006). there is an opportunity gap of inequitable access—to high quality rubel equity-directed practices journal of urban mathematics education vol. 10, no. 2 68 mathematics curricula, teachers, instruction, textbooks, technology, and more— which contributes to a further widening of broader social inequities (gutiérrez, 2008). as an example, consider how african american eighth-grade students are less likely to be recommended for algebra, even after controlling for performance in mathematics (faulkner, stiff, marshall, nietfeld, & crossland, 2014). because of the association between advanced mathematics coursework and wage earning (battey, 2013a), hindrance away from algebra bears clear economic significance for african americans (and other marginalized racial groups) and is part of the systematic maintenance of their subordination. more generally, whiteness tacitly positions white people, their experiences, and their behaviors as superior (battey & leyva, 2016; martin, 2009b), and it is supported by a set of corollary principles that function as “tools of whiteness” (picower, 2009, p. 204). for instance, the ideological principle of the united states as meritocracy is understood by many to be a central feature of american society, dictating that a combination of hard work and talent, or as cast in recent years, “grit” (duckworth, peterson, matthews, & kelly, 2007), yields success. equivalently, the principle of meritocracy also dictates that lack of success is a result of a lack of effort or ability (e.g., battey & franke, 2015; martin, 2009b). this principle functions as a tool of whiteness in how it ignores “systemic barriers and institutional structures that prevent opportunity and success” (milner, 2012, p. 704) as well as institutional structures that facilitate opportunities and the distribution of rewards not according to merit but instead according to race and social background (bowles & gintis, 2002; mcintosh, 1988). a second ideological principle that functions as a tool of whiteness is that of color-blindness (bonilla-silva, 2003; martin, 2009b; ulluci & battey, 2011). teachers who claim color-blindness—that is, they claim to not notice the race of their students—are, in effect, refusing to acknowledge the impact of enduring racial stratification on students and their families (martin, 2008). as ladson-billings (1994) contends: given the significance of race and color in american society, it is impossible to believe that a classroom teacher does not notice the race and ethnicity of the children she is teaching. further, by claiming not to notice, the teacher is saying that she is dismissing one of the most salient features of the child’s identity and that she does not account for it in her curricular planning and instruction. (p. 33) seemingly opposite to colorblindness is the “i can’t relate” principle (picower, 2009). unlike the colorblind stance, through this relate principle, teachers selfidentify as essentially different from their students. yet their constructions of those differences are typically cast in terms of deficit constructions about students, their places, and their families. all of these ideological principles—the myth of the meritocracy, colorblindness, and “i can’t relate”—function as tools of whiteness by ab rubel equity-directed practices journal of urban mathematics education vol. 10, no. 2 69 solving teachers from adopting new instructional practices that are proposed to further equity (e.g., stinson, 2006), engaging in processes of reflection about equity (e.g., rousseau & tate, 2003), and confronting fears of people of color to learn about students and their communities (e.g., aguirre, 2009). whiteness ideologies become amplified when an achievement lens is used to measure the quality of teaching in urban schools, yielding teacher caricatures in two forms (martin, 2007). “teacher as missionary” (martin, 2007, p. 13), promoted in popular depictions of urban schools, refers to the notion of the transformation or taming of struggling black and latinx students as a result of the sacrifices and efforts of a white teacher. a related caricature is “teacher as cannibal” (martin, 2007, p. 14), referring to a teacher who focuses solely on mathematics content, with little or no attention to relating to students or teaching mathematics that builds on or connects to students’ social realities. given this pair of undesirable caricatures, expanding the field’s understanding of quality mathematics teaching in hyper-segregated, urban secondary schools remains of pressing concern. next, i review four instructional practices that are advocated for mathematics teaching in (urban) schools. equity-directed instructional practices in mathematics i highlight equity-directed instructional practices from four models of progressive pedagogy that are typically recommended for the urban schools context: (a) standards-based mathematics instruction (sbmi; national council of teachers of mathematics, 2000), (b) complex instruction (ci; cohen, 1994; cohen & lotan, 1995; lotan, 2006), (c) culturally relevant pedagogy (crp; ladson-billings, 1994, 1995a, 1995b), and teaching mathematics for social justice (tmfsj; gutstein, 2003, 2006). here, and in the broader project through which the research was conducted, i identified one central instructional practice from each pedagogical model: (a) teaching for understanding from sbmi; (b) fostering multidimensional participation from ci; (c) connecting mathematics content to students’ experiences from crp; and (d) providing opportunities for using mathematics to read and write the world from tmfsj. in brief, i selected these specific practices because of the way that they illuminate the nested relationships among these pedagogies. all four of these pedagogies rely on teaching for understanding as a basic cornerstone. complex instruction then extends teaching for understanding with an articulated emphasis on participation. crp further extends complex instruction by including a focus on the cultural context of teaching and learning. finally, teaching mathematics for social justice accentuates crp’s engagement with cultural contexts but through a lens of power, using mathematics explicitly to analyze and respond to social injustices. in the discussion that follows, i situate each practice in the respective literature in terms of its particular theory of learning and emphasize the obstacles identified in the literature especially salient for the context of hyper-segregated urban rubel equity-directed practices journal of urban mathematics education vol. 10, no. 2 70 schools. finally, i utilize gutiérrez’s (2007a) equity framework to synthesize the four practices into a pair of dominant and a pair of critical equity-directed practices. sbmi – teaching for understanding. standards-based mathematics instruction (nctm, 2000) conceptualizes learning as engagement with mathematics that results in conceptual understanding. this orientation to teaching mathematics favors the understanding of mathematical concepts over mere fluency with algorithms and facts (carpenter & lehrer, 1999; hiebert & carpenter, 1992). mathematical understanding develops as part of social, discursive processes of conjecture, justification, and reasoning (gonzález & dejarnette, 2015; selling, 2016; sfard, 1998; zahner, velazquez, moschkovich, vahey, & lara-meloy, 2012). because of its dialogic nature, teaching for understanding implies that a classroom’s social culture is characterized by an emphasis on students’ mathematical thinking, students’ autonomy in choosing solution methods, the utilization of mistakes as learning opportunities, and intellectual authority residing in the mathematics itself (hiebert et al., 1997). to organize a classroom around understanding, a teacher must develop social and socio-mathematical norms that support and sustain student participation in such a social culture (bennett, 2014; cobb, yackel, & mcclain, 2000). the literature presents examples of teaching for understanding in mathematics in urban schools, characterized by high expectations for students (e.g., bonner, 2014; jamar & pitts, 2005; stinson, jett, & williams, 2013) and cognitively demanding mathematical tasks (e.g., boaler & sengupta-irving, 2016; kitchen, depree, celedón-pattichis, & brinkerhoff, 2007; silver, smith, & nelson, 1995; walker, 2012). even though it could be considered less accessible to students whose first language is not english, as long as the focus is on underlying mathematical ideas and not accuracy or fluency of linguistic production, teaching for understanding is advocated as especially productive for all students, including english language learners (moschkovich, 2013; zahner et al., 2012). a lesson’s mathematical task is essential to teaching for understanding, given that tasks directly determine the kinds of mathematical work the students will engage with and how (henningsen & stein, 1997; stein, grover, & henningsen, 1996). yet across classrooms in the united states, students spend more time on low-demand mathematical tasks—that is, learning mathematics by practicing procedures (boston & wilhelm, 2015; stigler & hiebert, 2004), as part of traditional instruction comprised of lecturing and drill with practice (mckinney, chappell, berry, & hickman, 2009). even when teachers have access to standards-based curricula, they do not always opt to implement the high-level tasks (boston & smith, 2009). especially relevant to the context of hyper-segregated urban schools, teachers’ views about students’ mathematical capabilities play a central role in their task selection and in the mathematical opportunities they provide (battey & franke, 2015; cobb & jackson, 2013; jackson, gibbons, & dunlap, 2017; wilhelm, munter, & jackson, 2017). teaching for understanding demands that teachers view rubel equity-directed practices journal of urban mathematics education vol. 10, no. 2 71 their students as possessing the prerequisite mathematical skills, literacy abilities, and problem-solving dispositions, a direct challenge to prevalent constructions of black and latinx youth. ci – multidimensional participation. a view of learning that emphasizes participation considers students as guided by the teacher as expert, knowledge as an aspect of practice, and knowing as inherently tied to participation (lave & wenger, 1991; sfard, 1998). such a perspective emphasizes that learners are always in a state of flux, that actions might be unsuccessful but are not tied to them as individuals or indicative of permanent, individual traits (sfard, 1998). the range of mathematical opportunities provided by the teacher, however, determines opportunities for participation, quality of student participation, and perceptions of competence (gresalfi, martin, hand, & greeno, 2009; wilhelm, munter, & jackson, 2017). instead of viewing diversity among students in terms of challenges it might present, ci reframes diversity as a resource, leveraging a construct known as “multidimensionality” (cohen & lotan, 1995). in a classroom with a range of mathematical opportunities, a wide array of mathematical practices is valued, such as asking good questions, employing different mathematical representations, explaining ideas, generalizing, justifying, or revising methods (boaler & greeno, 2000; boaler & staples, 2014). multidimensionality offers greater breadth for mathematical competence (gresalfi et al., 2009)—in terms of what students are accountable for and to whom (dunleavy, 2015) as well as what kinds of agency they can exercise (gonzález & dejarnette, 2015). essentially, multidimensionality supports engagement by broadening the ways that students can enact competence and creates pathways to success for more students (langer-osuna, 2016). a growing set of examples in the research literature ties mathematical achievement at urban schools to instruction that emphasizes multidimensionality (boaler & staples, 2008, 2014; dunleavy, 2015; horn, 2012; nasir et al., 2014). teachers’ views about black and latinx students as learners, their families, and communities impact not only the kinds of mathematical tasks they make available but also the kinds of mathematical participation they make available (battey, 2013b; battey & franke, 2015; cobb & jackson, 2013; grant et al., 2015; jackson, 2009; jackson et al., 2017; wilhelm et al., 2017). in addition, teachers’ beliefs about students and about learning underlie how they distinguish between mathematical and cultural activity, and how they determine which forms of participation will be considered suitable for their classroom (hand, 2012). teachers often view participation of marginalized students as off-task, unproductive, or distracting, even when it reflects students’ membership of and competence in another social context, unbeknownst to the teacher (hand, 2010). the perception of order as a prerequisite to learning constrains individual teachers’ views about classroom participation and leads to didactic teaching (golan, 2015; ladson-billings, 1997; nasir, hand, & taylor, 2008). school policies com rubel equity-directed practices journal of urban mathematics education vol. 10, no. 2 72 mon to urban schools in the united states such as uniform dress requirements communicate a view of students as needing to be controlled (battey & leyva, 2016). “no excuse” policies popular in urban schools include intensive discipline systems and have been described as militaristic in the way that they dictate parameters for student behavior. furthermore, overt messaging that announces to students that they are “smart” (battey & leyva, 2016) or the increasing branding of urban schools using names that include words like “success,” “achievement,” “aspire,” “strive,” or “ascend” communicate deficit views of students that likely impact teachers’ choices about the range of options they provide for classroom participation in mathematics. crp – connecting mathematics instruction to students’ experiences. cultural perspectives view learning as situated, as the “acquisition throughout the life course of diverse repertoires of overlapping, complementary, or even conflicting cultural practices” (nasir, rosebery, warren, & lee, 2014, p. 686). youth encounter a range of cultural practices in their school day, through their activities outside of school, and across the course of experiences in their lives (gutiérrez & rogoff, 2003). every cultural practice has its own stance and is governed by its own set of purposes, symbols, and discourses, which then demands negotiation (rogoff, 2003). teachers can recruit this negotiation process for school learning by building on or connecting to students’ cultural practices, which, without interrogation, are typically organized according to white, middle-class cultural practices (aguirre et al., 2013; leonard, 2008; matthews, 2003; tate, 1995; watson, 2012). crp emanates from a cultural perspective on learning, guided by the principle that curriculum and instruction must draw on students’ own cultural practices and not just the realities of others (ladson-billings, 1995a, 1997). emdin (2016) emphasizes the importance of integrating students’ out-ofschool experience into curriculum—an integration that he terms building “bridges to learning” by integrating students’ “contexts” with “content” using symbolic or tangible artifacts from places from which youth come as “anchors of instruction” (p. 291). as ladson-billings (1994) explains: culturally relevant teaching is a pedagogy that empowers students intellectually, socially, emotionally, and politically by using cultural referents to impart knowledge, skills, and attitudes. these cultural referents are not merely vehicles for bridging or explaining the dominant culture; they are aspects of the curriculum in their own right. (pp. 17–18) thus, a teacher might plan a lesson to focus centrally on specific cultural practices as objects of mathematical study (civil, 2002; kisker et al., 2012; mukhophadhyay, powell, & frankenstein, 2009; turner et al., 2012) or include representations that draw upon students’ cultural practices. for example, the algebra project curriculum (moses & cobb, 2001) builds on students’ experiences with public transit toward rubel equity-directed practices journal of urban mathematics education vol. 10, no. 2 73 developing understanding of integers, but the lessons were not organized to thematically investigate the local transit system. contextualizing mathematics in students’ cultural practices can lend a sense of practical utility to mathematics, often interests students, mediates between students’ formal and informal knowledge, and supports mathematics identity development (e.g., birky, chazan, & farlow morris, 2013; brenner, 1998; hubert, 2014; kisker et al., 2012; martin, 2000; vomvoridi-ivanović, 2012; walkington, petrosino, & sherman, 2013). however, superficial understandings of students’ cultural practices can backfire as an equity strategy when teachers do not anticipate how students will take up realistic aspects of contextualized mathematics (brenner, 1998; lubienski, 2000; tate, 2005). to connect mathematics to students’ experiences in productive ways, teachers need to develop ongoing practices around learning about their students—their students’ interests, everyday activities, heritage, home languages, and more (e.g., emdin, 2016; gay & kirkland, 2003; milner, 2003; vomvoridi-ivanović, 2012)—all the while embracing the tension that “one’s students can never be known” (gutiérrez, 2009, p. 13). emdin (2016) explains, “teaching more effectively requires embedding oneself into students’ contexts and developing weak ties with the community that will organically impact the classroom” (p. 139). he articulates three steps for teachers, beginning with being in students’ social spaces, engaging with those contexts, and then making connections between the out-of-school context and classroom teaching. entering and spending time in students’ spaces is known to be a central challenge for white teachers (chu & rubel, 2010), given that it requires negotiating commonly held fears of people of color and their spaces (picower, 2009), to conduct what are effectively “border crossings” (anzaldúa, 1987). this challenge explains the tendency of teachers to draw on their own cultural experiences when contextualizing mathematics in cultural experiences as opposed to their students’ (matthews, 2003; watson, 2012). the research literature describes interventions designed around supporting preand in-service teachers about how to connect mathematics instruction to students’ experiences or everyday practices (aguirre et al., 2013; moll, amanti, neff, & gonzalez, 1992; rubel, 2012; rubel & chu, 2012; taylor, 2012). observing students in out-of-school settings is time intensive for teachers (nasir et al., 2008). moreover, even in the context of professional development or teacher education interventions in which teachers have been supported in observing students in outof-school settings, their identification of embedded mathematical practices in those activities was rare, ostensibly because of the inherent complexity in doing so (bright, 2015; gainsburg, 2008; nicol, 2002; wager, 2012). tmfsj – critical mathematics. sociopolitical perspectives foreground learning as identity-work: “learners are always positioning themselves with respect to the doing of mathematics, their peers, their sense of themselves and their communi rubel equity-directed practices journal of urban mathematics education vol. 10, no. 2 74 ties, and their futures” (gutiérrez, 2013a, p. 53). key to viewing learning as identity-work is the recognition of how “knowledge, identity, and power are interwoven and arising from (and constituted within) social discourses,” social discourses which “privilege some individuals and exclude others” (gutiérrez, 2013a, p. 40). tmfsj is a pedagogical model drawn from sociopolitical perspectives about teaching and learning mathematics, guided by the goal that students be “prepared through their mathematics education to investigate and critique injustice, and to challenge, in words and actions, oppressive structures and acts—that is, to “read and write the world” (freire, 1970/1988) with mathematics” (gutstein, 2006, p. 4). contextualizing mathematics in sociopolitical terms has been found to interest students more broadly in mathematics (e.g., brantlinger, 2013; hubert, 2014; rubel, lim, hall-wieckert, & sullivan, 2016; winter, 2007). moreover, it has been shown to support students in learning to use mathematics to better understand the sociopolitical contexts of their own lives, so to be better able to effect change for themselves and for others (e.g., frankenstein, 1995, 2009; gutstein, 2003, 2006, 2016; leonard, brooks, barnes-johnson, & berry, 2010; rubel, lim, hallwieckert, & katz, 2016; rubel, lim, hall-wieckert, & sullivan, 2016; turner, gutiérrez, simic-muller, & díez-palomar, 2009). in so doing, students learn more mathematics: as gutstein (2007) explains, “the two sets of goals—mathematical and social justice—dialectically interact with each other” (p. 4). the dialectic between mathematical and social justice goals relies on leveraging and “channel[ing] students’ implicit critiques” of the social order that they (students) already have” (goldenberg, 2014, p. 126). research demonstrates the necessity of teachers’ knowledge of sociopolitical contexts and solidarity with students and their communities for critical mathematics (gutstein, 2003; terry, 2011), with key recommendations that social justice issues be selected in collaboration with students, as “generative themes” (e.g., gutstein, 2016). aside from the time demands posed by lesson planning for investigations that are local and context driven, mathematics teachers are typically inexperienced with teaching in this way (e.g., bartell, 2013; esmonde, 2014; gonzalez, 2009). furthermore, there is tension between such an approach and how contexts are typically employed in the teaching of mathematics. typically, lesson planning in mathematics is driven by a predefined, interrelated set of mathematical concepts and skills, and real-world applications of those skills are introduced as examples. planning a unit around a specific social justice issue is a different process, in which the issue and potential paths towards justice drive the mathematical content and not the other way around (frankenstein, 2009). the larger project in which this article is derived included activities to support teachers in studying their students’ communities in various ways toward generating hypotheses about issues of social justice for their students and their families that could be integrated with their curricular plans. analyses in the related literature at rubel equity-directed practices journal of urban mathematics education vol. 10, no. 2 75 tribute teacher reluctance about critical mathematics to perceived or experienced tensions with mathematical rigor (e.g., brantlinger, 2013; enyedy, danish, & fields, 2011; garii & rule, 2009; gutiérrez, 2009), or around the necessary simplification of both the social phenomenon in question and the related mathematics (dowling & burke, 2012). other analyses attribute teacher resistance to fears of overwhelming students (gainsburg, 2008), causing those who are marginalized or those whose families benefit from inequitable power relations to feel uncomfortable (aguirre, 2009; simic-miller, fernandes, & felton-koestler, 2015); or to disinterest in or ambivalence about explicitly introducing social issues into mathematics teaching (atweh, 2012; de freitas, 2008; simic-miller et al., 2015). mapping onto gutiérrez’s equity framework gutiérrez’s (2007a) equity framework in mathematics spatializes equity according to four dimensions along two axes. the framework’s dominant axis consists of access to and achievement in mathematics; dominant in its reflection of society’s status quo. access—to quality teaching, instructional resources, and a classroom environment that invites participation—is a precursor to mathematics achievement. along the second, critical axis, are identity and power in mathematics; critical in addressing students’ cultural identities and sociopolitical issues, from the perspective of marginalized groups (gutiérrez, 2007a). consideration of the role of identity in learning is a precursor to addressing issues of power as they relate to identity. teaching for understanding and multidimensionality are centrally and explicitly organized around increasing access and supporting achievement in mathematics and map onto the dominant axis in gutiérrez’s (2007a) framework. teaching for understanding and multidimensionality can be seen as implicitly addressing identity and power along equity’s critical axis as well. borrowing a distinction made by stinson and wager (2012), instructional practices that encourage opportunities for equitable participation can themselves be an avenue toward social justice, and therefore, a way to teach for social justice. instructional practices organized around increasing access often draw on processes that are endemic to students’ identities in terms of out-of-school or cultural activities and experiences, such as the apprenticeship (e.g., civil & khan, 2001; masingila, 1993) or assumptions of competence (e.g., nasir, 2005). likewise, an emphasis on understanding through multidimensional participation, relates implicitly to power in terms of questioning how smartness in mathematics gets constructed (hatt, 2007), whether mathematical ability is seen as innate or learned (dweck, 2006), and considering whose interests are being served when speed is privileged over communication; symbolic arguments are valued over visual ones; or a hierarchical, sequential view of mathematics is held over a connected, networked view (horn, 2012; martin, 2012). teaching for understanding and multidimensionality create space for students to develop “practice-linked rubel equity-directed practices journal of urban mathematics education vol. 10, no. 2 76 identities” as learners and doers of mathematics (nasir & hand, 2008), which can be especially significant for black and latinx students for whom such space can be new (e.g., martin, aguirre, & mayfield-ingram, 2013). there is a distinction, however, between how teaching for understanding and multidimensionality stop short at making matters of identity and power explicit in terms of mathematics content. boaler and staples (2008) amplify this distinction in clarifying that the mathematics content at railside, an urban school whose success they analyze, was not “sensitive to issues of gender, culture, or class” (and presumably race). they argue, “there is more than one road to equity” aside from “culturally sensitive materials” and that “such materials…. can be uncomfortable for teachers if they require cultural knowledge that they do not possess or if their classrooms are extremely diverse” (p. 640). instead, boaler and staples (2014) advocate for equity in terms of participation: railside students learned to appreciate the different ways that students saw mathematics problems and learned to value the contribution of different methods. ... as the classrooms became more multidimensional, students learned to appreciate and value the insights of a wider group of students from different cultures and circumstances. (p. 32) this “more than one road to equity” perspective contrasts with style’s (1988, p. 1) “windows” and “mirrors” metaphor that explains how education needs to include “window frames in order to see the realities of others” and “mirrors in order to see his/her own reality reflected.” as style (1988) claims: knowledge of both types of framing is basic to a balanced education which is committed to affirming the essential dialectic between the self and the world. ... school curriculum is unbalanced if a black student sits in school, year after year, forced to look through the window upon the (validated) experiences of white others while seldom, if ever, having the central mirror held up to the particularities of her or his own experience. (pp. 1–4) gutiérrez (2007a) elaborates: “the goal is not to replace traditional mathematics with a pre-defined ‘culturally relevant mathematics,’ but rather to strike a balance between the number of windows and mirrors provided to any given student in his/her math career” (p. 3). without such a balance, “when someone with the authority of a teacher, say, describes the world and you are not in it, there is a moment of psychic disequilibrium, as if you looked in the mirror and saw nothing (rich, 1986)” (p. 3). this kind of “psychic disequilibrium” can explain the potential conflict between “identities that students are invited to construct in mathematics class and the kinds of persons they view themselves to be” (cobb & hodge, 2002, p. 279), especially salient to dis-identification with classroom mathematical activity (cobb, gresalfi, & hodge, 2009; nasir, 2002; tate, 1995). moreover, without addressing identity and power in mathematics itself, mathematics is constructed as rubel equity-directed practices journal of urban mathematics education vol. 10, no. 2 77 neutral and as universal, a positioning that can be seen as parallel to colorblindness (battey & leyva, 2016). not only is the teaching of mathematics political (e.g., fasheh, 1982; feltonkoestler & koestler, 2017; gutiérrez, 2013b), but also learning is “always tied to culturally rooted perception of the learning settings” and to “one’s cultural relation to the content” (nasir et al., 2012). seen in this way, the critical dimensions in gutiérrez’ (2007a) classification are necessary, and demand connecting mathematics content to students’ experiences, their communities, and possibilities for social transformation through tmfsj’s critical mathematics (cobb & jackson, 2013). gutiérrez (2007a) explains: we must keep in mind all four dimensions, even if that means that at times one or two dimensions temporarily shift to the background. a natural tension exists between mastering the dominant frame while learning to vary or challenge that frame. as such, access, achievement, identity, and power are not going to be equally or fully present in any given situation ... the goal is to attend to and measure all four dimensions over time. (pp. 4–5) even the critical practices often not critical enough? despite ladson-billings’ (1995a) underscoring of the importance of teachers engaging in “cultural critique” of the political underpinnings of students’ social circumstances, the practice of connecting mathematics to students’ experiences from crp is largely taken up in terms of contextualizing mathematics in general, everyday experiences. typically, mathematics teachers select “real-world” contexts related to sports; middle-class leisure activities; or adult experiences related to home remodeling, shopping, banking, or budgeting (bright, 2015; gainsburg, 2008; matthews, 2003; wager, 2012; watson, 2012), or initiate situated classroom settings related to classroom experiments or field trips (wager, 2012). ladson-billings (2014) has observed: as i continued to visit classrooms, i could see teachers who had good intentions toward the students. ... they expressed strong beliefs in the academic efficacy of their students. they searched for cultural examples and analogues as they taught prescribed curricula. however, they rarely pushed students to consider critical perspectives on policies and practices that may have direct impact on their lives and communities. (p. 77) sealey and noyes (2010) underscore this distinction in their case study of three schools, in which mathematics was constructed in terms of its practical relevance, the transferability of its processes, or its professional exchange value, but absent was a sense of the political relevance of mathematics and how it relates to power. the ever-expanding role of state testing, prescribed curriculum, and increasing demands related to accountability undoubtedly create obstacles for teachers. but rubel equity-directed practices journal of urban mathematics education vol. 10, no. 2 78 there is a “natural tension” between “mastering the dominant frame” by striving toward equity through the practices on the dominant axis (“playing the game”) and “challenging the dominant frame” through the practices on the critical axis (“changing the game”) (gutiérrez, 2007a, p. 4). as gutiérrez (2013a) exacts, “without an explicit focus on issues of identity and power, we are unlikely to do more than tinker with the arrangements in school that contribute to the production of inequities in the lived experiences of learners and educators” (p. 62). here, i explore this tension by studying the difficulties for teachers in adopting the critical equity-directed practices of connecting mathematics instruction to students’ experiences from crp and critical mathematics from tmfsj. different from the collection of rich case studies in the literature that profile experienced mathematics teachers with a full set of well-developed, equity-directed practices (see, e.g., birky et al., 2013; bonner, 2014; bonner & adams, 2012; clark, badertscher, & napp, 2013; ladson-billings, 1995b, 1997; leonard, napp, & adeleke, 2009), and different from literature that documents failures of teachers in urban schools, i present these cases to serve as texts that illuminate “teachers’ attempts—with all their flaws and complexity” (dutro et al., 2008, p. 295). methods and context i draw on data gathered as part of a larger project aimed to design and study professional growth for secondary mathematics teachers around the four aforementioned equity-directed practices. participants consisted of 12 teachers from 11 secondary schools (ten in grades 9–12 schools and two in grades 6–12 schools), who were recruited through a local teacher organization, geographically dispersed across a large city in the united states. teacher demographic data are summarized in table 1. the participating teachers taught a range of secondary mathematics, in grades 6–12, at schools with higher percentages of low-income, black and/or latinx students than the aggregated city percentages. i identify as an ashkenazic jew, pass as a white woman, and was the project director and lead facilitator of the associated professional development (pd). the pd was initiated with an 8-day summer institute in the summer of 2012 focusing on the four equity-directed practices and continued with monthly 2-hour group meetings. in all sessions, teachers regularly engaged in collaborative mathematical problem solving, experiences which i used to then draw out overlapping themes of conceptual understanding, challenging mathematics, participation, and critical thinking about mathematics. ilana horn visited the summer institute and introduced teachers to the theme of multidimensional participation; teachers were provided with horn’s (2012) book strength in numbers: collaborative learning in secondary mathematics (see appendix a for focal topics of pd sessions). rubel equity-directed practices journal of urban mathematics education vol. 10, no. 2 79 table 1 demographic data of participating teachers variable n n gender certification women 10 alternative 7 men 2 traditional 5 racial/ethnic self-identification years teaching white 9 three or fewer 3 afro caribbean 2 four to six 4 mexican american 1 seven to nine 4 ten or more 1 over the course of the project, teachers participated in a sequence of activities (see rubel, 2012) designed to facilitate and support them to enter students’ spaces and to learn to identify students’ funds of knowledge (i.e., learn to make connections to their everyday experiences and prior knowledge). teachers conducted a series of various forms of community walks to learn about their students’ lived worlds and experiences. for example, during the fall semester, teachers were asked to complete the following assignment: to choose a place near their school and spend time observing the space, people, and activities. teachers could opt to stand in one place to observe, for example, in front of a building or inside a corner-store; or to traverse a neighborhood’s streets, perhaps pursuing a specific theme of interest, such as locations of play-spaces for youth. teachers shared their experiences and reflected together at one of the group meetings. across the pd, teachers and i shared an array of examples of resources around opportunities to think critically with mathematics, including using the activities and lesson ideas in gutstein and peterson’s (2005) edited volume rethinking mathematics: teaching mathematics for social justice. across the school year, the research team (consisting of me and a graduate research assistant) observed five lessons per teacher, with the same groups of students, in evenly spaced rounds across the school year, for a total of 58 observed lessons.3 the research assistant observed and took field notes during all 58 lessons; i co-observed a subset of 28 of the 58 lessons. after each observation, we classified the lesson’s main mathematical task, as implemented by the teacher and enacted by the students, using stein and colleagues’ (1996) levels of cognitive demand. we 3 two observations were cancelled because of a major weather event and the emergency relocation of a school. both involved extensive disruptions at respective school sites, which would have led to unrepresentative observations. rubel equity-directed practices journal of urban mathematics education vol. 10, no. 2 80 resolved any discrepancies in ratings through discussion and through coordination of the task’s narrative description with stein and smith’s (1998) rubric. the research assistant converted field notes into a detailed, time-indexed, narrative memo that included a description of the lesson’s tasks and materials, as well as a synopsis of the lesson’s structure, teacher moves, and students’ observable participation. i reviewed the narrative memo within a day or two of the observed lesson, and clarified any missing details. i interviewed teachers individually two to six times during the school year, using a face-to-face, semi-structured, traditional question-and-answer format (hollway & jefferson, 2000). finally, teachers wrote reflective statements that year and in the following school year, and we archived these statements as well as any additional email correspondence. we used the lesson narratives and field notes to divide each lesson into a set of time-indexed units of activities. the unit of activity was determined by a change in what the students were asked to do by the teacher or by a change in whether the students were directed to focus individually, in groups, or as a whole class (stapleton, lefloch, bacevich, & ketchie, 2004). we coded the participation structure of each activity according to what the students were asked to do by the teacher, similar to stodolsky (1988), but using top-level categories (as well as further refined subcategories) of listening; investigating or problem solving; discussing; reading, writing, or reflecting; using technology; or practicing skills (adapted from weiss, pasley, smith, banilower, & heck, 2003). students being asked to listen to lecture presentations, copy notes from the board, or practice skills on worksheets were categorized as passive. activities were categorized as active when students were, for example, asked to participate in discussions, investigate with technology, write about a solution strategy, reflect about a concept or process, explore mathematical concepts with manipulatives, prove a mathematical conjecture, or listen to a classmate’s presentation. next, for each lesson, we calculated the ratio of the difference between active and passive minutes to the total instructional time, denoted as “difference in participation proportion” (dpp; rubel & stachelek, in press). dpp ranges from –1 to 1, from a lesson whose participation structures are entirely passive to a lesson whose participation structures are entirely active. exactly three teachers (b mary, c tracy, and d molly) had high dpp measures and four or five observed lessons with tasks of high cognitive demand (as shown in figure 1). i used case study methodology (merriam, 1998) to create cases of these three teachers. data sources for the case studies include five narratives of classroom observations group meetings, and audio of four to six interviews per teacher. data were converted to text and uploaded to dedoose (research software), organized chronologically and by teacher. i used a grounded analysis approach (strauss & corbin, 1998) with the narratives of the classroom observations to iteratively categorize aspects of the case teachers’ instruction using a priori codes related to connecting to rubel equity-directed practices journal of urban mathematics education vol. 10, no. 2 81 students’ experiences and critical mathematics as well as a code for views about their students as learners of mathematics. in subsequent iterations, i created and used sub-codes as shown in table 2. this coding process enabled the development of descriptive and interpretive, written case summaries. next, i triangulated these findings using interview data and teacher written reflections, looking for confirming or disconfirming evidence. finally, i conducted member checking (lincoln & guba, 1985) with the case teachers to validate interpretations. figure 1. ddp and cognitive demand by teacher. all three case teachers identified as white women and had started teaching careers after having moved to the city as young adults. mary and molly had entered teaching through an alternative certification pathway in response to a local teacher shortage (boyd et al., 2012), and tracy, with an undergraduate degree in mathematics, had entered teaching through a traditional pathway, with teacher certification from another state. in the academic year of data collection (2012-13), mary taught 10th grade geometry; molly taught 6th grade mathematics; and tracy taught an 11th and 12th grade algebra ii class in the fall semester, and a 9th grade algebra i class in the spring. rubel equity-directed practices journal of urban mathematics education vol. 10, no. 2 82 table 2 sub-codes connecting to students’ experiences and critical mathematics parent code sub-code connecting to students’ experiences  nature of context  role of context in lesson  connections between mathematical and everyday meanings  input from students to launch task critical mathematics  actualized connections of content to social justice  potential connections of content to social justice views about students as learners of mathematics  positioning students as successful learners of mathematics  positioning students as mathematical resources for other students  normalizing struggle in mathematics findings i present an analysis of the difficulties for the case teachers, who demonstrated success with the study’s articulated dominant equity-directed practices, in adopting the critical equity-directed practices of connecting mathematics instruction to students’ experiences. these three teachers selected and implemented high demand tasks across all (or nearly all) of their observed lessons, in contrast with other teachers, who more often built their lessons around low-demand tasks (as shown in figure 1). all of molly’s, mary’s, and tracy’s observed lessons had high dpp values, meaning that they included significant durations of opportunities for active participation. in what follows, first, i provide more details about the three case teachers, in terms of their success with the dominant equity practices. then, i shift the results to focus on an analysis of their struggles with the critical equity-directed practices. success with dominant equity-directed practices mary: 10th grade geometry. mary was 30 years old and in her sixth year of teaching, all at one school. mary’s school was grades 9–12, with most students identified as hispanic or latino (69%) and the remainder as black or african american (29%), and most qualifying for free or reduced-price (frep) school meals (86%). because of a history and perception of violence, students at this school were required to take off their shoes and belts and pass through a metal detector, which was staffed and supervised by police, to enter the building. her 10th grade classroom was overflowing with colorful, hand-made posters highlighting geometric definitions and relationships, photographs of students, and reminders rubel equity-directed practices journal of urban mathematics education vol. 10, no. 2 83 about classroom rules and expectations. the classroom contained rectangular tables arranged for assigned groups of four students. mary typically welcomed students enthusiastically as they entered the classroom and encouraged them to sit down quickly and get started with a task displayed on the smartboard. mary moved energetically around the classroom and spoke at a rapid pace, conveying enthusiasm for mathematics and for the lesson. mary supported the development of students’ mathematical understanding with various tools, notably using her classroom’s smartboard to produce colorful, visual, dynamic, and interactive mathematical representations. for example, in a lesson on vertical angles, mary used the smartboard to measure and compare two congruent angles, but one with significantly longer rays than the other, in anticipation of a common misconception that angles with longer rays have a greater measure (fischbein, 1987). the dynamic functionality of the smartboard allowed mary to rotate, translate, and compare angles with the students. students were regularly invited to operate the smartboard, to explore conjectures, draw pictures to express an idea, or represent mathematical thinking. mary often incorporated physical materials as tools to represent and model mathematical relationships. for example, in a lesson about triangle similarity, mary built a model to use measures of individual students’ heights to indirectly compute other heights. students worked with physical measurement tools, a mirror and measuring tape, and used a model involving similar triangles to calculate the unknown (and theoretically not directly measurable) height of the classroom. this task afforded students the opportunity to discover that the room’s height could be measured indirectly and to explore how the model adapts when a different person’s height is used. even though mary’s student population was majority latinx, with 19% of these students classified as limited english proficient (the official federal classification, not mine), mary’s lessons included significant amounts of mathematical discussions (25% of lessons, on average). she supported student participation in mathematical discussions by launching lessons (see jackson, shahan, gibbons, & cobb, 2012) with activities that directed students to individually write down and then share their prior knowledge of mathematical meanings. for example, in a lesson about vertical angles, mary began the lesson with an individual, written prompt: “i think congruent means _____.” mary then wove together responses from several students in a whole class discussion to arrive at a shared understanding of congruent figures or angles, which would be crucial to the rest of the lesson (lesson, 10/24/12). it was commonplace for students, at the end of class, to express gratitude to mary for her teaching; at the end of one observed lesson, as she gathered up her things to leave the room, one student shared her feelings of gratefulness, “i appreciate all the teaching you gave us” (lesson, 3/13/13). rubel equity-directed practices journal of urban mathematics education vol. 10, no. 2 84 molly: 6th grade mathematics. molly, age 25, was in her fourth year of teaching. her school was grades 6–12 school in the same neighborhood as mary’s school. nearly all of the students were identified as black or african american (66%) or hispanic or latino (30%), and most qualifying for frep school meals (86%). entering students had standardized test scores slightly above the city average, and students were required to wear uniforms. students were required by the school to line up in the hallway before they could enter the classroom, a hallway with posters reminding students that “you are a mathematician” and “you are a problem solver.” inside the classroom were colorful bulletin boards showcasing students’ work. molly typically greeted her students at the start of the period, at the door, shaking hands with some students and welcoming many in by name. students sat in assigned groups of three at trapezoidal tables. molly demonstrated a friendly, if stern, playfulness. in general, many students enthusiastically volunteered responses to her questions by actively raising their hands. if students seemed distracted, molly would lead the class in coordinated clapping rhythms. in two of the observed lessons, molly initiated brief sets of calisthenics exercises that incorporated review of mathematical number facts as a way to refocus the group. molly’s lessons emphasized sense-making and connections between students’ solutions and ideas. she organized her lessons around recruiting multiple methods of solving problems, which she typically then shared and compared in whole-class discussions. for example, in a lesson about area, molly began with a gridded rectangle and a gridded nonstandard polygon, and asked students—first working individually, and later working in a group—to find at least two methods to find each area. molly organized a whole class discussion in which students explained and compared two methods, and molly highlighted a contrast between decomposition and composition approaches. the lesson continued by extending students’ understanding of area and of methods of finding area to derive with students a method of finding the area of any triangle by building it into a rectangle of known dimensions, toggling between whole class discussion and individual mathematical investigations. tracy: high school algebra. tracy was 27 years old and in her third year of teaching. tracy’s school was grades 9–12, with a central location that attracted students from diverse geographies. most of the students at tracy’s school were identified as hispanic or latino (73%), a smaller number of students as black or african american (14%), with 75% qualifying for frep school meals. incoming 9th graders’ test scores were higher than the city average, and students were required to wear uniforms. tracy taught algebra ii to 11th and 12th graders in the fall semester, and algebra i to 9th graders in the spring. her classroom was mostly barewalled, with few posters or visual displays and without smartboard technology. students were seated in individual desks and chairs that were placed side-by-side to create pairs. at times, tracy asked students to form larger groups, at which point rubel equity-directed practices journal of urban mathematics education vol. 10, no. 2 85 the students would push their desks together into groups of four. tracy spoke to students in her lessons in what seemed likely a deliberately slow pace, peppered with pauses that provided both the space for students to share their own thoughts, questions, or concerns. she primarily stood at the front of the classroom, except when circulating around during group work. in addition to her outwardly reserved manner, she regularly communicated a dry sense of humor, using gesture and tone, which seemed to appeal to the students. in her classroom, students appeared comfortable asking questions, commenting on their own confusion, or venturing to provide potential solutions to a given task. tracy’s lessons were typically organized around whole-class discussions that were layered around various types of individual or group-focused investigations, involving rigorous mathematical activity around high demand tasks and with supporting tools. one paradigmatic observed lesson began with a short, individual activity in which students were asked to write about their ideas as to the difference between volume and surface area. the subsequent discussion was organized around students’ sharing their ideas in reference to various cardboard models of threedimensional figures. once there were some agreed upon definitions and units of perimeter, surface area, and volume, tracy gave each pair of students several physical models of nonstandard three-dimensional figures, as well as plastic unit cubes, and students were asked to determine strategies for finding volumes. the class was later reconvened as a whole group to share their strategies, with tracy guiding them toward generalization. students were given a handout containing a set of word problems of varying difficulty to practice applying an understanding of volume and were directed to work on these in groups. the lesson concluded with a whole class discussion that featured the sharing of student solutions to three pre-designated problems. as stated previously, the focus here is on the difficulties for teachers in adopting the critical equity-directed practices of connecting mathematics instruction to students’ experiences called for in crp and critical mathematics called for in tmfsj. alternate analyses could probe the factors and micro-practices supporting or challenging aspects of these teachers’ successes with the teaching for understanding and multidimensionality. in the context of their success with the dominant equity-directed practices, the cases of mary, molly, and tracy are ideal terrain in which to analyze their difficulties with the critical practices. challenges in connecting to students’ experiences the three teachers made efforts to connect mathematics to out-of-school contexts, but these were limited to experiences assumed to be general or teacherinitiated classroom settings. molly and tracy contextualized story problems in terms of general out-of-school contexts related to sports, business, or personal finance but not with specific connections to students’ experiences. for example, tra rubel equity-directed practices journal of urban mathematics education vol. 10, no. 2 86 cy’s lessons included contextualized problems around basketball shots for the topic of binomial probabilities (lesson, 9/28/12), a business plan for a clothing company for the topics of normal distribution (lesson, 11/9/12), or compound interest to explore exponential functions (lesson, 11/30/12). similarly, molly made connections in her lessons to general experiences, such as drawing an analogy to the general experience of getting her height measured at the doctor’s office as a way to explain how the altitude or height of a polygon has perpendicularity as part of its definition (lesson, 3/8/13). in mary’s class, geometry, she oriented her lessons largely around teacherinitiated, classroom-situated geometric settings and not on geometry specifically connected to students’ physical spaces outside of school. for example, in a lesson about triangle similarity, mary’s lesson built on similarity relationships to enable students to use their own heights to indirectly measure the height of the classroom. students worked with physical measurement tools, a mirror and measuring tape, and used a model involving similar triangles to calculate the unknown (and theoretically not directly measurable) height of the classroom. lacking knowledge and fearing people of color and their spaces. one interpretation of the difficulty for the case teachers in connecting to or building on students’ experiences is a lack of knowledge about students’ experiences because of fear of entering the spaces in which their students live, shop, play, or worship (picower, 2009). indeed, molly noted and reflected a sense of inadequacy in connecting to her students’ experiences that she explained in terms of feeling like an outsider relative to her students and their communities: “as a young, white teacher, who doesn’t live in my school’s neighborhood or in the neighborhood where my students live, and who didn’t grow up in a city, what position am i in to connect to my students’ experiences?” (reflection, 5/2014) the case teachers were reticent to embed themselves in students’ spaces to learn about their students. molly’s question as to “what position am i in to connect to my students’ experiences?” is reminiscent of the “i can’t relate” tool of whiteness presented by picower (2009), and seemed to function as a kind of release for molly from the need to cross boundaries or confront fears of students’ communities to begin the never-ending process of learning about students. the case teachers’ difficulties in connecting to students’ experiences can be understood in the context of the group’s engagement with activities in the project designed to facilitate their presence in students’ spaces. at the monthly group meeting in which teachers were asked to share their experiences in visiting their students’ spaces, they do so in a silent, snowball activity in which they recorded their responses to the following prompts on a large whiteboard: where did you go? how did you observe? what did you notice? what did you learn? about half of the teachers in the group focused their observations around typical stereotypes of lowincome urban spaces, commenting on noticing “heavy police presence,” “vacant rubel equity-directed practices journal of urban mathematics education vol. 10, no. 2 87 lots,” “99 cent stores,” “paucity of transportation access,” and spaces that seemed “disused” with “creepy vibes.” one teacher noted having observed in and around a corner store, “observing owner (yemeni), workers (dominican), and customers (dealers & thieves).” another explained that she had traversed her selected area looking in particular at “housing types, conveniences, and cleanliness.” there were few comments that did not reflect deficit views of low-income people of color, such as a presence of multi-generational families, murals with positive messaging, community resources like churches and daycares, and evidence of gentrification (meeting, 10/2013). mary later reflected on the tension that she experienced while completing the activity, between her typical noticing of her students’ places and what the assignment prompted her to notice. she wrote: i first took note of the heavy new york city police presence, the run-down lots, the burnt cars and “we’ll buy your house” signs. i thought of everything i “knew” (assumed) about the area. but then, as we walked through the subway station, i noticed beautiful stained-glass windows with sunlight streaming through. in the sunlight, a man stood at his card table selling incense and oils, two women laughed as they waited for a train, a toddler held his father’s hand as they walked up the stairs. (reflection, 5/2014) mary’s reflection is confluent with other teachers’ responses during the activity debrief and demonstrates how in this case, visits to students’ spaces can re-inscribe prior deficit views about students and their communities. this result more broadly should generate cautions in terms of how these kinds of community walk activities organized for teachers might actually reproduce teachers’ deficit views instead of challenging them (philip, way, garcia, schuler-brown, & navarro, 2013; rubel, hall-wieckert, & lim, 2016). learning about students subverted. observations of the material conditions in students’ communities can contribute to a subversion of teacher to “teacher as missionary,” one of martin’s (2007) two teacher caricatures. in my own work with youth at a hyper-segregated urban school, for example, i organized a celebratory class trip to go out for ice cream. instead of making efforts to learn about what local treats they might enjoy or teach me about, i presumed that they would want to go to sample expensive ice cream at a new outlet in a nearby, but gentrifying neighborhood, its very presence a consequence of this gentrification. while some might argue that this was an act of kindness or generosity, i understand now how this effort likely made the students uncomfortable in how it was a “white savior” maneuver. tracy’s efforts to learn about her students led to a similar subversion. she described a “fun, kind of experiment” that she tried which she felt was “an amazing teaching moment.” she invested personally in red cross training and in special insurance and founded a weekend cycling club for her students. she described how rubel equity-directed practices journal of urban mathematics education vol. 10, no. 2 88 she had initially dreamed of riding with the students out of the city but had realized that “we’re never going to get over the bridge, but it doesn’t matter. ... it’s not about that, it’s about enjoying the scenery and being with my students” (meeting, 5/2013). this example shows tracy’s willingness to spend time outside of school with her students, generosity with her resources, and her commitment to getting to know her students better. however, this particular effort was organized to engage students in one of her hobbies, without a parallel effort to observe or learn about students by engaging in one of their interests. tracy’s approach was to note, among her students, the absence of one of her own essential activities, and then make extensive efforts to bring it to her students. for tracy, her appropriation of the cultural positioning of youth of color as living incomplete or unhealthy lives led her to connect to students through her own, white experiences. challenges to critical mathematics whiteness as blinding. mary and molly avoided addressing issues of power and social justice in the content of their mathematics lessons, even though their lessons included examples of real world contexts that could have easily lent themselves to critiques of power and social justice. for example, one of mary’s observed lessons, on the topic of geometric loci, was contextualized in terms of an unspecific investigation about home locations, relative to constraints like not living too close to a power plant or a busy road (lesson, 2/11/13). the lesson did not mention the high asthma rates in the school’s local neighborhood and relationships to types and rates of local environmental pollution that have been found to be worse in that lowincome neighborhood. similarly, molly organized a lesson around the context of the school being moved by the district to a new location the following school year (lesson, 11/16/12). the lesson focused on using map scale to convert between inches on a map and actual distance. students were tasked to create a walking route from the current location to the new location, and then use the map scale to determine the actual distance of the route. a range of questions could have been asked: does the new location bring the school closer to you? closer to most students? how does the planned move impact the student community in terms of length of commute or new walking routes? what categories of amenities are available at each location? are students happy about the change? molly’s lesson was strictly limited to interpreting the map’s scale in terms of actual distance. mary agreed, in hindsight, that this topic had the potential to provide opportunities for critical thinking with mathematics about placement and displacement, desirability and undesirability, and gentrification, topics that were pressing in her students’ environments. when asked about why she had not connected this topic to sociopolitical circumstances impacting her students’ lives, mary explained that at the time, she had not noticed these potential connections, explaining that “because rubel equity-directed practices journal of urban mathematics education vol. 10, no. 2 89 of my privileges, i have had blinders on to a way of seeing these things” (interview, summer 2014). false neutrality. different from molly and mary, tracy developed unit projects for her algebra ii course on what she called “social issues” and described her approach to designing these units in terms of being “neutral about issues” and organizing students to examine a particular theme from “two opposing points of view” (reflection, 6/21/13). tracy presented the students with a topic and expected them to develop their own arguments but also arguments that support dominant, hegemonic perspectives about that issue using mathematics. for example, tracy wanted students to learn how mathematics can be used to defend the stop and frisk tactic used by local police, a tactic that had serious implications for most of her students impacting their day-to-day mobility. tracy’s assumption was that learning to support both stances would support students in clarifying and strengthening their arguments and to practice using mathematics to support a range of arguments. furthermore, by presenting an issue and arguing both sides could seemingly be a safeguard against fears about bringing political issues into the classroom. tracy’s technique of framing the assignment around having students argue dominant and critical perspectives of an issue was intended to allow for students to develop their own opinions along a wider continuum and prepare them for advocating for their position by better understanding the opposing argument. but in framing the task for students to justify both positions, tracy inadvertently positioned the two perspectives as equally viable, or as matters of opinion. for example, in the case of the analysis of stop-and-frisk data, tracy seemed to be equating the position that black and latinx youth are stopped at greater rates than white people tactic and that this is inconvenient, degrading, unjust, and is part of a broader system of racialized policing practices with the position that black and latinx youth are justifiably stopped more often. in framing these positions as equally viable, tracy undercut her students’ sense of injustice about their own experiences and, therefore, failed to leverage their already existing critiques of the social order. tracy herself reported that framing the social justice tasks in this way, that she understood as neutral, indeed backfired with the students. however, she reverted to deficit views of her students in her interpretation. she said that she had “learned the hard way that not all students have the same experiences and you can’t assume they know both sides to every social issue” (reflection, spring 2013) and not on how her anti-critical framing of the issue in the tasks for students might explain their lack of effectiveness. teaching for social justice as tool of whiteness. stinson and wager (2012) distinguish between teaching for social justice—that is, providing students from underserved, marginalized groups access to challenging mathematics (equity’s dominant axis)—with the critical equity-directed practices, or teaching about social justice. in other words, dominant equity-directed practices do not challenge the sta rubel equity-directed practices journal of urban mathematics education vol. 10, no. 2 90 tus quo (gutiérrez, 2007b). however, insofar as they are understood as teaching for social justice, addressing equity through the dominant equity-directed practices can seem sufficient, allowing schools or teachers to evaluate their programs, policies, or instruction as addressing equity. however, as i, and others, have argued, the critical equity-directed practices are necessary, since supporting students to “play the game” is not equivalent to “changing the game” (gutiérrez, 2009, p. 11). molly’s case illustrates how emphasizing the dominant equity-directed practices can obfuscate, or even co-opt, the critical. her classroom featured a poster placed front and center above her whiteboard informing students that, “without struggle there is no progress” (douglass, 1857), a passage excerpted from frederick douglass’s 1857 emancipation speech, which he delivered in the context of the movement to abolish slavery in the united states. nearly all students at her school identify as black or african american, and molly’s choice to feature a poster quoting an african american thinker whose writing is tied to slavery and emancipation could be seen as setting the stage for critical mathematics, by positioning mathematics as a tool for civil rights. douglass’s words suggest that social progress requires struggle, and in the original text continue with: this struggle may be a moral one, or it may be a physical one, and it may be both moral and physical, but it must be a struggle. power concedes nothing without a demand. it never did and it never will. find out just what any people will quietly submit to and you have found out the exact measure of injustice and wrong which will be imposed upon them, and these will continue till they are resisted with either words or blows, or with both. (para. 8) displaying these words on the wall of a mathematics classroom could signal that mathematics could be useful in a struggle for justice. molly’s explanation, however, as to why she highlighted that particular quote, though, illustrates her alternate interpretation. she related her choice of douglass’s (1857) words to her desire that her students “have every possible opportunity and choice open to them.” molly explained that her selection of a passage around struggle refers to “the hard work, effort and confusion that i expect them to face during math class” and that progress refers to the promise of “advancement (that) does come from that work” (interview, spring 2014). indeed, molly’s instruction generally reflected her stated belief that “math is not something you are innately good or bad at, but rather that everyone is expected to and can excel in math through hard work” (reflection, summer 2012). molly’s interpretation of this douglass (1857) quote is consistent with her interpretation of equity in dominant terms of access and achievement. she consistently and notably demonstrated high expectations for students by teaching for understanding and routinely provided her students with multidimensional opportunities for participation in mathematics. through her consistency around these dominant rubel equity-directed practices journal of urban mathematics education vol. 10, no. 2 91 equity-directed practices, molly could be seen as an exemplar of teaching for social justice. at the same time, molly’s interpretation of struggle diverted from an understanding of struggle as resistance to injustice. her framing of progress sidetracked an understanding of progress in terms of concession of power. instead, her articulations of the importance of effort along with her affirmations around the belief that effort is always rewarded, more closely correspond to various tools of whiteness, like the myths of meritocracy and colorblindness. discussion aggregating instructional practices from four models of pedagogy in this study has highlighted nested relationships among these pedagogies and provides a mapping of a set of equity-directed practices from these pedagogical models onto gutiérrez’s (2007a) equity framework. here, i used that mapping and offered an analysis of challenges for white teachers who demonstrate success with dominant equity-directed practices around critical equity-directed practices. the found imbalances could be an outcome of all four dimensions of equity not being equally present in this particular set of observed lessons. indeed, the practices on the dominant axis are practices that lend themselves to daily instruction, whereas the critical practices arguably could be seen as practices that teachers engage with less regularity. alternatively, the difficulty for the case teachers with the practices on the critical axis could be attributed to the professional development program, in its attempt to work with teachers on such a wide span of instructional practices from four pedagogical models across a single year. perhaps teachers develop expertise with the practices on the dominant axis more readily than the practices on the critical axis. the case teachers identified as white. readers could reasonably infer that teachers of color enact or struggle with the critical equity-directed practices differently than white teachers. examining this particular question in depth is beyond the scope of this article, but it is important to note aspects unique to the teaching of the two teachers of color in this group of teachers because they add significant nuance to implications of the cases presented here. contexts of any kinds were absent in nearly all of the lessons of the two participating teachers of color (both taught firstyear algebra). however, different from the case teachers, harriet and teresa found ways to connect their instruction to students through language. teresa drew on strategically using spanish (her native language) with students, to translate mathematical terms, to check in with students about their emotional state, and to redirect behavior. she regularly referred to her students as her “loves” and typically interacted with every individual student at some point in the lesson. although her family had immigrated from a different part of the world than her students’ families, her regular use of spanish, especially as used to take care of the students, communicated a sense of familial care and connection. rubel equity-directed practices journal of urban mathematics education vol. 10, no. 2 92 harriet’s approach at connecting to her students occurred primarily through language as well. her instruction was entirely in english but she often used youth language in her lessons, reminiscent of the lee case described in clark, badertscher, and napp (2013). for instance, harriet would playfully ask the class, “can i mess with you now?” before presenting more challenging exercises, or challenge that she would like “to mess with you, but not yet” (lesson, 12/12/12). this language use created a playful atmosphere, and communicated both support and a sense of challenges. these examples suggest that the field expand our understandings of critical equity-directed practices, led through expertise of teachers of color. for example, the practice of creating hybridity between school mathematics and students’ out-of-school practices called for in crp can be expanded beyond the integration of real-world contexts from students’ experiences. fostering hybridity through language (e.g., lee, 1997; moschkovich, 2002; rosebery, warren, & conant, 1992) or other kinds of “disruptions” of traditional learning environments (ma, 2016) are avenues that need further exploration. one contributing factor to the difficulty for teachers in engaging in the critical equity-directed practices likely resides in the limited emphases of teacher education on developing content knowledge and knowledge of mathematics for teaching among teacher candidates. as its modifier implies, the dominant notion of equity dominates any discussions or interventions around equity in mathematics education, while themes that engage the interplay of race, whiteness, and social justice in the context of the teaching and learning of mathematics are downplayed or sidestepped. martin (2007) has set criteria for highly qualified teachers for african american children which highlight, identity and power: (a) developing deep understanding of the social realities experienced by these students, (b) taking seriously one’s role in helping to shape the racial, academic, and mathematics identities of african american learners, (c) conceptualizing mathematics not just as a school subject but as a means to empower african american students to address their social realities, and (d) becoming agents of change who challenge research and policy perspectives that construct african american children as less than ideal learners and in need of being saved or rescued from their blackness. (p. 25) the capacities outlined by martin might be viewed as desirable but are not viewed widely as necessary. martin, however, states unequivocally: “teachers who are unable, or unwilling, to develop in these ways are not qualified to teach african american students no matter how much mathematics they know” (p. 25). this analysis of teachers who showed success with dominant equity-directed practice yet struggled with practices that relate to identity and power demonstrates that greater attention and focus is required around supporting mathematics teachers in developing critical equity-directed instructional practices. mathematics teacher preparation needs to de-silence race (martin, 2009b, 2013), address whiteness and its role in rubel equity-directed practices journal of urban mathematics education vol. 10, no. 2 93 mathematics education (battey & leyva, 2016; martin, 2013), and foster the development of political knowledge among teachers around issues related to schooling, education, identity, power, and mathematics, as well as cultivating their abilities to act on such knowledge (gutiérrez, 2013b). rather than general education courses on race, culture, and diversity, this preparation could approach these areas of emphasis through the lens of mathematics. there is the potential to develop norms among teachers, in content and methods courses, for example, around the posing of critical questions about mathematics content: who created it, to answer what questions, to what ends or purposes, to whose benefit, and to whose demise (brelias, 2015); whose experiences are reflected and valued in certain mathematics content or tasks and whose interests are ignored (greer, mukhopadhyay, nelson-barber, & powell, 2009; gutiérrez, 2002, 2007a); as well as the related question of why certain mathematics content is taught in schools (gutiérrez, 2002, 2013a; nolan, 2009). content and methods courses could address the “formatting power” of mathematics, how it “colonizes part of reality and reorders it” (skovsmose, 1994, p. 36), how mathematics gets privileged over other ways of knowing (borba & skovsmose, 1997; gutiérrez, 2013a), and how it is used to intimidate others or to squelch debate (ewing, 2011). exploring the ways that mathematics interacts with identity and power through mathematics might be an effective way to further support teachers’ development of critical equity-directed practices, rather than taking a general, non-discipline specific approach. in closing, the findings i presented here add necessary nuances to an oversimplistic, master-narrative about a “pedagogy of poverty” in urban high schools (haberman, 1991) to instead, identify and better understand challenges for teachers, especially white teachers in hyper-segregated urban schools. the cases of three teachers—competent early-career teachers, who demonstrated excellence at dominant equity practices while struggling with critical equity practices—communicate “teachers’ attempts—with all of their flaws and complexity” (dutro et al., 2008, p. 295). on one hand, we can reaffirm a commitment to an orientation of “perspectives and insights of possibility” (milner, 2011, p. 88) around teachers and teaching. we can signal the notion of “yet” from dweck (2006) and horn (2012) to choose a growth mindset about teachers and their teaching and interpret demonstrated excellence with the dominant equity-directed instructional practices with optimism regarding further growth. on the other hand, we must challenge any preconceptions that dominant notions of equity alone, through its compelling pillars of access and achievement, will be sufficient to challenge the status quo. changing society guided by social justice demands that we find ways to better support mathematics teachers in developing and engaging critical notions of equity in and through their teaching. rubel equity-directed practices journal of urban mathematics education vol. 10, no. 2 94 acknowledgments this research was supported in part by the national science foundation under grant 0742614. any opinions, findings, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the national science foundation. the author thanks aj stachelek, haiwen chu, and scott monroe for their contributions to earlier versions of this article. the author thanks dan battey, mary foote, victoria hand, luis leyva, judit moschkovich, joi spencer, david stinson, and all of the anonymous reviewers for their feedback on earlier versions of this article. references aguirre, j. m. 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(2012). mathematics teaching practices with technology that support conceptual understanding for latino/a students. journal of mathematical behavior, 31(4), 431–446. http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/98/87 rubel equity-directed practices journal of urban mathematics education vol. 10, no. 2 105 appendix a contents and resources of professional development meetings month meeting topic associated resource september levels of cognitive demand task sorting activity described in smith, stein, arbaugh, brown, & mossgrove (2004) october community walks in school neighborhoods and debrief protocol for studying community november improving questioning boaler & humphreys (2005) december launching a mathematical task talk by k. jackson, reading: jackson et al. (2012) january preparing for teacher presentation at national conference on learning about students’ communities to inform teaching february racial microaggressions talk by battey, about battey & leyva (2016) march community based mathematics talk by remillard and lim, about ebby et al. (2011) april presentations by “veteran” teachers: 1. classroom discourse 2. critical about and with mathematics smith & stein (2011) may share out of classroom observation data and next steps microsoft word final sengupta-irving vol 7 no 2.doc   journal of urban mathematics education december 2014, vol. 7, no. 2, pp. 31–54 ©jume. http://education.gsu.edu/jume   tesha sengupta-irving is an assistant professor of education and affiliate of the department of gender and sexuality studies at the university of california, irvine, 3200 education, irvine, ca 92697-5500; email: t.s.irving@uci.edu. as a learning scientist and former secondary mathematics teacher, her research interconnects the study of mathematical learning, peer collaboration, and issues of power in culturally and linguistically diverse classrooms. affinity through mathematical activity: cultivating democratic learning communities tesha sengupta-irving university of california, irvine in this article, the author demonstrates how a broader view of what shapes affinity is ideologically and practically linked to creating democratic learning communities. specifically, the author explores how a teacher employed complex instruction (an equity pedagogy) with her ethnically and racially diverse students in the “lowest track” algebra i course. sociometric network analyses used to model peer relationships revealed an affinity among three students that could not be explained by shared attributes or history (e.g., race or gender). through field note analyses, the author argues these students’ affinity was forged through shared mathematical activity—what she terms a workship. this workship reflected equitable relationships born of diverse youth learning to work together by working together. the author discusses implications of the workship for teachers and researchers, as well as the constraints that stratified mathematics programs can place on classroom-based efforts to advance equity. keywords: cooperative learning, democratic learning communities, equity, mathematics education, tracking tructuring opportunities in which adolescents learn to treat all peers as mathematically capable can be a challenge; yet, it is central to cultivating democratic learning communities. academic and social hierarchies often limit what is possible for adolescent relationships in school (abraham, 1995; cohen, lotan, scarloss, & arellano, 1999; eckert, 1989). nonetheless, for educators, anticipating and encouraging diverse peer relationships is both ideologically and practically linked to the development of supportive learning environments. in this article, i examine how a broader view of what shapes adolescent affinity could be a promising pedagogical tool for cultivating democratic learning communities. specifically, i explore how shared mathematical activity can shape a student’s relationship to another, in contrast to the ways shared social attributes or history are currently understood to drive affinity among adolescents. prior research suggests that adolescent affinity stems from shared social attributes like race or gender (e.g., miell & macdonald, 2000; tatum, 2003) or a shared social history like friendship (azmitia & montgomery, 1993; strough, berg, & meegan, 2001; strough, swenson, & cheng, 2001). in contrast, the focal s sengupta-irving affinity through mathematical activity journal of urban mathematics education vol. 7, no. 2 32 group in this analysis emerged as a unique case of adolescent affinity in the “lowest” mathematics track of an ethnically and racially diverse and socioeconomically stratified, suburban/urban (i.e., urban sprawl) u.s. public school. within this setting, three students developed an affinity for one another unlike what is typically predicted—they were a mixed-gender collective, ethnically and racially different, not previously friends, did not share social circles, and had no other classes together. i argue that these students’ affinity was forged through shared mathematical activity—what i term a workship. exploring how shared mathematical activity can shape students’ relationships (in contrast to the way students’ relationships to others shapes activity; see, e.g., civil & planas, 2004) is generative for two reasons: (a) it supports teachers in identifying and nurturing “unexpected” peer relationships in a diverse setting, and (b) it conceptualizes a closer link between peer collaboration and the creation of democratic learning communities. there is, however, an important caveat to this narrative of possibility. the three students developed their affinity while simultaneously setting apart a fourth student, who was arguably the most academically vulnerable in the collaborative group. therefore, this case of unforeseen affinity also speaks to the complicated and delicate nature of developing equitable relationships within a single group, let alone a classroom community. literature review in this case study, i conceptualize a generative relationship between pedagogy, students’ mathematical activity, and the creation of democratic learning communities. i begin with a discussion of complex instruction (cohen & lotan, 1995; cohen et al., 1999), a theory of pedagogy to advance equity that was used in this case study. i then describe two aspects of the pedagogy that are central to the analysis: (a) attending to students in relation to peers and mathematical activity, and (b) developing relational equity (boaler, 2008) among learners. complex instruction complex instruction (ci) is a theory of pedagogy aimed at advancing equity that involves, in part, teachers addressing power relations among students (cohen & lotan, 2004; featherstone et al., 2011). ci is typically used in collaborative settings in which students actively contribute to a mutual learning goal through shared effort (teasley & roschelle, 1993). collaborative learning challenges the traditional conception of mathematical authority as concentrated in a teacher or textbook, and redistributes it among students and the teacher (gresalfi & cobb, 2006). shifting from teacher to students heightens the role of peer relationships in advancing students’ mathematical thinking (e.g., barron, 2003; engle, langer sengupta-irving affinity through mathematical activity journal of urban mathematics education vol. 7, no. 2 33 osuna, & mckinney de royston, 2008; kotsopolous, 2010; sfard & kieran, 2001) and in supporting students to develop identities as capable mathematics learners (e.g., boaler & greeno, 2000; esmonde, 2009; langer-osuna, 2011). attending to students in relation to peers and activity ci presupposes that differences in status among learners are consequential to learning. status in ci refers to “an agreed-upon social ranking where everyone feels it is better to have high rank within the status order than a low rank” (cohen, 1994, p. 27). social ranking, or status, is “agreed upon” in two ways: (a) through local institutional arrangements (e.g., high status in a performing arts magnet school would be different than in a mathematics and science magnet school); and (b) through salient social and cultural attributes like race, ethnicity, or gender. ci urges teachers to disrupt status by altering the expectations of competence or ability within the local learning environment. teachers do this by looking for instances where students of relatively low status perform well on some aspect of a task; the teacher then provides specific, positive, and public praise. this pedagogical move, called a status treatment (for more, see cohen & lotan, 1995), strategically elevates a student’s contribution to cast the student as competent and important within the learning community. ci has been shown to impact student learning and the development of students’ identities in diverse schools (see, e.g., boaler & staples, 2008; cabana, shreve, woodbury, & louie, 2014; civil, 2014; featherstone et al., 2011). understandably, much of the research on ci focuses on the teacher and on interactions between the teacher and students. in the analysis reported here, however, i examine peer interactions often occurring beyond the reach of a ci teacher’s pedagogical moves. although ci focuses on teachers’ actions and their impact on how students are positioned within a learning community, my analysis also recognizes how peers act as resources for being seen as competent and important (wortham, 2004a, 2004b, 2005). similar to others, i attend to student talk and interactions to understand how students position in relation to peers and the discipline (engle et al., 2008; esmonde, 2009; hand, 2012; turner, dominguez, maldonado & empson, 2012; wood, 2013). this approach has, for example, demonstrated how students become marginalized in relation to peers or the classroom community despite having relevant and complex ideas to share (e.g., kurth, anderson &palincsar, 2002; gresalfi, martin, hand, & greeno, 2009; langer-osuna, 2011; moschkovich, 1999). understanding students in relation to peers through changes in moment-to-moment interactions and activity differs significantly from research on adolescent affinity in collaborations, which identifies how social attributes like race or gender often foretell peer interactions. within this literature, homogeneity in social attributes (friendships or same-sex grouping) is often demonstrated as a positive good; while on the other hand, heterogeneity of social attributes are ar sengupta-irving affinity through mathematical activity journal of urban mathematics education vol. 7, no. 2 34 gued as recruiting issues of status and influence that can mitigate positive outcomes (see, e.g., miell & macdonald, 2000; strough et al., 2001; webb, 1984, 1991; wilkinson & fung, 2002). the analysis here stands to complicate such conclusions by exploring the potentially significant role of affinity developing among highly diverse youth, within the greater project of creating democratic learning communities. relational equity boaler’s (2008) introduction of relational equity brings together the pedagogical commitments of ci and the explanatory potential of studying affinity in situ. boaler (2008) defines relational equity as students learning to communicate effectively with one another, demonstrating appreciation for others’ perspectives, and engaging respectfully in mathematical practice together (p. 167). though coined nearly a decade after ci was first introduced, the idea of relational equity grows from boaler’s 4-year study of a mathematics department using ci in an ethnically and culturally diverse urban high school. at that school, the mathematics department was de-tracked and the teachers adopted ci (boaler, 2008; boaler & staples, 2008). boaler and colleagues found mathematics students from different cultural groups, social classes, ability levels, and genders all achieved at high levels, while also demonstrating a deep appreciation for learning (boaler, 2006a, 2008; boaler & staples, 2008). boaler argues that relational equity serves an important goal of public education: the creation of an enlightened, connected, and committed citizenry. under the rubric of relational equity, her work encourages teachers to create opportunities for students to “act equitably” (boaler, 2008, p. 168, emphasis added). that is, equity is actualized through the way students speak and act toward one another and one another’s mathematical contributions. this analysis leverages boaler’s conception of relational equity as a signal of democratic learning. my analysis, however, differs from boaler’s work in terms of the institutional context within which the teacher was attempting to advance equity. in this case study, the teacher was the only one to adopt ci within an “abilitybased” tracked mathematics program (as will be discussed in the methods section). my analysis therefore builds on previous research by considering the affordances (and limitations) of pedagogy in cultivating relational equity when there is little systematic institutional support for such efforts. methods the methods, data sources, and analyses described in this section were engaged to answer: how might a focus on students’ talk, interactions, and mathematical activity extend what is currently understood about adolescent affinity as sengupta-irving affinity through mathematical activity journal of urban mathematics education vol. 7, no. 2 35 primarily based on shared social attributes or history? i conjectured that in answering this question, the results would offer new insights on the importance of relational equity and the cultivation of democratic mathematics learning communities. data sources semi-structured interviews and sociometric network surveys. i conducted two semi-structured interviews with the teacher at the start and end of the school year to discuss the aims of her pedagogy and later, her reflections on them. most of what i know from and about the teacher, however, is gleaned from frequent informal conversations over the year, where we discussed her lesson plan intentions, reflected on her teaching, shared thoughts about particular students or the department, and so on. i also conducted semi-structured interviews with the students at the start, mid-point, and end of the school year. the initial interview established a baseline of social relationships by asking who they knew, who were friends, and so on. it also established students’ familiarity with collaborative learning in mathematics, their perceptions of mathematics and themselves as learners, and their career or college aspirations. the mid-year interview focused largely on roster-format sociometric network surveys. in the 1980s and 90s, sociometric network analysis gained considerable momentum in the social sciences because of its power to reveal relationships among social entities, and the implications of such relationships (wasserman & faust, 1994, p. 3). sociometric network surveys were thus a particularly useful tool in this analysis because they could capture affinity among students in the class. at the end of the first and mid-year interviews, i gave students a list of their classmates and asked, for example: whom would you identify as a friend? if given the opportunity to choose your own group for a project that would determine your grade, whom would you choose? why did you choose those names? coordinating network surveys with interviews allowed me to identify affinity among students in the class, while also capturing students’ explanations of those affinities. the final interview elicited students’ reflections on their performance during the year, and their experiences of group work. field notes. as part of the larger study, i observed 84 days of classroom instruction over 40 weeks during the school year. a typical observation cycle focused on a single group of students for the duration of their collaborations (10 days). this analysis draws from field notes taken over four days (approximately 4 hours) of the focal group. in handwritten field notes, i attended primarily to a group’s interactions and talk, but also documented whole-class or teacher interactions as needed. i also recorded the results of summative assessments. after each observation, i elaborated on the handwritten field notes and converted them to sengupta-irving affinity through mathematical activity journal of urban mathematics education vol. 7, no. 2 36 electronic field notes—that is, “research protocols” (hatch, 2002)—before the next observation. other written artifacts. at the start of the school year, the teacher had students complete a biographical intake form, and required students to write a letter of introduction sharing what they thought was important for her to know as their teacher. periodically, students also did quick writes and short surveys to reflect on their learning experiences. i collected these written artifacts, which are used to provide descriptive detail on the students in this analysis. setting and participants the setting and participants for the study followed from the purposive selection of the focal teacher. for the study, i sought recommendations from faculty at a private teacher education program with an explicit emphasis on pedagogies of equity, including ci. ms. baker,1 a graduate of the program 8 years prior, was among the top nominees for the study. ms. baker was a middle-class, white woman who had completed her student teaching in the same mathematics department that boaler and colleagues studied. after graduation, she began teaching and later became a trainer in the use of ci. when i met ms. baker, she had just accepted a position to teach the lowest track of algebra i at a public high school. after meeting and expressing my research interests, she consented to the study of her algebra i class for an academic year. redbird high school (rhs) was in an affluent suburb of northern california. it embodied many of the social and economic tensions crossing over what once constituted the geopolitical line dividing “urban” and “suburban” schools in the united states. rhs was the most ethnically diverse high school in its district. the school population (approximately 1,900 students) was 35% latin@, 30% white, 16% asian, 14% filipino/a, and 4% african american. fifty percent of students at rhs qualified for free and reduced price meals, and 30% were designated english language learners. in the mid-1980s a high school in the predominantly ethnic minority and working class part of the city was closed due to enrollment declines. students were then bussed to the predominantly middle and upper-middle class center of the city, and to rhs in particular. yearlong algebra i courses at rhs were part of a newly created districtwide program targeting lower performing students. yearlong courses were not the norm (semester-long courses were); structurally, yearlong courses could keep students from advancing to the highest level of mathematics offered. in contrast, all other departments were eliminating academic tracking at the time of the study. ms. baker’s yearlong algebra i class had 16 students from grades nine through twelve. all of her students were from the ethnic minority and working class part                                                                                                                 1 all proper names are pseudonyms. sengupta-irving affinity through mathematical activity journal of urban mathematics education vol. 7, no. 2 37 of the city. on open-response written surveys, 11 students identified as female and five as male. all of the students reported this as their first experience using group work daily in mathematics. the focal group the four students discussed in this analysis are vivian, katrina, lorenzo, and james (each are further described). this group’s collaborations represent a unique and revelatory case for exploring collaborative learning, relational equity, and the cultivation of democratic community. first, their interactions led to one of the only instances all year of ms. baker rearranging groups just halfway through the typical 10-day cycle of collaboration. after class on the fifth day, i asked her why she disbanded the group and she explained that katrina was being left out. despite status treatments to support katrina (as ci would direct), ms. baker’s professional judgment was to reshuffle the groups instead. the second reason they were unique in the class was that despite their exceptionally brief (and only) collaboration in the first month of school, vivian, lorenzo, and james independently and exclusively identified each other as ideal collaborative partners on written surveys in mid-year interviews 5 months later. vivian. vivian was half filipina, one-quarter hawaiian, and one quarter white, but identified primarily as filipina in her introductory letter and interviews. vivian was a sophomore in her first year at rhs and lived with her grandmother, sisters, and cousin. she was estranged from her mother, and her father was incarcerated. vivian aspired to attend community college and then a large state university to pursue a career in law or middle school mathematics teaching. in whole-class discussions, vivian regularly volunteered to do warm-up problems and was typically the first to reach the participation point limit for the week. ms. baker resisted calling on vivian so others could speak, but would frequently call on her when no one else had the answer. katrina. katrina was an african american sophomore in her second year at rhs, as identified in her introductory letter and interviews. she and vivian identified each other as very close friends and like vivian, lived with her grandmother. in class, katrina used humor regularly. for example, i once observed ms. baker nearly lose her temper and katrina, perhaps sensing the same, suggested she stroke her earlobes and chant “woo-sa! woo-sa!” this reference to a popular movie (bad boys ii) diffused the tension with laughter. katrina’s classmates noted her use of humor and sarcasm as well. for example, amina explained she liked working with katrina because, “katrina explains in a funnier way,” while selma said katrina made others laugh (mid-year interviews). mathematics was katrina’s least favorite subject but she explained it was necessary to fulfill her goal of playing college basketball at michigan state university (initial interview). during sengupta-irving affinity through mathematical activity journal of urban mathematics education vol. 7, no. 2 38 the study, katrina was ineligible to play basketball because of low grades in algebra i and history. lorenzo. lorenzo was mexican and lived with his parents and brother, which he explained in his introductory letter and initial interview. lorenzo was a freshman who aspired to attend a 4-year university to study engineering. he considered mathematics his favorite subject and was the highest achieving student both semesters. his peers generally described him as quiet, hardworking, and very smart; in a mid-year interview, a classmate called him a “quiet boss.” i observed lorenzo to be consistently on time, working on the warm-up problem when the bell rang, and ready to volunteer. in a year-end survey asking what it means to be good at mathematics, lorenzo wrote: “to be good in math you have to show the best you got.” at the end of the year, ms. baker nominated lorenzo for the outstanding algebra student award, and he was the sole recipient across both tracks of algebra (yearlong and semester). james. james identified as a half vietnamese, half white ninth-grade student in his introductory letter. he lived with his parents and four siblings. i did not gain permission to “formally” interview james but we often spoke before or after class about school, coursework, and friends. james wanted to design cars and pursue a “mechanical degree” at a junior college. in his letter of introduction, he explained regretting not paying closer attention in prior mathematics classes. he also explained that he had no problem asking questions but that because he is a “fun person,” he is likely to talk a lot in his groups. in class, james regularly volunteered to do problems at the board and readily offered thoughts on solutions or methods during discussions. the researcher i am a former high school mathematics teacher who now conducts research and teaches graduate students and pre-service secondary candidates at a public university. i am a woman of color and the daughter of immigrants: i recognize that my own commitment to democratizing educational opportunity informs much of my professional work. at some base level, i wanted to see ms. baker succeed in her aims. through systematic reflexive practice, however, i worked vigilantly to tame my subjectivity (peshkin, 1988). qualitative validation strategies in the present analysis include researcher reflexivity, triangulation of multiple sources (interviews, observations, surveys, written artifacts), and prolonged engagement at the site (creswell, 2007). data analysis in this section, i first explain the overall approach to data analysis that led me to identify this collaborative group as a revelatory case of adolescent affinity sengupta-irving affinity through mathematical activity journal of urban mathematics education vol. 7, no. 2 39 within the classroom community. i then discuss the specific analyses of sociometric network surveys and field notes. in general, i first identified three of the four students (vivian, lorenzo, and james) as potentially representing a revelatory case of adolescent affinity based on outcomes of mid-year sociometric network surveys. in those surveys, students identified and explained whom they saw as ideal collaborative partners. this led me to assess all data collected in relation to the three students from the start to the midpoint of the school year. these data included: (a) letters of introduction, (b) interview transcripts, and (c) field notes. the analyses i present draw primarily on field notes, which revealed the three students had worked together only once, and in collaboration with a fourth student (katrina). sociometric network surveys. the premise behind sociometric surveys is that the relationships individuals have to one another can be determined and modeled as a network of ties (e.g., buchanan, 2002; cross, parker & cross, 2004). once students completed the network surveys in their mid-year interviews, i developed a model of affinities (ties) within the community. the model i present in the results section depicts whom students identified on surveys as ideal peer collaborators. the model shows how frequently someone was nominated (ordered highest to lowest vertically), which gives a sense of her or his positioning in the class, and which students nominated which—that is, a way to identify relationships in the class. field notes. the purpose in analyzing field notes was to explore why the three students emerged as such a unique collective in network survey outcomes. most striking (aside from meeting only once for four days, five months prior) was that they had been working with a fourth student (katrina) that none identified in the survey. i therefore focused my analytic efforts on what transpired among the students over those 4 days by using an analytic framework created to identify when students name, arrange, shift, and reformulate relationships among themselves during small group activity. mcdermott, gospodinoff, and aron’s (1978) framework uses the term contexts to define moments when participants name, shift, or reformulate relationships to one another in a given activity. based on a microanalysis of students during small group reading instruction, the framework was intended to attune ethnographic researchers to the way physical and verbal interactions prove significant to understanding shared activity. since 1978, this framework continues to inform studies of classroom discourse and peer microinteractions (see, e.g., cazden & beck, 1988; kendon, 1990), microethnographic studies of schools and microgenetic-historic studies of learning (see, e.g., corsaro, 1985; roth, 2014), and has even contributed to the development of other frameworks that describe peer influence in collaborative mathematics problem solving (see, e.g., engle, langerosuna & mckinney de royston, 2014). and yet, i leveraged the original frame sengupta-irving affinity through mathematical activity journal of urban mathematics education vol. 7, no. 2 40 work here because of its four specific criteria for identifying the formulation and reformulation of relationships in a group (i.e., contexts). these criteria aptly allowed me to consider how specific moment-to-moment peer interactions mark the formulation or reformulation of adolescent relationships. thus this framework provided the crucial logic linking in situ peer interactions during mathematical activity to the idea of developing affinity. according to the framework, the formulation and reformulation of relationships (i.e., contexts) become apparent in four ways: (a) when participants name them; (b) by changes in participants’ positioning, which represents a negotiation of relations; (c) when participants struggle to preserve relational order; and (c) when participants hold one another accountable to fit particular behaviors of the activity (mcdermott et al., 1978, p. 274). thus, analyzing field notes meant identifying and coding when student talk and behaviors reflected a particular criterion. with this more structured and systematic approach to analyzing field notes, i could explore the contexts emerging and later account for the mutual affinity vivian, lorenzo, and james reported in network surveys. analysis and results there are three parts to what is reported here. first, i present the results of sociometric network surveys administered during mid-year interviews. second, i present a relatively brief narrative account of “what happened” over four days as an overview. third, i present results of field note analyses that led to the primary outcome of this work: identifying a form of peer affinity grounded in shared mathematical activity rather than shared social attributes or history. identifying affinity through network surveys in the mid-year interview, students were asked to complete a survey identifying preferable collaborative partners for a major project. figure 1 depicts only those ties (as arrows) between students who mutually identified in response to the prompt. the numbers on the left in figure 1 reflect how often peers identified a student as a preferable partner. for example, three students perceived yasmin as a good partner while no one perceived thomas or kaitlynn as such. of importance to this analysis, vivian, lorenzo, and james identified one another and only one another as preferred partners. in general, as figure 1 depicts, there were few mutual identifications of affinity within the community by mid-year: a dyad of ofelia and amina, and a triad of vivian, lorenzo, and james. amina was palestinian from jordan and ofelia was mexican from mexico. they shared social attributes (both identified as female, both were recent immigrants, and both were english language learners) sengupta-irving affinity through mathematical activity journal of urban mathematics education vol. 7, no. 2 41 and history (they were friends and regularly worked together in science and mathematics). thus, their relationship was a relatively known quantity in the study of adolescent affinity. in contrast to amina and ofelia, the expressed affinity between vivian, lorenzo, and james could not be explained through social attributes or shared history: they were two boys and a girl, they were ethnically and culturally different, they were never friends, they shared no other classes together, and they collaborated only once, 5 months earlier. as the research question reflects, the survey results led me to consider how talk, interactions, and activity among the students might extend what is typically predicted as the basis for adolescent affinity in schools. figure 1. mid-year network model of mutually identified ideal collaborators. a narrative overview on the first day, ms. baker introduced the class to algebra tiles and showed them how to build and represent expressions. she built an expression, which groups would rebuild before determining the symbolic equivalent. james and lo sengupta-irving affinity through mathematical activity journal of urban mathematics education vol. 7, no. 2 42 renzo discussed how to manipulate the tiles. vivian worked on problems independently and checked her answers with james and lorenzo. vivian also doodled, sang, and texted under the table with katrina. on the second day, the group worked with algebra tiles and mats to simplify expressions. this meant, for example, flipping and shifting tiles such that − (x−2) would become − x+2. ms. baker circulated among groups. after watching vivian work, she gave her a high five. ms. baker then noticed katrina had not done her work and said to vivian, “a good friend would keep another on track.” as ms. baker left, vivian checked on katrina’s progress but shortly thereafter, katrina was off-task again. vivian called to katrina a few times to get her attention; when ms. baker returned to see little progress from katrina, she told vivian she was not being a true friend unless she helped katrina focus. ms. baker assisted katrina and then watched vivian help katrina with the next problem. throughout the period, katrina would get distracted again and vivian would admonish her to stay focused. at the end of class, vivian and katrina were both giggling and shoving each other, when vivian abruptly stopped and asked james for help. james and vivian disagreed on their answer and james tried it again. james then asked lorenzo about it as vivian showed lorenzo her work. james, looking over lorenzo’s shoulder, pointed to vivian’s paper and said: “you’re not supposed to do that yet.” vivian replied, “look, exponents first then multiple [sic] that.” james thought for a moment and erased his work as class came to an end. on the third day, ms. baker asked students to take turns building, simplifying, and representing increasingly difficult algebraic expressions as a group. lorenzo built and solved the first expression. vivian pushed her paper toward katrina and showed her the answer. lorenzo and james built the second expression in two different (and incorrect) ways. james and lorenzo asked vivian to check their work, and she agreed with james. vivian then flipped and shifted the tiles as the boys looked on. katrina continued working alone on her paper and never touched the tiles. when vivian and james compared their written simplified expression with what they determined through manipulating the tiles, they found it did not match. ms. baker came over and looked at them in a slightly exasperated way. she took a breath and built the expression for them. katrina declared she had that solution and ms. baker asked the others if they listened to katrina. vivian, lorenzo, and james remained silent as ms. baker said, “what happened? what is happening here? i’m feeling bad energy here,” and then left. vivian continued to work with james and lorenzo. when an assistant teacher came to check on the group’s progress, james showed him that he was done. when the assistant asked if katrina was done, james replied, “no, you have to help her out.” vivian looked up and said she was just helping katrina a moment ago. as class ended on the third day, lorenzo, james, and vivian were discussing a task. as james began reading his answer aloud, vivian and lorenzo chimed sengupta-irving affinity through mathematical activity journal of urban mathematics education vol. 7, no. 2 43 in, in unison, to complete it. the three laughed and cheered in celebration. vivian then turned and read her answer to katrina. as katrina wrote it down, vivian looked up and noticed a male student passing by. “that guy is cute,” she said to katrina who looked up and stopped writing. vivian reached out and took katrina’s paper to finish the problem as katrina remarked, “you’re causing me to get a bad education!” on the fourth day, the students corrected homework and took an individual quiz on the week’s material. the quiz results were: james, 98%; lorenzo, 94%; vivian, 89%; and katrina, 87%. on the fifth day, ms. baker rearranged the groups, only halfway through the typical 10-day cycle. identifying affinity through talk, activity, and interaction these results are organized by criteria that focus on participant’s talk, actions, and interactions to identify the formulation and reformulation of relationships (i.e., contexts) in a group. the results extend conceptions of adolescent affinity as governed by shared social attributes or history to include affinity as forged through shared mathematical activity. criteria 1: contexts become apparent when named. the contexts of friendship and a separate relationship of mathematical activity were named at different times during the group’s collaborations. for example, on the second day, ms. baker’s use of friendship as the rationale for vivian to keep katrina on track named friendship as a salient context. on the third day, when ms. baker realized the others had been ignoring katrina’s mathematical work, ms. baker used the phrase “bad energy,” which recognized the triad as formulating a context apart from katrina. on that same day, when james told the teaching assistant to help katrina, james implicitly named two formulations within the group: katrina and we share our mathematical work. notably, vivian responds by saying they were just helping katrina, as though re-naming james’ formulations as: we share our mathematical work and i help katrina. at the end of that day, katrina told her friend vivian she was causing her to get a “bad education.” in so doing, katrina was naming friendship as a salient context within the group (just as ms. baker had on the first day), as a context apart from one that affords a “good education” (perhaps because it is not grounded in mathematical activity), and as a context that only she and vivian share. these examples collectively point to participants naming two different contexts in the group: friendship and one characterized by mathematical work. moreover, these moments of naming contexts simultaneously marked talk and interactions isolating katrina from the others. in that sense, as the triad formulated a relationship based on mathematical activity, they were also formulating a relationship to katrina that cast her as a problem. sengupta-irving affinity through mathematical activity journal of urban mathematics education vol. 7, no. 2 44 criteria 2: contexts become apparent through changes in participant positioning. over the course of 4 days, vivian physically and verbally formulated a relationship with james and lorenzo around mathematical activity. the first indication of vivian’s changing position toward lorenzo and james came at the end of the second day when she was interacting with katrina and abruptly stopped to ask james a mathematical question. that moment signaled a physical and verbal change in her position (i.e., she leaned in, addressed him directly, and eventually engaged lorenzo in the task). from then on, she increasingly turned to lorenzo and james to manipulate tiles, compare answers, debate disagreements and finally, on the fourth day, to shout happily in unison with them upon completing a task. the cascading effect of moments of repositioning toward lorenzo and james indicated a relationship formulating in mathematical activity. in contrast to vivian’s changing position toward lorenzo and james, she changed very little in relation to katrina. vivian began by working alone and behaving off-task with katrina on the first day, shoving and being silly with her on the second, and nudging her on the third to notice a “cute guy.” although she also passed, read, and eventually wrote answers for katrina, vivian could not formulate a relationship that bridged her activities with katrina and those with lorenzo and james. that is, she could not bring together behaviors of friendship with behaviors of mathematical work. this is also evident by virtue of vivian being most often blamed for katrina’s mathematical isolation—first by the teacher (“a good friend would keep another on track”) and later by katrina herself (“you’re causing me to get a bad education”). in such instances, vivian’s position on the border between contexts was treated as the cause of katrina’s isolation. in actuality, a lack of change in positioning by lorenzo and james could have been identified as similarly causing katrina’s isolation. the relationship james and lorenzo formulated and maintained apart from katrina was afforded (if not sanctioned) repeatedly over the 4 days. on the first day, for example, when ms. baker implored vivian to help katrina, she did not commensurately compel james and lorenzo to help. though ms. baker named friendship as a salient context in the group with her words, she may have also been inadvertently creating a same-sex context for vivian and katrina that allowed james and lorenzo to abdicate their responsibility to help. there were other opportunities for lorenzo and james to position and formulate a context that would incorporate katrina in mathematical activity. for example, when katrina had the correct solution and they did not (third day), they could have subsequently engaged katrina in the task by speaking to her or leaning in her direction—that is, by changing their position. additionally, when the teaching assistant checked in on the group, james reported katrina needed help but made no effort to help, thereby maintaining katrina’s position of isolation in a context apart from activity. thus despite opportunities for lorenzo and james to formulate or reformulate sengupta-irving affinity through mathematical activity journal of urban mathematics education vol. 7, no. 2 45 a relationship that incorporated katrina in mathematical activity, the opportunities were largely unrecognized, unnamed, or ignored. criteria 3: contexts become apparent in significant moments that maintain relational order. there were at least four significant moments that maintained relational order. because some of these examples were discussed previously, this discussion is brief. first, when ms. baker named vivian and katrina’s friendship, it communicated a relational order where friendship supersedes group membership as what compels helping others. second, it was a significant moment when katrina determined the correct solution because it challenged the relational order of being isolated from the context of mathematical work. whereas the triad could have cheered and folded katrina into their discussions and activities, they instead reconciled to the same relational order. third, when james told the teaching assistant to help katrina, he maintained a relational order by which he was not responsible for helping her. fourth and finally, it was a significant moment when vivian directed katrina’s attention to the “cute guy” while reading answers to her because it starkly contrasted with what happened moments previous: lorenzo, vivian, and james cheering for themselves in completing a task. the relational order embedded in such a contrast nearly speaks for itself: james, lorenzo, and vivian shared a relationship that celebrated mathematical activity, while katrina and vivian shared a relationships that was minimally mathematical and more focused on their shared social attribute (gender and [assumed] sexuality) and history of friendship. criteria 4: contexts become apparent when participants hold others accountable to activity. vivian made most apparent how accountability to behaviors of activity constituted two separate contexts in the group. vivian started by holding katrina accountable to work, telling her to “shut up” when she got off-task. vivian said this with humor and a smile, as one might see among friends. vivian’s attempts to hold katrina accountable on the first day gave way to formulating a different relationship in the group. vivian became increasingly accountable to james and lorenzo (and they to her) through questioning, comparing answers, and supporting each other’s understandings—all of which are grounded in shared mathematical activity. as vivian became accountable to james and lorenzo, she stopped holding katrina accountable for producing anything mathematical and instead, by passing answers to her, held her accountable only to the behaviors of mathematical work (i.e., filling in answers). this lack of accountability between vivian and katrina foreclosed any chance of reformulating their friendship into a relationship of shared mathematical activity. when katrina chastised vivian for her “bad education” on the third day, she showed how she implicitly held vivian accountable for what she could accomplish in the group. katrina did not, however, hold james, lorenzo, or herself explicitly accountable; she still perceived friendship as demanding accountability in a way that group membership did not. sengupta-irving affinity through mathematical activity journal of urban mathematics education vol. 7, no. 2 46 summary. in figure 2, i offer an illustration of the criteria-based analysis just presented. depicted visually, the two contexts first become apparent as parallel dyads, with one student intermittently engaging in both. then, the student becomes increasingly central to the other context while minimally engaging in the first. finally, a triadic relationship results with a student set largely apart. figure 2. depiction of two salient relationship formulations (contexts) operating in a group; network surveys suggested the final triadic partnership was seen as ideal. in figure 2, a solid line inscribes lorenzo (l) and james (j), denoting the mathematical focus they share throughout. a dashed line connects vivian (v) and katrina (k), denoting friendship, which is how they primarily engage throughout. katrina remained at the periphery of mathematical work and was therefore never inscribed in a solid line. vivian, in contrast, reached toward the other partnership when mathematically challenged (solid line), reformulated her position, and finally joined in the shared activities of lorenzo and james. discussion the outcome of this analysis is the identification and naming of the workship as an affinity forged through shared mathematical activity. the workship rhetorically and literally identifies how shared work forges new possibilities for peer relationships within a diverse learning community. the idea of workships comes in response to what teachers (and researchers) often treat as foreseeable or predictable adolescent affinities for collaboration (e.g., “girls are most comfortable with girls in math…” and “those two are friends...”). i contend that peer relationships forged during mathematical activity are as consequential to issues of equity as affinities we predict prior to mathematical activity. that is, i am identifying how shared mathematical activity shapes a student’s relationship to another and sengupta-irving affinity through mathematical activity journal of urban mathematics education vol. 7, no. 2 47 not how relating to one another shapes a student’s relationship to mathematics. therefore the appeal of workships within the greater project of cultivating democratic learning communities is that they represent generative relationships born of diverse youth learning to work together by working together. to be clear, whereas the contrast with friendship in this group made the workship even more apparent, such a contrast was coincidental and not necessary for recognizing or naming the workship in this analysis. workships and relational equity if the workship is to be considered a promising pedagogical tool for cultivating democratic learning communities, its relationship to relational equity must be interrogated. workships envision unlikely partnerships forged through shared mathematical activity, while relational equity assures those partnerships emerge as sites of appreciation, communication, and respect. in this case, the workship embodied relational equity: we saw communication, appreciation, and respect for mathematical thinking among lorenzo, vivian, and james. that could not be said, however, of the group as a whole. by formulating relationships apart (james and lorenzo versus katrina; james, lorenzo and vivian versus katrina), the group as a whole reflected limited communication, appreciation, or respect for one another, let alone shared mathematical activity: katrina never touched the tiles, she never asked nor was she asked a mathematical question, and she never offered nor received help from lorenzo and james. in fact, when ms. baker identified katrina as the only one with the correct solution on the third day, the respect and support for learning together was so skewed that silence prevailed instead of a newly created context of shared mathematical activity among them all. one way to account for the workship is that these were simply highperforming mathematics students seeking each other out. there is some evidence to support this view: lorenzo, james, and vivian achieved the highest grades at mid-year (99.2%, 95.5%, and 94.9%, respectively), though two other students also earned a’s. in contrast, katrina’s mid-year mathematics grade was 74.4%. when vivian was asked in her mid-year interview why she chose james and lorenzo, vivian explained lorenzo was someone who got good test scores and was a fast thinker. she similarly thought of james as smart and added, “he’s pretty up there with me on how we think.” lorenzo explained he chose vivian and james because: “we were getting kind of [the] same answer every time we’re together and we’re doing work…we were getting our work done. every—everything was perfect” (mid-year interview). though james’ guardians did not consent to him being interviewing, james wrote “would help me with the problems” on his survey to explain his choice of vivian and lorenzo. there is nothing in the conceptualization of a workship that would preclude perceptions of “ability” as partially driving affinity. that said, how students develop perceptions of one another’s sengupta-irving affinity through mathematical activity journal of urban mathematics education vol. 7, no. 2 48 ability is important. to be consistent with the vision of engendering relational equity, “ability” must be conceived of broadly. this broader vision occurs when, for example, teachers use open-ended tasks to elicit and reward a range of abilities (e.g., lotan, 2003), employ ci status treatments to help assign competence (cohen & lotan, 1995; webb, 2009), or, more generally, organize opportunities for multidimensional learning (boaler, 2006b). certainly teachers like ms. baker can organize opportunities for students to create and cultivate unlikely alliances through collaboration, but how students rise to those opportunities and with what consequences deserves more attention. in the context of this study, it means asking: how do we make sense of the workship in terms of katrina’s eventual isolation? katrina’s isolation in culturally or racially heterogeneous settings, we know access to learning opportunities can vary for students where some are more influential than others (e.g., engle et al., 2008) or are positioned as having more or less authority and ability (e.g., sengupta-irving, redman, & enyedy, 2013; gresalfi, martin, hand, & greeno, 2009; langer-osuna, 2011). often, those students who become marginalized are also those who come from non-dominant racial, cultural, linguistic, or gender groups (kurth, anderson, & palincsar, 2002). katrina was different from the others in multiple ways that positioned her from the start as most vulnerable. she had the lowest grade of the four students, was on academic probation, and was the only one to identify mathematics as her least favorite subject. katrina was also the only african american female student in the class—an otherwise underrepresented and marginalized community in stem (science, technology, engineering, and mathematics) education more generally (katehi, pearson & feder, 2009; ong, wright, espinosa, & orfield, 2011). ultimately, to make causal claims about katrina’s isolation is beyond the scope of this analysis. what can be argued is that her quiz score did not suggest she was mathematically “less able” than her peers. her understanding that mathematics was a gatekeeper to realizing her dreams of becoming a college athlete certainly counter any notion she lacked investment in her performance. moreover, her behaviors in the group do not suggest that she could not conceptually or practically participate in the mathematical activities or discussions of the workship. indeed, at one point katrina was the only one to get the correct solution, and yet she was never able to establish a prominent position in shared mathematical activity thereafter. most important, katrina’s isolation highlights how a definition of equity that is outcome-oriented (i.e., 87% on her quiz) fails to appreciate what relational equity captures. an outcomes approach to equity obscures how katrina was rarely respected, treated, or supported as a mathematically capable peer. in her mid-year interview, katrina confided that she saw herself as a good group partner because she understood what was going sengupta-irving affinity through mathematical activity journal of urban mathematics education vol. 7, no. 2 49 on most of the time, but that she also knew her peers did not see her that way. in her sociometric network survey, katrina identified only one student as an ideal partner: lorenzo. she explained, “because he doesn’t really mess around and stuff,” suggesting she was oriented toward work even if never incorporated into the workship. by the standards of adolescent affinity, one might argue that katrina’s friendship with vivian could have signaled greater possibilities of shared mathematical activity than were seen. indeed, vivian was unable to broker her friendship with katrina and the formulation of a workship within the group. common to both contexts, vivian was in many ways best positioned to bring about relational equity within the group. as the center image in figure 2 would suggest, vivian had opportunities to bridge between the friendship she shared with katrina and the workship she created with lorenzo and james, but instead moved from one toward the other. for vivian, similar to james and lorenzo, katrina was seen as a “problem” in the group even when that “problem” had the solution. wortham (2004a) similarly examines how teachers and students reformulate the identity of a student named tyisha from “good student” to “outcast,” where, by drawing multiple resources, her identity as an outcast thickens (holland & lave, 2001) over time. this analysis cannot speak to the thickening of katrina’s identity as an outcast (what might occur over time in a specific local context) but does indicate, as wortham (2004a) explains, how particular events can signal her positioning as a problem or outcast at a given time (p. 166). thus, unlike most studies of ci, this analysis attempts to explore how students may take up the aims of ci (relational equity, for example) as part of their interactions with others. after all, having three high-performing students declare an exclusive affinity for one another ran directly counter to ms. baker’s goals of equitable learning. and yet, i consider the workship a potential pedagogical tool of democratic learning communities because making peer collaboration a generative experience, making mathematical learning in a diverse community a generative experience, means recognizing that some of this work is beyond the direct management of teachers. or at a minimum, it takes recognizing the significant challenge for one high school teacher to pedagogically counter traditional “smart student” hierarchies established in many years of prior schooling. the limits of pedagogy in stratified settings under the rubric ci, the responsibility to fold katrina into the others’ activities lies primarily with the teacher and not with the students. ms. baker’s attempts at status treatments, asking if anyone listened to katrina for example, largely failed. she also inadvertently set up some of the prevailing dynamics, particularly between vivian and katrina. calling on friendship as what keeps people on track rather than, for example, the task itself or a commitment to the group overall, al sengupta-irving affinity through mathematical activity journal of urban mathematics education vol. 7, no. 2 50 lowed james and lorenzo to largely abdicate any responsibility in attending to katrina. this discursive move at the start—gendered and affinity-oriented— arguably set in motion the contexts as formulating apart rather than together. thus, she saw little pedagogical recourse for the dynamics prevailing in the group. as i identify the shortcomings of ms. baker’s pedagogical moves, i must also acknowledge how she used a pedagogy of equity within a highly stratified school environment. ci was not uniformly adopted in the mathematics program, and the messages of competition and hierarchy were embedded in how the students understood their current placement (i.e., lowest track of mathematics) and their future placement. by examining ci within a school setting that is institutionally different (though not necessarily demographically different) to that of boaler and colleagues (boaler, 2006a, 2008; boaler & staples, 2008) we begin to see the affordance and limitations of pedagogy in advancing relational equity when the institution may be organized to undermine such efforts. conclusion this case study was set in the lowest track of a racially and ethnically diverse and economically stratified suburban/urban high school. the teacher’s pedagogical aims included cultivating a democratic learning community in which students expressed relational equity—respect, appreciation, and communication through mathematical activity. at mid-year, a network survey administered during individual student interviews showed how shared social attributes or history, which are often used as a basis for predicting adolescent affinity, could not fully account for who students identified as ideal collaborative partners. through analyses of field notes, i explored why three students, whose triadic preferred partnership emerged as distinct from all others in the network, developed an affinity through mathematical activity—what i termed a workship. moreover, this workship reflected relational equity, a link that allows us to imagine a closer relationship between collaborative learning and the greater social project of cultivating democratic learning communities among diverse people. for educators, anticipating and encouraging diverse peer relationships is both ideologically and practically linked to the development of supportive learning communities. workships remind us that democracies are built on the notion that everyone can contribute and that everyone should be valued, with or without shared social attributes. the tension presented in this case is that the democratic appeal of workships, of cultivating diversity in how and why adolescents value one another, can emerge in such a way that fails to incorporate everyone in the collective. that is, achieving relational equity uniformly in a group, let alone a classroom community, is a significant challenge even for motivated and experi sengupta-irving affinity through mathematical activity journal of urban mathematics education vol. 7, no. 2 51 enced teachers like ms. baker. when considered in light of the institutional setting, we can see constraints on a teacher’s pedagogical efforts to develop democratic learning communities. nonetheless, educators must endeavor to support students in seeing the value of unlikely peer collaborators, of creating alliances for learning that challenge the logic of social and academic hierarchies in their schools and communities, because doing so is among the greatest tools in our arsenal for assuring the longevity of democratic opportunity. for researchers, this analysis offers a new vantage point for exploring adolescent affinity and mathematics learning, where the conditions for affinity are set in motion pedagogically, and the actualization of affinity may be expressed through shared activity. in terms of research on ci, the 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(2005). socialization beyond the speech event. journal of linguistic anthropology, 15(1), 95–112. journal of urban mathematics education july 2013, vol. 6, no. 1, pp. 6–18 ©jume. http://education.gsu.edu/jume beth herbel-eisenmann is an associate professor in the department of teacher education at michigan state university, 620 farm lane, erickson 316, east lansing, mi 48824; email: bhe@msu.edu. her research interests include drawing on discourse and critical discourse studies to work with secondary mathematics teachers, examining, in particular, issues of authority, positioning, and voice in mathematics classrooms and mathematics teacher professional development. tonya gau bartell is an assistant professor in the department of teacher education at michigan state university, 620 farm lane, erickson 116n, east lansing, mi, 48824; email: tbartell@msu.edu. her research focuses on the tools and experiences that can support teachers’ development of equitable pedagogical practices with explicit attention to social justice, culture, race, and power in mathematics education. strong is the silence: challenging interlocking systems of privilege and oppression in mathematics teacher education beth herbel-eisenmann michigan state university tonya gau bartell michigan state university m. lynn breyfogle bucknell university kristen bieda michigan state university sandra crespo michigan state university higinio dominguez michigan state university corey drake michigan state university in this essay, the authors provide a rationale for the need to break the silence of privilege and oppression in mathematics education. they begin by providing a brief rationale from their personal and professional perspectives, which includes background about planning and executing the privilege and oppression in the mathematics preparation of mathematics teachers educators (prompte) conference, which motivated this special issue of the journal of urban mathematics education. the authors then move into a (more typical) literature-based rationale for a focus on exploring and engaging with systems of privilege and oppression in relationship to themselves as mathematics teacher educators and in the preparation of new mathematics teacher educators. keywords: mathematics education, privilege and oppression, mathematics teacher educators athematics teacher educators (mtes) are often hesitant (or shy) about disrupting (or breaking) the silence that has taken a strong hold of the inequitable systems of privilege and oppression (e.g., racism, classism, sexism, heterosexism, ableism) within which we operate. in particular, although mtes have begun to talk about these issues in relation to the preparation of mathematics teachm herbel-eisenmann et al. privilege and oppression stinson, d. w., & spencer, j. a. (eds.). (2013). privilege and oppression in the mathematics preparation of teacher educators [special issue]. journal of urban mathematics education, 6(1). 7 ers (mts) and mathematics teaching, we rarely talk explicitly about them with respect to our own preparation as mtes and our work in preparing future mtes. a fruitful conceptual exploration might be to consider the ways in which identifying, confronting, and transgressing systems of privilege and oppression with mts may be similar to or different from the work we do ourselves as mtes and our work with future mtes with respect to these systems. such examinations can help mtes to further specify and conceptualize research agendas, frameworks, approaches, and strategies for taking action toward equitable systems within the programs in which we work (i.e., preparing future mtes and future mts). our hypothesis is that concentrated attention to thoughtful discussion and action related to identifying, understanding, and confronting the interlocking systems 1 of privilege and oppression can improve mathematics teacher education and, ulti 1 by “interlocking systems,” we mean that the oppression of some people does not exist without systems supporting the unearned privilege of other people. for example, racism does not exist without systems supporting white privilege. that is, “racism is understood to be widespread and ingrained in society, rather than manifested only in the actions of a few ‘irrational’ people” and, “through this perspective, racism is perceived as an entity that affects everyone in society, benefiting some and victimizing others” (s. marx, 2006, p. 5). m. lynn breyfogle, a former professor of mathematics, is associate dean in the college of arts and sciences at bucknell university, 113 marts hall, lewisburg, pa 17837; email: mbreyfog@bucknell.edu. her research has focused on changing preand in-service teachers' beliefs about mathematics and supporting effective classroom discourse. kristin bieda is an assistant professor in the department of teacher education at michigan state university, 620 farm lane, erickson 317, east lansing, mi 48824; email: kbieda@msu.edu. her research interests focus on the engagement of all students in mathematical practices such as argumentation, and preparing secondary mathematics teachers to develop a teaching practice where such engagement regularly occurs. sandra crespo is an associate professor in the department of teacher education at michigan state university, 620 farm lane, erickson 116p, east lansing, mi 48824; email: crespo@msu.edu. her research interests focus primarily on preservice elementary teachers and their development as learners of mathematics and mathematics teaching; her work crosses multiple boundaries as she conducts research in the united states, canada, and the dominican republic. higinio dominguez is an assistant professor in the department of teacher education at michigan state university, 620 farm lane, erickson 116k, east lansing, mi 48824; email: higinio@msu.edu. he participated as a post-doctoral fellow at cemela (center for the mathematics education of latinos/as) and is currently the principal investigator of the nsf-funded project titled: career: reciprocal noticing: latino/a students and teachers constructing common resources in mathematics. corey drake is an associate professor and director of teacher preparation in the department of teacher education at michigan state university, 620 farm lane, erickson 118a, east lansing, mi, 48824; e-mail: cdrake@msu.edu. her research interests include the design of teacher education experiences and contexts to support new teachers in learning to connect to students’ resources and knowledge bases, as well as the use of curriculum materials as learning tools for teachers. https://bl2prd0512.outlook.com/owa/redir.aspx?c=j70rxspyqkkvhcdc-ylxcx7uemm0wdai_n0bivxhntakoymlk4-yepepsqgy9zrzm10oat9xeho.&url=mailto%3ambreyfog%40bucknell.edu https://bl2prd0512.outlook.com/owa/redir.aspx?c=j70rxspyqkkvhcdc-ylxcx7uemm0wdai_n0bivxhntakoymlk4-yepepsqgy9zrzm10oat9xeho.&url=mailto%3akbieda%40msu.edu https://bl2prd0512.outlook.com/owa/redir.aspx?c=j70rxspyqkkvhcdc-ylxcx7uemm0wdai_n0bivxhntakoymlk4-yepepsqgy9zrzm10oat9xeho.&url=mailto%3acrespo%40msu.edu https://bl2prd0512.outlook.com/owa/redir.aspx?c=j70rxspyqkkvhcdc-ylxcx7uemm0wdai_n0bivxhntakoymlk4-yepepsqgy9zrzm10oat9xeho.&url=mailto%3ahiginio%40msu.edu mailto:cdrake@msu.edu herbel-eisenmann et al. privilege and oppression stinson, d. w., & spencer, j. a. (eds.). (2013). privilege and oppression in the mathematics preparation of teacher educators [special issue]. journal of urban mathematics education, 6(1). 8 mately, will impact mts’ and students’ learning experiences in mathematics classrooms, especially students who have been historically underserved in schools. we believe that we need to break this silence and provide venues in which to plan and take thoughtful action in relationship to systems of privilege and oppression, develop strategies for working on these systems amongst ourselves and with our graduate and undergraduate students, and invite others into such conversations. in this essay, we provide a rationale for the need to break this silence. we begin by providing a brief rationale from our personal and professional perspectives, which includes background about the planning and executing of the privilege and oppression in the mathematics preparation of mathematics teachers educators (prompte 2 ) conference, which motivated this special issue of the journal of urban mathematics education (jume). we then move into a (more typical) literature-based rationale for a focus on exploring and engaging with systems of privilege and oppression in relationship to ourselves as mtes and in the preparation of new mtes. breaking the silence our story(ies) this essay denotes, in many ways, the beginning of a discussion that we recently began amongst some mathematics education faculty in teacher education at michigan state university (msu) and m. lynn breyfogle at bucknell university. although we have each had interests in various aspects of equity in our scholarship and work with prospective and practicing teachers, prior to summer 2012 we had not come together to talk about this common commitment. an internal grant we proposed to msu’s new collaborative research in educational assessment and teaching environments for science, technology, engineering and mathematics (create for stem) institute provided a context in which to begin these conversations. we decided to focus the proposal on breaking the silence related to systems of privilege and oppression with respect to mtes and met to talk about our rationale for this work and for a focus on mtes. in our first meeting to discuss the issues put forth here, we shared examples related to the ways in which we have struggled with issues of privilege and oppression as mtes. our discussion was the first instance in which we collectively began to disrupt the silence by recalling tensions that occur for us in our work as 2 privilege and oppression in the mathematics preparation of teacher educators (prompte) conference (funded by create for stem institute through the lappan-phillips-fitzgerald cmp 2 innovation grant program), michigan state university, battle creek, mi, october 2012. any opinions, findings, and conclusions or recommendations expressed herein are those of the authors and do not necessarily reflect the views of the funding agency. herbel-eisenmann et al. privilege and oppression stinson, d. w., & spencer, j. a. (eds.). (2013). privilege and oppression in the mathematics preparation of teacher educators [special issue]. journal of urban mathematics education, 6(1). 9 mtes. here are some of our examples (some of which have also been reported in the literature):  how do i unpack my own privilege and what it does for the ways we interact and engage with things like readings for the course?  as a white teacher educator, i often find that white prospective teachers tend to just agree with me. how do i get them to engage more deeply with these issues? (see also gillespie, ashbaugh, & defoire, 2002.)  what can i do when my students resist my talk about race because they think i have an “agenda”? (see also aguirre, 2009.)  an issue i have run into is that mts want to jump in to “solve the problem.” that is, they want to be in charge of the solution rather than working carefully in partnership on solutions. what can i do to get them to sit with these issues and tackle them thoughtfully? how can i help them understand that potentially fast, careless, and well-intentioned contributions can lead to perpetuating imperialism?  there is such a lack of comfortableness with talking about issues of privilege. i’m not sure how to tackle that sometimes. for example, in one class a prospective teacher said something about a child and her parents not caring and other prospective teachers in the class pushed back, sometimes in good ways but at other times in potentially damaging ways. how do i get those good ways to happen more often?  as a white mte, i’m unsure how to handle it when discussing these issues in settings where there are students from many different racial backgrounds. for example, what do i do when a student of color voices some of the meta-narratives that indicate that outcomes are all about hard work and do not relate to things like race? as can be seen from how we engaged our voices to begin to break the silence, even when university faculty have been engaging in work related to equity, we do not have answers for the many dilemmas that we confront in this work. we talked about how these kinds of stories and experiences impacted our work with doctoral students, too, as we prepared them to be mtes. we also began to talk about our identities and experiences in order to better understand one another and ourselves. in particular, we talked about how these identities related to our memberships in “target” or “non-target” groups (batts, herbel-eisenmann et al. privilege and oppression stinson, d. w., & spencer, j. a. (eds.). (2013). privilege and oppression in the mathematics preparation of teacher educators [special issue]. journal of urban mathematics education, 6(1). 10 2002), based on whether a particular identity has been historically marginalized or not. for example, latina, woman, dis-abled, gay, and child would all be target groups as compared to white, man, able-bodied, straight, and adult. additionally, we recognized that the more non-target identity groups one belongs to, the better the odds are for positive life outcomes, due, in part, to unearned privileges. we have also come to talk about systems of privilege and oppression as acting on at least four levels: the personal, interpersonal, institutional, and cultural (batts, 1998, 2002; harro, 2000a, 2000b). the four levels have helped us to think about generating examples of how these systems might be instantiated, for example, in one’s beliefs, people’s interactions, institutional policies, and in our cultural images and stereotypes associated with different identity groups. after these initial discussions, we were left with many questions. for example: how might we better structure these conversations? what knowledge of systems of privilege and oppression is reasonable for new teachers and mtes to take with them into settings where there are multiple narratives about these systems? what are some reasonable action strategies for actually addressing these broader systems that prospective teachers and mtes can take with them when they leave our institution? we wanted to take these questions and others to a broader group of mtes to explore. we were fortunate to receive the internal grant, which enabled us to host a small conference in michigan in october 2012. about 40 mtes participated, representing about 25 different institutions across the united states. we chose to focus the conference on racism and classism as simultaneous systems that oppress some people while granting privileges to others. we selected these two systems to be illustrative of issues to consider when working on interlocking systems of privilege and oppression, more generally. the conference was dually structured to focus on reflection on our identities and systems of privilege and oppression and on actions we might take as mtes. the sessions examined the histories and dynamics unique to racism and classism as well as how these systems intersect and reinforce each other in the context of mathematics education. not all participants experience the interlocking systems of privilege and oppression in the same ways, of course. our identities occur at the moving intersections of various categories. in fact, as beauboeuf-lafontant (2002) contends, teachers (and we would add mtes) are both products of and culprits within an inequitable set of systems. core to these sessions was the assumption that we can become as passionate about dismantling the systems from which some unjustly benefit as we are about eradicating the systems that oppress us. the conference sessions sought to provide multiple contexts in which people challenged themselves and each other to ask deeply and continually: now that we see, what are we going to do about it? herbel-eisenmann et al. privilege and oppression stinson, d. w., & spencer, j. a. (eds.). (2013). privilege and oppression in the mathematics preparation of teacher educators [special issue]. journal of urban mathematics education, 6(1). 11 the experiential sessions and activities were led by facilitators from allies for change 3 and allowed us to reflect on systems of privilege and oppression, both personally and in the context of our work as mtes. we negotiated norms for having difficult conversations, for example, we agreed not to freeze each other in time and to consider both the intent and the impact of things we said and did. these kinds of conversations are personal and emotional, and learning to talk about interlocking systems of racism and classism in ways that break the silence involves a long-term process and vulnerability. they also require generosity, forgiveness, and support. we envisioned action in these sessions by doing things like practicing strategies for interrupting oppression, drawing on scenarios that participants had experienced in their mathematics methods courses that were similar to the ones we shared previously. the action-oriented sessions were structured to support taking action in concrete and specified ways by focusing on particular products and action plans to be accomplished. for example, this jume special issue, other co-authored publications and an edited book, conference presentation proposals, and grant proposals were discussed, conceptualized, and worked on in small groups. a literature-based rationale our literature-based rationale for the need to break the silence is primarily based on three important points: 1. although an increasing percentage of school children in the united states are children of color, poor, and from homes where family members speak languages other than english—all potential sources of privilege and oppression—mts and mtes remain fairly homogeneous along these demographic lines (hollins & guzman, 2005). 2. although the literature on preparing teachers in the united states to work in diverse classrooms, schools, and communities has recently been growing, there is a paucity of work on preparing mtes to facilitate this kind of work. this work includes not only preparing graduate students to be new mtes but also examining the work currently conducted by practicing mtes themselves (see mcleman, vomvoridi-ivanovic, & chval, 2012, for initial work examining the practice of mtes). 3 allies for change is a nationally recognized network of educators who share a passion for social justice and a commitment to creating and sustaining life-giving ally relationships and communities (see www.alliesforchange.org). http://www.alliesforchange.org/ herbel-eisenmann et al. privilege and oppression stinson, d. w., & spencer, j. a. (eds.). (2013). privilege and oppression in the mathematics preparation of teacher educators [special issue]. journal of urban mathematics education, 6(1). 12 3. although the growing literature on equity in mathematics education has been framed in various ways to address issues of oppression and (sometimes) emancipation (e.g., battey, 2013; gutiérrez, 2007; gutstein, 2006), we think that anti-oppression activism also requires confronting the privilege granted by institutions and society through addressing interlocking systems of privilege and oppression in order for our mathematics education community to thoughtfully avoid replicating imperialism (i.e., enabling the powerful to act and speak on behalf of the oppressed). below, we discuss, in turn, more about each of these points. diverse schools and relatively homogeneous teaching populations. a reality in mathematics education is that the teaching population in public schools and universities in the united states is fairly homogeneous in terms of race, class, and language facility (i.e., white, middle class, and english monolingual). for example, nearly 90% of teachers in the united states are white (national center for education information, 2005). available statistics about students in pre-k–12 classrooms, however, indicate that the students are much more racially, economically, and linguistically diverse than the teaching population. nationally, for example, 43% of students enrolled in public schools are students of color (fry, 2007). scholars have argued that these differences have serious implications for teaching and learning (e.g., ladson-billings, 1994; gay, 2010; larson & ovando, 2001). consider the issue of racial difference, for example. it is typical for white teachers to claim to be “color-blind” and treat all students the same (bell, 2002). this color-blindness, however, masks the inequities created by race, class, and power (johnson, 2002). without explicit attention to racial identity development in all mts and mtes, it is possible that white teachers, albeit unintentionally, could negatively impact the performance of students of color and undermine multicultural practices and policies (lawrence & bunche, 1996). research also suggests the following patterns of white people (including teachers) confronting race and equity issues: white elementary teachers are often ill-informed about racial inequality (king, 1991) and claim a color-blind approach (sleeter, 1992); if confronted with inequity, they feel blamed for injustices and can act defensively toward information on issues of social inequality and white privilege (gillespie, ashbaugh, & defoire, 2002); tend to approach issues of inequality from a personal perspective rather than as societal, systemic, and institutional manifestations (mcintosh, 1989; mcintyre, 1997); and want to be told what to do in a multicultural classroom, how to teach “others” rather than to explore the impact of their attitudes on multicultural teaching effectiveness (cooney & akintude, 1999). as herbel-eisenmann et al. privilege and oppression stinson, d. w., & spencer, j. a. (eds.). (2013). privilege and oppression in the mathematics preparation of teacher educators [special issue]. journal of urban mathematics education, 6(1). 13 taylor and kitchen (2008) stated, “it is well-documented that teachers hold lower expectations for students of color and those from poor families than they do for white middle class students (ferguson, 1998; grant, 1989; knapp & woolvertson, 1995; zeichner, 1996)” (p. 112). scholars have argued, in fact, that these lower expectations are not unique to white teachers. as bell (2002) pointed out: though teachers of color are less likely than their white peers to deny the existence of racism or to cling to dominant ideology about color blindness and dramatic social progress (l. bell, 2003; a. thompson, 1998), they may benefit from an opportunity to discuss and analyze their own experiences with racism in the broader society (bennett, 2002), the ways that racism is internalized by members of subordinated groups, and issues of collusion and horizontal oppression among different groups of color (hardiman & jackson, 1997). (p. 236) given the current underachievement in mathematics of many students of color and students who live in poverty, we need to collectively and individually, in thought and action, implicitly and explicitly stop the silence and address these issues in the mathematics education community, particularly among mtes. focusing also on mathematics teacher educators to create systems of equitable work. recent literature that considers how this fairly homogeneous teaching population works with students who are racially, economically, and linguistically different from them highlights the increasing attention to teachers and teaching in pre-k–12 public schools. yet, in order to create systems of equitable work, it is imperative that these issues be explored and considered in relationship to mtes. in the conference board of mathematical sciences report on u.s. doctorates in mathematics education, the following five “needs” were identified for the preparation of ph.d. students in mathematics education: 1. to learn about diversity/equity in all of their coursework and to develop national leaders in this area; 2. to learn “core knowledge” and have common experiences related to diversity/equity issues across institutions within doctoral programs in mathematics education; 3. to have professional experiences in a diversity of settings; 4. to develop an appreciation of diversity/equity issues even if diversity/equity is not central in the research they undertake; and 5. to develop an appreciation of theoretical frameworks related to diversity/equity and have knowledge of the research that has been undertaken that relates to diversity/equity in mathematics education (taylor & kitchen, 2008, pp. 112–114). each of these needs is important and requires careful consideration in order to prepare mtes to understand how to move beyond a “missionary or cannibal” ap herbel-eisenmann et al. privilege and oppression stinson, d. w., & spencer, j. a. (eds.). (2013). privilege and oppression in the mathematics preparation of teacher educators [special issue]. journal of urban mathematics education, 6(1). 14 proach (e.g., martin, 2007) and to understand what thoughtful collaboration to dismantle systems of privilege and oppression might look like. when mtes and mts have not critically examined their own place in the interlocking systems of privilege and oppression, they can bring a deficit model and exhibit behaviors that are patronizing because they view this work through a lens of charity rather than justice. in fact, even when mtes have begun to explore the complexity of some of these ideas, there is always more work to be done. we emphasize that this work must be an ongoing process to do well. understanding oppression and privilege as interlocking systems. equity research has become a growing line of research in the past two decades in mathematics education. in particular, the early and prevalent line of equity work focused on the “achievement gap” and access issues. there have been debates, however, about whether this is an overly limited or even an opposing way to consider issues of equity. in education research more generally, ladson-billings (2006) suggested that the achievement gap be re-named the “education debt.” by choosing to reframe the issue, she argued, the focus can shift from being only about individual student’s achievement on narrow standardized tests to also considering historical and systemic issues in the institution of schooling. as policy researchers have argued, how problems are framed shapes responses made by policy makers and mathematics educators (choppin, wagner, & herbel-eisenmann, 2011). if, for example, we also focused on the “education debt” rather than just the “achievement gap,” the manner in which changes are made would need to be different. for instance, we might examine and change policies and programs that support students and partner with communities to change schooling, rather than doing things like add test preparation to our curriculum. thus, the ways in which these issues are framed have ramifications for how students and families experience these realities. to broaden mathematics education’s view of equity, for example, gutiérrez (2007) offered a framework for equity that included achievement and access issues (which she calls the “dominant axis”) but pushed mathematics educators to consider issues of identity and power (which she called the “critical axis” of equity work). some mathematics educators and teacher educators have recently focused on issues of identity and power, often adopting frames like teaching mathematics for social justice (see the 2009 special issue of the journal of mathematics teacher education, for example; see also the 2013 special issue of the journal for research in mathematics education) or that of critical mathematics education. in these perspectives, the goal of education relates to emancipation and dismantling systems of oppression at the interpersonal, institutional, and cultural levels. gutstein (2006) draws on the work of paulo freire to teach students how to read and write the world with mathematics. that is, when he has students use mathe herbel-eisenmann et al. privilege and oppression stinson, d. w., & spencer, j. a. (eds.). (2013). privilege and oppression in the mathematics preparation of teacher educators [special issue]. journal of urban mathematics education, 6(1). 15 matics to analyze social, political, and economic situations that relate to issues of oppression, he teaches them to “read” the world with mathematics; when his students generate and engage in action related to these issues of oppression, he is teaching them to “write” the world with mathematics. in this literature, systems of oppression are explicitly named and critically challenged. to dismantle systems of oppression, however, we believe that the interlocking system of privilege must also be interrogated. yet, we find less attention to systems of privilege in mathematics education literature. a couple of exceptions to this in mathematics education include mtes who have used whiteness theory to explore aspects of their own identity in mathematics teacher education work (e.g., gregson, 2013; gutstein, 2003). more recently, battey (2013) showed “how color-blind ideology and whiteness produce material stratification through the institution of mathematics education” (p. 332) by analyzing national data sets to locate mathematics education within a broader racial context. his findings demonstrate the long-term economic advantages to whites due to differential access to mathematics as totaling hundreds of billions of dollars. if we look beyond mathematics education literature, however, systems of privilege are regularly examined and debated and are considered a valued form of scholarship. for example, there is a growing literature that uses whiteness theory to understand how prospective teachers work in diverse schools (e.g., cochran-smith, 1995; 2000; mcintyre, 1997; paley, 1979). using a mathematical analogy, we see privilege and oppression as complementary sets that must both be considered together to weaken the silence built around such an interlocking system. understanding and acknowledging privilege is not enough. in our identification of the problem, we stated that mtes are often silent about systems of privilege and oppression. yet, it is imperative that we:  deepen awareness of how oppression, privilege and power are at work in all relationships and organizations;  invite people with privilege to recognize and unlearn the habits and practices that protect their privilege;  nurture collaborative action and authentic relationships across differences of race, age, gender, dis/abilities, class, and sexual identity;  equip organizations—in this case, academic programs—to recognize, and then take action to decrease the disparity between their current practices and their inclusive ideals; and herbel-eisenmann et al. privilege and oppression stinson, d. w., & spencer, j. a. (eds.). (2013). privilege and oppression in the mathematics preparation of teacher educators [special issue]. journal of urban mathematics education, 6(1). 16  encourage mtes to explore and deepen their resources for social change and to connect our resources and the resources of mts and students. concluding thoughts it is time to break the silence. it is time to consider mtes’ knowledge and practice—their preparation and their research agendas, frameworks, approaches, and strategies for action toward equity in relation to the interlocking systems of privilege and oppression within which they (we) operate. one way to address the goals set forth here may be to engage mtes in both thoughtful reflection and action related to identifying, understanding, and confronting systems of privilege and oppression. the experience of people working together on issues of race and class can be profound and transformative and can result in deep and spreading changes in scholarship, teaching, and programmatic work that creates widening effects (apol, 2011; apol & herbel-eisenmann, 2012). the work that began at the prompte conference further illustrates how profound and transformative it can be for mtes to reflect and take action. as we move forward, we hope to expand the number of participating scholars in mathematics education engaged in identifying, understanding, and confronting systems of privilege and oppression both within and beyond mathematics education by engaging in further discussions. we see this jume special issue as one expansion and look forward to others. we encourage readers to use the essays of this special issue to engage their colleagues in contemplating the ideas put forth and hope they inspire further reflection and action. acknowledgements an earlier version of this paper was presented at the 7 th international conference on mathematics education and society, cape town, south africa, april 2–7, 2013. references aguirre, j. 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(1996). educating teachers to close the achievement gap: issues of pedagogy, knowledge and teacher preparation. in b. williams (ed.), closing the achievement gap: a vision for changing beliefs and practices (pp. 56–76). alexandria, va: ascd. journal of urban mathematics education july 2011, vol. 4, no. 1, pp. 15–22 ©jume. http://education.gsu.edu/jume joan kwako is an assistant professor of mathematics education in the education department in the college of education and human service professions, at the university of minnesota duluth, 125 bohannon hall, duluth, mn 55812; email: jkwako@d.umn.edu. her research interests include alternative assessment, in particular, collaborative testing and teaching mathematics with and for social justice. public stories of mathematics educators changing the balance in an unjust world: learning to teach mathematics for social justice joan kwako university of minnesota duluth n 2007, the first creating balance in an unjust world conference on mathematics education and social justice convened in brooklyn, new york (see http://creatingbalanceconference.org/). the conference was a forum for sharing ideas of social justice mathematics education. the organizers never intended it to be an annual event but due to the enormous amount of interest, they planned a second conference. the second conference was also held in brooklyn with nearly 400 attendees, including researchers, teachers, and students spanning all educational levels from 26 states and 3 countries. two preservice elementary teachers (maria and claire [pseudonyms]) and i, a mathematics methods teacher, traveled from minnesota to present at the conference. maria summarizes how attending the creating balance conference affected both her view of teaching and of life: the exposure to so many people, along with my presence at the creating balance in an unjust world conference, has given me a new perspective on people, education, and most importantly life. the visit has allowed me to see the importance and fragility of life. it has allowed me to see the importance of teaching the truth as well as teaching the gift of love, life, friendship, respect, and uniqueness. i was given the inspiration to teach these educational lessons whether they are life lessons, social lessons, or emotional lessons with passion, dignity, and genuineness. the conference was a deeply moving experience for all of us, one i wanted to capture by documenting the effect it had on maria and claire both initially and when they started teaching in their own classrooms. i contacted maria and claire 2 years after the conference to compare their expectations about teaching mathematics for social justice with their current teaching experiences. in this public story, i describe how teaching mathematics for social justice is expressed in the literature, explain how i engage students in teaching mathematics for social justice in my elementary mathematics methods class, and recount our experience of attending and presenting at the creating balance conference. i end i http://creatingbalanceconference.org/ kwako public stories journal of urban mathematics education vol. 4, no. 1 16 with stories of maria and claire’s experiences teaching mathematics for social justice (or not) in their current classrooms, 2 years later. teaching mathematics for social justice although it is not a new idea (see, e.g., freire, 1970/1993), teaching for social justice has become a growing focus in education (ayers, quinn, & stovall, 2009; darling-hammond, french, & garcia-lopez, 2002; mccoy, 2008; zajda, 2010) and, more specifically, in mathematics education (frankenstein, 1990, 1997; gau, 2005; gutstein, 2003, 2007; gutstein & peterson, 2005; murrey, 2008; peterson, 2005; stocker, 2008).definitions of social justice vary among teachers and researchers; the underlying ideas, however, are most often comparable. according to gau (2005), ―social justice in education [is] a process of analyzing oppression and critiquing inequities while helping students identify how those issues connect to their lives, and engaging them in purposeful action to challenge those inequitable structures‖ (p. 7). as an educator, one of my core beliefs is that education can achieve positive social change; i urge my students to see not only mathematics but also every subject as pathways to that influence. as i have researched the relationship between social justice and mathematics education, i have developed a pedagogical philosophy in which i first work to develop awareness among preservice teachers of the many factors that contribute to the inequities in society. this process begins with observation and acknowledgement and continues when preservice teachers incorporate social justice issues in their classrooms. when i began discussing social inequities with my students (future elementary teachers), i learned that many believe their students are unaware of the injustices of both their world and the world. the following statements exemplify many of my students’ initial reactions to discussing social justice issues with elementary students: we can’t tell them that! they’re too young! we need to take care of them, shelter them from the harsh realities, scary inequities and violence imposed on people—and children—in society. children are innocent and incapable of understanding big problems. my response to these students is, ―is that a child’s perspective, or your perspective? do you think that a child does not know that she or he is impoverished or homeless? is it better to ignore the discrimination due to economic standing, the color of their skin, or living conditions that to many children experience daily? or is it better to acknowledge the realities in which our children live, with the goal of changing those realities and expanding their possibilities?‖ when conventional mathematics curricula grow sterile and detached from the lives of students, stu kwako public stories journal of urban mathematics education vol. 4, no. 1 17 dents lose interest. mathematics lessons drawn from today’s headlines may not always be pleasant, but the savvy teacher can integrate them into lesson planning in age-appropriate ways. most of my students (preservice teachers) have not experienced homelessness, abuse, or poverty but, at some point in their careers, all will surely teach children who have. by raising my students’ awareness of the different realities and inequities that exist, they can create an environment in which their students are able to use the mathematics they learn through social justice contexts i provide to achieve social justice goals. my focus on social justice provides a context through which my students examine themselves and their roles as teachers and as members of our society. finally, i work to help my students understand that not only will they teach children who are victims of injustice but also that it is essential they recognize that these issues of injustice are not defining characteristics or qualities of the children, their families, or their culture. instead, the issues are the result of social, political, and economic forces in society that shape their lives. it is through these discussions and the design of mathematics lessons using a social justice context that i work to bring my students to a level of maturity and pedagogical awareness of social, political, and economic injustice. there is an obvious connection between social justice and social studies, between social justice and history, and even between social justice and literature. but mathematics? yes, mathematics! it is possible to teach mathematics using contexts that illustrate societal imbalances. in this way, mathematics teachers can serve as advocates for positive social change. introduction to social justice lessons i initiate discussions about social justice with my preservice teachers by modeling lessons that focus on various inequities in our society. the first lesson, ―unequal distribution of wealth in the united states‖ from eric (rico) gutstein and bob peterson’s (2005) edited volume rethinking mathematics: teaching social justice by the numbers, requires students to analyze data and create a threedimensional graph representing population and wealth distribution in the united states. groups of students use a 10 square cm sheet of grid paper and 100 centimeter unit cubes to graph the following data: 1% of u.s. households owns 39% of the wealth, 19% of u.s. households own 46% of the wealth, and 80% of u.s. households own 15% of the wealth (langyel, 2005). generally, groups create one very tall tower (39 cubes high) on one square, 46 cubes spread out among 19 of the squares, and 15 cubes spread out over 80 of the squares. as the tower on one square grows to 39 cubes, it becomes a very powerful and startling image of the imbalance of wealth in the united states. not all of the groups display the data the kwako public stories journal of urban mathematics education vol. 4, no. 1 18 same way; some assign different colored cubes a different value or use differentsized cubes to represent different amounts. any debate about which graph more accurately reflects the data only adds to a conversation about representation of wealth distribution. it also deepens the mathematical understanding of how the way data are represented can significantly change the message that data might convey. my students engage in and create lessons surrounding some disturbing topics, but i do not force any set doctrine; the numbers provide their own testimony. some lesson topics include the cost of wars in iraq and afghanistan, discrepancy between salaries of ceos of major u.s. retailers and salaries of those who actually make the products, racial profiling in arrests and the death penalty, and comparative wages of women and men in the same field. engaging students in relevant mathematics helps students to use classroom mathematics to critically analyze numbers in their lives outside of the classroom. creating balance in an unjust world conference for mathematics education and social justice the social justice lesson plans notwithstanding; it was the experience of presenting at the creating balance conference that had the most profound impact on the two students who presented with me. we organized our presentation into an interview format. i had a list of open-ended questions for claire and maria intended to elicit the challenges and successes of designing elementary mathematics lessons using a social justice context. although i gave them general questions ahead of time, we did not rehearse their answers; i wanted the presentation to be as authentic as possible. as such, conference attendees were encouraged to ask follow-up or clarifying questions; thus, the presentation became a live interview involving everyone in the room. immediately upon returning to minnesota, both claire and maria wrote about their experiences at the conference. below are their thoughts at that time, in 2008: claire: our visit to the site where the world trade centers once stood was very shocking and emotional. it is impossible to explain the feelings of sadness and anger i experienced when visiting the memorial museum and site. i was sixteen years old when the events occurred on september 11, 2001, and i know what i did learn about the event was taught by my parents, not through any formal education. this was the moment that i confirmed my belief that the truth about the world’s issues is not being taught enough in our schools today. claire realized that her only knowledge from one of the most devastating events in the last 10 years came from media, not from her teachers. had her parents not kwako public stories journal of urban mathematics education vol. 4, no. 1 19 shared with her what happened, her knowledge about the event would have been superficial at best. maria: teachers have to be very aware of the students, community, etc. when designing lessons using a social justice context. this is not always easy. as teachers, we do not want to single out someone or disrespect anyone. social justice topics can really take a toll on students’ emotions if it applies too directly to their lifestyle. we need to be respectful. all learners must be considered and incorporated into the design of the lesson. maria highlighted the care teachers must exercise when designing lessons that use delicate, unpredictable, and potentially controversial topics. for example, it would be a mistake to start discussing racial profiling in a multiracial classroom without significant conscious thought towards the impact the discussions will have on the children in the classroom, the school staff, and the parents. designing such lessons is difficult, and is more challenging without institutional support: maria: it can be difficult to find support from colleagues as you present social justice topics. not everyone wants to be involved because of the risk of getting fired or getting into trouble with administration. sometimes colleagues will frown upon your willingness to question society and support what is right. i struggled with this. i had a cooperating teacher who was concerned about my lesson on overcrowded classrooms. i thought it was a great way for students to understand the importance of building safety and to question places in their community that may or may not be safe, all the while teaching about area (italics added). by questioning dangerous places, my students learned that they can make a difference and help to create a safer, more just society. this is what is important. maria, and later claire, recognized the importance of having support from colleagues and administration when incorporating, in this case, an issue as seemingly uncontroversial as overcrowded classrooms: claire: through the entirety of the conference and our presentation, my views were dramatically altered about educating students on our world issues of social justice. as teachers, we need to be leaders in the classroom and empower students to become future leaders through education about the truth of the real world. we also need to teach and encourage other educators to research and use their knowledge of real life political, social, and economic issues in their everyday classroom lessons. each person at the conference shared their challenges in designing and implementing these lessons in the classroom, but it is the success of the students that is the most amazing. when social justice education is taught with another subject such as math, student interest and understanding rises. the lesson i learned is that as a teacher, i can provide my future students with the truth about today’s world issues while still providing an education in core subjects such as math. teaching students about social justice is crucial to the future of our world. kwako public stories journal of urban mathematics education vol. 4, no. 1 20 her enthusiasm for teaching for social justice was apparent in 2008. would it wane? 2 years later: 2010 our world changed significantly in 2 years. i contacted claire and maria to see if the experiences with teaching mathematics for social justice both in my class and at the conference really affected how they now taught. the following are their 2010 responses to one of the question originally asked in the presentation at the creating balance conference. joan kwako: i have come to realize that you can use mathematics to teach and learn about issues of social justice, and conversely, that such social issues can be the context to learn mathematics. did the fact that we focused on social, political, and economic issues in my class have an impact on your teaching? maria: if there is a current local state, national, or world topic that can relate to a math topic we are learning or have learned, i utilize it. students love to apply what they are learning to something that is meaningful. aside from teaching what is going on in our world, i am also consistently faced with relating the lives of my students to what we learn. i have students who are living in poverty, families with children who are first generation americans, and students with diverse backgrounds and life experiences…[which] allow my students to develop understanding, as many of them are english language learners. therefore, their own experiences allow them to make connections and learn. i am fortunate in my school to have flexibility in what i teach. (yet at times i am feeling the pressure of teaching to the test rather than relating topics to real life.) many of my students have a deep understanding of the social issues that arise in conversation. it allows them to make a connection between the issue and the content that is being taught. it is amazing to see the connections being made. this fall i gave my students information on haiti and the problems they have been facing to create a deeper understanding of the word community. we viewed pictures of what a community looks like as well as haiti and its community before, during, and after its disaster. we talked about what people have had to live without and how communities come together to help those when are in need. it has allowed many of my students to reflect on our learning community and what we stand for in our classroom. i have enjoyed seeing the change in some of my students’ behaviors. information on the past and current haiti community and our own learning community have proven to be not only effective, but meaningful. when students find meaning in what they learn, it is the best feeling in the world. claire: even though i have not taught math with social justice, i have had many students ask about social, political, and economic issues during my teaching. i find a resource and teach the answer using the resource, not using my opinion. oftentimes, i have had the students find the answer to their question on their own or by working with another student. students can often teach each other if they have experience in the social issue topic. i believe the focus on social, political, and economic issues has impacted my teaching, whether or not i have taught using social justice in my kwako public stories journal of urban mathematics education vol. 4, no. 1 21 curriculum or had the opportunity to design and teach my own lessons. social justice is always on my mind and i am constantly thinking about how it relates to my everyday life. i was reminded that diversity and social justice are two issues that are in our classrooms everyday and cannot be ignored. we should also remember that by incorporating social justice into our teaching that our students may be able to better relate, learn more about our world, and reach their full potential as learners. i’m teaching in [minnesota] as a long-term sub in second grade. i haven’t had the opportunity to really create my own curriculum or really teach anything using social justice yet. i don’t think there is a social justice component in either our math or literacy program. also, i’m in the midst of trying to secure a full-time job, and am obviously not tenured. i am working in a district where parents and administrators track me all of the time, and i can only teach what i’m assigned to teach. in claire’s last statement, she made it clear to me that she did not want her name, her school, or even her district mentioned, for fear of retribution. she clearly wants to incorporate issues of social justice in her classroom but does not feel she has the support to do so. maria had a similar concern in the earlier interview. how do we, as teacher educators, help them? we can model teaching as a means for positive social change, which will equip them with the knowledge necessary to change the status quo, especially when that status quo is so inequitable. however, having the knowledge to change the status quo does not imply that beginning teachers can always stand up to administrative forces that come against them. even so, as gau (2005) states, ―an important component in the literature on teaching mathematics for social justice…is that teaching mathematics for social justice is fundamentally about students learning mathematics‖ (p. 75). and isn’t our entire purpose of teaching mathematics for students to learn mathematics? although it would be nice to take credit for the change in my students’ perspectives, i know i was only a guide. i do believe that the focus on social justice in class and the opportunity to attend and present at the creating balance conference allowed my two students to recognize that real issues can provide rich contexts for learning mathematics. these contexts can serve to not only motivate students to learn and enjoy mathematics but also to expose them to—and thus work to change—the real social justice issues that exist in our country. it strikes me as counterproductive to separate mathematics from reality. if we choose to do so, we deserve the taunt that ―math doesn’t matter in their lives.‖ they are wrong; mathematics surrounds us. it is the invisible web that became the internet; it is the underpinnings of every economic transaction. numbers chart the heights of human achievement and illustrate the depths of human despair. we cannot ignore things in an effort to wish them away; to change inequities, we must first acknowledge them. lessons we choose for our students should reflect these intricate links to the real world. if we are to build enthusiasm, and at the same time, avoid aversion to mathematics, we need to connect it to what is real and important. as kwako public stories journal of urban mathematics education vol. 4, no. 1 22 teacher educators, it is our responsibility to prepare future mathematics teachers to teach for social justice and thus work to change the balance. references ayers, w., quinn, t., & stovall, d. (2009). handbook of social justice in education. new york: routledge. darling-hammond, l., french, j., & garcia-lopez, s. p. (2002). learning to teach for social justice. new york: teachers college press. frankenstein, m. (1990). incorporating race, gender, and class issues in a critical mathematical literacy curriculum. journal of negro education, 59, 336–347. frankenstein, m. (1997). in addition to the mathematics: including equity issues in the curriculum. in a. trentacosta & m. kenny (eds.), multicultural and gender equity in the mathematics classroom. reston, va: national council of teachers of mathematics. freire, p. (1993). pedagogy of the oppressed. new york: continuum. (original work published 1970) gau, t. r. (2005). learning to teach math for social justice (unpublished doctoral dissertation). university of wisconsin-madison, madison, wi. gutstein, e. (2003). teaching and learning mathematics for social justice in an urban, latino school. journal for research in mathematics education, 34, 37–73. gutstein, e. (2007). ―and that’s just how it starts‖: teaching mathematics and developing student agency. teachers college record, 109, 420–448. gutstein, e., & peterson, b. (eds.). (2005). rethinking mathematics: teaching social justice by the numbers. milwaukee, wi: rethinking schools. langyel, m. (2005). unequal distribution of wealth in the united states. in e. gutstein & b. peterson (eds.), rethinking mathematics: teaching social justice by the numbers (pp. 68–69). milwaukee, wi: rethinking schools. mccoy, l. p. (2008). poverty: teaching mathematics and social justice. mathematics teacher, 101, 456–461. murrey, d. (2008). making numbers count. teaching tolerance, 33, 50–55. peterson, b. (2005). teaching math across the curriculum. in e. gutstein & b. peterson (eds.), rethinking mathematics; teaching social justice by the numbers (pp. 9–15). milwaukee, wi: rethinking schools. stocker, d. (2008). maththatmatters: a teacher resource linking math and social justice. ottawa, canada: ccpa education project. zajda, j. (ed.). (2010). globalization, education and social justice. dordrecht, the netherlands: springer. journal of urban mathematics education december 2012, vol. 5, no. 2, pp. 87–111 ©jume. http://education.gsu.edu/jume eugenia vomvoridi-ivanović is an assistant professor in the department of secondary education at the university of south florida, 4202 e. fowler ave, edu105, tampa, fl 33620; email: eugeniav@usf.edu. her research interests include the mathematics education of language minority students, teacher development in informal mathematics learning contexts, and culturally responsive mathematics teacher education. “estoy acostumbrada hablar inglés”: 1 latin@ * pre-service teachers’ struggles to use spanish in a bilingual afterschool mathematics program eugenia vomvoridi-ivanović university of south florida in this article, the author explores factors that appear to contributed to bilingual latin@ pre-service teachers’ difficulties in using spanish as an instructional resource while working on mathematical activities with bilingual latin@ students in an urban afterschool mathematics program. qualitative analysis of the perservice teachers’ oral and written comments reveals two main patterns associated with their difficulties. the first relates to their schooling/academic experiences, which were predominantly in english. the second relates to their experiences with students in the afterschool program who showed preference in using english. the author discusses implications of the findings for mathematics teacher preparation. keywords: bilingual instruction, latin@ education, mathematics education, teacher education, urban education i always talk to them (the children) in english, when i realize it, i try to switch to spanish but then it just sounds weird and it does not sound natural. i always thought that spanish was my dominant language, but i guess not. uanita,2 a bilingual latina elementary pre-service teacher (pst), wrote the above statement after 4 weeks of working in an urban after-school mathematics program, los rayos, with children who were also latin@s and bilingual. 1 translation: “i am used to speaking in english.” 2 all names are pseudonyms. editor’s note: gutiérrez (2010) explains the use of the @ sign as a means: “to indicate both an ‘a’ and ‘o’ ending (latina and latino). the presence of both an ‘a’ and ‘o’ ending decenters the patriarchal nature of the spanish language where is it customary for groups of males (latinos) and females (latinas) to be written in the form that denotes only males (latinos). the term is written latin@ with the ‘a’ and ‘o’ intertwined, as opposed to latina/latino, as a sign of solidarity with individuals who identify as lesbian, gay, bisexual, transgender, questioning, and queer (lgbtq)” (p. 5). gutiérrez, r. (2010). the sociopolitical turn in mathematics education [special issue]. journal for research in mathematics education research, 41(0). j vomvoridi-ivanović estoy acostumbrada hablar inglés journal of urban mathematics education vol. 5, no. 2 88 throughout this time, she used little spanish in her instructional interactions with the children, even though the program was designed to promote mathematical biliteracy, in english and spanish. in fact, other bilingual latin@ psts, who worked in los rayos, also used almost exclusively english with the children. but why would someone whose home language is spanish and is fluent in spanish not use it as a resource when working on mathematical activities with spanish speaking children? why is this an issue for the mathematics education of latin@ students and what are some implications for the mathematics teacher preparation of latin@ psts? here, i address these questions and discuss results from a larger study, which examined bilingual latin@ psts’ use of language during their participation as facilitators in an urban after-school mathematics program (vomvoridi-ivanović, 2009). i have two purposes: (a) to understand the challenges that bilingual latin@ psts may face as they attempt to integrate their home language (spanish) in mathematics instruction and the underlying factors that contribute to these challenges, and (b) to draw implications for the mathematics teacher preparation of latin@ psts, and other psts who went through the educational pipeline as language minority students (lms).3 through the elevation of the voices of latin@ psts, i highlight the historical, sociopolitical, and linguistic factors culminating in bilingual latin@s’ struggles to incorporate and leverage spanish to maximize young latin@s mathematics learning. the findings i present, as well as their implications, are applicable to many urban contexts, nationally and internationally, where the languages of instruction and of use by students, families, and the wider community are different. conceptual framework literature on bi/multilingual mathematics learners considers students’ home language(s) as resources that teachers need to build on to support students’ learning of mathematics (e.g., adler, 2001; barwell, barton, & setati, 2007; fuson, smith, & lo cicero, 1997; gutstein, lipman, hernandez, & de los reyes, 1997; khisty, 1997; moschkovich, 2000; setati, 2005; also see the edited volume: tellez, moschkovich, & civil, 2011). this literature has taken a strong position for the use of the students’ home languages in teaching mathematics and has argued that to facilitate lms’ participation and success in mathematics, teachers should recognize and utilize their home languages as legitimate languages for mathematical communication. there is still a question, however, of how to pre 3 by using the term lms, i refer to those students whose linguistic and cultural backgrounds have not traditionally been considered as resources for academic learning. vomvoridi-ivanović estoy acostumbrada hablar inglés journal of urban mathematics education vol. 5, no. 2 89 pare those teachers who share similar linguistic backgrounds with lms4 to integrate their knowledge of student’s home language in mathematics instruction. the literature on teacher preparation for linguistically and culturally diverse students has been primarily concerned with english monolingual teachers (lucas & grinberg, 2008; villegas & davis, 2008) and has not paid adequate attention to the preparation, support, and empowerment of lmt, who may need different kinds of supports to incorporate their home language(s) into pedagogical practices in mathematics. this focused attention is particularly relevant to latin@ teachers as they are the fastest growing minority teacher group, especially in urban school districts where the latin@ student population is the highest (strizok, pitsonberger, riordan, lyter, & orlofsky, 2006). according to a report from the national center for educational statistics (nces, 2011), in 2009, latin@ teachers made up 8% of the u.s. teacher population and latin@ students made up 22% of the u.s. student population. these percentages have more than doubled during the past two decades. in addition, according to the same report from nces, in 2009, 21% of the u.s. student population spoke a language other than english at home, with spanish being the predominant language. because the majority of lms are latin@s (nces, 2011), latin@ teachers tend to teach latin@ students (villegas & davis, 2008), and many latin@ teachers speak spanish, it is vital for the field of mathematics teacher education to consider ways of helping latin@ preand in-service teachers to build on the unique strengths they bring into teaching mathematics, particularly their knowledge of spanish. literature in this area, however, is scant, as research in mathematics teacher preparation has been primarily conducted in english monolingual settings (clift & brady, 2005). in teacher education, typically, language is treated as a subject, and is separated from the content subjects. this situation is evidenced by the absence of substantial language and discourse content and on teaching practices appropriate for bi/multilingual classrooms in most mathematics teaching courses for psts (setati, 2005). a few studies have focused on mathematics teachers’ language practices in bi/multilingual contexts (e.g., adler, 2001; fabelo, 2008; khisty, 1995; setati, 1998, 2005; setati & adler, 2000; vomvoridi-ivanović, 2009; vomvoridiivanović & khisty, 2007). these studies highlight the complexity of using more than one language during mathematics instruction and point to the fact that simply knowing how to speak students’ home language does not ensure that the teacher will use that language appropriately, if at all, as a resource during mathematics discussions. khisty (1995), for example, found that in classrooms where students and teacher were bilingual and latin@, very little spanish was used in the mathematics context compared with other subjects such as reading and/or language 4 i will refer to these teachers as language minority teachers (lmt). vomvoridi-ivanović estoy acostumbrada hablar inglés journal of urban mathematics education vol. 5, no. 2 90 arts. in addition, very few whole thoughts were conveyed in spanish during mathematical explanations. spanish was used in a perfunctory manner as an “instrument” to discipline students, to call their attention to the subject of the lesson, or to punctuate a statement. finally, spanish was used primarily to give encouragement and to motivate the class. in other words, spanish was not used in the context of mathematics meaning making. fabelo (2008) also found that novice bilingual latin@ teachers had difficulties using academic spanish during mathematics instruction. she argued that bilingual latin@ teachers raised in the u.s. who teach mathematics encounter the same issues with the mathematics register (halliday, 1978) as they do with registers for other content areas in academic spanish. they have learned the mathematics register in an academic setting in english or informally with their families in spanish (khisty, 1995; ron, 1999). many do not have training in the technical language of mathematics in spanish and are left to develop it themselves (vomvoridi-ivanović & khisty, 2007). not knowing the language of mathematics in their home language, however, is only one factor that might influence a lmt’s choice of whether or not to integrate their home language in mathematics instruction. setati (2005) pointed to the fact that different infrastructures in and around classrooms make different demands on mathematics teachers and this affects their willingness to use students’ home languages for instructional purposes in mathematics. she noted, “to fully describe and explain the use of language(s) in multilingual mathematics classrooms we need to go beyond the pedagogic and cognitive aspects and consider the political role of language” (p. 464). though it is rarely made explicit, one of the most common distinguishing features in schools with large numbers of lms is their overwhelming press toward assimilation of students into mainstream cultural—including linguistic patterns (clayton, barnhardt, & brisk, 2008). clayton and colleagues argue that, on the surface, this cultural assimilation orientation, which is prevalent in most schools, seems to offer lms an opportunity to gain access to the skills and recourses necessary to participate in the larger society on equal terms with others. because english is seen as the dominant language associated with access to social mobility and success, this may influence lmt’s language choices in mathematics instruction. clayton and colleagues (2008) further argue that in most instances, the goals of schools are bound to universalistic intellectual or social functions associated with the dominant society. in other words, the basic thrust of schooling is toward the breaking down of particularistic orientations and developing in their place a more universalistic outlook. even where accommodations are made to include ethnic studies or bilingual education in the curriculum content, the structure, method, context, and processes through which the content is organized and transmitted are usually reflective of mainstream patterns and exert a dominant influ vomvoridi-ivanović estoy acostumbrada hablar inglés journal of urban mathematics education vol. 5, no. 2 91 ence on the students. schools are agents of the dominant society and, as such, they reflect the underlying cultural patterns of that society. as long as they reflect the structure and cultural fabric of the dominant society, they can be expected to perpetuate its values, attitudes, and behavior patterns associated within an explicit framework of assimilation. the assimilationist nature of lmt’s own schooling experiences results too often to the formation of certain cultural models (holland & quinn, 1987) for teaching and learning mathematics. holland and quinn defined cultural models as presupposed, taken-for-granted models of what is considered to be normal. in other words, they are shared conventional ideas about how the world works and “provide a framework for organizing and reconstructing memories of experience” (p. 4). cultural models link values, goals, motives, emotional states, and knowledge (of things and processes, classifications, social relations, etc.), as relevant, together in a conventional representation of behavior (kronenfeld, 2005). they are embedded in peoples’ words and practices, and are shared with others through the media, written materials, and through interaction with others in society (gee, 1996, 1999). the cultural model of what it means to be a mathematics teacher is initially passed on to future mathematics teachers while they are students and later passed on during teacher training and through the media (setati, 2005). in order to begin to think about how to help teachers who share similar linguistic backgrounds with their students develop and incorporate their knowledge of their students’ home language into pedagogical practices in mathematics we first need to better understand the historical, linguistic, and sociopolitical factors that influence their language choices and especially their difficulties in using their home language as an instructional resource in mathematics. in this article, i unpack several factors that contributed to four bilingual latin@ psts’ difficulties in using spanish as a resource while working on mathematical activities with bilingual latin@ students in an urban afterschool mathematics program. i now turn to describe the context of the study, the participating psts, and the methods employed. next, i discuss the factors that contributed to the psts’ difficulties in using spanish while working on mathematical activities with bilingual latin@ children. i close my discussion with some concluding thoughts related to the implications for mathematics teacher preparation. methods context the work presented here draws on a wider study that explored how latin@ psts used language and culture as instructional resources in mathematics vomvoridi-ivanović estoy acostumbrada hablar inglés journal of urban mathematics education vol. 5, no. 2 92 (vomvoridi-ivanović, 2009). it reflects current work carried out by a center for learning and teaching, funded by the national science foundation, which focuses on the research and practice of the teaching and learning of mathematics for latin@s in the united states through the integration of socio-cultural theory, language, and culture. this center for learning and teaching created afterschool projects at two of its sites, one of which, los rayos, is the context of the present study. los rayos was located in a large mid-western urban school district and in a community that has 93.5% of latin@s, predominantly of mexican heritage. the project took place in a neighborhood school that has a dual language (spanishenglish) program for all of its grades and predominantly serves the students of working class mexican families. this neighborhood is representative of many predominantly mexican and segregated neighborhoods in the city. this specific scenario of latin@ students from these neighborhoods, becoming teachers of bilingual latin@ youth deserves our attention if we want to improve what has been historically the mis-education of latin@s in our cities (and elsewhere). the afterschool program consisted of two parts: the actual informal learning environment, los rayos, where bilingual latin@ psts engaged in various kinds of mathematical activities with latin@ children, and a debriefing meeting where psts and researchers discussed the children, activities, and various aspects of mathematics and curriculum that arose during the afterschool sessions. all materials used in the program were written in spanish and english and all informational meetings with children were conducted in spanish first and then in english. psts and other afterschool personnel who were bilingual were encouraged to speak spanish as much as possible to provide children with university role models who also spoke spanish. furthermore, the afterschool program was housed in a school that strived for biliteracy and it was hoped that the program would build on and extend this idea to mathematics. the environment in which the psts worked and the activities they were engaged in offered them many opportunities to choose to use spanish, english, or a hybrid of the two. in other words, the project was conducted in a bilingual environment; bilingualism carried over to the debriefing meetings as well. as part of their participation in the afterschool program, psts took field notes and reflected on their interactions with the children in los rayos. after each session in los rayos, psts constructed descriptive and reflective field notes that focused on their own and the children’s use of language, the students’ mathematical strategies, their own assistance strategies, and the students’ interests. the same topics or items were discussed in the weekly debriefing meetings where the psts met with university researchers, whom i will refer to as fellows, and discussed what occurred in los rayos by reflecting on their interactions with the children. these discussions were open-ended in that psts could easily and naturally raise vomvoridi-ivanović estoy acostumbrada hablar inglés journal of urban mathematics education vol. 5, no. 2 93 questions, offer suggestions, and try to make sense of their own and their students’ mathematical behaviors as related to issues of language, culture, and identity, to name a few. some of the fellows that co-facilitated the meetings were native spanish speakers and the discussions in these meetings were bilingual. participants jose, juanita, maria, and lupe, the four psts who participated in this study were all undergraduate students at a large university in the midwest. at the time of the study, jose and lupe were elementary education majors and maria was a secondary mathematics education major. juanita was an undeclared major in her sophomore year and, at the time of the study, was strongly considering entering the elementary education program. (here, i refer to her as a pst even though she was not officially an education major during the time of the study.) all four psts’ home language is spanish and their parents are immigrants from mexico. jose and juanita learned english prior to attending elementary school. jose learned english from interacting with his english speaking babysitters, while juanita began to learn it in pre-school. maria and lupe on the other hand did not begin to learn english until they attended first grade. moreover, lupe was born in mexico and moved to the united states at age 5, while all other psts were u.s. born. all four psts were brought up in predominantly latin@ communities and were schooled in the same urban public school district and attended different bilingual/esl school programs. specifically, jose attended a “pull-out esl” program and juanita attended a “transitional bilingual” program until third grade after which they attended mainstream classrooms where instruction was in english only. lupe and maria, on the other hand, attended “maintenance bilingual” programs until sixth grade after which they also attended mainstream classrooms where instruction was in english only. all psts noted that during the time that they attended mainstream classrooms they were forbidden to speak in spanish at all times. data collection and analysis the four psts’ field notes and oral comments during the debriefing meetings are the primary data sources for this study. additional data sources used for triangulation are observations of the psts’ interactions with the students and my personal field notes. the psts were observed once per week for 4 weeks for approximately two hours each time as they participated in a debriefing meeting. these meetings ran concurrently with the afterschool sessions but on another day of the week. the meetings were videotaped, discussions were transcribed, and spanish talk was translated to english. in addition, study participants’ own field notes were collect vomvoridi-ivanović estoy acostumbrada hablar inglés journal of urban mathematics education vol. 5, no. 2 94 ed twice per week for 5 weeks. in their field notes, the psts were asked to address (a) the language(s) used and the contexts in which those were used during the afterschool session amongst their group, (b) the children’s mathematical strategies, (c) their own assistance strategies, (d) children’s interests, and (e) participants’ reflections on items 1 through 4. finally, psts were observed twice per week for 4 weeks of the afterschool program as each worked with a small group of two to five fifth-grade bilingual latin@ students. the afterschool sessions were videotaped, discussions were transcribed, and spanish talk was translated to english. grounded theory (strauss & corbin, 1990) methodology was employed to identify recurring themes in the data. first, i identified excerpts from the psts’ field notes and the transcriptions from the debriefing meetings where the psts referred to their own and the children’s language. then, i compiled a list of general framing codes, including participants’ challenges in using spanish during mathematical activities and psts’ explanations as to why discussions were english dominant. next, the data were coded, and the emergence of additional codes occurred through multiple passes of the entire dataset; four passes through the dataset were required before categories began to stabilize. the coding scheme aimed to characterize the nature and content of the psts’ comments when they addressed issues related to language use. finally, i identified the episodes from the afterschool sessions that the psts referred to in their comments and mapped them with the psts’ comments. findings data analysis revealed two main patterns associated with the psts’ difficulties with integrating spanish talk during mathematical activities. the first pattern relates to their experiences as lms in the u.s. educational system. the psts’ schooling/academic experiences were predominantly in english. specifically, their mathematics learning experiences were only in english at some point early in their schooling. as a result they lacked experience in talking mathematically in spanish and had come to associate academic (including mathematical) discourses and institutions as being english monolingual. the second pattern relates to their experiences with students in los rayos who showed preference in using english. this preference was evidenced in two ways: by students’ consistent use of english even when the psts addressed them in spanish, and by individual students demanding that the psts and other students use english. vomvoridi-ivanović estoy acostumbrada hablar inglés journal of urban mathematics education vol. 5, no. 2 95 psts’ academic experiences were in english lack of experiences speaking mathematically in spanish. all four psts expressed facing several challenges when attempting to facilitate mathematical discussions in spanish both during the debriefing meetings when they were asked to discuss mathematical activities in spanish, and when working with the students in los rayos. they attributed part of their challenges in facilitating mathematical discussions in spanish to the fact that they had never been taught mathematics in spanish and, hence, did not have experience speaking mathematically in spanish. after reflecting on his language use during his first session in los rayos, jose mentioned in his field notes that he felt more comfortable explaining himself in english rather than in spanish when doing mathematics. the other psts shared the same experience with jose, as can be seen in the following excerpt from one of the debriefing meetings. during this debriefing meeting, one of the fellows, salvador, presented a mathematical task dealing with proportions in spanish. he asked the psts to solve it using spanish only. after discussing and solving the task collaboratively, the psts were asked to reflect on the process of solving and discussing this task in spanish. one of the psts mentioned that it was difficult for her to discuss the task in spanish as she feels more comfortable expressing her thinking in english than spanish. jose and juanita shared the same feelings with that pst and added: jose: and for that reason, i’d really rather use english to express myself when it comes to math. um…i can hold the concepts in spanish…but if i wanted to explain a point i’ll go—and switch the spanish—i’ll do it in english. …i knew i had to speak in spanish because we were more pushed but i was thinking in english and was translating what i wanted to say from english to spanish. i just feel more comfortable explaining myself and it just clicked…but i used english in my mind to figure it out. …i’ve been taught math in english and not in spanish. juanita: i thought about it (the task) in english, and then tried to translate it in spanish. jose expressed his difficulty with thinking about mathematics in spanish and with explaining mathematical ideas in spanish. just like juanita, he talked about his need to think about a mathematical task and reason mathematically in english and then translate it to spanish if asked to explain his thinking in spanish. he attributed his difficulty in thinking and expressing mathematical ideas in spanish to the fact that he had not been taught mathematics in spanish. both jose’s and juanita’s comments underscore the importance of mathematics discourse in the learning of mathematics. vomvoridi-ivanović estoy acostumbrada hablar inglés journal of urban mathematics education vol. 5, no. 2 96 other psts, namely maria and lupe, claimed that they felt very comfortable expressing themselves in spanish. in fact, maria said that she always thinks in spanish, rather than english. still both maria and lupe mentioned that when it comes to talking about mathematics with the students in los rayos they would often resort to english due to their lack of the necessary mathematical terminology in spanish. they also expressed that many times, when they would read the spanish version of the activities they had to use the english version because they would not understand the terminology. during the same debriefing meeting maria and lupe reflected on their comfort level in using spanish when doing mathematics with the students or at the debriefing meetings: maria: it depends…on the vocabulary because sometimes we don’t know how to—we don’t know what one word means. lupe: i’m comfortable with both, but like the way maria said it depends on the vocabulary. cause if like some word is in spanish like for math that i’m not sure what they mean but if you tell it to me in english i might go like “yea, it’s this shape” or something. according to maria and lupe, not knowing mathematical terminology in spanish made their task of speaking mathematically in spanish more difficult. however, it remains unclear as to whether maria and lupe were really referring to their not knowing mathematical terminology in spanish or simply to the fact that they were simply not used to talking about mathematics in spanish because they had been taught mathematics in english. the pst who perhaps expressed most intensely her discomfort in facilitating mathematical discussions in spanish was juanita who repeatedly noted this in her field notes. she found it difficult to use spanish during mathematical discussions, seemingly because she lacked the specialized language of mathematics in spanish. this is evident in her field notes: i always talk to them (the students) in english, when i realize it i try to switch to spanish but then it just sounds weird and it does not sound natural. i always thought that spanish was my dominant language, but i guess not… when juanita was asked to elaborate on her comment that “it does not sound natural” she responded: it sounds as if i am trying too hard, and sometimes i do not use the right words. it’s like i don’t feel comfortable using spanish when i do math. i don’t know all the words and i can’t explain it in spanish. vomvoridi-ivanović estoy acostumbrada hablar inglés journal of urban mathematics education vol. 5, no. 2 97 juanita’ statement that she is unsure as to whether spanish is her dominant language as she had previously thought leads to her realization of the difference between conversational and academic fluencies. juanita, just like the other psts, were fluent in conversational spanish but not in academic (mathematical) spanish. this realization appeared to unsettle them, and they realized that to facilitate mathematical discussions in spanish they needed to learn the specialized language of mathematics in spanish and to practice doing mathematics in spanish. associating academic discourse and settings to english. another challenge that the psts appeared to face when trying to speak mathematically in spanish was that because of their educational experiences, they had come to associate academic discourse to english language rather than spanish. through reflecting on their personal histories as lms, they expressed that throughout their academic career they have made a very strong effort to develop english academic proficiency and they have come to associate english as being the language used in academic (including mathematical) discussions. also, they had associated academic institutions as being english monolingual establishments because that had always been their experience. they expressed that it felt “weird” to talk about mathematics in spanish in los rayos, which was housed inside a school, and they also felt “weird” using spanish during the debriefing meetings, which were housed inside the university (i.e., an academic setting where they typically used english). some of them shared experiences of having been forbidden to use spanish in school and even though they all remember this as being a painful experience, overtime they developed the notion that spanish is not the language used for academic discussions and in academic institutions. during a debriefing meeting, the psts reflected on their background experiences growing up as lms in a large urban mid-western school district. during this meeting we were discussing several issues related to using different natural languages in different contexts. maria, reflecting on her use of spanish and english inside and outside academic establishments commented: in general, i feel more comfortable speaking spanish but like when it comes to school it’s like you know, all of our lives we have been told “you have to do this right, you have to speak in english, you know it’s the language of america” and they are teaching it in school and you need to speak english to get a good job so we grew up feeling pressured to like speak in english. but once we are outside and we are free in the environment then we are more comfortable to speak however we want. maria’s comment points to the pressure these psts felt to use english inside academic institutions and to the fact that they grew up learning that english is the dominant language in the united states. they realized early on in their academic careers that to succeed in schooling they needed to learn how to speak english well. at the same time, they feel that outside academic institutions, and in situa vomvoridi-ivanović estoy acostumbrada hablar inglés journal of urban mathematics education vol. 5, no. 2 98 tions where they do not feel any type of pressure to use a certain cultural language, they feel comfortable using spanish as well as english. after discussing several factors that the psts felt affected their language choice in different contexts, including in the context of mathematics, jose summarized the discussion as follows: i think we were all saying that because when we’re feeling that we’re doing something that’s in school or is related to school because our school experience was in english that we have to speak in english. but if we are outside, then we don’t feel that sort of like pressure so it’s basically the environment. and then we were saying that we’ve been taught math in english so we might not know the terminology in spanish and in the school in general we use academic language and we do that for math too and this is not the language we use outside. so maybe it’s both, right? during this discussion, the psts noted that because they had been taught mathematics in english, they did not learn the specialized language of mathematics in spanish and at the same time they learned to associate english as the language used in mathematical discussions in schools. moreover, since their schooling has been in english, it is the language they are accustomed to using when they are discussing subjects related to school, including mathematics. during a later debriefing meeting, while the psts were reflecting on their use of spanish during a mathematical task with salvador and paco, the two spanish-speaking fellows, they revisited these issues: jose: it’s like all our lives we have been trying to speak proper english in school and to sound academic and then we come here and we try to talk about math in spanish and math is academic and it is very hard. now it feels weird to use spanish in here because this [math] is not something i am used to talking in spanish but in english only. it’s like i am not used to talking about things i learned in school in spanish because we always used english and i always tried hard to sound proper and here [at the university] we have to sound academic like when we write papers and it is all in english. juanita: yea it’s like we—all these years we had to, to use english in school and now using spanish is weird—i am not used to talking in spanish in school—i mean when i’m in class. so this here [doing math in spanish] is very different. lupe: i remember in 6th grade when the principal told us we couldn’t speak spanish any more and that it would all be in english and i remember that really hurt—i really remember that—and it was very hard but i guess after that i got used to it and now it feels weird talking in spanish when i am in school because i got used to the english but when i was little i didn’t want to use all english because it was hard for me and i would rather do it in spanish. whoa! it switched! i just thought of that now! vomvoridi-ivanović estoy acostumbrada hablar inglés journal of urban mathematics education vol. 5, no. 2 99 maria: i also had that thing about not talking in spanish anymore in 6th grade and it was hard for me too. but eventually you get used to the fact that english is the language for school and spanish for outside. all four psts had experienced one point in their schooling where they were forbidden the use of spanish in the classroom. whether this took place during the early grades, as in jose’s and juanita’s case, or during middle school, as in lupe’s and maria’s case, they all remember it as a painful experience. lupe realized that in some sense she felt a similar discomfort when discussing mathematical activities in spanish to the discomfort she had felt when she was forced to use english only in 6th grade. the fact that over the years the psts had gotten accustomed to using english when doing mathematics became an extra challenge for them when they tried to use spanish in their mathematical discussions. being bilingual does not automatically mean being an effective facilitator in two languages, and this is more nuanced than simply sharing the same home language. during the same debriefing meeting paco asked the psts why they switched from spanish to english half way through doing a mathematical activity. lupe and juanita said that they have become used to speaking in english to everyone other than their parents and therefore have become accustomed to speaking in english rather than spanish: lupe: yo pienso que primero tratamos de usar más el español pero yo estoy acostumbrada hablar en inglés. porque en el día estoy en la escuela y hablo inglés. en la casa nomás hablo español con mi mamá y mi papa. [i think that first we tried to use only spanish, but i am used to speaking in english. because during the day i am at school and i speak english. at home only i speak in spanish with my mother and father.] paco: pero es interesante porque yo no noto que ustedes estén luchando. [but it is interesting because i don’t notice that you guys are struggling.] lupe: yo no estoy diciendo que lucho, yo estoy diciendo que estoy acostumbrada hablar ingles. [i am not saying that i struggle, i am saying that i am used to speaking in english.] lupe explained that not only had she become accustomed to using english inside academic institutions but also she has become accustomed to using english everywhere other than when speaking to her parents. juanita shared the same experience and added that even with her parents she had to make an effort not to use english. lupe regularly noted in her field notes that because she spends most of her time at the university where she speaks “in english all day long,” it carries over to the afterschool without her realizing it. one thing that is interesting with lupe’s final comment is her statement about not struggling with speaking mathe vomvoridi-ivanović estoy acostumbrada hablar inglés journal of urban mathematics education vol. 5, no. 2 100 matically in spanish but rather simply being accustomed to using english. paco’s comment reveals that lupe displayed competence in using spanish when talking about mathematics and did so with ease. however, to lupe, it felt more natural to use english rather than spanish in mathematical discussions. students in los rayos resisted spanish the second pattern associated with psts’ difficulties in integrating spanish during mathematical activities involves’ the students’ language choice. during the sessions that the psts expressed that using spanish with the students was very difficult, the students’ talk was english dominant, even when the psts addressed them in spanish. in addition, individual students resisted the use of spanish by demanding that the psts and other students in their groups speak english. students’ resistance to spanish relates to the previous theme of associating academic (mathematical) discourse in english. both themes result from strong socialization patterns that lead to disassociating mathematics with spanish. student talk was english dominant. juan, maria, and lupe attributed their increased (and often exclusive) use of english during mathematical discussions to the fact that students spoke amongst themselves in english and, as a result, they felt it was more natural for them to use english as well. lupe reflected on this during one of the debriefing meetings: i go there…and before…we begin teaching or whatever i hear the kids talking in english so i don’t know i guess i just…automatically—because they’re doing it, you know, …i think about speaking to them in spanish but i do…catch myself, like i’m saying all of this in english and i see how…we could be speaking spanish but since they…hear me speaking english they’re not…speaking it either. lupe here explains that because she would usually hear the students chatting in english before the afterschool sessions began, she would instinctively talk to them in english as well and that resulted in english dominant dialogue that carries into the mathematical discussions as well. maria shared a similar experience as with lupe, as her students would regularly use english when chatting about various topics. in maria’s and juan’s case, however, even when some of their students spoke in spanish, they were “forced” to switch to english due to individual students in their groups who repeatedly demanded to use english. juanita also reflected on the few sessions where she had english dominant students in her group and explained that the fact that those students spoke english influenced her language choice as well: i hear them speak in english so i talk english and then i…notice myself speaking english so i try to switch it to spanish but then i somehow get back to english and i try both of it but spanish doesn’t come natural to me. …cause like if they ask me vomvoridi-ivanović estoy acostumbrada hablar inglés journal of urban mathematics education vol. 5, no. 2 101 something, they usually ask me in english so i respond in english like i didn’t think about the language. similar to the other psts, juanita realized that because the students talked to her and to each other in english, using spanish did not feel “natural.” when addressed to in spanish, students responded in english. in addition to the student’s predominant talk being in english, individual students would regularly show resistance towards using spanish. in most cases, this resistance would take the form of consistently responding in english when the psts would address them in spanish. but in other cases, students’ resistance to spanish was more aggressive and individual students would verbally ask and even demand english to be spoken in their group. jose, whose group was consistently english dominant, regularly described and reflected on the phenomenon of the first case in his field notes. he repeatedly noted that certain students in his group would use english even when he talked to them in spanish: today we had two boys who are english dominant they are andre and alfonso hence the dominant english speaking. i know that if necessary both andre and alfonso can respond in spanish. …i would get one-word responses from them both when i would ask simple questions in spanish, or they would simply respond in english, showing me they can fully comprehend what is being asked of them in spanish. jose realized that even though both andre and alfonso understood spanish, they chose to use english. he attributes the english dominance of the discussions during the afterschool sessions partly to the fact that certain students, such as andre and alfonso, consistently use english even when being addressed to in spanish. in later field notes, jose explains that this behavior has influenced his, as well as other students’, language choice: throughout the meeting with the boys we mainly spoke english. again i believe it has to do with the main students who shape the tone of the group which influence the english dominance. i began talking to them in spanish but i mainly got most of my responses in english so i think that is why eventually i ended up speaking english. even rodrigo spoke mainly english, which is funny because the first encounters i had with him he would mainly talk spanish. i wonder if it’s the others’ influence on him that just have gotten him used to talking in english. jose explained that not only is his language choice influenced by the fact that certain students consistently use english but also that other students, like rodrigo, who in the past have used a lot of spanish, began using english as a result. during a debriefing meeting, psts were asked to reflect on their observations that mathematical discussions in los rayos were being increasingly con vomvoridi-ivanović estoy acostumbrada hablar inglés journal of urban mathematics education vol. 5, no. 2 102 ducted in english. jose explained that when he first joined los rayos he was more conscious of his language choice and he was purposefully using a lot of spanish. however, as time went by and students kept responding to him in english, he eventually gave up using spanish: when i first…was there i was aware of my english and spanish…when i wanted to…speak but i think in time, because some of them are more english dominant students there…i felt that they kind of like pushed us more towards the english language. …and not that like i was aware of that but i would sometimes even ask questions in spanish to see if i can get like a response to it—and i would get some spanish responses but (most were in english and) i know it is dominant english, i know that we have gotten deeply into english speaking because of that. jose would initiate discussions in spanish or in both spanish and english but students would reply in english, which led him to use english as well: the boys mainly spoke english today even if i asked them questions about the problems in spanish they still would answer me in english. …i was hoping to spark a spanish conversation using spanish but it didn’t happen. even those other students who i know have great spanish spoke mainly english like rodrigo and arnoulfo. i’m sure it had to do with the fact that english has been this group’s main response language. jose realized that even students such as rodrigo and arnoulfo, who had used spanish in the past, began using english almost exclusively. in other words, jose realized that english had become the group’s dominant language. the other psts shared similar experiences with jose. they would often initiate discussions in spanish but these discussions in most cases would eventually become english dominant because of the students’ predominant use of english. even juanita, who was the pst who regularly included spanish in mathematical discussions, and was the pst that out of the four used the most spanish, also realized that having english dominant students during two sessions made it difficult for her to carry out mathematical discussions in spanish. reflecting on this experience during a debriefing meeting she said: even if i talk to them in spanish—when i talk in spanish i notice myself speaking in spanish and i’m waiting for them to like answer back in spanish but they don’t. they answer in english. even if i keep trying and trying they answer in english. she expressed that her attempts to include spanish were blocked by these students’ consistent responses in english. for juanita, it did not feel natural to talk to the students in spanish if they responded in english. in fact, during these two sessions that juanita is referring to, more than half of the times that she addressed the vomvoridi-ivanović estoy acostumbrada hablar inglés journal of urban mathematics education vol. 5, no. 2 103 students in spanish or in hybrid languages they responded in english thus making her task of including spanish talk more difficult. some students demanded english. the second form of students’ resistance to spanish was that of individual students’ demands for english to be used in all types of conversations within their groups. one of maria’s students, for example, monica, verbally demanded english every time maria or any of the other students in the group used spanish whether it was during mathematical discussions or when they were chatting about non-mathematical topics. every time monica demanded english, the group would switch to english and use english throughout the rest of the session. maria noted this switch in her field notes of the first afterschool session, writing: “i also tried to talk mostly in spanish through the process of the activities but monica would scream at me ‘english please!’” in fact, during this session that maria is referring to, monica demanded english by interrupting the rest of the group that was having a side conversation gossiping about a girl in the school in spanish and said: “en ingles por favor.” (in english please) and the group switched to english immediately after that. later, while working on a mathematical activity, maria’s attempt to include discussions in spanish was interrupted by monica’s request to use english. prior to the excerpt that follows, maria, griselda, and lisbeth were discussing one of the mathematics tasks in spanish when monica interrupted them: monica: wait! i don’t get this part. how can you go to the side up and then to the side if it’s already here? maría: ok, cuántas flechas hay aquí? [ok, how many arrows are there here?] monica: dos. [two] maría: ok monica: why don’t we talk about it in english? maría: ok six. six is right here, right? as evident, maria, the pst, attempted to respond to monica using spanish but monica asked her to use english. maria, in turn, immediately responded to monica’s request, switched to english, and continued the dialogue in english. in fact, every time monica requested english, the entire group switched to english. maria, however, tried to keep using spanish when talking to the other girls in the group individually. soon after, however, maria resorted to english and used only english every time monica was present: i feel like i need to speak more in english than spanish because every time i try to talk in spanish monica doesn’t want me to. she says that she doesn’t understand. she keeps saying, “english please.” now if one of the other girls that know how to speak spanish asks me a question in spanish then i will talk in spanish. but monica wants to know what’s going on. so i end up using english with everyone. it’s just easier that way. vomvoridi-ivanović estoy acostumbrada hablar inglés journal of urban mathematics education vol. 5, no. 2 104 maria felt that it was easier for her to speak to everyone in english since monica claimed that she did not understand spanish. in other words, she felt that english was the language that everyone in the group comprehended, and because spanish was not, she resorted to using almost exclusively english regardless of the other students’ english proficiency or whether they preferred spanish. in fact, in the sessions that followed, maria used only english even though she had students, lisbeth and miriam, who had displayed their stronger proficiency in spanish and their preference in using spanish. it must be noted that even though there were individual students who openly showed their preference to spanish by consistently choosing the spanish version of the activities, for instance, these students never demanded that spanish be spoken. although the psts had noticed and noted in their field notes those students whose preferred language seemed to be spanish, the psts still chose to use english with their groups. the assumption on their part appears to have been that because everyone understands english to a certain extent, and some students openly resist the use of spanish, by using english with everyone the sessions would run smoothly. these scenarios are commonplace—around the country, but especially in our cities—because of the increasing variations of language backgrounds and proficiencies. lisbeth, for example, was a student that maria had identified as being spanish dominant. lisbeth always chose the spanish version of the mathematical activity. when monica was not present, maria would converse with lisbeth in spanish. nevertheless, during a session in october where maria was working with both lisbeth and monica together, maria did not use one word of spanish with lisbeth. similarly, in the following session, where maria worked with monica, lisbeth, and miriam, even though both miriam and lisbeth chose the spanish version of the activity, maria did not use any spanish with them during the entire session. neither lisbeth, nor miriam complained or asked for spanish. this lack of a request is possibly due to pressures they may have felt to work within english, as the importance of english is often overtly conveyed. the fact, however, remains that in both instances, when monica was present, maria did not use spanish as a resource to support these girls’ understanding of the mathematics involved in the activities. jose shared similar experiences with another student who demanded english, namely alfonso. even though jose had noticed that alfonso understood spanish very well, he regularly demanded english be spoken in the group. jose dealt with these situations similarly to how maria did by talking to alfonso in english and using spanish to other students but eventually used english when addressing everyone. for example, during one session, alfonso had the english version of the mathematical activity in front of him while the other students in the group had had the spanish version in front of them. jose asked one of the students in the group, rodrigo, to explain the directions of the activity. when rodrigo vomvoridi-ivanović estoy acostumbrada hablar inglés journal of urban mathematics education vol. 5, no. 2 105 started reading the spanish version alfonso interrupted him and asked him to read it in english: rodrigo: (reading) que son comunes al círculo y al pentágono pero no en el triángulo o el rectángulo. [that are common to the circle and the pentagon, but not in the triangle or the rectangle] alfonso: do it in english! josé: he can do it however he wants. he can do it in english or spanish. alfonso: i can’t understand him. josé: (to rodrigo) can you tell him what you mean? even though alfonso had the english translation of what rodrigo was reading in front of him, he still demanded english and claimed that he did not understand. the conversation that followed afterwards was in english, except for one point later when jose briefly interacted with rodrigo in spanish; but again, jose was interrupted by alfonso’s request for english: josé: (to rodrigo) ok, let’s take a look at it. que son comunes al círculo y al…, ¿qué es eso? [what are common to the circle and to the…, what is it?] alfonso: read it in english? josé: (to alfonso) i will. (to rodrigo) pero no en el triángulo o el rectángulo. significa que está afuera o adentro del triángulo? [but not in the triangle or the rectangle. does it mean that it is outside or inside the triangle?] rodrigo: lo que está dentro del triángulo; las estrellas que están dentro del triángulo no se pueden contar. [whatever is in the triangle: the starts that are inside the triangle, you cannot count them.] josé: no se pueden contar esas. [you cannot count those] all right? (to alfonso) did you get that? after that short interaction with rodrigo in spanish, jose continues in english throughout the remainder of the session. this conversation is just one of the many examples in which alfonso demanded english, and soon after the discussions would become english dominant, even when there were students in the group who were identified by jose as being spanish dominant. discussion and implications the patterns described throughout suggest that the psts’ language choice was influenced by social definitions of spanish and english that permeate our society. historically, the education of lms in the united states has been dominated by the notion of a common language for unity and, ultimately, americanization (lippi-green, 2012). english enjoys hegemony over other languages. it is believed to be superior, desirable, and necessary (shannon, 1995). the sociopolitical context of language learning in the united states has created an environment vomvoridi-ivanović estoy acostumbrada hablar inglés journal of urban mathematics education vol. 5, no. 2 106 where maintaining a language other than english is considered an obstruction to developing proficiency in english (nieto, 2010). latin@s have been on the receiving end of a pattern of devaluation of their language and culture for generations, in both schools and society. spanish, the language spoken by the largest linguistic minority group in the united states, is too often associated with poverty and marginalization and is socially deemed an obstacle to academic success; it is too often undermined as public language, and is treated as fit only for family life. according to mainstream ideology, the message is that spanish holds latin@s back. the notion of a common language, english, as the language of power is the driving force of many of the present movements of hostility toward other languages and their speakers (nieto, 2010; cummins, 1986, 2000). many believe that learning english “is the solution to the problems of immigrants and ethnicity in modern u.s.” (garcía, 2001, p. 49). the belief is that once the public language is learned, normalization occurs, and having one language, english, makes us american. the cultural and ethnic diversity that is upheld as an example is seen as a threat (cummins, 2000). more times than not, when one speaks a different language, is from a different country, or belongs to a different ethnic group other than the anglo majority group, one is considered a foreigner, and linguistically and culturally inferior. learning english is a condition for belonging, for inclusion in our society. a major institution that perpetuates this belief is the nation’s educational school system as it transmits the culture of the dominant class (mclaren, 2006). although there is no official language in the united states, the primacy of english in all aspects of our society is well established (crawford, 2004). english is considered the language of political, economic, and social power. it is the language of prestige and now also the language of technology. too often, the mainstream message in the united states is that languages and cultures that are different from the norm are not welcomed or desired by its people. english is viewed as an essential instrument of opportunity and success, and schools reflect the power structure of the society (garcía, 1990). in social conditions of unequal power relations between groups, classroom interactions are never neutral with respect to the messages communicated to students about the value of their language, culture, and intellect. students in turn internalize these messages as the rejection by the more powerful culture. this internalization impacts their self-value (macedo, 2008). for example, when educators discourage or prohibit students from using their home language in the school, it echoes the societal discourse that proclaims bilingualism or multilingualism as undesirable. it also sends the message that the students’ home languages, if other than english, have no place in academic settings and are not fit to serve academic purposes. vomvoridi-ivanović estoy acostumbrada hablar inglés journal of urban mathematics education vol. 5, no. 2 107 it is within this larger context that the four psts in the study were raised and continue to live in. it should not be surprising then that their use of language seemed to reflect negative definitions of spanish as manifested in several experiences such as being forbidden to use spanish in school, or by being explicitly told that the only way for academic success is to learn to use english in certain ways and to assimilate in the dominant culture. these experiences are not direct negations of using spanish for learning mathematics, but they are subtle socially acceptable messages that get conveyed by the broader society, schools, media, political groups, and by interpersonal interactions that english is for mathematics. students’ reluctance to using spanish in los rayos also reflects these socially acceptable messages that have english as the language for success and for academic subjects, such as mathematics, and spanish as the language spoken at home. none of the children who worked with the psts in this study ever demanded the use of spanish in los rayos, seemingly as if they didn’t feel they had the right to do so. on the other hand, several students did not hesitate to demand that english be spoken in their groups and psts were quick to accommodate these students by using english only. given the negative social definitions of spanish and the high status associated with english, demanding english to be used, even in a bilingual setting such as los rayos was “naturally” considered to be a valid and socially acceptable request to make and the psts never seemed to question it until provoked to do so. psts’ personal histories and experiences within this larger sociopolitical context also led them to form certain cultural models (holland & quinn, 1987) for teaching and learning mathematics. the psts’ experience as students has been in english. they had never been taught mathematics in spanish after grade 3 where all their classes in mathematics were in english, their books were in english, and as they themselves noted, they were always expected to use english in academic settings. in addition, their teacher preparation classes were in english and the public discourse for teaching, as conveyed in the media, has always been in english. as a result they developed a habit of thinking, or a cultural model, that related academic settings and subjects to english. this cultural model for teaching and learning mathematics involved a subtle exclusion of spanish and related mathematics to school and to english. in other words, doing mathematics in english was “normal” while doing it in spanish sounds “weird” as one pst noted. such a cultural model makes the process of using spanish as a resource to facilitate mathematical activities not so straightforward. psts’ ability to use spanish as a resource while facilitating mathematical activities is not just a matter of having the desire to use it, or of simply translating, or of having bilingual materials. it is also a matter of changing the habit of mind (i.e., cultural model) that has english as the language of schooling and of mathematics. it is a matter of including spanish in the cultural model, which can be done through experience using vomvoridi-ivanović estoy acostumbrada hablar inglés journal of urban mathematics education vol. 5, no. 2 108 spanish for academic purposes, in academic settings, and to discuss mathematical ideas. it is also a matter of realizing how the sociopolitical context of language both in the macro level—as manifested through public discourse and legislation— and in the micro level—as manifested through social interactions and daily experiences—shapes these cultural models and as a result affects teachers’ language practices in bi/multilingual settings. several researchers indicate that teachers using a non-english language for instruction did not have the language skills or basic professional preparation to do so (e.g. fabelo, 2008; figueroa & garcía, 1994; waggoner & o’malley, 1984). fabelo (2008) and guerrero (2003) have argued that for in-service and pre-service latin@ bilingual teachers gaining access to academic spanish is the root of the problem. however, the findings of this study suggest that gaining access or learning academic spanish is only part of the picture as this will not necessarily change latin@ mainstream teachers’ cultural models, which relate mathematics and schooling to english. knowing academic spanish in itself is not enough for countering one’s habit of mind that has spanish as the language for family and social life and english as the language of schooling and mathematics. it also is not enough for countering socially accepted behaviors that subordinate spanish, and therefore, present themselves as additional obstacles when trying to use spanish for instructional purposes. for maria and lupe, two of the psts in this study, for example, it was not a matter of gaining access to academic spanish as they felt comfortable and were capable of using spanish when communicating mathematical ideas. nonetheless, students’ resistance to spanish as well as the fact that they had associated english with being the language of schooling and mathematics made it difficult for them to use spanish. so what does this all mean for the mathematics teacher preparation of latin@ and other language minority psts? i provide two recommendations for the preparation and support of latin@ psts in mathematics: (a) create spaces for latin@ psts and other language minority psts to share their unique insights; and (b) provide language minority psts with opportunities to do mathematics in their home language. if we want latin@s and other language minority psts to use their linguistic knowledge (i.e., home language) for pedagogical purposes in mathematics we need to provide them with experiences that will help them include their home language in their cultural model of teaching and learning mathematics. earlier i suggested that in order to include spanish in one’s cultural model of teaching and learning mathematics one needs experience using spanish for academic purposes, in academic settings, and to discuss mathematical ideas; this includes having access to academic spanish but the key is having the experience. not only should language minority psts have opportunities to use their home language while assisting children in mathematics but also they should experience speaking mathe vomvoridi-ivanović estoy acostumbrada hablar inglés journal of urban mathematics education vol. 5, no. 2 109 matically in their home language with other people who are academically proficient in mathematics in the university setting. these experiences could be in the form of taking a mathematics content course in their home language and discussing the various issues that emerge. this way, not only do latin@ and other language minority psts gain access to academic language but also they come to recognize their home language as a valid language that can be used for academic (mathematical) purposes and in academic institutions. moreover, language minority psts can gain knowledge related to issues of language in the teaching and learning of mathematics through their own experience as bilingual mathematics learners in this class rather than through readings only. furthermore, we need to provide language minority psts with spaces where they can reflect on the sociopolitical context of language and the schooling of lms in the united states and how it shapes instructional interactions in mathematics. such experiences could be implemented through a seminar where language minority psts engage in discussions around language, culture, and mathematics teaching and 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(1984). teachers of limited english proficiency children in the united states. journal of the national association for bilingual education, 9(3), 25–42. microsoft word 3 final timmons brown et al vol 9 no 1.doc journal of urban mathematics education july 2016, vol. 9, no. 1, pp. 19–47 ©jume. http://education.gsu.edu/jume stephanie timmons-brown is a research associate and the executive director of the maryland institute for minority achievement and urban education in the college of education, 4716 pontiac st., suite1102, college park, md 20740; email: stbrown@umd.edu. her research interests include college and career readiness for underrepresented minorities, experiential learning through stem, and school contextual factors. catharine warner is an affiliate of the maryland institute for minority achievement and urban education, 4716 pontiac st. suite 1102 college park, md 20740; email: cwarner3@umd.edu. her interests include teacher and leader effectiveness, educational equity, school contextual factors, and educational policy and analysis. she is also a senior study director at insight policy research. using a conference workshop setting to engage mathematics teachers in culturally relevant pedagogy stephanie timmons-brown university of maryland, college park catharine warner university of maryland, college park in this article, the authors explore using a conference workshop setting to engage mathematics teachers, who serve largely underserved student populations, in culturally relevant pedagogy (crp). the conference workshop encouraged the exchange of information among teachers of similar grade levels and classroom contexts. the authors’ analysis of the findings highlight improvements in teachers’ perceptions of their crp knowledge as well as beneficial features of the conference workshop. these features include the creation of networks among mathematics teachers and team leaders, new post-conference mathematics lessons to implement in the classroom, and encouragement for the expansion of relationships and engagement in the classroom. while some teachers found their new knowledge of crp served to validate current practices, others found that the conference workshop provided a language with which to integrate successful practices into the mathematics classroom. keywords: culturally relevant pedagogy, mathematics education, mathematics teacher professional development ulturally relevant pedagogy (crp) has the potential to empower students’ learning and to generate high levels of success among racially and ethnically diverse student populations (ladson-billings, 1994, 1995a, 1995b, 1997). the use of crp has been connected to increased student engagement (boutte, kellyjackson, & johnson, 2010) and has been widely considered as a promising approach to improve student learning for various cultural groups within mathematics as well as other disciplines (dallavis, 2013; darrow, 2013; delpit, 2006; enyedy & mukhopadhyay, 2007; griner & stewart, 2012; hill, 2009; howard, 2003; leonard, 2008; malloy & malloy, 1998; tate, 1995). the national council of teachers of mathematics has highlighted several components of a culturally relevant apc timmons-brown & warner conference workshop and crp journal of urban mathematics education vol. 9, no. 1 20 proach as central goals to achieve equity in mathematics teaching and learning, particularly the goal to explore the importance of immediacy and relevancy of mathematics to students’ lives (gutstein et al., 2005; waddell, 2010). although crp has been applied to mathematics and science (boutte et al., 2010; martin, 2010; tate, 1995), we still lack systematic approaches to distilling knowledge of crp among mathematics teachers. there is relatively little literature linking crp efforts in mathematics to specific professional development approaches (bartell, 2011; leonard, brooks, barnes-johnson, & berry, 2010; morrison, robbins, & rose, 2008). while some research provides lesson plans and specific practices for implementation in the classroom (e.g., gutstein & peterson, 2006; irvine & armento, 2001; strutchens, johnson, & tate, 2000), there is little research on broad methods to encourage crp among teachers of underserved student populations in particular (matthews, jones, & parker, 2013). in this article, we use preand post-conference survey results from a conference intended to foster knowledge of crp as well as in-depth interviews to understand how mathematics teachers benefited (or not) from professional development about crp and its potential to raise student engagement in mathematics. in particular, we consider how mathematics teachers translated greater awareness of crp into practice one year later. the conference, hosted by the maryland institute for minority achievement and urban education (the institute), represents an effort to provide mathematics teachers with professional development through a conference workshop setting, creating collaboration among teachers of students of similar ages and within similar school contexts. the goal of this article is to illustrate how a 2day workshop experience highlighting crp in mathematics yielded improvements to teachers’ perceptions of their effectiveness, classroom practices, and teachers’ relationships with students. while workshops offer an efficient means to share information to large groups and to establish networking opportunities, once teachers return to their individual school contexts, the implementation of change can pose challenges. despite steps to provide teachers with partners and teams with which to implement results, there are often barriers to the translation of workshop experiences to new classroom practices. one of the challenges when investigating teachers’ professional development experiences lies in linking professional development efforts to student outcomes (loucks-horsley & matsumoto, 1999). ladson-billings (2006) notes that preparing teachers in crp requires a compilation of curriculum requirements, teaching practices, social justice, and the ability to deconstruct knowledge. most importantly, ladson-billings explains that you cannot tell anyone how to “do” crp. it is dependent on the individual students and the home, school, and community contexts. teachers who attended the 2-day, professional development conference reported here participated in workshops that guided and instructed teachers to consider the social and family context of their students and how individual student experiences timmons-brown & warner conference workshop and crp journal of urban mathematics education vol. 9, no. 1 21 may influence how they teach. in this article, we identify several benefits of the workshop format for improving crp among a diverse group of teachers. using quantitative analyses of teachers’ self-reports and qualitative analyses of in-depth interviews, we document several ways in which teachers translate workshop experiences to classroom practices. culturally relevant pedagogy: mathematics, implementation, and professional development the literature on crp represents a wide range of practices and theoretical applications, making the implementation of specific culturally relevant practices difficult given the potentially vague understandings of the term. here, crp refers to the theory put forth by ladson-billings (1994; see also 1995a, 1995b, 1997) to describe a pedagogy that “empowers students intellectually, socially, emotionally, and politically” (p. 18) by using culture and rapport to exchange knowledge. ladsonbillings (1995a) highlights three broad components to crp: (a) setting high expectations of academic success, (b) developing cultural competence with greater understandings of students’ (and teachers’) identities, and (c) creating awareness of inequality and social justice issues. crp uses the community, knowledge, and experiences of the students to inform the teacher’s lessons and methodology, in addition to reflecting on inequalities and power relations in society. this approach emphasizes students’ own knowledge and encourages them to be active participants in shaping the daily material taught during classroom lessons (gay, 2000). crp and mathematics much of the research on the use of crp in the mathematics classroom stresses the importance of relationships and cultural awareness (dance, wingfield, & davidson, 2000; ladson-billings, 1997; leonard et al., 2010). relationships with students are a central component of teachers’ abilities to empower students intellectually and to implement curricular that reaches students in the classroom setting (morrison et al., 2008). in using a culturally relevant approach to mathematics teaching and learning, teachers intentionally embed the mathematics content in socially meaningful contexts that matter to students (herzig, 2005). to establish a sense of “everyday life” in the mathematics classroom, teachers must have a strong sense of the cultural components of students’ cognitive approaches to learning (albert, 2000; brenner, 1998; matthews et al., 2013; nasir, 2002; waddell, 2010). for example, everyday relationships to mathematics may be better understood through exposure to students’ out-of-school experiences (gonzález, andrade, civil, & moll, 2001; nasir, 2002). teachers must make a careful appraisal of a single student’s performance, school setting, and household responsibil timmons-brown & warner conference workshop and crp journal of urban mathematics education vol. 9, no. 1 22 ities, moving teachers out of the mathematics classroom into home and community (bartell, 2011). these relationships with students also extend to students’ senses of community in the classroom. teachers must intervene in classroom exchanges among students to ensure peer support for student learning (dance et al., 2000; morrison et al., 2008). in a description of culturally relevant teaching practices in middle school mathematics classrooms, waddell (2014) distinguishes between the learning environment, classroom climate, and classroom community. through each of these components, respectively, teachers establish high standards, create a safe environment for students, and consider supportive relationships between students and community members. waddell claims that it is only within these various types of relationships that crp occurs. finally, a successful crp approach also must incorporate social justice pedagogy, which requires strong relationships with students (leonard et al., 2010; morrison et al., 2008). for example, leonard and colleagues (2010) note the intersection of crp and a social justice approach in mathematics occurred as a teacher realized that a students’ classroom behavior stemmed from personal changes. in rethinking mathematics: teaching social justice by the numbers (gutstein & peterson, 2006) several educators highlight relevant social justice issues such as racial profiling (peterson, 2006) and environmental justice (tate, 2006) and demonstrate how they are used to teach various mathematics topics. crp and implementation despite the potential success of crp there are challenges to putting culturally relevant theories into practice (boutte et al., 2010; milner, 2011; young, 2010). some researchers argue that culturally relevant techniques must be developed over time, through experience, and within individual teachers (see, e.g., milner, 2011). others have shown case studies of new teachers in urban settings adopting a culturally relevant approach by generating dialogue among teachers, students, and classmates; establishing personal relationships; and developing a sense of community in the classroom (see, e.g., price-dennis & souto-manning, 2011). together, the dialogue, relationships, and sense of community created in the classroom allow teachers the space to address issues of injustice as well as improve the contexts in which students learn classroom material (price-dennis & souto-manning, 2011). nonetheless, teachers often find incorporating all three components of crp—setting high expectations, developing cultural competence, and creating social justice awareness—challenging (young, 2010). as a result, approaches to implementing crp too often appear “limited and simplistic” (sleeter, 2012, p. 568). generally speaking, there are several concerns about the successful implementation of crp in classrooms. timmons-brown & warner conference workshop and crp journal of urban mathematics education vol. 9, no. 1 23 first, the time available in the classroom for planning and implementation is a potential problem. teachers have limited time to reflect, develop, and integrate crp into their curricular and teaching strategies, particularly with the introduction of “common core” requirements and changing expectations across schools and states. additionally, teachers are challenged to find time to collaborate and consult with their colleagues about what is working well in their respective classrooms. but research on the success of professional development initiatives indicates that allowing planning time for teachers improves the level of classroom implementation (penuel, fishman, yamaguchi, & gallagher, 2007). moreover, there is concern among educators about how to incorporate crp tactics in the classroom. teachers may have access to suggested practices but remain unsure about how to implement these suggestions (waddell, 2014). practices may also require additional teaching time, while teachers stretch to meet existing mathematics learning objectives within expected timeframes (young, 2010). second, some teachers may be less willing to include issues of injustice and inequality, a critical component of crp, into seemingly “objective” subject matter like mathematics (sleeter, 2012; young, 2010). but research on social justice mathematics suggests that such an approach offers students—in particularly, students of color—the opportunity to form positive mathematics identities, and can produce mathematics knowledge that is powerful in everyday life (see, e.g., gutstein, 2003, 2006). without the social justice component, teachers may engage with students in simplistic ways, teaching about “culture” rather than using culture to elevate learning objectives, or reduce crp to “steps to follow” rather than a broad approach to understanding social justice issues (sleeter, 2012, p. 569). many social justice issues can be addressed through a mathematical perspective, but there is a standing concern that, absent a true social justice approach, teachers will use cultural deprivation or deficit approaches to understanding how children of color succeed in the classroom (schmeichel, 2012). the third challenge arises in teachers’ efforts to establish relationships with students in the classroom. this aspect of teaching can be equally challenging as creating active learning strategies and implementing new classroom practices. at the foundation of a culturally relevant approach, teachers must connect lesson plans to students’ daily lives in their communities, families, and leisure activities, and be willing to learn from students’ experiences outside the classroom (boutte et al., 2010). a caution, however, is that teachers might develop an overemphasis on understanding students’ home cultures only superficially rather than understanding institutional issues of power, privilege, and inequality in the school setting and society at large (young, 2010). this overemphasis can result in an essentializing of student behaviors substituting for the development of supportive and caring relationships. timmons-brown & warner conference workshop and crp journal of urban mathematics education vol. 9, no. 1 24 the workshops hosted by the institute during the 2-day crp conference were intended to address some of these challenges and to support teachers’ ability to integrate crp into mathematics classrooms. as noted by morrison and colleagues (2008), however, a culturally relevant approach is often counter to the organizational structural of schools, significantly limiting teachers’ ability to implement longstanding change. analyses of the findings discussed here address the specific types of practices implemented in the classroom and the benefits teachers’ received from the workshop approach to professional development. crp and professional development traditional paths to teaching focus on pedagogical content knowledge, generally gained through university education, internships, and work experience (baumert, kunter, blum, brunner, & voss, 2009). the research on professional development is broad and emphasizes multiple kinds of knowledge: content, student learning, teaching methods, and how to help individual students gain knowledge in the classroom (loucks-horsley & matsumoto, 1999). a focus on teaching methods rather than content is only one way to improve teacher knowledge, but it has met with success in the past. baumert and colleagues (2009) found that, while content knowledge is required for teachers to impart knowledge to students, pedagogical training has a larger effect on student progress, making additional training efforts such as the conference workshop discussed here important for improved educational outcomes. several aspects of professional development activities have positive effects on teachers’ self-reported increases in knowledge and skills. effective practices include long-term professional development programs, immersing participants in active learning efforts, providing curriculum strategies, creating collaborative networks, focusing on improvements to student inquiry, and using university-based partners for professional development (garet, porter, desimone, birman, & yoon, 2001; loucks-horsley & matsumoto, 1999; penuel et al., 2007). one professional development effort involving crp among middle school science teachers occurred over a period of 15 months and provided content training, curriculum instruction, regular classroom observations, and, most importantly, networking opportunities with fellow teachers and outside observers to discuss problems and solutions to incorporate components of crp (johnson & marx, 2009). as a result of the professional development, teachers reported closer relationships with students and colleagues as well as the implementation of new, successful teaching practices (johnson & marx, 2009). while such long-term initiatives are not easily established, some components of such successful endeavors may be used in a lower-cost setting that targets a large number of participants. workshops are a setting commonly used to share such knowledge with teachers outside the classroom, and while they can be effective in generating ideas for timmons-brown & warner conference workshop and crp journal of urban mathematics education vol. 9, no. 1 25 change, they may not provide teachers with the necessary resources to implement those changes in the classroom (garet et al., 2001; loucks-horsley & matsumoto, 1999). incorporating successful features of workshops, such as individual components used by johnson and marx (2009), however, may provide one avenue to professional development within existing structural constraints. using a national sample of mathematics and science teachers, garet and colleagues (2001) found that facilitating networking, grouping participants by grade level, using active learning strategies, and including multiple teachers from the same school led to greater gains in knowledge and changes in classroom practice. by including these networking opportunities and generating collaboration within schools, educators have a better chance of implementing change if they have someone with whom to collaborate once they return to their home school. crp is one theory that may be well applied in a workshop setting to give educators additional resources with which to create gains in student learning. teachers used as exemplars for culturally relevant approaches in the classroom are often part of the community in which they teach (ladson-billings, 1995b). participation in the community provides necessary knowledge (a type of training) to employ crp practices. participation in a broader community that supports crp reflects teachers’ beliefs in establishing student–teacher relationships; strengthening connections with students and their communities; and conceptualizing multiple forms of knowledge (ladson-billings, 1995b). waddell (2014) focuses on culturally ambitious teaching practices in mathematics accomplished through a community of teachers receiving weekly coaching; such practices were created with the intention they would be “starting points” for community conversations on classroom practices throughout the school year (p. 15). problem statement and research questions the extensive research on professional development for educators provides a number of best practices for improving student outcomes. however, we know little about the extent to which these practices prove effective for the multiple components necessary for crp. additionally, extensive professional development efforts can require significant time and resources less often available to teachers of underserved populations. as a result, it becomes important to prioritize components of professional development that have evidence-based findings to support teachers’ knowledge development. workshop settings are a positive means of introducing teachers to crp practices for mathematics classrooms because they offer opportunities for teacher communities to form, allowing teachers to grow supportive networks as they implement crp practices. the resources and networks teachers develop through crp workshops contribute to a greater chance of crp implementation, and, eventually, teachers’ development of improved relationships with stu timmons-brown & warner conference workshop and crp journal of urban mathematics education vol. 9, no. 1 26 dents through implemented practices. given the significant emphasis on forming relationships with students in a culturally relevant approach, this facet of professional development must be emphasized (brenner, 1998; matthews et al., 2013; nasir, 2002). complicating the implementation of crp following a conference workshop are the many challenges and components to achieve a mathematics curriculum that fully reflects crp, not the least of which is an extended period of time to establish relationships and improve practices. therefore, much of the function of a workshop setting on crp must be to introduce educators to the concept, instruct them on key practices, and provide them with resources to continue to develop their pedagogy. here, we focus our analyses on two key research questions: 1. can a 2-day conference workshop setting improve teachers’ perceptions of their crp knowledge specific to mathematics? 2. what benefits do teachers perceive as a result of a conference workshop setting focused on crp in mathematics classrooms? methods the 2012 conference “helping mathematics teachers become culturally relevant educators: new tools for a new generation” was designed for those who teach mathematics in the elementary grades and for mathematics teachers in middle and high school. this conference drew on successes from an earlier conference in 2010 that indicated teachers welcomed information and techniques that would help them address the cultural knowledge of their students. at the conclusion of the 2010 conference, conference attendees submitted post-conference comments that suggested teachers faced daunting barriers to implementation. some teachers wished that other teachers from their home school had attended so that they could share ideas. in response, the 2012 conference assessed here incorporated workshops that invited teachers to attend as school teams and provided tools and instruction for classroom use. in addition, there were three plenary speakers and three workshop facilitators invited to the 2012 conference. the plenary speakers were requested to provide attendees with two perspectives from which to gain a better understanding of crp from both a theoretical and a mathematics practice perspective. dr. jacqueline irvine (emory university) and dr. geneva gay (university of washington), notable researchers in the field, provided an overview of the state of research and current thinking of crp. dr. lawrence clark (university of maryland, college park), an experienced mathematics classroom teacher and researcher in mathematics education, provided the more classroom-specific content. over the 2-day conference, timmons-brown & warner conference workshop and crp journal of urban mathematics education vol. 9, no. 1 27 each teacher participated in at least six hours of workshop activities with their designated workshop facilitator. analyses presented here are based on a closed-ended survey and follow-up interviews with conference attendees 1 year after the conference workshop. the research process, therefore, is a sequential explanatory mixed methods model, first collecting the quantitative data and then collecting qualitative data to better answer research questions not adequately addressed through quantitative data (creswell, 2007). to evaluate respondents’ perceived gains in crp knowledge over the 2-day conference workshop, preand post-conference surveys were administered to attendees. the pre-conference survey was distributed and collected prior to the first workshop. attendees were asked to complete the post-conference survey after the last workshop, during lunch. sample questions from the survey are included in appendix a. a total of 81 individuals attended the conference; 76 respondents (93.8% response rate) completed the pre-conference survey, while 48 respondents (59.3% response rate) completed the post-conference survey. analyses used paired t-tests to measure significance (at p < .05) preand post-change, resulting in a sample size of 48. the paired t-test approach takes into account the shared error in respondent characteristics of the before and after survey sample, increasing the standard error in analyses to allow more conservative estimates of change. table 1 characteristics of all conference attendees compared to preand post-conference survey respondents all attendees (n = 76) all attendees sd paired sample (n = 48) paired sample sd race/ethnicity percentage percentage white 65.3 0.48 64.6 0.48 black 27.8 0.45 29.2 0.46 am. indian/ nat. hawaiian 2.8 0.17 2.1 0.14 asian 4.2 0.20 4.2 0.20 gender male 18.1 0.39 12.5 0.33 female 81.9 0.39 87.5 0.33 grade taught elementary 40.3 0.49 43.8 0.50 middle 44.4 0.50 37.5 0.49 high 13.9 0.35 16.7 0.38 middle /high 1.4 0.12 2.1 0.14 note: sd = standard deviation table 1 provides characteristics—race, gender, and grade taught—of the conference attendees. on most demographic characteristics, individuals that responded timmons-brown & warner conference workshop and crp journal of urban mathematics education vol. 9, no. 1 28 to both surveys (at the start and completion of the conference) are similar to the rest of the conference attendees, including those that did not respond. two exceptions are in respondent self-reported gender identity and the grade that the individual teaches. slightly more women are included in the results included here compared to those at the conference (88% compared to 82%) and more respondents to both surveys taught elementary school compared to total conference attendees (44% compared to 40%). in an effort to gain a more detailed account of how attendees implemented culturally relevant practices in the classroom, eleven interviews with conference attendees and their school principals were conducted. schools with multiple educators as attendees were targeted for the follow-up interviews. we also solicited respondents for the interviews using grade level and geographic location as guidelines, targeting teacher attendees from both elementary and middle schools as well as teachers from each of the three surrounding states represented by conference attendance. the proximity of surrounding states allowed for in-person interviews. the second author approached 16 teachers and instructional coaches at 11 elementary and middle schools to participate in the research. in-depth (in person and phone) interviews were conducted with five elementary school teachers and three middle school teachers at six schools. after speaking with the educators, the second author also contacted principals at those six schools to gather more information on school-wide priorities and policies regarding crp. three principals at these corresponding schools agreed to interviews. interviews were 15 minutes to 1 hour in length, with the principal interviews considerably shorter than the teacher interviews. table 2 provides summary information on teachers’ characteristics, detailing the educator’s role as a teacher or instructional coach, and primary classroom grade level. the schools at which the educators are employed vary considerably in their demographics. conference organizers solicited participation from schools with a substantial proportion of students of color, although in some cases principals also requested that their teachers have an opportunity to participate in the conference workshops. the racial and ethnic composition and free and reduced-price lunch percentages reported for the 2011–12 school year are also included in table 2. in-depth interviews addressed what respondents remembered about the conference workshop 1 year later, how they had used conference workshop information, and some of the attendees’ potential paths to classroom implementation. the timing for the interviews is important, as 1 year had passed before assessing how (and whether) participants used (or not) the conference workshop information. experience from a past conference on crp conducted by the first author indicated that teachers were best able to implement new teaching techniques when they attended the conference in teams. as a result, the institute solicited team attendance from schools for the 2012 conference. the interviews used the same basic interview guide, developed by the first author (available on request). all quotes have been timmons-brown & warner conference workshop and crp journal of urban mathematics education vol. 9, no. 1 29 edited for false starts, repeated words, and extraneous phrases (e.g., well, um, like, you know, kind of, i mean, whereas). ellipses note when some sentences or phrases have been cut to clarify the statement without changing the original meaning. table 2 qualitative interview sample characteristics pseudonym grade level % free or reduced-price meals % school racial composition title cathy jackson elementary 83% 63% black 21% white teacher diedre noon jennifer pence elementary 10% 6% black 77% white teacher teacher rosa daniels elementary 75% 88% black math coach tanya nash elementary 84% 98% black math coach beth idler russell gates middle 65% 53% black 38% white math coach teacher kendra burns middle 52% 25% black 62% white math coach findings survey findings survey findings were analyzed using paired t-tests of preand postconference evaluations. results from the evaluation show that participants increased their basic knowledge of key terms and concepts as they pertain to crp. given the large range of terminology that surrounds a culturally relevant approach, participants were asked to rate their familiarity with key terms at the start of the conference workshop and again at the close. attendees consistently rated their perceived knowledge of the many concepts by which crp may be known as improved. these results are indicated in table 3. as an example, familiarity with the term “crp” in the first row increased by a full standard deviation by the end of the conference workshop. this translates to a pre-conference survey average rating of being familiar with the term “to a small extent” to a post-conference survey average rating of being familiar with the term “to a moderate extent.” the aim of the conference workshop, however, was to improve both their perceived knowledge and anticipated crp practice in the classroom. a second set of questions tested the use of culturally relevant teaching practices in the 2011–12 school year prior to the conference (pre-test) and intended usage for the remainder of the 2011–12 school year following the conference (post-test). while many practices were used quite frequently prior to attending the conference workshop, there are several areas in which participants show likelihood of increased use of practices timmons-brown & warner conference workshop and crp journal of urban mathematics education vol. 9, no. 1 30 consistent with crp, as shown in table 4. specifically, participants noted a statistically significant increase in their intended use of the following practices: planning lessons toward a variety of abilities, making sure all students understood content before moving on, allowing students to share artifacts from their own cultures, using real-world examples, implementing strategies to ensure that teachers’ attention is equitably distributed, and reflecting on their own (teachers’) cultural biases. table 3 self-reported change in familiarity with culturally relevant terminology pre-test post-test mean sd mean sd “culturally relevant pedagogy” (crp) 2.42 0.89 3.29 * 0.65 “culturally relevant teaching” 2.64 0.87 3.52 * 0.58 “culturally responsive teaching” 2.44 0.87 3.44 * 0.58 “culturally sensitive teaching” 2.54 0.83 3.44 * 0.68 note: scores represent mean of the range where 1 = no extent, 2 = small extent, 3 = moderate extent, and 4 = large extent; only pairs are in the sample for score analyses (n = 48); * p < .05; sd = standard deviation table 4 respondents’ self-reported current and anticipated use of crp practices pre-test (current use) post-test (anticipated use) mean sd mean sd a. plan lessons toward a variety of abilities and needs 4.26 1.08 4.67 * 0.52 b. visit student families outside of schools 1.60 1.09 1.84 0.78 c. make sure all students understood the content before moving on 4.14 0.99 4.47 * 0.50 d. allow students to share cultural artifacts 2.70 1.36 3.85 * 1.00 e. use real-world examples 4.65 0.61 4.86 * 0.35 f. use strategies to ensure attention is equitably distributed 4.48 1.09 4.81 * 0.40 g. engage with students about their problems 4.67 0.61 4.60 0.49 h. explain concepts in different ways 4.81 0.55 4.84 0.37 i. reflect upon own cultural heritage and biases 3.58 1.26 4.26 * 0.83 note: scores represent the mean of the scale of use of the practice where 1 = never, 2 = quarterly /annually, 3 = monthly, 4 = weekly, and 5 = daily; only pairs with non-missing scores are included (n = 46); * p < .05; sd = standard deviation moreover, reviewing teachers’ anticipated change in practice could inform our understanding of how their actions align with other notable pedagogical theory. for example, items “a” and “e” in table 4 have gained particular attention in the timmons-brown & warner conference workshop and crp journal of urban mathematics education vol. 9, no. 1 31 teacher practice literature as teaching activities that influence classroom engagement—differentiated instruction (item “a”) and funds of knowledge (item “e”). as teacher attendees indicated that they anticipate employing more lesson plans that address a range of student abilities and ultimately more engaged learners and possibly increased academic achievement. for item “e”—use of real-world examples— the survey results increased from 4.65 to 4.86, which represents a statistically significant change and provides important implications for teacher practice. according to civil (2002) teachers’ use of real-world examples positively influences students’ interest in the lessons presented and their ability to comprehend and retain the information. for the most part, the anticipated increase in use of key practices was relatively small, with less than a standard deviation for all the practices. however, for five of the six practices, teachers left the conference workshop with the intention of engaging students in these approaches on a weekly basis, as opposed to a monthly or quarterly basis, engaging students more frequently than prior to attending the conference workshop. in response to the first research question—whether a conference workshop setting improves knowledge of crp—the survey data indicates that educators left the conference with an improved perceived understanding of crp and intentions to use related practices more frequently. given the limitations of the results collected at the conference site through preand post-conference surveys, we followed up with participants one year after the conference workshop to assess the extent to which crp had been integrated into their classrooms. these follow-up interviews indicate that participants found particular aspects of the conference workshop beneficial in changing classroom practices, with some limitations on the extent to which changes were successful when fully integrated, as outlined in the next section. interview findings the second research question addresses the particular components of professional development efforts that serve to change classroom practices, translating knowledge to practice. there are three key benefits that accrued to conference workshop participants: networking with others, sharing teaching practices, and developing teacher–student and student–student relationships. within each of these, there are additional benefits. for example, networking provided participants not only with new colleagues but also a new language for sharing culturally relevant teaching practices that they did not previously possess. networking with others. educators sought an opportunity to share experiences in the classroom, and the crp conference workshop provided the setting to interact with teachers in similar grade levels. for example, rosa daniels, a mathematics specialist in a title i elementary school, noted that the chance to share information with such a large number of colleagues was one of the conference highlights: timmons-brown & warner conference workshop and crp journal of urban mathematics education vol. 9, no. 1 32 what i remember most was the participants … i remember speaking with them and listening to how they were trying to transform the way they do mathematics in their school system. and what impacted me was them talking about no more procedural math being done until the fourth grade. so k–3 they were trying to do more exploration and more conceptual types of mathematics versus doing procedural type of mathematics. and they explained some of the things that they were doing in that county and how they were changing those things. … the discourse in the session was more powerful than anything else. networking opportunities provide language for positive efforts underway at individual schools and offer educators pedagogical support for current or imagined projects. tanya nash, a title i mathematics specialist at an elementary school, also felt that networking was one of the most useful components of the workshop. she noted: it’s not often that we get a chance to have professional dialogue. coming together and sharing. the fact that we were from different cultures, different backgrounds to share experiences. i thought that was a great form and a great way to come together. that’s what i really remember most. the networking. additionally, by allowing educators the opportunity to discuss practices that build relationships and provide a sense of understanding of students’ home and school environments, educators are given a language to highlight successful practices. educators referred to the break-out grade level sessions as “fantastic” and noted the many ideas that “flowed” through the session. kendra burns, a mathematics coach in a middle school, noted that the conference reinforced existing practices by providing a vocabulary for describing her efforts: i was doing a lot of it before; it was just never called “culturally relevant teaching”… but since the conference, i’m just more aware of it, and … so that when i come up with something, and i feel it’s pretty darn good, i’m looking around to see, what else is out there that might touch base with a student that maybe has a different background. in many ways, networking provided additional validation and support for teachers that often felt uncertain about how their classroom technique would be perceived by colleagues. additionally, by providing educators with new ways to understand their existing practices, participants were able to expand upon successful efforts using a culturally relevant approach. the networking available at the conference also validated some educators’ more creative efforts to capture children’s attention and build relationships. according to rosa daniels (an elementary instructional mathematics coach at a title i school): timmons-brown & warner conference workshop and crp journal of urban mathematics education vol. 9, no. 1 33 it comforted me to know that i wasn’t some wacky, crazy teacher out here trying to just do all these things … it really was something that was needed for our children to relate to, and then tying in economics with doing some addition and subtraction … i’ve always done stuff like that; it just helped me to understand that it was important, and it was needed. for many, networking at the crp conference workshop offered teachers a label for practices they may already employ. teachers that attended were more likely to work with underserved student populations. such teachers often found themselves seeking to establish relationships with students, to present the subject matter in an accessible manner, and to provide a fun atmosphere for learning combined with high expectations for students. however, it was not as often that these practices were given a name and status as “best practices” approaches to engaging with students. the language of the conference workshop offered teachers a way to communicate about their efforts in a professional setting, with the support and acknowledgement of experts. sharing teaching practices. similar to the networking opportunities, the workshop format modeled several best practices in culturally relevant teaching that could be implemented in the classroom. as participants came together, they were able to share practices that had worked well in their own classrooms. after the conference workshop, jennifer pence describes her elementary classroom as “a busy, busy place.” the classroom practices covered several areas of mathematics learning. we offer three examples of teachers we interviewed to demonstrate how they engaged in crp for mathematics. in the first example, mathematics teachers used a culturally relevant approach to foster student-to-student communication in the classroom. for example, russell gates, a middle school mathematics teacher, noted: i also remember one of the guys in the group made a real impression on you, because he used the respectful talk: “i respectfully disagree.” that was a great conversation that came up in our session … having the kids learn how to talk to each other, to work cooperatively in groups and having that level of respect. it doesn't always work, but i did implement it since that conference, day one. these kids are already shut down enough. the last thing we need is to shut them down even more by saying, “your answer is wacky,” which [other students] want to do. the practice of having students respectfully disagree with peers was successful in russell’s classroom and easily implemented. in this way, students had an opportunity to learn from each other, but were required to do so from a place of respect. in a second example, several teachers noted that an immediate (and relatively simple) improvement to classroom practices was the use of real-world examples for students. kendra burns, a middle school mathematics coach, gave one example— timmons-brown & warner conference workshop and crp journal of urban mathematics education vol. 9, no. 1 34 it’s just being more aware of pulling in different aspects of those real-world problems. so instead of having a problem that would predominantly be something that a white caucasian person would encounter, it’s more broad in general. and just being specific, for instance, we’ve done a lesson with music and math, and the title of the lesson was literally “can music kill you?” it was talking about the heart rate, and it was based on a lesson that was designed through a website, it’s called mathalicious. basically, we changed the music so that it kind of hit all different cultures, not just classical music or something of that sort. we’ve just incorporated more into what was not maybe a meaty math problem, but to make it relevant to the students. seven of the eight teachers interviewed provided examples of real-world applications they have used in mathematics. suggestions ranged from examples using sports percentages and shopping, to the use of manipulatives and mathematics applications that related to recent reading assignments. finally, in a third example, three respondents commented on their success with employing active learning practices in the classroom, which they learned at the conference workshop. these respondents were primarily teachers of elementaryage children, who perhaps had the most to benefit from providing students with a way to be actively involved in their mathematics lessons. cathy jackson, a second year general elementary teacher, described why after the conference workshop mathematics became the more successful lesson time in her classroom— math is so much more hands-on, and manipulative, and drawing and creating models and seeing what other people are doing and creating. the students just seem more like they’re into it with math, whereas reading—i hate to say this—i love reading and i’m a reading person, and i’m an avid reader and have been so those little readings just more sit and focus on what you’re doing, not a lot of opportunity to move and bounce around the room. with math, you have them working more collaboratively to achieve something. similarly, rosa daniels provided a specific example of how mathematics lessons could become more movement oriented in the classroom. she provided an example for a general elementary teacher at her school that would get students moving, but also communicate mathematics curriculum. she explained: so we’re running as fast as we can in place, and [we sing this song called] “5,280,” understanding that every time we hit our feet to the floor that means it’s a foot. so it’s, “5,280 5,280” and i’ll say, “what does that mean?” [and, the students say,] “fivethousand-two-hundred-eighty feet in a mile.” so she [the teacher] allows me to help her with things like that. and she’s actually come up with her own things to help engage them in that way too. so i think i inspired her by coming to the conference to do more things like that with the kids. the workshop approach not only provided educators with active instruction from an expert in the field but also with several opportunities to network with col timmons-brown & warner conference workshop and crp journal of urban mathematics education vol. 9, no. 1 35 leagues to learn of new practices that could be applied in classrooms. the approach to networking at the conference workshop sorted teachers according to grade level offered teachers several advantages: new peer networks and colleagues, a language for labeling successful teaching practices, validation of successful teaching practices, clear examples of new practices to employ in the classroom, and a broad network to share other teachers’ best practices. developing teacher–student and student–student relationships. in following up with educators one year after the conference workshop, they often focused on their efforts to establish relationships in the classroom. these relationships included educators’ desire to reach students on a personal level as well as their interest in facilitating the classroom relationships among students to promote cooperative learning. several participants highlighted these ideas when discussing how they employed knowledge from the conference workshop in the classroom. for example, one teaching team at an elementary school changed their classroom furniture from individual desks to small groups at tables in an effort to promote cooperative learning following the workshop. according to diedre noon, “we’ve gotten rid of the traditional desk and have tables where [students] are encouraged to be much more collaborative and share their thinking, and to really foster that sense of community and culture and care.” teachers also noted that the preparation in crp inspired them to be cognizant of the issues outside of school that may influence students’ learning. for example, an elementary teaching team that attended the conference workshop suggested that they are better able to consider each child as an individual person, rather than an individual learner. according to diedre noon, an elementary teacher— i think [the conference] just helped us to … look at each child as an individual. i mean we already do that when we differentiate, but we were looking at them rather than just on achievement, really, what the whole package was. what is that child all about and how can we plan lessons and how can we talk to them and how can we respond to them in ways that show that we understand all of that and we’re respectful of it. a culturally relevant approach requires that teachers develop relationships with children over time, and learning outcomes improve as teachers gain better knowledge of how to teach an individual child. diedre found that the conference workshop encouraged her to think beyond achievement and focus on understanding the whole child. this focus on the individual child also extends to the acknowledgement of a child’s home life as well as how they appear in the school setting. russell gates, a middle school mathematics teacher, described how important it is to consider why students may perform a certain way at school: timmons-brown & warner conference workshop and crp journal of urban mathematics education vol. 9, no. 1 36 the whole conference, the whole purpose was to try to understand where kids from different cultures, different levels of poverty are coming from, and how to accommodate. i think that’s what i got the most out of it. it’s just not, “you didn't turn in your homework? okay, ‘d.”’ [instead] you say, “maybe we can talk to this student and figure out is there something that we can do.” … their home lives are so different than what we’re probably used to. so, maybe they didn’t have time at home. you know, their parents aren’t home and they’re babysitting all afternoon, or they go to bed late, whatever the case may be. that’s what i got the most out of, just not judging, assuming right away. they didn’t turn in their homework. they don’t seem as happy today. did something happen to keep them from doing the homework assignment? … learning more about them, establishing that relationship, and it goes miles. classroom implementation of approaches to culturally relevant teaching may initially be limited to one aspect of crp, such as emphasizing cultural differences in the classroom as sleeter (2012) cautioned. our findings echoed this issue, as teachers focused on the family background of students as explanation for academic problems and sought to build relationships with those students. additionally, teachers might make assumptions that there are issues at home or that students are “going to bed late,” but not highlight personal relationships with students that would make clear this knowledge was entirely correct. overall, there was less emphasis on setting high expectations for academic success for students and providing sources of support within the school or of exposing oppressive power relations in the school or in the curricular content. in addition to creating teacher–student relationships, teachers sought to improve the relationships among students in the classroom to encourage cooperative learning. for example, diedre noon also noted: “in terms of grouping, [the speaker] especially talked a lot about cooperative groups and just using that to benefit you … so when i'm planning lessons … i'm thinking about how i'm going to put those kids together so that i'm drawing out on all of their strengths.” diedre’s quote is important, because ladson-billings (2006) notes that cooperative learning is not always an answer to social justice in the classroom. teachers must do careful thinking about the setting and be able to recognize when group activities will not generate learning outcomes. in other examples, participants noted how they guided more conversations in the classroom to foster religious and cultural awareness. in some cases, teachers highlighted each student’s contribution to class discussion as a way to develop a collaborative environment and identify ideas for future lessons. it is clear that relationship building may not come easily in the classroom, especially for teachers and students. crp may address some of those issues, but in other cases the teachers who need professional development may not seek it. according to rosa daniels (an elementary mathematics specialist at a title i school) establishing a rapport with the student can be one of the most challenging parts of teachers’ jobs: timmons-brown & warner conference workshop and crp journal of urban mathematics education vol. 9, no. 1 37 the teachers work so hard here to try to build these relationships with the students, and it’s such a disconnect. and i believe that they could use a whole session here, a series. so what i’ve been trying to do is to model for the teachers how to communicate with the students, how to teach them through doing lessons, showing them how to talk with the students. so my approach from what i got from your experience there is that i’ve been trying to live it as an example for the teachers, rather than trying to tell them this is what you should do or this is how you should handle it. in the end, student relationships are a core component of a crp, although they are not easily achieved. ladson-billings (2006) explains that before teachers can learn and implement crp, they must understand the ideology of the curriculum and the individual needs of their students. when an educator embraces the core components of crp and also assumes a lifelong interest in how their students are in 5 or 10 years and how education has shaped them, then they are “doing” culturally relevant teaching (ladson-billings, 2006). discussion despite the literature on the implementation of crp in mathematics classrooms, there is less attention to professional development efforts geared toward large groups of teachers. given the challenges associated with the implementation of crp, especially the numerous definitions of cultural relevancy and the structural constraints teachers face at individual schools, mathematics teachers would benefit from additional professional development efforts in crp. in response to these issues, the institute established a conference workshop setting for mathematics teachers at schools serving largely underserved student populations to improve their knowledge of crp and to create networks with which to build their classroom practices. here, we have highlighted three benefits to crp professional development that embraces a workshop setting approach. in addition to these benefits, we have also noted areas in which teachers’ application of professional development curriculum was lacking. these instances provide opportunities to refocus professional development efforts in crp in mathematics classrooms to address such challenges. in the first benefit, the 2-day conference workshop setting improved teachers’ confidence in their perceived understanding of knowledge of culturally relevant approaches to teaching and of grade-specific practices they might implement in their classrooms. in particular, teachers reported a better familiarity with key terms following the conference workshop as well as plans to increase the frequency with which they engaged in particular active learning strategies in the classroom as a result of attendance. it is important to note that the quantitative survey sample is a small sample, with significant sample loss between preand post-surveys. bias as a result of attrition and nonresponse is more likely related to geography than school or teacher characteristics. furthermore, given that teachers were not observed or timmons-brown & warner conference workshop and crp journal of urban mathematics education vol. 9, no. 1 38 evaluated on their understanding of crp or on the presence of particular classroom practices, results only indicate attendees’ perceived growth and confidence with key concepts and pedagogies. at the close of the conference workshop, quantitative results indicated that teachers intended to implement at least six practices related to crp on a more frequent basis, usually weekly instead of monthly. these practices included planning lessons geared to a variety of abilities, allowing students to share their culture in the classroom, using real-world examples, and reflecting on their own (teachers’) cultural heritage. with changes at less than 1 standard deviation, the increased use of crp in the classroom is not dramatic. nonetheless, any additional incorporation of these practices and other culturally relevant pedagogies are likely to facilitate relationships with students, especially in terms of teachers recognizing their place of privilege, power dynamics in the classroom, issues of social justice, and making the classroom content more specific to the daily contexts of students (herzig, 2005; price-dennis & souto-manning, 2011). we can draw on existing research for two of the practices teachers discussed using in the classroom to highlight instances where these improvements may yield additional results and align with notable pedagogical theories. the practice of teachers increasingly planning lessons for a variety of abilities aligns with the promotion of crp in mathematics classrooms. differentiation may be described as a group of common theories and practices acknowledging student differences in background knowledge, readiness, language, and learning style, and interests, inciting teachers to respond to individual student needs (tomlinson & kalbfleisch, 1998). teachers anticipated use of real-life examples aligns with another aspect of crp in that using examples from the students’ social context helps them connect to the academic concept. the concept funds of knowledge (cf., gonzález, moll, & amanti, 2005) highlights this practice; learning modules building on students’ local knowledge validate students’ experiences and backgrounds. teachers’ use of realworld examples positively impacts students’ interest in the lessons presented and their ability to comprehend and retain the information (civil, 2002). therefore, for the limited time in which teachers participated in the conference workshop and the practices they were able to learn over the course of 2 days, these improvements are relatively noteworthy and beneficial to students. quantitative findings show that the conference workshop fostered greater awareness of crp in mathematics and prompted such actions as those discussed, but teachers also needed additional concrete strategies and lesson time for implementation. as a result of the positive, but limited, quantitative results, we conducted indepth interviews with a convenience sampling of conference participants to better understand how crp professional development through a workshop setting informed mathematics practices. through these interviews, we found that teachers’ experiences supported the quantitative findings indicating improvements in under timmons-brown & warner conference workshop and crp journal of urban mathematics education vol. 9, no. 1 39 standing of crp, growth in professional relationships around crp, and new classroom applications. here, we note that the qualitative interview sample was selected from a group of educators already interested in improving their skills in reference to crp. furthermore, those educators that made time for the additional follow-up interview were more likely to have been pleased with their experience at the conference workshop. some survey respondents noted that one particular session was not useful for them, but we could not identify any of those attendees for interviews, as survey responses were anonymous. therefore, the educators participating in the interviews are more likely to look back favorably on their experience at the conference. in the second key benefit of using a conference workshop setting approach to increasing mathematics teachers’ confidence in crp, our analysis indicates that the workshop setting fostered mathematics teachers’ networks and enhanced their ability to communicate with others about their teaching efforts. several mathematics teachers interviewed lacked the language to articulate the emphasis they already placed on building relationships with students and sharing ideas. attending the conference workshop allowed these teachers an opportunity to further their interests, acquire new classroom practices, and generate a language to discuss their successes with other teachers at their home school. teachers who attended shared findings and best practices with other teachers at their school. teachers noted several practices during interviews that improved the exchange of mathematics knowledge in the classroom and facilitated the development of new classroom relationships, among both teachers and students. teachers focused on developing collaborative relationships among students and the use of realworld examples in mathematics lessons. teachers of elementary-aged students in particular, found that using active learning strategies during mathematics topics encourage student engagement and created a “busy” classroom. overall, the study participants are more likely to be educators associated with schools with fewer resources, larger proportion of students of color, and more socioeconomic inequality. the potential lack of resources at the school may make implementation of new practices more difficult, although several participants also noted that title i funding often helped them to access key resources for the classroom, such as manipulatives. while best practices in professional development indicate that efforts lasting over one year that take place in the classroom or school setting are key to improving teacher knowledge, such interventions are not always economically feasible. given the limited finances available to improve culturally relevant teaching practices in mathematics, the conference workshop achieved a number of positive results among participants. finally, results show that after attending the 2-day conference workshop, teachers returned with renewed energy for fostering classroom relationships. these efforts occurred in a number of ways, from helping students to communicate with timmons-brown & warner conference workshop and crp journal of urban mathematics education vol. 9, no. 1 40 each other in small group settings, changing the physical structure of the classroom, and encouraging other teachers to develop a greater rapport with students. given the emphasis on developing deep relationships with students in order to pursue a culturally relevant approach in the classroom (herzig, 2005; ladson-billings, 1997; leonard et al., 2010), teachers’ renewed interest in classroom relationships following the conference workshop is a positive result. research in mathematics diversity and success urges that a sense of belonging fosters student achievement (herzig, 2005), and as teachers in this study focused on making the classroom content more relevant for students they also succeeded in improving relationships and cultural awareness in the classroom. however, findings also indicate that developing relationships with students can be a time-consuming and complicated process, not always done successfully. unfortunately, teachers also indicated that in an effort to understand students’ lives, they might also oversimplify students’ experiences and generalize knowledge of one student’s home life to apply to another. in crp, teachers must not use their perceptions of student background to substitute for setting high expectations for academic success or to justify current institutional policies that may not benefit all students in the classroom. the conference workshop setting provided teachers with important and helpful information to move forward with changes to classroom practices and integrating all students more clearly into how key skills are developed. as noted by sleeter (2012), teachers are rarely able to embrace all tenets of a culturally relevant paradigm in a short period of time, such as 1 year. success with crp is one that comes slowly, over a period of time, as teachers develop relationships with students, become a part of the school community, and develop their knowledge of what social justice issues can inform their teaching practice (ladson-billings, 2006; milner, 2011). the teachers in this study displayed an active interest in these tenets and the perceived ability to expand their knowledge of crp. based on the affective responses from interviewed teachers and the changes they 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(2010). challenges to conceptualizing and actualizing culturally relevant pedagogy: how viable is the theory in classroom practice. journal of teacher education, 61(3), 248–260. timmons-brown & warner conference workshop and crp journal of urban mathematics education vol. 9, no. 1 44 appendix a sample questions from preand post-conference surveys pre-conference survey questions 1. please indicate your race/ethnicity. you may check more than one. € white € american indian or native alaskan € native hawaiian or other pacific islander € black or african american € asian € hispanic or latino 2. please indicate your gender. € male € female 3. please check the appropriate boxes to indicate what grade(s) you teach. € elementary € middle € high school € combination 4. what subject(s) do you teach? ____________________________ 5. to what extent are you familiar with the following terms and concepts: large extent moderate extent small extent no extent a. i am familiar with the term “culturally relevant pedagogy” ○ ○ ○ ○ b. i am familiar with the term “culturally relevant teaching” ○ ○ ○ ○ c. i am familiar with the term “culturally responsive teaching” ○ ○ ○ ○ d. i am familiar with the term “culturally sensitive teaching” ○ ○ ○ ○ timmons-brown & warner conference workshop and crp journal of urban mathematics education vol. 9, no. 1 45 6. during the current school year (2011-12), how often did you do the following? 7. the actions above are elements of a culturally relevant teaching (crt). collectively, how would you describe the extent to which you have attempted to apply these concepts in your classroom during the current school year? always often occasionally rarely never a. i have attempted to use crt ideas in my classroom in all subjects ○ ○ ○ ○ ○ b. i have attempted crt for most mathematics lessons ○ ○ ○ ○ ○ c. i have attempted to develop a multicultural curriculum in my classroom ○ ○ ○ ○ ○ daily weekly monthly quarterly/ annually never a. plan lessons geared toward a variety of student abilities and social needs. ○ ○ ○ ○ ○ b. visit student families outside of schools ○ ○ ○ ○ ○ c. make sure all students understood the content before moving on with the lesson plan ○ ○ ○ ○ ○ d. allow students to share cultural artifacts from home culture ○ ○ ○ ○ ○ e. use real world examples ○ ○ ○ ○ ○ f. use systematic strategies to ensure attention is equitably distributed to all students ○ ○ ○ ○ ○ g. engage with students about their problems or experiences not related to school ○ ○ ○ ○ ○ h. explain concepts in different ways to ensure all students understood the material ○ ○ ○ ○ ○ i. reflect upon your own cultural heritage and possible biases ○ ○ ○ ○ ○ timmons-brown & warner conference workshop and crp journal of urban mathematics education vol. 9, no. 1 46 post-conference survey 1. to what extent are you familiar with the following terms and concepts: large extent moderate extent small extent no extent a. i am familiar with the term “culturally relevant pedagogy” ○ ○ ○ ○ b. i am familiar with the term “culturally relevant teaching” ○ ○ ○ ○ c. i am familiar with the term “culturally responsive teaching” ○ ○ ○ ○ d. i am familiar with the term “culturally sensitive teaching” ○ ○ ○ ○ 2. would you be interested in another conference on a related topic hosted by (the same university)? if so, would you suggest any topics? _____________ 3. given the information gained from the conference, how often do you plan to use the following activities in the remainder of the school year (2011– 12)? daily weekly monthly quarterly/ annually never a. plan lessons geared toward a variety of student abilities and social needs. ○ ○ ○ ○ ○ b. visit student families outside of schools ○ ○ ○ ○ ○ c. make sure all students understood the content before moving on with the lesson plan ○ ○ ○ ○ ○ d. allow students to share cultural artifacts from home culture ○ ○ ○ ○ ○ e. use real world examples ○ ○ ○ ○ ○ f. use systematic strategies to ensure attention is equitably distributed to all students ○ ○ ○ ○ ○ g. engage with students about their problems or experiences not related to school ○ ○ ○ ○ ○ h. explain concepts in different ways to ensure all students understood the material ○ ○ ○ ○ ○ i. reflect upon your own cultural heritage and possible biases ○ ○ ○ ○ ○ timmons-brown & warner conference workshop and crp journal of urban mathematics education vol. 9, no. 1 47 7. the actions in item #6 are elements of a culturally relevant teaching (crt). collectively, how would you describe the extent to which you plan to apply these concepts in your classroom for the remainder of this school year (2011–12)? always often occasionally rarely never a. i plan to use crt ideas in my classroom in all subjects ○ ○ ○ ○ ○ b. i plan to use crt for most mathematics lessons ○ ○ ○ ○ ○ c. i have plans to use crt for all my mathematics lessons. ○ ○ ○ ○ ○ 8. to what extent has your knowledge of teaching changed by attending this conference? large extent moderate extent small extent no extent a. my knowledge of teaching from a multicultural perspective has been enhanced ○ ○ ○ ○ b. my knowledge of teaching by incorporating students’ cultural has been extended ○ ○ ○ ○ journal of urban mathematics education december 2013, vol. 6, no. 2, pp. 7–19 ©jume. http://education.gsu.edu/jume rochelle gutiérrez is professor of mathematics education in the department of curriculum and instruction and latina/latino studies at the university of illinois at urbana-champaign, 1310 south sixth street, champaign, il 61820; email: rg1@illinois.edu. her research interests include: equity in mathematics education, race/class/language issues in teaching and learning mathematics, the achievement gap, and teachers’ knowledge bases and dispositions to teach marginalized students. commentary why (urban) mathematics teachers need political knowledge rochelle gutiérrez university of illinois at urbana-champaign ublic school teachers everywhere are under attack. but “urban” mathematics teachers in public schools are in a particularly tough position of having to advocate for their students and themselves at a time when school reforms (e.g., common core state standards, high-stakes tests, new teacher evaluations systems, and changes in collective bargaining agreements) are stripping them of their ability to exercise professional judgment. the low status that urban teachers experience is inextricably linked to the low status of the historically underserved and/or marginalized youth (defined here as students who are black, 1 latin@, 2 american indian, and low income) that they serve. among other things, urban mathematics teachers must: (a) negotiate their practice with colleagues, students, parents, administrators, colleges, and members of for-profit organizations who may not agree with their definitions of “mathematics,” “education,” or “learning”; (b) work with fewer material and human resources than teachers in more wealthy school districts; (c) support their students to compete on an unfair playing field that constantly changes; and (d) buffer themselves from images of students as unmotivated, not having the proper amount of “grit,” lacking role models in their community, and having cultural and linguistic obstacles to overcome, as well as 1 i use the term black, as opposed to african american, to highlight the fact that many black students living in the united states have ancestry in the caribbean, south america, and asia, among other places. nonetheless, black students who attend schools and live in the united states are racialized in similar ways, regardless of country of origin. 2 i use the @ sign to indicate both an “a” and “o” ending (latina and latino). the presence of both an “a” and “o” ending decenters the patriarchal nature of the spanish language where is it customary for groups of males (latinos) and females (latinas) to be written in the form that denotes only males (latinos). the term is written latin@ with the “a” and “o” intertwined, as opposed to latina/latino, to show a sign of solidarity with individuals who identify as lesbian, gay, bisexual, transgender, questioning, and queer (lgbtq). p http://education.gsu.edu/jume mailto:rg1@illinois.edu gutiérrez commentary journal of urban mathematics education vol. 6, no. 2 8 images of urban teachers as slackers, saviors, or people who simply could not obtain work elsewhere. i have spent 15 years researching effective, urban high school mathematics departments that served black, latin@ and low-income adolescents (see, e.g., gutiérrez, 1996, 1999a, 1999b, 2000, 2002). these were schools where students took more mathematics than was required by their district; where english learners, recent immigrants, and students who juggled childcare took college preparatory mathematics courses; where historically underserved and/or marginalized students scored better than their peers on standardized tests of mathematical achievement; where a large percentage of seniors took calculus; and where high achieving students reflected the demographics of those who attended the school. as might be expected, their teachers presented engaging lessons where students: worked in groups, used rigorous texts and appropriate technology, worked in spanish and english, and had opportunities to do projects or problems that reflected their lives. but perhaps more important, their teachers also: met regularly (inside and outside of school) to discuss students, teaching, and mathematics; worked hard to recruit like-minded staff and to socialize new members into a strength-based perspective on students; strategized collectively to eliminate lowlevel courses; interpreted creatively (or simply bent) the rules to fit the long-term needs of their students; convinced colleagues of students’ capabilities; refused to enact discipline policies that kept students out of the classroom; and twisted students’ arms to take advanced mathematics courses. in essence, these teachers negotiated the politics of school reform, language, racism, and testing (gutiérrez, 1999a; gutiérrez, in preparation a; gutiérrez & morales, 2002). but more than just documenting these successful mathematics departments, i was eager to help build more places like the ones i had found. i was particularly interested in the kinds of experiences and knowledge bases teachers need in order to become advocates for all students to have a deep understanding of mathematics and to develop robust mathematical identities (mcgee & martin, 2011; stinson, 2008). yet, when i looked at the knowledge bases typically emphasized in educational literature and schools of education (see, e.g., ball, thames, & phelps, 2008; darling-hammond, bransford, lepage, hammerness, & duffy, 2005; hill, sleep, lewis, & ball, 2007), i did not see anything that acknowledged what i characterize as the political nature of teaching that the teachers i studied seemed to understand and possess. instead, there was and is an emphasis on many of the things that have taken center stage in schools of education for decades: content knowledge, pedagogical knowledge, and pedagogical content knowledge (shulman, 1987). recently, several researchers have made significant gains in extending these knowledge bases to better address the needs of black, latin@, american indian, and low-income youth, especially drawing on concepts such as: funds of knowledge (civil, 2002; gutiérrez commentary journal of urban mathematics education vol. 6, no. 2 9 foote, 2009; gonzalez, andrade, civil, & moll, 2001), culturally relevant/responsive teaching (aguirre & zavala, 2013; leonard, brooks, barnesjohnson, & berry, 2010; turner, drake, mcduffie, aguirre, bartell, & foote, 2012), ethnomathematics (d’ambrosio, 1985; knijnik, 2007; powell & frankenstein, 1997), and social justice mathematics teaching (turner & strawhun, 2005; frankenstein, 2005; gutstein, 2003). nevertheless, the political component of teaching is still largely missing from discussions of important knowledge that teachers need. in this commentary, i highlight some of the ways that both mathematics and mathematics teaching are political. i then argue that educators in general, and mathematics educators in particular, must expand what we consider to be necessary knowledge for teaching, adding political knowledge for teaching. finally, i share what i have learned from supporting mathematics teachers to develop political knowledge and to advocate for historically underserved and/or marginalized youth. mathematics and mathematics teaching is political one might laugh at the calvin and hobbes 3 comic strip in which calvin suggests that mathematics is a religion. but, similar to religion, there are many ways in which mathematics requires believing a particular paradigm and a way of doing things in the world. mathematics carries with it a set of values that are transmitted every time we engage in it (burton, 1994). and, while it may be easy to see how mathematics education is political (e.g., as a result of teacher beliefs, tracking, stereotypes, racism), i am arguing that mathematics, itself, is political. let us consider how. mathematics operates with a kind of formatting power on our lives. viewed as objective, unrelated to emotions or morals, mathematics is often seen as an arbiter of “truth” (christensen, skovsmose, & yasukawa, 2008; volmink, 1994). in fact, many college students choose to major in the discipline because they see it as black and white, involving one right answer, and giving them a sense of satisfaction at efficiently arriving at that answer. the terms “elegant” or “beautiful” mathematics can convey this phenomenon of presenting the simplest path to the correct solution. yet ask a mathematician whether mathematics is black and white and you will likely get an argument that highlights the uncertainty in mathematics (borba & skovsmose, 1997). in fact, as calvin in the comic strip suggests, many forms of mathematics require a leap of faith. for example, the mathematics community has verified neither kepler’s sphere-packing conjecture nor the classifica 3 calvin and hobbes by bill watterson, march 9, 2011. http://www.gocomics.com/calvinandhobbes/2011/03/09#.urcqvqvatwj gutiérrez commentary journal of urban mathematics education vol. 6, no. 2 10 tion of simple finite groups—yet mathematics continues its expansion. moreover, highly theoretical branches of mathematics rely on probability rather than certainty. yet we continue to view mathematics as truth, not social phenomena. when we operate with mathematics as an objective arbiter of truth, we maintain its status as pure, separate from and even superior to other fields. however, similar to whiteness, mathematics holds unearned privilege in society. 4 that is, we could have a society where being artistic or relating well to other people would be seen as signs of “intelligence.” instead, in much of the west and the colonized world, mathematical proficiency is used as a proxy for intelligence. yet there is nothing inherent in mathematics that qualifies it to deem one as intelligent. that is the myth we have constructed: some people are good at mathematics and some are not; therefore, some people possess intelligence and some do not. in general, we fail to question the unearned privilege that mathematics holds in society, in part, because we are convinced that it is merely a reflection of our natural world. ontologically, we see mathematical concepts as separate from humans. we point to fibonacci sequences in flowers, seeds, shells, animals, and music to confirm that mathematics reflects enduring truths, the way things were meant to be. presented as a mere reflection of the order in our universe, mathematics becomes a means to control (b. rotman, as cited in walkerdine, 2004). 5 one way that mathematics operates as a proxy of intelligence is through reasoning (walkerdine, 2004). based on a western conception of rationality, individuals are seen to progress through levels of reasoning until they reach the highest level of intellectual thought: abstracted logic. abstraction requires an absence of intimacy and humanity. in schools, the value placed on abstraction translates into an overreliance on algebra and calculus rather than other forms of mathematics and a privileging of the symbolic form, coming up with the correct equation or the most general rule, rather than understanding the meaning of variables or the context in which a mathematics problem occurs. when abstract thinking represents the highest form of intellect, those who deviate from that form are seen as primitive. hence, mathematics—by way of simultaneously being highly valued in society (conveying intelligence) and involving abstraction (requiring an individual to separate from one’s body and emotions)—can be viewed as a form of microaggression (solórzano, 1998; see also gutiérrez, in preparation a). i am not suggesting that we completely change the mathematics content we are teaching. 4 elsewhere, martin (2013) has argued that mathematics education (emphasis added) operates as white institutional space. i am arguing that mathematics itself operates as whiteness. 5 when mathematics does not coincide with the physical world, mathematicians claim it operates in a world unto itself. yet mathematics is always a form of recasting (panza, 2013) that presents slippage between object and form (landry, 2013). gutiérrez commentary journal of urban mathematics education vol. 6, no. 2 11 but as long as mathematics continues to convey status in society, we need to be more cognizant of what we are asking students to do in order to participate in mathematics classrooms. in fact, many teachers (and researchers) are unknowingly complicit in the reproduction of mathematics as a form of microaggression. this residue of participating in school mathematics continues into individuals’ adult lives (see, e.g., martin, 2006). by virtue of mathematics being political, all mathematics teaching is political. all mathematics teachers are identity workers, regardless of whether they consider themselves as such or not. they contribute to the identities students construct as well as constantly reproduce what mathematics is and how people might relate to it (or not). hence, any form of teaching that breaks with tradition can be seen as subversive. subversive mathematics teaching, among other things, creates a counter-narrative to the achievement gap discourse; questions the forms of mathematics presented in school; highlights the humanity and uncertainty of mathematics; positions students as authors of mathematics; challenges deficit narratives of students of color in need of mathematics; and recognizes that not all students aspire to (or should) become research mathematicians or scientists (gutiérrez, 2013c, in preparation b). knowledge(s) needed for teaching although i use the term “knowledge” in the title of this commentary, i prefer the term conocimiento (anzaldúa & keating, 2002). knowledge is the literal translation of the spanish word conocimiento, yet knowledge (in english) does not allow us to distinguish between things we know objectively versus subjectively. 6 for me, political conocimiento assumes clarity and a stance on teaching that maintains solidarity with and commitment to one’s students. among other things, political conocimiento involves: understanding how oppression in schooling operates not only at the individual level but also the systemic level; deconstructing the deficit discourses about historically underserved and/or marginalized students; negotiating the world of high-stakes testing and standardization; connecting with and explaining one’s discipline to community members and district officials; and buffering oneself, reinventing, or subverting the system in order to be an advocate for one’s students. 6 the distinction between conocimiento and knowledge is similar to the difference between the terms chicana and mexican. the choice to use chicana to define myself instead of mexican (or mexicana) is not just an indication that i have indigenous ancestry; it is a political statement about the indigenous people (now referred to as mexicans) who were on this land before the u.s.mexico border was created. gutiérrez commentary journal of urban mathematics education vol. 6, no. 2 12 for-profit corporations like pearson are moving forward with teaching materials, assessment items (e.g., partnership for assessment of readiness for college and career [parcc]), and evaluation procedures for credentialing teachers (e.g., teacher performance assessment [edtpa]), all with the argument that students across the nation need to be “college and career ready.” such goals sound lofty until one examines them more deeply. being ready for college does not necessarily mean one’s education has been meaningful, one’s cultural roots have been strengthened, or that one is ready to participate in a democratic society. it merely means one has learned to play the game of school and can decipher the materials necessary to gain acceptance into college. the meaning behind “career ready” is even murkier. presumably, industry officials were not contacted to develop test items that would indicate an individual is ready to enter a career. many organizations rely on conley’s (2010) definition of career readiness that suggests students will be prepared to enroll and succeed, without remediation, in a post-secondary certificate program that provides entry into a career pathway. what many miss by focusing almost exclusively on mathematics and english language arts content, however, is four key skills that conley also promotes: listening, speaking, research, and technology proficiency. these skills get at some of the analytic and interpersonal capacities that we need to participate in a democratic society. yet, in today’s political economy, corporations continue to narrowly define what counts as learning and high-quality teaching and, therefore, what counts as being an educational professional. therefore, mathematics teachers need to be able to do more than just construct good lesson plans that are inquiry-based or be prepared to develop meaningful relationships with their students. they must be able to deconstruct narratives being written about education in general (e.g., in movies like waiting for superman, reports about falling u.s. rankings on international tests of achievement, studies and initiatives funded by the bill and melinda gates foundation) and about black, latin@, american indian, and low-income youth in particular (e.g., media reports about the achievement gap, race to the top policies, the push for charter schools). elsewhere, i have argued: making dominant discourses more apparent is a necessary step toward both recognizing how those discourses dis/advantage individuals and in challenging those discourses and their associated practices so as to put new ones into place …deconstruction is a useful process, as it highlights the ways in which current realities are not necessarily the only, or the most natural, of those that could be constructed (e.g., we could have a very different evaluation system in place for students, teachers, and researchers). (gutiérrez, 2013a, p. 14) until teachers are given the proper time and support to reflect on broader social realities involved in schooling, mathematics, identity, and power, they are unlike gutiérrez commentary journal of urban mathematics education vol. 6, no. 2 13 ly to challenge the powerful messages and policies being enacted by those outside of education. one example is the “achievement gap” discourse. much of the discourse around closing the achievement gap implies the root of the problem is technical—how do we design curriculum and instruction so as to better motivate and engage students who are black, latin@, low income, or english learners (gutiérrez, 2008)? yet building a site that engages and advances such learners in meaningful ways is not the most difficult part; sustaining it is. studies of successful mathematics teachers and mathematics departments indicate that they experience backlash when they succeed with historically underserved and/or marginalized students. for example, several of the successful mathematics departments i studied did not receive praise or support from their principals. in fact, i often found myself in the position of touting their success or advocating for faculty. the problem is not technical; it is moral. do we have the desire and will to support (and extend to other schools) the pockets of success that already exist throughout the nation? a brief look at history tells us that we do not. union high, for example, was a chicago school with 87% of students qualifying for free or reduced-price meals, but where 45% of the senior class was taking calculus; however, its gains in students mathematical understanding and percentages of seniors taking calculus were derailed by district politics and a back-tobasics movement (gutiérrez, 1999a; 2000; 2002; strutchens, quander & gutiérrez, 2011; h. morales, personal communication, 2011). even though teachers were successful at making mathematics meaningful to students, getting students to take large numbers of mathematics courses (especially advanced courses), and to develop robust mathematical identities, administrators were intent on getting teachers to focus on test scores to the exclusion of all other efforts. some political knowledge extended teachers’ efforts for a few years. for example, when they were told to stop using interactive mathematics program (imp) 7 textbooks in favor of district assigned texts, they offered to become a “control group” for the district so they could be compared to other schools instituting the new textbooks. when they were eventually forced to give up the imp textbooks, they met outside of school to discuss how they could maintain the reform-oriented curriculum and the focus on group learning that had engaged their students. using imp-inspired principles and worksheets they had copied, they described their teaching as “imp in the closet.” eventually, fed up with the politics and stripped of their professional judgment, several teachers left the school. railside high, a northern california school so successful it has been studied by several teams of researchers (boaler, 2006; 2008; boaler & staples, 2008; hand, 2004; 2012; horn, 2005; jilk, 2010, in press) and used as a model for pro 7 for information about the interactive mathematics program (imp), see http://mathimp.org. http://mathimp.org/ gutiérrez commentary journal of urban mathematics education vol. 6, no. 2 14 fessional development (jilk, 2012; jilk & o’connell, in press), experienced an analogous phenomenon. similar to union high, the school served primarily latin@ students and students who qualified for free or reduced-price meals. after several years of success using integrated mathematics curriculum, group work, and a focus on assigning competence to students, district politics (e.g., mandated direct instruction) derailed teachers’ efforts and many effective teachers left the school. not only did the school fail to sustain its efforts but also jo boaler, who first published the success of the mathematics department, was publicly attacked for several years by mathematics professors who vehemently opposed reform mathematics (jaschik, 2012). moreover, teachers at the school were harassed in efforts to disclose the school sites and discredit the findings (l. m. jilk, personal communication, january 2012). clearly, getting more historically underserved and/or marginalized students to engage and perform well in mathematics is not a technical problem with a technical solution. preparing teachers to act on their political knowledge given the current context of high-stakes education, i seek to help teachers build the knowledge and stances required to creatively resist a definition of the profession that unnecessarily limits the relationship between mathematics and historically underserved and/or marginalized youth. elsewhere, i have described the process of creative insubordination whereby teachers find loopholes in policies or interpret rules and/or procedures in ways that allow them to advocate for historically underserved and/or marginalized students (gutiérrez, 2013a, 2013b, 2013c; gutiérrez & gregson, 2013). my research team 8 and i have worked to develop an equity-based teacher education program that aims to influence the knowledge bases, skills, and dispositions of pre-service secondary mathematics teachers (psmts) who prepare for and eventually teach historically underserved and/or marginalized students (gutiérrez, irving, & gerardo, 2013). the pre-service teachers are mathematics majors who receive a minor in education. during their 2 years as pre-service teachers, the psmts attend the regular teacher education program, which includes field observations, lesson planning, and portfolio development, among other things, for state-level credentials in grades 6–12 mathematics. they also participate in a 3-hour, biweekly seminar that focuses on issues of rigorous and creative mathematics, social justice teaching, as well as strategies for supporting historically underserved and/or marginalized youth and for negotiating teaching in an era of high stakes testing; conferences and movie viewings that provide deeper understandings of students in a diverse society; and meet every 8 members of the research team include the following graduate students: sonya e. irving, juan m. gerardo, and gabriela vargas. gutiérrez commentary journal of urban mathematics education vol. 6, no. 2 15 other week with our partner teacher who models creative insubordination in practice. the psmts also are required to develop activities for and volunteer in a weekly afterschool mathematics club that supports black and latin@ middle school students. finally, each psmt attends bi-weekly, hour-long mentoring sessions with me and/or a doctoral student. with respect to political conocimiento, pre-service teachers in our program learn to do more than just deconstruct racist comments and policies in their schools. through cycles of rehearsals and roleplays, they learn how to speak back to such policies and comments. the teachers with whom i have worked and who have developed political knowledge exhibit political clarity (bartolomé, 1994). similar to the black teachers in beauboef-lafontant’s (1999) study who knew how to advocate for their black students in segregated schools, the psmts know what they stand for and aim to have their everyday teaching practices mirror that stance. they recognize that all teaching is political and that their definitions of mathematics and learning affect their pedagogy, as well as who learns and what is learned in their classrooms. they can articulate to other teachers why having political clarity is important and can convince people around them that they also should take such a stance on teaching (gutiérrez, 2013b, 2013c). more than just profess the importance of political conocimiento, teachers in our project are able to take action. some of the things they have exhibited include:  instituting regular learning logs to help students get in touch with what they are learning and what they still need to learn, rather than just relying on test scores to indicate their achievement.  renaming a course to reflect the fact that it only covers western, euclidian geometry, not all geometries that are practiced in the world.  standing up to an administrator in a public meeting when he implied that black student culture was the cause of the achievement gap at their school.  challenging a school dress-code policy that disproportionately penalized black students for wearing “sagging pants.”  refusing to go along with procedures at a workshop that asked teachers to publicly advocate for the common core state standards in mathematics.  convincing a co-teacher that the mathematics being taught needed to reflect a more rigorous curriculum so that students understood why particular procedures worked.  helping lead a professional development workshop so that local teachers could reflect on how their definitions of mathematics influenced who did well in their mathematics classes. gutiérrez commentary journal of urban mathematics education vol. 6, no. 2 16 these acts of teaching are not easy. nonetheless, our psmts suggested that some of the things which supported them to take risks in their teaching and in their interactions with colleagues include having opportunities to reflect on the nature of mathematics (i.e., what is mathematics?), deconstruct prevailing discourses in education within a community of teachers who seek to reclaim the profession and humanize the mathematics classroom, and interact with more experienced teachers who model political conocimiento. i have argued that in addition to mathematics teaching, mathematics itself is political. as a result, i have suggested that all mathematics teachers are identity workers that need to develop political conocimiento, involving the interdependent relationships necessary to deconstruct the images of mathematics, public education, teaching, and learning that circulate in mainstream society. given how easily school politics have derailed highly successful mathematics departments, it is important that we prepare urban mathematics teachers for the kinds of scenarios they will face and give them opportunities to practice taking a stand in high-needs schools. expanding the knowledge base we consider necessary for teaching to include political knowledge is one step in the right direction. with greater awareness of the unearned privilege that mathematics holds in society, teachers are better prepared to rethink their role in how mathematics is carried out in school and society. if we intend to transform learning and our relationship to mathematics and each other on this planet, political knowledge needs to be taken as seriously as notions of content knowledge, pedagogical knowledge, and pedagogical content knowledge. as the nation pushes to have more students master school mathematics and enter stem (science, technology, engineering, and mathematics) related fields, we must consider how this emphasis will influence the kinds of citizens and, in the end, the kinds of humans, we create. references aguirre, j., & zavala, m. 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(1994). reasoning in a post-modern age. in proceedings of the 5 th international conference on thinking: international interdisciplinary perspectives, (pp. 116–126). melbourne, australia: hawker brownlow. journal of urban mathematics education december 2015, vol. 8, no. 2, pp. 27–43 ©jume. http://education.gsu.edu/jume john bragelman is a doctoral student in the department of curriculum and instruction at the university of illinois at chicago and a mathematics professor at harold washington college; email: jbrage2@uic.edu. his research interests include identity and self-efficacy in developmental mathematics education, restorative practices, teaching for social justice, and the morality of mathematics education. public stories of mathematics educators praxis as dialogue: teacher and administrator john bragelman university of illinois at chicago n the spring of 2014, i accepted an administrative position at a community college in chicago. i struggled with the decision because it meant leaving the mathematics classroom, a space i call home. as a critical educator, i had the opportunity to watch my students become critical participators in their communities, readers and writers of their world (gutstein, 2006). i chose to transition to administration because i hoped for a similar narrative on a broader scale, impacting thousands instead of hundreds, but i worried that my values would be negatively influenced. i worried that by accepting this “power” that i was somehow placing myself in a position in which i would unintentionally reify systemic oppression rather than doing my part to dismantle it. that spring, i encountered bell hooks’ (1994) book teaching to transgress: education as the practice of freedom, and it spoke to me deeply. in teaching to transgress there is a section where hooks presents a dialogue between herself and her writing voice. this writing style inspired the dialogue presented in this public story between me, the mathematics educator (i.e., “john the teacher”), and a recently adopted identity, me, the administrator (i.e., “john the administrator”), to make my fears salient and to confront the effects my new experiences were having on my values. i worked in administration for a year before returning to the dialogue, reflecting on the effects of the position and its power. there is no conclusion, as my identities are still under negotiation, the dialogue ever continuing. john the administrator (ja): before we get into why i’ve asked for this dialogue, could you tell us a little about yourself? john the teacher (jt): i’ll be happy to. you see, education wasn’t my first career. i entered the corporate world at a young age of 20, bouncing between school and corporate positions until i realized in 2005 that i wouldn’t ever be happy in that context. in 2006, i began a master’s of education degree program at georgia state university. it was there that i was first exposed to i http://education.gsu.edu/jume mailto:jbrage2@uic.edu bragelman public stories journal of urban mathematics education vol. 8, no. 2 28 freire (e.g, 1970/1993b), ladson-billings (e.g., 1994, 1995), ladsonbillings and tate (e.g., 1995), woodson (e.g., 1933/2006), tate (e.g., 1994), and others. it was there that i found my calling as an educator. during the program, i tutored at a high school on the south side of atlanta, and after i completed the program, i taught in and around that area for 3 years. in 2010, i transitioned to teaching in a college while i worked on a doctorate. and you? ja: as you know, i’m new in the sense that this is my first administrative role; it’s the first time in 7 years that i’m not in the classroom. i recently accepted a position as director of developmental education at a community college in chicago. i find myself overwhelmed, lost, and without guidance. ninety percent of our students place in developmental courses, and almost all come to our college having endured systemic injustices (lipman, 2011). as mathematics is the gatekeeper for college (martin, 2000), i’m the person who works the gatehouse. frankly, i need help. i’ve always considered you a critical educator. you taught for social justice through culturally relevant lessons (e.g., gutstein, 2006, 2008; gutstein & peterson, 2006; ladsonbillings, 1995), and you were always an advocate for students. it was because of you that i took this position. can you help me? jt: yes, but understand i can only reflect on our past experiences. freire (1970/1993b) would say it’s up to you to continue the dialectical cycle of action and reflection, of praxis. ja: thank you! i want to bring the same critical emphasis to this new position that you had as an educator. who or what is a critical administrator? does such a thing exist? administrating for social justice? jt: i would say that a critical administrator takes a broader perspective of a critical educator. a critical educator focuses on the dialectics of teacher-andlearner, theory-and-practice, taking a problem-posing approach to education mediated through the students’ lived realities (freire, 1970/1993b). in pedagogy of hope (1994/2004), freire argues: the problem facing the leaders is: they must learn, through the critical reading of reality that must always be made, what actions can be tactically implemented, and on what levels they can be so implemented. in other words, what can we do now in order to be able to do tomorrow what we are unable to do today? (p. 188) bragelman public stories journal of urban mathematics education vol. 8, no. 2 29 notice that freire speaks of the same praxis for administrators as he does for educators all the while broadening the perspective to include tactical choices within a long-term strategy. in education and democracy (del pilar o’cadiz, wong, & torres, 1998), the purpose of his administration is made explicit: the orientation of freire’s administration, as we have tried to argue and exemplify, was towards passionately and slowly building a social movement responsive to the educational needs of popular communities rather than coldly and efficiently developing coordinated curriculum packages to be identically replicated in the city’s 691 schools. (p. 247) the intentionality behind these perspectives is unpacked in pedagogy in process (freire, 1978) and literacy: reading the word and the world (freire & macedo, 1987). ja: in letters to mario cabral about advising guinea-bissau in adult literacy education, freire (1978) mentions traveling to guinea-bissau to experience the lived realities of the popular classes. is this what he means by “being responsive to the educational needs of popular communities?” jt: i’m not freire, and i can’t directly speak for him. with that said, yes, i think so. and similarly in literacy (1987), when he presents the work he did with the republic of são tomé and the príncipe, in the shared construction of the exercise workbook and the second popular culture notebook. in both cases, freire and his administration work with. ja: and by work with, what do you mean exactly? jt: do you remember some of my failed lessons when i was teaching high school? during those first few months, i brought lessons in, ones carefully crafted during the master’s program, ones that integrated technology and mathematics, and ones that aligned perfectly with what i thought teaching should look like, because that’s how it looked like when i was in high school. ja: and ones that left the students disengaged? jt: exactly! none of those lessons derived from the students, from their popular knowledge and lived experiences. even when i tried using lessons created by social justice educators (see, e.g., gutstein & peterson, 2006), i found lit bragelman public stories journal of urban mathematics education vol. 8, no. 2 30 tle success. the lessons were about my reality or, worse yet, on the assumptions i was making about the students’ realities. i was centering my educational practice “exclusively on … the educator” rather than understanding “educational practice in terms of the relationship obtained among its various components” (freire, 1994/2004, p. 93). when i remembered this, when i remembered what culturally relevant pedagogy really meant (ladsonbillings, 1995), i threw my lessons away and sat down with the students. i let them inform what we would learn and how we would learn it. ja: so, you mean when you constructed the basketball statistics lessons because a few of the young ladies in your classes played on the varsity team, you were building lessons with them? jt: yes. their realities, their acts of sharing informed my pedagogy. and through this sharing we moved toward critically examining those realities through problem-posing pedagogy (freire, 1970/1993b); we discussed sports as a means of class mobility. freire (1998) echoes how i felt perfectly in pedagogy of freedom: reflecting on the duty i have as a teacher to respect the dignity, autonomy, and identity of the student, all of which are in process of becoming, i ought to think also about how i can develop an educational practice in which that respect, which i know i owe to the student, can come to fruition instead of being simply neglected and denied. (pp. 62–63) so, how would this perspective play out as a critical administrator? ja: a critical administrator would privilege the realities and voices of her educators, students, and staff in forming and administrating policy, using respect for the faculty and student as a lens. for example, in education and democracy (del pilar o’cadiz et al., 1998), “the overwhelming majority of teachers we spoke with asserted that the one outstanding feature of the pt administration was that for the first time in their professional lives they were afforded the opportunity to have a voice” (p. 245). administrating with is an activity that exhibits respect. jt: exactly. always “with the people, teaching and learning mutually” (freire, 1978, p. 9). the dangers of your position are your academic knowledge and the privileges it brings. “the relations among world-consciousness-practicetheory-reading-of-the-world-reading-of-the-word-context-text, the reading of the world cannot be the reading made by academicians and imposed on bragelman public stories journal of urban mathematics education vol. 8, no. 2 31 the popular classes” (freire, 1994/2004, p. 90). you must remain constantly vigilant against that privilege (freire, 1970/1993b). ja: but how? jt: by remembering what a critical administrator is not. at my second high school, the principal spent the majority of her time in the office; she spent little to no time in dialogue with the students and faculty, choosing to spend the majority of time in dialogue with other administrators and the school board. this choice increased the distance between the administration and the students rather than decreasing it. my first principal was in the hallways constantly, always in dialogue with the students. her evaluations of us as educators came first from the students. ja: and yet she spent little time with the faculty. jt: true. and that is why being a critical administrator is difficult, finding that balance in the dance of dialectics. ja: how do you frame, navigate, and privilege the dialectics between administrator and faculty, administrator and students, and administrator and staff? jt: this is something difficult for me to answer. because navigating those dialectical relationships while privileging them equitably, not necessarily equally, was and is something i consider myself poor at accomplishing. if freire were here, i think he would tell us that there is no single solution, that only through praxis, dialogue, and the theory–practice dialectic can the dance be performed, finding some balance in the tension. in pedagogy in process (1978), he says: in a certain moment it becomes true that one no longer studies in order to work nor does one work in order to study; one studies in the process of working. there comes about, thus, a true unity between practice and theory. (p. 21) there is a glimmer of an answer in this quote, the notion of “studying in the process of working,” that supports my point. i don’t think it’s possible to separate those dialectical relationships and navigate them separately. the focus should not exclusively be on any single dialectic or group, and dialogue in conjunction with practice is an inoculation against his path that can often lead to oppression. bragelman public stories journal of urban mathematics education vol. 8, no. 2 32 ja: but which group do you support as an administrator, and when? i’m attempting to build a safe space for faculty and students to implement critical theory and practice, and it seems one group would receive the most emphasis. jt: take care with your framing. to name a thing is to give it power, and thoughtless naming invites domination and oppression. you are not building a safe space; you are co-creating a safe space. with, not for. earlier in pedagogy in process, freire (1978) states: authentic help means that all who are involved help each other mutually, growing together in the common effort to understand the reality in which they seek to transform. only through such praxis … can the act of helping become free from the distortion in which the helper dominates the helped. (p. 8) you see the difference? ja: yes, i do. my best work in this role has derived from faculty and staff’s reflections on their practice. the programs i’m implementing have originated in these dialogues. jt: what about the students? how is your dialogue with them? ja: to be honest, there hasn’t been much dialogue with students, but that’s slowly changing. i’m interacting more with students as new programs begin, and i’ve spoken with each who’ve stopped by. but you’re right. it’s something that requires more attention. it seemed easier being a critical educator. jt: haven’t you begun yet to see that it’s the same, just more? that being a critical administrator requires the same epistemological and ontological values? that it requires negotiating the same dialectical tensions? reflect for a moment. perhaps freire doesn’t emphasize critical administration in his work because he doesn’t perceive critical administration to be any different than critical pedagogy; that even as an administrator, there is still educator and educand (freire, 1994/2004). in literacy: reading the word and the world (freire & macedo, 1987), he alludes to this lack of difference: to live or embody this obvious confrontation [that we are not alone in this world], as an educator, means to recognize in others, whether they are becoming literate or are participants in university courses, students of primary schools or members of a public assembly, the right to express their thoughts, bragelman public stories journal of urban mathematics education vol. 8, no. 2 33 their right to speak, which corresponds to the educator’s duty to listen to them. (p. 40) yes, teachers as educators are positioned in such a way to foster and cocreate liberatory movements, and because of this power to cocreate, freire emphasizes educators’ roles and devotes the majority of his life to serving them (freire, 1978, 1970/1993b, 1994/2004). freire’s texts emphasize so often teaching with, learning with. he demonstrates this teaching/learning with by example through his work with educators and with educands. is there a need to create this false dichotomy? aren’t educators and educands the same persons? ja: i share the same perspective, educators and educands as one. but few teachers do. so often at work i see faculty buying in to a “banking system of education” (freire, 1970/1993b) and presenting content in some extreme acquisitionist form (sfard, 1994). how do you shift pedagogical philosophies? jt: how did i shift them as an educator? i’ve often worked with traditional educators, and through my dialogue with students i came to know how their pedagogies were affecting our students. it was the first time my pedagogy extended out of the classroom. my reasoning aligns with freire (1998) in pedagogy of freedom: whether the teacher is authoritarian, undisciplined, competent, incompetent, serious, irresponsible, involved, a lover of people and of life, cold, angry at the world, bureaucratic, excessively rational, or whatever else, he/she will not pass through the classroom without leaving his or her mark on the students. (p. 64) i chose not to “fold my arms fatalistically in the face of misery” (p. 72). and even if my intentions weren’t to follow a freirian framing, it still happened. i, you, shift pedagogical philosophies by doing with, or in this case, teaching with. i openly shared my pedagogical philosophies, discussing the theories i had read, sharing texts. i co-taught with other teachers. in my courses, students actively taught lessons. my classroom was a transparent place where power dynamics were constantly upended. but none of that happened in one week, or even in one month or one year. freire would tell you that you must learn the community first, as he said to mario cabral (1978) or as he showed in education and democracy (del pilar o’cadiz et al., 1998) and literacy (freire & macedo, 1987). bragelman public stories journal of urban mathematics education vol. 8, no. 2 34 ja: but that takes so much time. students are finishing courses next week. almost no changes have been made to the developmental program that i direct. students are failing while i take the time to learn the community. jt: the sense of urgency you feel is good, but don’t let it lead to impatience. dwell within a space of urgent patience, but heed freire’s counsel (1978) against moving from that space: “breaking the tension between patience and impatience, under such circumstances, inevitably leads to teaching without dialogue” (p. 64). and teaching without dialogue, administrating without dialogue, is just oppression. ja: it’s been difficult, learning the culture of the faculty here. i hadn’t anticipated how distrustful they would be of administration … … … jt: was i not the same? often i acted in self-perceived defiance and subversion all the while being unwilling to engage in real dialogue with the administration. it’s entirely possible my administrators were always aware, and they let me keep these pretenses of rebellion because it was what i needed at the time. ja: yes, you’re right; i need to be more patient. show them through actions that i can be trusted, by listening (freire, 1998) and through respect (freire, 1994/2004). “what ought to guide me is … respect, at all costs, for all those involved in education” (freire, 1998, p. 101). jt: and show them through dialogue. learn who your faculty and students are. freire speaks on this extensively. in pedagogy of hope (1994/2004), he discusses the importance of learning and privileging their culture and language: it was by attempting to inculcate a maximal respect for the cultural differences with which i had to struggle, one of them being language—in which i made an effort to express myself, as best i could, with clarity—that i learned so much of reality, and learned it with chileans. (p. 34) i did the same as an educator. my students taught me about their language, their culture, and their communities, and together we taught and learned mathematics. the goal is to begin with the students’ and faculty’s realities, not yours: even though the educator’s dream is not only to render his or her ‘here-andnow’ accessible to educands, but to get beyond their own ‘here-and-now’ with them, or to understand and rejoice that educands have gotten beyond their bragelman public stories journal of urban mathematics education vol. 8, no. 2 35 ‘here’ so that this dream is realized, she or he must begin with the educands’ ‘here,’ and not with her or his own. (freire, 1994/2004, p. 47) from their realities, you work toward accessing their dreams. those realities are the key. their realities are the mediators for their learning, and they’re the fountain of their knowledge. later in pedagogy of hope (1994/2004), freire speaks of the importance of cultural activities as part of those realities: educators need an understanding of the meaning their festivals have as an integral part of the culture of resistance, a respectful sense of their piety in a dialectical perspective, and not only as if it were a simple expression of their alienation. their piety, their religiousness, must be respected as their right, regardless of whether we reject it personally (and if so, whether we reject religion as such, or merely do not approve the particular manner of its practice in a given popular group). (p. 91) it’s difficult sometimes to reject those assumptions and beliefs you hold most closely when you’re faced with the lived realities of others, especially when they’re the antithesis of your own. it was through dialogue with my students that i learned to be more human, and they learned how to critically read and write their world with math (gutstein, 2006), as i “discuss[ed] with the students the logic of these kinds of knowledge in relation to their contents” (freire, 1998, p. 36). it was through dialogue my students discovered and developed counter narratives to the master narratives (lyotard, 1984; martin, 2000; nelson, 2001). ja: so dialogue helps the community move toward progress. what will that look like? how do you create a critical curriculum and pedagogy? jt: for the most part, it sounds as if you’re working toward that point. what has never worked in the past, at least in terms of students’ empowerment and success, is homogenized, standardized curricular packages. freire (1994/2004) speaks of this in pedagogy of hope: the fundamental problem—a problem of a political nature, and colored by ideological hues—is who choses the content, and in behalf of which persons and things the ‘chooser’s’ teaching will be performed—in favor of whom, against whom, in favor of what, against what. (p. 94) a homogenized curriculum only serves the dominant group through and by normalization (foucault, 1975/1995; leonardo, 2009). he exemplifies his bragelman public stories journal of urban mathematics education vol. 8, no. 2 36 own suggestion during the implementation of the inter project under his administration: the orientation of freire’s administration, as we have tried to argue and exemplify, was towards passionately and slowly building a social movement responsive to the educational needs of popular communities rather than coldly and efficiently developing coordinated curriculum packages to be identically replicated in the city’s 691 schools. (del pilar o’cadiz et al., 1998, p. 247). the popular communities are the students and the faculty, the people you serve. developing a coordinated curriculum means working with students and faculty to develop a curriculum for your college, by your college, with your college. ja: similar to the programs i’m currently working with faculty to build, curriculum informed by and created with the students is trickier. that will take faculty buy-in; it’s a different type of pedagogy than what they’re used to. jt: it will be, and there will be challenges associated with it, just as there were with the inter project; challenges like teacher frustration and disenchantment from no standardized curriculum (del pilar o’cadiz et al., 1998). the faculty will require significant support. ja: how? jt: professional development. how many times have i attended professional development that models the most traditional of classrooms, some “specialist” standing at the front of the room, the sage on the stage, imparting her or his knowledge to me? you need to implement supportive professional development that privileges the difficulties and rigor a curriculum like this demands. ja: what kind of development would that entail? the school has already begun implementing some, including a set of development sessions on proactive restorative practices (sugai, o’keeffe, & fallon, 2011) that the faculty and i are co-developing. it brings intentionality to creating communities within classes, and it scaffolds students’ mathematics identity (martin, 2000). jt: but you have it! professional development follows the same processes as any other educational endeavor. it should be problem posing, rooted in praxis, privileging both theory and practice, and foremost it should derive from bragelman public stories journal of urban mathematics education vol. 8, no. 2 37 the realities of the faculty. freire (1978) speaks of his vision in pedagogy in process: another thing the administration has to do in it’s dealings with the faculty and staff and their respective roles is to think, organize, and create programs of permanent staff development counting on the help of those scientists with whom we have until now been working. a permanent staff development must be based, above all, on reflection about practice. it is through thinking about his or her practice, it is through confronting the problems that will emerge in his or her daily practice, that the educator will transcend his or her difficulties with a team of specialists who are scientifically qualified. (p. 20) freire even models an implementation of his vision, exhibiting the importance of privileging theory and practice in professional development detailed in education and democracy: it was necessary to reorient the policy for staff training and development by overcoming the traditional holiday courses that insisted on a discourse about theory, thinking that afterwards the teachers would put directly into practice that theory which was discussed through the practice of discussing practice. such courses were not developed through an efficacious form of living a dialectic unity between theory and practice. (del pilar o’cadiz et al., 1998, p. 249) these two excerpts suggest that you will have to change the culture of development at the school. have you considered something that encourages active reflection? ja: i’ve submitted a proposal to implement a reflective model for teaching and learning. faculty would videotape classes and reflect on their practice in small sessions with other faculty. initially, administration would have to be left out of the reflection process until more trust is garnered. jt: it’s reflective, yes. it even encourages faculty to consider the dialectic of theory and practice. but is it deriving from the faculty or from you? ja: that’s a good question. there are faculty members who’ve approached me about improving their pedagogy, but they’re unsure how to go about the process. i posed this as a suggestion, and the faculty seemed interested. jt: excellent. also freire, i think, would suggest development oriented on the realities of the students. this orientation could be theoretical or practical. bragelman public stories journal of urban mathematics education vol. 8, no. 2 38 ja: i can see the practical perspective. based on dialogue i’ve had with faculty members, i don’t think they fully understand chicago public schools (cps) or the communities the students call theirs. i want to bring educators from cps to campus to speak with our faculty about the realities of teaching in a system plagued by neoliberal oppression (lipman, 2011). afterwards, i was hoping to send faculty to cps high schools, to see what teaching and learning look like in that context. i think their realities—the faculty here and at cps—are more similar than they perceive. jt: if possible, it would be worth taking faculty members to the students’ actual communities, not just their old schools. i held so many assumptions and generalizations of the communities on the south side of atlanta, and all were shattered within days of teaching there. my students spoke different than i, and they came to school with a different knowledge than i ever held. it was something i never knew or understood until they taught me the knowledge of their realities. i was determined to respect all aspects of my students, and this respect forged a freirian philosophy, albeit unintentionally. he speaks extensively of these types of knowledges in many of his books, but pedagogy of hope (1994/2004) resonates with me the most: i have argued the need we progressive educators have never to underestimate or reject knowledge had from living experience, with which educands come to school or to informal centers of education … a respect for both knowledges— a respect of which i speak so much—with a view to getting beyond them, must never mean, in a serious, radical, and therefore critical, never sectarian, rigorous, careful, competent reading of my texts, that the educator must stick with the knowledge of living experience. (pp. 71–72) here, he privileges the importance of both types of knowledges, the popular knowledge of the people and the formal, hegemonic knowledge of the academy. he also warns of over-privileging one type, here the popular. macedo (1991) makes a similar warning in the context of english language learners, over-privileging the home language to the detriment of the students as they’re not exposed then to english, and as the primary tool by the dominant class, students are highly served by its access. the same must be true of any curriculum you develop and perspective you take, a balance in the tension of the dialectic, of both practice and theory, of both popular knowledge and academic knowledge. ja: how will i know if i’ve found that balance? can you assess it? so much assessment at this institution is quantitative, and while i’ve pushed against this bragelman public stories journal of urban mathematics education vol. 8, no. 2 39 belief, there is still an inordinate amount of pressure to show results quantitatively, especially significance testing. jt: i can’t give much guidance on this question, as i have little experience in program analysis. the majority of my assessment experience is in the classroom, in a formative context. freire (1993a) warns of assessment through traditional, dominant methodologies in pedagogy of the city: the evaluation criteria the school uses to measure students’ knowledge— intellectualism, formal, bookish—necessarily helps these children from the so-called privileged social classes, while they hurt children from poor and low socioeconomic backgrounds. … the experience of children from the middle class results in the acquisition of a middle-class vocabulary, prosody, syntax, in the final analysis a linguistic competence that coincides with what the school regards as proper and correct. the experience of poor children takes place not within the domain of the written word, but within direct action. (pp. 16–17) this passage suggests assessment of a critical curriculum combines measuring the learning of both popular and academic knowledges in the context of the students’ realities. freire (1994/2004) confirms this interpretation in pedagogy of hope: [it is necessary to find] a critical comprehension of how university arts and sciences ought to be related with the consciousness of the popular classes: that is, a critical comprehension of the interrelations of popular knowledge, common sense, and scientific cognition. (p. 169) if you consider freire’s framework, every perspective, decision, and ideology is rooted in the dialectic. ja: so then, i move in the “right” direction, positioning myself in the tension between qualitative and quantitative methodologies? jt: i think so. in my readings, freire (1993a) speaks of this only briefly. for instance, in pedagogy of the city, he writes, “it is important also to point out that a critical politics of education cannot mechanically understand the relationship between these deficits—quantitative and qualitative—but it must understand them dynamically and contradictorily” (p. 15). this statement, i believe, verifies your rightness. ja: how will i know if i’m successful? bragelman public stories journal of urban mathematics education vol. 8, no. 2 40 jt: the students and faculty will be your lens. you’re successful when they’re successful, as they define the measure of success. and i don’t mean successful as high grade point averages or retention rates or graduation rates. i mean successful critically: in other words, it is a process of knowing with the people how they know things and the level of that knowledge. this means challenging them, through critical reflection, regarding their own practical experience and the ends that motivate them in order, in the end, to organize the findings, and thus to replace mere opinion about facts with an increasingly rigorous understanding of their significance. (freire, 1978, p. 25) ja: how do i know if i’m doing the right thing? jt: because you’re not “evading [your] responsibility, hiding behind lukewarm, cynical shibboleths that justify [your] inaction because ‘there is nothing that can be done.’ the exhortation to be more a spectator; the invitation to (even exaltation of) silence, which in fact immobilizes those who are silenced” (freire, 1993a, p. 72). you are struggling with. ja: but how will i know when i’m a critical administrator? jt: this dialogue is a step toward the answer. one year later in the spring of 2015, i returned to this dialogue to reflect on my growth and the challenges i faced as a mathematics teacher turned administrator, the impact this new identity has had on my practice, and the criticality of my work. jt: you’ve worked as director of developmental education for over a year now. how does it feel? ja: i miss the classroom. every single day, i miss it. i question whether this was the right choice, this path as an administrator. i do good work in this role, but i’m overwhelmed. the struggles i faced as an educator are magnified as an administrator. i don’t have 120 students in a semester; i have 2,000. jt: what struggles? bragelman public stories journal of urban mathematics education vol. 8, no. 2 41 ja: as a teacher, my students, their communities, grounded me. the relationships i developed affirmed my choices, teaching for justice, teaching with. it took this past year to realize that i was missing these relationships with students as an administrator. i work in broad swaths. when i uncover need, i build programs to support that need, but programs target hundreds. this is the good that i do. i’ve been given the power to enact change at the school level, but i lose reflexivity without these relationships. it’s so easy to lose the with. i felt this happening when i first took this position, so now i build relationships with students at this school, engaging them in dialogue to learn of their lived experiences and the impact (or lack thereof) my programs have on their lived experiences (freire, 1994/2004). jt: what do you mean? ja: i sat with a student a few weeks ago in my office. she alternated between tears of frustration and, in her words, militancy. battered by her reality and the sociopolitical forces that buffet her on a daily basis, she was losing hope. she is taking our gatekeeper math class for the second time, and at midterms, she was failing. the school has no additional services to support her. so i tutor her now, two to three times a week. teaching with. learning with. struggling with. jt: it sounds like you are still a teacher at heart. ja: it’s my most salient identity. it’s the truth of me, and it guides my decisions still. i feel as an administrator that i should question if devoting time to one or three students is beneficial to the whole college, but the teacher part of me, you, demands i invest in the individual. jt: why not return to the classroom then? to teaching? to what you love? ja: i … i can’t. not yet. despite not wanting this, despite missing the classroom, i can do too much good in this position. and if i left, who would replace me? i don’t trust anybody else. jt: so you continue on this journey? after all of you gone through, do you consider yourself a critical administrator? ja: freire (1970/1993b) would say i’m in the process of becoming. and part of me thinks he would be correct. i navigate political pressures, especially pol bragelman public stories journal of urban mathematics education vol. 8, no. 2 42 icy and how it is enacted. i work with both faculty and students, and i strive to build a program that serves the “educational needs of [the] popular communities rather than coldly and efficiently developing coordinated curriculum packages to be identically replicated” (del pilar o’cadiz et al., 1998, p. 247). but no, despite this, i’m not an administrator. i’m a math teacher. references del pilar o’cadiz, m., wong, p., & torres, c. a. (1998). education and democracy: paulo freire, social movements, and educational reform in sao paulo. boulder, co: westview. foucault, m. (1995). discipline and punish: the birth of the prison. westminster, md: vintage. (original work published in 1975). freire, p. (1978). pedagogy in process: the letters to guinea-bissau. new york, ny: the seabury press. freire, p. (1993a). pedagogy of the city. new york, ny: continuum. freire, p. (1993b). pedagogy of the oppressed. new york, ny: continuum. (original work published in 1970). freire, p. (1998). pedagogy of freedom: ethics, democracy, and civic courage. new york, ny: rowman & littlefield. freire, p. (2004). pedagogy of hope: reliving pedagogy of the oppressed. new york, ny: bloomsbury academics. (original work published in 1994). freire, p., & macedo, d. (1987). literacy: reading the word and the world. westport, ct: bergin & garvey. gutstein, e. (2006). reading and writing the world with mathematics: toward a pedagogy for social jusice. new york, ny: routledge. gutstein, e. (2008). building political relationships with students. in e. de freitas & k. nolan (eds.), opening the research text: critical insights and in(ter)ventions into mathematics education (pp. 189–204). new york, ny: springer. gutstein, e., & peterson, b. (eds.). (2006). rethinking mathematics: teaching social justice by the numbers. milwaukee, wi: rethinking schools. hooks, b. (1994). teaching to transgress: education as the practice of freedom. new york, ny: routledge. ladson-billings, g. (1994). the dreamkeepers: successful teachers of african american children. san francisco, ca: jossey-bass. ladson-billings, g. (1995). toward a theory of culturally relevant pedagogy. american educational research journal, 32(3), 465–491. ladson-billings, g., & tate, w. f. (1995). toward a critical race theory of education. teachers college record, 97(1), 47–68. leonardo, z. (2009). race, whiteness, and education. new york, ny: routledge. lipman, p. (2011). the new political economy of urban education: neoliberalism, race, and the right to the city. new york, ny: routledge. lyotard, j. f. (1984). the postmodern condition: a report on knowledge. minneapolis, mn: university of minnesota press. macedo, d. (1991). english only: the tongue-tying of america. journal of education, 173(2), 9– 20. bragelman public stories journal of urban mathematics education vol. 8, no. 2 43 martin, d. b. (2000). mathematics success and failure among african-american youth: the roles of sociohistorical context, community forces, school influence, and individual agency. mahwah, nj: erlbaum. nelson, h. l. (2001). damaged identities, narrative repair. ithaca, ny: cornell university press. sfard, a. (1994). on two metaphors for learning and the dangers of choosing just one. educational research, 27(4), 4–13. sugai, g., o’keeffe, b. v., & fallon, l. m. (2011). a contextual consideration of culture and school-wide positive behavior support. journal of positive behavior interventions, 14(4), 197–208. tate, w. f. (1994). race, retrenchment, and the reform of school mathematics. phi delta kappan, 75(6), 477–483. woodson, c. g. (2006). the mis-education of the negro. san diego, ca: book tree. (original work published in 1933). microsoft word final confrey vol 3 no 2.doc journal of urban mathematics education december 2010, vol. 3, no. 2, pp. 25–33 ©jume. http://education.gsu.edu/jume 
 jere confrey is the joseph d. moore distinguished university professor of mathematics education at the friday institute for educational innovation, college of education, north carolina state university, 1890 main campus drive, raleigh, nc 27606; email: jere_confrey@ncsu.edu. her research focuses building diagnostic assessments for rational number reasoning using wireless technologies and on policy involving standards and assessments. response commentary “both and”—equity and mathematics: a response to martin, gholson, and leonard jere confrey north carolina state university n the commentary “mathematics as gatekeeper: power and privilege in the production of knowledge,” this issue of jume, authors danny martin, maisie gholson, and jacqueline leonard (2010) react to editor kathleen heid’s (2010) statements in the march 2010 issue of the journal for research in mathematics education (jrme), and to a panel presentation during the 2010 national council of teachers research presession (harel, ball, battista, thompson, & confrey, 2010). i was the discussant for that panel. in summary, martin et al. argue that by posing the question “where’s the math in mathematics education research?” the scholars on the panel and the editor of jrme marginalize scholars who wish to study deep and complex factors that concern issues of identity, language, power, racialization, and socialization in mathematics education. they state that a call for more attention to mathematics content in mathematics education research is closely aligned with a “back to basics” agenda. warning that focusing too heavily on the mathematics may sustain a tradition of deficit thinking about minority and indigent children, they intone that this focus fails to acknowledge the complex factors that limit these children’s access to rich opportunities to learn mathematics with the cultural resources that they bring to school. further, martin et al. (2010) observe the limited diversity in terms of race and to a lesser degree, gender, in many areas of mathematics education such as among the scholars who have focused on research on whole number concepts and operations (nesher, 1980); rational number concepts (confrey, 1988, 2008; empson &turner, 2006; lamon, 2007; steffe, 2002); proportional reasoning concepts (behr, harel, post & lesh, 1992; confrey, 1995; hart, 1988; lamon, 1993; lesh, behr, & post, 1987; karplus, pulos, & stage, 1983; noelting, 1980); algebra, problem solving, and proof (harel & sowder, 1998, 2007); and geometric and spatial thinking (barrett & clements, 2003; barrett, clements, klanderman, pennisi, & polaki, 2006; battista, 2007; clements & sarama 2009; clements, wilson, & sarama, 2004; hollebrands, 2002). they seem plainly offended by and critically challenge harel’s assertion that the questions addressed by the presession panel were apolitical and neutral; they characterize the questions instead as i confrey response commentary 
 journal of urban mathematics education vol. 3, no. 2 26 “displays [of] power and privilege” (p. 13). in their conclusion, they warn that to interpret their criticisms as personal rather than professional “would be disingenuous and misses the point” (p. 21). i welcome martin et al.’s (2010) highlighting these issues, and their challenge of whether a call for a focus on mathematics content results in a call for the exclusion of broader issues of equity. i acknowledge some of the points raised by the authors and disagree with others. first, i outline my points of agreement with the authors: 1. equity is not a “special issue” facing us in mathematics education; it is “the central issue” (darling-hammond, 1994, 2010; bennett et al., 2004). in my opinion, the math wars (wilson, 2002) have been a distraction from these fundamental issues just as the current media focus on the culture wars often distracts from more fundamental underlying perspectives on the distribution of wealth, corporate influence over democracy, and everyday injustices to people. 2. the tendency for educational professionals to treat children of color and of poverty solely in deficit terms is a devastating problem that severely constrains our discourse and understanding of how to generate robust solutions to differences in performance and high failure rates in mathematics. 3. to address the magnitude of the inequity, one must understand how the entire education system, as a component of the larger system of human services, fails to provide fair and equal opportunities to learn for large segments of the population. marilyn frye (1983), a feminist scholar, describes a bird cage analogy, noting that if one looks at any single wire of a bird cage, one wonders why the bird does not fly away. only by observing the whole configuration of wires does the inherent confinement of the cage become apparent. 4. the field of mathematics education needs to be more diverse—who does the scholarship does matter—due to differences in experiences, priorities, interpretive frameworks, and identification. when i recently hosted a discussion of martin et al.’s (2010) commentary, of the women who participated (african american and caucasian, and all them professionals in educational research), more than half, including myself, had been at one time or another explicitly counseled not to study issues of race or gender as a scholarly enterprise for risk of being pigeonholed, and hence restricted in our subsequent professional opportunities. therefore, i agree that the marginalization of scholarship referred to in the commentary is a widespread and unfortunate phenomenon reinforced by different forms of mentoring. confrey response commentary 
 journal of urban mathematics education vol. 3, no. 2 27 the concerns raised by martin et al. (2010) clearly have merit. however, there is also merit in the perspective that the field of mathematics education research has drifted too far from close and careful attention to the mathematics we seek to have students learn. my advice to scholars interested in equity agendas can be communicated in relation to four particular issues; each focuses on the centrality of the mathematics in the agenda. 1. it is essential to stay current with research on cognition to be certain that the students one seeks to help are getting the most up-to-date access. for example, the algebra project (moses & cobb, 2001) quite properly and significantly asserted that algebra is a civil right; under the leadership of robert moses, the algebra project created community-based approaches to improving access and success. however, their curricular materials were slow to transition to a functions-based approach to algebra, and instead concentrated on algebra as expressions and equations. i would argue that the delay in updating the algebra project’s approach constituted a disadvantage to students when the functions-based approach gained purchase in many introductory college mathematics courses. 2. serving the needs of all students requires one to stay abreast of, and make one’s opinions known concerning, the full range of mathematics that is of most importance and value to the communities one serves— from topics of basic arithmetic proficiency, to competence with fundamental tools (excel, statistical software, graphing software and hardware), to critical maths, to “career and college readiness,” and to twenty-first century skills (confrey, 2009). during the development of the voluntary-state common core state standards for mathematics (ccss, 2010), a highly political decision was made to drop probability and statistics from the elementary curriculum. as a member of the national validation committee for the common core state standards, i had access to many, but not all, of the comments. i witnessed a strong response from the learning scientists and the curriculum writing communities: vigorous evidence-based arguments to strengthen the statistics and probability content in the early grades—that research demonstrates children’s early rich probabilistic awareness, that these topics must be developed over extended time, and that probability and statistics are critical mathematical topics for the 21st century. to what degree did the scholars on critical theory register concerns that the ccss writers’ decision would increase students’ alienation from mathematics and weaken students’ ability to reason critically and quantitatively about the world around them? it is not confrey response commentary 
 journal of urban mathematics education vol. 3, no. 2 28 clear, for all comments were not made available, even to the validation committee. the writers’ final decision, however, to reduce the opportunity for all children to learn that they live in a world of probability, to incorporate probability into their mathematical reasoning, and that fair opportunity is an issue of improving one’s odds, seems to me to reinforce an implicit impression communicated within the ccss that mathematics is apolitical and objective. scholars to whom martin et al. (2010) refer, including m. frankenstein, a. powell, r. gutiérrez, e. gutstein, s. lubienski, and others have raised active concerns about the role of critical mathematics and the effects of reform programs in high poverty settings. martin et al. ask the question: “is this the mathematics to which heid, harel, and perhaps the panelists might be referring?” (p. 14). my answer would be yes, this and more. a continuation of the lines of research represented by those scholars, research that involves direct attention to the interaction of the mathematics and the lived experience of the student, is imperative. the work of some of the centers for learning and teaching (clt) such as the dime (diversity in mathematics education) center and cemela (center for the mathematics education of latinos/as) keeps these linkages central to their work. it is also essential to recognize that the mathematics one learns, the tools one has with which to learn, and the expectations and opportunities expressed in the schools are part of the lived experience of the student, and matter greatly. my question therefore is whether, in the research on increasing students’ success in mathematics and lessening the achievement gap, has there been enough attention paid to: (a) ensuring increased awareness of learning technologies and twenty-first century skills, (b) the development of career awareness and preparation, and (c) the intensification of topics like statistics and modeling? i emphasize the importance of this development, not so much for the identification of research mathematicians, but for the education and empowerment of resourceful mathematicians (confrey, 2007, 2010). to my mind, these topics are consistent with a call for more focus on the mathematics in mathematics education and are also central to a more effective equity agenda. 3. good teaching that fosters rich classroom interactions and engagement is a key to a successful equity agenda. many mathematics educators share with me the belief that good teaching and fair access to qualified teachers is the key route to increased equity. while improved teaching entails policy dimensions and actions at the systemic level, it also requires significant work in understanding teaching, interpreting classroom interactions, and improving professional development. confrey response commentary 
 journal of urban mathematics education vol. 3, no. 2 29 much of the work around creating and supporting discourse communities (hufferd-ackles, fuson, & sherin, 2004; thomas, 2010), building teaching capacity (allen et al., 2006; nptars, 2005), and strengthening curricular implementation (grouws, reys, papick, tarr, & chavez, 2010; mcnaught, tarr, & sears, 2010) can transform students’ school learning experiences and enhance their opportunities to learn. this work requires precise, creative, and extended attention to what happens within mathematics classrooms and how students develop their understandings over long periods of time. it could be further strengthened, to the benefit of students and teachers, by increased attention to the cultural resources students bring to school, sensitivity to differences in linguistic nuances of region, culture, and class, and leveraging these characteristics for improved student mathematical understanding. this is an arena in which scholars across mathematics education communities could help to produce massive change in children’s experiences in schools. similarly, related development of uses of assessment to promote fair and engaging opportunities to learn would benefit from a more active role from the larger community committed to increased equity (black & wiliam, 1998). formative assessment (assessment for learning) is one classroom process in which the imperatives of mathematical content, student engagement, and student ownership of their own learning intersect to the benefit of all students, and thus could itself be a productive focus for collaboration of mathematics educators of various subspecialties and perspectives. for instance, mcmanus (2008) studied how to instill formative assessment practices in low-achieving schools. she identified three essential conditions for successful implementation of formative assessment: high level of content and pedagogical knowledge by the teacher, students trusting that they are partners in learning in the classroom environment, and dialogic mathematical content discourse. these elements, along with the essential steps of formative assessment, were the basis of successful formative assessment leading to increased student motivation, ownership of their own learning, and improved self-efficacy. my own research team is undertaking designing new forms of classroom-embedded diagnostic assessment based on learning trajectories to help all students to participate more fully in classroom activities (confrey & maloney, in press). 4. a lack of significant attention to mathematics in mathematics education research results in the marginalization of the entire field, threatening the preparation of the next generation of scholars and teachers. while i fully agree with martin et al.’s (2010) view that harel’s characterization of the issues as apolitical (during the panel) was in error, i do believe confrey response commentary 
 journal of urban mathematics education vol. 3, no. 2 30 that the advice of the collective group of mathematics education scholars— to pay closer attention to the mathematics—represents wise and pragmatic counsel to which all members of the mathematics education must attend. mathematics educators have been effectively sidelined in many policy decisions and related expert panels. the worst example was reflected in the national mathematics advisory panel report (national mathematics advisory panel, 2008), in which the chapter on learning and serving the needs of special education students virtually mandated direct instruction; the whole document severely limited attention to context, technologies, and modeling (kelly, 2008). some of those problems may turn out to be amplified in the common core state standards (ccss, 2010), which virtually eliminated statistics and algebra in the early grades, and in which the consequent intensity of the mathematics assigned to middle school is likely to have negative consequences for impoverished schools that lack sufficient teaching capacity. scholars as well as policy specialists need to keep a very close eye on this development. one can argue that the displacement of the mathematics education community by mathematicians and cognitive scientists (primarily experimental psychologists) was simply a product of the prior federal administration’s biases. but this marginalization continues in the current administration, and attacks on colleges of education continue. in light of these developments, the counsel to keep mathematics central in our work seems to me to be wise counsel. concluding remarks we must not underestimate the value of the type of debate and discussion that martin et al.’s (2010) commentary facilitates. the group mentioned previously who took part in the discussion at north carolina state university left in a state of excitement and an increased sense of community, which resulted from spirited debate and attempts to discuss the “wicked problems”1 raised by the commentary. the exercise of debate, strong calls for self-examination, and emphasis on inclusion are key values, particularly in a national culture in which enlightened thought and debate seem to be increasingly attenuated, and local policies that increase segregation, attacks on public schooling, bullying on sexuality, and excessive blaming of teachers for societies’ woes are rampant. i can understand the undertone of anger, urgency, and critical concern in martin, gholson, and leonard’s commentary. however, i would have preferred to have seen less 























































 1 for an explanation of wicked problems, see http://en.wikipedia.org/wiki/wicked_problem#formal_definitions. confrey response commentary 
 journal of urban mathematics education vol. 3, no. 2 31 rhetorical devices, such as the link to “back to basics” which was relatively unfounded and implies an unduly conservative agenda, and more emphasis on the fact that the statement represents differences in opinions among scholars all of whom are exercising leadership. while critical theory is essential in raising consciousness, there is also a need for heavy lifting on the practical problems that cause inequity to persist. my voice in this debate is to call for “both and.” we certainly need attention to both equity and mathematics, and in some critical areas with the potential to improve the situation, we have too few hands to lift and too few scholars gaining the technical know-how of methods and technological advances to be keeping pace with the opportunities that are arising. a new synthesis of research methodologies that combines attention to equity and mathematics has the potential to benefit more children and teachers as well as to improve the strength of mathematics education. acknowledgments i acknowledge the contributions of alan maloney, allison mcculloch, and kenny nguyen in reviewing and commenting on this commentary. all opinions expressed in the paper are solely the responsibility of the author. references allen, m., coulter, t., dwyer, c. a., goe, l., immerwahr, j., jackson, a., johnson, j., oliver, r. m., ott, a., et al. (2006). america’s challenge: effective teachers for at-risk schools and students. washington, dc: national comprehensive center for teacher quality. barrett, j. e., & clements, d. h. (2003). quantifying path length: fourth-grade children’s developing abstractions for linear measurement. cognition and instruction, 21, 475–520. barrett, j. e., clements, d. h., klanderman, d., pennisi, s., & polaki, m. v. (2006). students’ coordination of geometric reasoning and measuring strategies on a fixed perimeter task: developing mathematical understanding of linear measurement. journal for research in mathematics education, 37, 187–221. battista, m. t. (2007). the development of geometric and spatial thinking. in f. k. lester (ed.), second handbook of research on mathematics teaching and learning (pp. 843–908). charlotte, nc: information age. behr, m., harel, g., post, t., & lesh, r. (1992). rational number, ratio and proportion. in d. grouws (ed.), handbook of research on mathematics teaching and learning (pp. 296–333). new york: macmillan. bennett, a., bridglall, b. l., cauce, a. m., everson, h. t., gordon, e. w., lee, c. d., mendozadenton, r., renzulli, j. s., & stewart, j. k. (2004). all students reaching the top: strategies for closing academic achievement gaps. naperville, il: national study group for the affirmative development of academic ability. black, p., & wiliam, d. (1998). inside the black box: raising standards through classroom assessment. phi delta kappan, 80,(2), 139–148. ccss (2010). common core state standards for mathematics, retrieved from http://www.corestandards.org/assets/ccssi_math%20standards.pdf confrey response commentary 
 journal of urban mathematics education vol. 3, no. 2 32 clements, d. h., & sarama, j. (2009). learning and teaching early math: the learning trajectories approach. new york: routledge. clements, d. h., wilson, d. c., & sarama, j. (2004). young children’s composition of geometric figures: a learning trajectory. mathematical thinking and learning, 6, 163–184. confrey, j. (1988). multiplication and splitting: their role in understanding exponential functions. in m. behr, c. lacompagne, & m. wheeler (eds.), proceedings of the 10th annual meeting of the north american chapter of the international group for the psychology of mathematics education (pp. 250–259). dekalb, il: northern illinois university press. confrey, j. (1995). student voice in examining “splitting” as an approach to ratio, proportions and fractions. proceedings of the 19th annual meeting of the international group for the psychology of mathematics education (vol. 1, pp. 3–29). recife, brazil: universidade federal de pernambuco. confrey, j. (2007). tracing the evolution of mathematics content standards in the united states: looking back and projecting forward. paper presented at k–12 mathematics: what should students learn and when should they learn it?, arlington, va. confrey, j. (2008). a synthesis of the research on rational number reasoning: a learning progressions approach to synthesis. paper presented at the 11th international congress of mathematics instruction, monterrey, mexico. confrey, j. (2009). steering a course for preparing students for the mathematical sciences in the 21st century. paper presented at the annual meeting of the research in undergraduate mathematics education conference, raleigh, nc. confrey, j. (2010). what now? priorities in implementing the common core state standards for mathematics. paper presented at the kick-off for the institute for research in mathematics and science education, washington, d.c. confrey, j., & maloney, a. p. (in press). next generation digital classroom assessment based on learning trajectories in mathematics. in c. dede & j. richards (eds.), steps toward a digital teaching platform. newyork: teachers college press darling-hammond, l. (1994). performance-based assessment and educational equity. harvard educational review, 64, 5–29. darling-hammond, l. (2010). the flat world and education: how america’s commitment to equity will determine our future. new york: teachers college press. empson, s. b., & turner, e. (2006). the emergence of multiplicative thinking in children’s solutions to paper folding tasks. journal of mathematical behavior, 25, 46–56. frye, m. (1983). the politics of reality: essays in feminist theory. trumansberg, ny: the crossing press. grouws, d., reys, r. e., papick, i., tarr, j. e., & chavez, o. (2010). comparing options in secondary mathematics: investigating curriculum (cosmic). retrieved from http://cosmic.missouri.edu/. harel, g., ball, d. l., battista, m. t., thompson, p., & confrey, j. (2010). where’s the mathematics in mathematics education? paper presented at the research presession of the annual meeting of the national council of teachers of mathematics, san diego, ca. harel, g., & sowder, l. (1998). students’ proof schemes: results from exploratory studies. in a. schoenfeld, j. kaput, & e. dubinsky (eds.), research in collegiate mathematics education iii (pp. 234–283). providence, ri: american mathematical society. harel, g., & sowder, l. (2007). toward a comprehensive perspective on proof. in f. lester (ed.), second handbook of research on mathematics teaching and learning (pp. 805–842). charlotte, nc: information age. hart, k. a. (1988). ratio and proportion. in j. hiebert & m. j. behr (eds.), number concepts and operations in the middle grades (pp. 198–219). mahwah, nj: erlbaum. confrey response commentary 
 journal of urban mathematics education vol. 3, no. 2 33 heid, m. k. (2010). where’s the math (in mathematics education research)? journal for research in mathematics education, 41, 102–103. hollebrands, k. (2002). the role of a dynamic software program for geometry in high school students’ developing understandings of geometric transformations. in d. mewborn (ed.), proceedings of the 24th annual meeting of the north american chapter of the international group for the psychology of mathematics education (pp. 695–706). columbus, oh: eric clearinghouse for science, mathematics, and environmental education. hufferd-ackles, k., fuson, k. c., & sherin, m. g. (2004). describing levels and components of a math-talk learning community. journal for research in mathematics education, 35, 81– 116. karplus, r., pulos, s., & stage, e. (1983). proportional reasoning of early adolescents. in r. lesh & m. landau (eds.), acquisition of mathematics concepts and processes (pp. 45–90). orlando, fl: academic press. kelly, a. e. (ed.). (2008). special issue on foundations for success: the final report of the national mathematics advisory panel [special issue]. educational researcher, 37(3). lamon, s. j. (1993). ratio and proportion: connecting content and children’s thinking. journal for research in mathematics education, 24, 41–61. lamon, s. j. (2007). rational numbers and proportional reasoning: toward a theoretical framework. in f. k. lester (ed.), second handbook of research on mathematics teaching and learning (pp. 629–668). charlotte, nc: information age. lesh, r., behr, m., & post, t. (1987). rational number relations and proportions. in c. janiver (ed.), problems of representations in the teaching and learning of mathematics (pp. 41– 58). hillsdale, nj: erlbaum. martin, d. b., gholson, m. l., & leonard, j. (2010). mathematics as gatekeeper: power and privilege in the production of knowledge. journal of urban mathematics education, 3(2), 12–24. retrieved from http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/95/51. mcmanus, s. (2008) a study of formative assessment and high stakes testing: issues of student efficacy and teacher views in the mathematics classroom. unpublished doctoral dissertation, north carolina state university, raliegh. mcnaught, m., tarr, j. e., & sears, r. (2010). conceptualizing and measuring fidelity of implementation of secondary mathematics textbooks: results of a three–year study. paper presented at the annual meeting of the american educational research association, denver, co. retrieved from http://cosmic.missouri.edu/aera10/. moses, r. p., & cobb., c. e. (2001). radical equations: math literacy and civil rights. boston: beacon press. national mathematics advisory panel (2008). foundations for success: the final report of the national mathematics advisory panel. washington, dc: u.s. department of education. nesher, p. (1980). the stereotyped nature of school word problems. for the learning of mathematics, 1, 41–48. noelting, g. (1980). the development of proportional reasoning and the ratio concept, part i – differentiation of stages. educational studies in mathematics, 11, 217–253. nptars (2005). qualified teachers for at-risk schools: a national imperative. washington, dc: national partnership for teaching in at-risk schools. steffe, l. p. (2002). a new hypothesis concerning children’s fractional knowledge. journal of mathematical behavior, 102, 1–41. thomas, s. (2010). examining the impact of the north carolina integrated mathematics (ncim) project professional development on two teachers’ instructional practices: a case study. unpublised doctoral dissertation, north carolina state university, raleigh. wilson, s. (2002). california dreaming: reforming mathematics education. new haven, ct: yale university press. microsoft word final truxaw et vol 7 no 2.doc journal of urban mathematics education december 2014, vol. 7, no. 2, pp. 21–30 ©jume. http://education.gsu.edu/jume mary p. truxaw is an associate professor of mathematics education in the department of curriculum and instruction in the neag school of education at the university of connecticut, 249 glenbrook road, unit 3033, storrs, ct 06269; email: mary.truxaw@uconn.edu. her research interests include discourse to enhance mathematical meaning making in linguistically diverse mathematics classrooms, pre-service and in-service mathematics teacher education, and mathematics teacher collaborative leadership. eliana d. rojas is an assistant professor of bilingual education in the department of curriculum and instruction in the neag school of education at the university of connecticut, 249 glenbrook road unit 3033, storrs connecticut 06269; email: eliana.rojas@uconn.edu. her research and practice concentrate on the ecology of processes involved in teaching and learning mathematics within socio-culturally and -linguistically diverse contexts. she is the director and pi of math-lead a professional development research grant that promotes configuring a transdisciplinary approach to the development of mathematics discourses.   public stories of mathematics educators challenges and affordances of learning mathematics in a second language mary p. truxaw university of connecticut eliana d. rojas university of connecticut prologue n this public story, we explore challenges and affordances of learning mathematics when the learner’s primary language is not the primary language of instruction. we, the authors, are mary truxaw, a predominantly monolingual (english, with limited spanish) mathematics educator and eliana rojas, a bilingual (spanish and english) bilingual/multicultural educator with expertise in mathematics education. we tell this story predominantly as a first-person narrative from mary’s point of view; however, cogenerative dialogue (tobin & roth, 2005) between us informs the story throughout. setting up the journey i (mary truxaw) came to this journey with a research background focusing on language as a mediator of meaning (vygotsky, 2002) in mathematics classrooms (e.g., truxaw & defranco, 2008). in recent years, i had recognized that linguistic diversity was—and still is—growing in u.s. schools (national clearinghouse for english language acquisition, 2011) and that there was evidence that u.s. schools, overall, are not adequately supporting english language learners (ells) (national center for education statistics [nces], 2012).1 further, i knew                                                                                                                 1 for example: the 2011 national assessment of educational progress (naep) results reported that nationally only 14% of fourth-grade english language learners (ells) (as compared to 40% i truxaw & rojas public stories journal of urban mathematics education vol. 7, no. 2 22 that, other than english, spanish is the language spoken most frequently in the united states (united states census bureau, 2013). i had all of these things in mind when i had the opportunity to take a sabbatical from my university faculty position in 2012. i decided to use my sabbatical to enhance my understanding of linguistically diverse mathematics classrooms and, in particular, classrooms where some or all of the students had spanish as a home language. i did not want to employ a deficit model for teaching mathematics to bilingual students, but i also recognized that, as an english-speaking person in an english-dominant culture, i could not fully appreciate the challenges or affordances involved in trying to learn mathematics in a second language. i wanted to combine formal research with a more personal journey and, therefore, set up experiences including an intensive immersion spanish language experience in guatemala (in order to develop greater spanish language fluency that would allow me to participate more actively in linguistically diverse schools for research and professional purposes). i also conducted observations and data collection in three types of mathematics classrooms: classes taught in spanish in guatemala, classes taught in spanish in the western united states, and classes taught in english using strategies designed to support ells. the larger study (truxaw, 2014) includes formal discourse analysis of dialogue within the linguistically diverse mathematics classrooms, but this article focuses on the story of the more personal journey to better understand challenges and affordances related to learning mathematics in a second language. as i was planning these experiences, i had ongoing conversations with my colleague and co-author, eliana rojas, who has expertise in both bilingual education and mathematics education. eliana provided professional support such as helping me to make connections with a member of the guatemalan ministry of education in order to set up observations in schools. she also provided scholarly support related to bilingual education and shared personal perspectives from the point of view of someone whose first language is spanish rather than english, who came to the united states as a graduate student, taught mathematics at college and high school levels, and, for years, has followed the educational experiences of mathematics teachers working with ells. eliana encouraged me to do a self-study (loughran, 2007) of my own experiences in guatemala as an englishspeaking mathematics educator immersing myself in a primarily spanishspeaking culture and its schools. related to this idea, she encouraged me to spend time in mathematics classrooms while my spanish language was still very much developmental. this experience helped me to appreciate what it might feel like to try to learn mathematics in a second language. to support this process, eliana                                                                                                                                                                                                                                                                                                               of non-ell fourth-grade students) were at or above proficient levels for mathematics and that nationally only 5% of eighth-grade students (as compared to 35% of non-ell eighth-grade students) were at or above proficient levels for mathematics (nces, 2012). truxaw & rojas public stories journal of urban mathematics education vol. 7, no. 2 23 generously offered to participate in “cogenerative dialogue” with me before, during, and after my time in guatemala (usually via email). cogenerative dialogue involves reflection where members refer to the same set of events and explanations are cogenerated, thus supporting reflection on experiences and co-generation of perspectives (tobin & roth, 2005). the focusing question for my self-study and for this public story follows: what can i, as a monolingual (english-speaking) mathematics educator, learn about relationships of language to mathematics education through an immersion experience where spanish is the language of mathematics instruction? the journey to help me make sense of this question, i observed mathematics classrooms, audio or video recorded classroom dialogue, took field notes, kept journals documenting and reflecting on classroom observations and experiences, and participated in cogenerative dialogue with eliana. our cogenerative dialogue reinforced our contention that language is critical to understanding mathematics. further, my experiences in spanish-language classrooms, coupled with cogenerative dialogue with eliana, helped me to gain greater appreciation of challenges and affordances inherent in learning mathematics in a second language. to illustrate themes and issues of this journey, i highlight observations and reflections related predominantly to one mathematics lesson in a second grade classroom in an all-girls public school in guatemala. there were 23 students, 1 teacher, and 6 practicantes (high school students practicing to be teachers; 2 men, 4 women). the students were seated in desks arranged in rows. as i entered the class, the students stood up and offered a choral greeting. the teacher introduced me and i briefly explained (in spanish) that i was from the united states and was interested in seeing their classroom. the teacher showed me to a desk in the back of the room where i could observe. all dialogue took place in spanish. the lesson seemed to be in progress when i entered as the teacher referred to representations showing circles, lines, and numbers on a whiteboard (see figure 1). the teacher asked students to come to the board to complete parts of the representation and to explain their work. i noticed that the verbal exchanges followed a typical triadic structure with the teacher initiating, the student responding, and the teacher evaluating (cazden, 2001). it seemed that the class was reviewing previously learned skills. i sat in the back of the room (with notebook and audio recorder), trying to make sense of the language, the representations, and the mathematics. truxaw & rojas public stories journal of urban mathematics education vol. 7, no. 2 24 figure 1 representation from second-grade classroom in guatemala. following are excerpts from my journal: even with some spanish language skills and, hopefully, math skills (for secondgrade math!)…i did not know what was happening most of the class. it became a puzzle for me as i copied every example, as best i could, to see if i could figure out what was happening my initial thoughts as i copied example after example from the board was that they were doing some form of decomposition of numbers. after reviewing several examples on the whiteboard, the students (with the help of the practicantes) were directed to cut out paper circles and strips. also, they each folded a paper in two parts to create a “mat” for placing the circles and strips. then, the teacher asked students to form numbers on their mats using the circles and strips—for example, “forma el número cuarenta y siete” (see figure 2). figure 2 example representation for “forma el número cuarenta y siete” [form the number 47]. truxaw & rojas public stories journal of urban mathematics education vol. 7, no. 2 25 as the students represented the numbers on their mats using the circles and strips, the practicantes walked around to correct and to help the students. for example, i heard a practicante count by 20s with a student (providing a hint that 20s were important). the following excerpt is from my journal: i kept thinking that i had figured it out, trying to guess the correct picture before looking at student work. sometimes i’d get it right and sometimes wrong. one thing that i didn’t figure out until the end was that the lines represented 5 times the place value. my scribbly notes include things like, “boy am i lost…i don’t understand… some kind of decomposition…i don’t understand why they are using 20s instead of 10s.” it wasn’t until the end of the class when homework was written on the board that i was able to figure out what they had been doing. the teacher wrote, “tarea: formar los siguientes números utilizando la numeración maya en su cuaderno” [homework: form the following numbers using the mayan numeration system in your notebook] “aha!” i said to myself. “mayan numbers!”…suddenly there was a context. there was a potential reason for using 20 as a place value—a different number system. having a context made a difference, but i still needed time to think (in english— my primary language), drawing on notes taken during class. eventually, i was able to figure out the number system,2 but not before recognizing the impact of language, representation, and context on my ability to learn and perform mathematically: i had to ask myself, if i, a university mathematics educator, was confused in a second-grade math class, how would a second-grade student in similar circumstances feel? cogenerative dialogue with eliana helped to further unpack the experiences. eliana suggested that “live” experiences (such as what i was doing) help to provide some sense of the challenges involved with learning and doing mathematics in a second language (for me, spanish) and/or within an unfamiliar culture (e.g., mayan number system). eliana asked me to think about how experiences trying to learn mathematics in such a context might impact students’ attitudes and “appre                                                                                                                 2 in figure 1, the numbers to the left show the place value. the top box has a place value of 20. one circle represents 1 x 20 = 20. the bottom box has a place value of 1. four circles represent 4 times the place value or 4 x 1 = 4. a line represents 5 times the place value. three lines represent 3 x 5 = 15. the total value is 20 + 4 + 15 = 39. truxaw & rojas public stories journal of urban mathematics education vol. 7, no. 2 26 ciation of mathematics” (national council of teachers of mathematics, 2000, p. 15) and, in turn, their self-efficacy3 (bandura, 1986). eliana reminded me that successful (or unsuccessful) learning experiences could impact attitudes, selfefficacy, and performance. eliana stressed that involvement in the mathematical activity requires a disposition where one “dares to do it” (atreverse in spanish). she also noted that lack of language to communicate one’s understanding might hinder attempts to solve a problem. to push the point a bit more, eliana asked me if i had been confronted with having to answer a mathematics question publicly in spanish. i had been asked to introduce myself in spanish, but that was a practiced recitation using everyday language. i had not been asked to answer any questions related to the mathematics lesson itself or to use academic language. with eliana’s question in mind, i imagined myself as a second-language student in this classroom. although the lesson included strategies identified as helpful for teaching second language learners— for example, use of visuals, hands-on activities, and interactions (e.g., echevarría, vogt, & short, 2010)—i could still picture myself as a student in this class trying to shrink down to avoid public participation. understanding, self-esteem, and performance would have been issues for me if i had been a student in this classroom trying to learn in a second language. this classroom experience uncovered several challenges: basic (i.e., second-grade) academic language in a second language, lack of opportunity to ask questions or make sense in my first language, unfamiliar representations, unfamiliar mathematical content/concepts, and cultural differences. in subsequent observations in this same classroom, i had some opportunity to see how decreasing one or more of the challenges could impact understanding of the classroom practices and the mathematics involved. for example, i found it easier to follow along during the next class that i observed. on that day, one of the practicantes was demonstrating adding oneand two-digit numbers using a base ten chart—a representation that was familiar to me. the vocabulary was in spanish (e.g., “centenas, decenas, y unidades” instead “hundreds, tens, and ones”), but the numerals and the setup of the place value chart were familiar (see figure 3).                                                                                                                 3 self-efficacy is defined as “people’s judgments of their capabilities to arrange and execute courses of action required to attain designated types of performances” (bandura, 1986, p. xii). self-efficacy impacts the things we do, our efforts toward them, and how long we persist in working out solutions to problems.   truxaw & rojas public stories journal of urban mathematics education vol. 7, no. 2 27 figure 3 place value chart used to represent addition. c = centenas (100s), d = decenas (10s), u = unidades (1s). in all of the classes observed, my limited spanish language proficiency made it difficult to follow verbal directions. however, when the representations were familiar and when i already had background knowledge, it made it more possible for me to perform familiar arithmetic processes. an excerpt from a related journal entry follows: the math here was much more familiar to me than the mayan numbers from the last observed class. familiarity with the same system certainly helps. also, having done the same math for 50+ years helps. in thinking about the mayan numbers from the last time, i’m wondering if students unfamiliar with base 10 number system might be just as lost (or more lost) than i was last time. the mayan system that i saw used just 2 symbols within each place value—1s and 5s. the base 10 system has 10 different digits. if a student didn’t have the numerical background, it could be very confusing. if, however, the student had learned to add numerals in spanish, then adding in english would be easy enough, i think—though the names for the numbers would take time to process. what would get lost, i think, is moving beyond computation to concepts. for example, if a student were asked in a second language to explain something related to zeros, they might be able to do so in their native language, but doing so in another language would be challenging. the how and why questions would be difficult. as i compared experiences across the lessons, i found myself agreeing with theories and research suggesting the importance of opportunities to reason and make sense of new concepts in one’s first language in order to support learning and engagement with a second language (e.g., alanís & rodríguez, 2008; cummins, 2000, 2005). truxaw & rojas public stories journal of urban mathematics education vol. 7, no. 2 28 implications and recommendations our personal experiences and cogenerative dialogue agree with research that suggests cognitive advantages of speaking more than one language including cognitive flexibility, better problem solving, higher order thinking skills, and linguistic resources for managing demands of group mathematics discussions (anhalt & rodriguez pérez, 2013; hakuta, 1986; howard, christian, & genesee, 2004; zahner & moschkovich, 2011). however, we also agree that there are compound challenges when using a developmental second language while trying to learn mathematics—especially for a student who is a beginning speaker and if the mathematics and/or cultural representations are unfamiliar (moshchkovich, 2007; rojas, 2005, 2010). examples of issues and recommendations related to learning mathematics in a second language that emerged from my experiences, eliana’s experiences, and our cogenerative dialogue include: • academic language is much more challenging than conversational language; it is more abstract, more contextualized, more specific, and more culturally determined. • working to understand even basic academic instructions in a second language is challenging and exhausting. • asking or answering meaningful questions in a second language is intimidating and difficult; one may choose not to publicly participate when learning in a second language. • one is likely to appear (and feel) less intelligent than one really is. • unfamiliar representations and contexts may cause confusion and present additional challenges. • second language learners may be ignored or called on less frequently than others in order to avoid communication challenges. encouragement to speak in either language is important. simple gestures like eye contact and smiling can make a difference. • visual representations can help, but are not sufficient—especially when the purposes of the representations are not clearly identified. • lack of opportunity to reason in one’s primary language can hinder sense making. • providing opportunities to reason in one’s primary language can support sense making. • when purposely facilitated by the instructor, wait time to support reasoning in one’s primary language can bolster self-esteem and support sense making. • personal experience as a second-language learner can be painful but enlightening. the issues and recommendations presented are consistent with existing literature (e.g., alanís & rodríguez, 2008; anhalt & rodríguez pérez, 2013; cummins, 2000, 2005; echevarría, et al., 2010; moschkovich, 2002, 2007, 2013; rojas, 2010; thomas & collier, 2002). however, they are more poignant when personally experienced. truxaw & rojas public stories journal of urban mathematics education vol. 7, no. 2 29 there are potential implications for mathematics teaching and learning in a second language. language is a mediator of meaning (vygotsky, 2002) that is fundamental for learning mathematics (moschkovich, 2002; truxaw & defranco, 2008). language comprehension may impact students’ attitudes toward mathematics and, as a result, their self-efficacy. because self-efficacy is associated with effort, persistence and resilience, it may, in turn, impact academic performance (bandura, 1986, 2001). considering relationships of learning, self-efficacy, and performance, teachers need to work to better understand both challenges and affordances inherent in trying to make sense of mathematics when language and/or representations and/or cultural contexts are unfamiliar. through this public story, we encourage mathematics teachers to put themselves in positions where they experience affordances and challenges related to learning mathematics in a second language. awareness is an important first step, but more needs to be done to make sound recommendations for supporting students whose 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(2013). k–8 teachers’ concerns about teaching latino/a students. journal of urban mathematics education, 6(2), 42–61. retrieved from http://edosprey.gsu.edu/ojs/index.php/jume/article/view/158 bandura, a. (1986). social foundations of thought and action: a social cognitive theory. englewood cliffs, nj: prentice-hall. bandura, a. (2001). social cognitive theory: an agentive perspective. annual review of psychology, 52, 1–26. cazden, c. b. (2001). classroom discourse: the language of teaching and learning (2nd ed.). portsmouth, united kingdom: heinemann. cummins, j. (2000). language, power and pedagogy: bilingual children in the crossfire. buffalo, ny: multilingual matters. cummins, j. 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(2014, april). lessons learned from linguistically diverse mathematics classrooms. paper presented at the 2014 annual meeting of the american educational research association, philadelphia, pa. retrieved from http://www.aera.net/publications/onlinepaperrepository/aeraonlinepaperrepository/tab id/12720/owner/230321/default.aspx truxaw, m. p., & defranco, t. c. (2008). mapping mathematics classroom discourse and its implications for models of teaching. journal for research in mathematics education, 39(5), 489–525. united states census bureau. (2013). language use in the united states: 2011. american community survey reports. retrieved from http://www.census.gov/prod/2013pubs/acs-22.pdf vygotsky, l. s. (2002). thought and language (13th ed.). cambridge, ma: mit press. zahner, w., & moschkovich, j. (2011). bilingual students using two languages during peer mathematics discussions: ¿qué significa? estudiantes bilingues usando dos idiomas en sus discusiones matemáticas: what does it mean? in k. téllez, j. moschkovich, & m. civil (eds.), latinos/as and mathematics education: research on learning and teaching in classrooms and communities (pp. 37–62). charlotte, nc: information age. microsoft word final mcqueen et al vol 03 no 2.doc journal of urban mathematics education december 2010, vol. 3, no. 2, pp. 47–56 ©jume. http://education.gsu.edu/jume mekyah q. mcqueen is a high school mathematics teacher at westlake high school, 2400 union road, sw atlanta, georgia 30331; email: mmcqueen1@student.gsu.edu. her research interests include the effects of problem-based electronic simulations anchored in situated learning theory on student mastery of mathematics standards. curtis v. goings is a fourth-year graduate student in the division of educational studies of emory university in atlanta, georgia; email cgoings@emory.edu. he has taught high school mathematics in the dekalb county school system. his research interests include informal ways of knowing and teaching mathematics, and the ways that african american elementary students construct mathematical competence. stanley f. h. shaheed is a high school mathematics teacher at cross keys high school, 1626 north druid hills road, ne, atlanta, ga 30319; e-mail: sshaheed1@student.gsu.edu. his research interests include ethnomathematics as well as multinational and multicultural students who have achieved local success in mathematics and the mathematical sciences. iman c. chahine is an assistant professor in the department of middle-secondary education and instructional technology in the college of education, at georgia state university, p.o. box 3978, atlanta, ga 30302; email: ichahine@gsu.edu. her research interests include ethnomathematics, situated cognition, problem solving in nonconventional settings, and multicultural mathematics. public stories of mathematics educators voices, echoes, and narratives: multidimensional experiences of three teachers immersed in ethnomathematical encounters in morocco mekyah q. mcqueen westlake high school stanley f. h. shaheed cross keys high school curtis v. goings emory university iman c. chahine georgia state university or reasons that perhaps only phenomenological methodologies can illustrate clearly and precisely, reflecting upon our sensitively subjective lived experiences in a foreign culture is nearly an impossible endeavor. to attempt an “honest” description for any particular event we encountered while being acculturated into a rich, vibrant context, falling victim to our preconceived notions and encountering the constraints of textualizing the meanings of our experiences seem inevitable. in describing lived experience, van manen (1997), quoting dilthey, writes: a lived experience does not confront me as something perceived or represented; it is not given to me, but the reality of lived experience is there-for-me because i have reflexive awareness of it, because i possess it immediately as belonging to me in some sense. only in thought does it become objective. (p. 35) f mcqueen et al. public stories journal of urban mathematics education vol. 3, no. 2 48 the following narratives (i.e., storytelling) represent our best attempt to describe the individual and collective lived experiences of a group of mathematics educators—three mathematics education graduate students (and their professor)— engaged in an ethnomathematical cultural immersion experience. here, we chose storytelling as a somewhat natural method to recount our lived experiences and to create a reasonable order out of the many vivid encounters we witnessed while culturally immersed in morocco. our narratives emerged in attempt to grasp subjective essences of our collective experiences as we embarked on making meaning of our exposure to ethnomathematical practices. while narrating each of our lived experiences separately, our collective reflections are meant to be teleological—that is, a modest lexicon of purposeful achievements that consolidate to give a single perspective of what it meant to be acculturated into the moroccan culture. our cultural immersion experience was part of a graduate-level, mathematics education course that we were enrolled in during the summer semester 2010 at georgia state university, atlanta. the course provided an introduction to ethnomathematics, which combines aspects of mathematics, mathematics education, sociology, psychology, anthropology, and linguistics (d’ambrosio, 2001). the cultural immersion component of the course included a site visit to the city of fez in morocco to conduct field observations to explore the ethnomathematical ideas that transpire in the daily practices of craftspeople and practitioners on the streets of the old city. fez, the oldest city in morocco, differs from other cities by its divided metropolis, which includes new fez (fez-el-djedida or ville nouvelle) and old fez (fez-el-bali or medina). new fez was built in the 14th century, while old fez was founded in the 9th century by the first muslim dynasty to rule morocco, the idrissids. most people of fez continue to live in the medina, fez-el-bali instead of moving to the ville nouvelle, which is more modern and urban. within the medina is an awe-inspiring marketplace—a maze of narrowing, cobblestoned streets lined with small shops and street merchants—selling anything from fresh moroccan spices to hand-made cedar wood mirror frames. in the medina, we were allowed to choose an ethnomathematical context as an observation setting such as a tile factory, metalwork gallery, embroidery atelier, or a carpentry shop in which to conduct our observations. our journeys began as we witnessed breaths of vibrant cultural nuances that flooded our senses and rendered visible deep personal experiences. three narratives we’re in africa! it has been several weeks since i returned from my first trip to the land of my genetic and spiritual ancestors. when i first learned of an opportunity to visit mcqueen et al. public stories journal of urban mathematics education vol. 3, no. 2 49 africa and participate in an ethnographic study abroad that investigated ethnomathematics, i knew that such a prospect does not happen often for a high school mathematics teacher. as far back as i can remember my interests have been global and international. nonetheless, i never became aware of a program that would satisfy my academic, cultural, historical, ancestral, and spiritual needs while attending classes during my summer break. for me, this trip became a high priority on my to-do-list. as an african american, words are inadequate to describe the thrill and the sense of privilege to land and walk on the continent of my fore-parents. i consistently reminded my classmates—as i reminded myself—with a steady periodic refrain of: “we’re in africa. we’re in africa. we’re in africa!” although my words cannot adequately describe this indelible experience, my senses were flooded with the architecture, sounds, and smells of a place that fulfilled a longing deep within my soul. my challenge was to stay focused and remind myself that this trip was about conducting research that would be meaningful as well as purposeful to mathematics education and ethnomathematics in particular. the real coursework data collection did not start until we reached fez; our flight landed in casablanca. while in casablanca, i relaxed a bit and simply became a sponge and absorbed. i was able to pray in the largest mosque that i have ever visited, reflecting on the fact that i was making prayer in africa. i could touch and read some of the calligraphy decorating the tile work of masjid hassan el thani. after misplacing and recovering my digital camera, an experience that reminded me of the core goodness that still exists in human beings, i was ready to settle down, steady my focus, and prepare myself for fez where the physical and mental work would be done. good people were the norm in fez. the family that hosted us, the professional tour guide, our awesome driver, and the wonderful family that made us feel like family as we visited, ate, prayed, and talked, all played an essential role in enhancing my experience as well as enriching my spoken arabic vocabulary. the consecutive days of dry, mosquito-free heat were priceless. the shards of ice in the sidi ali bottled water actually converted a devout juice drinker like myself to the tasteless refreshing benefits of pure african water. as a muslim, the seamless transition from tile work (the site of my ethnomathematics observations) to prayer and returning to work is a freedom that mere words cannot convey, particularly when compared to experiences in the united states where explanation and justification often become linked to performance of the afternoon prayers. in summary, i would encourage any and all students to participate in a study abroad experience. subsequent writings will delve more into the mathematics and the ethnomathematical aspects of my ethnographic study. for this narrative, i will simply state, as i did on all of my post cards sent to the united states, i am so glad that my first trip to africa was in morocco. mcqueen et al. public stories journal of urban mathematics education vol. 3, no. 2 50 found in translation as i reflect upon my summer experience, i sometimes find myself tempering my retellings with calm and reserve for fear of being labeled as “weird” or “strange” by my listeners. even my most impassioned communications fall fumbled and inadequate. the dilemma for me is that, while i believe that any expression—whether in photographs, conversations, or pages—is in a number of ways inadequate, i cannot escape the compulsion to share my experience in fez, morocco. although i had travelled internationally, i had never visited morocco or any other african country. nor did i know a great deal about the country or its cultures. my ignorance fueled some level of apprehension and anxiety. how would i manage in a foreign setting? the answer would be provided within a few short days. during our first tour in the medina, we witnessed artisans and craftspeople diligently and meticulously working in their ateliers. after the tour, i narrowed my focus on a small carpentry shop and instantly decided to conduct all my observations there. the selection was interesting in that i could hardly communicate with my hosts. the two master carpenters and their five young apprentices spoke arabic and, to some degree, french. i speak no arabic. additionally, the only french to which i had been exposed is that which fulfilled my foreign language requirement in high school and in college over twenty years ago—and none since. despite this obstacle, i decided that i would observe the activities in the carpentry shop of ahmad and hisham. this decision was influenced by the mastery in their (hand) craftsmanship. more so, it was inspired by their hospitality. i sensed that they were open to my presence and that they would attempt to provide access to how they were thinking and conceptualizing. indeed, my hosts were generous in their attempts to share whatever they could of their craft and skill, and in their embracing me as a guest. yet, i felt what might be every ethnographer’s consideration: “my presence is an imposition.” without the tools of common language, i became particularly vulnerable and grew slightly uneasy. would they rescind their invitation? would they regret their hospitality? these concerns were alleviated as i was invited to enjoy and participate in their humanity. i was given space and offered the only seat in an already crowded setting. as i witnessed a team—master carpenters and their apprentices—work, their concentration was palpable. despite their focus on their work, i never was ignored. through the broken french that i would try to revive, we spoke of poverty, income, occupation, family, and education. and when language failed, patient gestures often accompanied and sometimes supplanted our words. i did not forget my primary purpose of examining the mathematical ideas that emerged or were accessed in the shop. those conversations followed more of an expected and traditional trajectory. mcqueen et al. public stories journal of urban mathematics education vol. 3, no. 2 51 although a language barrier muted a number of verbal expressions, our attempts to reach out, to inquire, to engage, and to share were never attenuated. on one day, master carpenters ahmad and hisham bought and shared their meal of fish, bread, sauce, and melons with me before any of the apprentices were allowed food. we ate and laughed. i was no longer an intruder—at least not at this moment. and this moment was among the many that i would remember most fondly. it would be a gesture that i was resolved to reciprocate. a couple of days later, i would provide fruit and water to the people at the shop. this gesture was neither an obligation nor a burden, but a commitment to a community that had opened itself to me. that commitment and bonding continue in my trajectory as a mathematics educator and as a mathematics education doctoral student. in the aftermath of this experience, i have become more sensitive to the power of difference and its influence upon students and their teachers in the mathematics classrooms in the united states. the conspicuous phenotypic differences of complexion, and hair texture contributed to my feelings of apprehension. with every distinctly american utterance and through my african american presence, i was ever mindful of the portraits—both as american and as african american—that i presented. differences in ethnicity, ideology, and religion became pronounced as cultural markers became highlighted. i tried to be careful of the characterizations that my actions, gestures, and speech might reify. i also came to consider the characterizations that i had unconsciously imposed on my surroundings. these are among the issues that mathematics educators must contemplate, for these considerations present and unmask significant implications. implicit in my attempts to negotiate what i perceived to be stereotypes of me were reflections of my own preconceptions about my new environment and its inhabitants. in other words, the fact that i tried to manage my behavior and speech while in a foreign culture implies that i hold certain conceptualizations of how one may perceive my actions. this type of dilemma in some ways parallels the experiences of students and educators who have dissimilar cultural heritages, yet perhaps comparable goals for student outcomes. in the midst of recognizing such discontinuities, both teachers and students risk a paralyzing fear of making mistakes and the vulnerability of having those mistakes exposed. although many of these mistakes occur because of sincere efforts or the lack of them to bridge communication between persons or groups of people, the far greater error lies in either side remaining comfortably immobile. everything i know, i learned from a student since i became a teacher, people often ask me if i will pursue a doctoral degree. i never thought about the answer to this question; i simply responded with “no.” when asked “why not?” my response has always been, “because i’m al mcqueen et al. public stories journal of urban mathematics education vol. 3, no. 2 52 lergic to research,” and with a chuckle and a wink i’m able to end the “dr. mcqueen” conversation right there. until last summer, i was quite comfortable being disillusioned with the subjectivity of writing, and i had not yet been inspired by a topic i deemed worthy of the work that any substantive research commands. then i registered for a class that was dedicated to the discipline of ethnomathematics and during the course of the semester i became inspired in ways i truly never imagined. i learned of the ethnomathematics course through the advertisement of an opportunity to spend a week in fez, morocco, which was all the information i needed to register. even after i realized that those who took the trip would be conducting research, the excitement i felt about the chance to travel to africa fully eclipsed any misgivings i had toward participating in research. more than anything, i thought about the tremendous opportunity to set foot on african soil— an experience that many african americans wish for but too often never have. traveling to the continent of my ancestors has always been a bullet item on my bucket list, yet i was unsure if i would ever be emotionally prepared to make such a personally profound journey. part of me saw the trip to morocco as a way to visit the motherland without risking the emotional breakdown that a trip to senegal or ghana may trigger. to my surprise, the experiences i had in morocco left impressions on my mind and spirit that reshaped the passion and level of dedication i have to the practice of teaching. a 13-hour layover in a foreign country can seem tedious and overwhelming. fill the time with rich and impromptu teaching and learning experiences and you wind up wondering where the time went. one such experience took place as we were in line waiting for the commuter plane that would take us from casablanca to fez. my colleagues and i had spent several hours basking in all that casablanca had to offer. we ate lunch less than 100 feet from the atlantic ocean, spent time at the breath-taking masjid hassan el thani, and bargained with merchants along the city streets. once we returned to the airport, a colleague and i had a passionate debate about the current state of task-based mathematics standards we are now mandated to follow, a conversation that i continue to ponder to this day. the four of us (my two classmates, our professor, and me) were so engrossed in the discourse that we barely heard the boarding call. nevertheless, we eventually did respond to the boarding call, gathering our luggage and filing into the line with the other passengers bound for fez. as we waited, to the left of me was a group of adolescent boys, ages 10–13, who were sitting on the floor playing a card game. the boys and the game immediately caught my attention and before i knew it i had put down all of my belongings and approached the circle to watch. within minutes, i was asking questions about card choice and strategy and after a few questions they invited me to play. i was a bit reluctant to join initially because i was unsure of the gender dynamic at mcqueen et al. public stories journal of urban mathematics education vol. 3, no. 2 53 play. i was afraid that my presence as an adult woman would be offensive to any parents of the boys whose circle i had joined. i noticed an older man close by who watched my interaction with the boys. eventually, he made his presence known as the father to one of the boys and later came to serve as a translator only when there was a communication barrier between the group playing cards and me. learning how to play a card game with a group of young moroccan boys was an extremely valuable experience for several reasons. the random experience took place at the heels of a well-meaning, yet rather intense intellectual debate about the current state of mathematics curriculum in georgia and its impact on students. this incident ignited within me a predisposition to believe that we all possess the innate ability to learn and to teach; that the teaching and learning of children occur best when adults meet them where they are and are not afraid to be taught something themselves. one of the boys in particular absolutely came to life when i asked if he could show me how to play the game. although he was a little hesitant because of the language barrier, he took great care and pride in making sure i understood each play and constantly gave me praise when i made the right move; by watching him, the other boys began to engage with me the same way. it struck a chord with me that allowing the boys the opportunity to become the “teacher” could have been a very empowering experience for them. i was awfully touched and encouraged by the experience at the casablanca airport because it reminded me that teaching and learning might take place anywhere, with anyone. the notion that being willing to engage in a teachable moment is perhaps the only prerequisite for learning to take place, and that the role of teacher and student can be interchangeable within a learning experience resonated with me on the flight to fez. i realized that i was becoming more inspired by what my stay in fez had to offer and shockingly found myself anticipating the work in research that awaited me in the medina. while in fez, i chose to do my ethnomathematical field experience in two sites: a bronze shop and an embroidery atelier that is housed inside an embroidery school. the maison du bronze, a one-room shop, larger than most in the medina, displayed and sold finely carved bronze, silver, and wood crafts. the walls of maison du bronze were covered from floor to ceiling with metal artwork, bronze and silver mirrors, and plates of all sizes. most of these items were made of bronze, while some, such as the tea kettles, were made using nonshoor, a blend of copper, silver, and tin. the two show tables erected in the middle of the shop held items made of bronze and patina, a substance used by the artisans to give the items a more antiquated look. behind the tables was an open case filled with berber daggers and berber syouf (swords). the artisans of maison du bronze were two male arabs. during the observation, both bronze artisans were working on the same large bronze plate in a small corner of the workshop. the artifacts used by both artisans were a matara mcqueen et al. public stories journal of urban mathematics education vol. 3, no. 2 54 kat takleedia (hammer), needle, and chisel. the needle was to create filikian work, or designs made by a series of tiny dots; the chisel was used to make groove designs. both artisans described that there were two types of patterns usually created for bronze work inspired by either berber or arab designs. later, they explained that the arab designs represent nature, as they take on a floral design full of connected curves, many deriving from the arab henna designs. the berber motifs are more geometric in nature and highly symmetric. a second site that i observed was the only embroidery school in the medina, a two-story space with the showroom and atelier on the bottom floor and the school on the second floor. the showroom was visible upon entrance and displayed neatly folded and hand-made tablecloths, napkins, handkerchiefs, and pillows. to the side of the showroom was the artisans’ workspace where more work was displayed on the walls, a large glass table in the middle of the workspace, two wall shelves holding inventory, and an l-shaped bench used by the women who worked on the embroidery. both embroidery artisans, smehan and kinza, began embroidery work at the age of seven. sitting against a small pillow on a cloth-covered bench, both were working on the border of a large tablecloth using a murma (circular, wooden hoop used to hold the white cotton cloth in place), a needle, and ketha demse (thread made of cactus leaves). kinza was creating the main motif of her tablecloth, a design originating from fez called zushlena. the expertise demonstrated by kinza and smehan, neither of who finished high school, was a testament to the vocational learning and knowledge that are celebrated in other countries. the concepts of symmetry and geometric dilations were continuously stressed when describing the nature of the designs depicted in the bronze and embroidery work. the intricate and highly graphic and geometric designs executed by kinza and smehan were phenomenal with miniscule stitches, only a few millimeters in size. young girls, who are taught the art of embroidery at the school, are taught to count each thread. this is an extremely time-consuming technique, which requires mathematical precision and much concentration and patience, as there is no reverse side to the design. the vivid learning experiences and phenomena i encountered and lived during my time in morocco not only resonate with me as fond memories but also have ignited a new passion. i returned to the united states an advocate for authentic situated mathematics learning experiences having observed and witnessed firsthand rich and insightful results. instructional activities designed so that students are actively learning within an authentic situation are most ideal to ensure that concept retention and mastery take place. i will be forever grateful to my gracious young teachers at the airport in casablanca—kinza, smehan, and abdul— for exposing me to such authentic situated learning experiences, as i am henceforth committed to create as many as i can for my students. mcqueen et al. public stories journal of urban mathematics education vol. 3, no. 2 55 collective closing thoughts our lived experiences yielded three important insights into the ethnomathematical practices of the craftspersons and artisans in the medina. first, there is a condensed amount of tacit knowledge gained and used by each artisan. tacit knowledge involves learning a skill but not in a way that can be written down. each artisan described that they ascertain the knowledge needed for their craft through their lived personal experience. as there are no patterns used, and very little direct instruction given, there is a tacit aspect of their mathematical knowledge as well. with tacit knowledge, people are not often aware of the knowledge they possess or how this knowledge can be valuable to others. second, the cultural connection proliferating in the craftwork is undeniable. in most art forms, there is a clear influence from intimate aspects of the moroccan culture on the motifs and patterns that thrive in the design of most of the handmade crafts. in a berber motif, any sign has a magical or prophylactic meaning. the women, who would use them, originally when weaving rugs, to protect their feminine mores that they essentialize as being distinct and uncontrolled by masculine polity, always conceal the real meaning. these secrets are transmitted from the mother to the daughter over hundreds of generations. within berber motifs, the art of making symmetrical combination from simple forms emerges to personify the “world” that is depicted in whatever is created in a harmoniously balanced order. finally, the connection to and motivation drawn from the craftspersons’ spiritual practices are noteworthy. many of the same designs used in crafts are also found on walls and ceilings of the mosques within the city, symbolizing the continuity of faith and its appearance in the everyday lives of muslims. moreover, the way in which the artisans appeared effortlessly excellent in their craftwork can be explicably tied to the never-ending desire to perfect their work as an act of sincere obedience and unconditional worship of god. as we contemplate these personal reflections, the mathematics education community wrestles with its own perceptions and conceptions. such understandings are linked to contested issues of what counts as mathematics, mathematical activity, and mathematical ideas—along with who decides. as western educators, we are knowingly confined by mandates and guidelines intended to aid us in adequately exposing our students to quality mathematics teaching and learning experiences. many of us cry out for more time to be able to introduce our pupils to alternate theories and practices, with hopes that even a glimpse may ignite a spark within some of our struggling students or fuel the flame carried by those already excelling. as we reflect on our collective experience, we are compelled to contemplate how each aspect has seared an impression on our teaching practices and what we deem as meaningful mathematical learning. in addition, we can allow mcqueen et al. public stories journal of urban mathematics education vol. 3, no. 2 56 each separate and specific aspect of our encounters in morocco to guide us in how we motivate our students in constructing their own sense-making experiences in the classrooms. whether it is the risk-taking journey to new soil, the patience and perseverance needed to communicate through language barriers, or a commitment to unparalleled work ethic and pride, we are now the beholders of many precious nuggets that can be shared with our students as a means to help them become better learners. experiencing the ethnomathematics of the craftspeople consecrated in context allowed us to fathom the value and richness of knowledge as gestalt inextricably weaved into people’s everyday life, fervently evolving and collectively maintained in the cause of survival. we are evermore committed to finding and creating as many teachable moments as possible through which to share our vivid encounters with such knowledge and constantly look forward to our opportunities to do so in the future. references d’ambrosio, u. (2001). ethnomathematics: link between traditions and modernity. rotterdam, the netherlands: sense. van manen, m. (1997). researching lived experience: human science for an action sensitive pedagogy. albany, ny: state university of new york press. jumesubmission.docx journal of urban mathematics education july 2013, vol. 6, no. 1, pp. 19–27 ©jume. http://education.gsu.edu/jume joel amidon is an assistant professor in the department of teacher education at the university of mississippi, p.o. box 1848, university, ms 38677; email: jcamidon@olemiss.edu. his research interests include advancing theories of teaching and learning and the improvement of mathematics pedagogy to address issues of equity and diversity. teaching mathematics as agape: responding to oppression with unconditional love joel amidon university of mississippi in this essay, encouraged by the critical examination of mathematics education and mathematics teacher education at the privilege and oppression in the mathematics preparation of teacher educators conference, the author asks the question: what do i do from a position of power and privilege to interrupt oppression and enable everyone the opportunity and expectation of success in mathematics and life? the author proposes a response with agape (pronounced ägäpā), or unconditional love. starting with the question what would it mean to teach mathematics as an act of unconditional love? the author theorizes an ideal relationship between students and mathematics that is functional, communal, critical, and inspirational, generated from wanting to teach mathematics as agape. keywords: equity pedagogy, mathematics education y decision to pursue a career in mathematics education was immediately affirmed by the images of all my white, middle-class, male, mathematics teachers who looked just like me, even down to the thick-rimmed glasses, and the occasional use of a pocket protector. given that inequity exists in the world, there is no denying that i am sitting on the side of privilege. in response to this realization and encouraged by the critical examination of mathematics education and mathematics teacher education at the privilege and oppression in the mathematics preparation of teacher educators (prompte 1 ) conference, i ask the question: what do i do from this position of power and privilege as a mathematics teacher, researcher, and teacher educator to interrupt oppression and enable everyone the opportunity and expectation of success in mathematics and in life? in this essay, i propose to respond with agape (pronounced ägäpā), or unconditional love. i theorize an ideal relationship between students and mathematics that is functional, communal, critical, and inspirational, starting with the question: what would it mean to teach mathematics as an act of unconditional love? 1 privilege and oppression in the mathematics preparation of teacher educators (prompte) conference (funded by create for stem institute through the lappan-phillips-fitzgerald cmp 2 innovation grant program), michigan state university, battle creek, mi, october 2012. any opinions, findings, and conclusions or recommendations expressed herein are those of the authors and do not necessarily reflect the views of the funding agency. m amidon teaching mathematics as agape stinson, d. w., & spencer, j. a. (eds.). (2013). privilege and oppression in the mathematics preparation of teacher educators [special issue]. journal of urban mathematics education, 6(1). 20 agape some may ask if agape is appropriate for a field such as mathematics education, or education in general. in response, i quote paulo freire (1998): “we must dare in the full sense of the word, to speak of love without the fear of being called ridiculous, mawkish, or unscientific, if not antiscientific” (p. 3). agape is one of the many greek words for love, more specifically “unconditional love” (wivestad, 2008, p. 307), which can be better understood by examining another greek word for love, eros. eros is “a love of the worthy” and “desires to possess” (morris, 1981, p. 128). agape is in direct opposition to eros, it is a love that is “given irrespective of merit” and “seeks to give” (morris, 1981, p. 128). turning back to the guiding question of this essay—what would it mean to teach mathematics as an act of unconditional love?—would imply the teacher “seeks to give” knowledge of mathematics. ladson-billings and tate (1995) describe mathematics as “intellectual property” all students should have access to, but obviously do not. this notion of “intellectual property” can be likened to callan’s (1995) notion of “common education,” which he defines as “a range of educational outcomes—virtues, abilities, different kinds of knowledge—as desirable for all members of the society” (p. 252). using this definition, it would be accurate to label mathematics as “common property,” something “desirable for all members of the society” (p. 252). denying students access to the common property of mathematics has been equated with being denying access to society—mathematics is a “gatekeeper for citizenship” (moses & cobb, 2001, p. 14). this denial leads to the question what does it mean to gain access, or learn mathematics? learning as building relationship the traditional mathematics classroom, described by palmer (1998) as “the dominant model of truth-knowing and truth-telling” (p. 100), is where students are not in a direct relationship with mathematics, but are merely passive receivers of information from the teacher as expert. it is this model that dominates the majority of classrooms and is where the teacher controls access to the common property of mathematics. mathematics, in the form of procedures and examples, is distributed to the students in static, regulated doses dictated by what is on the next page of the textbook and students are passive receptacles for such doses (see freire, 1970/2000, for a detailed critique of this model of education, which he calls the “banking concept” of education). thus, the working assumption of this essay is to reject “the dominant model” and presume the classroom as a relational space with the key players as the students, the teacher, and mathematics. a classroom as relational space is reflected in lampert’s (2001) description of the “problem space” of teaching, and what lave and wenger (1991) describe amidon teaching mathematics as agape stinson, d. w., & spencer, j. a. (eds.). (2013). privilege and oppression in the mathematics preparation of teacher educators [special issue]. journal of urban mathematics education, 6(1). 21 as a “community of practice.” a comparable perspective is what palmer (1998) describes as a “subject-centered” classroom where the teacher and students are in a direct relationship with mathematics and each other. learning within this space would be described as moving more central within the community of practice by strengthening the relationship between the subject and the knower (lampert, 2001; palmer, 1998). this model suggests that the teacher and students are constantly interacting with mathematics as an entity to relate to and understand, and not just a set of examples and procedures. this relationship with mathematics is thus a product of the processes and products (udvari-solner, villa, & thousand, 2005), which a teacher initiates in the problem space of teaching (lampert, 2001). teaching mathematics as agape in this section, i attempt to provide a probable answer to the driving question using the lens of agape, and the idea that learning mathematics is about developing a relationship with mathematics. in my search for a response, several equity pedagogies were reviewed and organized into four emerging facets of the ideal relationship that teaching mathematics as agape might promote: functional, communal, critical, and inspirational. relationship is functional to proclaim to teach mathematics as agape implies promoting a relationship between students and mathematics that is functional, meaning students can work with mathematics to achieve success as defined by society. this “success” can be equated to scoring well on high-stakes tests (gutstein, 2006), graduating from high school, being accepted to college, and/or being hired in a mathematically related profession (frankenstein, 1990; gutstein, 2006). the label functional is borrowed from north’s (2009) investigation of social justice teachers, where she defines “functional literacy” as the competencies that students need to access the opportunities of society. gutstein (2006) also defines functional literacy as “the various competencies needed to function appropriately within a given society” (p. 5). in addition, gutstein describes “classical knowledge” as “specific competencies students need to pass gate keeping tests and to pursue advanced mathematics and mathematically related careers” (p. 203). other labels for this facet are “dominant mathematics” (gutiérrez, 2007) as “aligning with society” (p. 40), “math literacy” (moses & cobb, 2001), and “academic achievement” (ladson-billings, 1994, 1995). what these labels have in common is the demand that an approach to teach mathematics must facilitate students’ success as society has defined it. the teacher does not have to agree with this definition, but as ladson-billings (1995) states: “students must achieve. no amidon teaching mathematics as agape stinson, d. w., & spencer, j. a. (eds.). (2013). privilege and oppression in the mathematics preparation of teacher educators [special issue]. journal of urban mathematics education, 6(1). 22 theory of pedagogy can escape this reality” (p. 475). but what if society’s definition of success does not align with the student’s definition of success? for instance, carraher, carraher, and schliemann’s (1985) classic study describes children from the streets of recife, brazil who were able to accurately execute computations in the streets selling fruit, but when asked to do similar mathematical practices in a school environment they were unable to demonstrate the same proficiency. the children’s relationship with mathematics in the streets could be described as functional, but in the classroom it was not functional. one solution to this problem would be to do as lampert (2001) suggests and explicitly teach them how to participate in the classroom environment. alternatively, there can be movement toward connecting the students’ ways of participating in the world with how they participate within the mathematics classroom. to teach mathematics as agape, it would be logical to embrace who the student is and the communities and cultures that they participate in as assets for instruction, and not deficits to overcome. this perspective calls for another facet to the relationship between students and mathematics. relationship is communal to proclaim to teach mathematics as agape implies promoting a relationship between students and mathematics that is communal, meaning students can work with mathematics in the contexts and through the practices of the students and their communities. this facet goes far beyond finding engaging contexts for “word problems” but, as paris (2012) describes with respect to culturally sustaining pedagogy, supporting students in “sustaining the cultural and linguistic competence of their communities while simultaneously offering access to dominant cultural competence” (p. 95). the belief that the classroom should incorporate the day-to-day lives of students to bring relevance to educational objectives and activities can be found in several places within the literature (e.g., civil, 2007; emdin, 2013, ladsonbillings, 1994; paris, 2012; udvari-solner, villa, & thousand, 2005). as a part of culturally relevant pedagogy, ladson-billings (1995) describes “cultural competence” as creating a classroom environment where a student can achieve academically without having to sacrifice their cultural identity. gutiérrez’s (2007) definition of equity calls for a coordination of “efforts to get marginalized students to identify with “dominant mathematics” (p. 38). a similar notion can be found in teaching mathematics for social justice as promoting “community knowledge” that is defined as “knowledge of … community life in all its complexity, and of perspectives and interpretations of the world” (gutstein, 2006, p. 201; also see 2003). this communal knowledge also can be equated with acknowledging and employing the “funds of knowledge” that exist in the community, and to use this amidon teaching mathematics as agape stinson, d. w., & spencer, j. a. (eds.). (2013). privilege and oppression in the mathematics preparation of teacher educators [special issue]. journal of urban mathematics education, 6(1). 23 knowledge or competence as context and motivation for facilitating the use and development of other types of knowledge (gonzales, moll, & amanti, 2005). to facilitate a relationship between students and mathematics that is communal would not be limited to utilizing community contexts, but, as alluded to, would include connecting the students’ ways of participating in the world with valued ways of participating in the classroom community of practice. boaler (2007) compared two schools with different pedagogical approaches, one used an “open-ended, project-based approach” and the other used a “traditional, procedural approach” (p. 29). when comparing outcomes from the two schools, the female students scored significantly lower than the male students at the school with the traditional, procedural approach. in addition, female students at the school using the open-ended, project-based approach were “significantly more positive and confident” (p. 29) than the female students at the other school. such results could suggest that using the traditional, procedural approach (equated with the commonly understood mathematics classroom) would not be teaching mathematics as agape. if teachers are not actively looking for ways to incorporate students’ means of participation into the set of valued classroom practices, then they are ignoring how students participate and/or may perceive students as not participating. the literature describes segments of the student population (specifically african american students) as disproportionally represented in the special education population, segregated from the regular education classroom, and further denied access to the common property of mathematics (blanchett, 2006). this denial is in direct opposition to the work of emdin (2013) who names hip-hop cultural practices that are in direct alignment with the valued practice of scientific argumentation. this contradiction further necessitates that teaching practice be shaped to facilitate a communal relationship between students and mathematics to counteract documented inequities and sustain cultural practices (paris, 2012). it would be a significant accomplishment to facilitate a relationship between all students and mathematics that is functional and communal. but if all that is accomplished is more people are inserted into a system that produces inequities, then we are just perpetuating the current system (apple, 1992) or doing something that “serves the reproductive purposes (i.e., maintaining the status quo) of the dominant interests in society” (gutstein, 2006, p. 5). perpetuating a system that marginalizes people would fall short of teaching mathematics as an act of unconditional love that seeks to give the common property of mathematics to all students. the system needs to be changed, which calls for another facet to the relationship between students and mathematics. relationship is critical to proclaim to teach mathematics as agape implies promoting a relationship between students and mathematics that is critical, meaning students can work with math amidon teaching mathematics as agape stinson, d. w., & spencer, j. a. (eds.). (2013). privilege and oppression in the mathematics preparation of teacher educators [special issue]. journal of urban mathematics education, 6(1). 24 ematics to analyze and question the world. this facet of the relationship between students and mathematics suggests that nothing escapes the assessment, including the students, the mathematics, and the system that they are a part of (gutiérrez, 2007; martin, 2003). using mathematics to question and analyze the world is found in several places within the literature. earlier, gutiérrez’s (2007) definition of “dominant mathematics” was presented as “aligning with society” (p. 40). the counterpart that aligns with this facet of the relationship between students and mathematics is “critical mathematics,” which is about “exposing and challenging society” (p. 40). the “critical” component of culturally relevant pedagogy is achieved when teachers prompt students to “recognize, understand, and critique current social inequities” (ladson-billings, 1995, p. 476). similarly, north (2009) would describe this critical component as “critical literacy,” and gutstein (2003, 2006) as “critical knowledge,” a component of teaching mathematics for social justice, which is “knowledge of how to read the world with mathematics … knowledge beyond mathematics that students need to understand their sociopolitical context” (2006, pp. 202–203). for example, wager (2010) presented the story of caroline, a teacher concerned with teaching mathematics more equitably, who had reservations about teaching mathematics with a critical and/or social justice context. caroline stated: “‘i think that the thing about not presenting our world as a big problem is so important’” (p. 88). caroline ties this statement to her own practice by relating the story of a student who began crying after completing a project on global warming (wager, 2008). was the global warming project strengthening the relationship between this particular student and mathematics? will students who work with mathematics to expose the problems of the world continue to work with mathematics? some students who are confronted daily by the problems of the world may find the opportunity to work with mathematics to understand their own struggles liberating. however, the account from caroline’s teaching practice (i.e., the student who cried after the global warming project) suggests the opposite for students who may be sheltered from such problems. either way, promoting a relationship between students and mathematics that is critical needs to be balanced with something. the aim is not to generate students (or teachers) who are disillusioned or frightened by the inequities and problems of the world, but rather students (and teachers) who are confident that change can occur, and to equip them to be instruments for such change. so, given the brokenness of the world, and the litany of problems that can be identified, what is the means for fueling the effort to keep moving forward? this calls for a final facet to the relationship between students and mathematics. amidon teaching mathematics as agape stinson, d. w., & spencer, j. a. (eds.). (2013). privilege and oppression in the mathematics preparation of teacher educators [special issue]. journal of urban mathematics education, 6(1). 25 relationship is inspirational to teach mathematics as agape implies promoting a relationship between students and mathematics that is inspirational, meaning students can work with mathematics to vision and move toward a better world. gutstein (2006) describes “writing the world with mathematics means using mathematics to change the world” (p. 27), but change to what? if mathematics is used to analyze and critique society, then a vision is needed of an ideal society, and mathematics needs to be part of that vision. north (2009) calls this “visionary literacy,” which she describes as “developing a story for our personal lives and the world that we can not only tolerate but also desire: doing our best to realize that story through concrete, human, and therefore, imperfect actions” (p. 151). as a world that we can not only tolerate but also desire, gutiérrez (2007) offers what equity could look like:  being unable to predict students’ mathematics achievement and participation based solely upon characteristics such as race class, ethnicity, gender, beliefs, and proficiency in the dominant language. (p. 41)  being unable to predict students’ ability to analyze, reason about, and especially critique knowledge and events in the world as a result of mathematical practice, based solely upon characteristics such as race, class, ethnicity, gender, beliefs, and proficiency in the dominant language (p. 45).  an erasure of inequities between people, mathematics, and the globe. (p. 48) gutiérrez has named the target of her concrete, human, and therefore, imperfect actions by working with mathematics to define her ideal. this perspective is an example of how a relationship between students and mathematics that is inspirational can be used to vision, and move toward, a better world. conclusion i began this essay with the realization of the fact that students need mathematics to have access to academic and economic opportunities, the problem of segments of students being denied access to the common property of mathematics, the inspiration of the proceedings and participants of the prompte conference, and the question: what would it mean to teach mathematics as agape? teaching mathematics as agape implies a desire to give access to the common property of mathematics in the form of a relationship with mathematics. what emerged were four facets to that relationship, calling for students to work with mathematics to achieve success as defined by society (functional), in the contexts and through the practices of the students and the students’ communities (communal), to analyze amidon teaching mathematics as agape stinson, d. w., & spencer, j. a. (eds.). (2013). privilege and oppression in the mathematics preparation of teacher educators [special issue]. journal of urban mathematics education, 6(1). 26 and question the world (critical), and to vision and work toward a better world (inspirational). similar to most approaches to teaching more equitably, they remain conjecture until they are attempted in the classroom, and the associated practices can be studied and compared to what is described as the ideal. an appropriate step would be to study the facilitation of the described relationship, and the associated facets, thus the next logical question: what does teaching mathematics as agape look like in practice? stay tuned… references apple, m. w. (1992). do the standards go far enough? power, policy, and practice in mathematics education. journal for research in mathematics education, 23, 412–431. blanchett, w. j. (2006). disproportionate representation of african american students in special education: acknowledging the role of white privilege and racism. educational researcher, 35(6), 24–28. boaler, j. (2007). paying the price for “sugar and spice”: shifting the analytical lens in equity research. in n. s. nasir & p. cobb (eds.), improving access to mathematics: diversity and equity in the classroom (pp. 24–36). new york, ny: teachers college press. callan, e. (1995). common schools for common education. canadian journal of education, 20, 251–271. carraher, t. n., carraher, d. w., & schliemann, a. d. (1985). mathematics in the streets and in schools. british journal of developmental psychology, 3, 21–29. civil, m. (2007). building on community knowledge: an avenue to equity in mathematics education. in n. s. nasir & p. cobb (eds.), improving access to mathematics: diversity and equity in the classroom (pp. 105–117). new york, ny: teachers college press. emdin, c. 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(1995). toward a theory of culturally relevant pedagogy. american educational research journal, 32, 465–491. ladson-billings, g., & tate, w. f. (1995). toward a critical race theory of education. teachers college record, 97, 47–68. lampert, m. (2001). teaching problems and the problems of teaching. new haven, ct: yale university press. lave, j., & wenger, e. (1991). situated learning: legitimate peripheral participation. cambridge, united kingdom: cambridge university press. martin, d. (2003). hidden assumptions and unaddressed questions in mathematics for all rhetoric. the mathematics educator, 13(2), 7–21. morris, l. (1981). testaments of love: a study of love in the bible. grand rapids, mi: wb eerdmans. moses, r. p., & cobb, c. e. (2001). radical equations: math literacy and civil rights. boston, ma: beacon press. north, c. e. (2009). teaching for social justice?: voices from the front lines. boulder, co: paradigm. palmer, p. 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(2008). the educational challenges of agape and phronesis. journal of philosophy of education, 42, 307–324. journal of urban mathematics education december 2016, vol. 9, no. 2, pp. 122–152 ©jume. http://education.gsu.edu/jume mary candace raygoza is a ph.d. candidate in urban schooling and collegium of university teaching fellow at the university of california, los angeles, 1320 moore hall, los angeles, ca 90095, usa; email: mary.candace.raygoza@ucla.edu. her research interests include mathematics learning, teaching, and teacher education; equity and social justice in mathematics education; critical and culturally relevant/sustaining mathematics pedagogy; and transformative school change. striving toward transformational resistance: youth participatory action research in the mathematics classroom mary candace raygoza university of california, los angeles in this article, the author contributes to the growing body of scholarship on critical mathematics pedagogy. in particular, the author advances this scholarship by outlining how critical pedagogy in the mathematics classroom can support students to engage in transformational resistance. using a critical practitioner research approach, the author retells (some of) her experiences as a high school mathematics teacher of ninth-grade latin@ students in an algebra i classroom. beginning the course with activities to build a beloved community and connecting mathematics with social justice issues, the author strived to facilitate a learning space that supported transformational resistance. through a culminating youth participatory action research project, students developed a critique of societal oppression, a motivation for social justice, and critical mathematical literacy. keywords: critical pedagogy, teaching mathematics for social justice, transformational resistance, youth participatory action research he sun beamed through our east los angeles classroom windows as my 23 algebra i students excitedly entered data from a school-wide student survey on school food (in)justice issues which they had designed and conducted the weeks prior. students were still learning how to use data software on the classroom laptops, when one student noticed that a column for data entry was missing. i showed her how to insert a new column. she titled it, paused, and then said, “this column is a variable, right? yeah, yeah, that’s a variable.” her tone was as if something spinning around in her mind for a while, or perhaps since she first took algebra i the year prior as an eighth grader (and “failed”), just settled into place. this moment encouraged me to conclude class that day with a discussion of the meaning of a variable, something that the students, instead of only saying “a letter that represents a number,” now attached real-world significance to as they were defining, measuring, representing, and making claims about variables related to a social justice issue in their lives. as a teacher, i sought to develop a critical pedagogy to support students to understand and transform the world (freire, 1970/2007). witnessing students’ t http://education.gsu.edu/jume mailto:mary.candace.raygoza@ucla.edu raygoza transformational resistance in mathematics journal of urban mathematics education vol. 9, no. 2 123 excitement as they came to understand the meaning of a foundational component of mathematical knowledge in the context of conducting their own research and activism related to school food (in)justice, affirmed for me the promise of teaching mathematics for social justice (see, e.g., gutstein, 2003) and integrating mathematical investigation in youth participatory action research (terry, 2011; yang, 2009). my aim with this article is to contribute to the growing body of scholarship on critical mathematics pedagogy (see, e.g., bacon, 2012; bartell, 2013; brantlinger, 2013; gonzález, 2009; gregson, 2013; gutstein, 2003, 2006; gutstein & peterson, 2013; terry, 2011; wager & stinson, 2012; yang, 2009). in particular, i advance this body of work by outlining how critical pedagogy in the mathematics classroom can support students to engage in transformational resistance (cf., solórzano & delgado bernal, 2001). using a critical practitioner research approach, i retell (some of) my experiences as a high school mathematics teacher of ninth-grade latin@ students’ in our algebra i classroom in east los angeles. beginning the course with activities to build a beloved community (king, 2010) and facilitating lessons throughout the course that connected mathematics with social justice issues (gutstein & peterson, 2013), i strived to foster the classroom as a space where developing transformational resistance was possible. through a culminating course youth participatory action research (ypar) project, students conducted a quantitative study on school food (in)justice. students developed a critique of societal oppression, a motivation for social justice (solórzano & delgado bernal, 2001), and critical mathematical literacy (gutstein, 2006). i argue that ypar involving quantitative investigation can support students to develop positive mathematical and researcher identities and contribute to change for social justice. conceptual framework teaching mathematics for social justice striving to teach mathematics for social justice involves engaging students in critical quantitative thinking around issues of social (in)justice that are relevant to their lives and daily experiences (see, e.g., frankenstein, 1983; gutstein, 2006; skovsmose, 1994). freire (1970/2007) argues, “the great humanistic and historical task of the oppressed” is “to liberate themselves and their oppressors as well” (p. 44). students of color and economically marginalized students bring valuable life expertise from their own experiences and backgrounds into the classroom, and can develop critical literacies to examine the unjust world around them so that they may both understand it and change it. freire posits that a problem-posing pedagogy in which students pose problems and develop solutions as they co-create knowledge is necessary for the liberation of all peoples. both teachers and students engage in cy raygoza transformational resistance in mathematics journal of urban mathematics education vol. 9, no. 2 124 cles of praxis, or the continuous process of “reflection and action upon the world in order to transform it” (freire, 1970/2007, p. 51). powell (2012) outlines the historical development of critical perspectives on mathematics education. teaching mathematics for social justice, also referred to as critical mathematics or criticalmathematics education in different regions and time periods—and also deeply intertwined with ethnomathematics—has origins from long ago, as national and global struggles for social justice and struggles for mathematics education have always been intertwined. conferences convened by organizations such as the african mathematical union and talks such as d’ambrosio’s 1986 plenary address introducing ethnomathematics were turning points in the mathematics education field. a number of conferences beginning in the late 1980s called for a sociopolitical look at mathematics and the teaching of mathematics, including the mathematics and society special programme of the international congress on mathematical education in 1998, the political dimensions of mathematics education: action and critique conference in 1990, and the critical mathematics education: toward a plan for cultural power and social change conference in 1990, which gave rise to the criticalmathematics educators group (cmeg). in a 1990 cmeg newsletter, frankenstein, volmink, and powell offer a definition of a criticalmathematics educator, offering insight into how those who strive to be a criticalmathematics educator may think about commitments as a mathematician, as a teacher, and as a concerned active citizen. criticalmathematics educators, as re-printed by powell (2012)— view the discipline as one way of understanding and learning about the world … as knowledge constructed by humans … as one vehicle to eradicate the alienating, eurocentric model of knowledge … listen well (as opposed to telling) and recognize and respect the intellectual activity of students … and have a relatively coherent set of commitments and assumptions from which they teach, including an awareness of the effects of, and interconnections among racism, sexism, ageism, heterosexism, monopoly capitalism, imperialism, and other alienating totalitarian institutional structures and attitudes. (pp. 26–27) drawing on frankenstein’s (1983) argument about the importance of mathematical literacy for gaining power in society over economic, political, and social structures and tate’s (1995) argument for culturally relevant pedagogy in mathematics, which includes the study of issues relevant to students’ lives, as well as other mathematics education scholars committed to education and change, gutstein (2006) argues that social justice mathematics prepares students “to investigate and critique injustice, and to challenge in words and actions, oppressive structures and acts” (p. 4). merging mathematics and social issues is not primarily for the purpose of understanding mathematical concepts but for using mathematics to create a more just world (frankenstein, 2010). raygoza transformational resistance in mathematics journal of urban mathematics education vol. 9, no. 2 125 stinson and wager (2012) define teaching mathematics for social justice as “the underlying belief that mathematics can and should be taught in a way that supports students in using mathematics to challenge injustices of the status quo as they learn to read and rewrite their world” (p. 6). they also define teaching with social justice as implementing equitable pedagogical practices, and teaching mathematics about social justice as teaching lessons with critical (and often controversial) social issue contexts. examples of social issue contexts that connect with mathematics detailed in rethinking mathematics: teaching social justice by the numbers (gutstein & peterson, 2013) include: racial profiling, how the unemployment rate is determined, global poverty, school evaluation, and media and body image. scholars argue that students may benefit from teaching mathematics about, with, and for social justice in the following ways:  developing a positive mathematics identity – students overcome fears in mathematics by building a positive mathematical disposition rooted in their culture and community (gutiérrez, 2012). gutiérrez contends: black and latino/a adolescents, like all young people, reap the benefits of programs that attend to their academic
 and their social/emotional needs. learners show more confidence and are better able to find an answer—and they can reflect on how reasonable that answer may be when they have opportunities to…use mathematics to analyze social injustices. (p. 35)  reading and writing the world with mathematics – students consume and produce texts from a critical standpoint, making sense of data and the social context behind numbers, and developing social agency (gutstein, 2006; yang, 2009). gutstein describes students’ critical questioning as he quotes one student, marisol, reflecting on a racial profiling unit: “i think i am better able to understand the world now using math…as soon as i finished the reading i already knew there was a problem there” (p. 67). yang describes how students produced their own version of a school accountability report card with measures that the students themselves felt best evaluated their school.  building mathematical power – students understand mathematical concepts, engage in complex mathematical tasks, and communicate ideas in ways that allow students to access spaces in which “dominant mathematics” is used, such as high school and university courses as well as in science, technology, engineering, and mathematics professions (gutiérrez, 2012; gutstein, 2006). raygoza transformational resistance in mathematics journal of urban mathematics education vol. 9, no. 2 126 while critical mathematics education scholars emphasize the possibilities and promise of teaching and investigating connections between mathematics and social change, there are also “limitations of the knowledge we gain from mathematical analyses of our world” (frankenstein, 1994, p. 56). brelias (2015) found that as high school students who were engaged in social justice mathematics lessons reflected back on mathematics as a tool for social inquiry, while they argued for its transformative power, they also argued mathematics can be “reductive and impersonal,” “irrelevant for moral arguments,” “inaccessible to the general public,” and can provide “inadequate explanations for problems” (p. 7). the students demonstrate a “reflective knowing,” as skovsmose (1994) describes, evaluating the use of mathematics. scholars have laid the groundwork in theorizing teaching mathematics for social justice and translating critical theory to practice mathematics in schools, but this body of work must be further informed by more accounts of the affordances and challenges of integrating critical social justice issues in the mathematics classroom (gutstein, 2006). duncan-andrade and morrell (2008) note: very little empirical work has been done that theorizes the possible translation of principles of critical pedagogy into practices, and even less work has been done that evaluates the outcomes of these practices in pushing forward the development of grounded theories of practice. (p. 105) youth participatory action research and transformational resistance as with critical pedagogy, conceptualizations of participatory action research (par) are influenced by critical theory developed in the frankfurt school. mctaggart (1991), drawing on carr and kemmis (1986/2004), asserts that par is “motivated by a quest to improve and understand the world by changing it and learning how to improve it from the effects of the changes made”; it “treats people as autonomous, responsible agents who participate actively in making their own histories and conditions of life” (p. 181). freire (1973) writes, “critical understanding leads to critical action” (p. 44). par can be understood as a cyclical process: those in oppressed conditions are engaged in research to understand a critical issue that they themselves identify as key to their freedom; they develop a plan for social action to challenge the inequity presented in that issue; and finally, they implement a plan for social change that they themselves developed. par methodology recognizes people as experts of their own knowledge and lived experiences. those experiencing oppression, therefore, must be leaders of the research-action to understand and challenge it. educators throughout the nation are bringing par into schools or doing par with youth outside of schools, adding a “y” in front, to explore how this actionresearch can be done specifically with youth (cammarota & fine, 2008). ypar raygoza transformational resistance in mathematics journal of urban mathematics education vol. 9, no. 2 127 engages youth as researchers on societal or school (in)justice and supports youth to take action connected to their findings. ypar is a problem-posing approach to education that challenges inequity by viewing youth as experts of their lived experiences and places power in their hands. cammarota and fine (2008) argue, “ypar teaches young people that conditions of injustice are produced, not natural; are designed to privilege and oppress; but are ultimately challengeable and thus changeable” (p. 2). they define ypar as a process in which “young people resist the normalization of systemic oppression by undertaking their own engaged praxis— critical and collective inquiry, reflection and action focused on ‘reading’ and speaking back to the reality of the world, their world” (p. 2). duncan-andrade and morrell (2008) put forth the argument that students in urban schools are the ones best equipped to be agents of change for equity and justice in urban education: as educators and as advocates for educational justice, we must understand that youth are much-needed collaborators with valuable experiences and energy to add to our movements. we firmly believe that youth participatory action research can ultimately develop the academic capabilities of students and, equally important, that youthinitiated research can help adult researchers and advocates to better confront the seemingly intractable problems of urban education. (p. 106) in this way, ypar challenges the notion of research as an objective endeavor and challenges traditional conceptions of who can become a researcher. additionally, ypar offers opportunities for young people to think about and partake in the ways in which research can be connected to resistance against oppression. ypar fosters students to resist injustice in a transformative way (cammarota & fine, 2008). transformational resistance, as defined by solórzano and delgado bernal (2001), encompasses a motivation for social justice and a critique of societal oppression. solórzano and delgado bernal provide the 1968 east los angeles student walkouts and the 1993 university of california, los angeles student strike for chicana/o studies as examples of transformational resistance, as secondary students and college students alike in each of these historic periods held critiques of oppressive societal conditions and were motivated to make change for greater social justice. cammarota and fine call for further documentation of how young people engage in transformational resistance—in and out of classrooms and related to various social injustices—and how educational processes can foster resistance. supporting students to bring a social justice motivation to their mathematical coursework and to critique societal oppression through mathematics is undertheorized and further examples of such praxis are needed. terry (2011) and yang (2009) each present theoretical arguments and empirical evidence to assert that young people can foster critical mathematical literacy and move that literacy to action as they engage in ypar. terry (2011) argues that ypar in mathematics can support students to engage in mathematical counterstory-telling, as he shares the raygoza transformational resistance in mathematics journal of urban mathematics education vol. 9, no. 2 128 ways in which black male youth identified and contradicted a dominant narrative and framed access to freedom, in the context of an after-school program in south los angeles. constructing mathematical counterstories can support students to selfidentify as doers of mathematics and ultimately involve “more transformative forms of resistance” (p. 43). yang (2009) addresses the quantitative aspect of a multiyear ypar study in which he and collaborators supported 30 high school youth to create their own school accountability report card. instead of accepting measures the state selected for school accountability, the youth decided what should be measured, including attributes like culturally relevant teaching. they conducted survey and interview research to discover if the school was meeting students’ needs and then reported that work to the student body and school leaders. students “critically consumed” and “critically produced” texts, thus engaging in the process of reading and writing the world with mathematics. however, “math education has not realized its full potential in developing youth researchers capable of producing critical texts at the level of public intellectuals” (p. 102). the study presented here further explores the possibility of quantitative action-research projects, examining the implementation of ypar within the context of an algebra i classroom. overall, i sought to provide students with the opportunity to engage in ypar in our mathematics course, after we had laid a foundation for transformational resistance by building community and completing units connected to social (in)justice issues. i wondered: how can teaching students to be critical quantitative scholar-activists be included as a meaningful part of teaching a mathematics course? methods the voices of teachers in national conversation and research are essential for advancing change for educational equity and justice (cochran-smith & lytle, 1999; oakes & rogers, 2006). practitioner research is a “promising way to conceptualize the critical role of teachers’ knowledge and actions in student learning, school change, and educational reform” (cochran-smith & lytle, 2009, p. 5). critical, qualitative methodologies (kincheloe & mclaren, 2002; steinberg & cannella, 2012) shaped the design of this practitioner research investigation. in particular, this study is informed by practitioner research that emphasizes “equity, engagement, and agency,” as cochran-smith and lytle (2009) identify as a more recent turn in practitioner research, pointing to books on ypar and critical pedagogy such as duncan-andrade and morrell’s the art of critical pedagogy (2008) and cammarota and fine’s revolutionizing education: youth participatory action research in motion (2008). they argue that such critical practitioner research has potential to question the goals of schooling, raise questions about power and whose voices are raygoza transformational resistance in mathematics journal of urban mathematics education vol. 9, no. 2 129 heard, bring meaning to equity in local contexts, and link teacher and student inquiry as interconnected. research questions the research questions that guided this investigation were: 1. how can a mathematics classroom develop as a beloved community lay a foundation for transformational resistance? 2. how can ypar in a mathematics classroom support students to engage in transformational resistance as they build critical mathematical literacy? these questions serve as the focus of this article because integrating ypar in the teaching of mathematics for social justice within mathematics courses is understudied. teachers’ stories that shed light on possibility and challenges of teaching mathematics for social justice are necessary to advance understandings of how social justice may come to fruition in a multitude of ways in the mathematics classroom (bullock, 2014). i do not assert that the questions i pose can be fully addressed within this practitioner research examination, but i hope that striving to share an indepth, critical practitioner research study can join in dialogue with other accounts of mathematics teaching (such as bacon, 2012; brantlinger, 2013; gutstein, 2003; terry, 2011; yang, 2009) and offer implications for future work. teacher and researcher positionality like all educational research, mathematics education research is political and non-neutral (d’ambrosio et al., 2013; gutstein, 2003). my experiences reflect the political nature of mathematics education research in my choice to be a part of social justice mathematics communities of educators, my belief that educational research can and should contribute to a more socially just world, and my choice to follow in the footsteps of scholars who examine their own positionality as researchers. i entered teaching with an understanding that being a continual learner is an essential aspect of critical pedagogy and teaching mathematics for social justice. i believe it important for teachers—and for myself as a white female teacher and a teacher of stem (science, technology, engineering, and mathematics)—to constantly work to become more aware and knowledgeable about the historical and present day intersections of oppression based on race, ethnicity, class, gender, sexual orientation, religion, age, special need, legal status, language, and so on. as a mathematics teacher, i strive to understand the ways in which stem education perpetuates yet can intervene to challenge oppression. i also believe it important to raygoza transformational resistance in mathematics journal of urban mathematics education vol. 9, no. 2 130 build with other educators, both new and veteran, to learn from and alongside each other. as a white woman, i came into this research with the same consciousness that i attempted to bring into my classroom teaching: an understanding that white women are the most represented demographic group in the teaching profession. voices from this group, therefore, are often centered and privileged in (and out of) schools. moreover, the teaching profession is looked down on as “women’s work”; a devaluating i often felt as a teacher and experienced the seeds of growing up as a young woman. interrogating my own positionality and pedagogy is a complex, lifelong process that can never be completed. amidon’s (2013) argument on teaching mathematics with agape (a koine greek word, often translated to “unconditional love”), pushes me to think about the following questions, which i adapted from his work: what can i do from a position of power and privilege to interrupt oppression and be a part of supporting students to have the opportunity and expectation of success in mathematics and life? how can i learn from and alongside teachers from similar and different backgrounds who are working to support students to succeed in mathematics (according to traditional and critical ways of thinking about success)? in my teaching of mathematics to latin@ students in an economically marginalized community, i sought to interrupt power, privilege, and oppression (amidon, 2013); develop a “critical care praxis,” challenging a colorblind approach of caring for students (rolón-dow, 2005); and express to students a political and radicalized love (darder, 2002). below i detail the ways in which i attempted to bring these ideas to life in my pedagogy, while acknowledging my continuous need for critical self-reflection. classroom and study context the public school in which i taught was one fought for and created by community teachers and organizations in east los angeles, including the organization innercity struggle (see http://innercitystruggle.org). it was one of the first public high schools built in the larger community in 80 years, after community organizing pushed for new schools due to over-crowding. this activism was by no means new in east los angeles. home of the 1968 east los angeles school walkouts (solórzano & delgado bernal, 2001), the community’s historical and present-day struggles for educational justice laid a significant and meaningful historical and cultural foundation for the new school. i had hoped to teach in a public school at a time when few jobs were available in public schools, so i felt fortunate for the opportunity to teach in this school, fought for by latin@ students, families, and community organizations. i believed that our mathematics classroom community could attempt to learn from and draw on the historical legacy of social justice efforts by and for the community. the algebra i class i chose to study for this investigation consisted of students who took and had previously “failed” (or were failed by) the course the prior http://innercitystruggle.org/ raygoza transformational resistance in mathematics journal of urban mathematics education vol. 9, no. 2 131 year as eighth graders in middle school, and mostly students who were not tracked into algebra in eighth grade at all—the highest track in the feeder middle schools being geometry for eighth graders. because of this “low” placement, the students in this class were positioned as the least knowledgeable/capable students in the school. the textbooks in the feeder middle schools as well as the textbooks at the high school where i taught, like many traditional texts from large textbook publishing companies, marched through mathematics procedures and included a long list of problems for which students were expected to apply procedures, with a couple “critical thinking” word problems at the end of each section. the textbooks did not arrange the mathematics from an integrated mathematics perspective but rather as separated subjects: algebra i, geometry, algebra ii, trigonometry/pre-calculus, and calculus. the books did not include relevant, real-world, current or historicized issues. the content area covered in the algebra i textbook, the same as the content covered on the high-stakes state exams, amounted to: properties of real numbers; solving, graphing, and writing linear equations and linear inequalities; systems of equations and inequalities; exponents and exponential functions; quadratics; polynomials; and rational equations and functions. i believed this class was a good fit to practice ypar because i hoped to provide the space for students to get excited about mathematics, which, for most students, had already become a despised school subject. for most students, this course was the first time they had access to algebra i, which tends to be a gate-keeping course. while i hoped to integrate ypar into multiple class periods in the same school, i recognized as i began to plan the introduction of the unit that i did not feel prepared to facilitate multiple different action-research projects at once, so i chose one class period of algebra i. data collection and analysis data sources included:  my teaching journals, in which i reflected on my pedagogical choices and took note of student leadership, participation, presentations, and specific contributions students made in the class (i put student statements i remembered verbatim in quotation marks);  preand post-questionnaire (see figure 1) with students on their beliefs about what it means to do research;  math autobiographies students wrote at the beginning of the course as well as an end-of-course reflection on mathematics; and  student artifacts, including a final paper the class collectively wrote on their ypar study and a video based on their quantitative data analysis and a collection of photos of the school food. raygoza transformational resistance in mathematics journal of urban mathematics education vol. 9, no. 2 132 to analyze the data, i developed (and triangulated) a coding schema across the data sources. i took note of reoccurring themes as i taught throughout the school year, using the constant comparative method for data analysis (bogdan & biklen, 1982). in this article, i discuss my pedagogical decisions and reflections on those decisions, incorporating some of the ways in which students engaged in the course and in the culminating ypar project. figure 1. preand post-questionnaire about research. developing foundations for transformational resistance beginning the algebra i course joining other mathematics educators across the nation, i attempted to create a social justice mathematics class to provide the space for students to feel empowered to read and write the world with mathematics (freire, 1970/2007; gutstein, 2006). from the first day of school, i sought to lay the foundations to critique societal oppression and explore possibilities for social justice within my classroom. the social issue-related units and ultimately the ypar project i facilitated in the classroom did not occur in a vacuum, but rather surfaced out of practices i had used and activities students had engaged in at the beginning of the school year. just prior to asking students to share more detailed information about themselves to me in a student information sheet (see appendix a), i briefly introduced myself and my history, opening up to students as a person. first, i told them about my name, mary candace. mary is my grandmother’s name and candace was her name: date: period:______________ research questionnaire – explain all your answers 1. what is research? give your own definition! 2. give an example of research. 3. what are the different steps of research? (make a guess if you’re not sure! come up with at least four steps.) 4. who does research in our society? 5. what skills do you think a researcher needs? do you have any of those skills? 6. do you think research is important for society? why or why not? raygoza transformational resistance in mathematics journal of urban mathematics education vol. 9, no. 2 133 sister’s, or my great aunt’s name. i go by “ms. candace” with students because of the kind of person my great aunt was. she suffered a heart condition and passed at a young age. throughout her life, she devoted herself to the happiness of others and to supporting people with what they needed. for example, she always remembered everyone’s birthdays and made elaborate cakes. after being diagnosed with her heart condition, she started a chapter of an organization in buffalo, new york called mended hearts (see http://mendedhearts.org), which brought together patients who had previously undergone heart surgery and patients who were about to go through it. the goal was to have conversations that may ease fear before experiencing the surgery. i told students that i hoped to bring my great aunt’s spirit with me to guide me in my teaching, and that i will always remember and celebrate each of their birthdays in honor of her. i told students a story of taking algebra i at a title i middle school in eighth grade and getting as, but when my parents transferred me to a high school out of our neighborhood, i got an 11% on the “how much of algebra do you remember?” test on the first day of school. i remained in geometry and did well, ultimately making it to calculus, but doubted myself and had always felt inferior to the other students at school because of that initial test. i told another story of intending to major in statistics in college but barely passing the first upper-division course because i was intimidated by what i perceived as the cut-throat, competitive nature of the class. i also felt i was at a disadvantage because i was one of only a few women enrolled in the course. i shared with the students that my experiences do not mean that i understand their experiences—acknowledging that the students have a wide variety of experiences with mathematics. i also recognized that although i, as a white person from a middle class background, have benefited from many privileges, i could, to certain extents, relate to feeling disempowered in mathematics as well as empowered. i asserted that to push back on disempowerment in mathematics, we must co-create a space in which we collaborate to build a beloved community. in “letter from a birmingham jail,” dr. martin luther king, jr. (1963) wrote: “injustice anywhere is a threat to justice everywhere. we are caught in an inescapable network of mutuality, tied in a single garment of destiny. whatever affects one directly, affects all indirectly” (p. 2). i first learned of dr. king’s concept of the beloved community as applied to the classroom when i was an undergraduate at the university of california, berkeley and took a course entitled “june jordan’s poetry for the people” in which we read and wrote poetry on systemic oppression, resistance, humanization, and liberation. i understood building a beloved community in the classroom to mean developing a space in which both the students and the teacher see the humanity in one another as they work toward a more socially just world. i told my students that each of them is each other’s greatest asset for learning, for making it to graduation and beyond, and for changing the world—that they are connected to and dependent on each other. http://mendedhearts.org/ raygoza transformational resistance in mathematics journal of urban mathematics education vol. 9, no. 2 134 during the first week of class, i facilitated class dialogues in which students declared their rights in the mathematics classroom, the demands they have of themselves and each other, and the demands they have of me as their teacher (see appendix b). i told students that i am a teacher because i believe they will be the ones to change the world. i also told students that i would try to make the class center around their own self-improvement and building confidence in mathematics as we use mathematics to understand and change the world. i then provided the syllabus to the class—the title of the course matched what i told students it would be about: “viewing and changing the world through mathematics.” underneath the title, i placed a picture of a mural in the housing projects of east los angeles,1 as gutstein (2006) writes about teaching about after he read an article of marilyn frankenstein’s (1997) that included a photo of the mural. some of the students in our classroom lived in this housing project, so i felt it was especially relevant. alongside che guevara is the message: “we are not a minority!!” i then facilitated a scaffolded class discussion, where students shared how they, as people of color, are not minorities—according to the mathematical definition and because they are not “less than” other people. i also began with a math autobiography writing assignment (cf., peterson, 2013), asking students to write me a letter telling me their stories (see appendix c). i structured the assignment with many questions about their past experiences with school in general and mathematics specifically, and why they feel mathematics is important to learn. the most common response to the latter question: “we need math to count change at the grocery store.” their responses motivated me to want to expose the students to how powerful a tool mathematics can be for understanding racial, social, and economic injustice in education, health, incarceration, and so on, and for constructing arguments to fight for justice through mathematics. i explained that certainly mathematics is important and essential for everyday calculations, but that i firmly believe students have the power to use mathematics in ways that connect more deeply to their lives. my role was to provide them the support and context for this learning. one of the first activities i implemented with students was an “icebreaker” similar to one from my teacher education program. students wrote down on an index card three “variables” about themselves: i = an interest they have, c = a characteristic or personality trait about themselves, and e = a unique event or story from their past. the whole class stood up as i began to read aloud a card. students remained standing until i read all three statements, and they guessed, from the group who was left standing, the person to whom the card belonged. students were excited to learn about each other’s “variables.” this activity built community by getting to know each other. 1 see http://www.muralconservancy.org/murals/we-are-not-minority. http://www.muralconservancy.org/murals/we-are-not-minority raygoza transformational resistance in mathematics journal of urban mathematics education vol. 9, no. 2 135 as a pre-service teacher, i dreamed of creating an algebra i course that consisted only of social issue-related and other real-world units. as i entered the classroom, i began to more fully understand the pressures on teachers to teach all of the content standards that bacon (2012) and gregson (2013) detail in case studies of social justice mathematics teaching. for algebra i, this includes a long list of many fundamental concepts necessary to accessing the “rest” of mathematics. these pressures come from a multitude of directions and, in my case, included:  “value-added” measures imposed by the school district that rate teachers based on student test score improvement,  standardized tests such as the high school exit exam, and  administrators that put pressure on teachers for students to perform “or else” (see gutiérrez, 2013 for many other examples). i realized that, while i would not compromise my plan to teach social-issue related units, i would have to create them in a way that was strategic, so that the activities i designed addressed a variety of content standards in a rigorous enough way to address these demands. i believed that covering content standards and teaching social justice mathematics did not have to be mutually exclusive; however, i had not yet conceived how i could tie much of the mathematical content to real-world contexts. across all mathematics lessons, i sought to draw on culturally relevant approaches, which i knew would require me to change the culture of my classroom to be different than how mathematics classes looked like for me as a student. a daily pattern of whole class instruction where students follow along passively, copying problems the teacher solves, and then working on a set of similar problems alone following the lecture does not call on students to communicate almost at all orally or build on each other’s ideas (tate, 1995). in my classroom, i worked toward supporting students to discover concepts in mathematics (as opposed to teaching procedures) and having students work in groups collaboratively to validate and build on the knowledge of one another (including seating the students in small circle groups). this classroom set up assisted me in providing the space to meaningfully connect mathematics and social justice issues whenever possible. teaching mathematics about, with, and for social justice one of the first tasks in algebra i is to review computation and the order of operations. students are expected to have mastered fractions prior to taking the course, and this often can be students’ least favorite element of a mathematics problem. i thought about how i could re-introduce fractions and start off my class with real-world connections that would show students how important fractions can be. i raygoza transformational resistance in mathematics journal of urban mathematics education vol. 9, no. 2 136 designed a lesson using the video if the world were 100 people2 (see appendix d), which students and i viewed together. in the video, statistics report that if the world was shrunk down to 100 people, for example, 12 people would have a computer and 21 would live on less than $1.25 per day. before watching the video, i asked the students to guess what they thought the numbers would be for all of the statistics, and, while watching the video, they then recorded the data. some students were shocked at the differences between their guesses and the actually reported statistical data. following this activity, i asked students to write the data as fractions out of 100 (e.g., 12/100), then reduce the fraction, if possible, write it as a decimal, and write it as a percent. we discussed the importance of being able to do so when making arguments in different contexts; that it can be powerful to express parts of a whole in different ways.3 for homework that night, i gave students a chart entitled “if east l.a. were 100 people,” with data i had taken from the census (we acknowledged and problematized that it often does not include people who are undocumented), and their task was to find the fraction, decimal, and percent equivalents and then write concluding observation sentences about what they noticed. my goal was for students to gain confidence in mathematics as they read the mathematical world around them. some students, accustomed to more traditional courses and skeptical of change, wondered out loud, “why are we talking about community in math class?” early in the algebra i course, it is typical to study inequalities, and i seized the opportunity to plan a unit that connected the way we discuss inequalities in mathematics with inequalities (or inequities) in society (see appendix e). to get a stronger sense of what issues students were feeling particularly interested in, i presented them with several graphs relating to topics such as race and incarceration, incomes of people with and without disabilities, child poverty before and after economic recessions, and health vs. wealth. in teams, students chose a graph to study as the starting point in their inequality project. they answered several questions about the graph (e.g., questions designed to invite a range of participation from all group members, such as what stands out to them the most) and wrote out mathematical statements of inequality, using less than, greater than, less than or equal to, and greater than or equal to—using variables and numbers—in ways that were meaningful to them. following this activity, students did additional background research on that inequality, created a poster, and presented findings to the class, in 2 see http://www.miniature-earth.com; the video is based on the book if the world were a village: a book about the world’s people (smith, 2011). 3 this was a brief moment in the lesson that i could have elaborated on more, by, for example, calling on students to identify instances of how parts of a whole have been communicated in the media and speculating together why they were communicated in such a manner. http://www.miniature-earth.com/ raygoza transformational resistance in mathematics journal of urban mathematics education vol. 9, no. 2 137 cluding questions that arose as they explored the topic. one of the students, roberto, who was having challenges in most classes because of issues outside of school, spoke during this presentation. at the moment that he began to break down the importance of looking at percent difference as opposed to numerical difference in the context of racial underrepresentation and racial overrepresentation in prisons, the principal walked in to our classroom. she asked him a question, then another, and another. this ninth grader, whom she had only seen in her office for misbehavior, confidently addressed all her questions related to this inequality and why it was an important one to understand. following his presentation, he expressed that he felt validated as an intellectual; that he had something to say about social issues and mathematics. my last example of a social justice mathematics lesson is on the topic of slope. often, textbooks and traditional curricular materials introduce slope as being equal to (y2 – y1) / (x2 – x1). this formula can be challenging for students to understand, especially if not introduced to the conceptual meaning of slope. i do not argue that students should never be introduced to such a formula. but rather, when they are, it should come after interacting with slope in more meaningful and conceptual ways so that students can understand that slope captures how much one variable changes with respect to another. without mentioning the word “slope” in class, i presented two graphs to students of data with almost linear relationships, the first being a graph of asthma rates over time in a california city and the other family income vs. student sat math scores. we discussed the first graph together as a class and the students studied the second graph together in groups. i asked students to share everything they noticed about the graphs. then, i asked students to focus on the x-variable (horizontal axis) and then on the y-variable (vertical axis), for each identifying how much that variable changes from one point to another point. next, i asked students to write this ratio out as a sentence (e.g., for every $20,000 more a family makes, the average student’s sat math score is higher by 14 points). after students identified this relationship, i said that we call this relationship “slope.” for many students who had already taken algebra i in eighth grade and “failed” (or were failed) they responded with statements, such as: “oh! why didn’t they just tell us that before?!” or “that’s it!? man i just couldn’t remember those formulas!” after students saw real-world context examples of slope, in subsequent exercises the majority of students excelled at finding and interpreting δy/δx as they discussed the meaning of a graph’s slope. i saw ways in which i believed students were striving to “read the world” with mathematics in my classroom. in addition to sharing their (our) learning on social issues, how could i support them to further “write the world” with mathematics? raygoza transformational resistance in mathematics journal of urban mathematics education vol. 9, no. 2 138 developing transformational resistance and critical mathematical literacy through youth participatory action research thinking of how to introduce the ypar unit, i returned to an article i had introduced to students earlier that year: solórzano and delgado bernal’s 2001 germinal article on transformational resistance. i had used the article earlier to introduce the cartesian coordinate plane (see figure 2). students are often introduced to the coordinate plane as an abstraction, and plotting points becomes a mechanical procedure without meaning (e.g., 5 over and 2 down). i had remembered that solórzano and delgado bernal’s article contained a quadrant showing four types of resistance, each quadrant representing the presence or absence of a variable, x – motivation by social justice, and y – critique of societal oppression. i put blue painters’ tape on the floor of the classroom to make a coordinate plane, asking students to move about it representing the presence or absence of two variables (first starting with variables such as their preferences on two different sports rivalries, and then moving to examples of high school student resistance that students could relate to, mapping them out on the coordinate plane according to solórzano and delgado bernal’s conceptualization of resistance). figure 2. defining the concept of resistance. from “examining transformational resistance through a critical race and latcrit theory framework: chicana and chicano students in an urban context,” by d. solórzano and d. delgado bernal, urban education, 36(3), p. 318. copyright © 2001 by corwin press, inc. reprinted by permission of sage publications, inc. to kick off the ypar project, i invited professor solórzano to come to my classroom and speak to the students about transformational resistance (my contact raygoza transformational resistance in mathematics journal of urban mathematics education vol. 9, no. 2 139 with him and his class visit are recounted in solórzano, 2013). professor solórzano shared with the class examples of transformational resistance and specifically spoke about the ethnic studies ban in arizona and upcoming film precious knowledge (mcginnis & palos, 2011). the students were energized following his presentation and excited to engage in action-research themselves. the students chose to study school food injustice, as health injustice related to race and class. they were passionate about this topic following the inequalities project. before we dove into designing the study, we watched the film unnatural causes (adelman, 2008), which weaves together biology, public health, sociology—and even mathematics, with data and graphs throughout—to understand how wealth disparities lead to poorer health. students chose to look at school food because: (a) it affects them every day, and the district had recently changed the food in such a that they were unpleased; and (b) after intense class debates about how to make change and what it means to be a leader, students felt that they had a realistic chance of making some concrete changes on this issue. as we embarked on the study design, students piloted survey questions with peers in other classes before refining a survey protocol, asking their peers to talk through how they understood each question and what should be modified for clarity or added. additionally, during lunchtime in the first couple weeks of the project, students talked with their peers and took pictures of school food to deepen their understandings of the student body’s greatest concerns. they also shared stories in class of their own experiences with school food. while i guided students toward a school-wide survey so that we might analyze large-scale quantitative data, we discussed strengths and weaknesses of quantitative and qualitative research. the informal conversations students led with their peers and the photo element of the investigation gave them a sense of various research methods (e.g. interviews, photo voice) and how the stories behind numbers are necessary to uncover what they want to know. that said, i acknowledge that in the spirit of ypar it is problematic to point youth in particular directions, but i chose to make the exception of guiding them toward quantitative investigation, while still highlighting the limits of doing so, because of my desire to integrate ypar in the mathematics class. the class then designed and implemented a school-wide survey with closedended and open-ended questions on the school food as well as demographic information, which they administered to over 400 of their peers. students analyzed the survey data and created graphs using microsoft excel, which they had not used before, and presented their findings to school food officials. in each phase of the action research, different student facilitators led the class (after i had a planning meeting with them on facilitating for that particular class segment). in summary, the class had five findings: (a) students overwhelmingly gave the school food the lowest rating; (b) the predominant reason students reported eating the school food was because they were hungry and did not have other options; (c) raygoza transformational resistance in mathematics journal of urban mathematics education vol. 9, no. 2 140 students had a desire for all components of the school lunch to be tastier, healthier, and in larger portions; (d) students wanted to exercise more choice in their meal options; (e) and all but eight students surveyed said they received “lunch tickets,” meaning they qualified for free lunch. one of the biggest shifts with respect to mathematics i observed was students’ use of mathematical discourse as they communicated with one another (and me) while entering, analyzing, and displaying their data. for example, students were regularly using the terms such as variables (as described in the anecdote in the introduction of this article), relationship, correlation, x-axis, y-axis, coordinate, data point, pattern, outliers, respondents, bar graphs, pie graphs, frequency, intervals, best-fit line, mode, mean, range. prior to the ypar unit, we covered linear equations beginning with the investigation of slope (as previously discussed). i had hoped that more students would apply the algebraic concepts behind linear equations as they engaged in this research, but few did, due to, i believe, a lack of support on my behalf. i would have liked to have done more, and at the same time i made the choice not to push certain aspects of the mathematics, assessing that students were gaining non-algebraic, statistical skills and ways of thinking and confidence in their participation as students in mathematics class. following analysis and writing a summary research report as a class, the students called for a meeting with the cafeteria manager to share their research findings and ask about the school lunch program. they concluded with recommendations to have daily comment cards available for all students so as to quickly give feedback on the food (and for these comments to make their way to those who prepare the food); to “change our menu into food that helps us be healthy—food that does not have an exaggerated amount of grease, fat, sugar, etc. so we can adapt to the habitat [sic] of being able to eat healthy”; to offer larger portions; to include more fruit and vegetable options; to offer water and fewer sugary drinks; to distribute a monthly menu; and to eliminate the lunch tickets given that almost the entire student body qualified for them. as other practitioner researchers striving to teach mathematics for social justice have observed, i believed that students were demonstrating more positive mathematical identities as they began to engage in the process of reading and writing the world with mathematics, solving rigorous problems, and building mathematical literacy. in their math autobiographies at the end of the year, students’ views on the importance of mathematics and why we must learn it shifted from being important in specific situations and because it will be useful in the future to an understanding of mathematics being an important tool for viewing and changing the world around them. most students wrote that they began to feel this way about mathematics as they came to believe they have the capacity to study and change their school, using mathematics. raygoza transformational resistance in mathematics journal of urban mathematics education vol. 9, no. 2 141 returning to the story at the beginning of this article, students often made meaning with mathematics that they previously had not experienced. as students were creating representations of their data, another student, recognizing how data representations could help tell a powerful story, said: “once they see our data, they will have to listen.” this statement demonstrates her perception of the power of critical mathematical literacy. reflecting on her statement and similar statements by other students, i am glad that the students held firm beliefs in the power of their work to tell a story and demand attention and action. but i wish i had done more to create space to address the complexity of speaking their truth to systems of power, the struggle of working toward social justice for the long haul, and community organizing tactics for making change. exploring such topics in the mathematics classroom, highlighting the role of mathematics, statistics, and numeracy in organizing for change, would have been powerful. students did begin to uncover the challenges behind making change toward greater social justice after meeting with the school’s cafeteria manager and learning that decisions about school food were not local school decisions but rather large district-wide ones, and after meeting with the healthy school food coalition4 and learning of persistent efforts by that organization over time, in solidarity with students, to win actionable issues. students also demonstrated a significant shift in their interpretation of the meaning of research. before they engaged in ypar, most students defined research as looking up a topic of choice online and did not view research as a process they could conduct by posing original questions and collecting data on a topic about which they were passionate. (figure 3 provides students’ statements after the ypar project.) prior to the ypar project, i hoped to lay a foundation in the classroom where our work would be motivated by social justice issues and to critique societal oppression. i do not believe working toward these goals in the ypar project would have been possible had we not taken time to build community and participate in social justice lessons throughout the course. i observed that a motivation for social justice and a critique of societal oppression were built over time, especially when fostered within a subject like mathematics where the connections are not often made. relating to the other forms of resistance solórzano and delgado bernal (2001) discuss, too often students in mathematics class express “reactionary behavior,” acting out against teachers and classmates without a critique of social, historical, or political factors influencing their behavior; or they engage in “self-defeating resistance,” understandably questioning the relevance of mathematics to their lives; or in “conformist resistance,” doing their work knowing that it will help them access college but not challenging the abstract, procedural, or context-void teachings of mathematics. critical pedagogy in mathematics can support more students to re 4 see http://www.oxy.edu/urban-environmental-policy-institute/programs/food/healthy-school-foodcoalition. http://www.oxy.edu/urban-environmental-policy-institute/programs/food/healthy-school-food-coalition http://www.oxy.edu/urban-environmental-policy-institute/programs/food/healthy-school-food-coalition raygoza transformational resistance in mathematics journal of urban mathematics education vol. 9, no. 2 142 sist in mathematics in transformative ways. during the ypar project, students resisted with both a motivation for social justice—to get tastier and healthier food for all at their school—and with a critique of societal oppression—that people of color and people living in poverty are often denied access to healthy food and many other benefits to overall health, and that school food should not exacerbate that inequity but rather fight against it. figure 3. student statements on the meaning of research after the ypar project. after representations from the healthy school food coalition visited our class, students wrote thank you notes to them. one student included the phrase “victory will be ours!” on the cover of his note. earlier in the project, this same student had been skeptical, questioning how students could really be school leaders who fight for change. another student’s note read: i am so glad and i feel so supported by a group of adults that actually care about what we care about, which is the school food. i feel motivated to want to learn more about this and make a change knowing that no one else would do it, but the ones that care. i feel like we can really make a change as long as we keep on trying.  thank you so much for hearing us out and listening to our thoughts and opinions. i really hope our work pays off in the end. i hope to stay in touch so that together we can also make a difference once we’ve done it at our school with the cafeteria manager first!  i have learned about research that when we do research it’s not only “copy, paste, and print,” we have to read it and understand it and do it. research is to find the information you want on your own.  research is to take time to look for information about something that you’re looking to uncover, to find facts and explanations about it. anyone can do it as long as you know what you’re looking for. i learned that creating and giving surveys you can really collect a lot of information and different opinions of many students. you get to learn what they think and feel on what you’re asking them.  there are explanations and steps to do research. research means to find out something but more detailed, like getting to the bottom of it.  it is very hard to come up with a simple survey. it takes a lot of patience. first we came into teams to construct each category. anyone can make a survey, just takes patience and research about the topic of the survey. what i learned is that you could get a lot of info in a couple simple questions, and we worked days on a survey and people finish it in minutes.  we can use research to find out patterns of info for the surveys.  to find out all of the information and put the information together and analyze and understand it. raygoza transformational resistance in mathematics journal of urban mathematics education vol. 9, no. 2 143 discussion in this article, i aimed to contribute to scholarship that imagines, builds, and investigates spaces where critical mathematical literacy and social action are intertwined from the perspective that the work of engaging in critical praxis is a lifelong process. as sleeter (2015) and ladson-billings (2015) contend, scholar-activism not only conceptualizes possibilities of teaching for social justice within inequitable schooling contexts but also works to alter such contexts. questions that arose for me in reflecting on this ypar practitioner research are: how can schools and teacher education be altered to foster social justice mathematics? how can social justice mathematics itself be a part of pushing for change to challenge structural inequities that persist within and outside of mathematics education? what role does or can critical mathematical literacy have in developing “justice-oriented citizens” (westheimer & kahne, 2004) who engage in transformational resistance as they envision and lead social movement? this investigation sheds light on the promise of ypar in the mathematics classroom to open up spaces for students to develop as subjects of transformation as one avenue of teaching mathematics about, with, and for social justice (cf. stinson & wager, 2012). school change efforts and research can further explore the expansion of ypar as a normalized part of students’ learning experience at school and the multiple critical literacies students can build as they engage in research and create change. westheimer and kahne (2004) argue that one way schools can teach for democracy is to develop students as community activists who analyze root causes of societal inequities and explore how human rights and social justice can be achieved as a result of collective action. critical pedagogy can be supplemented with “strategies used by community and education organizers” (anyon, 2005, p. 179). critical mathematics teachers and their students who engage in ypar can be positioned as mentors to pre-service teachers and, in teacher education, teachers can learn how to form educational justice-oriented groups with other teachers, such as critical inquiry groups (nieto, gordon, & yearwood, 2010) and nepantla circles (gutiérrez, 2012). while ypar in the mathematics classroom has the potential to open up space for students to write the world (the action in action-research can be thought of as writing the world), there are a multitude of ways in which young people can write the world with mathematics that can occur within or outside the context of ypar (e.g., [re]defining what it means to do mathematics such as bringing in cultural practice to mathematics, creating texts and media, sharing work with family and peers), and ypar itself does not demand particular actions. future research can capture a variety of ways in which young people can develop transformative resistance through or drawing on mathematics and continue to question the possibilities for and extent to which ypar and transformational resistance can be fostered raygoza transformational resistance in mathematics journal of urban mathematics education vol. 9, no. 2 144 within the classroom walls—within school, district, and larger social-political constrains that often run counter to the spirit of ypar. because i guided students toward quantitative investigation, placed a time limit on the project, and did not cover some of the “dominant mathematics” concepts i was supposed to address according to the content standards (polynomials and rational equations and functions), i need to complicate my own claims about ypar and transformational resistance in the mathematics classroom. these are all areas i would strive to improve on, think harder about, and talk with others about before future implementations. as studies by terry (2011) and yang (2009) do, future research can also include in-depth examinations of how students demonstrate their learning of mathematics in the context of ypar and following their engagement with ypar, as well as the connection between students’ mathematics learning and mathematical identities. i do not link students’ mathematical identities to their mathematical performance in terms of performance on tests or success with future mathematics access. in one sense, i believe this is a limitation of the study. at the same time, i argue that placing high value on and focusing in on how students perceive mathematics and how students perceive research in the context of participating in ypar are just as important areas of focus for research as other conceptions of what it means for students to “perform” mathematically in the context of ypar or social issue-related learning. pushing for change in schools and teacher education can foster environments where social justice mathematics can grow and, in turn, ypar connected to quantitative inquiry can work toward creating change itself. as stinson (2014) asserts, in times of great societal injustices and large-scale organizing against such injustices (e.g., the black lives matter and occupy wall street movements), we can call on mathematics education to contribute in greater ways to understanding and changing our society. references adelman, l. 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(1995). returning to the root: a culturally relevant approach to mathematics pedagogy. theory into practice, 34(3), 166–173. terry, c. l., sr. (2011). mathematical counterstory and african american males: urban mathematics education from a critical race theory perspective. journal of urban mathematics education, 4(1), 23–49. retrieved from http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/98/87 wager, a. a., & stinson, d. w. (eds.). (2012). teaching mathematics for social justice: conversations with educators. reston, va: national council of teachers of mathematics. westheimer, j., & kahne, j. (2004). what kind of citizen? the politics of educating for democracy. american educational research journal, 41(2), 237–269. yang, w. k. (2009, fall). mathematics, critical literacy, and youth participatory action research. new directions for youth development, 123, 99–118. http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/98/87 raygoza transformational resistance in mathematics journal of urban mathematics education vol. 9, no. 2 148 appendix a student information sheet student information sheet my full name is: i prefer to be called / nickname: my birthday is (month date, year): gender pronouns: do you need to sit near the front of the room to hear or see better? my home phone number is: family member’s name and their relationship to you (mom, tío, etc.): the music i like to listen to is: the foods i like to eat are: the language(s) i speak at home is/are: i am talented at: i am from: i am: i would like to get better at: i come to school because: in the future, i would like to: in the past, i have had good experiences in school when: i have had bad experiences in school when: something that gets on my nerves is: something i would like ms. candace to know about me is: i would like to change the world by: the questions i have about this class are: thank you for filling this out! i look forward to getting to know more about you! raygoza transformational resistance in mathematics journal of urban mathematics education vol. 9, no. 2 149 appendix b declared students’ rights raygoza transformational resistance in mathematics journal of urban mathematics education vol. 9, no. 2 150 appendix c mathematics autobiography use the following guidelines to write a six-paragraph letter to me explaining your her/history and experience with mathematics. this letter is an opportunity for you to explore your identity as a math student (and growing mathematician!). please neatly write or type your final letter. stick to the letter format as shown below, and indent for every paragraph. august , dear ms. candace: paragraph 1: my name is . i am years-old, and i am a th grader at [our school name]. add a few more sentences about yourself here! paragraph 2: what are your strengths and weaknesses as a student? for example, strengths are things you’re good at, part of your personality that you are proud of, things people compliment you on, and so on. weaknesses are areas where you want to learn more, get stronger, places where you struggle as a student, and so on. paragraph 3: what do you think about math? do you like/love/enjoy math? why or why not? explain. paragraph 4: what have your math classes been like in the past? how did the last school year of math go for you? do not only write about your grades but your experiences learning and with your teacher. paragraph 5: why do you think math is important for you to learn? think of all the reasons you can. do you believe that math can be used to understand and change the world? why or why not? paragraph 6: what are your goals for math class this school year? list all of them and explain why you are reaching for those goals. explain who will help you to reach your goals. include anything else at the end of the letter that you would like to include. sincerely / peace / your student (pick one!), (your signature) raygoza transformational resistance in mathematics journal of urban mathematics education vol. 9, no. 2 151 appendix d work sheet – if the world were 100 people name: date: period: if we could turn the population of the earth into a small community of 100 people, keeping the same proportions we have today, the village would look something like this … # of people who … your guess the data fraction (out of 100) reduce fraction (if you can) decimal percent are asian 61 61/100 61/100 0.61 61% are european 12 12/100 3/25 0.12 12% are north american 8 are south american/caribbean 5 are african 13 are oceanian 1 are women 50 are men 50 live in urban areas 47 are disabled 12 are christians 33 are muslims 21 are hindus 13 are buddhists 6 are sikhs 1 are jews 1 are non-religious 11 practice other religions 11 are atheists 3 live without basic sanitation 43 live without an improved water source 18 own 75% of the entire world income 20 are hungry or malnourished 14 can’t read 12 have a computer 12 have an internet connection 8 lives with hiv/aids 1 live on less than $1.25 per day 21 raygoza transformational resistance in mathematics journal of urban mathematics education vol. 9, no. 2 152 appendix e inequalities in society microsoft word 2 final bullock vol 7 no 2.doc journal of urban mathematics education december 2014, vol. 7, no. 2, pp. 6–8 ©jume. http://education.gsu.edu/jume     erika c. bullock is an assistant professor of mathematics education in the department of instruction and curriculum leadership in the college of education, health and human sciences, at the university of memphis, 419a ball hall, memphis, tn, 38152; e-mail: erika.bullock@memphis.edu. her research interests include exploring urban mathematics education curriculum and policy from a critical postmodern and historical perspective. she is associate to the editor-in-chief and public stories of mathematics educators section editor of the journal of urban mathematics education. editorial public stories of mathematics educators: an invitation to tell erika c. bullock university of memphis he work of urban mathematics education is not exclusively an academic exercise; it belongs equally to mathematics education researchers, teachers, students, administrators, parents, and community members. unfortunately, there are three significant gaps that characterize discussions of urban mathematics education. first, there exists a divide between scholarly journals and non-academic audiences, which is an ongoing challenge in education research to demonstrate a bidirectional relationship between the practitioner and scholarly communities (langrall, 2014). the second gap separates the realities of education practice from the processes that govern decision-making about education practice. mathematics holds a central position in education policy discussions (steiner, 1987), but policy decisions remain prescriptive for teaching and learning (lortie, 2002) and do not take into account the lived realities of mathematics classrooms and the teachers and students that inhabit them. finally, there is a gap between generalized conceptualizations of mathematics education and the nuances of mathematics education in urban spaces. at jume, we see these gaps as opportunities for mathematics educators1 to participate in public discourse related to mathematics teaching and learning. in volume 2 issue 1, dr. lou e. matthews (2009), founding editor-in-chief of jume, issued a call to urban mathematics educators to construct and share public narratives. this call inaugurated the public stories of mathematics educators section in volume 2 issue 2. this section is a forum for mathematics educators of all stations to move outside of traditional academic discourse in an effort “to define a more people-centric mathematics education” (matthews, 2009, p. 3). matthews argued that challenging times in urban education require transformative mathematics education leaders to frame our individual stories into public narratives that deliberately work to connect us with ourselves (stories of self), with                                                                                                                 1 the term mathematics educators refers to those who study mathematics teaching and learning. this group includes, but is not limited to, mathematics education researchers, mathematics teacher educators, and mathematics teachers at all levels (pre-k−graduate). t       bullock editorial journal of urban mathematics education vol. 7, no. 2 7 others (stories of us), and with our work (stories of now). contributions to this section have ranged from teachers interrogating policy and practice (e.g., hennings, 2010), to teachers using poetry to examine the complexities of teacherstudent relationships in mathematics classrooms (e.g., ball, 2012), to teacher educators reflecting upon successful pedagogies (e.g., truxaw & rojas, 2014). the public stories section is a space for all of us to share our good work, our lingering questions, and our responses to enduring challenges. these stories address a critical question raised in the formation of jume as a scholarly space: “how should we give ‘voice’ to the complex dynamics of change within the urban domain?” (matthews, 2008, p. 2). i am writing this editorial as an invitation to our readers to join with jume in an effort to shape the discourse about urban mathematics education through public stories. our students, the teachers with whom we work, and we ourselves have stories that can contribute to larger understandings of urban mathematics education. through these stories, we reveal who we are and how our identities shape and complicate our engagement with mathematics teaching and learning. however, it is important to note that public stories are not simply reflexive exercises; in order for these stories to connect, “we will have to be deliberate about (a) our intentions to do so, (b) what we choose—or choice points—to share, (c) what the moral will be, and (d) our audience” (matthews, 2009, p. 3). deliberatelyconstructed public narratives both problematize and enrich our understandings of urban mathematics education. jume was founded in response to marginalization of urban mathematics education research in mainstream scholarly outlets (matthews, 2008). keeping with a mission to resist such marginalization within mathematics education research, the public stories of mathematics educators section stands in the gaps between research and public narrative, policy and practice, and generalized and experiential scholarship. we invite you to tell your public stories and to encourage your students and other mathematics educators to do the same. together, we can use this space to challenge dominant discourses through the transformative power of stories that support “a discourse agenda that addresses urban complexities, challenges, and excellence” (matthews, 2009, p. 1). references ball, t. n. (2012). i am from…. journal of urban mathematics education, 5(2), 53−54. retrieved from http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/175/120 hennings, j. a. (2010). new curriculum: frustration or realization?. journal of urban mathematics education, 3(1), 19−26. retrieved from http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/90/44 langrall, c. w. (2014). linking research and practice: another call to action?. journal for research in mathematics education, 45(2), 154−156.       bullock editorial journal of urban mathematics education vol. 7, no. 2 8 lortie, d. c. (2002). schoolteacher (2nd ed.). chicago, il: university of chicago press. matthews, l. e. (2008). illuminating urban excellence: a movement of change within mathematics education. journal of urban mathematics education, 1(1), 1–4. retrieved from http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/20/9 matthews, l. e. (2009). identity crisis: the public stories of mathematics educators. journal of urban mathematics education, 2(1), 1–4. retrieved from http://ed-osprey.gsu.edu/ojs/index.php/jume/article/viewfile/41/11 steiner, h.-g. (1987). philosophical and epistemological aspects of mathematics and their interaction with theory and practice in mathematics education. for the learning of mathematics, 7(1), 7–13. truxaw, m. p., & rojas, e. d. (2014). challenges and affordances of learning mathematics in a second language. journal of urban mathematics education, 7(2), 21–30. retrieved from http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/233/160 journal of urban mathematics education december 2011, vol. 4, no. 2, pp. 15–25 ©jume. http://education.gsu.edu/jume virginia (ginny) keen is an outreach mathematician at the university of dayton, 300 college park drive, dayton, oh 45469-2316; e-mail: vkeen1@udayton.edu. she teaches content courses in early and middle childhood mathematics education. her research interests include alternative assessments, error diagnosis, and the use of video and other technologies to support mathematics teaching and learning. amber rose is the director of education at the salvation army ray & joan kroc corps community center, 1000 n. keowee street, dayton, oh 45404; e-mail: amber.rose@use.sal vationarmy.org. her research interests include elementary education, experiential learning, and practical application of research that positively impacts students; she has a particular interest in integrating standards with kinesthetic learning to reach students and families who face socioeconomic challenges and help them to be successful both in the classroom and in society. public stories of mathematics educators the pen pal partnership project: connecting theory to practice virginia keen university of dayton amber rose the salvation army ray & joan kroc corps community center n this public story, we describe the design and implementation of an interdisciplinary pen pal project that pairs university students 1 taking a mathematics course for prospective teachers with local elementary children. in its first 2 years, the project involved pairing students with children in a rural school district and, in subsequent years, it has included children in an urban school district. the geographical location of the children appears to have had little effect on the goals of the project; the students and their pen pals have enriched, learning experiences each time they get together. broad support and enthusiasm for the project in the urban schools, however, have produced some remarkable and somewhat unexpected effects. while the project focuses on supporting the learning of mathematics, it also offers social development opportunities for children. the children have the opportunity to practice writing a letter to an actual person and many are exposed to college for the first time. through the project, children in urban (and rural) schools are also connected to mathematics learning in ways that supplement what the public system is able to consistently offer. in addition, the university students apply and enhance their language arts skills as they extend their own mathematics learning to include teaching and learning mathematics by and with children. in the story that follows, we detail how my (virginia’s) plan for enhancing students’ learning of mathematics initiated a community-based partnership that continues to flourish today. 1 throughout this article, the word student refers to college-age students and the word children refers to students in pre-kindergarten to grade eight, unless a part of a quote. i keen & rose public stories journal of urban mathematics education vol. 4, no. 2 16 the pen pal partnership project what began as a fairly simple alternative assessment for students in a mathematics for prospective elementary teachers course has grown to offer many enhanced learning opportunities for children, students, classroom teachers, and me, virginia (a university professor). one of the projects i assign is to create a children’s book based on the mathematical principles studied in the course and to share these books with local school children. i was working at a rural university; therefore, the local schools the students and i visited were also rural schools for which there were few extra services or opportunities for children to interact with college students. the school visits became the highlight of the course for my students. i found the visits well worth the effort as the students benefitted from opportunities to see and hear how the children understood and responded to their books. students gained a much better sense of how important clear communication of mathematical ideas is in order for children to be able to make sense of them. when i moved to my current position at the university of dayton (an urban university), i wanted to continue using the book-creation assignment. in our teacher education program, students majoring in early childhood education and intervention services (special education) take two courses in mathematics for prospective teachers, often with the same instructor. in my course sections, i continue to use the children’s book assignment. in the first course, students create a book related to learning how to meaningfully count. they visit a local preschool, read their books to preschoolers, and, in the process, learn more about what it means to count and how young children make sense of the counting process. as the preschool is associated with the university, many of the students have spent time in the facility prior to the course and are comfortable in the setting. in the second course, students create a children’s book that focuses on geometric concepts and is suitable for a first or second grader that focuses on geometric concepts. students again visit children at local schools to see first hand how their books address the needs of young learners of mathematics. as i began to familiarize myself with the university of dayton, i discovered that there is an organization on campus, the fitz center for leadership in community, 2 whose focus is building and supporting partnerships in the dayton community. one arm of this mission involves strong connections to a set of urban schools through the development of neighborhood school centers with site liaisons. thus, when i began looking to develop relationships in local schools and to find teachers who would be willing to participate in a university–school partnership, i contacted the fitz center. their staff distributed a request to partnership schools inviting those interested in engaging in a pen pal project with a university 2 see http://www.udayton.edu/artssciences/fitzcenter/. http://www.udayton.edu/artssciences/fitzcenter/ keen & rose public stories journal of urban mathematics education vol. 4, no. 2 17 mathematics class for prospective elementary teachers. two of the site liaisons, including my co-author (amber), responded quickly and the three of us began conversations about how we might carry out the project. as a way to build a relationship between my students and the children, i suggested that i have my students write two pen pal letters with mathematics activities or questions included. the letters would add another medium for exhibiting the students’ understanding of both the mathematics being studied in our class and the appropriate level of questioning for children at the elementary level. thus the pen pal partnership project was born. logistical issues such as transportation to the school sites are handled through the fitz center while the site liaisons deliver the letters and arrange the visits. the two schools that participate in this project are neighborhood school centers, part of a move back to neighborhood schools in dayton that expands the role of the school to that of neighborhood community center. the site liaisons serve the school centers under the auspices of the university of dayton, the salvation army of the greater dayton area, and the ymca of greater dayton. both of these pre-k–8 facilities receive title i funds and serve a diverse urban population. the pen pal partnership project introduces students to urban school children in their school settings. in this urban school district, funding woes prohibit most small-group tutoring opportunities during the school day and one-on-one volunteers are in short supply. this project addresses this need as each child receives one-on-one attention while partnered with a college student who reads her or his book aloud to the child. the student can help the child read the book aloud and complete the tasks woven throughout. the service-learning component that unfolds through the project reinforces the first three of the four content areas studied in the mathematics course: statistics, probability, geometry, and measurement (in order). the students’ experience of creating a written product in which they clearly present mathematical concepts and support mathematical reasoning in the form a children’s book acts as an instructional tool for the students as well as the children. adding the tasks of composing two pen pal letters not only builds connections between students and children prior to meeting but also gives the students an opportunity to practice posing problems and articulate mathematical ideas, important experiences in their mathematics teacher preparation. pen pals are assigned with consideration for matching gender and any special needs of the child. students prepare two letters, one while studying statistics and the other while studying probability, which are mailed to an elementary school pen pal prior to the school visit. each letter includes a mathematical task or experiment for the pen pal to carry out. keen & rose public stories journal of urban mathematics education vol. 4, no. 2 18 pen pal letters the students’ first letter includes a brief personal introduction and a task that requires the child to collect data from her or his classmates on a question posed in the letter, such as, “what is your favorite flavor of ice cream out of vanilla, chocolate, and strawberry?” the question is multiplechoice with fewer than six choices identified for the child to keep the focus on data collection and avoid the distraction of a large number of choices. it also allows for more reasonable graphing of the data, should the classroom teacher choose to have the children graph the results using a graphing method appropriate to their grade level, according to state standards (state board of education and ohio department of education, 2001a, 2007). students are also encouraged to keep the common core state standards for mathematics (ccssm) (common core standards initiative [ccsi], 2010) in mind. the grade 2 ccsi standards include: “draw a picture graph or a bar graph (with single-unit scale) to represent a data set with up to four categories” (p. 20). the children that participate are typically in firstand second-grade classrooms, making this connection to standards significant. although not a requirement of the project, classroom teachers have developed additional in-class activities motivated by the student letters and the technology available for children to vote and create graphs each week during the project. two teachers extended the project to their language arts curriculum by having their children write letters back to their university pen pals that describe their findings using an appropriate informal letter format. considering the children’s return letters, an important issue came to light. i realized that the children were being taught to use a standards-based informal letter-writing format but the college students were not following the same format. knowing that my students needed to model good letter writing, i altered the assignment to include instructions for writing an informal letter in the assignment description based upon the concept of a friendly letter as outlined in the state standards (state board of education and ohio department of education, 2001b). i keen & rose public stories journal of urban mathematics education vol. 4, no. 2 19 also altered the rubric so that the grading of the pen pal letters now takes students’ letter-writing format into consideration. when the students introduce themselves to their pen pal in the form of a friendly letter, it reinforces the state standards and generates a positive mentor–mentee relationship. this exercise is one of the first opportunities for the prospective teachers to see standards use from the teacher’s perspective and highlights the importance of “thinking like a teacher” about the standards for various grade levels. this integration of language arts also underscores the interdisciplinary work required of the classroom teacher. students write the second pen pal letter while studying probability and chance. because this letter may arrive at the school only a short time prior to our visit, the experiments they ask the children to complete can be easily done with a partner, preferably a family member. not assuming that the children have easy access to coins, dice, or spinners, the students include the tools necessary for carrying out the experiment in the envelope with the letter. this inclusion raises an additional question that the prospective teachers often have not considered: what size coin should be included for a coin experiment? to avoid any problems in which children are unhappy with the coin in their letter, students are asked to include only pennies for coin experiments. creative students design experiments using cutouts and colorful objects that the children can use. some teachers use this as an opportunity to have pairs carry out the experiments in class and record their results to share with their pen pal when they meet. introducing children to the concepts of likelihood and uncertainty through simple experiments like tossing a coin or rolling a die several times and then asking children to make predictions or look for patterns in the results can serve to introduce them to probabilistic thinking and develop mathematical habits of mind such as predicting, conjecturing, revising, and considering alternative explanations. this process fits well with the grades 1 and 2 standard for probability and data analysis (state board of education and ohio department of education, 2001a, 2007). students are encouraged to include thoughtful questions about the keen & rose public stories journal of urban mathematics education vol. 4, no. 2 20 possible results of the experiments that they can ask their pen pals when they visit. pen pal visit once the students have written their letters and completed their work in statistics and probability, they move on to develop a book based on the state standards and geometry concepts covered in the course including the extended van hiele theory of geometric reasoning that is used as the framework for the children’s book on geometry. 3 students use a variety of modes for planning and creating the books, such as index cards, electronic storyboards, scrapbooking materials, and movable pieces, to make the books meaningful, content-rich, and interactive. 4 for example, one student used magnetic sheets cut into tangram pieces with magnetic pages on which the child was to create various figures in the story. another student included a plastic reflector in her book on symmetry. some students use their pen pals as the central figures in their books, personalizing the books as a way to deepen their connection to the children. after students develop their books, we take a fieldtrip to the school to meet their pen pals. the van ride to the school is a quiet one, with students silently wondering what their pen pal is like and if the pen pal will like them. they need not worry; even the most difficult child in a classroom is thrilled to see her or his college pen pal. the groundwork laid by the site liaison ensures a smooth flow through the building to the classroom. the classroom teacher beams as the children find their pen pals with the assistance of the nametags that they have provided. each student is matched with at least one pen pal. the noise level increases as the children and students exchange greetings and find a place for their conversation. some prefer sitting on the floor, others find room on the class carpet. once, we needed to spread out on the school stage; another time, we took over the school cafeteria because we had too many students and pen pals to accommodate 3 the van hiele theory posits that students progress through levels of reasoning that can be described as: level 0 – pre-recognition level, where the student does not reliably distinguish between classes of figures; level 1 – recognition or visual level, where the student recognizes shapes holistically; level 2 – analysis level, where the student can describe attributes of a figure; level 3 – relationship or informal deduction level, where the student recognizes interrelationships of figures and their properties; level 4 – deductive reasoning level, where the student can use deductive reasoning in proofs; and level 5 – rigor level, where the student recognizes that different axiomatic systems result in different geometries. (see van hiele, 1959 or clements & battista, 1992 for description of this theory.) 4 see keen (2004, 2007) for a description of the children’s book assignment and examples of student work. keen & rose public stories journal of urban mathematics education vol. 4, no. 2 21 in a single classroom. during each visit, the classroom teachers, site liaison, and i zigzag among the groups, observing the pairs as they turn pages and explore the ideas presented on each page. the atmosphere is always full of enthusiasm with a wide range of discussions as the students share their books with the children. the children ask, “did you make this?” “was it hard?” “do you go to college?” “would you like me to read the book?” after the book has been read and reread, some pairs exchange books with others, some visit special places in the room, and some carry out the probability experiment from the second pen pal letter. 5 as the meeting progresses, the students begin to think about the mathematics from a different perspective, seeing it through the eyes of their pen pals. students are encouraged to pay special attention to the ways the children demonstrate their mathematical understanding or their level of geometric reasoning. occasionally, the nature of the conversations surprises the students. for example, one student was talking with her pen pal about the angles in a rectangle as right angles. the child joyfully walked around the classroom identifying right angles, until she came to a “left angle.” the first grader was associating her learning of left and right, as directions, to the angles that she saw. this association was quite surprising to the student and prompted her to think about mathematics vocabulary and everyday language in a deeper way. departure time comes quickly; we say our goodbyes and return to the van. the trip back has a very different mood. the students excitedly share their experiences with their peers, laughing and talking about what they learned from their pen pal. they finger through their books, pointing out things that surprised or delighted them. i remind them to be sure to write about all that they discovered about how children learn and understand mathematics through the whole experience in their biweekly journal entry. the journal entries show the significant “in-person” value of our visit. for example: student 1: i learned a lot while at [*] elementary school, and while constructing my children’s book. first, i learned that you have to be very simple and precise when teaching children new mathematical concepts. all of the drawings in my 5 for a more comprehensive visual sense of the experience, see the book sharing collage in appendix a. keen & rose public stories journal of urban mathematics education vol. 4, no. 2 22 book had to be very accurate so not to confuse the student i was teaching these new shapes to. student 2: i learned a lot about math learning while i was there. i realized that you can’t just show a student a picture of a shape once and expect them to know it the second time around. it takes time and practice to learn a mathematic concept. learning mathematics isn’t that easy of a task. a task that may seem easy for us is not easy for a 1st grader and that is a very important concept to keep in mind. (emphasis in original) student 3: about math learning, i learned that the proper terminology in geometry was not being taught. while reading my book, christina kept calling an ellipse an oval. i know many kids also call kites diamonds. in my opinion, wouldn’t it be easier to teach kids the proper term in the beginning instead of trying to change names in later grades? student 4: as a teacher, i learned that if the student can teach themselves the concept, or find their own conclusion, it is very beneficial. that is, my student was able to find that a 5-sided figure was a pentagon, because i told her pent meant 5. then i said if hex means 6, then what is a 6-sided figure? she was able to make her own conclusion, and these were the two new shapes she remembered the second time we went through the book. student comments focus on the mathematical thinking and mathematics learning of their pen pals through their interaction, which is appropriate for their journal, but the classroom visit also opens the door to the world of urban education for many of the students. the university offers an urban teacher academy that works to prepare preservice teachers to work in an urban environment. a positive first experience in an urban classroom prior to the application period for the urban teacher academy increases the likelihood that a student will apply to the program. for the university, this program is a positive way to support a significant community need for future teachers. even if the students decide not to apply to the urban teacher academy, this experience offers valuable learning in an urban environment that the students may not receive through their formal placements. the pen pal partnership project worked so well at one of the schools that the teachers in the upper grades of the school set up a second pen pal project with black action thru unity, a campus-based organization for african american students. this organization works to be a positive influence on campus and in the keen & rose public stories journal of urban mathematics education vol. 4, no. 2 23 community, so developing a mentor–mentee relationship through the process of being pen pals also fits their goals. being a catalyst for this offshoot reinforces the concept that urban children are often in need of and benefit from mentor–mentee relationships and that universities can create meaningful service-learning opportunities within local school districts through interdisciplinary prospective and preservice teacher projects. conclusion considerations for improving the pen pal partnership project include the introduction of students to children prior to their meeting to share the books. technology opportunities that exist for teleconferencing at the school and university level are a consideration for further developing this program. the enthusiasm and support offered by site coordinators dedicated to school and community engagement allow follow-up volunteer opportunities that increase student learning while providing a valuable service to the children and school. with the opening of the salvation army ray and joan kroc corps community center in downtown dayton (amber is the center’s director of education), we have a new site for contributing to children’s opportunities to learn mathematics through the books the students create, with some producing extra copies for the center. the kroc center includes an education center with a dropin care facility, technology resources (including 60 laptops and 4 gaming systems), and a library of resources for families to use. an afterschool program at the kroc center, “authors and illustrators,” has used children’s enthusiasm in reading pen pal books to guide second and third graders in the creation of books for their own families. for the college student, this project serves as their initial exposure to the world of urban education, and, in some cases, the first opportunity to develop a written lesson (in the form of the book) that also serves as a valuable piece for their professional portfolio. as their journal entries indicate, the students gain insight into the importance of representations and clear, accurate language in support of meaningful mathematics learning. for the professor, the project allows for experiential learning and real-life challenges that supplement coursework while requiring students to apply the statistics, probability, and geometry content included in the course. the children and school reap some excellent benefits as well. through the project, the elementary teacher is able to develop cross-curricular activities incorporating mathematics and language arts while encouraging the children’s enthusiasm about college. the children look forward to their next mathematics challenge and are proud to do the work asked of them and learn graphing techniques. their dedication to impressing their college student pen pal is evident in their behavior keen & rose public stories journal of urban mathematics education vol. 4, no. 2 24 and sometimes-meticulous completion of assignments. when they write the prospective teachers, the children must answer the mathematical challenges, write and format a friendly letter, and show good penmanship for their letter to be approved by the teacher, thus reinforcing various interdisciplinary standards. classroom teachers report that the children in the grade below theirs come to ask if they will be able to have a college pen pal the next year, indicating that children throughout the school prize the experience. as an extension, i have presented the project at a conference with one of the teachers and students (sally kleiner and kim smethurst, respectively). i anticipate doing more of this sort of outreach beyond our partnership. the value of community partnerships for offering opportunities to build mathematical foundations for children should not be underestimated. while this is an annual, smallscale project involving two urban schools, two classes of children, two classes of college students, two classroom teachers, two school liaisons, and a college instructor, its effects are cumulative over the years, building a strong sense of collaboration and shared responsibility for the mathematics education of all children. references clements, d. h., & battista, m. t. (1992). geometry and spatial reasoning. in d. a. grouws (ed.), handbook of research on mathematics teaching (pp. 420–464). reston, va: national council of teachers of mathematics. common core standards initiative. (2010). common core state standards for mathematics. retrieved from http://www.corestandards.org. keen, v. (2004). recapturing the “disenchanted:” orienting prospective primary teachers toward problem posing and deeper understanding of the mathematics they will teach. discussion paper for dg 18: current problems and challenges in primary mathematics education, 10 th international congress on mathematics education, copenhagen, denmark. retrieved from http://www.icme-organisers.dk/dg18/. keen, v. (2007). penpals, children’s books, and learning mathematics. in d. k. pugalee, a. rogerson, & a. schinck (eds.), proceedings of the 9th international conference mathematics education in a global community (pp. 353–358). charlotte, nc: center for mathematics, science & technology education, university of north carolina-charlotte. retrieved from http://math.unipa.it/~grim/21_project/21_charlotte_keenpaperedit.pdf. state board of education and ohio department of education (2001a). ohio academic content standards for mathematics. columbus, oh: ohio department of education. state board of education and ohio department of education (2001b). ohio english language arts academic content standards. columbus, oh: ohio department of education. state board of education and ohio department of education. (2007). ohio early learning – primary content standards for mathematics. columbus, oh: ohio department of education. van hiele, p. (1959). levels of mental development in geometry. retrieved from http://www.math.uiuc.edu/~castelln/vanhiele.pdf. http://www.corestandards.org/ http://www.icme-organisers.dk/dg18/ http://math.unipa.it/~grim/21_project/21_charlotte_keenpaperedit.pdf http://www.math.uiuc.edu/~castelln/vanhiele.pdf keen & rose public stories journal of urban mathematics education vol. 4, no. 2 25 appendix a book sharing collage microsoft word final tate2 jume vol 1 no1.doc journal of urban mathematics education december 2008, vol. 1, no. 1, pp. 5–9 ©jume. http://education.gsu.edu/jume 
 william f. tate is
chair of the department of education, director of the st. louis center for inquiry in science teaching and learning, and edward mallinckrodt distinguished university professor in arts & sciences at washington university in st. louis, one brookings drive, campus box 1183, st. louis, mo 63130-4899; e-mail: wtate@wustl.edu. he is immediate past president of the american educational research association. tate’s interdisciplinary scholarship concentrates on two main areas: mathematics, science, and technology education, specifically, in metropolitan america; and the social determinants of education and health disparities. commentary putting the “urban” in mathematics education scholarship william f. tate washington university in st. louis how many candidates running for local, state, or federal office in the 2008 elections in the united states highlighted the importance of urban america as a site of opportunity, or even challenge? briggs (2005) argued that the geography of opportunity in education, employment, safety, health, and other vital areas of the next generation are invisible in the nation’s public life and agenda. in her classic book titled the death and life of great american cities (1992), the late jane jacobs argued a successful city neighborhood is a place that is sufficiently aware of its problems so it is not defeated by them. in contrast, an unsuccessful neighborhood is a place that is engulfed by its deficiencies and is increasingly more powerless before them. she argued we americans are poor at managing city neighborhoods as documented by the long collection of failures. her treatise is one of numerous scholarly projects that underscore the unique importance of recognizing the urban context as a powerful influence on human development broadly defined (orfield, 2002; pattillo, 2007; rusk, 2003). the purpose of this commentary is to serve as a warning that developing and testing theories is central to making urban mathematics scholarship a visible research enterprise. more specifically, i will argue that there are lessons to be learned from the social sciences literature that can inform the advancement of a robust, theoretically based, empirical project in urban mathematics education research. in addition, these fields of social science are part of the rationale for why putting the “urban” in mathematics education scholarship is important. perhaps there are some scholars who accept the notion of research focused on the urban context as relevant and of great consequence. they understand that urban cities and communities are unique contexts that require research and policy evaluation to support their governance function. not everyone accepts this notion. is there a growing research literature related to urban communities in mathematics education? unfortunately, the answer is clear. too many education researchers ignore tate commentary journal of urban mathematics education vol.1, no.1 6 geospatial considerations. my hope is that the journal of urban mathematics education will create a new marketplace where theories related to urban cities and metropolitan regions across the world can be empirically tested and evaluated. a major point of emphasis for the scholar interested in urban mathematics education is theory building and empirical evaluation. if there are no theories (small or grand) to test and evaluate akin to the efforts in other social science research, then the field will yield little more than polemic and empty ideology. to date, both are plentiful. a brief review of the social sciences literature will illustrate the importance of geography as part of theoretical construction and testing. urban economics is a branch of microeconomics that examines urban spatial structure and the location of households and firms (o’flaherty, 2005). the urban economics literature includes the study of industrial clusters and technologybased hubs in metropolitan communities across the globe (gordon & mccann, 2000; sorenson, 2003). how industries cluster is directly related to a range of social factors including employment opportunities and tax capacity. incidentally, these two factors influence the quality and financial support for education (orfield, 2002). employment rates and tax capacity are important constructs in school finance. moreover, employment opportunities and tax capacity are a part of an expanding literature in urban sociology. the point is that economic theories make it possible to test the nature and extent of relationships within economics and across fields of study including sociology. urban sociology is the scientific study of social relations, human life, and human behavior in metropolitan areas. in this field, the chicago school has been a major influence. for example, both social disorganization theory and the spatial mismatch hypothesis have been tested and studied as a part of this urban research tradition (bursik, 1988; fosterbey, 2006; wilson, 1996). the point here is not to review these two theories; rather, the intent is to make clear that there are important theoretical projects being tested and vigorously debated. if urban mathematics education research is to be taken seriously, this kind of theoretical and empirical interaction should be the norm. theory-driven, empirical research is the norm in other fields of social science as well. the political science literature includes a sub-field in urban politics (brunori, 2003; judd & swanstrom, 2008). urban regime theory is prominent in this field (stone, 1993). the urban public health literature includes the examination of medical resources, risk factors, and disease (airhihenbuwa & liburd, 2006; douglas, esmundo, & bloom, 2000; jones-webb & wall, 2008). epidemiological theories and method are central to urban public research. the literatures of community psychology and the developmental sciences examine child and adolescent development and cognitive outcomes in a variety of urban settings (lee, 2008; spencer, 2008; spencer, dupree, cunningham, harpalani, & muñoz-miller, 2003). in a range of research fields the study of social interaction tate commentary journal of urban mathematics education vol.1, no.1 7 in the urban metropolitan regions of the world is ongoing. this study of social interaction has been the case in education research as well. journals such as urban education and education and urban society have articles that are retrievable in electronic databases dating back to the 1960s. urban education published a special issue focused on mathematics education (tate, 1996). in sum, there is a long history of research that has taken seriously urban geography and related social interactions. this history suggests there is an intellectual space for urban mathematics education research. this intellectual space calls for scholars to fill the void. urban mathematics education is a rich topic with significant policy implications. during the 1980s and 1990s, both the ford foundation and national science foundation invested in mathematics education reform efforts and related evaluation studies in cities across the united states (campbell, bowden, kramer, & yakimowski, 2003; kim, crasco, blank, & smithson, 2001; silver & stein, 1996; webb & romberg, 1994). these large-scale interventions and evaluation studies brought attention to the topic of research and urban mathematics education. there is other mathematics education research focused on coursetaking, teacher quality, and assessment practice that has a spatial dimension (anderson & tate, 2008). the geography of opportunity has been central to the mathematics education research involving urban communities. however, there are two interrelated challenges that must be addressed if this scholarship is to flourish going forward. the first challenge involves theory. there is a need for theory building, testing, revision, and retesting. there are important lessons to be learned from closely examining the history of research in urban economics, urban sociology, urban health, urban politics, and community psychology. a second challenge is related to collective cognition. in their award winning book titled building civic capacity: the politics of reforming urban schools, urban regime theorists, stone, henig, jones, and pierannunzi (2001) contended that collective cognition matters when the goal is to take on the task of problem solving in urban school reform. to this end, i have argued elsewhere that urban communities are in desperate need of research consortiums where the distinguishing features are comprehensive data archives that provide sustained opportunities to study and learn about human development in the region (tate, 2008). the data archives should include at minimum the theoretically important measures related to urban mathematics education. in addition, this intellectual space is where researchers and practitioners should test and retest the theoretical project and push the boundaries of new knowledge. the challenge is to build theories and models that realistically reflect how geography and opportunity in mathematics education interact. if this challenge is addressed, the field will be one step closer to making scholarship in urban mathematics education visible. tate commentary journal of urban mathematics education vol.1, no.1 8 acknowledgments a special thank you is extended to celia keiko anderson, debra barco, richard milner, and dorothy white for their feedback on this commentary. this article is based on research and development supported by the national science foundation under award no. esi-0227619. any opinions, findings, and conclusions or recommendations expressed here are those of the author and do not necessarily reflect the views of the national science foundation. references airhihenbuwa, c. o., & liburd, l. (2006). eliminating health disparities in the african american population: the interface of culture, gender, and power. health education behavior, 33(4), 488–501. anderson, c. r., & tate, w. f. (2008). still separate, still unequal: democratic access to mathematics in u.s. schools. in l. english (ed.), international handbook of research in mathematics education (pp. 299–318). london: taylor and francis. briggs, x. (ed.). (2005). the geography of opportunity: race and housing choice in metropolitan america. washington, dc: brookings institution press. brunori, d. (2003). local tax policy: a federalist perspective. washington, dc: urban institute press. bursik, r. (1988). social disorganization and theories of crime and delinquency. criminology, 26(4), 519–551. campbell, p. f., bowden, a. r., kramer, s. c., & yakimowski, m. e. (2003). mathematics and reasoning skills (no. esi 95-54186). college park, md: university of maryland, mars project. douglas, l., esmundo, e., & bloom, y. (2000). smoke signs: patterns of tobacco billboard advertising in a metropolitan region. tobacco control, 9, 16–23. foster-bey, j. a. (2006). did spatial mismatch affect male labor force participation during the 1990s expansion? in r. b. mincy (ed.), black males left behind (pp. 121–147). washington, dc: urban institute press. gordon, i. r., & mccann, p. (2000). industrial clusters: complexes, agglomeration, and/or social networks? urban studies, (37)3, 513–532. jacobs, j. (1992). the death and life of great american cities (2nd ed.). new york: vintage books. jones-webb r., & wall, m. (2008). neighborhood racial/ethnic concentration, social disadvantage, and homicide risk: an ecological analysis of 10 u.s. cities. journal of urban health, 85(5), 662–675. judd, d. r., & swanstrom, t. (2008). city politics: the political economy of urban america. new york: pearson. kim, j. j., crasco, l. m., blank, r. k., & smithson, j. (2001). survey results of urban school classroom practices in mathematics and science: 2000 report. norwood, ma: systemic research. lee, c. d. (2008). the centrality of culture to the scientific study of learning and development: how an ecological framework in education research facilitates civic responsibility. educational researcher, 37(5), 267–279. o’flaherty, b. (2005). city economics. cambridge, ma: harvard university press. tate commentary journal of urban mathematics education vol.1, no.1 9 orfield, m. (2002). american metropolitics: the new suburban reality. washington, dc: brookings institution press. pattillo, m. (2007). black on the block: the politics of race and class in the city. chicago: university of chicago press. rusk, d. (2003). cities without suburbs (3rd ed.). washington, dc: woodrow wilson press. silver, e. a., & stein, m. k. (1996). the quasar project: the “revolution of the possible” in mathematics instructional reform in urban middle schools. urban education, 30(4), 476– 521. sorenson, o. (2003). social networks and industrial geography. journal of evolutionary economics, 13(5), 513–527. spencer, m. b. (2008). lessons learned and opportunities ignored since brown v. board of education: youth development and the myth of a color-blind society. educational researcher, 37(5), 253–266. spencer, m. b., dupree, d., cunningham, m., harpalani, v., & muñoz-miller, m. (2003). vulnerability to violence: a contextually-sensitive, developmental perspective on african american adolescents. journal of social issues, 59(1), 33–59. stone, c. n. (1993). urban regimes and the capacity to govern: a political economy approach. journal of urban affairs, 15(1), 1–28. stone, c. n., henig, j. r., jones, b. d., & pierannunzi, c. (2001). building civic capacity: the politics of reforming urban schools. lawrence, ks: university of kansas press. tate, w. f. (ed.) (1996). urban schools and mathematics reform: implementing new standards [special issue], urban education, 30(4). tate, w. f. (2008). “geography of opportunity”: poverty, place, and educational outcomes. educational researcher, 37(7), 397–411. webb, n. l., & romberg, t. a. (eds.) (1994). reforming mathematics education in america’s cities: the urban mathematics collaborative project. new york: teachers college press. wilson, w. j. (1996). when work disappears: the world of the new urban poor. newyork: knopf. microsoft word final stinson vol 3 no 2.doc journal of urban mathematics education december 2010, vol. 3, no. 2, pp. 1–11 ©jume. http://education.gsu.edu/jume 
 david w. stinson is an associate professor of mathematics education in the department of middle-secondary education and instructional technology in the college of education, at georgia state university, p.o. box 3978, atlanta, ga, 30303; e-mail: dstinson@gsu.edu. his research interests include exploring socio-cultural, -historical, and -political aspects of mathematics and mathematics teaching and learning from a critical postmodern theoretical (and methodological) perspective. he is a co-founder and current editor-in chief of the journal of urban mathematics education. editorial how is it that one particular statement appeared rather than another?: opening a different space for different statements about urban mathematics education david w. stinson georgia state university ver since i was provided the learning opportunity as a doctoral student of mathematics education at the university of georgia to explore the philosophy of of the french postmodern1 philosopher, sociologist, and historian michel foucault (see, e.g., 1966/1994, 1969/1972, 1975/1995, 1976/1990),2 i have been seduced into an intellectually reasoned different way of thinking about the human project of science and specifically, the human project of mathematics education science. the seduction, in actuality, was not all that difficult, given that in many ways the science of mathematics provides the roots of postmodern thought (tasić, 2001). postmodern thought in general, and foucault’s philosophyscience (no hyphen, intentionally) in particular, for me, brings clarity to the fact that the project of science is “always already entangled” with philosophy (st. pierre, in press)—a project that becomes increasingly dangerous when attempts are made to deentangle (or make invisible) this inextricable entanglement. this clarity of entanglement, in due course, brings an understanding that all science (social or otherwise)—thus, all knowledge—is a discursive formation (foucault, 1969/1972). “not an atemporal form, but a schema of correspondence 























































 1 often the words postmodernism and poststructuralism are used interchangeably in the literature, but there are acknowledged differences in the terms (for a brief discussion see peters & burbules, 2004). here, i use the term postmodern as an umbrella term for postmodernism and poststructuralism. it is also important to note that the term postmodernism “is what the french learned the americans were calling what they were thinking” (rajchman, 1987, p. 49). foucault (1983/2003), for example, in an interview once remarked, “i have never been a freudian, i have never been a marxist, and i have never been a structuralist” (p. 84). and later in the same interview he sarcastically asked, “what are we calling postmodernity? i’m not up to date” (p. 92). 2 see baker and heyning (2004) and walshaw (2007) for discussions of the uses of foucault in education. e 
 
 
 stinson editorial journal of urban mathematics education vol. 3, no. 2 2 between several temporal series” (p. 74), discursive formations are the dispersion and redistribution of statements—“neither visible nor hidden” (p. 109)—that bring into existence the possibilities (and impossibilities) to speak particular knowledge discourses. but within a foucauldian perspective, discourses are not the mere intersections of things and words that might be heard or read but rather discursive practices rendered irreducible to language or to speech that systematically form the possibilities and impossibilities of the objects, identities, activities, and so forth which might be (are) brought into existence (p. 49). knowledge (“scientific” or otherwise), therefore, no longer maintains its privileged status as an “objective” reality but rather is subjected to and limited by the socio-cultural, -historical, and -political assumptions, conditions, and power relations (foucault, 1976/1990)—in general, the discursive practices—available within a particular episteme (foucault, 1966/1994). by episteme, foucault (1969/1972) meant the set of relations that unite the discursive practices of a given period which give rise to the existence and operation of epistemological figures, sciences, and, at times, formalized systems. it, however, is not to be understood as a form of knowledge or type of rationality that crosses boundaries of various sciences but rather as “the totality of relations that can be discovered, for a given period, between the sciences when one analyses them at the level of discursive regularities” (p. 191). through his archaeological analyses of discursive regularities, foucault, asked: “how is it that one particular statement appeared rather than another” (p. 27, emphasis added)? foucault described his analysis of statements as a historical analysis, but one that avoids all interpretation: it does not question things said as to what they are hiding, what they were ‘really’ saying, in spite of themselves, the unspoken element they contain, the proliferation of thoughts, images, or fantasies that inhabit them; but on the contrary, it questions them as to their mode of existence, what it means to have them to have come into existence, to have left traces, and perhaps to remain there, awaiting the moment when they might be of use once more; what it means to them to have appeared when and where they did—they and no others. (p. 109) the surveilled and disciplined (foucault, 1975/1995) discursive formation mathematics education research, i believe, is riddled with statements of fictions, fantasies, and plays of power (walkerdine, 2004) that are brought into existence, appear and reappear, and leave traces that too often lead to devastating consequences for children. anyone who teaches (or has taught) mathematics to girls (or female students of any age) bears witness to the effects on children (and adults) of the lingering traces of the forever appearing and reappearing statement: sex differences in achievement in and attitude toward mathematics result from superior male mathematical ability. likewise, for those who teach (or have taught) mathematics in racially, ethically, linguistically, and/or socioeconomically diverse urban schools (or schools in general) bears witness to the effects on children (and 
 
 
 stinson editorial journal of urban mathematics education vol. 3, no. 2 3 adults) of the lingering traces of the forever appearing and reappearing statements: white, non-hispanic children and asian children usually enter kindergarten with greater mathematical knowledge than black and hispanic children and/or although low-income children have pre-mathematical knowledge, they do lack important components of mathematical knowledge. the three aforementioned statements are direct quotes from highly influential journals, policy documents, and handbooks; i, however, do not provide the customary citations of the statements here (see martin, 2009a, 2009b, 2010 for an analysis of statements such as the latter two). my intention is not to indict those who spoke/wrote these particular statements; they did not bring these statements into existence.3 but rather to use these particular statements as an illustration of how statements within the episteme of modernity morph and proliferate, appear and reappear, leaving traces not only within the discursive practices of mathematics education research and policy and mathematics teaching and learning but also, and more importantly, within society at large. but what happens when different statements, different discursive regularities are made available that might betoken different possibilities for the discursive practices of mathematics education research, specifically, within an urban context? opening such possibilities for difference within urban mathematics education research and, in turn, mathematics education policy and mathematics teaching and learning was the chief motivating factor behind the development of the journal of urban mathematics education (jume). this issue of jume marks the close of its third year. as the editorial team4 looks forward to many more productive years, to commemorate the occasion, i summarize the scholarship made available over the past 3 years here. first, however, it is important to note that each member of the editorial team works within 























































 3 for instance, consider another such similar statement spoken/written in the early 1780s by thomas jefferson: “comparing them by their faculties of memory, reason, and imagination, it appears to me, that in memory [blacks] are equal to the whites; in reason, much inferior, as i think one could scarcely be found capable of tracing and comprehending the investigations of euclid, and that in imagination, they are dull, tasteless, and anomalous” (as cited in tate, 1995, p. 191). it is important to note that by quoting thomas jefferson, i am not suggesting that he brought the statement into existence—he did not—but to demonstrate that the appearance and reappearance of such statements are part of the consequentially limiting discursive formation of the “white male math myth,” which has become a meta-narrative (cf. lyotard, 1979/1984) within modern western culture (stinson, 2010a). 4 the original editorial team included lou matthews, the founding editor-in-chief, and associate editors pier junor clarke, ollie manley, christine thomas, and myself. in the summer of 2009, i became the editor-in-chief; lou matthews continues to serve on the advisory board. in the spring of 2010, erika bullock and christopher jett joined the team as assistant editors, and later in the fall of 2010, nermin bayazit, stephanie behm cross, and iman chahine joined the team as associate editors. 
 
 
 stinson editorial journal of urban mathematics education vol. 3, no. 2 4 different philosophical/theoretical paradigms; therefore, the aforementioned postmodern perspective regarding philosophyscience is my own, and not necessarily shared by all members of the team. but what we do share unanimously is a desire to assist in providing different statements about and, in turn, the possibilities of different discursive practices for the children and adults who populate urban mathematics classrooms in the united states, and throughout the world. lou matthews (2008), in his inaugural editorial, described the painstaking deliberations during the nearly two-year developmental stages of jume. deliberations that, although smaller in scale, appear to be similar to those during the development of the journal for research in mathematics education (jrme) in the late 1960s (johnson, romberg, & scandura, 1994)—nearly 40 years later, the flagship journal of mathematics education research. nevertheless, as we worked through the start-up logistics of a peer-reviewed, online journal; speculated about its long-term sustainability; and struggled with the multiple meanings of “urban”; we also decided on a mission statement that guided our work, and continues to do so today: to foster a transformative global academic space in mathematics that embraces critical research, emancipatory pedagogy, and scholarship of engagement in urban communities. but before i summarize the scholarship within its online pages, i provide a description of jume by the numbers to illustrate its growth. in five issues (including this issue), we have published, in total, over 500 online pages of scholarly editorials, commentaries, public stories, research articles, and book reviews. registered users5 of jume have grown from 226 at the end of 2008 (the inaugural issue)6 to over 700 at the end of 2010, with nearly 200 registered reviewers7 and authors. and the total number of web views of jume content is approaching 12,000 as of december 2010. since the initial call for manuscripts in january 2008, jume has received over 60 research manuscripts for double-blind peer review; 16 have been published, providing a research manuscript acceptance rate around 25%. in addition 























































 5 initially, to gain access to jume content, users had to complete a free registration process. beginning in the summer of 2009, however, registration was no longer required. this change in registration process facilitated jume content being available in google scholar web searches. nevertheless, the number of registered users continues to grow. 6 it is interesting to note that the sample journal research in mathematics education, the prototype for jrme, was distributed freely to 150 persons by u.s. postal mail in 1967 (johnson, romberg, & scandura, 1994). 7 see reviewer acknowledgement: january 2008–december 2009, vol. 2, no. 2, pages 72–73. 
 
 
 stinson editorial journal of urban mathematics education vol. 3, no. 2 5 to these 16 research articles, jume has published 5 commentaries, 4 public stories, and 3 book reviews; these manuscripts (solicited and unsolicited) go through an open peer-review process conducted by members of the editorial team. in total, jume has published 33 manuscripts, including the editorial of each issue. as jume has continued to grow, we have added sections in hopes of increasing both the number of readers and those who might contribute manuscripts, adding the public stories of mathematics educators section with the 2009 fall/winter issue, and the response commentary section with this current issue.8 as mentioned previously, jume has published editorials (matthews, 2008, 2009; stinson, 2009, 2010b), commentaries (gutstein, 2010; martin, gholson, & leonard, 2010; tate, 2008), response commentaries (battista, 2010; confrey, 2010) public stories (dawson, 2009; hennings, 2010; mcqueen, shaheed, goings, & chahine, 2010; williams, 2009), and book reviews (jett, 2009; lemonssmith, 2010; wamsted, 2010). these scholarly contributions have been instrumental in the continued growth and success of jume as an outlet of engagement for urban mathematics educators. here, however, i summarize only the 16 research articles, authored by 37 education and mathematics education researchers, illustrating how these researchers, individually and collectively, provide different statements and, in turn, the possibilities of different discursive practices for students and teachers in urban mathematics classrooms. by and large, theoretically and methodologically, the research articles can be characterized as being in the social-turn (mid 1980s–) or sociopolitical-turn (2000s–) moments of mathematics education research (stinson & bullock, 2010).9 lerman (2000) described the social turn in mathematics education research as signaling something different; namely, the emergence of theoretical perspectives that “see meaning, thinking, and reasoning as products of social activity” (p. 23, emphasis in original). and more recently, gutiérrez (2010) marked the sociopolitical turn in mathematics education research as signaling “the shift in theoretical perspectives that see knowledge, power, and identity as interwoven and arising from (and constituted within) social discourses,” asserting that researchers “who have taken the sociopolitical turn seek not just to better understand mathematics education in all of its social forms but to transform mathematics education in ways that privilege more socially just practices” (p. 4, emphasis in original). 























































 8 to learn more about the types of manuscripts jume accepts for publication consideration, see “section polices” under “about” on the home web page: http://education.gsu.edu/jume. 9 erika bullock and i recently identified four distinct, yet overlapping and simultaneously operating, historical moments in mathematics education research: the process-product moment (1970s–), the interpretivist-constructivist moment (1980s–), the social-turn moment (mid 1980s–), and the sociopolitical-turn moment (2000s–). each continuing moment—hence, no end date—explores mathematics teaching and learning from different theoretical perspectives and employs different methodological procedures derived from a variety of academic disciplines. 
 
 
 stinson editorial journal of urban mathematics education vol. 3, no. 2 6 the social and/or sociopolitical turns of mathematics education are clearly evident in the four research articles in the december 2008 inaugural fall/winter issue of jume. davis and martin (2008) highlight how the test-driven instructional practices of the no child left behind act simultaneously reflect wellintentioned efforts of teachers and construct racial hierarchies that posit african american children as lacking in mathematical ability. skovsmose, scandiuzzi, valero, and alrø (2008) explore the mathematics learning experiences of five adolescents from a brazilian favela, asserting that understanding learning as relating to both students’ past (background) and future (foreground) acknowledges that students’ sense making of schooling in general, and of mathematics education in particular, is not only cognitive but also sociopolitical. leonard and evans (2008) demonstrate that community-based field experiences in pre-service mathematics teacher education positively influences the beliefs of pre-service teachers about urban children and communities and increases their capacity to teach urban children in culturally sensitive ways. and paek (2008) provides a review of 22 practices worthy of attention in her search for existence proofs of promising practitioner work of mathematics teachers in urban secondary schools. the 2009 volume 2 of jume marked the first year in which both spring/summer and fall/winter issues were published; a total of six research articles are available. rousseau anderson and powell (2009) employ critical race theory in an analysis of two school districts—one urban and the other rural—and suggest a “metropolitan perspective” within urban mathematics education research and policy that takes into account the interrelationships between urban cities and their suburban or rural neighbors. gonzalez (2009) reports on the everdeveloping identities of seven new york city high school mathematics teachers within a community of practice that focused on learning to teach mathematics for social justice. kitchen, cabral roy, lee, and secada (2009) conduct interviews with 32 fourth-grade teachers from two urban school districts to determine what distinguishes “highly effective” from “typical” elementary schools by examining teachers’ conceptions of mathematics and student diversity. pourdavood, carignan, and king (2009) describe the voices and practices of four mathematics teachers in a k–7 “coloured” township school in the eastern cape of south africa, underscoring the complexities between mathematics education reform and socio-cultural and -historical contexts. esmonde, brodie, dookie, and takeuchi (2009) investigate the racialized and gendered nature of groupwork in a heterogeneous urban high school mathematics classroom where students named interactional style, mathematical understanding, and friendships and relationships as the most influential factors in determining if a group works well. and capraro, young, lewis, yetkiner, and woods (2009) extend the “achievement gap” conversation by suggesting that investigating early trends in mathematics growth enhances knowledge about the achievement of black and hispanic students. 
 
 
 stinson editorial journal of urban mathematics education vol. 3, no. 2 7 the spring/summer and fall/winter issues of the 2010 volume 3 of jume contain three research articles each. staples and truxaw (2010) report on a mathematics learning discourse project in urban schools that increases students understanding of mathematics through emphasizing classroom discourse and higher-order thinking skills. mosqueda (2010) examines how disparities in the mathematics performance of latina/o students are exacerbated by the track placement of native and non-native latina/o english speakers. chahine and covington clarkson (2010) describe the cyclical process of a collaborative evaluation inquiry project that enhances urban elementary teachers’ opportunities to make informed decisions about their mathematics teaching practices based on the skillful use of data. chu and rubel (2010) open their conversation—between mathematics teacher and teacher educator—to a broader audience as they untangle the threads of their interwoven narratives about the development of culturally relevant mathematics pedagogy. nzuki (2010) explores the racial, mathematical, and technological identity constructions of five african american high school students, revealing that the positioning and authoring of identities are influenced by how students negotiate and interpret the constraints and affordances in the figured worlds in which they participate. and waddell (2010) investigates african american children’s convergent and divergent engagement with a standardsoriented mathematics curriculum, illustrating that students’ practices converge when teachers’ practices reflect the african american cultural dimension of social/affective interactions and diverge when students enact practices that reflect expressive creativity and nonverbal interactions. overall, the authors of the aforementioned research articles and of the commentaries, public stories, and book reviews noted earlier, as well as the editorial team, reviewers, and registered readers—a group that represents over 700 educators, from the novice to the accomplished—have opened up a different space in mathematics that embraces critical research, emancipatory pedagogy, and scholarship of engagement in urban communities. in short, individually and collectively, we have brought into existence the journal of urban mathematics education. through its online pages, jume makes available different research and scholarship and ultimately, different statements—thus, opening the possibilities of different discursive practices for urban mathematics education research and policy and, in turn, urban mathematics teaching and learning. in many ways, the people who have brought and continue to bring jume into existence have adopted “a degree of social consciousness and responsibility in seeing the wider social and political picture (gates & vistro-yu, 2003, p. 63) as they make (or did so long ago) the sociopolitical turn in mathematics education (gutiérrez, 2010). that is to say, most jume contributors and readers, i believe, not only reflect on the social and political discursive practices and subsequent consequences of mathematics education but also take action through their work as teachers, teacher 
 
 
 stinson editorial journal of urban mathematics education vol. 3, no. 2 8 educators, and/or education policymakers, researchers, and scholars in transforming mathematics education into more just and equitable humanizing possibilities. in a word, jume contributors and readers, i like to think, engage in praxis: “reflection and action upon the world in order to transform it” (freire, 1970/2000, p. 51). and given that “scholarship is activism” (e. a. st. pierre, personal communication, june 2001), we, the editorial team of jume, look forward to publishing more transformative “scholarly activism” (g. ladson-billings, personal communication, june 2010) in the years to come. because at the end of the day, the discursive practices of the philosophyscience of mathematics education are always already open to radical and uncertain humanizing transformations. references baker, b. m., & heyning, k. e. (eds.). (2004). dangerous coagulations?: the uses of foucault in the study of education. new york: peter lang. battista, m. t. (2010). engaging students in meaningful mathematics learning: different perspectives, complementary goals. journal of urban mathematics education, 3(2), 34–46. retrieved from http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/115/58. capraro, r. m., young, j. r., lewis, c. w., yetkiner, z. e., & woods, m. n. (2009). an examination of mathematics achievement and growth in a midwestern urban school district: implications for teachers and administrators. journal of urban mathematics education, 2(2), 46–65. retrieved from http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/33/20. chahine, i. c., & covington clarkson, l. m. (2010). collaborative evaluative inquiry: a model for improving mathematics instruction in urban elementary schools. journal of urban mathematics education, 3(1), 82–97. retrieved from http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/44/38. chu, h., & rubel, l. h. (2010). learning to teach mathematics in urban high schools: untangling the threads of interwoven narratives. journal of urban mathematics education, 3(2), 57– 76. retrieved from http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/50/60. confrey, j. (2010). “both and”—equity and mathematics: a response to martin, gholson, and leonard. journal of urban mathematics education, 3(2), 25–33. retrieved from http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/108/53. davis, j., & martin, d. b. (2008). racism, assessment, and instructional practices: implications for mathematics teachers of african american students. journal of urban mathematics education, 1(1), 10–34. retrieved from http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/14/8. dawson, i. d. (2009). how did i get this way? how bad is the damage? and how do i fix it? journal of urban mathematics education, 2(2), 12–17. retrieved from http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/52/34. esmonde, i., brodie, k., dookie, l., & takeuchi, m. (2009). social identities and opportunities to learn: student perspectives on group work in an urban mathematics classroom. journal of urban mathematics education, 2(2), 18–45. retrieved from http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/46/35. foucault, m. (1972). the archaeology of knowledge (a. m. sheridan smith, trans.). new york: pantheon books. (original work published 1969) foucault, m. (1990). the history of sexuality. volume i: an introduction (r. hurley, trans.). new york: vintage books. (original work published 1976) 
 
 
 stinson editorial journal of urban mathematics education vol. 3, no. 2 9 foucault, m. (1994). the order of things: an archaeology of the human sciences. new york: vintage books. (original work published 1966) foucault, m. (1995). discipline and punish: the birth of the prison (a. sheridan, trans.). new york: vintage books. (original work published 1975) foucault, m. (2003). structuralism and post-structuralism (j. harding, trans.). in p. rabinow & n. s. rose (eds.), the essential foucault: selections from essential works of foucault, 1954–1984 (pp. 80–101). new york: new press. (original work published 1983) freire, p. (2000). pedagogy of the oppressed (m. b. ramos, trans., 30th ann. ed.). new york: continuum. (original work published 1970) gates, p., & vistro-yu, c. p. (2003). is mathematics for all? in a. j. bishop, m. a. clements, c. keitel, j. kilpatrick, & f. k. s. leung (eds.), second international handbook of mathematics education (vol. 1, pp. 31–73). dordrecht, the netherlands: kluwer. gonzalez, l. (2009). teaching mathematics for social justice: reflections on a community of practice for urban high school mathematics teachers. journal of urban mathematics education, 2(1), 22–51. retrieved from http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/32/13. gutiérrez, r. (2010). the sociopolitical turn in mathematics education research. journal for research in mathematics education. retrieved from http://www.nctm.org/eresources/article_summary.asp?uri=jrme2010-06-1a&from=b. gutstein, e. (2010). the common core state standards initiative: a critical response. journal of urban mathematics education, 3(1), 9–18. retrieved from http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/88/43. hennings, j. a. (2010). new curriculum: frustration or realization? journal of urban mathematics education, 3(1), 9–26. retrieved from http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/90/44. jett, c. c. (2009). mathematics, an empowering tool of liberation? [review of the book mathematics teaching, learning, and liberation in the lives of black children, by d. b. martin (ed.)]. journal of urban mathematics education, 2(2), 66–71. retrieved from http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/48/21. johnson, d. c., romberg, t. a., & scandura, j. m. (1994). the origins of the jrme: a retrospective account. journal for research in mathematics education, 25, 560–582. kitchen, r. s., cabral roy, f., lee, o., & secada, w. g. (2009). comparing teachers’ conceptions of mathematics education and student diversity at highly effective and typical elementary schools. journal of urban mathematics education, 2(1), 52–80. retrieved from http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/24/14. lemons-smith, s. (2010). the nuts and bolts [review of the book culturally specific pedagogy in the mathematics classroom: strategies for teachers and students, by j. leonard]. journal of urban mathematics education, 3(1), 98–103. retrieved from http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/87/50. leonard, j., & evans, b. r. (2008). math links: building learning communities in urban settings. journal of urban mathematics education, 1(1), 60–83. retrieved from http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/5/5. lerman, s. (2000). the social turn in mathematics education research. in j. boaler (ed.), international perspectives on mathematics education, (pp. 19–44). westport, ct: ablex. lyotard, j. f. (1984). the postmodern condition: a report on knowledge (g. bennington & b. massumi, trans.). minneapolis, mn: university of minnesota press. (original work published 1979) martin, d. b. (2009a). liberating the production of knowledge about african american children and mathematics. in d. b. martin (ed.), mathematics teaching, learning, and liberation in the lives of black children (pp. 3–36). new york: routledge. 
 
 
 stinson editorial journal of urban mathematics education vol. 3, no. 2 10 martin, d. b. (2009b). researching race in mathematics education. teachers college record, 111, 295–338. martin, d. b. (2010). not-so-strange bedfellows: racial projects and the mathematics education enterprise. in u. gellert, e. jablonka, & c. morgan (eds.), proceedings of the sixth international mathematics education and society conference (vol. 1, pp. 42–64). berlin, germany: freie universität berlin. martin, d. b., gholson, m. l., & leonard, j. (2010). mathematics as gatekeeper: power and privilege in the production of knowledge. journal of urban mathematics education, 3(2), 12– 24. retrieved from http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/95/57. matthews, l. e. (2008). illuminating urban excellence: a movement of change within mathematics education. journal of urban mathematics education, 1(1), 1–4. retrieved from http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/20/9. matthews, l. e. (2009). identity crisis: the public stories of mathematics educators. journal of urban mathematics education, 2(1), 1–4. retrieved from http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/41/11. mcqueen, m. q., shaheed, s. f. h., goings, c. v., & chahine, i. c. (2010). voices, echoes, and narratives: multidimensional experiences of three teachers immersed in ethnomathematical encounters in morocco. journal of urban mathematics education, 3(2), 47–56. retrieved from http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/110/55. mosqueda, e. (2010). compounding inequalities: english proficiency and tracking and their relation to mathematics performance among latina/o secondary school youth. journal of urban mathematics education, 3(1), 57–81. retrieved from http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/47/48. paek, p. l. (2008). practices worthy of attention: a search for existence proofs of promising practitioner work in secondary mathematics. journal of urban mathematics education, 1(1), 84–107. retrieved from http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/6/1. peters, m., & burbules, n. c. (2004). poststructuralism and educational research. lanham, md: rowman & littlefield. pourdavood, r. g., carignan, n., & king, l. c. (2009). transforming mathematical discourse: a daunting task for south africa’s townships. journal of urban mathematics education, 2(1), 81–105. retrieved from http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/18/15. rajchman, j. (1987). postmodernism in a nominalist frame: the emergence and diffusion of a cultural category. flash art, 137(november-december), 49–51. rousseau anderson, c., & powell, a. (2009). a metropolitan perspective on mathematics education: lessons learned from a “rural” school district. journal of urban mathematics education, 2(1), 5–21. retrieved from http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/34/12. skovsmose, o., scandiuzzi, p. p., valero, p., & alrø, h. (2008). learning mathematics in a borderland position: students’ foregrounds and intentionality in a brazilian favela. journal of urban mathematics education, 1(1), 35–59. retrieved from http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/4/4. st. pierre, e. a. (in press). post qualitative research: the critique and the coming after. in n. denzin & y. lincoln (eds.) handbook of qualitative research (4th ed.). thousand oaks, ca: sage. staples, m. e., & truxaw, m. p. (2010). the mathematics learning discourse project: fostering higher order thinking and academic language in urban mathematics classrooms. journal of urban mathematics education, 3(1), 27–56. retrieved from http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/74/49. 
 
 
 stinson editorial journal of urban mathematics education vol. 3, no. 2 11 stinson, d. w. (2009). mathematics teacher educators as cultural workers: a dare to those who dare to teach (urban?) teachers. journal of urban mathematics education, 2(2), 1–5. retrieved from http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/54/28. stinson, d. w. (2010a). negotiating the “white male math myth”: african american male students and success in school mathematics. journal for research in mathematics education. retrieved from http://www.nctm.org/eresources/article_summary.asp?uri=jrme2010-062a&from=b. stinson, d. w. (2010b). the sixth international mathematics education and society conference: finding freedom in a mathematics education ghetto. journal of urban mathematics education, 3(1), 1–8. retrieved from http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/91/47. stinson, d. w., & bullock, e. c. (2010). critical postmodern theory in mathematics education research: transforming the discipline of mathematics into a humanizing discipline of freedom. manuscript submitted for publication. tasić, v. (2001). mathematics and the roots of postmodern thought. oxford, united kingdom: oxford university press. tate, w. f. (1995). economics, equity, and the national mathematics assessment: are we creating a national toll road? in w. g. secada, e. fennema, & l. b. adajian (eds.), new directions for equity in mathematics education (pp. 191–206). cambridge, united kingdom: cambridge university press. tate, w. f. (2008). putting the “urban” in mathematics education scholarship. journal of urban mathematics education, 1(1), 5–9. retrieved from http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/19/2. walkerdine, v. (2004). preface. in m. walshaw (ed.), mathematics education within the postmodern (pp. vii–viii). greenwich, ct: information age. waddell, l. r. (2010). how do we learn? african american elementary students learning reform mathematics in urban classrooms. journal of urban mathematics education, 3(2), 116– 154. retrieved from http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/62/65. walshaw, m. (2007). working with foucault in education. rotterdam, the netherlands: sense. wamsted, j. o. (2010). a mathematics teacher looks at mathematics educators looking at mathematics education [review of the book culturally responsive mathematics education, by b. greer, s. mukhopadhyay, a. b. powell, & s. nelson-barber (eds.)]. journal of urban mathematics education, 3(2), 155–159. retrieved from http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/104/67. williams, c. l. (2009). my intimacy with pedagogy of the oppressed. journal of urban mathematics education, 2(2), 6–11. retrieved from http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/53/29. journal of urban mathematics education july 2011, vol. 4, no. 1, pp. 1–6 ©jume. http://education.gsu.edu/jume david w. stinson is an associate professor of mathematics education in the department of middle-secondary education and instructional technology in the college of education, at georgia state university, p.o. box 3978, atlanta, ga, 30303; e-mail: dstinson@gsu.edu. his research interests include exploring socio-cultural, -historical, and -political aspects of mathematics and mathematics teaching and learning from a critical postmodern theoretical (and methodological) perspective. he is a co-founder and current editor-inchief of the journal of urban mathematics education. editorial “race” 1 in mathematics education: are we a community of cowards? david w. stinson georgia state university though this nation has proudly thought of itself as an ethnic melting pot, in things racial we have always been and continue to be, in too many ways, essentially a nation of cowards. – eric holder jr. (2009), attorney general of the united states he title of this editorial is evidently inspired by attorney general holder‘s statement, extracted from a speech he delivered on february 18, 2009, during the u.s. department of justice‘s african american history month program. 2 the specific phrase ―a nation of cowards‖ quickly became a sound bite on television news stations from cnn to fox news and a headline in newspapers from the new york times to the wall street journal. the media frenzy surrounding the phrase continued for days and even weeks within the op-ed pages of newspapers, postings on internet blogs, and the sound waves of talk radio. many people—from political pundits to everyday citizens—who responded to holder‘s (2009) remarks claimed that his use of the word cowards was too harsh and ultimately, divisive. but others, including president obama, argued that 1 it is important to note that i use the term race not to mark some biological taxonomy—no such taxonomy exists—but rather as a powerful socio-cultural and -political discursive formation (cf. foucault, 1969/1972) or worldview invented to assign some groups to perpetual low status, while others were permitted access to privilege, power, and wealth. the tragedy in the united states has been that the policies and practices stemming from this worldview succeeded all too well in constructing unequal populations among europeans, native americans, and peoples of african descent. (american anthropological association, 1998, ¶ 12) for the american anthropological association‘s complete ―statement on ‗race,‘‖ as adopted by the executive board may 17, 1998, see http://www.aaanet.org/stmts/racepp.htm. 2 for attorney general holder‘s complete remarks as prepared for delivery, see http://www.justice.gov/ag/speeches/2009/ag-speech-090218.html. t http://www.aaanet.org/stmts/racepp.htm http://www.justice.gov/ag/speeches/2009/ag-speech-090218.html stinson editorial journal of urban mathematics education vol. 4, no. 1 2 although it was perhaps not the ―best‖ word choice, that the attorney general did have a point (as reported in the new york times): president barack obama has chided his attorney general, eric holder jr., for describing america as a ―nation of cowards‖ when discussing race, wading into a tumult that flared over holder‘s indictment of the way this country talks about ethnicity. ―i think it‘s fair to say that if i had been advising my attorney general, we would have used different language,‖ obama said in a mild rebuke from america‘s first black president to its first black attorney general. in an interview with the new york times on friday, the president said that despite holder‘s choice of words, he had a point. ―we‘re oftentimes uncomfortable with talking about race until there‘s some sort of racial flare-up or conflict,‖ he said, adding, ―we could probably be more constructive in facing up to sort of the painful legacy of slavery and jim crow and discrimination.‖ (cooper, 2009, ¶ 1–4) but what does holder‘s (2009) statement have to do with mathematics education? and why even bring up holder‘s 2009 statement here? isn‘t that just old news? is it to incite divisiveness? or, is it to pose a question regarding an issue that we are too often uncomfortable talking (and researching) about? given the mission of the journal of urban mathematics education—to foster a transformative global academic space in mathematics that embraces critical research, emancipatory pedagogy, and scholarship of engagement in urban communities—the intent is evidently the latter. because, at the end of the day, even within the euphoric ―post-racial‖ sentiments surrounding the obama presidency (bonilla-silva & ray, 2009), race still matters (west, 1994), if we wish to hope that democracy still matters (west, 2004). for over 3 decades now, a growing number of mathematics education researchers and scholars have put forth reasoned arguments for researching and theorizing race in mathematics education, linking such arguments to goals of equity (for some of the earliest arguments see, e.g., matthews, 1984; reyes & stanic, 1988; secada, 1992; tate, 1994). 3 and more recently, martin (2009b) ad 3 matthew‘s (1984), in her review of ―minorities‖ in mathematics education, marked 1975 as the starting point of researching (theorizing?) race in mathematics education. but even as she did so, she noted several problems that limited the ―usefulness and appropriateness‖ of these early studies: one problem is that most reports of the studies are either unpublished papers or final reports to funding agencies and therefore are relatively inaccessible. another problem is that some of the findings could be fortuitous in that neither the original nor the primary focus of the study was on minorities. more often than not, the study concerned sex-related differences, and race was included as a background variable. inadequate reasons are then given to explain any race effects. (p. 84) for some of the earliest arguments for theorizing race in u.s. education in general, see woodson (1933/1990) and du bois (1935); evidently, calls for theorizing race in education began long before the 1970s. stinson editorial journal of urban mathematics education vol. 4, no. 1 3 dresses, directly, the task of theorizing race in mathematics education; he asks us to consider mathematics education as an institutional space of whiteness (martin, 2010) and mathematics learning (and teaching) as racialized experiences (martin, 2009a). therefore, in some ways, one might claim that we, as a community of researchers and scholars, have made progress in ―understanding‖ race in mathematics education over the past 3 decades or so. certainly, as a community, we have grown to understand how race, racism, and white supremacy function in education policy and within our communities, schools, and classrooms, and specifically, in our (urban) mathematics classrooms, haven‘t we? that is to say, we have become more informed, haven‘t we? here, i do not attempt to provide answers to these questions but rather briefly revisit some old data and present some new data that might shed some light on the more general question: just where are we in regards to race in mathematics education? the data are extracted from two analyses of mathematics education literature: lubienski and bowen‘s 2000 journal for research in mathematics education (jrme) brief report ―who‘s counting? a survey of mathematics education research 1982–1998‖ and parks and schmeichel‘s 2011 american educational research association paper ―theorizing of race and ethnicity in mathematics education literature.‖ 4 both analyses were conducted, in part, to determine the larger, ―mainstream‖ mathematics education community‘s commitment (or lack thereof) to issues of race/ethnicity by determining—through eric database searches between the years 1982–1998 and 1999–2010, respectively—the number of peer-reviewed journal articles published that contained both mathematics education and race/ethnicity descriptors. 5 moreover, both analyses, as mentioned, focused solely on peer-reviewed journal articles with the assumption that they ―reflect the interests and values of ‗mainstream‘ research communities more closely than books‖ (lubienski & bowen, 2000, p. 627). hence, the combined analyses provide if not an exact picture, 6 then certainly a detailed sketch, of the larger, mainstream mathematics education community‘s interests and values around issues of race/ethnicity over the past 3 decades. that is to say, if you believe in 4 parks and schmeichel (2011) extended their search to include two other methods, providing a fine-grain analysis (e.g., searching every jrme article published between 2008–2011). 5 lubienski and bowen‘s (2000) descriptor search included those pertaining to gender, ethnicity, class, and disability; parks and schmeichel‘s (2011) included those pertaining only to race and ethnicity. in other words, lubienski and bowen‘s analysis explored ―equity‖ research in mathematics education generally, whereas parks and schmeichel‘s zeroed in on race/ethnicity specifically. 6 due to the sheer size of the eric database, an exact counting is somewhat unproductive. in other words, analyses such as these are not an ―exact science.‖ stinson editorial journal of urban mathematics education vol. 4, no. 1 4 numbers, and if you believe that what we write about reflects what we value (e.g., the greater the number, the greater the interest and value). so, by the numbers, what might the two analyses say about the mathematics education community‘s interests and values in addressing the complexity of race/ethnicity? lubienski and bowen‘s (2000) search resulted in 112 articles with descriptors of ethnicity, published largely in non-mathematics education journals, out of 3,011 total mathematics education articles, or 3.7%. parks and schmeichel‘s (2011) search resulted in 403 articles with descriptors of race or ethnicity, again, published largely in non-mathematics education journals, for the years 1999–2010, and specifically, for the years 2005–2010, they identified 320 articles out of 8,326 total mathematics education articles, or 3.8%. thus, the actual count of articles between the two searches is encouraging, nearly a four-fold increase (112; 403). but as the percentages reveal, in real gains, this numerical increase is deceiving. in short, percent wise, there has been virtually no increase (i.e., for nearly 3 decades, the percentage of peer-reviewed journal articles that address mathematics education and race/ethnicity has stayed constant [3.7%; 3.8%]). so, is addressing the complexities of race/ethnicity of only 4% interest and value to the mathematics education community? or, more generally, one might infer that issues of equity are of only 4% interest and value to the mathematics education community. either way, even this constant of roughly 4% is dangerous, as parks and schmeichel (2011) note: taken as a body of work, the density of studies that conceptualized race as a variable contributed to a discourse of race as primarily an easily-defined (or often notdefined-at-all) category to which one belongs and to which particular traits or outcomes can be assigned. this way of thinking about race is quite different from thinking about it in the ways common in socio-political theories, where it can be analyzed as a performance, which allows researchers to study ―the enormous number of effects race thinking (and race acting) have produced,‖ (omi & winant, 2004, p. 9) or as a construct implicated by history and racism as critical race theory and latcrit theory allow scholars to do. (p. 8) these comments, in part, echo lubienski and bowen (2000), who a decade earlier noted: because the majority of articles on ethnicity or class pertained to student achievement whereas only a few related to educational environment, students in classrooms, assessment, or teacher education, one gets the impression that researchers look primarily at outcomes of these equity groups and rarely examine how schooling experiences contribute to these outcomes. (p. 631) based on these arguments, it appears that not only has the interest and value of race/ethnicity stayed constant over nearly 3 decades but also the ways in which it stinson editorial journal of urban mathematics education vol. 4, no. 1 5 is conceptualized in mathematics education research has stayed more often than not constant: ―as primarily an easily-defined (or often not-defined-at-all) category to which one belongs and to which particular traits or outcomes can be assigned‖ (parks & schmeichel, p, 8, 2011). this dangerously limiting, one-dimensional perspective of race/ethnicity as a category has resulted in a proliferation of ―gapgazing‖ research studies over the past several decades where the ―achievement outcomes‖ of african american and latina/o children are compared to their european and asian american counterparts (gutiérrez, 2008). these studies, taken to their extreme, offer little more than a static picture of the schooling inequities experienced by children, capturing neither the history nor the context of learning that produced such outcomes (gutiérrez). nevertheless, here i neither wish to discuss the pros and cons of gap-gazing research in mathematics education—that has been done effectively elsewhere (see, e.g., lubienski, 2008; lubienski & gutiérrez, 2008; gutiérrez, 2008)—nor to argue the vital importance of capturing the history and context of student learning—that too, has been done effectively elsewhere (see, e.g., martin, gholson, & leonard, 2010). but rather thoughtfully return to and focus on the 4%! – are we a community of cowards? references american anthropological association. 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(2009a). little black boys and little black girls: how do mathematics education research and policy embrace them? in s. l. swars, d. w. stinson, & s. lemons-smith (eds.), proceedings of the 31st annual meeting of the north american chapter of the international group for the psychology of mathematics education (pp. 22–41). atlanta, ga: georgia state university. martin, d. b. (2009b). researching race in mathematics education. teachers college record, 111, 295–338. martin, d. b. (2010). not-so-strange bedfellows: racial projects and the mathematics education enterprise. in u. gellert, e. jablonka, & c. morgan (eds.), proceedings of the sixth international mathematics education and society conference (vol. 1, pp. 42–64). berlin, germany: freie universität berlin. martin, d. b., gholson, m. l., & leonard, j. (2010). mathematics as gatekeeper: power and privilege in the production of knowledge. journal of urban mathematics education, 3(2), 12– 24. retrieved from http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/95/57. matthews, w. (1984). influences on the learning and participation of minorities in mathematics. journal for research in mathematics education, 15, 84–95. parks, a. n., & schmeichel, m. (2011). theorizing of race and ethnicity in mathematics education literature. paper presented at the annual meeting of the american educational research association, new orleans, la. reyes, l. h., & stanic, g. m. a. (1988). race, sex, socioeconomic status, and mathematics. journal for research in mathematics education, 19, 26–43. secada, w. g. (1992). race, ethnicity, social class, language, and achievement in mathematics. in d. a. grouws (ed.), handbook of research on mathematics teaching and learning (pp. 623–640). new york: macmillan. tate, w. f. 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(original work published 1933) http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/95/57 highly effective teachers’ conceptions of mathematics education journal of urban mathematics education july 2009, vol. 2, no. 1, pp. 52–80 ©jume. http://education.gsu.edu/jume richard s. kitchen is an associate professor of mathematics education at the university of new mexico; departments of educational specialties and mathematics & statistics, msc 05 3040, albuquerque, nm 87131; email: kitchen@unm.edu. his research interests include mathematics education at highly effective schools that serve poor communities and assessment of english language learners. francine cabral roy is a mathematics education consultant and assistant adjunct professor at the university of rhode island, kingston, ri 02881; e-mail: franroymathed@gmail.com. her research interests include urban education, equity in mathematics education, and teacher learning. okhee lee is a professor in the school of education at the university of miami, 1551 brescia avenue, coral gables, fl 33146; e-mail: olee@miami.edu. her research areas include science education, language and culture, and teacher education. walter g. secada is senior associate dean of the school of education and chair of the department of teaching and learning at the university of miami, 5202 university drive, coral gables, fl 33146; email: wsecada@miami.edu. his research interests include mathematics education, teachers' professional development, school restructuring, equity in education, bilingual education, student engagement, and curricular reform. comparing teachers’ conceptions of mathematics education and student diversity at highly effective and typical elementary schools richard s. kitchen university of new mexico francine cabral roy university of rhode island okhee lee university of miami walter g. secada university of miami in this study, the authors examined what distinguished highly effective from typical elementary schools in mathematics by examining the conceptions of fourthgrade teachers with regards to mathematics education (curriculum, instruction, and assessment) and student diversity (ability, culture, language, and socioeconomic status). the study was conducted in two large urban areas with high proportions of racially/ethnically and linguistically diverse student groups. interviews were conducted with 32 fourth-grade teachers from 16 elementary schools, including 10 highly effective and 6 typical schools in the two areas. compelling evidence was found that teachers at highly effective schools had better developed and better articulated conceptions of mathematics education and student diversity. while similar findings were found across the two areas (e.g., teachers’ beliefs about the influence of high-stakes assessment and the academic ability of lowachieving students), differences were also found across the two areas (e.g., teachers’ beliefs about the importance of implementing multiple instructional strategies and expectations for college attendance). keywords: conceptions of mathematics, highly effective schools, student diversity, urban schools kitchen et al. comparing teachers’ conceptions journal of urban mathematics education vol. 2, no. 1 53 the purpose of this study was to examine what distinguishes highly effective from typical elementary schools in mathematics achievement by examining the conceptions (knowledge and beliefs) of fourth-grade teachers. specifically, we compared teachers at highly effective schools with teachers at typical schools with regards to their conceptions in mathematics education (curriculum, instruction, and assessment) and student diversity (ability, culture, language, and socioeconomic status). it is clear from research that teachers‘ conceptions of mathematics content, instruction, and assessment are strongly linked to classroom practice and student learning in interesting and complex ways. what remains under researched in the literature on teachers‘ conceptions and practices is the role of race/ethnicity, culture, and language in mathematics education (rodriguez & kitchen, 2005; rousseau & tate, 2003). in this study, we pursued an integrated research project that examined the influence of teachers‘ conceptions with regards to both mathematics education and student diversity on student learning and achievement. two specific research questions guided the study: 1. how do teachers at highly effective schools articulate their conceptions of mathematics education compared to their colleagues at typical schools? 2. how do teachers at highly effective schools articulate their conceptions regarding student diversity compared to their colleagues at typical schools? another distinguishing feature of this study is that we searched for characteristics that differentiated schools rather than classrooms. this perspective necessitates that we consult the effective schools literature. much of this research identifies factors other than instruction as distinguishing more from less effective schools (edmonds, 1979; good & brophy, 1986; martin, mullis, gregory, hoyle, & shen, 2000; purkey & smith, 1983). these factors include school-level characteristics such as principal leadership, orderliness and safety, or opportunities at home such as access to reading materials. interestingly, the third international mathematics and science study (timss) results indicate that although the nature of mathematics and science instruction was important, it was not a defining characteristic distinguishing high from low performing schools (martin, et al., 2000). in light of increasing pressure being placed on schools to enhance student learning and achievement in mathematics, there is a danger that policymakers view the effective schools literature as suggesting that instructional quality fails to distinguish more from less effective schools. we sought to contest this potential pitfall by providing empirical evidence to address the gap in the research literature on the link between school-level effectiveness and student learning and achievement. to accomplish this goal, we investigated teachers‘ conceptions of and explanations for their practices in mathematics with diverse student groups. kitchen et al. comparing teachers’ conceptions journal of urban mathematics education vol. 2, no. 1 54 conceptual framework we examined the conceptions of teachers from multiple school districts across two of the usa‘s major metropolitan areas. teachers‘ responses were examined through a complex lens that includes constructs related to school organization, mathematics education, and multicultural education. the corresponding areas of literature that provided the conceptual framework for this study are: (a) school restructuring, (b) teachers‘ conceptions in mathematics education, and (c) teachers‘ conceptions in addressing student diversity. school restructuring in research on school restructuring and its relationship to student performance on high-level tasks, newmann and associates (1996; newmann, secada, & wehlage, 1995; newmann & wehlage, 1995) found that students enrolled in schools where the curriculum content and instruction focused on depth over coverage, analytic reasoning over memorization, and the construction of value over doing tasks as ends in themselves out performed students whose classrooms lacked these instructional features. lee and smith‘s (2001) study of secondary schools supports newmann et al.‘s findings. at secondary schools where mathematics and science course offerings were predominantly academic and where teachers as a whole tended to report instruction that focused on depth and higher order thinking, students evidenced higher achievement in mathematics and science and began to close the achievement gap based on social class, compared to schools where these instructional characteristics were lacking. in this study, we sought to examine whether teachers in highly effective schools were more likely than their counterparts at typical schools to articulate their goals to implement mathematics curricula and instruction as reported in newmann and associates (1996) and lee and smith (2001). teachers’ conceptions in mathematics education in the past several decades, notions about teachers‘ conceptions (knowledge and beliefs) of mathematics content (ma, 1999), instruction (ball & cohen, 1999), and assessment (kulm, 1994) have dramatically changed. current reforms in mathematics education stress the need for problem-solving approaches to promote students‘ reasoning and communication skills (national council of teachers of mathematics [nctm], 1989, 2000; national science foundation [nsf], 1996). furthermore, the mathematical processes of making connections, communicating, representing, and problem solving are no longer viewed as ends in themselves but as means to learning mathematics with understanding (hiebert & car kitchen et al. comparing teachers’ conceptions journal of urban mathematics education vol. 2, no. 1 55 penter, 1992; hiebert, et al.1996). the incongruity between this vision and the conceptions of many practicing mathematics teachers (e.g., mathematics as a sequential, rule-based discipline in which memorizing facts and following procedures are highly valued) has been well documented (e.g., romberg, 1992). three perspectives on teachers‘ conceptions of mathematics education are described next. ernest (1988, 1991) identified three possible views of the nature of an academic discipline. the first, called instrumentalist, views a discipline as an isolated body of discrete skills. the second, called platonist, regards a discipline as a body of connected and unified knowledge. the third, called problem-solving, deems a discipline as a process of inquiry that is continuously expanded by human creation. these three distinctions have also been cast as a duality between absolute (e.g., instrumentalist) and fallible (e.g., problem-solving) views that take shape in beliefs about mathematical knowledge and legitimate mathematical activities (romberg, 1992; thompson, 1992). teachers who embrace an instrumentalist view often look at mathematics as a sequence of fixed skills or concepts. mastery of pre-requisite skills is deemed necessary for subsequent learning. this view assumes that ―rules are the basic building blocks of all mathematical knowledge and all mathematical behavior is rule governed‖ (thompson, 1992, p. 136). thompson (1992) argued that there is a consistent relationship between teachers‘ beliefs and instructional practices in mathematics. because mathematics is characterized as static and predetermined in the instrumentalist philosophy, those who adhere to this view emphasize mathematical facts and pursue the drill-and-practice approach to teaching (schifter & fosnot, 1993). in contrast, teachers who adhere to a platonist view of mathematics emphasize the logic that connects concepts. these relationships are assumed to be fixed and often require explanations by the teacher. this view models a top-down approach in which instruction begins with the knowledge of the expert, rather than that of the learner (hiebert & carpenter, 1992). an example of teachers who embrace the platonist view of mathematics is provided in liping ma‘s book, knowing and teaching elementary mathematics (1999). a theme that runs throughout ma‘s book is that teachers must possess a deep and broad knowledge of mathematics to make conceptual connections between mathematical ideas. on the other hand, teachers who adopt a problem-solving or inquiry view of mathematics see their role as posing questions and challenging students to think and reason. teachers whose primary objective is to advance mathematical problem solving advocate that students are the ones ―doing mathematics‖ (davis & hersh, 1980; ernest, 1991; lakatos, 1976; tymoczko, 1986). from this view, ―learning is primarily a process of concept construction and active interpretation—as opposed to the absorption and accumulation of received items of infor kitchen et al. comparing teachers’ conceptions journal of urban mathematics education vol. 2, no. 1 56 mation‖ (schifter & fosnot, p. 8, 1993). pedagogy inspired by this view engages students in posing and solving problems, making and proving conjectures, exploring puzzles, sharing and debating ideas, and contemplating the beauty of ideas in an academic discipline. alternative or multiple assessment formats that do not simply assess facts and skills in isolation, but that also require students to apply their knowledge in real-life contexts align more with the platonist or problem-solving view. according to kulm (1994): alternative assessment approaches that include open-ended questions, presentation of solutions in both written and oral form, and other performances send very different messages to students about what is important in mathematics learning….the shift from an emphasis on producing correct answers to the expectation that students think and communicate is a major one for many students and teachers. (p. 6) this shift of emphasis corresponds to the philosophical change necessary for an instrumentalist to become a platonist or to promote problem-solving. such a shift is supported by researchers who advocate revising assessment practices to bring about changes in instruction based on how children learn (o‘day & smith, 1993). in addition, alternative assessment approaches and the use of multiple assessment formats require students to communicate their thinking and elicit a range of student responses (wiggins & mctighe, 1999). it is important to note that teachers‘ conceptions may not be easily classified into solely instrumentalist, platonist, or mathematical problem-solving perspectives and that their classroom practices may not align well with their stated beliefs. kitchen, depree, depree, celedón-pattichis, and brinkerhoff (2007) found that teachers at highly effective schools in low-income communities often expressed multiple perspectives on the teaching and learning of mathematics and their classroom practices reflected these multiple perspectives. according to thompson (1992), teachers may hold multiple perspectives that helps explain why they express one philosophical view while practicing another. teachers’ conceptions of student diversity the existing knowledge base for promoting academic achievement with a culturally and linguistically diverse student population is limited and fragmented, in part because disciplinary knowledge and student diversity have traditionally constituted separate research agendas (lee, 1999). in recent years, studies considering the interaction between academic disciplines and students‘ linguistic and cultural experiences have begun to emerge in mathematics education (e.g., civil, & andrade, 2002; gutiérrez, 2002). in this review of the research literature, two kitchen et al. comparing teachers’ conceptions journal of urban mathematics education vol. 2, no. 1 57 areas related to student diversity in mathematics education are considered: (a) teacher expectations and (b) culturally relevant instruction. teacher expectations. teachers hold varied conceptions pertaining to students, including students‘ potential for success, cultural backgrounds, strengths and weaknesses, and future placement in society. high expectations are lauded as a key to successful teaching of students often underserved by schooling (lipman, 1998; zeichner, 1996). zeichner (1996) argued that high expectations are a necessary attribute of classrooms that have the potential to narrow the achievement gap in urban schools. the research literature contains a growing consensus about what teachers need to be like, to know, and to be able to do in order to teach all students to high academic standards. this vision rests on teachers believing that all students can learn and taking the responsibility for this task regardless of students‘ economic circumstance or skin color. the belief that ―all children can learn‖ and advocacy for high expectations have become part of mainstream educational rhetoric. although high expectations are necessary for effective teaching, they are far from the norm—especially with racial/ethnic minorities and low socio-economic status (ses) students. teachers hold lower expectations for students of color and the poor than they do for white middle-class students (ferguson, 1998; grant, 1989; knapp & woolverton, 1995; zeichner, 1996). low expectations are believed to be at the root of ineffective pedagogy with students of color and the poor. low expectations appear to result in transmission, authoritative, and controlled forms of pedagogy emphasizing basic skills (haberman, 1991; knapp & woolverton, 1995). ladson-billings (1994) described how some teachers expect african american students to be more difficult to control, and therefore work harder at restraining these students in the classroom. in their review of social class and schooling, knapp and woolverton (1995) claimed that controlled forms of instruction teach low-ses students that little is expected from them except compliance to a rigid classroom environment. this ―pedagogy of poverty‖ (haberman, 1991) is prevalent in urban and/or low-ses schools. culturally relevant instruction. teachers who believe that all children can learn may have high expectations but may not necessarily provide culturally relevant instruction if they teach only dominant content but ignore issues of culture. notions of culturally relevant instruction embrace the significance of high expectations but move beyond it to highlight the significance of culture and power as both a means to an end and goals in and of themselves. culturally relevant instruction views student culture and cultural experiences as a strength and use this knowledge as a bridge to learning (ladson-billings, 1994, 1995). kitchen et al. comparing teachers’ conceptions journal of urban mathematics education vol. 2, no. 1 58 because mathematics tends to be presented as a set of objective and universal facts and rules, it is often viewed as ―culture free‖ and not considered as a socially and culturally constructed discipline (banks, 1993; peterson & barnes, 1996; secada, 1989). instructional practices have traditionally relied on examples, analogies, and artifacts that are often unfamiliar to non-mainstream students (ninnes, 2000). recent efforts to provide culturally relevant instruction indicate that when their cultural and linguistic experiences are used as intellectual resources, non-mainstream students are able to engage in academic learning and show significant achievement gains. for example, the algebra project (silva & moses, 1990) uses student knowledge of the subway system as a basis for understanding operations with integers. the focus on student strengths contrasts the remediation model of teaching urban students, in which curriculum and instruction are predicated on what students do not know and often emphasize rote skills (haberman, 1991; oakes, 1990). methods research setting and teacher participants district selection. the study was conducted in two areas that had previously received urban systemic initiative 1 funding from the national science foundation, ―president city area 2 ― and ―arbor city.‖ because president city consisted of three adjacent school districts with similar characteristics, we refer to it as the ―president city area.‖ the arbor city school district and three contiguous president city area school districts were chosen to participate in this study because they served large low-income and minority student populations; had on-going mathematics education reform efforts underway; adhered to standards in mathematics, science, and language arts; and gave standardized tests to 4 th graders in mathematics and language arts. all four districts used standardized criterion-referenced tests 3 (henceforth referred as the ―state assessment‖) to evaluate 4 th graders in mathematics and language arts. 1 the national science foundation launched its urban systemic initiative (usi) program in 1994, applying lessons learned from its earlier statewide systemic initiative program to the problems of inner-city school systems. the usi program was offered to 25 major cities with the largest number of k–12 students living in poverty. 2 names of participating cities are pseudonyms. 3 in arbor city, 4 th graders were tested with the stanford achievement test-9 th (sat9) edition: intermediate i. fourth graders in the president city area were tested with the state assessment of academic skills. kitchen et al. comparing teachers’ conceptions journal of urban mathematics education vol. 2, no. 1 59 arbor city, one of the nation‘s oldest cities, contained over 170 elementary schools. at the time the study was undertaken in 2000, the majority of the public school district‘s more than 200,000 students were african american (65%), with approximately equal numbers of white (16%) and hispanic/latino (13%) students. students in the public schools were predominantly low-income; 80% were eligible for free or reduced-price lunch. in arbor city, 62% of the teachers were white, 36% african american, and 1% hispanic/latino. the president city area schools, comprised of three contiguous districts, contained 116 elementary schools. of over 103,000 public school students, 85% were hispanic; 11% white; and 3% african american. over 26% of these districts‘ students were limited english proficient/english for speakers of other languages (lep/esol); nearly 70% qualified for free or reduced-price lunch. at all but one of the president city area schools, the majority of teachers were hispanic/latino/a (60% or more). at all but this one school, 49 to 74% of the teachers reported the ability to speak fluent spanish. school selection. to be selected for inclusion in this study, elementary schools had to meet two sets of criteria. first, in terms of demographic criteria, an elementary school had to serve predominately minority, low-income students. at least 50% of the students of a school‘s population had to be low-income as defined by the district. second, in terms of academic criteria, at schools that were designated as highly effective across the four participating districts, 90% or more of students were tested with the state assessment. additionally, student test scores in mathematics and science at the fourth-grade level from 1996 through 1999 were either increasing or were consistently high. this requirement reduced the likelihood that a school considered highly effective one year would fall below the threshold the next year—a common finding in the effective schools literature (good & brophy, 1986). finally, highly effective schools had to be among those with the highest percentage of 4 th grade students scoring at basic or above in reading in their district in 1999. in arbor city, criteria to select highly effective schools also included:  66% or more of the 4 th graders tested scored at basic or above in mathematics, and  30% or more of the 4 th graders tested scored at proficient or above in mathematics. in the president city area, criteria to select highly effective schools also included:  90% or more of the 4 th graders tested scored at passing or above in mathematics,  50% or more of the 4 th graders tested scored at proficient or above in mathematics, and kitchen et al. comparing teachers’ conceptions journal of urban mathematics education vol. 2, no. 1 60  30% or more of a school‘s students were limited english proficient (lep). based on the two sets of criteria described above, three schools were selected as typical schools matched to the five highly effective schools in both arbor city and in the president city area, respectively. typical schools were in the same district-administrative unit, called a cluster, as a highly effective school to ensure that the schools were in the same geographic region of the city and had similar access to professional development activities. they enrolled approximately the same percentage of low-income students and had approximately the same racial and ethnic breakdown. to the degree possible, typical schools and highly effective schools enrolled a similar number of students. one school that was initially selected as a highly effective school in arbor city was re-designated as a typical school because in 2000 their sat9 scores plummeted and remained very low in 2001. all selected schools were given a $600 incentive for participation. teacher selection. we selected fourth grade as the target because restricted resources limited us to studying a single grade. schools had been selected as highly effective or typical based on the achievement of their fourth graders, and fourth is also the grade when national assessment of education progress (naep) testing is done. principals at participating schools were asked to make a list of the fourth grade teachers in their schools who taught both mathematics and science. then, they ranked the teachers from ―most effective‖ to ―least effective‖ according to some explicit criterion, such as a recent job evaluation. in both the president city area and arbor city, two teachers per school were selected to participate in this study based on their being at neither extreme of the principal‘s ranking. thus, a total of 32 teachers took part in the study (16 from each city), including 20 from highly effective schools and 12 from typical schools across the two cities. data collection the data collection team consisted of three mathematics educators, including one who was involved in the development of the classroom observation instrument and one who had extensive experience in using this instrument. no one associated with this study was aware of which schools were highly effective and which were typical until the results were aggregated at the level of school effectiveness. data collectors visited each school twice during a single year—once in the fall and once in the spring. during each visit, each participating teacher was observed teaching twice—two distinct lessons on consecutive days, for a total of four observations of each of the 32 participating teachers in the sample. kitchen et al. comparing teachers’ conceptions journal of urban mathematics education vol. 2, no. 1 61 we conducted semi-structured interviews with each teacher immediately after the two classroom observations. each of the 32 teachers in the sample was to have been interviewed twice; however, due to teacher absenteeism, we finished 56 of 64 possible interviews for the study. we obtained copies of academic tasks (e.g., handouts) during classroom observations, and then used these tasks and classroom activities as the context to examine teachers‘ conceptions during interviews. in addition, the concrete and immediate nature of the tasks and activities facilitated interviews with the teachers. to examine teachers‘ conceptions (knowledge and beliefs) of mathematics education with student diversity, our semi-structured interview protocols were adapted from existing instruments (newmann, secada, & wehlage, 1995). during fall 2000, interviews were designed to examine teachers‘ (a) general perceptions of the lessons, (b) conceptions of the nature of mathematics, (c) conceptions of student diversity, and (d) conceptions of integrating mathematics with student diversity (see appendix i for fall 2000 teacher interview protocol). during spring 2001, interviews were designed to examine teachers‘ (a) general perceptions of the lessons, (b) influences of high-stakes assessments and accountability on classroom practices, (c) understanding of individual students‘ strengths and weaknesses, and (d) use of instructional strategies to meet students‘ learning needs in mathematics (see appendix ii for spring 2001 teacher interview protocol). each question was followed with probes. each interview lasted about 45 minutes to 1 hour. all the interviews were audio-taped and transcribed. data analysis the teacher interview transcripts were initially read as a complete dataset and coded for structural categories according to classroom data, interview questions, individual questions, students, and schools. with the use of nud*ist, this structure provided the initial database, allowing for flexible grouping and coding of teachers‘ conceptions. data were sorted by interview questions to examine (a) teachers‘ conceptions of curriculum, instruction, and assessment in mathematics and (b) teachers‘ conceptions of student diversity in mathematics. then, for each of (a) and (b), the data were sorted by district (president city area and arbor city). the data subsets were analyzed using interpretive methods (erickson, 1986). each data subset was read as a whole, followed by a period of open coding to allow for the emergence of themes. an iterative process of coding, memo writing, focused coding, and integrative memo writing followed (emerson, fretz, & shaw, 1995). creation of the codes went through multiple revisions, as the data were repeatedly read to check the consistency of themes. this process continued until either no new categories were developed or consistency was achieved. after we kitchen et al. comparing teachers’ conceptions journal of urban mathematics education vol. 2, no. 1 62 obtained the set of themes from the dataset, we searched for commonalities and differences in teachers‘ responses between highly effective and typical schools with respect to each of the themes. we sought both confirming and disconfirming evidence by searching for supportive and non-supportive evidence (erickson, 1986). we considered teachers‘ conceptions in each city to be replication of the other. results the results about teachers‘ conceptions are grouped into (a) mathematics in terms of curriculum, instruction, and assessment and (b) student diversity in terms of teachers‘ expectations and understanding of culture. frequencies (%) of teachers‘ responses for each key theme are presented; on occasion, an illustrative example is presented to clarify a theme. teachers’ conceptions of mathematics curriculum, instruction, and assessment we investigated elementary teachers‘ conceptions of mathematics curriculum, instruction, and assessment between highly effective and typical schools. findings are presented in table 1. teachers’ conceptions of mathematics curriculum. we identified two themes that differentiated teachers at highly effective and typical schools across the two cities: (a) student knowledge of addition and subtraction facts and (b) teaching more content than simple problem solving and basic facts. teachers at highly effective schools were less concerned about their students knowing the basic addition and subtraction facts (i.e., could recall sums and differences of two single-digit numbers). in contrast, the majority of teachers at typical schools included basic addition and subtraction as among the most important mathematical ideas that they believed they should teach. in the president city area, a higher percentage of teachers from typical schools (67%) prioritized teaching students basic addition and subtraction skills compared to only 30% of teachers from highly effective schools. in arbor city, all but one of the teachers from typical schools (93%), but only 25% of teachers from highly effective schools identified the importance of students knowing their basic addition and subtraction facts. because we did not study students‘ prior educational experiences in mathematics, we do not know if this difference can be attributed to students at highly effective schools having access to a more challenging mathematics curriculum. kitchen et al. comparing teachers’ conceptions journal of urban mathematics education vol. 2, no. 1 63 table 1 teachers’ conceptions of mathematics curriculum, instruction, and assessment theme school type president city area arbor city curriculum basic skills highly effective typical 30% 67% 25% 93% teaching more than problem solving and basic facts highly effective typical 90% 67% 50% 67% instruction make meaning of mathematics highly effective typical 40% 17% 7% 0% multiple instructional strategies highly effective typical 30% 33% 50% 0% meeting students‘ affective needs highly effective typical 40% 17% 50% 17% assessment influenced by test highly effective typical 100% 100% 63% 100% teach beyond test highly effective typical 63% 0% 11% 0% many teachers cited problem solving and basic facts (i.e., being able to operate on two single-digit numbers with the four basic operations) as among the three most important mathematical ideas to teach during fourth grade. yet, there were noticeable differences between highly effective and typical schools in the president city area, although such differences were not as evident in arbor city. in the president city area, teachers at typical schools were generally not particularly articulate about what their students needed to learn in mathematics. in contrast, 90% of the teachers from highly effective schools talked at great length about going beyond teaching only problem solving and the basic facts and identified additional mathematical ideas that they wanted their fourth graders to learn. for example, mr. bruno 4 from a highly effective school in the president city area discussed the importance of his students analyzing the reasonableness of their solutions and learning algebra, geometry, and measurement: 4 all names of participating teachers are pseudonyms. kitchen et al. comparing teachers’ conceptions journal of urban mathematics education vol. 2, no. 1 64 reasonableness is the first thing. i want them to be able to relate to reality. the second one would be being able to work as a whole, different kinds of math problems. multiplication, division, subtraction, even algebra, if possible, so that when they go up to higher grades or even on to college, they‘ll be able to perform mathematical situations in an easier way for them. i‘ve seen many students struggle with that, so i want to make it easier for them. so, the second one is to work problems. the third one is for them to get to know measurement in every aspect—weights, length, depth, volume. lastly, they need to get to know geometry. all those things together will make a mathematician. (bruno, highly effective, president city area) teachers’ conceptions of mathematics instruction. our analysis identified two themes that differentiated teachers at highly effective and typical schools across the two cities: (a) making meaning of mathematics and (b) using multiple instructional strategies. teachers from highly effective schools were more likely to express the importance of helping their students make meaning of mathematical ideas by engaging them in mathematical investigations. as shown in table 1, teachers from highly effective schools across both participating cities were more likely than their counterparts at typical schools to describe helping their students make meaning of mathematics as one of their instructional strategies. ms. james, a teacher from a highly effective school in arbor city described a multiplication algorithm that she taught and believed helped her students develop an understanding of multiplication: but multiplication is where they get a new algorithm; they don‘t do it the traditional way. this one, they do it according to an array that they draw and according to how many sections there are in the array. that is the way the algorithm is broken down and they have to explain each section. so this algorithm is much longer and it‘s detailed but it shows you exactly what steps you went through…. and it helps you understand the value of the number as opposed to ‗why am i doing this‘ or just going through a process and not understanding why you‘re doing it. (james, highly effective, arbor city) in arbor city, a significantly higher percentage of teachers from highly effective schools (50%) than teachers from typical schools (0%) discussed the importance of implementing multiple instructional strategies. this finding was not replicated in the president city area. ms. birge, another teacher from a highly effective school in arbor city, explained how her instruction changed as the academic year progressed to emphasize students justifying their ideas while problem solving in groups: i just keep plugging at it. every lesson that i teach, tell me how you came up with that. well, the answer is wrong. it doesn‘t matter; what made you say that? they also have math journals, which you didn‘t get a chance to see. in their math journals, they‘ll start to write more about how to solve problems and what they took from their kitchen et al. comparing teachers’ conceptions journal of urban mathematics education vol. 2, no. 1 65 problem and, more importantly, how it relates to everyday life…. they don‘t even do as much group work now as they will in the coming months. and that‘s when i really get to see how you think. so right now, all of them are doing relatively well but this is skill stuff. this is not really digging in—the meat and gravy of it, just the top layer. (birge, highly effective, arbor city) teachers’ conceptions of mathematics assessment. the teachers in this study were forthright about the influence of standardized testing on their mathematics curriculum and instruction. our analysis identified two themes that differentiated teachers at highly effective and typical schools across the two cities: (a) influenced by test, and (b) teaching beyond test. in the president city area, all of the teachers described how their mathematics instruction was strongly influenced by the mathematical content of the state assessment. they were candid that their assessment goal was to learn what mathematical topics students needed to study to prepare for the state assessment. they were expected to get specific training to prepare their students to be successful on the state assessment. this expectation appeared to be particularly true for teachers from typical schools, most of whom described attending the evaluation program workshops focused exclusively on preparing students for success on the state assessment. interestingly, 30% of teachers from highly effective schools, but no teacher from a typical school, challenged using only the evaluation program curriculum. a significant finding was that none of the teachers from the typical schools in the president city area discussed how their mathematics instruction went beyond simply preparing their students to be successful on the test. in contrast, 63% of teachers from highly effective schools stated that they were either more concerned about teaching to the state curriculum framework than preparing their students to be successful on the state assessment or that they tried to do more than simply teach to the test. ms. padron described how she taught to the state curriculum framework: i always use my bible, which is my state curriculum framework, and that‘s aligned to the state assessment….it‘s our state curriculum and we have to go by that. and that‘s what the state assessment also tests, but sometimes....i tell them that we‘re not learning just because of the state assessment….we go beyond that. we do more than that. that‘s just the minimum that you should know. (padron, highly effective, president city area) the arbor city teachers devoted considerable time to developing and implementing lessons to prepare students to be successful on the district‘s test. few teachers alluded to how they worked to teach beyond the learning objectives of the test. kitchen et al. comparing teachers’ conceptions journal of urban mathematics education vol. 2, no. 1 66 summary. the main themes that emerged in our analysis of the participating teachers‘ conceptions of mathematics curriculum, instruction, and assessment involved: (a) the mathematics content that should be taught, (b) the instructional strategies that should be used for lessons, and (c) the influence of standardized testing. teachers from highly effective schools across the two cities expressed more elaborate conceptions (i.e., conceptions aligned with recommendations found in the mathematics education reform literature) with respect to these themes and others (see table 1). though teachers across both cities were under pressure to prepare students to be successful on a standardized test in mathematics, teachers at highly effective school expressed more elaborate beliefs and knowledge about mathematics, placed less emphasis on preparing their students for the test (in the president city area), and implemented more diverse instructional strategies (in arbor city) than their colleagues from typical schools. teachers’ conceptions of student diversity in this study, we also examined and compared teachers‘ conceptions with regards to student diversity across highly effective schools and typical schools. findings are summarized in table 2. teacher expectations. four themes differentiated teachers at highly effective and typical schools across the two cities with regards to teacher expectations: (a) high-level thinking, (b) academic excellence, (c) academic potential of a low achieving student, and (d) expectation of attending college. first, high-level thinking (hlt) in mathematics includes reasoning, problem solving, evaluating, predicting, synthesizing, making connections, building models or simulations, and communicating about the content (nctm, 2000; nsf, 1996). the emphasis is on students engaging in these high-level cognitive processes as they make sense of mathematics by connecting new to existing knowledge and learn content with understanding (see, e.g., hiebert & carpenter, 1992; lampert, 1990). teachers‘ comments about high-level thinking fell into three sub-themes, including student generated solutions, mathematical inquiry, and logical or mathematical reasoning. the more common of these responses was student-generated solutions; some teachers described students deriving alternative solution strategies that had not been taught by the teacher: sometimes it takes awhile and then they have a whole different way of doing it. i will show them the math way and they would show another way and i would be like wow! i never thought of it that way then again i don‘t know how a 9-year-old mind works so yeah, they surprised me. it is nice when it happens you know. (haney, typical, president city area) kitchen et al. comparing teachers’ conceptions journal of urban mathematics education vol. 2, no. 1 67 table 2 teachers’ conceptions of student diversity theme school type president city area arbor city teacher expectations high level thinking highly effective typical 89% 20% 30% 33% academic excellence highly effective typical 78% 60% 60% 83% academic ability of low-achieving student (low) highly effective typical 22% 40% 0% 33% college attendance highly effective typical 100% 100% 57% 17% culture cultural connection highly effective typical 60% 50% 50% 17% multicultural curricula highly effective typical 60% 33% 30% 33% not culture highly effective typical 50% 83% 70% 83% hlt was a strong distinguishing factor between teachers from highly effective and typical schools in the president city area. most of the teachers from highly effective schools (89%) saw hlt as strengths compared to 20% of teachers from typical schools. hlt was less prominent with the arbor city teachers, with about one-third of the teachers from both groups citing hlt. second, teachers‘ efforts to promote learning or raise achievement beyond what was expected in the standard curriculum were coded under academic excellence. this term is consistent with ladson-billings‘ (1994) notion of academic excellence as a core component of culturally relevant pedagogy. strategies that teachers stated they used to promote academic excellence included (a) differentiating instruction, (b) pushing students to strive academically, and (c) referring students to gifted and talented programs. similar to the hlt analysis, there were differences with regards to academic excellence across districts. in the president city area, teachers from highly effective schools (78%) were more likely to promote academic excellence than teachers from typical schools (60%). the opposite was true for arbor city, with 60% from highly effective schools and 83% from typical schools respectively. kitchen et al. comparing teachers’ conceptions journal of urban mathematics education vol. 2, no. 1 68 the third signifier of teachers‘ expectations involved their beliefs about student ability. teachers who exhibit high expectations believe that students have the potential to succeed (ladson-billings, 1994). the results show some variation in teachers‘ expectations of a low-achieving student with regard to student behavior including good or poor academic performance, potential for success, and descriptors of the student (e.g., smart). of these three, the most frequent comments involved academic performance. across the participating districts, teachers at typical schools were more likely than their counterparts at highly effective schools to state that students they had designated as ―low‖ had less potential for academic success than their other students. finally, an analysis of the data shows differences in expectations for college attendance between the two cities. all teachers from the president city area (highly effective and typical) stated that they believed a majority, if not all, of their students would attend college. such was not the case with arbor city. instead, teachers often cited the negative influences of students‘ neighborhoods and lack of college-educated role models as barriers to college attendance. nevertheless, teachers from highly effective schools (57%) were more likely to see college as an attainable goal for many of their students compared to teachers from typical schools (17%). culture. the culture analysis is an examination of teachers‘ conceptions of students‘ cultures (including race, ethnicity, language proficiency, and social class), the influence of culture on teachers‘ practices, and the relationship of culture to achievement. three themes emerged from our analysis: (a) cultural connection, (b) multicultural curricula, and (c) not culture. with the exception of multicultural curricula, these themes were not specific to mathematics but encompass education issues generally. first, some teachers commented on sharing cultural characteristics with students. the teachers described cultural connections in terms of race/ethnicity, shared language (spanish), and common life experiences: one of the ways that i win the parents over is by having a conference with the family. once they see that i am interested in their culture—it‘s my culture too. it‘s my background too. i let them know how i learned spanish because i had a blind grandmother and she just corrected me as i went along....i think that i‘ve been able to help the students not only be successful, but i‘ve been able to reach them….i‘m very much a part of them. (mercado, highly effective, president city area) across the two cities, teachers from highly effective schools were more likely than teachers from typical schools to mention connecting culturally with students. the difference among teacher groups was greater in arbor city (50% versus 17%) than the president city area (60% versus 50%). kitchen et al. comparing teachers’ conceptions journal of urban mathematics education vol. 2, no. 1 69 second, some teachers talked of incorporating culture into the teaching of mathematics. the most common strategy was situating mathematics within contexts that were familiar to students from either a personal or cultural perspective. the purpose of this strategy was to promote meaning and understanding of the content: my example that i give them, when we got to mexico. because the kids have been to mexico. is there only one way to get to mexico? well, no. we go by the freeway, we go through the downtown, we take this bridge, so i say how many ways is that? like five. that‘s the way you do the math. there are different ways to get to the answer. when we get to mexico, we just don‘t go the same way. there are different ways and you can still get to mexico. so in math, there are different ways to get to the answer. so they laugh about it. but they remember it. (rincon, highly effective, president city area) another strategy was to infuse into the curriculum current professional or historical figures in mathematics who come from diverse backgrounds. in president city, teachers from highly effective schools were almost twice as likely (60%) to embed multicultural curricula aspects into their instruction of mathematics than teachers from typical schools (33%). this difference was not found in arbor city where the rates were relatively equal at 33%. finally, a major theme involved teachers‘ statements that culture does not play a significant role in learning in general or the teaching and learning of mathematics (i.e., ―not culture‖). as alternatives, teachers offered ―kids are kids,‖ or the universality of mathematics as significantly affecting achievement. across both cities, teachers from typical schools were more likely than teachers from highly effective schools to state that culture was not a significant influence on learning (83% versus 50% in president city area, and 83% versus 70% in arbor city). however, when we examined subthemes, the ―kids are just kids‖ subtheme emerged as the most significant: i really try hard not to make it an issue of english and spanish or hispanic or nonhispanic. that is not the issue. the issue is that you are a child in this world that has value. (james, highly effective, arbor city) summary. the principal themes that emerged in our analysis of the participating teachers‘ conceptions of student diversity involved: (a) teachers‘ academic and future expectations for their students, and (b) teachers‘ conceptions of students‘ cultures. teachers from highly effective schools across the two cities expressed more elaborate conceptions (i.e., conceptions aligned with recommendations made by researchers in multicultural education) with respect to these themes and others (see table 2). teachers at highly effective schools across the two districts were more likely to state that all their students had potential for academic kitchen et al. comparing teachers’ conceptions journal of urban mathematics education vol. 2, no. 1 70 success. in addition, teachers from highly effective schools were more likely than teachers from typical schools to report sharing a cultural connection with students and this difference was greater in arbor city than in the president city area. lastly, while multicultural curricula did not distinguish the teacher groups in arbor city, teachers at highly effective schools in the president city area were almost twice as likely as teachers at typical schools to state that they used cultural contexts or cultural icons to teach mathematics. discussion and implications a guiding principle for this study was that teachers at highly effective schools possess desirable characteristics or engage in desirable instructional practices more so than teachers at typical schools. while other research has focused on non-instructional, school-level characteristics (edmonds, 1979; martin et al., 2000; purkey & smith, 1983), in this study we focused on teacher conceptions of mathematics and student diversity. following the literature on school restructuring (lee & smith, 2001; newmann and associates, 1996), we set out to examine how teachers in highly effective schools articulated their learning goals in mathematics for diverse student groups in comparison to their colleagues at typical schools. major findings and relevant findings the research findings help to explain the differentiation between highly effective and typical schools. finding #1: we found consistent evidence that teachers at highly effective schools expressed more elaborated conceptions of mathematics education than teachers at typical schools. across the two cities, teachers at effective schools stated that mathematics is more than just addition and subtraction, that instruction should help students make sense of mathematics, and that they employ multiple instructional strategies. while all teachers acknowledged that their curricular decisions were heavily influenced by high-stakes mathematics tests, the effective schools‘ teachers were more likely to state (very eloquently, at times) that they tried to teach beyond the test. furthermore, teachers at effective schools in the president city area tried to incorporate more advanced mathematics content in their fourth-grade curriculum. these teachers‘ conceptions were more aligned with the platonist view (hiebert & carpenter, 1992) or the problem-solving view (ball & cohen, 1999) of mathematics than the instrumentalist view. kitchen et al. comparing teachers’ conceptions journal of urban mathematics education vol. 2, no. 1 71 finding #2: we found evidence that teachers at highly effective schools possess more elaborated conceptions involving student diversity than teachers at typical schools. for instance, teachers at highly effective schools across the two cities advocated for explicitly including multicultural referents in the curriculum, whereas teachers at typical schools expressed ideas to the effect that mathematics is a culturally neutral subject. interestingly, but unexpectedly, teachers‘ conceptions of student diversity varied by city. in the president city area, teachers at highly effective schools couched their conceptions in terms of teacher expectations of their students. the teachers emphasized students‘ higher order thinking in mathematics, tried to improve academic excellence in mathematics, saw potential in low-achieving students, and expected the majority of their students to go to college (lipman, 1998, zeichner, 1996). in arbor city, teachers at highly effective schools couched their conceptions in terms of their own and their students‘ cultural backgrounds. the teachers stated that they shared similar cultural backgrounds (either through race/ethnicity or upbringing) with their students, and noted that their students brought knowledge acquired in their homes and communities that could enrich or make the teaching of mathematics more meaningful (banks, 1993; ladsonbillings, 1994, 1995). discussion of the two findings: we found compelling evidence that teachers at highly effective schools have more elaborated conceptions of both mathematics education and of student diversity. in general, teachers at highly effective schools articulated more often than their counterparts at typical schools conceptions that align with recommended reforms in mathematics education and conceptions with respect to student diversity that parallel scholars‘ recommendations related to the constructs of teacher expectations and culturally relevant instruction. because we did not study students‘ prior educational experiences, we do not know if these differences could be explained in terms of students at highly effective schools having already learned and achieved at higher levels prior to the study. moreover, these differences did not exist for each category (e.g., teachers at typical schools in arbor city were more likely to promote academic excellence than their counterparts from highly effective schools). nevertheless, given the care that was taken to differentiate highly effective schools from their typical counterparts, our findings do demonstrate that differences existed between these two groups of schools with regards to teachers‘ conceptions of mathematics education and of student diversity. teachers at highly effective schools believed in the importance of teaching more than problem solving and the basic facts and stated that they used a variety of instructional strategies to help students make sense of mathematics. teachers at highly effective schools also emphasized how they taught their students more than kitchen et al. comparing teachers’ conceptions journal of urban mathematics education vol. 2, no. 1 72 what was covered on the state assessment or standardized tests. while instruction as described in the reform literature (see, e.g., nctm, 2000; nsf, 1996) is very important, it is also the case that teachers do more than simply teach. beyond teaching, teachers can exhort students to do well, show care for students and their parents, and provide students with reasons for staying in school and trying hard for their future lives. all of these features would seem to rely on the kinds of expectations and cultural knowledge that the teachers in our highly effective schools expressed. thus, it seems reasonable to conclude that teachers‘ conceptions of student diversity, as well as their conceptions of mathematics education, both influence student learning (rodriguez & kitchen, 2005; rousseau & tate, 2003). our results are certainly consistent with such a view. highly effective teachers who participated in this study not only expressed high expectations for their students and/or valued their students‘ cultural backgrounds, they also talked about their desire to utilize mathematics curricula and implement instructional strategies aligned with reforms in mathematics education. it was beyond the scope of this study to determine whether causality existed between differing teachers‘ conceptions, such as how teachers‘ conceptions about student diversity may influence their conceptions about mathematics education. nevertheless, this study motivates questions about the interaction between teachers‘ conceptions about mathematics education and student diversity. the results also suggest the need for a more nuanced analysis of how variation in student backgrounds could result in variations in teachers‘ responses to those backgrounds. in this study, teachers at highly effective schools in the president city area, many of whom shared the same cultural backgrounds as their students, said more about their expectations for their students than their shared backgrounds and shared experiences. in contrast, teachers at highly effective schools in arbor city reported relying on their shared cultural backgrounds more than they discussed the kinds of expectations they had for their students. both groups of teachers, however, were very similar in their more elaborated conceptions of the teaching of mathematics with diverse student groups than teachers at typical schools. implications for teaching and teacher development to promote reform oriented instruction in mathematics, professional development efforts should involve helping teachers develop deep understanding of content knowledge, foster students‘ initiation and exploration in problem solving, identify students‘ mathematical conceptions and misconceptions, and implement content-specific teaching strategies (garet, porter, desimone, birman, & yoon, 2001; richardson & placier, 2001). this need is great with elementary teachers in kitchen et al. comparing teachers’ conceptions journal of urban mathematics education vol. 2, no. 1 73 mathematics, because they are generally not adequately prepared with content knowledge or effective teaching practices. by their comments, many teachers in the study displayed limited knowledge of how to incorporate students‘ linguistic and cultural diversity into mathematics instruction. some teachers, especially in typical schools, considered mathematics as ―culture-free.‖ in contrast, teachers in highly effective schools made statements about student diversity that could serve as a basis for the provisio n of professional development. that is, professional development should involve enhancing teachers‘ knowledge of student diversity and the intersections of culture and language with teaching practices and student achievement. such an effort may involve diversifying and tailoring professional development efforts to the specific challenges that different groups of teachers face with their students. for example, ―pedagogy of poverty‖ (haberman, 1991) may need to be addressed in arbor city, whereas issues of english language learners should be integrated in the president city area (moschkovich, 1999). recommendations for further research this study provided an in-depth analysis of fourth-grade teachers‘ conceptions across schools in geographically distinct areas. the focus on differences in teachers‘ conceptions affords one way to examine why some schools are more effective than others. our data, however, did not provide the means to understand how significant a role teachers‘ conceptions actually are in differentiating highly effective schools from their more typical counterpart. research needs to be undertaken to understand the significance of teachers‘ conceptions in distinguishing schools. furthermore, research needs to be carried out to examine the relationship between teachers‘ conceptions about mathematics education and student diversity. if teachers have more elaborate conceptions about the teaching and learning of mathematics, are they more likely to develop more elaborate conceptions towards student diversity and vice versa? effective instruction requires both content coverage and reform-oriented teaching practices. teachers at highly effective schools repeatedly stated that they taught more than what was covered on the state assessment or standardized tests, and their conceptions of the mathematics curriculum were more elaborate than those of typical schools‘ teachers. the results of this study suggest that future research should examine teachers‘ content coverage and seek to tie the content coverage to their specific teaching practices. high-stakes testing had a strong influence on teachers‘ conceptions of the teaching and learning of mathematics with diverse student groups. the results suggest that teachers believe that high-stakes tests press them to focus on teaching limited content to the exclusion of other content and constrain the instructional kitchen et al. comparing teachers’ conceptions journal of urban mathematics education vol. 2, no. 1 74 practices that teachers believe they can employ. further research on how highstakes testing influences teachers‘ conceptions and curriculum content coverage and how it also influences student learning and achievement seems warranted. this research topic is urgent as high-stakes assessment and accountability policy impacts more students, especially those who have traditionally been underserved in mathematics. the research findings described here combine constructs of effective pedagogy found in mainstream mathematics education with multicultural education. although research on academic content and student diversity rarely overlap, both emerged as significant in our data. these findings then beget the need for researchers, policymakers, and teacher educators to draw from multiple knowledge bases. yet, little is known about how the constructs inform and interact with one another at the student, teacher, or school level. if research is to improve education, there is a need to develop a research agenda that combines the expertise of mainstream academic content and multicultural education researchers. acknowledgments this study is part of a larger research project that compared mathematics education between highly effective and typical elementary schools in two cities that had participated in an urban systemic initiative (usi). both, the larger and this study, were supported by a grant from the national science foundation (nsf) to the urban institute in washington, dc. we would like to thank beatriz (toni) clewell and patricia campbell for their assistance and feedback on this manuscript; our research methodology section draws heavily from their book on the larger study (see clewell, campbell, & perlman, 2007). the bulk of this work was done while walter secada was on faculty at the university of wisconsin–madison. secada received additional support during his sabbatical year from the uw–madison graduate school and school of education, by the national center for improving student learning in mathematics and science (funded by the u.s. department of education office of educational research and improvement) and by diversity in mathematics education/center on learning and teaching (funded by nsf). additional support was provided to okhee lee by a grant from the national science foundation. all findings and opinions are the authors‘. no endorsement by any of the above organizations or funding agencies should be inferred. references ball, d. b., & cohen, d. k. 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(1990). opportunities, achievement, and choice: women and minority students in science and mathematics. in c. b. cazden (ed.), review of research in education, vol. 16 (pp. 153–222). washington, dc: american educational research association. o‘day, j. a., & smith, m. s. (1993). systemic school reform and educational opportunity. in s. h. fuhrman (ed.), designing coherent educational policy: improving the system (pp. 250– 311). san francisco: jossey-bass. peterson, p. l., & barnes, c. (1996). learning together: the challenge of mathematics, equity, and leadership. phi delta kappan, 77(7), 485–491. purkey, s., & smith, m. (1983). effective schools: a review. the elementary school journal, 83(4), 427– 452. richardson, v., & placier, p. (2001). teacher change. in d. v. richardson (ed.), handbook of research on teaching (4 th ed., pp. 905–950). washington, dc: american educational research association. rodriguez, a. j. & kitchen, r. s. (eds.) (2005). preparing mathematics and science teachers for diverse classrooms: promising strategies for transformative pedagogy. mahwah, nj: lawrence erlbaum associates. romberg, t. a. (1992). problematic features of the school mathematics curriculum. in p. w. jackson (ed.), handbook of research on curriculum (pp. 749–788). new york: macmillan. rousseau, c. & tate, w.f. (2003). no time like the present: reflecting on equity in school mathematics. theory into practice, 42(3), 210–216. http://timss.bc.edu/timss1995i/timsspdf/t95_effschool.pdf kitchen et al. comparing teachers’ conceptions journal of urban mathematics education vol. 2, no. 1 77 schifter, d., & fosnot, c.t. (1993). reconstructing mathematics education: stories of teachers meeting the challenge of reform. new york: teachers college press. secada, w. g. (1989). educational equity and equality of education: an alternative conception. in w. g. secada (ed.), equity in education (pp. 68–88). philadelphia, pa: the falmer press. silva, c. m., & moses, r. p. (1990). the algebra project: making middle school mathematics count. journal of negro education, 59(3), 375–391. thompson, a. (1992). teachers‘ beliefs and conceptions: a synthesis of the research. in d. grouws (ed.), handbook of research on mathematics teaching and learning (pp. 127–146). new york: macmillan. tymoczko, t. (1986). new directions in the philosophy of mathematics. boston, ma: birkhauser. wiggins, g.p., & mctighe, j. (1999). understanding by design handbook. alexandria, va: association for supervision and curriculum development. zeichner, k. m. (1996). educating teachers to close the achievement gap: issues of pedagogy, knowledge, and teacher preparation. in b. williams (ed.), closing the achievement gap: a vision for changing beliefs and practices (pp. 56–76). alexandria, va: association for supervision and curriculum and development. kitchen et al. comparing teachers’ conceptions journal of urban mathematics education vol. 2, no. 1 78 appendix i teacher interview protocol fall 2000 at the completion of two consecutive visitations with teachers at participating schools, the interviewer/observer conducts an interview with the teacher. a primary purpose of interviews is to understand teachers‘ conceptions (beliefs and knowledge) about mathematics teaching and learning with culturally and linguistically diverse students. a secondary purpose is to extend the information about mathematics to other subject areas. a third purpose is to use interview responses in verifying or triangulating classroom visitations in their classrooms as well as survey responses of the entire teacher group in the research. interviews examine teachers‘: (a) general perceptions of the lessons, (b) conceptions of the nature of mathematics, (c) conceptions of students, and (d) conceptions of integrating mathematics with students. each question is followed with probes. after a classroom visitation, the observer obtains academic tasks, such as handouts or overhead transparencies, that are used as the context for the interview. to contextualize the interviews in relation to the lessons, both the interviewer/observer and the teacher use examples of salient events during the lessons. teachers’ general perceptions of the lessons 1. do these lessons give us as a good sense of the kind of work that you typically have this group of students do in mathematics? probe: if yes, how is it typical? if no, what is typical? probe: use specific incidents to probe teacher‘s perceptions. 2. what were the mathematical ideas that you were trying to cover during these lessons? probe: ask the teacher to define terms used, such as ―topics‖ and ―skills.‖ topics – e.g., measurement, geometry, fractions skills (basic, low-level) – e.g., computational algorithms processes (higher-level) – e.g., observe, describe, classify, infer, conclude teachers’ conceptions of the nature of mathematics 3. what are three most important mathematics ideas for your students to learn in general? probe: why? 4. what determines the mathematical content that you cover during the year? probe: to what extent do you rely on: textbooks, curriculum guides, standardized tests, new or innovative curricular, the standards, personal expertise, and/or beliefs about mathematics kitchen et al. comparing teachers’ conceptions journal of urban mathematics education vol. 2, no. 1 79 note: ask the teacher to explain each of these items. probe: how do the lessons fit into what you just said is important? teachers’ conceptions of the students 5. are there some students who aren‘t doing as well in mathematics as they should? probe: anyone who surprised you? anyone who disappointed you? 6. why do you think some students are doing better than you expected? why do you think some are not doing as well as they should? probe: probe for teachers‘ accounts of high achievement probe for teachers‘ accounts of low achievement 7. what are you doing to meet the learning needs of these students? probe: does your instruction change during the academic year to meet your students‘ learning needs? if so, how? probe: probe for teacher efficacy (i.e., how teachers work to actually meet needs of students). what is the impact of your efforts on these students? teachers’ conceptions of integrating mathematics with the students 8. is math ―culture free,‖ or does culture have any influence on math? probe: give an example of how students‘ backgrounds (such as their home culture and gender) have any influence on mathematics. probe: the education literature says that students from certain cultural backgrounds do not question adults at home and, therefore, do not ask questions to the teacher. what do you think about the influence of your students‘ cultural backgrounds on math learning? 9. do you consider your students‘ culture, language, and gender in your teaching in order to promote their math learning? probe: give an example of how you used your students‘ cultural experiences during the lesson (if possible). (if not), give an example from your teaching at other times or during other lessons during this year of culture, gender, english language proficiency, socioeconomic status, disability. kitchen et al. comparing teachers’ conceptions journal of urban mathematics education vol. 2, no. 1 80 appendix ii teacher interview protocol spring 2001 1. teachers’ general perceptions of the lesson what were the mathematical (scientific) ideas (broad, generic sense) that you were trying to cover during these lessons? probe: probe distinctions among topics, skills, or processes (a) topics – e.g., measurement, geometry, fractions, biology, ecology, physical science (b) skills (basic, low-level) – e.g., measurement, computational algorithms (c) processes (higher-level) – e.g., measure, observe, describe, classify, infer, conclude 2. testing during our last visit, many teachers spoke about the impact standardized testing was having on instruction. in what ways, if any, do you feel your math (science) lesson today was influenced by state/district standardized tests? 3. for the two students who are the focus of the math (science) interview could we talk for a few minutes about student a? (note: it will be hard for this to happen without teacher using student name.)  did students a do anything that surprised you during your lesson? if yes, what did the student do? why did that surprise you?  how well do you usually expect student a does in math (science) in general? how well does student a perform on math (science) tests?  what strengths does student a bring to math (science)? what weaknesses?  which, if any, strategies work particularly well with student a in math (science)? why do you think student a responds well to these strategies?  which, if any, strategies don‘t work particularly well with student a in math (science)? why do you think student a doesn‘t respond well to these strategies?  what, if any, in and out-of-class resources do you have that can help you better teach student a mathematics (science)? could we talk for a few minutes about student b? go through the same procedure for student b journal of urban mathematics education december 2008, vol. 1, no. 1, pp. 1–4 ©jume. http://education.gsu.edu/jume lou edward matthews is an assistant professor of mathematics education in the college of education at georgia state university, 30 pryor street, atlanta, ga, 30303; e-mail: lmatthews@gsu.edu. his research focuses on how teachers incorporate visions of culturally relevant teaching into practice, as well as with the critical examination of mathematics reform ideology, black masculinity and schooling, and teacher experiences in the context of reform. dr. matthews is currently serving as the 2007–2009 president of the benjamin banneker association, a national non-profit organization advocating excellence in mathematics for black students. dr. matthews is also co-founder and currently inaugural editor-in-chief of the journal of urban mathematics education. editorial illuminating urban excellence: a movement of change within mathematics education lou edward matthews georgia state university over a year ago my colleagues and i embarked on an unchartered quest to ―open up‖ within the mathematics education community a scholarly space that could honor—not marginalize—the professional work in the domain we characterized as urban. we sought to open up a space in mathematics education that would honor and enrich the work in this domain which had become central to our endeavors as reformers. admittedly, with only one tenured professor in our group of six, the catalyst for this risky endeavor laid in our own frustrations within the academy to gain access to, and collectively synthesize, the complexities of mathematics reform taking place in urban schools. our initial ―conversational‖ surveys of the landscape of mathematics education discourse in the fall of 2007 (e.g., ―top-tier‖ mathematics education journals and research conference offerings) revealed a suspicious absence of urban scholarship. in addition, access to existing voices outside of the community has been significantly restricted with limited access to the eric database, the proliferation of pay-to-read scholarship, and narrowly defined notions of what counts as ―scientific‖ research. after months of painstaking deliberation, our efforts culminated in the launching of this journal, the journal of urban mathematics education (jume), on january 15, 2008 with a national call for manuscripts and our home webpage: http://education.gsu.edu/jume. the following mission statement heralds this initiative: to foster a transformative global academic space in mathematics that embraces critical research, emancipatory pedagogy, and scholarship of engagement in urban communities. as we met to frame the components of this statement, several tensions surfaced, centering around three crucial questions that i wish to discuss: (1) how should we define ―urban‖ in mathematics education? (2) how should we orient ourselves http://education.gsu.edu/jume matthews editorial journal of urban mathematics education vol.1, no.1 2 toward work in the urban domain? and (3) how should we give ―voice‖ to the complex dynamics of change within the urban domain? when we say urban while engaging in our discussion around what happens in the urban domain, we found ourselves often at odds as to how to define the domain. in one sense, it was obvious that urban is used to define a particular geographical space, for example ―metro atlanta.‖ yet, geography alone was not enough to draw attention to the various complexities regarding the racial and ethnic makeup of schools and communities, degree of economic hardships, and neighborhood and community traditions. by default, many educators forgo delineating these complexities, focusing only on specific groups within urban schools and communities. in this negligent manner, the term urban is often relegated to an umbrella term used indiscriminately to denote african american, hispanic, immigrant, or low-income students. furthermore, given that mathematics ―achievement gaps‖ are popularly depicted in terms of race, ethnicity, and/or income, the term urban is often utilized as an all encompassing deficit term. not wanting to continue the status quo exercised in practice, we settled on the following definition of the urban domain, which will no doubt undergo extensive, ongoing reflection and refinement: here, the view of the urban domain extends beyond the geographical context, into the lives of people within the multitude of cultural, social, and political spaces in which mathematics teaching and learning takes place. with this definition, we set a standard that all scholarship, which has the urban domain as its primary focus, should give a thorough accounting of the complexities of cultural, social, and political elements through which mathematics teaching and learning and mathematics education reform is experienced. excellence as a frame of reference the focus of jume is aligned with a counter-trend of equity-focused organizations (e.g., benjamin banneker association’s national leadership summit on the mathematics education excellence of black children) to replace the standard practice of ―gap-gazing‖ as a catalyst for action with what we call ―illuminating excellence.‖ why do this? certainly, this is not to ignore the strained efforts of educators working to meet yearly nclb progress goals, counter budgets cutbacks, and develop and implement new mathematics standards for teaching and content quality. these concerns are not to be ignored. we are aware that within matthews editorial journal of urban mathematics education vol.1, no.1 3 georgia there is growing concern that the implementation of new standards in mathematics has contributed to an alarming numbers of students who have not passed recent standardized tests in mathematics at the upper elementary and middle grades (georgia department of education, 2008; strepp, 2008a, 2008b). we were equally concerned about the invisibility of exemplary teachers and administrative practice facilitating mathematics success like the kind we encounter in our everyday duties. an example of this kind of scholarship that positions excellence as a starting point from which to examine urban mathematics reform is gutiérrez’s (2000) ―urban youth in mathematics: unpacking the success of one math department.‖ we have all borne witness to the underutilized, hidden wisdom found in some of our partner schools and classrooms. while this wisdom of teaching, curriculum management, culturally relevant pedagogy, collaboration, and local action has provided powerful counter narratives to the diatribe on urban achievement in mathematics, exclusion of such narratives in mainstream mathematics education journals relegates them to largely ―asystemic‖ to the community of mathematics education reformers. the reporting of excellence within the urban domain has been suspiciously underreported in top-tier mathematics education journals. the existence of this work outside of the espoused canons of mathematics education literature is cause for significant concern (see, e.g., gutiérrez, 2000). for one, it calls into question whether mathematics educators consider the urban domain as relevant engagement (i.e., beyond a sample size comparison), or, even more ominous, whether our ―major‖ scholarship is relevant for truly reforming urban practice. the latter question of relevance for practice, interestingly enough, was the primary agenda topic at a research agenda conference hosted by the national council of teachers of mathematics in hyacinth, maryland. the conference gathered 60 to70 national representatives from the research community for the primary purpose of developing an agenda for research that could be used to inform practice to a greater extent than is currently seen. in lay terms, researchers pondered the relevance of current research to practice as we asked: do practitioners not use our work? the notion of relevance is an important standard for the professional lives of mathematics educators. without such a notion, there is little real community to speak of beyond the ivory towers of academia. urban change as a movement of people we choose solidarity as an important focus for the work of mathematics education reformers. in that, another decision was made to honor the way in which urban groups move amidst the aforementioned complexities of urban mathematics reform. a commitment to excellence meant broadening a definition of mathematics reform to include the social movement of people. the prevailing matthews editorial journal of urban mathematics education vol.1, no.1 4 view of reform in mathematics posits that true change vis-a-vis improvement in learning experiences is accomplished by increasing the content and pedagogicalcontent knowledge of practicing teachers. less attention is given to the racialized, cultural, and political experiences within the urban domain that influence people to move urban practice. the prevailing view essentially sidesteps any critical analysis of race, culture, and/or policy constraints that have been documented by a substantial number of equity researchers in mathematics education (albeit, seldom in mainstream mathematics education journals). social aspects of human development have been largely ignored in mathematics education research. although we are taught that the discipline lies at the nexus of social change (along with changes in psychology and mathematics), this change is most often articulated in terms of shifts in government ideologies (e.g., sputnik) and economic trends (e.g., ―mathematically literate workers‖). mathematics reform has scarcely been defined in terms of ―ground up‖ movements of people. little is documented about the local actions of school and community families to ―right‖ the inequities espoused in mathematics reform. social movements of equity such as the civil rights movements for gender and racial equality have been scarcely emphasized as critical to mathematics reform success. capturing the excellence of local groups as they author change has the potential to connect mathematics education scholarship to the very communities it intends to serve. what this look likes as a base of scholarship remains to be seen, but in jume we open this space. the articles in this inaugural issue range in nature in the ways in which the scholars tackle the questions we, ourselves, have struggled through; in that, they extend the very definition and context of urban, challenge racist conventions of urban schooling, and offer insights for finding excellent practice. we trust that you will be challenged as you join us on this journey in which old questions are explored (differently) and new questions formulated. this space is open and freely accessible to all who have as their primary interest the illumination of urban excellence. references georgia department of education. (2008). scores rise, gap closes on new crct. retrieved december 10, 2008, from http://www.doe.k12.ga.us/pea_communications.aspx?viewmode=1&obj=1635 gutiérrez, r. (2000). advancing african-american, urban youth in mathematics: unpacking the success of one math department. american journal of education, 109(1), 63–111. strepp, d. (2008a). failed math tests = swollen summer classrooms [electronic version]. the atlanta journal-constitution. retrieved december 10, from http://www.ajc.com/metro/content/metro/stories/2008/05/23/summerskl_0525.html strepp, d. (2008b). unhappy students: classes start right away for those failing crct [electronic version]. the atlanta journal-constitution. retrieved december 10, from http://www.ajc.com/search/content/metro/stories/2008/06/01/summer_school_crct.html http://www.doe.k12.ga.us/pea_communications.aspx?viewmode=1&obj=1635 http://www.ajc.com/metro/content/metro/stories/2008/05/23/summerskl_0525.html http://www.ajc.com/search/content/metro/stories/2008/06/01/summer_school_crct.html microsoft word final lemon-smith vol 3 no 1.doc journal of urban mathematics education july 2010, vol. 3, no. 1, pp. 98–103 ©jume. http://education.gsu.edu/jume shonda lemons-smith is an assistant professor in the department of early childhood education in the college of education, at georgia state university, p.o. box 3978, atlanta, ga 30302; email: slemonssmith@gsu.edu. her research focuses on mathematics education in urban contexts, specifically, teacher development, issues of equity, and culturally relevant pedagogy. book review the nuts and bolts: a review of culturally specific pedagogy in the mathematics classroom: strategies for teachers and students1 shonda lemons-smith georgia state university ince the passage of the no child left behind act of 2001 (nclb),2 increasing attention has been given to the academic achievement of students in u.s. public schools, particularly historically marginalized populations such as students of color, students living in poverty, and students who are english language learners. this unprecedented era of high-stakes accountability tasks teachers and school personnel with ensuring that all students attain academic success (albeit, most often through the misguided measures of standardized tests). gone are the days where it was commonplace for school and school district personnel to show limited—or, dare i say, superficial—concern for the lack of achievement of certain student populations. students’ race, ethnicity, socioeconomic status, gender, language, or other attributes can no longer be posited as explanatory variables and detractors from accountability measures. instead, student attributes of difference shift from functioning as barriers to teaching and learning to being a statement of fact. while scholarship related to culturally specific (or relevant, responsive, etc.) teaching was delineated in the field of education generally, and mathematics education specifically, long before the enactment of nclb (see, e.g., ladsonbillings, 1995a, 1995b), its visibility has increased substantially as school and school district personnel grapple with how to promote the academic excellence of all students. jacqueline leonard’s book culturally specific pedagogy in the mathematics classroom: strategies for teachers and students (routledge 2007) is indeed an invaluable resource for mathematics teachers and school personnel as 1 leonard, j. (2007). culturally specific pedagogy in the mathematics classroom: strategies for teachers and students. new york: routledge. 207 pp., $43.95 (paper), isbn 978-0-8058-6105-1 http://www.routledge.com/books/details/9780805861051/ 2 no child left behind act of 2001, public law 107-110, 20 u.s.c., §390 et seq. s lemons-smith book review journal of urban mathematics education vol. 3, no. 1 99 they engage in the day-to-day work of realizing that admirable and attainable goal. in chapters one through three, leonard articulates a theoretical backdrop for culturally specific teaching. chapters four through six provide the reader with concrete examples of what culturally specific teaching “looks like” in the mathematics classroom. chapters seven and eight collectively conclude the book; these chapters discuss empowerment in diverse mathematics classrooms and race and achievement in mathematics, respectively. intertwining theory and strategies throughout chapter one, “culture, community, and mathematics achievement,” opens pointedly with leonard recalling an experience she had with a young african american man at a bus stop where the discussion of whether a person needs to know mathematics beyond the “basics” was the topic of a chance, casual conversation. after the conversation, leonard pondered why the young man did not “see” algebra, statistics, and other domains of mathematics as relevant to his everyday life. the inclusion of this story, i believe, introduces the fundamental premise that culturally specific mathematics teaching promotes connections to the real world and fosters positive beliefs about the need for mathematics. this chapter highlights the recent trends in mathematics and science study and national assessment of educational progress datasets, and u.s. school demographic data in general. using these datasets, leonard critiques educational reform for its inability to achieve equity in education for the majority of students of color and students living in poverty, a sentiment shared by numerous other scholars, whose research and scholarship she amply cites throughout to support her critique. leonard not only provides a critique of the current state of education in chapter one but also outlines the theoretical framework guiding culturally specific pedagogy: critical race theory (crt). she articulates how crt has been adapted and used by scholars and outlines its six major themes: crt (a) recognizes that racism is endemic to american life; (b) expresses skepticism toward dominant legal claims of neutrality, objectivity, colorblindness, and meritocracy; (c) challenges ahistoricism and insists on a contextual/historical analysis of the law; (d) insists on recognition of the experiential knowledge of people of color and their communities of origins in analyzing law and society; (e) supports an interdisciplinary perspective; and (f) works toward the end of eliminating racial oppression as part of the broader goal of ending all forms of oppression. she shares prior research on culturally based education such as work done with american indian/alaska native and native hawaiian children. leonard also includes a poignant discussion of teachers’ beliefs about culture and learning mathematics, the culture of power, and the role of mathematics identity and socialization. here the widely held notion of america as a “melting pot” is explored through excerpts of lemons-smith book review journal of urban mathematics education vol. 3, no. 1 100 teacher comments from a graduate mathematics education course. leonard delineates a solid argument for why framing instruction within the context of diverse students’ culture is a well-founded paradigm for mathematics teaching and learning. chapter two, “cognition and cultural pedagogy,” addresses cultural transmission and cultural capital, and situates culturally specific pedagogy within the context of cognitive theory. leonard highlights several cognitive theorists including saxe and vygotsky and the extent to which culture plays an important role in learning mathematics. saxe’s belief that individuals create new knowledge while participating in culturally mediated activities is explicitly tied to culturally specific teaching. similarly, vygotsky advocated that cultural artifacts might be used to facilitate children’s learning. this chapter is vital in that it responds to critics of culturally specific pedagogy who often assert that it lacks substantial theoretical grounding. that is to say, in this chapter, leonard provides a context for centering the work, making sound connections to broader paradigms such as social constructivism. to further support the theoretical grounding of cultural specific pedagogy, leonard reports on (and makes connections to) a number of other more familiar studies that examined children’s cognition, including cognitively guided instruction and the quasar (quantitative understanding: amplifying student achievement and reasoning) project. reported are the research studies of malloy and jones (1998) who examined how african american middle school students engaged in mathematical problem solving, and ku and sullivan (2001) who explored the problem solving strategies of taiwanese students. these studies are valuable to the chapter because they explicitly attend to the cognitive activities of diverse student populations. chapter three, “cultural pedagogy,” delineates the various domains of cultural pedagogy including culturally relevant teaching, cultural brokering, border crossing, culturally responsive teaching, culturally specific pedagogy, and diversity pedagogy. leonard offers clear, concise definitions of the various types of cultural pedagogy. furthermore, vignettes are presented to help the reader understand how a teacher might engage in culturally specific pedagogy. the inclusion of the vignettes is crucial because it introduces practical examples of how culturally specific pedagogy plays out in the classroom. in courses, workshops, and conferences the question of what culturally mediated mathematics teaching actually “looks like” persists. leonard further illuminates this question by outlining teachers’ pedagogical practices and presenting examples and counterexamples of instructional interactions. for example, she shared the counterexample of ms. harding who attempted to implement a culturally relevant activity by asking her students, who were hispanic, to make tortillas. this activity met resistance from some of the students’ mothers regarding their sons’ participation. ms. harding had not anticipated this cultural conflict and leonard uses this situation to high lemons-smith book review journal of urban mathematics education vol. 3, no. 1 101 light the importance of using parents and other community members as a resource when planning culturally relevant activities as they can offer insight into cultural norms and values. it should be noted, however, that these examples and counterexamples are not intended to be a recipe, but rather a purposeful lens for thinking about teacher and student instructional interactions. while the first three chapters articulate a theoretical backdrop for culturally specific teaching, chapters four through six provide the reader with classroombased, concrete examples of culturally specific teaching. chapter four, “problem solving, problem posing, multicultural literature, and computer scaffolding,” discusses the problem-solving process and the role culture plays in how students approach a problem. one example included is the widely cited bus problem shared by tate (1994): it costs $1.50 each way to ride the bus between home and work. a weekly pass is $16. which is the better deal, paying the daily fare or buying the weekly pass? this problem, used on a district-wide assessment, highlights the disconnect that often exists between test developers and test takers. in the case of the bus fare problem, the test developers assumed that students would choose to pay the daily fare because it was cheaper. that assumption, however, was not consistent with the lived experiences of many of the students taking the assessment. students indicated that the weekly pass was cheaper because it could be used more than once a day and shared with other members of the family. a “typical” test developer is likely to be middle-class, own a car, and work only one job. thus, her or his perspective is markedly different from many students of color living in urban areas and significantly influences the mathematical lens through which the problem is seen. leonard also refers to the use of multicultural literature and computerassisted instruction to facilitate students’ problem posing and problem solving. to this end, she highlights the intelligent computer-assisted instruction program—a benjamin banneker project that examined the engagement and strategies of elementary african american students as they read culturally relevant stories and solved problems on the computer. vignettes and excerpts from teacher interviews from a similar project at parker charter school in new york are highlighted as well. chapter five, “the underground railroad: a context for learning mathematics and social justice,” discusses notions of equity and social justice. leonard describes in detail a thematic unit about the underground railroad that was used to engage students in culturally specific pedagogy. two teachers, ms. baker and ms. cho, are highlighted and vignettes of their classroom instruction analyzed. examples of students’ journal writings maintained during the unit are also shared. collectively, the vignettes and writings provide a thick description of the lives of classroom teachers implementing culturally specific and empowering mathemat lemons-smith book review journal of urban mathematics education vol. 3, no. 1 102 ics instruction to diverse student populations. the chapter is indeed powerful in that it brings tangibility to the construct of culturally specific pedagogy. similarly, chapter six, “women in aviation and space: the importance of gender roles in mathematics education,” delves into gender and academic achievement in mathematics education. here leonard explores gender differences in standardized testing, the gender imbalances of advanced mathematics courses, and teacher beliefs of mathematics as a male domain. she reports on projects aimed at counteracting these trends. for instance, the bessie coleman project, which aimed to foster female and african american students’ achievement and positive attitudes toward mathematics, is highlighted along with space links, a similar program, that integrated space science and mathematics. chapter seven, “learning mathematics for empowerment in linguistically and culturally diverse classrooms,” addresses language acquisition and parental involvement, and discusses projects aimed at fostering the mathematics learning experiences of students who are english language learners. teacher beliefs and reflections on practice are illuminated through vignettes. the chapter also illustrates examples of multicultural literature that can be used to engage linguistically and culturally diverse students in mathematical discourse. for example, leonard highlights several texts, including: grandfather tang’s story, one grain of rice, the spider weaver, the three little javelinas, sadako, first day in grapes, harvesting hope, and a migrant child’s dream: farm workers adventures of cholo, vato, and pano. she provides the reader with a brief synopsis of these texts and underscores embedded mathematical content. in the final chapter, “race and achievement in mathematics,” leonard provides a historical perspective on race and schooling, the achievement gap, the mathematics socialization and identity of african american students, and links to everyday mathematics. leonard uses martin’s (2000) scholarship to explore the salience of race and its role in the underachievement and low achievement of african american learners. though a complex construct, the implications of race are stated in an easily understood manner and links are made to principles of culturally specific pedagogy. an indispensable staple culturally specific pedagogy in the mathematics classroom: strategies for teachers and students is a must-have resource for mathematics teachers, teacher educators, and school personnel who serve (diverse) student populations. collectively, the chapters articulate a theoretical backdrop for culturally specific teaching and illuminate concrete examples of what culturally specific teaching “looks like” in the mathematics classroom. the book is filled with vignettes, teacher reflections on practice, and other instructional artifacts that provide the reader with lemons-smith book review journal of urban mathematics education vol. 3, no. 1 103 the “nuts and bolts,” so to speak, of culturally specific teaching. the text is solid in scope and depth, yet easy to read. it introduces individuals who might be novices to culturally specific pedagogy, but crystallizes the knowledge base for those who are more advanced. this book indeed moves the field of mathematics education forward and should be an indispensable staple on the bookshelves of preservice and inservice mathematics teachers and mathematics teacher educators. references ku, h., & sullivan, h. j. (2001). effects of personalized instruction on mathematics word problems in taiwan. paper presented at the 24th national convention of the association for educational communications and technology, atlanta, ga. ladson-billings, g. (1995a). making mathematics meaningful in a multicultural context. in w. g. secada, e. fennema, & l. b. adajian (eds.), new directions for equity in mathematics education (pp. 126–145). cambridge, united kingdom: cambridge university press. ladson-billings, g. (1995b). toward a theory of culturally relevant pedagogy. american educational research journal, 32, 465–491. leonard, j. (2007). culturally specific pedagogy in the mathematics classroom: strategies for teachers and students. new york: routledge. malloy, c., & jones, m. g. (1998). an investigation of african american students’ mathematical problem solving. journal for research in mathematics education, 29, 143–163. martin, d. b. (2000). mathematics success and failure among african-american youth: the roles of sociohistorical context, community forces, school influence, and individual agency. mahweh, nj: erlbaum. tate, w. f. (1994). race, entrenchment, and the reform of school mathematics. phi delta kappan, 75, 477–484. microsoft word final oppland-cordell vol 7 no 1.doc journal of urban mathematics education july 2014, vol. 7, no. 1, 19–54 ©jume. http://education.gsu.edu/jume     sarah b. oppland-cordell is an assistant professor of mathematics at northeastern illinois university, 5500 north st. louis avenue chicago, il 60625, e-mail: s-cordell@neiu.edu. her research interests include examining the identity development of marginalized students to understand how they succeed in mathematics, exploring relationships between marginalized students’ identity development and participation in collaborative mathematics classrooms, and drawing on marginalized students’ experiences to create equitable mathematics teaching and learning environments. urban latina/o undergraduate students’ negotiations of identities and participation in an emerging scholars calculus i workshop sarah b. oppland-cordell northeastern illinois university in this article, the author presents a qualitative multiple case study that explored how two urban latina/o undergraduate students’ emerging mathematical and racial identity constructions influenced their participation in a culturally diverse, emerging scholars program, calculus i workshop at a predominately white urban university. drawing on critical race theory and latina/o critical theory, cross-case analysis illustrates that participants’ emerging mathematical and racial identities—co-constructed with their other salient identities—contributed to positively shifting their participation by: (a) changing their perceptions of their and peers’ mathematics abilities, (b) allowing them to challenge racialized mathematical experiences, and (c) strengthening their comfort levels in the workshop environment. the latina/o participants’ counter-stories support that the sociopolitical nature of identity development and participation in mathematical learning contexts should be embraced because it provides additional knowledge regarding how and why latina/o students attain mathematical success. keywords: collaborative learning, identity, latina/o students, mathematics education, race anessa and immanuel1 were latina/o freshman2 enrolled in an emerging scholars program (esp) calculus i workshop at hall university. in this optional mathematics workshop, which ran parallel to vanessa’s and immanuel’s required calculus i course, culturally diverse peer groups collaboratively solved challenging calculus i problems. this workshop consisted of 27 students: approx                                                                                                                           1 vanessa, immanuel, and the names of all people and places used in this article are pseudonyms. 2 the classifications freshman, sophomore, junior, and senior refer to describing participants (and workshop students they interacted with) as first, second, third, and fourth year students, respectively, at hall university. these classifications are also used to describe participants’ year in high school. v oppland-cordell urban latina/o students in calculus i journal of urban mathematics education vol. 7, no. 1   20 imately 41% latina/o (4 women and 7 men), 30% asian (5 women and 3 men), 22% white (2 women and 4 men), and 7% african american (1 woman and 1 man), and 44% women. i was not only the researcher but also the instructor of the calculus i workshop. in her interviews, vanessa revealed that her interactions with ms. johnson, an african american woman, her sophomore (geometry) and senior (precalculus) high school mathematics teacher, were relevant to understanding her participation in the calculus i workshop at hall university. vanessa believed her latina status contributed to the low mathematical expectations and negative treatment she received from ms. johnson. such treatment included having her preferred problemsolving strategies (e.g., translating mathematical symbolic notation into words) criticized: “i have a tendency of writing a lot. i write words and sentences and with that teacher she’s like, ‘no, no, no. math is all about no words’” (interview 1, january 25, 2008). because of such interactions, vanessa believed it was inappropriate for her to apply some of her preferred problem-solving strategies to learn workshop material. however, as vanessa participated in the workshop she realized that such strategies were appropriate: “i use words to read math and read math by using words [laughs], so that helped a lot too. we would do that a lot in the workshop” (interview 4, may 28, 2008). vanessa’s workshop experiences, which included encountering opportunities to resist negative high school mathematics experiences, aided her in constructing a strengthened self-perception as a latina mathematics learner: i was like, wow! i guess it’s good to see how other hispanic people are so good at doing math…. it’s not what people usually think of. i think it makes me proud that there’s a chunk of us, i’ll put myself in that group, that are willing to do whatever to be good at math or to excel in math. (interview 3, may 12, 2008) vanessa’s and immanuel’s stories, which the readers will learn more about in the findings section, provide examples of not only how aspects of latina/o students’ mathematical and racial identities can shape participation in a collaborative mathematics learning context but also how elements of these identities can positively merge as a result of this participation. for example, on one hand, vanessa’s negotiations of ms. johnson’s perceptions of her mathematics ability initially contributed to limiting her workshop participation. on the other hand, however, her participation in the workshop’s mathematical practices allowed her to develop tools to resist harmful racialized mathematics experiences3 (e.g., chal                                                                                                                           3 in this article, racialized mathematics experiences refer to participants’ mathematics experiences that are structured by societal meanings for race, racial identity, or racial stereotypes (martin, 2009). in this article, i also refer to mathematics learning contexts as racialized (martin, 2009) given that i perceive students’ racialized mathematics experiences as influencing how they participate in and learn mathematics within such settings. oppland-cordell urban latina/o students in calculus i journal of urban mathematics education vol. 7, no. 1   21 lenging ms. johnson’s negative ascriptions to her mathematical and racial identities), to build on positive aspects of her latina and mathematical identities, and to positively merge her latina and mathematical identities. as revealed in the findings section, positive mergings of vanessa’s and immanuel’s mathematical and racial identities, including their connection to other salient identities and broader contexts in which meanings for these identities became relevant, contributed to positive convergences among the various strains of their identity development and participation. such stories about latina/o students are rare despite evidence that students’ practice-linked identities⎯“identities that people come to take on, construct, and embrace that are linked to participation in particular social and cultural practices” (nasir & hand, 2008, p. 147)⎯inform, and are informed by, intersecting aspects of their identities in mathematics learning contexts (esmonde, brodie, dookie, & takeuchi, 2009). the ways in which latina/o students’ identities and participation can positively influence one another over time as they engage in collaborative mathematics learning contexts leads one to question (a) what additional knowledge has yet to be uncovered about these relationships and (b) how such knowledge can contribute to understanding their mathematical success. because mathematics classrooms are racialized (martin, 2009), it is also important to understand how latina/o students make sense of, narrate, perform, and negotiate mathematical and racial identities, and to understand how these identities inform how they make sense of, narrate, perform, and negotiate practice-linked (or disciplinary) identities in collaborative mathematics learning contexts. here i illustrate how two urban latina/o (and mexican american4) undergraduate students negotiated ways of more productively participating in the workshop via their emerging mathematical and racial identity (re)constructions. this study occurred in an esp mathematics workshop: a collaborative mathematics learning environment that has been nationally recognized for effectively supporting underrepresented students in succeeding in mathematics. while the majority of research conducted on esp mathematics settings has focused on participants’ static achievement outcomes instead of forces that influence these outcomes (an exception includes asera, 1988), this study uses an identity lens to examine latina/o students’ perceptions of their participation within an esp cal                                                                                                                           4 in the united states, mexican americans constitute approximately 65% of the latina/o population (motel & patten, 2012). due to the essentialization of latinas/os’ experiences, aspects of mexican americans’ histories, culture, and immigration experiences that differ from those of other latina/o cultural subgroups are often disregarded (telles & ortiz, 2008). historically, mexican americans have experienced significant obstacles linked to colonization, immigration, segregation, and discrimination, including within educational contexts (rodriguez, 2000). given mexican americans’ unique histories and experiences in the united states, it is critical that mathematics education scholars take into consideration the sociopolitical nature of mexican american students’ mathematical lives.   oppland-cordell urban latina/o students in calculus i journal of urban mathematics education vol. 7, no. 1   22 culus i workshop. a calculus i workshop was selected because calculus functions as a gateway course for stem5 majors, making it an interesting setting to explore the role of identity, power, and broader contextual experiences. i focus on how urban latina/o students constructed their identities in relation to their participation in this environment with the aim of deepening the mathematics education community’s understanding of how identity development contributes to underrepresented (and represented) students’ mathematical success. drawing from a larger study on latina/o students’ identity construction and participation in an esp calculus i workshop (oppland, 2010), this study addresses the following questions: 1. how did participants negotiate their emerging mathematical and racial identities over time as they participated in the calculus i workshop?   2. what changes, if any, occurred in how participants’ negotiations of their emerging mathematical and racial identities influenced their participation over time in the calculus i workshop? identity, race, and mathematics learning social identities are constructed as individuals engage in social interactions tied to societal norms (johnson, 2001). there are numerous social identities that latina/o individuals may construct at any given moment (e.g., racial, gender, class). because all latina/o americans are exposed to racial socialization (helms, 1994), this study aims to examine how urban latina/o students construct their emerging mathematical and racial identities. in this study, emerging mathematical and racial identity constructions (emrics) refers to participants’ negotiations of privately and socially constructed meanings of the terms latina/o, race, racism, and racial inequality in relation to their workshop participation. privately constructed meanings refer to meanings for such terms that are developed within the individual, while socially constructed meanings refer to meanings for such terms that are developed jointly with other individuals. because such meanings are (re)constructed through social interactions, i consider participants’ selfperceptions, their perceptions of others, and perceptions they believe others impose on them. i also take an intersectional approach (crenshaw, 1991) by exploring how their emrics intersect with their other salient identities (e.g., gender, class). i aim to gain a deeper understanding of latina/o students’ perspectives of how they negotiated their emrics as they participated in the workshop and how such negotiations might have caused changes in their participation. because this study aims to capture latina/o students’ perceptions of the historical, sociopoliti                                                                                                                           5 stem is an acronym for science, technology, engineering, and mathematics. oppland-cordell urban latina/o students in calculus i journal of urban mathematics education vol. 7, no. 1   23 cal, and situational nature of how their emrics influenced their participation, i draw on both sociocultural and sociopolitical perspectives of learning. sociocultural and sociopolitical learning theories from a sociocultural theoretical standpoint, learning refers to becoming more central and competent participants in the valued practices of communities of practice (cop), including by adopting trajectories of participation and constructing identities that contribute to this convergence (lave & wenger, 1991; rogoff, 2003; wenger, 1998). as individuals engage in a cop, which is “a history collapsed into a present that invites engagement” (wenger, 1998, p. 156), they continuously negotiate their identities, identities belonging to others, and meanings projected onto their identities by others (rogoff, 2003; wenger, 1998). as individuals’ identity constructions and participation continuously inform one another, these dialectics can support or impede the creation of opportunities that allow them to participate in cops’ valued practices. it is in this sense that identity, participation, and learning are inextricably linked. drawing on this theory, the present study viewed participants’ emrics as strengthening their participation if these constructions contributed to their convergence towards becoming more central and competent participants in the workshop’s valued practices (e.g., strengthened engagement in mathematical group work with a culturally diverse student population). wenger (1998) describes the complexity associated with becoming a more central and competent participant in the valued practices of cops. as individuals engage in such practices, this process can involve transformations in how identities are (re)negotiated, how various degrees of participation and non-participation are adopted, and how identity constructions and participation relate amid broader social structures involving power relations. drawing on this perspective, i anticipated that examining participants’ emrics in relation to their participation might involve changes in how they constructed these particular identities, how they adopted ways of participating and not participating, how their identities and participation related over time, and how such changes were connected to power relations and identity constructions linked to broader contexts. although participants’ various forms of identification became salient as they described their workshop experience, this study focused on relationships between their emrics and participation. by analyzing the latina/o participants’ perspectives of these relationships, i illustrate how esp-type environments can “hold the key to real transformation” in terms of possibly helping students negotiate what it means for them to be mathematics learners of a particular cultural group in ways that support them in becoming more central participants in valued mathematical practices (wenger, 1998, p. 85). oppland-cordell urban latina/o students in calculus i journal of urban mathematics education vol. 7, no. 1   24 mathematics education research studies have drawn on sociocultural learning theories to explore students’ mathematics participation and learning processes in cops (e.g., boaler, 2002; boaler & greeno, 2000; gresalfi & cobb, 2006; martin, 2000, 2006). collectively, this body of research asserts that using an identity analytic lens provides an effective way to capture how students construct identities and participate in the practices of mathematical cops. this research also indicates that dialectics exist between how students construct identities as mathematics learners and participate in the practices of mathematics learning communities. on one hand, students’ participation in mathematical practices influences aspects of their mathematics identity constructions (boaler, 2002; boaler & greeno, 2000; martin, 2000, 2006). these constructions include how students negotiate sociohistorical forces (i.e., lived experiences tied to broader communities and schools) and their agency as they engage in mathematics classroom practices (martin, 2000, 2006). on the other hand, however, aspects of students’ mathematics identity constructions shape how they engage in mathematical practices (gresalfi & cobb, 2006). mathematics education research that has drawn on sociocultural theories indicates how students’ identities as mathematics learners are constructed in social interactions (i.e., practice-based interactions). however, mathematics education scholars have indicated that a weakness of sociocultural research is its tendency to adopt color-blind paradigms (esmonde et al., 2009; gutiérrez, 2013). that is, this research often ignores the intersectional nature of identity development and its influence on practice-linked identity constructions (esmonde et al., 2009), the voices of marginalized students (gutiérrez, 2013), and the roles of politics and power relations (gutiérrez, 2013). color-blind approaches are particularly damaging to underrepresented students (martin, 2009) because they mask how inequities functioning in their histories, experiences, and mathematical lives impact how they negotiate identities in relation to their participation in mathematical cops. in the case of latinas/os (and, in particular, mexican americans), it is imperative that the “social, political, and historical patterns of exclusion, degradation, and racism” they have experienced in education (gutiérrez, willey, & khisty, 2011, p. 28) be recognized when investigating how their identities and participation inform one another in mathematical learning contexts. critical race theory and latina/o critical theory. because of the limitations associated with sociocultural mathematics education research, i also draw on sociopolitical perspectives of learning (gutiérrez, 2013). sociopolitical perspectives of learning view “knowledge, power, and identity as interwoven and arising from (and constituted within) social discourses” (gutiérrez, 2013, p. 40). from this scholarship, i draw on the theoretical frameworks critical race theory (crt) and latina/o critical theory (latcrit). crt in education refers to a “set of basic insights, perspectives, methods, and pedagogy that seeks to identify, analyze, and oppland-cordell urban latina/o students in calculus i journal of urban mathematics education vol. 7, no. 1   25 transform those structural and cultural aspects of education that maintain subordinate and dominant racial positions in and out of the classroom” (solórzano & yosso, 2002, p. 25). crt emphasizes the centrality of race and racism and their intersectionality with other forms of subordination. latcrit complements crt, while also offering an important lens to discuss transformational resistance specifically for latinas/os (solórzano & bernal, 2001). latcrit emphasizes that latinas/os’ identities are multidimensional and addresses “the intersectionality of racism, sexism, classism, and other forms of oppression” that may impact latinas/os specifically, such as language, ethnicity, and culture (bernal, 2002, p. 108). crt and latcrit also both emphasize the centrality of marginalized students’ experiential knowledge (solórzano & bernal, 2001). crt and latcrit also encourage examining marginalized students’ experiences through counter-stories (bernal, 2002; solórzano & yosso, 2002), which, in this study, refer to participants’ stories that challenged dominant ideologies that disregard how social constructions of latina/o, race, and racism shape their mathematical experiences. this study contributes to mathematics education equity scholarship embracing the sociopolitical turn in mathematics education (gutiérrez, 2013). applying both sociocultural and sociopolitical perspectives to examine bidirectional relationships between latina/o students’ emrics and participation allows for viewing identity development complexly⎯as involving multiple perceptions (selfperceptions, perceptions of others, imposed perceptions); as (re)constructed in relation to engagement in social practices within local contexts situated in broader sociohistorical and sociopolitical contexts; as being intersectional in nature; and as connected to complex issues, such as race, class, gender, language, agency, hegemony in society, and power dynamics. from crt, i emphasize the centrality of race and racism in participants’ mathematical experiences and how such meanings intersect with other forms of oppression (e.g., gender, class). from latcrit, i emphasize what it means for participants to be latina/o mathematics learners, including how they manage forms of oppression that may be particularly relevant for latinas/os (e.g., language). from both crt and latcrit, i stress the intersectional nature of participants’ emrics, the centrality of their experiential knowledge, and the importance of capturing their counter-stories. drawing on both sociocultural and sociopolitical theoretical frameworks allows for examining what it means for participants to be latina/o mathematics learners and how their negotiations of such meanings related to their workshop participation. methods in this study, i applied qualitative multiple-case study methodology (miles & huberman, 1994; yin, 2009) to holistically explore latina/o students’ perspectives of their emrics as they engaged in the workshop. here multiple-case study oppland-cordell urban latina/o students in calculus i journal of urban mathematics education vol. 7, no. 1   26 refers to the fact that i address the research questions for two cases. my goal was to provide thick descriptions (geertz, 1973) of each case and to identify emergent themes (miles & huberman, 1994) across both cases. thick descriptions refer to rich, detailed portrayals of participants’ experiences situated within broader sociopolitical contextual layers related to the phenomena investigated. therefore, i used multiple data sources, including interviews, a questionnaire, reflections, and direct classroom observations, to construct an in-depth case study for each participant that described their perspectives of how their transforming emrics impacted their participation. i sought to capture participants’ perceptions of significant avenues they experienced, obstacles they faced, and critical transformations that occurred as they engaged in the workshop, specifically related to their negotiations of privately and socially constructed meanings of latina/o, race, racism, and racial inequality. the units of analysis for each of the two in-depth case studies were the individual students; each case presents an inimitable description of how a latina/o participant’s negotiations of their emrics informed (or did not inform) their participation. the subsequent sections (i.e. research context, participants, data collection, and data analysis) provide detailed descriptions of the methods. research context this study is part of a larger study on latina/o students’ identity constructions and participation in an esp calculus i workshop at hall university, a predominately white, 4-year research university in chicago, illinois (oppland, 2010). the university’s socioeconomically diverse undergraduate student population (~16,000 students) included approximately 45% caucasians, 23% asians, 16% latinas/os, 9% african americans, <1% native american, 2% international students, and an approximately equivalent number of male and female students. latinas/os were significantly underrepresented within hall’s undergraduate student population (16%); a claim supported by the fact that latinas/os represent approximately 45% of the chicago public schools (cps) student population (“stats and facts,” 2014). historically, a significant percentage of latina/o students also have struggled to successfully complete their mathematics requirements at hall university, challenging their efforts to obtain stem degrees. emerging scholars program. prior to fall 1989, data collected at hall university confirmed that a considerable number of latina/o and african american students were failing precalculus and calculus (brugueras, hernández-gonzález, & libgober, 2005). for example, it was discovered that over 55% of latina/o and african american students were failing precalculus (brugueras et al., 2005). to improve latinas/os’ and african americans’ precalculus and calculus achievement levels, hall’s mathematics department initiated esp math workshops in fall 1989. esp at hall aims to increase the number of underrepresented students who oppland-cordell urban latina/o students in calculus i journal of urban mathematics education vol. 7, no. 1   27 successfully complete introductory mathematics and physics courses. at the time of this study, esp mathematics workshops were offered for precalculus, calculus i, calculus ii, calculus iii, and introduction to advanced mathematics. these optional workshops ran parallel to students’ required mathematics courses, encouraged culturally diverse peer groups to collaboratively solve challenging worksheet problems, were held for 2 hours either once or twice per week, and were worth one credit hour (graded as satisfactory or unsatisfactory). while all students were encouraged to participate, underrepresented students were recruited more heavily. hall university’s mathematics workshops are modeled after uri treisman’s doctoral dissertation work (treisman, 1985), which aimed to discover explanations for why african american students were struggling to learn calculus at the university of california (uc), berkley in the mid-1970s. treisman analyzed differences between twenty african american and twenty chinese students’ study habits for their calculus courses. while both groups practiced the mathematics study habits advised by their institution’s study skill courses (e.g., studying individually for several hours each week), several chinese students’ study habits differed from the african american students (e.g., discussing problem solutions with peers) (treisman, 1992). these findings led to the implementation of the mathematics workshop program at uc berkeley in 1978, which offered “minority” students enrolled in first-year calculus the opportunity to collaboratively solve challenging mathematics problems with peers (asera, 1988). the program received national recognition as participants exceeded non-participants in mathematics achievement and persistence levels (fullilove & treisman, 1990). the workshops have proliferated in universities across the united states where they continue to be recognized for assisting students in attaining mathematical success. to gain in-depth information about how latina/o students’ identities and participation can positively inform one another as they engage in mathematical learning contexts, i argue it is important to study this development in mathematical learning contexts that have been recognized as effective. this study occurs in an esp mathematics workshop because esp environments have been recognized as effective. first, esp’s implementation of group work has been nationally identified as a powerful pedagogical tool for increasing marginalized students’ mathematics achievement levels (hsu, murphy, & treisman, 2008). second, esp workshops typically incorporate practices recognized in mathematics education literature as effectively supporting latina/o students in improving their mathematics achievement levels, such as “math-based enculturation and socialization experiences” (martin, 2004, p. 11). third, as an esp precalculus workshop instructor prior to this study taking place, i was inspired by positive changes i witnessed related to how culturally diverse workshop participants embraced mathematics as the semester advanced, including students who had expressed skepticism about oppland-cordell urban latina/o students in calculus i journal of urban mathematics education vol. 7, no. 1   28 the workshop’s benefits earlier in the semester. for these reasons, i argue that esp mathematics workshops provided an appropriate setting for investigating positive relationships among latina/o students’ identity development, mathematics participation and learning, and success. calculus i esp workshop and course. in this study’s calculus i workshop, students solved worksheet problems within peer groups. worksheet problems were related to, but often more challenging than homework problems assigned in students’ required calculus i course. examples of acceptable social norms within peer groups included discussing problem-solving processes, supporting peers in understanding mathematical explanations, and presenting alternative strategies. students were also encouraged to present their strategies and solutions on chalkboards, which sparked whole class discussions. calculus i workshop students were simultaneously enrolled in a required calculus i class, which consisted of a lecture and discussion section. during lectures, students listened to an instructor present the material in a large lecture hall. during discussion sections, a graduate teaching assistant guided students in understanding homework problems and students completed quizzes. all calculus i sections followed a common schedule that was posted on a main course webpage. the course covered chapters 2 through 5 of calculus, early transcendentals by jon rogawski (2008). the required calculus i course grade was based on the total points earned from two hourly exams, homework, quizzes, and a final exam. with respect to identity issues, the esp mathematics workshops’ unique features may have affected marginalized students’ identity development differently than experiences they negotiated in hall university’s required mathematics classes. for example, while esp mathematics workshop facilitators aim to support culturally diverse student populations in attaining mathematical success, instructors of required mathematics courses may or may not have had this goal in mind. although all mathematics learning contexts are racialized (martin, 2009), meanings for race may become more salient for underrepresented students in an esp mathematics workshop through their consistent interactions with a culturally diverse student population and an instructor aware of equity and diversity mathematics education issues. researcher identity and positionality. conducting this research involved carefully balancing my roles as workshop facilitator and researcher of vanessa and immanuel workshop experiences. as the facilitator, i influenced workshop students’ engagement. for example, my identity development as a mathematics learner (e.g., challenges i faced as a female student) prompted me to encourage students to incorporate their strengths (e.g., mathematical, cultural, linguistic) while engaging in problem-solving practices. however, to observe how students’ participation progressed “naturally,” i avoided interfering with their engagement, oppland-cordell urban latina/o students in calculus i journal of urban mathematics education vol. 7, no. 1   29 when possible. my responsibilities and goals included: crafting worksheets that contained challenging calculus i problems that could be solved with minimal instructor guidance, motivating students to engage in mathematical discourses with peers, addressing students’ mathematical inquiries without providing complete solutions (i.e. socratic methods), and urging students to validate students’ preferred strategies and ways of engaging with mathematics. as the facilitator, i often played a supportive, behind-the-scenes role as students primarily constructed solutions with peers; this role provided ample opportunities for me to observe students. as a white middle-class woman investigating latina/o students’ mathematics experiences, this research involved entering into unfamiliar territory in many respects. although i may relate on some level to latinas who face mathematical boundaries related to their gendered identities, my race grants me advantages that individuals from other cultural backgrounds, such as this study’s participants, may not experience. although my identities, including my white background, informed all facets of this study, i used strategies to attempt to mitigate the effect of my race on influencing the study’s construction in ways that would “reproduce white privilege”6 or take away from participants’ perspectives of their experiences (gordon, 2005, p. 300). such strategies included bringing up race directly during interviews, seeking to capture participants’ counter-stories, and member checking. while the former two provided participants with a forum to challenge dominant perspectives that disregard the role of race, racism, and whiteness in latina/o students’ mathematics learning, the latter provided participants with the opportunity to critique, confirm, or challenge my interpretations of their mathematical lives. participants the participants are vanessa, an 18-year-old, female, second-generation mexican american immigrant, and immanuel, a 19-year-old, male, firstgeneration mexican american immigrant. the majority of k−12 schools the participants attended were public schools located in chicago, illinois with predominately latina/o and/or african american student populations. vanessa, a chemistry major, graduated salutatorian from a third-ranked, public high school in chicago, illinois. immanuel, who intended to major in computer engineering,7 graduated from a public high school in a small suburb near chicago, illinois. both                                                                                                                           6 white privilege is “a system of opportunities and benefits conferred upon people simply because they are white” (solórzano & yosso, 2002, p. 27). 7 at the time of this study, immanuel had not declared his major. however, in his interviews, he indicated that he planned to major in computer engineering. oppland-cordell urban latina/o students in calculus i journal of urban mathematics education vol. 7, no. 1   30 participants were collegiate freshmen, middle class, and bilingual (english and spanish). table 1 summarizes participant background information taken from questionnaire and interview data. the larger study in which these two case studies were situated (oppland, 2010) was started by inviting all 27 students in my spring 2008 calculus i workshop to participate. after completing data collection with nine latina/o students, i selected vanessa and immanuel for case study analysis because they provided powerful, yet different, accounts of how their emrics and participation informed one another.8 i also selected a latino and a latina for this study to explore similarities and differences in how their other salient identities (e.g., gender) might intersect (or not intersect) with their emrics. table 1 background information for case study participants background vanessa immanuel university classification freshman freshman age 18 19 gender female male race hispanic, latino, or latino-american; mexican-american hispanic, latino, or latino-american; mexican-american ses middle middle language bilingual (english and spanish) bilingual (english and spanish) birth country united states mexico major or intended major chemistry (major) computer engineering (intended major) data collection multiple data collection strategies were used to understand how participants negotiated their emrics and how these negotiations related to their participation. in-depth interviews were the primary data source; a questionnaire, four reflections, and direct observations were secondary data sources. interviews. three individual interviews were designed to capture participants’ perceptions of their mathematics experiences and histories (including retrospective accounts), how their participation developed, and how their                                                                                                                           8 providing diverse accounts of how latina/o students’ identities and participation inform one another is critical because mathematics education research too often essentializes latina/o students’ mathematics experiences.   oppland-cordell urban latina/o students in calculus i journal of urban mathematics education vol. 7, no. 1   31 emrics shaped their participation. interview 1 focused on the first category; interviews 2 and 3 focused on the latter two. the semi-structured interviews were conducted at the beginning, middle, and end of the semester. participants were asked directly about race, gender, and class issues related to their mathematical trajectories and to elaborate on these phenomena (e.g., do you feel that race/ ethnicity plays a role in achieving in math? why or why not? describe any experiences where your race/ethnicity played a role in the workshop). after completing individual case study drafts, each participant participated in a followup interview to provide feedback on their case studies, which were then edited based on their comments (eight total interviews; average 1.6 hours each for vanessa; 2 hours each for immanuel). interviews were audio-recorded and promptly transcribed to allow for any necessary modifications to the interview protocols. questionnaire and reflections. the purpose of the questionnaire was to collect personal background information (e.g., racial/ethnic affliation), perceptions of collegiate mathematics experiences prior to participating in calculus i (e.g., mathematics coursework completed), and perceptions of the calculus i course and workshop (e.g., motivation for participating in the workshop) for each of the 27 workshop students, including vanessa and immanuel. four written reflections were used to capture participants’ more immediate perceptions of their workshop experience, including how they perceived their participation as developing over time. direct observation. direct observations of vanessa and immanuel in the workshop also were collected. during each of the 28 workshop sessions, i used observation protocols to record detailed fieldnotes about each participant; a videocamera to record participants’ engagement; two audiorecorders located in the classroom to capture participants’ conversations; another audiorecorder located in my pocket to capture my interactions with participants; and photographs of participants’ board work to capture another aspect of their engagement. special attention was paid to how participants engaged with the environment, the instructor (me), peers, and the mathematics, and comments they made that shed light on their emrics. this analysis used audio and video data from three class periods—one each at the beginning, middle, and end of the semester—as time markers to capture how participants’ identities and participation changed during the semester. as a teacher-researcher, i had an insider perspective, including an understanding of the general nature of the workshop context, participants’ inter-actional tendencies with students and me, and participants’ behaviors that they may not have described in interviews and reflections. data was organized in a case study database that contained notes, documentation, tabular materials, and narratives (yin, 2009). classroom observations were oppland-cordell urban latina/o students in calculus i journal of urban mathematics education vol. 7, no. 1   32 gathered in one file; separate files, one for each participant, included the questionnaire, reflections, and interview protocols. a primary computer file contained a spreadsheet with organized questionnaire data and separate secondary files for each of the following: photographs of boardwork, audio files, typed classroom observations, and interview data. the interview data file contained separate files for each participant, each of which contained transcripts, interview data tables, and narratives. i also converted all videotape data onto cds (videotapes were not transcribed). table 2 illustrates the data collection methods. table 2 summary of database methods january 2008−may 2008 data collected interviews* • interview 1: beginning of semester • interview 2: middle of semester • interview 3: end of semester • interview 4: follow-up interview after the semester ended • audiotaped and transcribed interviews • 8 total interviews • average 1.6 hours each (vanessa) • average 2 hours each (immanuel) questionnaire • one time at the beginning of semester • hard copy of questionnaire reflections • four times during the semester; once per month • hard copies of four reflections classroom observations • each workshop meeting throughout the semester • this study uses data collected during three class periods-one at the beginning, middle, and end of the semester • field notes on 28 workshop meetings (roughly 56 hours of observation) • audiotape and videotape of 28 workshop sessions (roughly 56 hours) • photographs of boardwork * some interviews were conducted on different days. data analysis data collection and case study analyses occurred simultaneously. the goal was to identify emergent themes related to how vanessa’s and immanuel’s management of their emrics related to their participation. keeping this goal in mind, the initial codes described in table 3 were used to analyze interview transcripts. oppland-cordell urban latina/o students in calculus i journal of urban mathematics education vol. 7, no. 1   33 table 3 codes for interview analysis code description background information evidence that revealed information about participants’ mathematics experiences in various contexts throughout their lives (e.g., societal, community, family, and school contexts) broader mathematical and racial identity constructions evidence that revealed how participants constructed their mathematical and racial identities in relation to broader contexts (e.g., societal, community, family, and school contexts) workshop mathematical and racial identity constructions evidence that revealed how participants simultaneously constructed their mathematics and racial identities in relation to the workshop workshop participation evidence that revealed information about how participants’ participated in the workshop context after completing data collection, interview data for each participant was categorized by the initial codes into data tables. an iterative coding scheme (miles & huberman, 1994) was then applied to carefully scrutinize interview data within the data tables, which involved developing and refining codes. for the second and third codes in table 3, i identified (as best i could) identities that intersected with participants’ mathematical and racial identity constructions (e.g., gender). after completing this iterative coding process, the interview text placed under the third and fourth codes in table 3 in the data tables were put in chronological order according to interview dates to help recognize any changes that occurred over time. next, i meticulously analyzed and compared all data sources multiple times. through this review and comparison, themes emerged for each participant regarding (a) how they negotiated their emrics (e.g., negotiated racialized experiences); (b) changes that occurred in how they negotiated their emrics (e.g., challenged racialized experiences they had managed earlier); and (c) how their participation evolved (e.g., became stronger participants in the workshop’s valued practices). after identifying these themes, i investigated relationships between changes that occurred in how they negotiated their emrics and how their participation evolved. to understand these connections, in addition to drawing on crt and latcrit to capture participants’ perspectives of meanings for the terms latina/o, race, racism, and their possible intersection with other forms of oppression, i used a grounded theory approach (corbin & strauss, 2008), allowing relevant themes to emerge in the data. to gain a deeper understanding of students’ emrics (including how this development transformed), i also drew on interview data to compare how they negotiated these identities in the workshop versus within and across broader contexts (e.g., societal, community). oppland-cordell urban latina/o students in calculus i journal of urban mathematics education vol. 7, no. 1   34 next, while keeping the research questions in mind, i conducted case analysis using three stages recommended by miles and huberman (1994): within-case analysis, data reduction, and cross-case analysis. first, within-case analysis involved developing theories about how each participant’s emrics related to their participation, testing these theories against evidence collected for that particular student from various data sources, and deciding whether to accept, modify, or discard the theories. second, data reduction involved constructing a case study narrative for each student. as the instructor of the workshop, my knowledge of the classroom and the participants aided me in constructing detailed contextualized narratives. additionally, each participant also confirmed that the final version of their case study accurately portrayed their experiences. third, after this member checking occurred, the final step of the data analysis involved conducting a cross-case analysis on the two in-depth case studies. that is, after identifying themes for each participant, i compared the two case study narratives to identify similar themes across the participants. participants did not always talk explicitly about meanings related to latina/o membership, race, and racism as mathematics learners. therefore, i took into consideration when they spoke about issues that might be relevant for understanding their emrics (e.g., diversity issues related to their mathematical experiences). as mentioned above, i also took into account how these meanings intersected with other social categories, such as gender. although the social categories addressed in the interview questions were limited to race, gender, and class, i did not assume that gender and class identities were the only identities that might intersect with participants’ emrics. instead, i aimed to capture the complex social categories that were relevant for understanding what it means for participants to be latina/o mathematics learners participating in the workshop. case study findings in this section, i first provide a detailed description of vanessa’s and immanuel’s case studies. each case study illustrates how a latina/o participant’s emrics and participation informed one another over time. when describing each case, i attempt to highlight how the identity constructions examined intersected with participants’ other salient identities, how such constructions were related to their negotiations of experiences in broader sociopolitical contexts, and how their workshop experiences influenced their negotiations of their identities and participation. after presenting the two case summaries, i present a cross-case comparison that reveals common themes, and also some variations, in how aspects of participants’ emrics shifted their participation. although the findings presented within each case study are based on the unique experiences of a particular latina/o participant, the main themes reported in the cross-case comparison were similar across both cases. oppland-cordell urban latina/o students in calculus i journal of urban mathematics education vol. 7, no. 1   35 when presenting each case summary, i refer to table 4 and figure 1. table 4 contains background information for all students in figure 1. figure 1 provides snapshots of the workshop during three meetings—at the beginning (january 15, 2008), middle (march 20, 2008), and end (april 10, 2008) of the semester. the workshop met in three different classrooms during the semester. from january 15, 2008 until february 26, 2008, the workshop met in classroom 1, a small classroom that included several small desks, a large desk, and boards along the front and back walls. because classroom 1 felt cramped and provided limited board space, i relocated the workshop to classroom 2. from february 28, 2008 until the end of the semester (except on april 1, 2008), the workshop met in classroom 2, a spacious meeting room that included several small desks, a large table, a free-standing board positioned in a private location, a significant amount of board space along two walls, and additional space for peer groups to spread out in the room. due to a university activity occurring in classroom 2 on april 1, 2008, the workshop met in classroom 3, which was similar to classroom 1. during each meeting, students had the freedom to self-select into peer groups, change peer groups, interact with students in multiple groups, and move desks and the free-standing board. in addition, the workshop student population did not always remain consistent during each meeting. table 4 name, year, and race/ethnicity of students identified in figure 1 name year race/ethnicity agustin freshman latino angel freshman latino anthony freshman african american carlos freshman latino cathy freshman white dante freshman latino debbie freshman asian dustin junior white eduardo freshman latino gail freshman asian hank freshman asian immanuel freshman latino julian freshman latino lala freshman latina lilliana sophomore latina matt freshman white molly freshman asian nancy freshman asian peter freshman white ryan freshman white sonja freshman african american tuan freshman asian vanessa freshman latina oppland-cordell urban latina/o students in calculus i journal of urban mathematics education vol. 7, no. 1   36 snapshot of classroom 1 on january 15, 2008 snapshot of classroom 2 on march 20, 2008     snapshot of classroom 2 on april 10, 2008 figure 1: snapshots of the workshop context during three workshop meetings.   chalkboard a chalkboard b desk         sonja                       gail vanessa lilliana immanuel dustin agustin carlos angel hank matt eduardo ryan cathy julian nancy   tuan debbie peter molly instructor   chalkboard a free standing chalkboard c hank c h a l k b o a r d b c lilliana immanuel ryan debbie peter tuan anthony dustin vanessa dante molly angel agustin nancy cathy julian   instructor side b side a table side b indicates the individual mainly sat at a desk during this meeting indicates the individual mainly stood during this meeting indicates individuals vanessa interacted with during this meeting indicates individuals immanuel interacted with during this meeting indicates individuals both participants interacted with during this meeting   chalkboard a free standing chalkboard c hank dante vanessa anthony debbie ryan lilliana instructor molly dustin lala agustin nancy cathy julian eduardo table tuan side b side a immanuel   peter c h a l k b o a r d b angel side b oppland-cordell urban latina/o students in calculus i journal of urban mathematics education vol. 7, no. 1   37 what it means for vanessa to be an urban latina workshop student vanessa is an outgoing, first-generation, latina freshman with long brown hair, brown skin, and a contagious smile. although vanessa had confidence in her mathematics ability throughout her academic development, she experienced a decrease in her mathematical confidence in high school and college. vanessa revealed complex experiences she negotiated as a latina that acted as avenues and barriers for her mathematical success, including those involving race, gender, culture, language, and power. as the following discussion illustrates, vanessa’s history as a latina significantly impacted how her emrics influenced, and were influenced by, her participation over time. negotiating mathematical and racial identities: avenues and barriers for participation. when the workshop relocated from classroom 1 to classroom 2 (see figure 1), vanessa’s participation shifted in some critical ways. for example, she shifted from solving problems mainly with her initial group in classroom 1 while seated to solving problems with predominately latina/o peer groups at board c in classroom 2 while standing. vanessa revealed how her emrics supported this shift. vanessa indicated that her preference for working with latinas/os in “segregated” classroom 2 was tied to her perception that she shared “culture” and “language” backgrounds with latina/o peers. for instance, encountering opportunities to occasionally speak “math in spanish” while engaging with latina/o peers made her feel comfortable:9 it’s funny how we are all segregated. i think it’s just nature. everywhere i’ve been in school the mexicans or the hispanics will always be together no matter what. it’s just the fact that you see similar things between each culture, like the language especially. we talk sometimes math in spanish, so it’s just like you can really tell them a little bit more than the other people. (interview 2, march 11, 2008) vanessa also indicated that she felt comfortable working with latina peers because of their shared gender backgrounds. although her emrics contributed to creating these avenues, she also revealed several barriers that limited her participation, including three that i identified as related to her emrics: (a) perceiving differences between her and peers’ cultural and language backgrounds; (b) peers’ cultural and gender backgrounds contributing to her perceiving differences between her and peers’ mathematical abilities; and (c) negotiating racialized and gendered experiences. first, vanessa’s awareness of differences between her and peers’ cultural and language backgrounds alienated her from some classmates. for example, in                                                                                                                           9 although vanessa did not believe practicing mathematics in spanish strengthened her comprehension of mathematics, she did believe that, for other latinas/os, “language can be a barrier” for understanding mathematics. oppland-cordell urban latina/o students in calculus i journal of urban mathematics education vol. 7, no. 1   38 the last sentence of the previous excerpt, she revealed that speaking mathematics in spanish allowed her to “tell” latina/o peers “a little bit more” than “other” students (who were predominately asian and white). she also believed peers who shared culture and language backgrounds could comfortably practice mathematics together. this shared background made it “awkward” for her to join groups where members possessed cultural and language backgrounds that differed from her own because she did not want to “break something that’s out of the norm” and “mess everything up” (interview 3, may 28, 2008). in fact, during much of the semester, vanessa rarely interacted with students who typically participated in predominately asian and white peer groups (e.g., molly, peter, anthony). the ways in which peers’ cultural and language backgrounds both encouraged and limited vanessa’s participation may have been related to her prior experiences, which included forming her closest friendships with latinas/os and experiencing a lack of exposure to cultural diversity throughout her life. second, peers’ cultural and gender backgrounds contributed to vanessa perceiving differences between her and peers’ mathematical abilities. for example, vanessa viewed asian and white students as mathematically superior because of their racial/ethnic backgrounds. the last sentence of the following quotation reveals vanessa’s perception that asian students possessed stronger mathematical abilities “because they’re asian.” she also expressed a desire to interact with asian students (including over mexican students) because she viewed them as “smarter” and “wittier”: vanessa: you right away see who has more capacity for understanding sort of things and you just go to them. i don’t know. maybe their race like maybe ‘cause they’re asian they’re smarter and they’re wittier…if i were to go to the classroom and i would have to choose one partner that i think would help me understand a math problem i would definitely go to an asian person. i wouldn’t go to a person that’s mexican as much as i would feel a comfort zone with them. sarah: can you say more about that? why? vanessa: why? i think that people look up to the asian countries…. that’s sad to say i wouldn’t go to one of the people that are mexican. i’m not saying that they’re dumb or anything. i mean i’m mexican too, but it’s just like you tend to go to someone that has more of a reputation. yeah. sarah: so would you have to know that they’re smart before you go to them or would you just kind of assume that because they’re asian… vanessa: because they’re asian they’re smart. (interview 3, may 12, 2008) oppland-cordell urban latina/o students in calculus i journal of urban mathematics education vol. 7, no. 1   39 even though vanessa expressed this viewpoint, she rarely interacted with asians, due to cultural and language differences that limited her interactions with them (as previously mentioned). students’ genders may have also impacted how vanessa assessed peers’ mathematical abilities. she tended to perceive men as mathematically superior: “males have a tendency of knowing more about math” (interview 1, january 25, 2008). blending this belief with her perceptions of how peers’ cultural backgrounds related to their mathematics abilities may have made her more prone to develop an identity of marginalization in relation to students positioned externally to her peer groups who possessed both the cultural and gender statuses she associated with mathematical superiority. for example, she described white male peers positioned externally to her peer groups (such as peter in the following quote) as “intimidating” due to their strong mathematics abilities, which led her to “tend to go to other people”: i don’t think he means to be intimidating, but he just knows exactly what to do and it’s kind of scary. you just kind of tend to go to other people, not the people that know less, just the people that are more laid back about it. i think he’s a little too intimidating, too hyper, too excited about it, which is nothing bad. that’s just not how i am. (interview 3, may 12, 2008) the ways in which peers’ cultural and gender backgrounds contributed to limiting vanessa’s interactions with peers may have been related to her prior experiences, including managing societal, community, and institutional messages that supported the notion of a racial and gender hierarchy of mathematical ability with asians, whites, and men positioned as superior to latinas/os, african americans, and women. third, vanessa negotiated racialized and gendered experiences. in the previous descriptions, vanessa managed some racialized experiences (e.g., a lack of exposure to cultural diversity, stereotypes about cultural groups’ mathematical abilities). in addition, she managed racialized and gendered experiences related to her interactions with ms. johnson (previously mentioned teacher). vanessa described numerous ways ms. johnson mistreated her, including criticizing her preferred mathematical problem-solving strategies (e.g., constructing step-by-step solution processes, converting mathematical symbols into words), rejecting her mathematical ideas that contrasted with instructional approaches, and intimidating her as she solved problems on boards positioned in front of the class. in the following excerpt, vanessa revealed how “racism” and her latina background contributed to her feeling marginalized by ms. johnson: sarah: do you want to say anything more about differences you observed in high school about how certain races were encouraged in math and others weren’t? oppland-cordell urban latina/o students in calculus i journal of urban mathematics education vol. 7, no. 1   40 vanessa: well, it was sex and it was the difference in the color of your skin…. this was precalc last year with the certain teacher that i had. she was african american and she was kind of light and she would emphasize and encourage the students that were light skinned instead of the other, and then the hispanics i guess she just based it on this whole sex thing. she would literally go into the room and say, “okay, my yellow and brown children.” …yeah. she would say that and i’m like i’m not yellow. it’s like what’s the whole point? i’m a little brown…she would somehow integrate race, integrate it somehow…i guess she went through a lot of racism and stuff, but she’s doing the same thing… (interview 3, may 12, 2008) vanessa believed ms. johnson viewed her mathematics ability as inferior to “light skinned” students because of her “brown” skin and inferior to men (among the latina/o students in the class) because of her gender. in the following quotation, vanessa revealed that her experiences with ms. johnson, and in particular ms. johnson’s criticism of her problem-solving strategies (e.g., “writing” mathematics), “slowed” down her mathematical development. in the workshop, vanessa seemed reluctant to apply problem-solving strategies that ms. johnson had criticized and she was surprised to witness me translating mathematical notation into words when providing mathematical guidance to students: i think what you have is you write a lot and for awhile i stopped writing everything for math and that kind of slowed me down and so now i have to write everything now…. that’s why when the first couple of weeks that you were writing a lot it was weird ‘cause i thought you were gonna be what my teacher said, “higher math, you need less words the better.” and i’m like wow she’s writing... (interview 1, january 25, 2008) negotiating mathematical and racial identities: critical transformations. as the semester progressed, vanessa’s participation changed in some significant ways. for example, she contributed her ideas to mathematical discussions more frequently, she regularly solved problems with a more culturally diverse student population at board c in classroom 2, and she asked me (the instructor) mathematical questions about her work on board c in classroom 2. as these changes occurred, she challenged the aforementioned barriers tied to her emrics that restricted her participation. first, perceiving differences between her and peers’ cultural and language backgrounds acted as less of a barrier. as indicated in the following passage, although vanessa maintained the belief that interacting with students with similar cultural and gender backgrounds was “important” due to the “comfort zone” this connection helped create, she also indicated that it was important for her “to learn how to get along” with students regardless of their cultural backgrounds: oppland-cordell urban latina/o students in calculus i journal of urban mathematics education vol. 7, no. 1   41 sarah: is it important to you that there are people of your own gender and/or ethnicity in the workshop or do you not care? vanessa: i think it is important and it’s important to have some people that are from your same racial background. not only is it an ego booster, you just feel a certain comfort zone like i said. but then at the same token, i wouldn’t really care if there wasn’t anyone with the same background. you just have to learn how to get along with other people and just work with them too. (interview 3, may 12, 2008) as time evolved, vanessa did gradually interact with a more culturally diverse student population compared to earlier in the semester (e.g., such as anthony on april 10th in figure 1 and molly later in the semester). as this change occurred, she recognized the value of participating in mathematical discussions “with everyone else to learn about how other people study, how other people work the math problems ‘cause everyone has their own different thing” (interview 3, may 12, 2008). second, transformations occurred regarding how peers’ cultural and gender backgrounds contributed to her perceiving differences between her and peers’ mathematical abilities. as vanessa engaged with a broader peer population, she realized she possessed mathematical strengths that some “advanced” students did not (e.g., asian students), she observed latina/o students and women (including herself) exhibiting strong mathematics abilities, and she witnessed students she had initially framed as “advanced” encountering mathematical setbacks. for example, in the following excerpt, she discussed her realization that asian students encountered similar mathematical challenges, which made her “feel good”: they’re struggling the same way i am and they’re asian…. they probably need a lot more help than i do and that makes me feel good. sometimes you have to see. i mean the majority they’re really smart, but some of them are struggling like any other person that’s here…i always thought that they were smart until i came here and i realized not everyone is as smart as they look you can say. [laughs] (interview 3, may 12, 2008) these changes contributed to strengthening vanessa’s confidence as a mathematics learner and allowed her to positively merge aspects of her latina, gender, and mathematics identities. for example, vanessa grew to “accept” who she was “as a math student,” to perceive women (including latinas) as “smarter” and “better” in mathematics, and to feel “proud” to perceive latinas/os (including herself) as “good at doing math” (see the quotation in the introduction). third, vanessa challenged racialized and gendered experiences. in the previous descriptions, vanessa challenged some racialized experiences (e.g., comfortably practicing mathematics with a culturally diverse student population despite having a lack of exposure to cultural diversity, challenging socially and pri oppland-cordell urban latina/o students in calculus i journal of urban mathematics education vol. 7, no. 1   42 vately constructed meanings for white, asian, and latina/o mathematics learners). additionally, she challenged racialized and gendered high school experiences involving ms. johnson. for example, she practiced mathematics comfortably at the boards without being criticized and recognized that her preferred problemsolving approaches were valued. the photograph in figure 2 displays an example of how vanessa used preferred problem-solving strategies that ms. johnson had criticized to attack a problem (i.e. breaking solution processes down into step-bystep formats, writing steps in words).   figure 2. example of vanessa sharing her preferred ways of engaging with mathematics in the workshop context (boardwork, april 8, 2008).10 what it means for immanuel to be an urban latino workshop student immanuel is a sociable, first-generation, latino freshman with short brown hair, brown skin, and a friendly demeanor. due to a lack of mathematical opportunities in late middle school and high school, immanuel experienced a decrease in his mathematical confidence. immanuel revealed complex experiences he negotiated as a latino that acted as barriers and avenues for his mathematical success, including experiences involving race, class, culture, language, and power. as the following discussion illuminates, immanuel’s history                                                                                                                           10 the photograph depicts a solution for a practice exam problem. in the first photograph, vanessa wrote: (1) find f ’(x), (2) equal to zero (two binomials), (3) do chart to find max min (c.p. stands for critical points), (4) second derivative = 0, and (5) use the both charts to graph, which describes her steps for finding and classifying critical points, finding inflection points, and graphing the function. problem-solving steps written by vanessa oppland-cordell urban latina/o students in calculus i journal of urban mathematics education vol. 7, no. 1   43 as a latino critically impacted how his emrics influenced, and were influenced by, his participation over time. negotiating mathematical and racial identities: avenues and barriers for participation. when the workshop relocated from classroom 1 to classroom 2 (see figure 1), immanuel shifted towards frequently engaging in predominately latina/o peer groups at board c. he revealed this shift was related to aspects of his emrics, including his perception that it was “easier to get along” with latina/o peers because they were “culturally the same” and could “speak spanish”: i feel when you’re culturally the same it’s easier to get along for some reason, but i’ve only seen it with [vanessa] and [dante] and everyone because we all speak spanish and just come culturally from the same background… (interview 3, april 30, 2008) however, immanuel perceived speaking spanish around peers who did not understand the language as “rude” because he did not want to “leave people out.” therefore, he did not want his use of spanish to limit his interactions with peers. immanuel also believed that cultural differences between him and his peers did not create levels of discomfort or obstruct his practice with peers: “there’s no barriers. we say hi to each other and we know if we need help who knows this, who knows that” (interview 3, april 15, 2008). in fact, he believed asian and white workshop students who tended to congregate together did so because they had established a close rapport with one another, not because they shared similar cultural backgrounds. he also cited friendly and unpretentious personalities, rather than cultural similarities, as a more dynamic factor that encouraged him to engage with peers. immanuel’s perception that cultural differences did not impede his participation may have been related to his frequent exposure to culturally diversity in his prior schools, his closest friendships being formed with both latina/o and white students throughout this life, and his belief that it was important for him to assimilate into american culture, including “to feel more equal to the rest of my peers.” regarding this latter factor, he mentioned that latina/o workshop students who tended to distance themselves from other students may have been “less assimilated to the american culture” (reflection 4, april 15, 2008). although immanuel indicated that there were “no barriers” with peers, he did describe two barriers that limited his participation that i identified as related to his emrics: (a) negotiating racialized and ses experiences, and (b) perceiving differences between his and peers’ mathematical abilities due to his and peers’ cultural backgrounds. first, immanuel negotiated racialized and ses experiences in the workshop. for instance, he managed the lack of mathematical opportunities he had encountered in 8th through 12th grade. this lack of opportunity was related to various complex factors, including limited funding/funding cuts in his predominately la oppland-cordell urban latina/o students in calculus i journal of urban mathematics education vol. 7, no. 1   44 tina/o and/or african american schools, frequent interactions with unmotivated latina/o and african american peer populations, high school mathematics teachers who held low expectations for latina/o and african american students, and authority figures and unfair tracking practices that prevented him from participating in honors high school mathematics classes. for example, as a freshmen and sophomore high school student at hook career academy, he interacted with academically unmotivated african american and latina/o peers and mathematics teachers he perceived as uninterested in teaching their predominately african american and latina/o students, which created significant barriers for his mathematical success: from my experiences with the school, [hook career academy], it was half black and half hispanic, they didn’t want to learn the majority of them. there was like one or two, but the teachers didn’t care about the students because half of the class didn’t want to learn…. i mean, if the students wanted to learn over there at [hook career academy], they still wouldn’t teach you because they knew that it was gonna go down the drain anyways. that’s just how i perceived it. (interview 1, january 24, 2008) immanuel described how the negative mathematical experiences he negotiated from 8th to 12th grade, and particularly those at hook, had a negative domino effect on his mathematics development, culminated in his decision to not enroll in a mathematics class his senior year of high school, and harmed his mathematical confidence: “i never had good experiences with learning math and i just feel like i’m not supposed to be here” (interview 1, january 24, 2008). the aforementioned lack of mathematical opportunities impacted immanuel’s workshop participation in terms of how he positioned his mathematics ability in relation to peers’ mathematics abilities. for instance, he often described himself as “behind” and as “a little fish compared” to “smarter” workshop students, which limited his participation in terms of him refraining from sharing his mathematical knowledge in some instances and letting students that knew “how to best do it do it”: i’m always behind…some people are really on this and i’m trying to catch up to that…. i don’t want to seem like a little fish compared to them…i’m like, yeah i don’t know how to get past the first step, which is fine because i’ll catch up eventually, but i mean i’m gonna let the person that knows how to best do it do it. (interview 2, march 6, 2008) second, peers’ cultural backgrounds contributed to immanuel perceiving differences between his and peers’ mathematical abilities. his frequent encounters with academically unmotivated latina/o and african american peers in k−12 contexts contributed to him assuming that latina/o and african american work oppland-cordell urban latina/o students in calculus i journal of urban mathematics education vol. 7, no. 1   45 shop students would also be less academically motivated, including in mathematics. although he regularly interacted with latina/o workshop students, this belief may have been related to his infrequent interactions with anthony during much of the semester (the only african american student who persisted in the workshop). further, he believed asians and whites tend to be strong in academics, including mathematics, due to, for instance, asian students’ strong work ethic and white students’ parents passing on their knowledge of the american educational system to their children (this latter factor contrasted with his family experience). in fact, when asked to describe workshop students he perceived as mathematically strong he mentioned two white men (peter and matt). although he indicated there were “no barriers” with students and he engaged with some asian and white peers, his perception that asian and white students tend to possess strong mathematics abilities appeared to contribute to him perceiving himself as less knowledgeable and refraining from sharing his mathematical knowledge in some instances. the ways in which peers’ cultural backgrounds appeared to impact immanuel’s participation may have also been related to his management of societal, community, and institutional messages throughout his life that supported the notion of a racial hierarchy of academic and mathematical ability with latinas/os and african americans positioned below asians and whites. negotiating mathematical and racial identities: critical transformations. as the semester progressed, immanuel’s participation changed in some critical ways. for example, he contributed his ideas to mathematical discussions more frequently, he solved problems with a more culturally diverse student population at boards a and c in classroom 2, and he asked me (the instructor) mathematical questions about work he completed on boards a and c in classroom 2. as these changes occurred, he challenged some of the aforementioned barriers tied to his emrics that limited his participation. first, immanuel challenged racialized and ses experiences. for instance, immanuel often spoke about how the ample mathematical learning opportunities the workshop provided contrasted with the lack of mathematical opportunities he experienced in k−12 contexts (e.g., interacting with supportive and academically motivated individuals who were not “looking down” on his mathematics ability, identifying himself as a mathematical resource for students, viewing peers and the instructor as mathematical “resources,” encountering opportunities to solve mathematical problems at his “own pace”). immanuel discussed a subset of these experiences when asked to describe which workshop experiences, if any, had affected his motivation to learn mathematics: the difference that made me feel like this is the fact that i’m really grasping the concepts and i feel like it’s not being shoved down my throat. i feel like it’s given to me on a plate and there telling me to take my time. it’s okay if you’re slow. just take your time. you’ll get there, as opposed to all my high school math classes, i feel like oppland-cordell urban latina/o students in calculus i journal of urban mathematics education vol. 7, no. 1   46 (a) there wasn’t enough resources to get extra help and (b) most of the teachers weren’t that helpful…i mean it just doesn’t work for me like that…. and i feel like it’s right to be wrong because people are there to help you and somebody’s gonna be wrong at one time or another and they’re gonna be happy that you’re there to help them. (interview 2, march 6, 2008) in addition to encountering resources he could draw on to learn mathematics, he identified himself as a mathematical resource for students, which was a particularly powerful experience (and change) for him. the photograph in figure 3 illustrates how immanuel used the boards to encourage students to use online mathematical youtube videos (as indicated by “math tv”) because they helped strengthen his understanding of content outside of class. some peers adopted his recommendations, which strengthened his confidence and encouraged him to continue to recommend videos: “feels good…i’ve been telling people about the youtube thing and they’ll be like, ‘oh thanks, [immanuel]. that really helped’” (interview 3, april 15, 2008). this sharply contrasted with how he typically interacted with most peers in high school, particularly latinas/os and african americans at hook career academy. figure 3. example of how immanuel used the boards to act as a mathematical resource for peers (boardwork, april 8, 2008). second, changes occurred regarding how peers’ cultural backgrounds contributed to him perceiving differences between his and peers’ mathematical abilities. as immanuel engaged with a more culturally diverse peer population, he indicated that the “communication barrier with everybody was broken” so he felt more comfortable asking a wider peer population mathematical questions (interview 2, march 6, 2008). for example, in the following quotation, he described how he left his peer group to interact with anthony, an african american male student he rarely interacted with earlier in the semester:   mathtv written by immanuel oppland-cordell urban latina/o students in calculus i journal of urban mathematics education vol. 7, no. 1   47 sarah: do you feel that you belong to a certain group? immanuel: sometimes i do. sometimes i don’t. sometimes i like to get out of that group ‘cause they talk too much…i need help with this math problem, so i’ll stand up and go to the board like the other one in this room. and today [anthony] was having the same problem with number one…and i was like, “damn, see that’s why you’re up there at that board, [anthony]. you know what you’re doing.” and then he did all of it and he did it in his head and i was like, “how’d you know that?” and he’s like, “well, i took calc in high school.” …yeah, he did it really fast, but he explained it to me, and then [vanessa] and them were still on that graph problem. i didn’t feel like doing it. i wanted to do the integrals ‘cause they seemed cool. (interview 3, april 15, 2008) as immanuel engaged with a broader peer population (e.g., anthony, hank), transformations occurred regarding how he viewed his and peers’ mathematical abilities. for example, he perceived peers’ mathematics abilities as more equal and advanced over time; he observed an african american male student (anthony) and latinas/os exhibiting strong mathematics abilities (which contrasted with some prior experiences); and he observed students he had framed as mathematically advanced (e.g., asians) encountering mathematical challenges and “not looking down” on his mathematics ability. these transformations contributed to strengthening immanuel’s confidence as a mathematics learner and allowed him to positively merge his latino and mathematical identities. for example, the workshop aided him in attaining a mathematical appropriation level that he had not experienced in his prior mathematical pathway: “i just get it now after all this work we’ve been doing. i get all this other stuff that in my life i’ve never really gotten” (interview 3, april 15, 2008). he also gained motivation due to perceiving latinas/os (including himself) as “breaking stereotypes” about latina/o students’ mathematics abilities: [strengthened mathematics identity as a latino mathematics learner] immanuel: i never met hispanic or latino people that are this high up like me…i don’t know. it’s perplexing me because if you’re used to something all your life, like i was expecting to come to [hall university] and not be similar to people in that cultural background because you never hear of a lot of black or latino people going to college. but it’s funny because this is one of the only classes where it’s like wow you guys want to be engineers…. it’s like that’s really cool. sarah: does that help motivate you? oppland-cordell urban latina/o students in calculus i journal of urban mathematics education vol. 7, no. 1   48 immanuel: yeah, it does because i don’t feel alone…most of the people you see at [hall university] are white or asian or something like that. it’s weird not to have anybody from your background that’s somehow connected in a different way. that knows what’s going on…my econ class was all white. i was like, “wow, i’m the only hispanic person interested in econ.” but then when you get to this class it’s like that’s cool man…you and i are breaking stereotypes. (interview 3, april 15, 2008) in sum, analyzing vanessa’s and immanuel’s perspectives of their workshop experience through sociocultural, crt, and latcrit lenses reveals that their negotiations of their emrics (and their intersection with their other identities, such as gender in vanessa’s case and ses in immanuel’s case) contributed to creating significant avenues and barriers for their participation. however, as the semester progressed and their participation continued to influence how they negotiated these identities, they were able to challenge barriers they had managed earlier, which influenced some profound positive transformations in their participation. cross-case comparison data analysis for this study involved first constructing the above case study narratives for each participant and then comparing the findings across both cases. to complement the above findings, i now discuss important cross-case findings to illuminate how two latina/o students negotiated their emrics in relation to their participation. three main (overlapping and positive) themes emerged regarding how participants’ emrics shifted their participation over time. first, participants’ perceptions of their and peers’ mathematics abilities changed in ways that strengthened their participation. for example, both participants challenged privately and socially constructed meanings of what it means to be latina/o, asian, and white mathematics learners (and an african american learner in immanuel’s case) and they constructed strengthened self-perceptions as latina/o mathematics learners. second, both vanessa and immanuel challenged racialized experiences tied to prior school experiences in ways that strengthened their participation. for example, vanessa challenged how ms. johnson had constructed her preferred mathematical strategies as deficits, including due to her latina status, which encouraged her to apply these methods in the workshop. immanuel challenged his prior interactions with academically unmotivated african american and latina/o peers, including by witnessing mathematically talented african american and latina/o students use mathematical resources he recommended, which encouraged his participation. third, both participants established strengthened comfort levels in the workshop not only with peers but also with me (the instructor) and the mathematics, oppland-cordell urban latina/o students in calculus i journal of urban mathematics education vol. 7, no. 1   49 which encouraged their participation. for example, interacting with peers and an instructor that supported their preferred ways of engaging with mathematics (e.g., writing mathematics in words in vanessa’s case and comfortably sharing links to mathematical videos in immanuel’s case), which contrasted with prior racialized experiences they managed, strengthened vanessa’s and immanuel’s comfort levels and participation. discussion this study presented two case studies that illustrated how latina/o students’ participation through their emrics critically influenced positive shifts in their participation in a culturally heterogeneous, collaborative esp calculus i workshop. vanessa’s and immanuel’s voices not only revealed intimate relationships regarding how their complex negotiations of their emrics and their participation informed one another but also how these relationships became strengthened over time. although participants’ negotiations of their emrics created some obstacles for their participation, they participated through their salient identities (e.g., disciplinary, racial, gender, class) in ways that allowed them to challenge these barriers and experience positive shifts in their participation over time, which ultimately strengthened relationships among their identity development and participation. the findings indicate that understanding how vanessa’s and immanuel’s identities and (non)participation evolved and related could not be separated from the social, political, and historical contexts in which the local workshop was situated. a significant implication that this understanding has for research on students’ participation in mathematics is that we must not only consider the sociopolitical nature of interactions and (non)participation in mathematics classroom contexts but also how participation in local classroom contexts relates to the broader sociopolitical environment. sociocultural mathematics education studies that investigate students’ participation without using a sociopolitical lens would not reveal the sociopolitical nature of participation, including how students’ social identities relate to their practice-based identity constructions. the case studies reveal the complexity, diversity, and significance of the sociopolitical nature of the co-construction of identity development and participation for vanessa and immanuel in the workshop context. relationships that emerged between participants’ emerging mathematical and racial identity (re)constructions and their participation processes were connected to various interrelated issues related to their histories and lives (e.g., meanings for latina/o, race, racism, language, culture, gender, class, power relations, stereotypes), multiple contextual layers (e.g., society, community, family, school, peer groups), the classroom environment (e.g., interactions with supportive peers and the instructor, classroom layouts, pedagogical practices), and their agency and resilience. as oppland-cordell urban latina/o students in calculus i journal of urban mathematics education vol. 7, no. 1   50 pects of participants’ emrics that powerfully influenced the evolution of their participation aligned strongly with factors recognized as informing latinas/os’ racial identity constructions. such factors include meanings for racial/ethnic group memberships, racial stereotypes, treatment by teachers and/or peers in school contexts, and interactions with and degree of exposure to racial/ethnic groups (ancis, sedlacek, & mohr, 2000; barajas & pierce, 2001; ferdman & gallegos, 2001). for example, interactions with encouraging dominant and nondominant students allowed participants to challenge stereotypes about latina/o, african american, asian, and white students’ mathematics abilities and contributed to strengthening their self-perceptions as latina/o mathematics learners. therefore, similar to esmonde and colleagues (2009), this study found that aspects of marginalized students’ various identities can be related to how broader contextual power relations become manifested in mathematics classrooms through collaborative classroom practices, which influence the mathematical learning opportunities they experience. regarding the classroom environment, particular classroom characteristics that supported strengthened relationships among identity and participation included: collaborative peer groups, self-selecting and changing peer groups, supportive instructors and peers, writing mathematics on privately located boards (i.e., not positioned in front of the class) in a spacious classroom, and “the valorization of multiple methods for thinking about and doing school mathematics” (jilk, 2007, cited by civil, 2008, p. 17). as the instructor, my actions contributed to encouraging participation patterns that supported positive identity development, including by treating students as valuable resources and giving them the freedom to engage with mathematics in ways that supported them to challenge prior experiences with authority figures and/or peers that had threatened facets of this development. for instance, participants were encouraged to use various languages, write mathematics in words, use step-by-step strategies, and share mathematical videos. participants also exercised their personal agency to actively negotiate aspects of their emrics in ways that influenced supportive shifts in their participation in terms of becoming more active members of the workshop cop. transformations in vanessa’s and immanuel’s participation (as previously described) occurred, in part, because of their willingness to take risks. for example, they shifted towards recording problem solutions on boards, sharing their solutions with peers and the instructor, repositioning themselves in the classroom including by engaging with a more culturally diverse peer population, and resisting cultural mathematical stereotypes. as their participation evolved, their negotiations of their emrics contributed to them responding to barriers they had faced earlier in the semester in diverse ways, which led to some critical transformations in how their participation developed. oppland-cordell urban latina/o students in calculus i journal of urban mathematics education vol. 7, no. 1   51 findings for both cases also resonate with mathematics education research indicating that the intersectionality of underrepresented students’ mathematical and other salient identities can inform, and be informed by, their participation in mathematical learning communities (esmonde et al., 2009; nasir, 2002). a combined crt and latcrit lens revealed that avenues, barriers, and supportive shifts for participation were related to how meanings for latina/o, race, and racism complexly and diversely intersected with covert sociopolitical constructs (i.e., gender, ses, language, culture). similarities and differences emerged regarding how intersections among participants’ salient identities and emrics informed their participation. for example, sharing cultural and language backgrounds with latina/o participants increased both participants’ comfort levels. however, unlike immanuel, vanessa’s negotiations of the nexus of her mathematics, racial, and gender identities contributed to creating significant avenues, barriers, and productive shifts for her participation (e.g., witnessing latinas’ and women’s mathematical talents allowed her to challenge the notion of a gender hierarchy of mathematical ability, which strengthened her desire to participate). unlike vanessa, immanuel’s management of his collective mathematics, racial, and class (ses) identities played powerful roles in creating barriers and productive shifts for his participation (e.g., encountering numerous mathematical opportunities in the workshop allowed him to challenge prior k−12 experiences, which strengthened his participation). on the other hand, however, shifts in vanessa’s and immanuel’s participation also strengthened aspects of their emric’s (e.g., constructed stronger perceptions as latina/o mathematics learners). at a time when latina/o students’ mathematical experiences are often (negatively) essentialized in mathematics education scholarship, this study supports that the complex intersectional nature of latina/o students’ emrics in mathematics classrooms must be acknowledged. employing crt and latcrit also allowed for capturing participants’ counter-stories, which reveal how societal stereotypes, low expectations, and barriers they faced in their mathematical trajectories, including those involving racialized, gendered, and classed meanings, impacted how their emrics and workshop participation related. such stories offer possible explanations for latina/o students’ lower mathematics achievement and participation levels (chapa & de la rosa, 2006; ortiz-franco & flores, 2001) and why some latinas/os struggle to identify with mathematics. however, the findings also reveal how latinas/os resisted and overcame oppressive experiences to participate, learn, and succeed in mathematics. such findings suggest that to better support underrepresented students in succeeding in mathematics it is imperative to recognize how interlocking variables such as race, gender, and ses function in their mathematical lives and to eliminate barriers they face linked to structural, institutional, and everyday racism. oppland-cordell urban latina/o students in calculus i journal of urban mathematics education vol. 7, no. 1   52 conclusion addressing the mathematics education community, gutiérrez (2013) stated: if, as a field, we are not willing to recognize the political nature of mathematics education or the fact that teaching and learning are negotiated practices that implicate our identities, we might as well give up on all of this “talk” about equity. (pp. 62–63) assuming this standpoint, this study supports that the mathematics education community must embrace the sociopolitical nature of how latina/o students coconstruct their identities and participation in mathematical learning contexts because this more expansive theoretical lens can provide additional knowledge regarding how and why latina/o students attain mathematical success. this study also addresses the importance of capturing students’ perspectives that may be invisible but also may be critical for understanding their mathematical success. future research should use multiple data sources to examine contextualized interpretations of the meanings students give to their participation in effective mathematics classrooms. such research will not only help to reveal the complexities, avenues, and transformations that support students’ mathematical success but also will expand knowledge about how to design identity-affirming, equitable mathematical learning environments. acknowledgements i wish to thank dr. danny martin from the university of illinois at chicago for his ongoing guidance, support, and useful recommendations on this work. i also would like to thank vanessa and immanuel, the participants in this study, for their commitment to this project, their valuable comments on this work, and their willingness to share their mathematical stories. finally, i wish to thank my family for its support and encouragement. references ancis, j. r., sedlacek, w. e., & mohr, j. j. 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(2009). case study research: design and methods. thousand oakes, ca: sage. journal of urban mathematics education july 2015, vol. 8, no. 1, pp. 31–61 ©jume. http://education.gsu.edu/jume keith e. howard is an associate professor in the college of educational studies at chapman university, one university drive, orange, ca, 92866; email: khoward@chapman.edu. his research interests include equity in mathematics access and instruction, technology integration into k–12 instruction, and statistical analyses of large-scale data sets. martin romero is an assistant professor in the department of mathematics at santa ana college, 1530 w. 17th street, santa ana, ca, 92706; email: romero_martin@sac.edu. his research interests include equity in mathematics education, teaching for social justice, and developmental mathematics education at post-secondary institutions. allison scott is a director of research and evaluation at the level playing field institute, 2201 broadway, oakland, ca 94612; email allison@lpfi.org. her research interests include examining structural and social/psychological barriers facing underrepresented students of color in pursuing and persisting in science, technology, engineering, and mathematics (stem) fields, and the effectiveness of interventions and initiatives to improve stem outcomes among students from underrepresented backgrounds. derrick saddler is a visiting mathematics instructor in the college of arts and sciences at the university of south florida sarasota-manatee, 8350 n. tamiami trail c263, sarasota, fl 34243; email: dsaddler@usf.edu. his research interests include curriculum effectiveness in mathematics education, organization of mathematics content, students’ learning of algebra, and analysis of large-scale data sets. success after failure: academic effects and psychological implications of early universal algebra policies keith e. howard chapman university martin romero santa ana college allison scott level playing field institute derrick saddler university of south florida in this article, the authors use the high school longitudinal study 2009 (hsls:09) national database to analyze the relationships between algebra failure, subsequent performance, motivation, and college readiness. students who failed eighth-grade algebra i did not differ significantly in mathematics proficiency from those who passed lower-level courses, but initially demonstrated significantly lower mathematics interest, mathematics utility, and mathematics identity. both groups were less likely than the general population to meet college requirements in the eleventh grade, although students who passed a lower-level mathematics course fared better than those who failed algebra i. implications for policies addressing mathematics course enrollments are discussed. keywords: ability grouping, educational policy, equity, longitudinal studies, mathematics education ublic education in the united states was ushered in with a notion that allowing equal access to education for all children would make the american dream possible for anyone who was willing to work for it. a good education was presumed to p http://education.gsu.edu/jume mailto:khoward@chapman.edu mailto:romero_martin@sac.edu mailto:allison@lpfi.org mailto:dsaddler@usf.edu mailto:dsaddler@usf.edu howard et al. success after failure journal of urban mathematics education vol. 8, no. 1 32 be both a mechanism for ensuring equal opportunity to upward mobility and a means of raising the acumen and contributions of the populace at large. horace mann tabbed the common school as the “great equalizer,” suggesting that its establishment would drive down poverty and crime while equipping an intelligent populace to develop the natural and material resources to benefit society as a whole (cremin, 1957/1979). but, if education were to operate today as an equalizer that affords every citizen an opportunity through academics, then providing equal access to course sequences that chart a path to college would seem to be an approach without controversy. however, a long and historical debate over whether high schools should provide the same “college preparatory” curriculum to all students, as opposed to a curriculum tailored to students’ individual interests, abilities, and needs, complicates the issue (mirel, 2006). nowhere is this debate more evident today than in the controversy surrounding mathematics instruction in secondary schools in the united states. the sequence and level of mathematics courses taken in secondary school are critical factors that influence access to higher education. algebra in particular has been identified as a key factor in academic trajectory, mainly for its perceived role as a “gatekeeper” course into higher education (national mathematics advisory panel, 2008; walston & mccarroll, 2010). research findings support this notion of gatekeeper as being more than just rhetoric; when taken in eighth grade, algebra has been shown to be associated with positive long-term outcomes for students, including increased mathematics test scores (attewell & domina, 2008; gamoran & hannigan, 2000); enrollment in advanced high school mathematics and science course-taking sequences (paul, 2005; stein, kaufman, sherman, & hillen, 2011); and greater rates of college application, acceptance, and attendance (atanda, 1999; spielhagan, 2006). both access to the algebra course and the grade in which it is taken are important factors in shaping a student’s academic trajectory. taking algebra in the eighth grade is crucial in order to allow students the time to complete a 4year, college preparatory sequence in high school (loveless, 2008), which in turn enhances their chances of college acceptance (attewell & domina, 2008), and subsequent entrance into potentially high paying science, technology, engineering, and mathematics (stem) fields (evan, gray, & olchefske, 2006). the potential benefits of algebra enrollment in eighth grade, combined with past racial and socioeconomic disparities in access to algebra courses (adelman, 2006; silva, moses, rivers, & johnson, 1990; stein et al., 2011) have fueled major initiatives proposing early algebra-for-all students. the central idea is that providing the opportunity for all students to take algebra in eighth grade will ensure that they are not excluded from college opportunities by course sequences that fall short of college admission policy requirements (liang, heckman, & abedi, 2012). there is some evidence that nationally, placement into algebra is far less likely if a student howard et al. success after failure journal of urban mathematics education vol. 8, no. 1 33 is black or hispanic, low ses, male, or from a single-parent household (walston & mccarroll, 2010). although the “top-down” approach to mathematics equity by requiring all students to acquire the same content has been criticized as lacking relevance to the everyday lives of marginalized students (martin, 2003), some policy advocates have identified mathematics access as a civil rights issue, suggesting that mathematics proficiency is a necessary component of literacy in the computer age in order to fully participate in society (moses, 1995; moses & cobb, 2001). in this study, we compare students who performed poorly in eighth-grade algebra to students who passed a lower-level, eighth-grade mathematics course, and examine the academic and psychological and motivational factors associated with success and failure. such comparisons can be problematic due to a myriad of factors that may contribute to a given student’s success or failure in a course. for that reason, in our analyses, we utilize propensity score matching to simulate random assignment and to reduce possible selection bias on several demographic factors that may influence student outcomes. specifically, our analyses address the following research questions: 1. how does the algebra proficiency level of students who took algebra in the eighth grade and failed (defined in this study as receiving a grade of “f” or “d”) compare to that of students who took a lower-level mathematics course in eighth grade and passed (receiving a grade of “c” or higher)? 2. how does the mathematics identification of students who failed algebra in the eighth grade compare to the mathematics identification of those who passed a lower-level, eighth-grade mathematics course? 3. how does the mathematics utility value (perceived usefulness) of students who failed algebra in the eighth grade compare to the mathematics utility value of those who passed a lower-level, eighth-grade mathematics course? 4. how does the mathematics interest of students who failed algebra in the eighth grade compare to the mathematics interest of those who passed a lower-level, eighth-grade mathematics course? 5. what is the relationship between eighth-grade course enrollment and eleventh-grade “college readiness”? overview of the literature as a result of recent policy changes and initiatives, secondary students are enrolling in advanced mathematics at levels never before seen in u.s. public education. an examination of national longitudinal transcript data revealed an in howard et al. success after failure journal of urban mathematics education vol. 8, no. 1 34 crease in the number of mathematics courses taken from 2.7 total credits in 1982 to 3.6 credits in 2004 (dalton, ingels, downing, & bozick, 2007, p. iv). over that same period, there was a threefold increase in the number of students taking precalculus and calculus, and a 50% drop in the number of students finishing high school with algebra i or less as their highest course taken (p. 12). data from the early childhood longitudinal study (ecls-k) reveal that by the 2006–07 school year, 39% of the study cohort was taking algebra i or higher in the eighth grade (walston & mccarroll, 2010). in california, the percentage of eighth graders enrolled in algebra increased from 32% in 2003 to 59% by 2011, following the implementation of policies designed to ensure that all eighth graders take algebra (liang et al., 2012). moreover, there is some evidence that groups previously underrepresented in higher mathematics courses have started to make inroads as a result of the mandates in many urban districts requiring all students to be enrolled in algebra in the eighth grade. a study of academic curricular intensity using transcript data from the national educational longitudinal study (nels:88) found that asians and whites were taking the most demanding curricula overall. however, once ses and prior performance were controlled, results revealed that traditionally underperforming black and hispanic students had begun to take a more demanding curriculum than white students (attewell & domina, 2008). the shift towards increased student access to higher mathematics courses is not without criticism. significant debates have ensued among policymakers and educators about the benefits and consequences of increasing early algebra readiness and course-taking (allensworth, nomi, montgomery, & lee, 2009; national council of teachers of mathematics, 2008; rosin, barondess, & leichty, 2009). allensworth and colleagues examined outcomes from a chicago initiative mandating universal college preparatory coursework and framed the debate in terms of a continuum, which has “constrained curriculum” advocates at one end and “social efficiency” supporters on the other. proponents of a social justice approach to curriculum (a position that allensworth and colleagues term “the constrained curriculum”) contend that all students should experience a rigorous curriculum that prepares them equally as well for both the workforce and higher education. on the other end of the continuum, the social efficiency argument suggests that, because students have different intellectual capacities, skills, and aspirations, “schools have a duty to sort and match students to their future places in the social and economic system” (p. 368). the social efficiency position seems to redefine the role of public education as the “great sorter” rather than the “great equalizer.” but from a social justice perspective, an important issue is whether this sorting is based on academic merit as opposed to other factors. howard et al. success after failure journal of urban mathematics education vol. 8, no. 1 35 placement factors and algebra i outcomes although more students are now being enrolled in higher-level mathematics courses, some critics argue that all students might not be best served by taking algebra i early. loveless (2008) maintains that the impetus for universal eighth-grade algebra is an equity focus rather than one based on empirical evidence. however, some empirical evidence suggests that certain groups have not received the same access to algebra, even when their academic proficiency would seem to have predicted otherwise. an examination of course placement practices utilizing early childhood longitudinal study (ecls-k) data revealed significant disparities in algebra course placements for black students (faulkner, stiff, marshall, nietfeld, & crossland, 2014). high achieving black students’ odds of being placed in algebra were two-fifths lower than their white peers. an earlier examination of the ecls-k revealed that even though schools were more likely to place their students with the strongest mathematics skills into courses leading to algebra by the eighth grade, only 35% of black students who scored in the highest two nationally normed quintiles on a fifth-grade mathematics test were placed into algebra in the eighth grade, compared to 63% of whites, 68% of hispanics, and 94% of asians (walston & mccarroll, 2010). higher achieving male students moved on to eighth-grade algebra at lower rates than female students (56% vs. 70%). in every quintile, students placed in algebra had higher subsequent mathematics test scores than their counterparts placed in lower mathematics courses, although the difference in the lowest quintile was not statistically significant. an earlier study by stone (1998) revealed that high ses students in a large urban district who scored in the upper quartile on a standardized test of academic ability were three times more likely to be enrolled in algebra i than their low ses counterparts scoring in the same quartile. the disparity in algebra access for high achieving students raises a question as to whether mathematics ability should be the sole determinant of access to college preparatory coursework. stereotypes and perceived abilities may play a role in placement decisions that are not consistent with demonstrated performance. welldocumented disparities in african american disciplinary actions, as well as their overrepresentation in “judgment categories” of special education (t. c. howard, 2010, p. 20), point to possible social and behavioral factors contributing to evaluations of competence. the influence of such nonacademic factors in placement decisions cannot be completely dismissed, especially for groups of students whose test scores do not correlate with their opportunities. evaluations of mathematics competence have also been found to be related to language proficiency levels, which can be exacerbated by placement into tracked courses (mosqueda, 2010). for english language learners, bilingualism could be utilized as a resource as some have suggested (téllez, moschkovich, & civil, 2011), rather than a barrier to access. howard et al. success after failure journal of urban mathematics education vol. 8, no. 1 36 in an examination of national assessment of educational progress (naep) data, loveless (2008) highlighted the lack of correlation between enrollment in advanced courses and naep test scores, as well as a small decline in scores for students in advanced courses over a period when more students were being granted access to algebra. from among the students placed in advanced courses, he examined the characteristics of the bottom 10% of participating students in terms of performance on the naep (extrapolated to represent approximately 120,000 students nationwide) whom he refers to as the “misplaced” students. he found that a large percentage of these students were poor minorities, with parents who did not graduate from college, and who had less qualified teachers while attending large urban schools. he also noted that the large urban schools tended to shun tracking practices in their course assignments, suggesting that such sentiments led to the misplacement of these low-achieving students in the name of equity. based on his analyses of the students performing at the very bottom, he appeared to call for putting the brakes on the algebra-for-all thrust: research exists showing that knowledge of algebra is essential for entry into occupations earning middle class wages. no evidence exists that it matters whether algebra is learned in eighth grade or later, and some students may need more than a year to learn the subject. (p. 12) one can hardly argue with the assertion that students who are several years behind grade level and enrolled in advanced courses are misplaced, especially if they are given no support beyond that received by students performing at grade level. policies that place students into courses for which they are underprepared certainly do not serve the ends of social justice, but rather potentially set up the students for failure and make the work of teachers and other students more difficult. however, a possible limitation in this argument is that the policy is being judged based on the characteristics of a group of students that represented the lowest performing 120,000 out of approximately 4.2 million eighth-grade students nationwide. by contrast, high achieving students who are denied access to algebra i when other similarly proficient students are granted access are arguably misplaced as well (walston & mccarroll, 2010). in both cases, it appears that the students are not being well served by their placements. course placement decisions that are not consistent with students’ demonstrated content knowledge give rise to policies designed to ensure each student has access to courses of critical importance. however, in the case of universal algebra access, the policy must be judged by its effects on all student groups. the research is mixed on whether low-performing students benefit from being placed in higher mathematics courses. data from the nels:88 revealed that students at all levels of performance benefitted from taking algebra in the eighth grade; howard et al. success after failure journal of urban mathematics education vol. 8, no. 1 37 however, those in the lowest quintile benefitted the least (gamoran & hannigan, 2000). achievement scores of ninth-grade chicago students were unaffected following the enforcement of a policy mandate that required low-ability students to be enrolled in algebra i; however, over the long term, students in the lowest ability groups were more likely to earn credits for upper-level mathematics courses than they did prior to the implementation of the policy (allensworth et al., 2009). a review of 44 studies published from 1995 onward provides some indication as to the causes for differential access; stein and colleagues (2011) concluded: “our data suggest two major reasons for these demographic imbalances: underpreparedness and subjective placement factors” (p. 460). the fact that many students are not prepared to take on algebra in eighth grade supports some critics’ argument that simply placing students in higher-level mathematics courses without changing their k–7 educational experiences is a recipe for failure (liang et al., 2012; schmidt, 2004; williams, haertel, kirst, rosin, & perry, 2011). nonetheless, when students are among the highest achievers in their respective courses and still find themselves denied access to this gatekeeper course, questions of equity in terms of course placements naturally arise. failure associated with raising the bar another major criticism of universal access mandates is the potential for increased failure rates as a byproduct of moving students into algebra irrespective of their proficiency levels. a mandate requiring all chicago ninth-graders to take algebra i resulted in increased failure rates among students moved into algebra as a result of the mandate (allensworth et al., 2009). dropout rates, however, did not increase and mathematics scores on proficiency tests taken at the end of ninth grade were unaffected. increased algebra enrollment rates have been found to be associated with higher rates of failure in those, and subsequent mathematics courses, as well as higher dropout rates later in high school (silver, saunders, & zarate, 2008; waterman, 2010). an examination of u.s. census data for individuals who graduated between 1980 and 1999 found that in states where mathematics and science course graduation requirements were mandated, dropout rates increased for all groups except hispanics, although black and hispanic male students were most severely impacted. the dropout rate increase overall was less than a percentage point (0.82%), but the increases for black (1.88%) and hispanic (2.58%) male students were the greatest (plunk, tate, bierut, & grucza, 2014). these course graduation mandates were also associated with a decrease in the likelihood for black women and hispanic men to enroll in college, but no statistically significant impact to college enrollment was observed on the overall sample. howard et al. success after failure journal of urban mathematics education vol. 8, no. 1 38 given that placing more students into a higher, more difficult mathematics course results in more students failing the course, some have suggested placing lower achieving students on a slower path (liang et al., 2012; loveless, 2008). however, increased access to algebra has also resulted in more students scoring proficiently on algebra knowledge measures, potentially leaving the gate open for college access. for example, in california, the proportion of eighth-grade students enrolled in algebra i increased from 32% to 57% between 2003 and 2010, as did the proportion of students scoring “proficient” or higher on the algebra i california standardized test (cst) (39% to 46%) over that same period (williams et al., 2011). many students are taking advantage of the chance to succeed provided by policy-mandated algebra i access. the difficult decision arising from these circumstances is determining whether a policy designed to afford every student with an opportunity for college access should be scrapped despite the advantages it provides to many students. conversely, a policy requiring all students to take algebra despite data indicating that many are not prepared to succeed at it is difficult to defend from a social justice perspective. if universal access to algebra, regardless of prior achievement, is used as a vehicle to ensure college preparation opportunities, some authors have contended that it should include additional supports to scaffold learning for low achievers, as well as support for teachers in learning how to handle heterogeneous student-ability groupings. nomi (2012) found the lack of these additional supports in the chicago algebra-for-all policy to be a “critical limitation” in its implementation (p. 501). she asserted that such supports were instrumental in a somewhat successful “doubledose” approach employed by nomi and allensworth (2009). in that approach, ninth graders with lower test scores from the previous year were enrolled in both a regular algebra course and an algebra support course. nomi suggested, as an alternative, homogeneous groupings could be used while providing additional instruction time for struggling student groups. in either grouping option, the additional support does not necessarily need to be in the form of more time utilizing the same teaching methods. technology-based supports have shown promise in supporting underlying basic mathematics skills in the context of a traditional mathematics classroom setting (k. e. howard, 2012) and can be utilized for those students who demonstrate a need for them. addressing universal access with homogeneous groupings has been viewed as problematic by critics who allege that lower achievers may be placed in courses that are algebra in name only, providing a watered down curriculum that does not prepare students as purported (schneider, 2009). in that scenario, the same subjective placement factors that have caused potentially algebra-ready students to be placed in lower-level mathematics courses (stein et al., 2011) may continue to limit opportunities to learn for capable students. in addition, given the negative academic con howard et al. success after failure journal of urban mathematics education vol. 8, no. 1 39 sequences and unequal access that have been associated with tracking practices in the past (mosqueda, 2010; oakes, 2005), policies that separate students solely on the basis of achievement levels have fallen out of favor. conversely, mixed-ability groupings present their own set of equity issues. although such groupings may address the needs of capable students who would have been denied access to algebra under previous policies, inclusion of lower-achieving students, particularly those at the lowest levels, may restrict the pace of education for higher achieving students. research is mixed as to the impact that heterogeneous ability groupings have on the most advanced students. some researchers’ findings have suggested that the performance of higher achieving students is compromised by de-tracking lower performing students into more advanced courses (allensworth et al., 2009; loveless, 2009; nomi, 2012), presumably due to the need for teachers to slow the instructional pace or otherwise adjust content delivery for a wider range of proficiency. other researchers suggest that not only do lower-achieving students benefit from detracked courses, but also higher-achieving students have been found to achieve at even higher levels in mixed-ability groups than in high-level tracked groups (boaler, 2011). however, as discussed next, there is some evidence that not all students in higher achieving homogeneous classrooms benefit from this arrangement. comparing data across two chicago initiatives, nomi and allensworth (2014) found that although test scores were higher on average when course enrollments were sorted by student skill levels, grades and pass rates in high-skill courses actually declined. nomi and allensworth suggested that these declines were due to grading practices (e.g., on a curve), higher demands, and high-achieving students finding themselves below average compared to their classroom peers. consequently, students with skills near the bottom of their respective courses had higher failure rates. on the other hand, low-skilled students’ mathematics scores declined slightly after sorting when no changes were made to curriculum, but increased considerably, along with grades and pass rates, when more classroom time and teacher professional development resources were provided. in addition, low-skilled students were more likely to experience disruptive classroom environments due to more behavioral problems when sorted by skill level. whether students are sorted by skill levels or placed in mixed-ability groupings, raising the bar on the difficulty level of eighth-grade courses will likely result in increased failure rates in the short term, especially when the structure of the preceding k–7 instruction has not yet been adequately redesigned to better prepare students for the increased demands (schmidt, 2004). if students, for whatever reason, find themselves unprepared to pass algebra in the eighth grade, would it be better to place those students in a lower-level mathematics course instead? are there academic benefits to at least exposing eighth graders to algebra regardless of their preparedness, even if they fail the course? in other words, can we find a measure of howard et al. success after failure journal of urban mathematics education vol. 8, no. 1 40 success associated with course failure? what about the psychological and motivational implications of such an approach? research indicating the relationship between higher-level course-taking and higher levels of mathematics achievement has not often examined the impact of failure in comparison to successfully completing a lower-level course, nor has it specifically focused on the psychological outcomes associated with success and failure. a comparison of proficiency rates among californian, ninth-grade students who failed (scored less than proficient) the previous year’s algebra i cst to those who passed the cst for general mathematics in eighth grade (scored proficient or above) revealed stark differences in the two groups: the students who had failed the algebra i cst had 69% less of a chance of passing it in the ninth grade than those who had passed the general mathematics cst in the eighth grade (liang et al., 2012). at first glance, this would seem to support placing more students in general mathematics courses; however, upon closer examination only 27.5% of the students in general mathematics passed the cst in the eighth grade (as opposed to a 41.8% pass rate for algebra i). the group of students who passed the general mathematics cst likely included some misplaced students, who may have scored well enough to be placed in algebra but were sorted out by subjective factors previously discussed. nonetheless, the high rate of repeated failure (5 out of 6) for those who had failed the algebra i cst the previous year cannot be ignored. this rate indicates that the vast majority of students repeating algebra did not benefit enough from the previous year’s exposure to score proficiently the second time around. an alternative explanation is that many were not motivated enough to fully apply themselves either time they took the course, which may be the case for students who have chronically failed mathematics during their short k–8 experience but continued to advance due to social promotion practices (zinth, 2005). the psychology of failure examining the impacts of failing algebra or passing a lower-level course, all other factors being equal, is an essential element to assessing the wisdom of an algebra-for-all policy. the direct academic impacts of such a policy may include lower test scores and the narrowing of the window available for completion of a college preparatory course sequence. possible indirect psychological impacts are less obvious, but can be just as detrimental to students’ future academic success. these psychological effects may include identifying less with the mathematics domain, having a lower perceived value for mathematics, and experiencing a decline in overall interest in mathematics (crocker, major, schmader, spencer, & wolfe, 1998; keller, 2007; spencer, steele, & quinn, 1999). several psychological theories have addressed identification with mathematics among underrepresented groups by illuminating coping responses to stigmatiza howard et al. success after failure journal of urban mathematics education vol. 8, no. 1 41 tion and societal stereotypes (major & o’brien, 2005; steele & aronson, 1995). socially stigmatized groups have exhibited patterns of underperformance when group membership was made salient in testing environments—a phenomenon referred to as “stereotype threat” (steele, 1997). the effects of stigma, or an attribute or identity that is devalued or undesirable in a social context, can impact the psychological, educational, and social outcomes of stigmatized group members through confirming negative stereotypes associated with group membership (steele, 2003; steele & aronson, 1995), disengagement or “disidentification” (crocker et al., 1998), and distancing one’s identity from the target domain (crocker, major, & steele, 1998). if disidentification with the mathematics domain were to occur following failure in eighth-grade algebra, it would likely influence the students’ psychological dispositions towards mathematics courses in subsequent years, offering some explanation for the observed repeated failure rates. identifying with a domain is not, in and of itself, sufficient to motivate students to engage in the work necessary to excel in that domain. wigfield and eccles examined children’s mathematical beliefs through their expectancy-value theory (wigfield, 1994; wigfield & eccles, 2000), and posited that expectancy and task value are the two most influential predictors of achievement behavior. the expectancy component refers to a student’s beliefs as to whether they are able to perform a task (expectation level for success), whereas the value component addresses their beliefs about why they should perform the task at all (pintrich & schunk, 2002; schunk, pintrich, & meece, 2008). the influence of negative stereotypes on student expectancy has been demonstrated as early as middle school with black, latina/o, and low ses students (k. e. howard & anderson, 2010), the very groups that have been found to be most frequently represented among the lowest performing students in advanced courses (loveless, 2008). the value students ascribe to the mathematics domain is an important marker of student engagement insofar as it is a strong predictor of whether students decide to apply themselves to their mathematics courses. in the expectancy-value model, the value component has two subcomponents (among others) that may be affected by failing a course: intrinsic value and utility value (schunk et al., 2008). intrinsic value is the student’s subjective interest or enjoyment in doing a task, which is influenced by past performance in that task or domain (eccles & wigfield, 2002). utility value is the perceived usefulness of a task in helping the student to meet his or her future goals; these goals can be adjusted downward as a result of selfregulatory processes following failure (schmitz & perels, 2011). these two subcomponents of task value are of particular interest in this study as they may help to illuminate possible indirect effects of universal access policies that may not be evident in immediate academic outcomes. understanding how these variables are im howard et al. success after failure journal of urban mathematics education vol. 8, no. 1 42 pacted by placement decisions, as well as by subsequent success or failure, can clarify the wisdom and consequences of enacting particular course-taking policies. it has been said that a society can be judged by how it treats its most vulnerable members. policymakers must weigh the impacts of a mandate that requires all students to follow a particular curriculum against those of a policy that does not, particularly on students whose futures are impacted most severely as a result. thus, research is needed to examine the impact of failure on students’ longitudinal mathematics achievement and attitudinal outcomes, and to determine if higher-level course placements benefit students from all levels of attainment. this study is an examination of the impact of eighth-grade mathematics course enrollment and success and failure on subsequent algebra proficiency, college readiness, and psychological and motivational dispositions. methods and data a nationally representative sample was used for this study (ingles et al., 2014; national center for education statistics, 2011) to examine the relationships among mathematics course success, failure, and achievement outcomes, as well as to examine the levels of domain identification, utility, and interest associated with students who fail eighth-grade algebra i. the high school longitudinal study of 2009 (hsls:09) was conducted by the u.s. department of education national center for educational statistics. we examined base-year student data, which were collected from ninth graders during the fall of the 2009–10 school year, and the first follow-up wave of data collected in 2012 when most of the students were in the spring semester of eleventh grade. the hsls:09 data were derived from a sample consisting of more than 21,000 students from 944 public, charter, and private schools in the united states. each wave of data collection included a mathematics assessment of algebraic reasoning and a computer-based survey addressing various psychological and motivational constructs (ingels, dalton, holder, lauff, & burns, 2011; ingels et al., 2014). data sources all of the analyses in this article were based on data obtained from the student-level public use files for the hsls:09 (base year) and the hsls:12 (first follow up) data sets. the hsls:09 variable identifying the highest course each participant took in the eighth grade (s1m8) includes nine self-reported options, including courses more advanced than algebra i. we were only interested in comparing students who took a version of algebra i with those who took a lower mathematics course; therefore, we initially filtered the 21,444 cases in the complete data set to include only the students who reported taking algebra i (including ia and ib; n = howard et al. success after failure journal of urban mathematics education vol. 8, no. 1 43 6,675) in group 1, and those who reported taking math 8, advanced/honors math 8 (not including algebra), or pre-algebra (n = 12,254) in group 2. the hsls:09 variable identifying the self-reported grade each participant received in their highest level mathematics course taken in the eighth grade (s1m1grade) was used to filter group 1 to only include the students who received a grade of “d” or “f” in algebra i, which resulted in 337 students (5.1%). group 2 was filtered to include only the students who received a grade of “c” or higher in a course lower than algebra i (i.e., math 8, advanced or honors math 8, or pre-algebra), which totaled 11,000 (89.8%). finally, both groups were trimmed to exclude any cases which had missing data for any of the variables being considered, which resulted in a group 1 total of 274 and a group 2 total of 8,504. propensity score matching procedures given that our data were compiled from non-randomized cases, propensity score matching was utilized to achieve balance on several observed covariates (stuart, 2010) and to reduce the impact of treatment-selection bias in the estimation of treatment effects. our final data set (before matching) contained 274 cases of students who received a d and/or f in their respective algebra i course, and 8,504 cases of students who received an a, b, or c in their lower level mathematics course in eighth grade. the covariates selected for generating the propensity scores were the hsls:09 student-level data file variables for students’ sex (x1sex), race (x1race), language status (x1duallang), locale (x1locale), region (x1region), socioeconomic status quintile (x1sesq5), along with a dummy variable for participant’s test date. the language status variable identifies the participant’s first language as either english, non-english, or equally english and a non-english language. the locale variable identifies the participant’s address as city, suburb, town, or rural. the region variable identifies the participant’s location in the country as northeast, midwest, south, or west. the ses quintile variable is a composite variable derived from the parent/guardians’ education, occupation, and family income. the test date variable identifies the date when the study-administered test of algebraic reasoning was completed. the tests were administered on three dates during the 2009–10 school year, providing participants with different levels of exposure to their ninth-grade curriculum prior to taking the test. this variable was included in the matching procedure to control for the amount of time elapsed between the date when the participants received their letter grade in the eighth grade and the date when they took the test in the ninth grade. the propensity score matching procedure was conducted in spss 21, using an “r” plugin, which allows estimation of propensity scores using logistic regression (r development core team, 2008). the propensity scores represent the probability that a person or case will land in the treatment condition given the values on all of the co howard et al. success after failure journal of urban mathematics education vol. 8, no. 1 44 variates. in this study, the propensity scores represent the probability that a student would fail algebra i, given the values on the seven included covariates. once the propensity scores are computed, the program utilizes three “r” packages to perform the procedure: “matchit” (ho, imai, king, & stuart, 2007), “ritools” (bowers, fredrickson, & hansen, 2010), and “cem” (iacus, king, & porro, 2009). the matching technique utilized in this study was 1:1 nearest neighbor matching (without replacement), meaning each “treated” participant was matched to a single “untreated” participant who had the propensity score that was closest to an exact match. we used a caliper of .2 of the standard deviation of the logit of the propensity score to ensure good matches (thoemmes & kim, 2011). the caliper defines the maximum allowable difference in estimated propensity scores for matches. cases outside of the common support area (region of distribution in which cases from both groups are observed) were discarded for the larger (untreated) group only, in order to maintain a sufficient sample size for subsequent statistical analyses. after propensity score matching, the treatment and control groups each consisted of n = 274 students. an overall imbalance measure (hansen & bowers, 2008) was not significant, x2(7) = 1.35, p = .99, indicating good balance after matching. the histograms of the standardized differences depicted in figure 1 illustrate greatly improved covariate balance after matching as compared to the unmatched sample. the matched sample histogram is heavily centered on zero, indicating no systematic differences existed after matching. figure 2 depicts the standardized mean differences for each of the covariates before and after matching, illustrating that the magnitude of the standardized differences overall were greatly improved after matching. the standardized mean differences of all covariates were close to 0 after matching. figure 1. histograms with overlaid density estimates of standardized differences before and after matching. howard et al. success after failure journal of urban mathematics education vol. 8, no. 1 45 figure 2. dotplot of standardized mean differences (cohen’s d) for all covariates before and after matching. measures algebra proficiency an online mathematics test of algebraic reasoning was completed by each of the hsls:09 participants during the fall semester of the ninth grade and during the spring semester of the eleventh grade. the instrument used to measure algebra proficiency was developed and validated by the american institutes of research (ingels et al., 2011). the test and item specifications addressed six algebraic content domains (the language of algebra; proportional relationships and change; linear equations, inequalities, and functions; nonlinear equations, inequalities, and functions; systems of equations; and sequences and recursive relationships) and four algebraic processes (demonstrating algebraic skills; using representations of algebraic ideas; performing algebraic reasoning; and solving algebraic problems). item analyses were conducted to compile a pool of optimally performing items, including differential item functioning (dif) statistics to detect potential racial/ethnic and gender biases. base year. the base year assessment administered to the participants contained 40 items and employed an adaptive design. there were 72 unique items used to compile all of the variations of the test. the item response theory (irt) estimated reliability of the base year instrument was 0.92 after applying sample weights (ingels et al., 2011). the hsls:09 base year variable x1txmscr represents an irt-based estimate of the score for each participant on the full set of 72 items. each participant howard et al. success after failure journal of urban mathematics education vol. 8, no. 1 46 completed the assessment on one of three different test dates during the fall term of his or her ninth-grade school year (september 2009, october 2009, or january 2010). the hsls:09 base year theta score variable x1txmth is an ability estimate score based on the same metric as the irt item-level difficulty parameters. it provides a summary measure appropriate for longitudinal analyses to measure growth in achievement over time. first follow-up. a second algebra assessment addressing the previously listed six algebraic content domains and four algebraic processes was administered during the spring semester of the study cohort’s junior year. the follow-up theta score variable (x2txmth) was used to compare participant scores over both waves as the common items between waves allowed for equating the irt scores across waves. the follow-up assessment contained a 69-item pool, including 23 common items across the two waves. this item pool also included 20 new items that were field tested and added to include higher difficulty items and guard against a ceiling effect. the irt-estimated reliability of this follow up assessment was 0.92 after sample weights were applied (ingels et al., 2014). psychological and motivational scales several scales were created for the hsls:09 and included in the field-testing for the algebra proficiency variables (ingels, herget, pratt, dever, & copello, 2010). three of those scales were examined as part of our analyses over both waves of data. all of the data for the scales were reported in the hsls:09 in standardized form (with a mean of 0 and a standard deviation of 1) to allow for combining variables with different scales during analyses. a threshold of .65 was used for inclusion of each of the scales as an adequate measure of internal consistency and each of the scales performed above this threshold in both waves of data collection. table 1 provides coefficients of reliability for each of the scales examined over each wave of data (ingles et al., 2011; ingels et al., 2014). mathematics identity. the hsls:09 mathematics identity scale score was used to measure students’ mathematics identity during both waves of data collection (x1mthid/x2mthid; cronbach’s alpha .84 and .88, respectively). this scale measures the participant’s level of agreement from 1 (strongly agree) to 4 (strongly disagree) on two statements: “you see yourself as a math person” and “others see me as a math person,” with higher values indicative of higher mathematics identity. mathematics utility. the hsls:09 mathematics utility scale score was used in both waves of data collection to measure participants’ beliefs in the utility value of learning mathematics (x1mthuti/ x2mthuti; cronbach’s alpha .78 and .82, respectively). this scale measures the participant’s level of agreement from 1 (strongly agree) to 4 (strongly disagree) on three statements about the importance of howard et al. success after failure journal of urban mathematics education vol. 8, no. 1 47 mathematics for their future: “my math class will be useful for college”; “my math class will be useful for everyday life”; and “my math class will be useful for a future career.” higher values indicate higher mathematics utility. mathematics interest. the hsls:09 mathematics interest scale score was used for both waves of data collection to measure participants’ interest in their current mathematics course (x1mthint/x2mthint; cronbach’s alpha .75 and .69, respectively). this scale includes two questions inquiring as to the students’ favorite and least favorite (wave 1 only) school subjects, followed by four statements about their current grade mathematics course assessing their level of agreement from 1 (strongly agree) to 4 (strongly disagree): “you are enjoying this class very much”; “you think this class is a waste of your time”; “you think this class is boring”; and “you really enjoy math.” higher values indicate higher mathematics interest. the first year follow-up instrument for mathematics interest was revised to eliminate a question asking for the student’s least favorite subject, but still showed adequate internal consistency. college readiness variables three variables measuring college readiness were examined by group and compared to the entire hsls:12 sample. each variable corresponds to a single question that was part of the questionnaire in the follow-up data collection in the fall semester of the participants’ eleventh-grade school year. community college. the hsls:12 variable s2req2yr corresponds to the question “by the summer of 2013, do you think you will have met the minimum requirements needed for admission to a 2-year community college?” the response options were: yes, no, and don’t know. typical four-year college. the hsls:12 variable s2reqty4yr corresponds to the question “by the summer of 2013, do you think you will have met the minimum requirements needed for admission to a typical 4-year college?” the response options were: yes, no, and don’t know. selective college. the hsls:12 variable s2reqsel4yr corresponds to the question “by the summer of 2013, do you think you will have met the minimum requirements needed for admission to a highly selective 4-year college such as harvard university?” the response options were: yes, no, and don’t know. results descriptives statistical analyses were performed on the final base-year set of 548 participants, split evenly between students who took and failed algebra in eighth grade howard et al. success after failure journal of urban mathematics education vol. 8, no. 1 48 and those who took and passed a lower-level mathematics course in eighth grade. as illustrated in table 1, the groups were well balanced through propensity score matching on gender, race, language status, and socioeconomic status. the balance on these important variables was maintained for the participants in the matched group who provided responses in the follow-up data collection. there was a response rate of 83.8% (n = 459) for the follow-up wave mathematics assessment for this group of participants, which compared favorably to the overall unweighted follow-up response rate of 84.3%. little’s mcar statistic (spss missing values 22.0) revealed that the missing data met the assumption of mcar, 2(39) = 52.84, p = .07. there were no systematic patterns of missing data when compared to the observed values for all of the matched covariates, the prior mathematics assessment and psychological measure scores, and the college-bound variables. table 1 post-matching demographic characteristics of the treatment (algebra fail) and control (lower-level math pass) groups base year mathematics assessment follow up treatment n = 274 control n = 274 treatment n = 228 control n = 231 gender male 62% 62% 62% 63% female 38% 38% 38% 37% race/ethnicity white 47% 49% 45% 50% black 10% 10% 9% 10% latino 26% 20% 26% 19% asian 3% 8% 3% 8% other 14% 13% 17% 13% first language english only 80% 78% 79% 78% non-english 13% 14% 14% 12% english and non-english 7% 9% 7% 10% ses quintile first 24% 25% 25% 23% second 22% 19% 21% 18% third 20% 24% 21% 26% fourth 19% 18% 19% 17% fifth 15% 15% 14% 17% howard et al. success after failure journal of urban mathematics education vol. 8, no. 1 49 algebra proficiency to examine for algebra proficiency differences between to two eighth-grade course conditions (failing algebra or passing lower-level course) and any longitudinal differences, a mixed between-within subjects analysis of variance was conducted. the participants’ theta scores on the test of algebraic reasoning were compared across the two waves of data. there was no significant interaction between condition and time: pillai’s trace = .002, f(1, 457) = .80, p = .37, partial η² = .002. as expected, there was a significant main effect for time: pillai’s trace = .32, f(1, 457) = 218.50, p < .001, partial η² = .32. both groups showed statistically significant increases in scores over time. the main effect comparing the two course conditions was not statistically significant: f(1, 457) = .92, p = .34, partial η² = .002. whereas both groups significantly increased their mean theta scores from the base year to the follow-up data collection, there were no statistically significant differences in scores between the two groups. psychological and motivational outcomes the study participants were measured on psychological and motivational scale variables of interest at each of the two waves of data collection. repeatedmeasures manova analyses revealed a significant multivariate effect for coursetaking group: pillai’s trace = .03, f(3, 362) = 3.86, p = .01, η² = .03; and a significant interaction effect between course assignment and time: pillai’s trace = .03, f(3, 362) = 3.37, p = .02, η² = .03. univariate tests revealed significant interaction effects between course assignment and time for mathematics identity f(1, 364) = 7.05, p < .01, η² = .02; mathematics interest f(1, 364) = 4.86, p = .03, η² = .01; and mathematics utility f(1, 364) = 4.11, p = .04, η² = .01. these were small effect sizes. to further examine the interaction effects, the data set was split and manovas were performed for each wave of data (time 1 in ninth grade; time 2 in eleventh grade) separately. there was a statistically significant difference between the groups on the combined dependent variables for the base year data: f (3, 544) = 14.38, p < .001; pillai’s trace = .07; partial η² = .07. the eta squared of .07 indicates a medium effect size (cohen, 1988). the difference on each of the three dependent variables reached statistical significance when the data were separately analyzed. the students who took algebra i and failed reported lower ratings in the ninth grade on mathematics identity f(1, 546) = 23.44, p < .001, partial η² = .04; mathematics interest f(1, 546) = 24.17, p < .001, partial η² = .04; and mathematics utility f(1, 546) = 31.88, p < .001, partial η² = .06 than students who had passed a lower-level, eighth-grade mathematics course. the mean scores for the differences by group are displayed in table 2, along with the weighted mean scores for students howard et al. success after failure journal of urban mathematics education vol. 8, no. 1 50 who passed algebra i in the eighth grade. the effect sizes for mathematics identity and mathematics interest were small, whereas the effect size for mathematics utility was moderate (cohen, 1988). table 2 means and standard deviations for psychological and motivational dependent variables for treatment group (algebra fail), control group (lower level math pass), and algebra pass group treatment (failed algebra) n = 274 control (passed lower math) n = 274 passed algebraa n = 6240bc dependent variable m sd m sd m sd mathematics identity -.44 .99 -.03 .98 .32 .93 mathematics interest -.33 1.05 .11 1.05 .14 .98 mathematics utility -.26 1.11 .23 .92 -.09 .97 note. scale values were standardized to a mean of 0 and standard deviation of 1. ano statistical comparison analyses were performed using the “passed algebra” group due to unequal sample sizes; statistics for this group are provided only for reference purposes. bcases weighted by base year student weight (w1student) to adjust for oversampling of specific subgroups. cthis group was matched on covariates, therefore no filtering of the data occurred for these students. data from the second wave were examined on the same variables. the missing/non-response levels for mathematics identity, interest, and utility in the followup wave of data were 18.1%, 32.3%, and 18.8%, respectively. the non-response level for interest was higher due to the fact that interest in current mathematics course did not apply to all respondents (i.e., this item was skipped for any student who was not enrolled in any mathematics course at the time). although the missing data were found to be non-systematic in the mcar analysis previously noted, response rates under 80% fail to meet researchers’ recommended acceptance levels of ignorable missing data (sterner, 2011; vriens & melton, 2002). for such instances, imputation methods represent a more favorable approach than traditional methods such as deletion (casewise or pairwise) or mean substitution (acock, 2005). imputation allows the researcher to analyze a complete data set and prevents the loss of statistical power that results from the elimination of data. the missing values in this data set were replaced using multiple imputation. the analyses were performed us howard et al. success after failure journal of urban mathematics education vol. 8, no. 1 51 ing the original data set, but were repeated using the imputed data sets for confirmation purposes. one-way between-groups multivariate analyses of variance (manovas) were performed on both the original data (with casewise deletion) and on the full imputed data sets to examine the group differences on the psychological and motivational scale variables of interest for the second wave of data collection. for the original (non-imputed) data set, there was no statistically significant difference between the groups on the combined dependent variables: f(3, 362) = .84, p = .47; pillai’s trace = .01; partial η² = .007. summary mean statistics of the five complete data sets generated using multiple imputation also revealed no statistically significant differences in the two groups: f(3, 546) = 1.74, p = .20; pillai’s trace = .02; partial η² = .009. figures 3, 4, and 5 visually depict the changes in students’ mathematics identity, interest, and utility from the base year data collection in ninth grade to the first follow-up wave in eleventh grade using the original data set. whereas students who failed algebra i in eighth grade exhibited lower scores on psychological and motivational scale variables than those who passed a lower course in wave 1 (ninth grade), these differences were no longer present by the second wave (eleventh grade). figure 3. line plot of change in mathematics identity from base year to first follow up. howard et al. success after failure journal of urban mathematics education vol. 8, no. 1 52 figure 4. line plot of change in mathematics interest from base year to first follow up. figure 5. line plot of change in mathematics utility from base year to first follow up. howard et al. success after failure journal of urban mathematics education vol. 8, no. 1 53 college readiness data analyses data from the follow-up survey were examined to determine whether there were differences between course enrollment groups in their expected college qualification rates. the nonresponse rate for the three college readiness variables ranged from 18.2% to 19.3%. chi-square tests of independence were performed to examine the relation between eighth-grade course enrollment and reported on-track status for meeting college requirements in the second semester of eleventh grade. the relation between course enrollment and meeting requirements for a 2-year college was not statistically significant: 2(1, n = 548) = .01, p = .93, phi = -.01. the relation between course enrollment and meeting requirements for a typical 4-year college was statistically significant: 2(1, n = 548) = 7.52, p < .01, phi = -.12. students who failed eighth-grade algebra were less likely to report being on track for meeting the requirements to attend a typical 4-year college by the end of their senior year than students who passed a lower-level, eighth-grade course. the relation between course enrollment and meeting requirements for a selective 4-year college was not statistically significant: 2(1, n = 548) = 1.2, p = .27, phi = .06, although relatively few students from either group are represented in this category. table 3 displays the percentages (by college type) of each group that expected to meet college requirements by the end of their senior year. the same statistics are provided for students who passed algebra i in the eighth grade, as well as for the entire hsls:09 sample. table 3 percentage of students who will meet college requirements group 2-year college typical 4-year selective 4-year failed algebra i 70.9 44.8 9.1 passed lower mathematics course 77.7 57.3 5.1 passed algebra i 90.2 80.7 16.8 entire hsls:09 samplea 79.8 65.4 13.0 note. cases weighted by first follow-up student weight (w2student) to adjust for oversampling of specific subgroups. athis group was matched on covariates; therefore, no filtering of the data occurred for these students. discussion and implications the hsls:09 data show that students who failed algebra i in eighth grade did not score significantly differently in ninth grade on a test of algebraic reasoning from students who passed a lower-level mathematics course in the eighth grade, howard et al. success after failure journal of urban mathematics education vol. 8, no. 1 54 when matched on several covariates. when compared to the matched sample, the students who failed algebra i in the eighth grade did not demonstrate any immediate proficiency benefits from their exposure to algebra i content. this finding is consistent with previous studies’ findings that enrollment in college preparatory mathematics courses did not impact academic test outcomes (e.g., allensworth et al., 2009; loveless, 2009). some prior research suggests that residual benefits from exposure to algebra content may not become evident until after subsequent algebra course-taking has occurred. for example, gamoran and hannigan (2000) observed achievement growth between grades 8 and 10 from exposure to algebra, and attewell and domina (2008) found an advanced high school curriculum to be associated with higher test scores in the twelfth grade. however, the participants in this study who had been enrolled in eighth-grade algebra did not demonstrate any residual performance benefits by the eleventh grade, as they did not perform statistically differently from those who had been enrolled in a lower eighth-grade mathematics course. it should be noted that, without having access to performance data for our sample prior to the eighth grade, we could not state definitively that the groups did not differ in proficiency level prior to receiving their failing or passing grades. nonetheless, their measured proficiency rates did not differ in the ninth and eleventh grades. although the students’ exposure to eighth-grade algebra i did not affect test scores, failing the course was associated with lower levels of mathematics identity, interest, and utility in the following year as compared to students who passed the lower-level course. however, by the second half of eleventh grade when the second wave of data was collected, these two groups were no longer statistically different on these psychological measures. although it might be encouraging to attribute this outcome to resiliency on the part of the failing students, an examination of the univariate outcomes suggest that this inference is only partly accurate. whereas the mathematics identity scores for the failing students rebounded to be statistically at the same level as those who had passed a lower course, the attenuation of the gap in scores for interest and utility was more the result of declines on the part of the students who had passed the lower level course. it is important to note that it cannot be inferred that the more negative psychological scores were necessarily caused by the failing grades. it is feasible that the students failed because they already had more negative dispositions toward mathematics, or conversely that negative dispositions emerged following the failure. regardless of the direction of causality (if any), the existence of more negative dispositions towards mathematics indicates that psychological dispositions are somehow associated with mathematics performance in the eighth grade, and placing these students in advanced mathematics courses without addressing these dispositions may be inadequate in terms of the support they may need. the results also provide howard et al. success after failure journal of urban mathematics education vol. 8, no. 1 55 an argument against placing students in the same course the year following a failure without attempting to address the motivational characteristics that may have contributed to their failure. one interpretation for these findings is that the misplacement of students in eighth-grade algebra i may have detrimental psychological effects on underprepared students (loveless, 2008). this interpretation has implications for policy and practice, especially considering the fact that there does not appear to be any academic advantage gained by such placements after enduring the adverse psychological effects. rationale for placing students in courses in which they are (might be) unable to pass would need to be weighed against the possible impact of failure on future interest in mathematics, and ultimately on mathematics performance. to that point, the higher mathematics utility and interest ratings for the students enrolled in lower mathematics courses did not persist, but rather declined to a level not statistically different from the students who failed algebra i by the time they reached eleventh grade. a possible silver lining is that with respect to mathematics identity, any impact from failure does not appear to be permanent as the failing algebra i students recovered by the time the second wave occurred. on the critical issue of college readiness the results were somewhat mixed. the groups did not differ statistically in terms of being on track to meet the requirements of a 2-year college, which generally speaking, usually equates to having a high school diploma or equivalent (college board, 2012; learn.org, 2015). over 70% of students from each group reported meeting the requirements to attend a 2year college. in contrast, students who failed algebra i were significantly less likely than those who passed a lower mathematics course to be on track for meeting the requirements of a typical 4-year college by the end of their senior year. the students in this study were the highest performing among those placed in lower mathematics courses and the lowest performing of those placed in algebra i. these results are consistent with nomi and allensworth’s (2014) findings of better college readiness for students just below the cut-score for sorting into different courses, as opposed to those just above it. the results of this study are also consistent with the gatekeeper notion that exiting eighth grade without passing algebra, whether due to a failing grade or nonenrollment in the course, negatively impacts one’s chances for meeting the requirements for admission into college (atanda, 1999; spielhagan, 2006). the higher college readiness rates for students who passed a lower-level course would seem to provide some support for putting underprepared eighth-graders on a slower path (liang et al., 2012; loveless, 2008), except for the fact that in both cases they fell far short in college readiness of both the national average and the students who passed algebra i. whereas passing the lower course provides better odds than failing algebra i, it seems to represent a band-aid solution for a much larger issue— howard et al. success after failure journal of urban mathematics education vol. 8, no. 1 56 inadequate preparation for the required intellectual demands of algebra during these students’ k–7 experiences. it apparently does matter whether algebra is learned in the eighth grade or later. moreover, to engage in practices or enact policies that delay the acquisition of content that predicts college readiness is to, in effect, build inequality into the educational structure. in urban schools with large populations of african american, latina/o, and lower ses students, perceptions about ability may lead well-intentioned policymakers to give students the “gift of time” by delaying their entry into algebra, but the end result of such a gift may be a life trajectory absent of higher education and all the benefits that accompany it. the results on the psychological and motivational scales do not appear to support the idea of placing some students on a slower track as a long-term solution either. the differences between the groups on these measures all but dissipated by the time they reached eleventh grade, and neither group approached the college readiness rate that students passing algebra i attained. given that eighth-grade algebra i success equates to significantly better odds of college readiness, an approach systematically denying some students access by building slower paths into the educational structure is not supported by our findings, especially when others have found increased access to result in more students passing algebra (williams et al., 2011). the focus in this study has been placed on the algebra i students who failed, but we must keep in mind that 95% of the students taking algebra i reported passing the course, putting them among the group that reported a 4-year college readiness rate of 80%. a limitation that must be acknowledged in this study is the fact that the propensity score matching technique utilized only accounts for the selection bias on variables observed in the hsls data set, but not for any unobserved variables that may have influenced the outcomes analyzed. in addition, our focus on those students who self-reported failing their courses severely limited our sample size to a fraction of the overall study sample. as a result, the impact of failing algebra observed for this subsample on subsequent measures cannot be generalized to the vast majority of students who were not analyzed in this study. it is plausible, and even likely, that psychological measures for students who passed their courses would be impacted differently had these students experienced failure in the course, especially if course failure is not something they have encountered previously. these limitations notwithstanding, in order for policymakers to respond to critics that point to increased failure rates and the resulting psychological effects as justification for eliminating universal algebra access policies, understanding the possible impacts of these failures is critical. in addition, understanding the effect that placing struggling students in lower mathematics courses can have on college-readiness informs decisions on whether to support universal access for all students. howard et al. success after failure journal of urban mathematics education vol. 8, no. 1 57 the algebra-for-all policies that have been implemented in recent years appear to have arisen from the hope that providing access to this important course will move our educational system a little closer to horace mann’s notion of great equalizer for some of our most vulnerable students. given the results of this study and others, it would appear that the most prudent approach to attaining such a goal is a multifaceted one. a policy that moves students into a higher-level course at eighth grade would likely have a much better chance for success if it were implemented over multiple years, starting with changes in the curriculum prior to eighth grade. moreover, if policymakers find it imperative to implement such a policy at a given point in the education path (i.e., eighth grade), an indispensable component should be mandated additional support for students who do not handle the shift to higher ground with the same proficiency. in the final analysis, students in this study who were assigned to algebra i in eighth grade did not fare as well as students taking a lower mathematics course in terms of 4-year college readiness by the eleventh grade, but readiness for 2-year and selective 4-year colleges did not differ significantly. more importantly, neither group fared well in comparison to the national average nor compared to those who passed algebra i. we submit that the best solution may not be to figure out where it is best to place underprepared students, but rather to address the lack of preparedness in a systematic manner throughout students’ elementary and middle school experiences. to do otherwise would be to resign to the position that some students cannot or will not acquire prerequisite knowledge by eighth grade that has been proven to be instrumental to college access. acknowledgments this research 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(2005). student promotion/retention policies. denver, co: education commission of the states. microsoft word final hennings vol 3 no 1.doc journal of urban mathematics education july 2010, vol. 3, no. 1, pp. 19–26 ©jume. http://education.gsu.edu/jume jacqueline a. hennings is a high school mathematics teacher at woodland high school, 800 north moseley drive, stockbridge, ga 30281; e-mail: jkeuler@student.gsu.edu. her research interests include the effects of curriculum reform on teachers and how to prepare teachers to be effective with diverse student populations. public stories of mathematics educators new curriculum: frustration or realization? jacqueline a. hennings woodland high school he first day of school is rapidly approaching and the calm, stress-free life of summer is fading away. as the time draws near when voices fill the hallways signifying the start of a new school year, excitement and anxiety start to creep into the psyche. but this is not an ordinary start. this customary excitement and anxiety quickly turn into fear and pressure because this year will be different. this year is the first year the new mathematics curriculum is implemented in the ninth grade. even after the training provided by the state and school district, many teachers, including me, feel overwhelmed by the expectations the new school year and the new curriculum bring. the aforementioned is a brief description of how i felt at the start of the 2008–2009 academic year when the new, state-mandated, high school mathematics curriculum, the georgia performance standards1 (gps), was to be implemented. even after attending almost all the professional development sessions supplied by the state and school district and learning about the task-based curriculum while obtaining my master’s degree in secondary mathematics education, i still had a very uneasy feeling in my stomach. i thought the feeling would subside, but as my fellow colleagues and i tried to implement the new curriculum, we were faced with many challenges and questions. this uneasiness would turn into a long trek of reevaluating my pedagogy to better understand my students and their mathematical experiences as well as combating traditional norms set by society with 1 the mathematics curriculum mandated by the georgia performance standards is an integrated curriculum in which the content, knowledge, and skills introduced in the traditional courses of algebra, geometry, trigonometry, and data analysis and statistics are developed throughout an integrated mathematics sequence of courses: math i, math ii, math iii, and math iv. each course in the sequence uses mathematical tasks to model “real-world” scenarios in an attempt to connect mathematics to students’ lives; the tasks increase in mathematical complexity as students progress through the course sequence. on the whole, the curriculum stresses student-centered, collaborative groupwork where the teacher is a facilitator in the mathematics classroom as students, individually and collectively, work through the mathematical tasks. for complete information regarding the georgia performance standards, see www.georgiastandards.org. t hennings public stories journal of urban mathematics education vol. 3, no. 1 20 with respect to the teaching and learning of mathematics. here, in this public story, i describe some of the frustrations and realizations i have come across while implementing the georgia performance standards high school mathematics curriculum. reflecting on teaching the fear and anxiety subsided as i began teaching the new curriculum, but the challenges of change could be heard loud and clear. reflection became my mantra because it was something i had to do constantly under the weight of expectations from the state and school district. going from a “traditional” mathematics classroom where the teacher was the presenter of knowledge (hiebert, 2003) to a standards-based, student-centered classroom was exciting—and scary. given that i had to teach differently, i developed new strategies to use in my classroom. using the implications of existentialism, i came to the realization that, to change my pedagogy, i must change myself as a teacher (feldman, 2003). i must let go of a quiet, inactive classroom where each student is working individually and move toward a noisy, active classroom where students’ mathematical discussions and debates are occurring. mathematics exploration replaced mathematics demonstration. that is, “teaching” was no longer providing numerous demonstrations of similar mathematics problems with little, if any, mathematical understanding from the students. it has been a struggle because i was taught mathematics in the traditional way, but i have learned much from my students with respect to their mathematical abilities and frustrations in my attempts to provide a different learning environment. not every student learns mathematics the same way, so it is important to diversify teaching strategies to promote understanding and yearning for knowledge. as i engaged in a self-study or self-reflection, the importance of aligning my pedagogical philosophy with actions (i.e., teaching strategies) in my classroom became apparent (loughran, 2007). when one becomes more familiar with oneself and develops a multifaceted philosophy of teaching or pedagogy, one is better able to develop an understanding of others (ellis & bochner, 2000). in the discussion that follows, i highlight aspects of my attempt to reflect upon my developing pedagogy as well as the growth i have encountered while teaching the gps curriculum. learning to adapt strategies to meet the needs of my students, as well as myself, has been an eye-opening experience. frustration: combating traditional norms it has been a long, hard struggle over the past 2 years to understand the framework of the gps curriculum and to implement it in its intended, non hennings public stories journal of urban mathematics education vol. 3, no. 1 21 traditional way. how can teachers be successful in changing their pedagogical style in a classroom where a standards-based approach is expected when there is so much resistance from students, parents, and even teachers? this resistance has been the most frustrating part of implementing the gps curriculum and subsequent pedagogical strategies. all students have the right to learn mathematics, and it is the teachers’ job to differentiate their strategies in acknowledgement of this right. promoting student voice and democracy (dewey, 1937/1987) regarding student learning in our schools can open doors to possibilities for all children. the educational system in the united states, i fear, has gotten away from promoting democracy in the schools by forcing students to become robots who regurgitate material for the teacher and perform on standardized tests. i believe that the gps curriculum, however, is trying to give each student a voice in the mathematics classroom. as the gps and subsequent classroom practices are an unfamiliar approach to mathematics education, i have to constantly inform parents about the changing curriculum and explain the reasons behind the change (kilpatrick, 1992). change is difficult for everyone to endure, but it is necessary to provide a more thoughtprovoking curriculum and create better problem-solvers to fill the needs in our society (brownell, 1947/2004). in the 2 years that i have taught the gps curriculum, i have found that many parents and some teachers were not informed of the motives behind the new curriculum; thus, it has been a struggle to adjust their way of thinking about how mathematics might be taught and learned in schools. parents, and too many teachers, are often so consumed by the traditional style of teacher lecture followed by skills practice that they do not see the potential for students to become great thinkers of mathematics while coming to a deeper understanding of the discipline (sfard, 2003) through a more active, engaged approach to mathematics teaching and learning. we have to transfer the responsibility for learning from the teacher to the student because they are creators of their own knowledge (steffe & kieren, 1994/2004). all teachers can really do is plant the seed and watch it grow. i am passionate about allowing students to carve out their own space to develop their mathematical ideas and come to a deeper understanding of the discipline. but, i am not perfect. i still fall into the rut of lecturing and giving examples even when i see students “spacing out” and not paying attention. i know firsthand the struggles that a teacher must endure to let students work on their own while, at the same time, making sure they are being productive. it takes great effort to find the balance of student-centered learning with 30 or more students each bringing their own experience to the table (lerman, 2001). i know parents, students, and fellow teachers sense my enthusiasm when i discuss the gps curriculum, but they still want mathematics to be taught the traditional way because that is how they learned it in schools. where are the text hennings public stories journal of urban mathematics education vol. 3, no. 1 22 books? where are the examples? where are the practice problems? are just some of the persistent questions and reminders to me that changing teaching strategies will be a long, difficult journey (hiebert, 2003). i believe, however, that parents and teachers can all come together to look past the obstacles to promote mathematical and democratic growth for all students. realization: changing teaching strategies in the past, i prided myself in finding ways to connect mathematics to my students’ daily lives and get them excited about mathematics. one way was to develop “catchy phrases” for my students to remember certain mathematical properties. for instance, regarding the property of negative exponents, i taught my students to “drop it like it’s hot.” this phrase followed students to their next mathematics class but, unfortunately, most did not understand the mathematics behind the phrase. similarly, i believe the gps curriculum attempts to connect mathematics to the “real world,” but i do not think it effectively connects to the students’ real world. that is to say, the gps curriculum, i believe, unintentionally makes the same mistake that i have made over the past 7 years in my teaching: thinking that my world is the same as the students’ world. although promising, the gps curriculum has still left me with troubling questions: why do students need to learn the topics “covered” in the curriculum? where do basic life skills such as balancing a checkbook, saving, and purchasing a home come into play? where does critical mathematics literacy (gutstein, 2006) fit in? i know that i cannot change the eurocentric perspective which has dominated mathematical discourse for the past 200 years (ernest, 2009), but how can i change my teaching strategies to encourage my students to become critical, independent, life-long learners interested in mathematics while being forced to perform on standardized tests? one important aspect of answering this question for me has been to gain students’ trust by connecting with them on a human level (gutstein, 2006). i now open up to my students and let them know that i am more than just their teacher; i have my own struggles and triumphs and do not just sit at home grading papers. making visible my human side has been important for my students to see, as they begin to relate to me more and develop a level of respect for me both inside and outside the classroom. this visible human side has assisted in the development of a reputation of respect and caring amongst students and parents—i show them respect and let them know that i care. nevertheless, it takes more than respect and caring to create a positive atmosphere where students want to learn mathematics and excel to their utmost potential; it also takes a curriculum and a teaching atmosphere that intrigues the students and makes them want to learn more. hennings public stories journal of urban mathematics education vol. 3, no. 1 23 over the past 7 years of teaching, 2 of which focused on implementing the gps curriculum, i have found that students “feel that mathematics is cold, hard, uncaring, impersonal, rule-driven, fixed and stereotypically masculine” (ernest, 2004, linking philosophies of mathematics and mathematical practice, ¶ 12)— not to mention, useless in today’s society. despite my good rapport with students and my endless efforts to make mathematics “fun,” i often miss the most important aspect of good mathematics teaching: utility. i have struggled only to find that utility is absent in the gps curriculum even with its real-world tasks because those tasks are not part of the students’ world. they are interested in technology, sports, social activities, and so forth, not about “paula and her peaches” or “pete’s parking deck dilemma” (two of the gps mathematical tasks). i have tried to adapt my teaching style along with the new curriculum to help students see the power mathematics has to offer, but i usually fall short of gaining their enthusiasm because the material does not intrigue them. in the future, i plan to adjust the tasks to scenarios the students might have an interest in, but for now, i feel that i need to go through the 4-year curriculum once to have an overview of what is expected of my students and me. nevertheless, i have changed the way i introduce the material over the past 2 years through reflection on my teaching, my students’ learning, and the curriculum in general. it is always a new day in my class, even though there are routines. i require explanations on all assignments and assessments in order for students to develop their mathematical understanding and communication. depending upon the content to be taught, i shift between teacher centered and student centered, group discussion and class discussion, individual assessment and group assessment. i try to balance teaching the intended content and getting my students engaged in their own learning, which is not easy. i stress to my students that we are going to try multiple ways to get the knowledge and skills across and that their input is invaluable. believe me, they are not hesitant to let me know if they do not like doing something a certain way. with these changes comes more student confidence (frankenstein, 2005). they feel comfortable talking about ideas in class and discussing different problem-solving strategies. i believe they are developing a deeper understanding of the mathematics behind the tasks and are able to make connections amongst some concepts. my students of the past 2 years are the first group to go through the gps curriculum, and the process has been a slow one, probably too slow for their parents. parents are frustrated with the lack of multiple homework problems every night along with the absence of textbooks containing examples that explain procedures. parents often feel helpless in assisting their child because the structure of the curriculum is so different from the one they knew. i am hopeful that, over time, the frustration will subside and the mathematical abilities of the students will soar. hennings public stories journal of urban mathematics education vol. 3, no. 1 24 indeed through time, i have found using multiple strategies of teaching and assessing not only allows students freedom to express themselves but also various ways to do so. parents, too, seem to value the idea of collaborative groupwork as long as their child’s grade is not affected by someone else’s performance, or lack thereof. overall, parents and students appear to understand the importance of collaboration in gaining an understanding of mathematics. collaboration has also been a tool to address the different learning styles of the students in my classroom and has been a positive component of the gps curriculum overall. realization: growth of students i have decided to go through all 4 years of the gps curriculum to see how the state has sequenced the mathematics content. my students understand that i am in the trenches and will experience the new curriculum with them. (i am attempting to loop through all 4 years of the curriculum with the same students.) over the past 2 years, the changes i have witnessed in my students’ reaction toward mathematics have been encouraging. at the beginning of their freshman year, they were typical rowdy teenagers who wanted to be anywhere other than a mathematics classroom. they complained about doing tasks and constantly mimicked the sentiments of their parents that this “new way” of doing mathematics did not make sense. they gave up frequently and it was hard to keep them focused and excited. i grew extremely frustrated during that first year, and my discussions during lunch in the faculty workroom were very negative. i found myself putting so much time and effort into something that i was not getting much out of with respect to student learning and motivation. i complained about the students and their lack of mathematical ability. i found comfort in my colleagues because they were experiencing the same lack of motivation, basic skills, and work ethic from students that i was. as hard as it was that first year (i.e., math i), i did see change. toward the end of their freshman year, many students stopped complaining (all the time) about working tasks. they began to communicate their thought processes and ask thoughtful questions. students started to put the pieces together and inquired about more advanced mathematical ideas. i was proud of my freshmen when they argued for a particular approach to solve a problem, only to determine that there were multiple approaches. they stopped fighting me when i placed them in groups, and they learned how to collaborate with their peers instead of always asking me to do the mathematics. during my second year, the students’ sophomore year (i.e., math ii), they knew what to expect from me at the very beginning. we covered some difficult topics and made connections to their previous learning in math i. i found that they reminded each other to explain their reasoning. they spent more time on tasks hennings public stories journal of urban mathematics education vol. 3, no. 1 25 instead of giving up as soon as it was given to them. all in all, students took on challenges with more ease. now this change could be because they were a year older and a little wiser regarding the new curriculum, but, even if this was the case, they got away from the traditional “drill and kill” notion that has plagued mathematics for too long. it was exciting to hear them work on a mathematical task together and use the language of mathematics in their discussions. it was also interesting to see how students were learning to negotiate their positions and perspectives during collaborative groupwork while still keeping their beliefs intact. maintaining or breaking the status quo it has been a difficult road to conquer in trying to implement a new method of mathematics teaching and learning with all the resistance from students, parents, and teachers, but i have found it rewarding for student growth, confidence, and understanding. i have invested much time and effort into this new method, and i believe my students are responding well because they are able to tackle more advanced problems and ideas for longer periods of time. in the classroom, “if students are not able to transform their lived experiences into knowledge and to use the already acquired knowledge as a process to unveil new knowledge, they will never be able to participate rigorously in a dialogue as a process of learning and knowing” (macedo, 2000, p. 19). even though currently the tasks do not address the students’ interests per se, i believe i can adapt them in the future to garner students’ curiosity while teaching the intended mathematics content. i have had to learn to negotiate amongst the varying views of students and parents as well as combat the traditional stance taken by many of my colleagues. nonetheless, i labor to negotiate the gps curriculum because i believe education can either persist in a cyclical pattern of maintaining the status quo or it can break the pattern and transform the world (shaull, 1970/2000). it is up to teachers, students, parents, and other stakeholders to decide how the future will look with respect to mathematics education. i feel this new approach to mathematics will give the students a voice and a better understanding of the world in which they live. it will take some time to develop, but in the long run, i hope and trust, students will become active in their communities. they will not conform to the traditional views of education that left so many behind; 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(1992). a history of research in mathematics education. in d. a. grouws (ed.), handbook of research on mathematics teaching and learning (pp. 3–38). new york: macmilliam. lerman, s. (2001). a cultural/discursive psychology for mathematics teaching and learning. in b. atweh, h. forgasz, & b. nebres (eds.), sociocultural research on mathematics education: an international perspective (pp. 3–17). mahwah, nj: erlbaum. loughran, j. (2007). researching teacher education practices: responding to the challenges, demands, and expectations of self-study. journal of teacher education, 58, 12–20. macedo, d. (2000). introduction to the anniversary edition. in p. freire, pedagogy of the oppressed (pp. 11–28). new york: continnuum. sfard, a. (2003). balancing the unbalanceable: the nctm standards in light of theories of learning mathematics. in j. kilpatrick, w. g. martin, & d. schifter (eds.), a research companion to principles and standards for school mathematics (pp. 353–392). reston, va: the national council of teachers of mathematics. shaull, r. (2000). forward. in p. freire, pedagogy of the oppressed (pp. 29–34). new york: continnuum. (original work published 1970) steffe, l., & kieren, t. (2004). radical constructivism and mathematics education. in t. p. carpenter, j. a. dossey, & j. l. koehler (eds.), classics in mathematics education research (pp. 68–82). reston, va: the national council of teachers of mathematics. (original work published 1994) 404 not found microsoft word 8 final sheldon vol 7 no 2.doc journal of urban mathematics education december 2014, vol. 7, no. 2, pp. 97–103 ©jume. http://education.gsu.edu/jume     james sheldon is a sally cassanova predoctoral scholar conducting research on complex instruction and discourse at the university of arizona, 1430 e. second street, tucson, az 85721, email: jsheldon@email.arizona.edu. his research draws upon queer theory and disability studies to explore how students are constructed as low-achieving and how teachers can move from looking at individual deficits to models of curriculum and pedagogy based on complex instruction. he is a special education mathematics teacher at the california virtual academies, a virtual title i public charter school.   book review transforming from the bottom up: a book review of mathematics for equity: a framework for successful practice1 james sheldon the university of arizona athematics for equity: a framework for successful practice is the latest edited volume in james banks’s multicultural education series published by teachers college press. mathematics for equity is a collection of 14 chapters, 4 are revisions of previously published work and 10 are new chapters, written specifically for this volume. collectively, the 14 chapters tell the story of railside high school, a school where mathematics teachers successfully transformed their mathematics department to create classrooms in which all students could successfully tackle challenging mathematics. the book is divided into four parts: part i: the railside approach, part ii: student experiences at railside, part iii: teacher learning and professional community, and part iv: moving on and looking forward. the book, in many ways, offers a story of inspiration and hope, grounded in research, for secondary mathematics teachers who wish to transform their own mathematics departments. railside high school’s journey into mathematics reform railside high school is a school that has taken on an almost mythic status within mathematics reform and has even become a target for those who seek to discredit such reform (see boaler, 2012 for a discussion about professional and personal attacks regarding railside high school). jo boaler (see boaler and staples, 2008, 2014) named this specific school railside because it was a school that was on the “wrong side of the tracks”: low-income students of color lived on one                                                                                                                           1nasir, n. s., cabana, c., shreve, b., woodbury, e, & louie, n. (eds.). (2014). mathematics for equity: a framework for successful practice. new york, ny: teachers college press. pp. 288, $44.95 (paper), isbn 0807755419 http://store.tcpress.com/0807755419.shtml m sheldon book review   journal of urban mathematics education vol. 7, no. 2 98 side of the railroad tracks while privileged families lived on the other. in the mid1980s, during an otherwise routine accreditation process, students at railside expressed to the visiting team of accreditors their frustrations that “they were not learning mathematics” (tsu, lotan, & cossey, 2014, p. 129). the accreditors’ final report, however, did not make any specific recommendations for improving mathematics instruction, leaving the mathematics department to develop its own plan to redesign mathematics teaching and learning at railside. teachers in the department began working with ruth cossey, who, at the time, was studying complex instruction (ci) while beginning her doctoral studies at stanford university. ruth introduced the railside mathematics teachers to ci, and it became the framework for curriculum and instructional redesign. ci is an instructional approach developed by elizabeth cohen, rachel lotan, and colleagues at stanford university.2 as a sociologist, cohen was particularly interested in the ways in which classrooms tended to divide students into status hierarchies based on their perceived competence. cohen noticed that in classrooms, teachers’ and students’ perceptions were that some students are competent at mathematics and others are not. moreover, competence tended to be ascribed to students who fell into socially privileged categories, such as whiteness and/or maleness. these perceptions of competence often led to self-fulfilling prophecies: those who are perceived as competent demonstrate mathematical competence; those who are perceived as lacking competence struggle mathematically. cohen proposed that teachers address this issue of status by placing students in groups to work on what lotan (2003) later named “group-worthy tasks.”3 groupworthy tasks facilitate students’ interdependence by foregrounding multiple abilities and multiple representations, requiring students to work together in solving complex mathematical problems. these tasks involve sufficient interdependence and challenge; even those students who are perceived as “advanced learners”4 often experience difficulty completing the tasks on their own. thus, students in a classroom based on ci are compelled to work together in productive ways to complete tasks. in the end, ci became particularly significant in the mathematics reform work that transpired at railside high school.                                                                                                                           2 see http://cgi.stanford.edu/group/pci/cgi-bin/site.cgi. 3 lotan (2003) used a hyphen between group and worthy, but this volume follows the current convention of eliminating the hyphen, thereby spelling groupworthy as one word. 4 by advanced learner, i am referring more to the process by which the teacher, the student, and classmates attribute mathematical competence to the learner than to actual mathematical competence. sheldon book review   journal of urban mathematics education vol. 7, no. 2 99 uc berkeley researchers team up with railside teachers mathematics for equity comes out of a study that university of california, berkeley researchers na’ilah suad nasir and nicole louie began in 2009. nasir and louie teamed up with three railside mathematics teacher leaders (carlos cabana, barbara shreve, and estelle woodbury) to study how the mathematics department at railside was changing as the district “derailed railside”5 by doing away with block scheduling, re-tracking classes, and terminating mathematics teachers. these changes became too much for these teacher leaders to deal with and they left railside at the end of the 2009–2010 school year. however, they continued to work closely with nasir and louie and the group collaborated to write mathematics for equity. connecting with other researchers and railside alumni, they constructed an anthology that explores the salient features of the railside approach and shows how ci can be implemented department wide to create equity-based mathematics classrooms in which all students are positioned as mathematically competent. the book examines how the teachers in the mathematics department took the initiative to collaborate in order to create classrooms where all students were viewed as possessing intellectual and academic competence in a context where other academic departments in the school were not fully engaged in making similar changes. nasir, cabana, shreve, woodbury, and louie (2014) created a volume that expands earlier work by boaler and staples (2008), horn (2005), and little and horn (2007); revised versions of these articles on railside are included in the book and provide context and historical framing. rather than merely compiling existing research, the authors built upon the prior work in three chapters they wrote specifically for the anthology: “working towards an equity pedagogy,” “building and sustaining professional community for teacher learning,” and “derailed at railside.” the first original chapter offers a vignette from a railside classroom as an overview of the core principles of the railside approach to mathematics. the second chapter discusses how the department built a professional community and asserted its own autonomy by taking responsibility for things that ordinarily might be within administrators’ purview (e.g., hiring teachers and newteacher induction). and the third original chapter tells the story of how top-down changes at railside derailed the use of ci and led to 4 out of 10 teachers in the mathematics department resigning by the end of the 2009–2010 school year. other perspectives from former railside students and teachers were incorporated into the anthology in part ii. railside alumnus maria velazquez worked with nicole louie to conduct a focus group with former students to examine how mathematics classes led students to recognize that everyone possessed compe                                                                                                                           5 the metaphor of “derailing” railside features prominently in the collective narrative of the anthogloy, particularly in chapter 12, which is titled “derailed at railside.” sheldon book review   journal of urban mathematics education vol. 7, no. 2 100 tence in mathematics (velazquez & louie, 2014). victoria hand, who studied with boaler at stanford, contributed a chapter exploring two case studies of students actively shaping their own environment within the mathematics classroom (hand, 2014). lisa jilk, a former railside teacher, contributed two chapters. one is entitled “everyone can be somebody” and features a case study of a student who, although possessing racial, cultural, and gender identities, insisted that her key identity was being “somebody” who can contribute to the world and how this “somebody” identity was manifested in her experiences in a railside mathematics classroom (jilk, 2014). together, these chapters complement existing research on railside by demonstrating how railside teachers’ use of ci in their mathematics classrooms gave students mathematical agency. is equity always about race? at first glance, a book titled mathematics for equity sounds like one that would be primarily about race or culture, especially given that the volume is part of james banks’s multicultural education book series. these two constructs, however, do not even appear in the index. the contributing authors throughout the volume use different language around the issues of equity. for example, instead of speaking about race, the authors speak about students “taking up their space” and “being somebody”; these phrases are different ways of speaking about race and culture that were derived from the authors’ experiences at railside. despite the lack of fleshing out concepts like race or culture specifically, a close reading reveals that railside teachers did indeed engage in culturally sensitive instructional practices. furthermore, although the authors suggest that railside teachers did not organize their work around race, the fact that they created classrooms where everyone was viewed as smart challenged the racialized hierarchies of perceived mathematical competence too often found in classrooms and in society at large. railside alumni confirm that in their mathematics classes (unlike their other classes) their fellow students perceived everyone to be intellectually competent. is it a book for practitioners? the back cover of mathematics for equity sells the book as “invaluable reading for teachers, schools, and districts” interested in pursuing a focus on equity in their mathematics programs. the book offers useful suggestions and resources for practitioners, particularly those working in “urban” mathematics classrooms by demonstrating how railside mathematics teachers used groupworthy tasks in ways that valued what all students brought to the table both culturally and mathematically. as a high school mathematics teacher, i refer to the book frequently, both as a source of inspiration and as a guidepost as i refine my own sheldon book review   journal of urban mathematics education vol. 7, no. 2 101 teaching practice and interact with my colleagues as we confront issues of equity in our school. nevertheless, i do want to caution those looking for a “how-to” guide about ci. although the book offers many useful resources for practitioners, it is a collection of research studies, not a how-to book. mathematics educators, administrators, and other educational leaders looking to immediately start using these techniques in mathematics classrooms might be better served reading a practical guidebook like featherstone and colleagues’ (2011) smarter together! or cohen and lotan’s (2014) designing groupwork—both provide a more nuts-and-bolts approach. once teachers have advanced to using ci in their classrooms, mathematics for equity offers a useful resource for fine-tuning their instructional practices and for adjusting ci to specifically meet the needs of their students. mathematics for equity demonstrates how to use ci in a way that takes into account multiple identities without subsuming them under a singular category such as race. it offers both a vision and a pragmatic agenda for those who believe that groupworthy tasks are not just for those with privilege in society and that every student, regardless of their background, is able to tackle and should be provided access to challenging mathematics. what this book offers to urban practitioners that other books do not is a sense of what ci “looks like” beyond an individual classroom. mathematics for equity paints a picture of what these instructional practices look like when they are implemented in an entire department, and discusses how railside teachers built structures of collaboration and support as they made these changes in their department. rather than simply making their classroom tasks groupworthy, the teachers learned how to make their own departmental conversations groupworthy. they worked together to solve problems of mathematics instruction, structure, and curriculum. this collaboration required interdependence and intentionality in combining their multiple abilities in department meetings. this collaboration paralleled what they expected their students to do in the context of their mathematics classes. applying ci to the problems that a department encounters is something unique and intriguing about the railside approach. the chapters in part iii on teacher learning and professional community offer critical insights to practitioners about how this groupworthy decision-making process played out in the mathematics department at railside. an inspiration for bottom-up change i came to this book from a fairly unique perspective; i teach pre-algebra and algebra to special education students. seeing the perpetual struggles my students have had with mathematics and how many of them were failing to succeed, i real sheldon book review   journal of urban mathematics education vol. 7, no. 2 102 ized something needed to change. approximately four years ago, i started to look at how special education students were being denied opportunities to critically engage with mathematics problems and to learn how to develop their own solutions. i studied the literature related to mathematics reform and special education and had a vague sense that what we needed was more “problem solving” for students with dis/abilities, but i was not sure how to go about it. mathematics for equity’s vision of a mathematics curriculum in which all students can find their own entry points into problems has inspired me as i have worked to find ways to focus on my students’ strengths rather than on their dis/abilities. using this book as a guide, i have begun to transform my own teaching and to incorporate groupworthy tasks into my instruction. although this has not always gone as smoothly as i might have liked, my students have risen to the challenge and worked to tackle the problems i give them in their groups. inspired by the railside teachers’ assertions of their own departmental autonomy, i have worked with other teachers in my department to make similar bottom-up changes within our school. together, teachers at my high school have made strides in moving the discourse from a narrative of “here’s what administrators would like us to do to work toward equity” to a narrative of “here’s what we would like to do to work toward equity.” what i take away from mathematics for equity, therefore, is a continued faith in the ability of teachers to create bottom-up change. there is no one-sizefits-all solution to reforming a mathematics department. these changes at railside were not the result of someone telling teachers what to do but rather railside teachers had to confront the challenge and invent a mathematics department from the ground up. if we are to create (urban) mathematics classrooms and (urban) schools that truly enable all students to develop mathematical competencies, it will have to come from the bottom-up. using this book as a guide, mathematics educators will need to accept the challenge and engage in perhaps the most groupworthy task of all: working within their departments to design mathematics curricula and instructional practices that enable all of their students to succeed mathematically. references boaler, j. (2012). jo boaler reveals attacks by milgram and bishop: when academic disagreement becomes harassment and persecution. retrieved from http://www.stanford.edu/~joboaler/ boaler, j., & staples, m. (2008). creating mathematical futures through an equitable teaching approach: the case of railside school. teachers’ college record, 110(3), 608–645. boaler, j., & staples, m. (2014). creating mathematical futures through an equitable teaching approach: the case of railside school. in n. s. nasir, c. cabana, b. shreve, e. woodbury, & n. louie (eds.), mathematics for equity: a framework for successful practice (pp. 11– 34). new york, ny: teachers college press. cohen, e., & lotan, r. (2014). designing groupwork: strategies for the heterogeneous classroom, (3rd ed.). new york, ny: teachers college press. sheldon book review   journal of urban mathematics education vol. 7, no. 2 103 featherstone, h., crespo, s., jilk, l. m., oslund, j. a., parks, a. n., & wood, m. b. (2011). smarter together! collaboration and equity in the elementary math classroom. reston, va: national council of teachers of mathematics. hand, v. (2014). “talking up our space”: becoming competent learners in mathematics classrooms. in n. s. nasir, c. cabana, b. shreve, e. woodbury, & n. louie (eds.), mathematics for equity: a framework for successful practice (pp. 91–106). new york, ny: teachers college press. horn, i. s. (2005). learning on the job: a situated account of teacher learning in high school mathematics departments. cognition & instruction, 23(2), 207–236. jilk, m. l. (2014). “everybody can be somebody”: expanding and valorizing secondary school mathematics practices to support engagement and success. in n. s. nasir, c. cabana, b. shreve, e. woodbury, & n. louie (eds.), mathematics for equity: a framework for successful practice (pp. 107–125). new york, ny: teachers college press. little, j. w., & horn, i. s. (2007). resources for professional learning in talk about teaching. in l. stoll & k. s. louis. (eds.), professional learning communities: divergence, detail and difficulties. london, united kingdom: open university press. lotan, r. a. (2003). group-worthy tasks. educational leadership, 60(6) 72–75. nasir, n. s., cabana, c., shreve, b., woodbury, e., & louie, n. (eds.). (2014). mathematics for equity: a framework for successful practice. new york, ny: teachers college press. tsu, r., l., lotan, r. a. & cossey, r. (2014). building a vision for equitable learning. in n. s. nasir, c. cabana, b. shreve, e. woodbury, & n. louie (eds.), mathematics for equity: a framework for successful practice (pp. 129–144). new york, ny: teachers college press. velazquez, m. d., & louie, n. (2014). what you can’t learn from a book: alumni perspectives on railside mathematics. in n. s. nasir, c. cabana, b. shreve, e. woodbury, & n. louie (eds.), mathematics for equity: a framework for successful practice (pp. 77–89). new york, ny: teachers college press.   journal of urban mathematics education december 2015, vol. 8, no. 2, pp. 119–126 ©jume. http://education.gsu.edu/jume jessica hopson burbach is a mathematics teacher at an alternative school, portland youthbuilders, and a doctoral student in the educational leadership program at portland state university; email: jhopson@pdx.edu. her research interests include culturally responsive and social justice mathematics teaching, the experiences of youth who are pushed out of school, and youth participatory action research. book review self-empowering urban students and teachers: a book review of math is a verb: activities and lessons from cultures around the world1 jessica hopson burbach portland state university inally, there is a practical book providing specific lesson plans concerning teaching mathematics in a culturally inclusive manner! barta, eglash, and barkley’s (2014) math is a verb: activities and lessons from cultures around the world presents eleven lessons that demonstrate how mathematics influences culture and how culture influences mathematical practices. each of the eleven chapters is focused on a specific location and cultural context and outlines three lesson plans based on different grade bands: grades k–3, 4–8, and 9–12. every lesson includes the objectives, materials, and content standards aligned to the common core state standards (ccss) and the national council for teachers of mathematics (nctm) standards (national governors association center for best practices & council of chief state school officers, 2010; nctm, 2000). within each lesson, students learn about the cultural context; explore the cultural context using a variety of mathematical skills; and create a product, including designs, patterns, data tables, algorithms, and mathematical models. at the end of every lesson, there are ideas for further application and extension and discussion questions that serve to formatively assess student learning, pushing students to think of mathematics beyond what is traditionally taught in u.s. classrooms. similar to the work of other mathematics education researchers (e.g., aguirre, mayfield-ingram, & martin, 2013; aguirre & zavala, 2013; gutiérrez, 2009; gutstein, 2006; martin, 2000), barta and colleagues provide concrete examples of how teachers and students can challenge traditional school mathematics by asking three important questions: (1) what is mathematics? (2) who does mathematics? (3) for what purpose is mathematics done? 1 barta, j., eglash, r., & barkley, c. (2014). math is a verb: activities and lessons from cultures around the world. reston, va: the national council of teachers of mathematics. 163 pp., $29.85 (paper), isbn 9780873537070 http://www.nctm.org/store/products/math-is-a-verb-activities-and-lessons-from-cultures-around-the-world/ f http://education.gsu.edu/jume mailto:jhopson@pdx.edu http://www.nctm.org/store/products/math-is-a-verb--activities-and-lessons-from-cultures-around-the-world/ http://www.nctm.org/store/products/math-is-a-verb--activities-and-lessons-from-cultures-around-the-world/ burbach book review journal of urban mathematics education vol. 8, no. 2 120 what is mathematics? traditionally, mathematics is viewed as culturally neutral and “objective”; math is a verb represents a direct challenge to this perspective. the lessons presented in the book encourage teachers to approach mathematics with their students from an ethnomathematical lens. ethnomathematics studies how different cultures have various ways of thinking, learning, and doing mathematics (d’ambrosio, 1997). barta and colleagues (2014) suggest that mathematics is not fixed or rigid, but rather that it “is a process shaped and influenced by its use and by the culture of those using it” (p. v). in other words, mathematics is influenced by and also influences cultures. barta and colleagues (2014) present a broader definition of mathematics that goes beyond the mathematics that is taught in school. for example, they recognize the mathematical practices in people’s everyday lives. some researchers have referred to these everyday practices as “funds of knowledge” (gonzález, moll, & amanti, 2005), which are based on the idea that all people gain knowledge through their life experiences. these researchers recognize that people’s cultural, community, and household tools and practices are valuable and valid knowledge that influence how people think and develop (gonzález et al., 2005). as an example of honoring students’ funds of knowledge, chapter 3 explores nonstandard forms of measurement as students learn about how a historic city in brazil was built using measurements based on the emperor’s own body parts. students are also encouraged to think about how their own family members use nonstandard forms of measurement. the definition of mathematics that barta and colleagues (2014) present in math is a verb is also influenced by bishop’s (1988) six universal actions in mathematics performed by people throughout the world: counting, measuring, designing, locating, explaining, and playing. for example, chapter 11 focuses on the mathematical probabilities involved in playing the potawatomi two-sided dice game. as students play the game and analyze probabilities, they are asked to think about early applications of mathematics given that gaming and gambling have been a pastime of people since the dawn of civilization. students also get to design and explain the mathematics found in graffiti in urban areas around the world in chapter 7. these lessons show students how activities they enjoy and encounter in their own communities are mathematical in nature, thereby expanding their definition of what mathematics is. within their definition of mathematics, barta and colleagues (2014) include culturally responsive mathematics, which is the process of reciprocal interaction between mathematics and culture. an example of culturally responsive mathematics is found in chapter 4 where they suggest that students analyze the mathematics in navajo beading and weaving patterns. as students learn about navajo values of burbach book review journal of urban mathematics education vol. 8, no. 2 121 beauty, harmony, and balance, they explore the geometric patterns, shapes, and frequency of different colors found in navajo beadwork. students are asked to consider how the beadwork illustrates shapes and images valued in the navajo community and whether they think that navajo craftspeople believe mathematics is beautiful. eventually, students design their own beadwork pattern. in essence, students are enacting the connection between mathematics and culture. barta and colleagues (2014) quote educator and member of the navajo nation, clayton leong, “mathematics and life are one for the navajo people…traditional arts and crafts…illustrate our constant use of mathematics shaped by who we are, the navajo!” (p. 48). thus, mathematics is influenced by culture, is alive, is changing, and is responsive to those who use it. equally evident in leong’s quote is another theme found in the book—the interconnectedness of identity and mathematics; discussed in the next section. who does mathematics? barta and colleagues (2014) posit that identity and a person’s relationship to mathematics are fundamentally important to learning mathematics. other researchers have suggested that in order to build robust mathematics identities, students must “see themselves as legitimate and powerful doers of mathematics” (aguirre et al., 2013, p. 14). mathematics educators must support students to develop strong mathematical identities by affirming their students’ racial, gendered, cultural, and academic identities while countering the negative views of these identities (aguirre et al., 2013). for students to develop a positive relationship with mathematics, they need to “dance with the numbers” (barta et al., 2014, p. 4). in other words, students need to have meaningful interactions with relevant mathematical activities. for this reason, all of the lessons in math is a verb invite students to interact with the mathematics with hands-on, inquiry-based, or technology-infused activities that allow them to design, create, and model. additionally, the lessons create spaces for students with diverse backgrounds to experience the joy of mathematics as embedded in cultural practices. there are examples of lessons in which students “dance with the numbers” throughout the book. in chapter 1, students use a web-based application to study the geometry of cornrow braiding and to create their own braid design. students in grades 9–12 can use the same web-based application to plot a logarithmic spiral to fit the curvature of a cornrow braid. consequently, students see people from contemporary urban communities and african communities since 500 bce that participate in cornrow braiding as important doers of mathematics (barta et al., 2014). throughout the lessons, barta and colleagues emphasize the ways in which people from various cultures and various occupations engage in the practice of mathematics. when students integrate their cultural heritage into the mainstream pedagogy of burbach book review journal of urban mathematics education vol. 8, no. 2 122 mathematics, they are more likely to see themselves as doers and part of the world of mathematics. identity also plays an important role in the teaching of mathematics. math is a verb serves to educate teachers about the vast array of cultural mathematics practices and, hopefully, shift their own definitions of mathematics and their conceptualizations of who does mathematics. barta and colleagues (2014) assert, “we do this to encourage teachers to further investigate their own understanding of culture and its influence on how they teach an increasingly diverse student population” (p. vi). teachers’ approaches and practices are affected by their beliefs about who they believe can do mathematics. overall, the authors assist teachers in deconstructing their beliefs about mathematics and in developing pedagogical approaches that will benefit (most likely) the growing number of culturally and linguistically diverse students in their classrooms. for what purpose is mathematics done? the purpose of learning mathematics in many u.s. schools is based in the achievement of key competencies for standardized tests. instead, barta and colleagues (2014) show how mathematics can be used to investigate social issues and can be a tool for problem solving. hence, students are taught that the purpose of mathematics is more than just learning knowledge and skills needed to graduate high school. social justice advocate and consultant for math is a verb, dan lyles, believes that students need to see the purpose that mathematics plays in their own lives and in the world. he argues: we, as a culture, are obsessed with improving math education, teaching it, and we all wish we were a little better at it. however, we’re not spending the critical time talking to each other about what we intend to do with that math or what wonderful things we can create in the world. when working with kids, they’re thankfully too honest to do what they’re supposed to. if you can’t connect that math education to something that means something to them and gives them room to participate both in the process and in the world, then you’re going to lose them. (as cited in barta et al., 2014, p. 87) as a result, mathematics education environments have the potential to be spaces where students are self-empowered to use mathematics as a tool for understanding and confronting issues in their communities and in the world. there are many examples of how mathematics can be used to analyze social issues throughout math is a verb. one particular lesson in chapter 2 illustrates how students might use mathematics to problem solve about social issues in a guatemalan farming community. students learn about the corn growing practices of the maya, which are still practiced in the highlands of guatemala. farmers are constantly battling crop loss due to disease and insects. in the lesson, students create mathe burbach book review journal of urban mathematics education vol. 8, no. 2 123 matical models to determine whether farmers should continue to use their traditional planting practices or explore a new method. at the end of the lesson, students are asked, “how does using mathematics as a tool for investigation leading to social change affect the way we think about mathematics as a subject and about ourselves as students of mathematics?” (barta et al., 2014, p. 33). through this lesson, students are exposed to the idea that mathematics can be a tool for affecting pro-social change. this lesson in particular lays the groundwork for teachers to present social justice mathematics lessons where mathematics is used to reveal and respond to contemporary issues in students’ own communities. conclusion: self-empowering urban students and teachers as a secondary mathematics teacher who works at an urban alternative school, i am inspired by how math is a verb respects my students’ intellects and my own agency as a classroom teacher. although my students are what some consider high school “dropouts,” most have a story about how the comprehensive high school system pushed them out or failed to meet their needs. i hypothesize that this pushing out might be because the mathematics curriculum was neither relevant nor reflective of their everyday lives. similarly, barta and colleagues (2014) do not blame students for developing a negative relationship with mathematics. instead, they question whether there are other ways to teach and engage students with mathematics. they argue for making mathematics “a bridge to their own heritage culture” (p. 1). my urban alternative school serves a disproportionately large number of students of color and students from working class families. some researchers suggest that these are students who believe that mathematics is developed and owned by a community that they do not belong to (barta, cuch, & exton, 2012). delpit (2012) argues that one of the biggest barriers to learning for students of color in traditional school systems is that students do not see themselves or their own life experiences reflected in the curriculum, especially in mathematics. it also may devalue their home cultures and may disengage students from learning the content. as a result, it can be more difficult for students of color to build strong mathematics identities. ultimately, by presenting mathematics in culturally situated contexts, math is a verb honors the intellect and capacity for mathematical thinking of our urban students, especially those who are culturally and linguistically diverse. the lessons throughout math is a verb create spaces for urban students to tell counternarratives (solórzano & yosso, 2009) about the mathematics knowledge within their communities and their own potential for obtaining mathematical excellence. hence, students who live in urban environments are self-empowered because they see themselves and members of their community as powerful doers of mathematics. beyond the rhetoric of providing urban students greater access to mathe burbach book review journal of urban mathematics education vol. 8, no. 2 124 matics, the lessons seek to change the definitions of mathematics and mathematics education to include the knowledge and practices of urban communities. these lessons serve to strengthen urban students’ mathematics identities and beliefs in their ability to use mathematics as a tool for exploring injustices in their communities. thus, mathematics becomes a space influenced by and within the lives of urban students, instead of one to which urban students need access. in a mathematics education system of increasing standards and scripted curricula, there is a sense of distrust in the ability of teachers to design lessons that are responsive to the specific needs and backgrounds of students while, at the same time, challenging students with rigorous content. barta and colleagues (2014) assert that that is simply not true. they show how executing culturally situated mathematics lessons do not require sacrificing rigorous mathematics. for example, lessons have students use logarithmic functions to model the curvature of a cornrow braid and curves in graffiti art, create a mathematical model to explore the relationship between corn planting and production, and create an algorithm for designing cobblestone streets like those in southeastern brazil, to name a few. furthermore, the authors show that examples of advanced mathematics and mathematical applications can be found in the everyday practices of communities around the world. to achieve equity in mathematics education, teachers need to prepare students not only to play the game of mathematics education but also to change the game of mathematics education (gutiérrez, 2009). math is a verb shows teachers how to do both. to play the game of mathematics education, teachers can easily justify their use of these culturally responsive lessons by citing their alignment to specific standards as advocated by both the ccss and the nctm. on the other hand, teachers can change the game of mathematics education by using the lessons to promote a vision of mathematics that is culturally situated and privileges the knowledge and practices of urban communities and of the students themselves. math is a verb acknowledges that we, as practicing teachers, are working within a system; however, the text encourages us to challenge the mental models that influence our thinking about what is mathematics, who does mathematics, and for what purpose mathematics is done. in addition, barta and colleagues (2014) respect my agency as a teacher to design my own lessons using those in the book as a model. in fact, they express a hope that teachers “will realize that they too can create personalized math lessons specifically developed for those that they teach” (p. vi). hence, they honor the knowledge i have about my students and self-empower me to incorporate that knowledge into lessons i design. for example, i plan to create a new lesson that combines my favorite elements from each of the different grade-brand lessons in a single chapter from the book. math is a verb inspires me, and i look forward to trying these lessons with my students to see how they respond to cultural ways of doing mathematics. teachers can examine the lessons, and connect a lesson to a burbach book review journal of urban mathematics education vol. 8, no. 2 125 standard to introduce and deepen students’ understanding of the mathematical concepts within that standard. as barta and colleagues (2014) suggest, by employing an ethnomathematics perspective of how culture and mathematics influence each other in my mathematics classroom, i have begun to create lessons involving issues of social justice. using the work of social justice mathematics educator eric (rico) gutstein (see, e.g., 2013), i have developed a series of lessons in which students analyze police traffic stop data for evidence of racial profiling. the lessons are relevant, engaging, and create spaces where students can share their experiences of being racially profiled. students also begin to see how mathematics can be used as a tool for revealing and exploring social justice issues. gay (2010) states, “empowerment translates into academic competence, personal confidence, courage, and the will to act” (p. 34). when students find connections between mathematics and their everyday lives, it validates their experiences in their communities and can increase their engagement with the mathematical concepts. as a result, students can build their confidence and competence in mathematics. additionally, social justice mathematics lessons create spaces where students can enact their agency and respond to the injustices in their communities. my personal experience tells me that culturally situated and social justice oriented mathematics lessons are important for the education of urban youth; it is inspiring to see my experience validated by other mathematics educators in math is a verb. while nctm has been critiqued for ignoring culturally responsive mathematics education (martin, 2015), math is a verb represents a nod toward embracing a culturally responsive approach to teaching mathematics. it gives me hope for the future of mathematics education—a future in which urban students see themselves and their experiences reflected in the mathematics taught in schools. references aguirre, j., mayfield-ingram, k., & martin, d. (2013). the impact of identity in k–8 mathematics: rethinking equity-based practices. reston, va: national council of teachers of mathematics. aguirre, j., & zavala, m. (2013). making culturally responsive mathematics teaching explicit: a lesson analysis tool. pedagogies: an international journal, 8(2), 163–190. barta, j., cuch, m., & exton, v. n. (2012). when numbers dance for mathematics students: culturally responsive mathematics instruction for native youth. in s. t. gregory (ed.), voices of native american educators (pp. 145–166). lanham, md: lexington books. barta, j., eglash, r., & barkley, c. (2014). math is a verb: activities and lessons from cultures around the world. reston, va: national council of teachers of mathematics. bishop, a. (1988). mathematical enculturation: a cultural perspective on mathematics education. norwell, ma: kluwer. d’ambrosio, u. (1997). ethnomathematics and its place in the history and pedagogy of mathematics. in a. b. powell & m. frankenstein (eds.), ethnomathematics: challenging eurocentrism in mathematics education (pp. 13–24). albany, ny: state university new york press. burbach book review journal of urban mathematics education vol. 8, no. 2 126 delpit, l. (2012). “multiplication is for white people”: raising expectations for other people’s children. new york, ny: the new press. gay, g. (2010). culturally responsive teaching: theory, research, and practice (2nd ed.). new york, ny: teachers college press. gonzález, n., moll, l., & amanti, c. (2005). funds of knowledge: theorizing practices in households, communities, and classrooms. new york, ny: routledge. gutiérrez, r. (2009). embracing the inherent tensions in teaching mathematics from an equity stance. democracy & education, 18(3), 9–16. gutstein, e. (2006). reading and writing the world with mathematics: toward a pedagogy for social justice. new york, ny: routledge. gutstein, e. (2013). driving while black or brown: a mathematics project about racial profiling. in e. gutstein & b. peterson (eds.), rethinking mathematics: teaching for social justice by the numbers (2nd ed., pp. 16–18). milwaukee, wi: rethinking schools publication. martin, d. b. (2000). mathematics success and failure among african-american youth: the roles of sociohistorical context, community forces, school influence, and individual agency. mahwah, nj: erlbaum. martin, d. b. (2015). the collective black and principles to actions. journal of urban mathematics education, 8(1), 17–23. retrieved from http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/270/169 national council of teachers of mathematics. (2000). principles and standards for school mathematics. reston, va: national council of teachers of mathematics. national governors association center for best practices, & council of chief state school officers. (2010). common core state standards for mathematics. washington, dc: council of chief state school officers. solórzano, d. g., & yosso, t. j. (2009). critical race methodology: counter-storytelling as an analytical framework for educational research. in e. taylor, d. gillborn, & g. ladson-billings (eds.), foundations of critical race theory in education (pp. 131–147). new york, ny: routledge. http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/270/169 journal of urban mathematics education december 2012, vol. 5, no. 2, pp. 28–43 ©jume. http://education.gsu.edu/jume dante abdul-lateef tawfeeq is an associate professor in the department of mathematics and computer science at john jay college of the city university of new york (cuny), 524 west 59th street, new york, ny 10019; email: dtawfeeq@jjay.cuny.edu. his research interests include the preparedness of african/black american and latina/o students for the learning of mathematics at the collegiate level, project/inquiry based learning of calculus, and the intellectual identity of african/black american male students. paul woo dong yu is an associate professor at grand valley state university in allendale, mi; email yupaul@gvsu.edu. his research interests include lesson study as a form of professional development with pre-service teachers and issues of semiotic discourse and culturally responsive pedagogy in the mathematics classroom. public stories of mathematics educators developing a socio-cultural pragmatic mathematics methods course danté a. tawfeeq john jay college paul w. yu grand valley state university ulturally responsive pedagogies that develop students’ construction of mathematical knowledge, in conjunction with a mathematics curriculum that is student centered and promotes positive learning practices, are vital to the development of a skillful quantitatively thinking population (mathematical association of america conference board of the mathematical sciences, 2001; national council of teachers of mathematics [nctm], 2000). however, in spite of attempts by many mathematics teachers, mathematics teacher educators, and mathematics education researchers to provide all students with an “equitable,” meaningful, and high-quality mathematics learning experience (nctm, 2000), historically marginalized segments of the u.s. population continue to experience injustices in terms of learning opportunities and education resources (e.g., african american/black and latina/o students 1 and students of poverty). while the potential systemic solutions to racial and economic inequity are complex, the issues that reproduce the injustices are foundational in nature: lack of local and state funding in school districts with a high percentage of minority students (center for understanding race and education, 2009; stony brook university center for survey research, 2008), under-qualified content-specific teachers (oakes, joseph, & muir, 2001), and a general marginalization of equity issues in mathematics education (martin, 2003). in an attempt to explicitly confront these persistent inequities, a secondary mathematics education methods course was developed and taught with the explicit goal of bringing issues of race and eq 1 throughout this public story, african american and black are used interchangeably as well as brown and latina/o and/or hispanic. c tawfeeq & yu public stories journal of urban mathematics education vol. 5, no. 2 29 uity in mathematics education to the forefront. during the methods course, the pre-service teachers were exposed to literature on culturally responsive pedagogy (among other things) and provided with the opportunity to teach a culturally responsive mathematics micro-lesson to black and brown students from a “highneeds” urban high school. in this public story, 2 we—paul w. yu, the second author, and i, danté a. tawfeeq—a black mathematics education professor and the first author—discuss how i instructed a small cohort (n = 5) of white pre-service secondary mathematics teachers. this public story occurred when i was an assistant professor of mathematics education at a small liberal arts university in suburban new york. this school is one of the largest producers of mathematics teachers in metro new york city; the majority of the students in the mathematics education program are white. many of these pre-service teachers have stated that they would prefer future teaching positions in schools that most resemble those which they attended: majority white and middle class schools. despite their preference, many of the available teaching positions are in metro new york city schools that serve majority black and brown student populations. i have observed that these pre-service teachers seem to have had limited life experiences with racial and cultural diversity. furthermore, in class comments made by my pre-service teachers often reflect “stereotypes” about black and brown children and youth, and black and brown communities in general. as a black professor, one of my course goals was to expose my students to research-based literature on race, culture, and mathematics education (e.g., martin, 2003, 2009; tate, 1997) to help these students grow in their understanding of how to engage students from diverse racial backgrounds. here, we first describe the theoretical foundation for the socio-cultural pragmatic mathematics (s-cpm) methods course as well as the course background, rationale, and students (i.e., participants). we then discuss the structure and implementation of the course curriculum and field-based experience. next, we reflect on the course outcomes, and conclude with a brief discussion about both the limitations and implications of the course. 2 the data for this public story were derived from teacher-researcher reflections, two post-course interviews with the course instructor, post-course student surveys administered one semester after the course, and post-course correspondences with some of the student participants. because i, danté (i.e., the course instructor), served as a facilitator and sense-maker of the student discussion of issues regarding race, culture, and mathematics instruction, the use of teacher-researcher reflections provided insight into both the classroom discussion and the post-course follow-up survey and correspondences (ball, 2000; confrey & lachance, 2000; simon, 2000). additionally, the second author (i.e., paul) conducted two semi-structured interviews with me. the interview protocols were based on the post-course student surveys; they were transcribed and coded. tawfeeq & yu public stories journal of urban mathematics education vol. 5, no. 2 30 culturally pragmatic diversity: a theoretical framework issues related to facilitating the learning of mathematics in pre-k–12 mathematics curricula converge during pre-service mathematics methods courses (hill, rowan, & ball, 2009; shulman, 1986). one purpose of mathematics methods courses is to help pre-service teachers develop the means to facilitate the learning of mathematics. these methods courses and the programs supporting such courses vary among institutions in philosophy and structure; however, mathematics teacher education programs tend to emphasis the development of both mathematics content knowledge and pedagogical knowledge—what shulman (1986) described as pedagogical content knowledge (pck). within mathematics education, pck has been extended to include mathematical knowledge for teaching (mkt), which includes “explaining terms and concepts to students, interpreting students’ statements and solutions, judging and correcting textbook treatments of particular topics, using representations accurately in the classroom, and providing students with examples of mathematical concepts, algorithms, or proof” (hill, rowan & ball, 2005, p. 373). responding to calls to weave culturally based knowledge throughout teacher education programs (see, e.g., gay, 2000), the s-cpm course extended notions of mkt to explicitly consider knowledge of the socio-cultural factors that affect the teaching and learning of mathematics (leonard, brooks, barnesjohnson, & berry, 2010). our belief is that mathematics content courses and methods courses, when properly integrated with resources reflective of culturally diversity (e.g., culturally responsive teaching), provide pre-service teachers with a point of view that considers mkt issues within teaching experiences that reflect cultures other than their own. specifically, we framed the s-cpm course using jia’s (2007) concept of culturally pragmatic diversity, which refers to differences in values, norms, or social conventions from culture to culture and questions the tendency for people to judge or evaluate the behavior of others by their own cultural standards. these differences often shape the way one determines what is appropriate or inappropriate speech or behavior in a given situation. in the mathematics classroom, when the teacher and her or his students are from different socio-cultural backgrounds, it is an unawareness of this culturally pragmatic diversity that, we suggest, may lead to misunderstandings in the mathematics learning process. two components of cultural pragmatism framed the s-cmp course.  culturally pragmatic diversities may lead to misunderstandings and communication breakdowns in the course of classroom communication. these misunderstandings have practical consequences. tawfeeq & yu public stories journal of urban mathematics education vol. 5, no. 2 31  pragmatic conventions or norms as to what is appropriate behavior and what is not in a given mathematics classroom may either have to do with cultural values, beliefs, or with situational factors (jia, 2007). the s-cpm course was designed to provide pre-service teachers with an experience in which exposure to culturally responsive teaching was supported by a field-based experience that reflected the diverse culture of their course readings and discussions. furthermore, the s-cpm methods course content provided a socio-cultural perspective for which the pre-service teachers could consider how to navigate issues of mkt in a teaching context with students of color. this culturally pragmatic frame was used to clearly differentiate this course from more “traditional” mathematics education methods course—courses that too often present or discuss vignettes of pedagogical interest in a culturally free or neutral manner. developing and implementing the course course background the general purpose of the s-cpm methods course was to expose five (four female and one male) graduate pre-service secondary mathematics teachers to issues regarding instruction and assessment in mathematics education and, in particular, to issues regarding teaching racially and culturally diverse students in high-needs schools. this emphasis on issues pertaining to race and culture in teaching mathematics was important because all of the participants were raised in white, middle class communities and had limited experience working with racially and culturally diverse students in urban settings. however, their university required a portion of their student teaching to be completed in a “culturally diverse” school. while populations in certain metro new york schools are, at times, weighted towards a particular racial and/or ethnic group, the culturally diversity among most racial groups is broad. moreover, based on the university’s most recent data regarding teaching positions acquired by recent graduates, these preservice teachers would most likely find themselves teaching in a high-needs urban school. in essence, the s-cpm methods course was created to address the sociocultural disconnect between white and middle-class pre-service teachers’ past experiences and their current and future teaching contexts in diverse racial and cultural school settings. the course was one of only two mathematics methods courses designed for secondary mathematics education majors. this upper-undergraduate and graduate-level methods course was part of the university’s 5-year bachelor of science (bs)/master of arts (ma) teacher certification program, and was designed for students who had earned their bs in mathematics and were seeking an ma in tawfeeq & yu public stories journal of urban mathematics education vol. 5, no. 2 32 mathematics education. the five students in this course were required to take 25 field clinical hours as part of the 100 total field clinical hours required by the state of new york prior to student teaching. the 75 remaining hours were to be completed during the second mathematics methods course and other education courses. course rationale the development of the s-cpm methods course was both extrinsically and intrinsically motivated. extrinsically, the motivation for this course was as a result of the class discussion during the first session: the pre-service teachers agreed that, while they learned a great deal of “potentially” useful material in all of their educational courses, they felt that the material was not presented to them in a way that would help them teach mathematics in the classroom. furthermore, they felt they were unable to make connections between concepts investigated in their previous educational coursework and their experiences during the clinical field placements. on the first day of class, roslyn (a pseudonym, as are all participant’s names) commented: they [previous education courses] stuck me out there in my field observations [in an urban school] and i didn’t even know what to look for [in field placements in high needs school]. how can i tell what is good or bad teaching? my professor said i should look for this or i should look for that…and if i see this, then that means there is a good teacher. that makes no sense. …not worth my time. intrinsically, i, the first author, was motivated by my own racial experiences and academic upbringing. when asked directly about my motivation behind the course, i stated: “because i’m a black teacher that is concerned with black and hispanic students in urban environments…and that’s where the jobs [in urban schools] will be for these kids [college students].” to address the critical sentiment of the students toward previous education courses and my desire to expose the students to issues of culturally responsive teaching in the mathematics classroom, certain curricular and practical structures were implemented to make the connections between the methods course and other education courses meaningful to their required field-based experience. below are brief descriptions of the five students from the course and an account that illustrates the curricular structure and practical implementation of the s-cpm methods course that provides some cohesion between methods courses and field-based experiences. course participants roslyn was a 23-year-old, white female raised in a middle-class, suburban area. she was in her fifth year of the bs/ma program. demi was a 24-year-old, tawfeeq & yu public stories journal of urban mathematics education vol. 5, no. 2 33 white female also raised in a middle-class, suburban area. unlike roslyn, demi had completed her degree in mathematics at another university and was enrolled in the graduate teaching certification program. naomi was a 24-year-old, indian american, yet considered herself to have been socialized into white, middle-class culture. she also received her degree in mathematics at another university and was enrolled in the graduate teaching certification program. morrison was a 28year-old, white male student raised in a middle-class, suburban area. while receiving a bachelor’s degree at another university, morrison worked in business for a few years prior to enrolling in the graduate teaching certificate program. finally, ruth was a 49-year-old, white woman raised in a middle-class, suburban context. she had a business degree, but worked in the home to raise her children; after such, she returned to school to pursue a teaching certificate. course structure on the first day of the course, students were asked to articulate what they thought should make up a quality mathematics teacher education program. after some negotiation and modification, the list was narrowed down to two experiences that this group of pre-service teachers desired: (a) teaching in a classroom environment with school-aged students and (b) reviewing collegiate mathematics that mirrors high school mathematics. these two student-generated objectives, in conjunction with my objective to include culturally responsive pedagogy, became the foundation for the s-cpm methods course. the first goal of the course was to allow the pre-service teachers’ to apply what they learned in the methods course in a classroom environment. i sought support from the institute for student achievement (isa), a new york-based, non-profit school improvement organization. this support allowed for transportation to the university for approximately twenty black and latina/o ninththrough eleventh-grade students from a high-needs high school in new york city that was 99% black and latina/o. the students came to the university weekly over 14 weeks after their regular school day for mathematics instruction and support during the first hour of the two-hour s-cpm methods course. the high school students understood that their participation in the afterschool mathematics program provided opportunities to supplement their high school mathematics coursework. the second goal of the course was to allow the pre-service teachers to review collegiate mathematics as reflected in high school mathematics coursework. as many mathematics teacher education programs require coursework in calculus, abstract algebra, number theory, and applied mathematics, pre-service teachers are presumed to be highly knowledgeable in mathematics by the end of their undergraduate education. while these required courses would expose pre-service teachers to increasingly abstract notions of the mathematics they encountered in pre-k–12, this exposure alone does not necessarily make them better potential tawfeeq & yu public stories journal of urban mathematics education vol. 5, no. 2 34 mathematics teachers. according to hill and colleagues (2005), specialized mathematical knowledge and skills used in teaching, or pedagogical content knowledge, positively impacts students’ learning of mathematics. the abstraction of collegiate mathematics in such areas as advanced calculus, number theory, and abstract algebra can potentially move pre-service teachers away from high school mathematics. such a claim does not mean that we object to pre-service teachers completing high-level mathematics; on the contrary, we encourage the infusion of rigorous mathematics courses in pre-service teachers of mathematics programs. nevertheless, we also consider the possibility that procedures and concepts needed to successfully engage high school level mathematics become increasingly embedded in the concepts and procedures needed to successful engage collegiate mathematics. for example, the simple procedure of factoring in order to better manage an integration problem in calculus is one of many examples of an embedded mathematics procedure within high-level mathematics. furthermore, while factoring is extensively practiced in the average high school algebra class, this mathematics procedure may become so overly nuanced when doing mathematical problemsolving exercises in collegiate mathematics courses that the importance of reflecting on these procedures as discrete bundle of pertinent strategy (phrase used by in s-cpm methods course) is mitigated. these pre-service teachers begin to take simple procedures and concepts for granted because while they continuously move through more rigorous mathematics, many of these strategies become a second nature reaction during problem solving. in other words, some of these strategies become trivialized during the thought process of pre-service teachers in their mathematics courses they take as part of their program requirements. for this reason, the pre-service teachers wanted to revisit much of the lower division college mathematics to re-familiarize themselves with these strategies in the context of high school mathematics. the review of this lower-level collegiate mathematics parallels much of the mathematics found at the high school level (lutzer, maxwell, & rodi, 2002). finally, the third goal, to expose the pre-service teachers to curricular, pedagogical, and assessment issues that address race and culture in the mathematics classroom was accommodated by assigning course readings that reflected a broad overview of some of the current equity research in mathematics education. for example, topics explored included teacher preparation and knowledge when teaching in high-needs schools (e.g., oakes, joseph, & muir, 2001; obidah & howard, 2005), learning styles and support of racially diverse students (e.g., berry, 2003; walker, 2005), and issues of race, culture, and social justice in mathematics education (e.g., burton, whitman, yepes-baraya, cline & myun-in kim, 2002; martin, 2003, 2009; moses & cobb, 2001; tate, 1997). the classroom dis tawfeeq & yu public stories journal of urban mathematics education vol. 5, no. 2 35 cussion and implementation of ideas found in such readings were framed by the two components of cultural pragmatism previously outlined. course implementation the five pre-service teachers worked as a team when developing the teaching schedule and curriculum for the high school students. the class was held in a large stadium-seating classroom that could seat about eighty students. this arrangement allowed the five teachers and the instructor to view lessons from multiple points in the classroom without interfering with the learning process. the room had an lcd projector, a white board, and a chalkboard, which allowed the pre-service teachers latitude when presenting their lessons. each pre-service teacher taught approximately three times during the semester and was required to follow the curriculum that they designed as a team. after the high school students left the classroom to return to their school, the pre-service teachers evaluated the performance of that day’s instructor as a group. the discussion topics included, remediation, management of manipulatives, mathematical representation, inquiry learning of mathematics, and teaching techniques within a culturally diverse setting. subsequently, the pre-service teachers, after critiquing the lesson, adjusted their instruction so as to better facilitate learning. the following five objectives guided the post-lesson discussions  have open discussions about mathematical knowledge relative to teaching;  have a free exchange of ideas, which provide an arena to explore personal beliefs about the learning of mathematics;  have open debate about learning issues related to sociopolitical and socioeconomic aspects of teaching mathematics;  provide a considerable amount of dialogue regarding techniques used during their lessons; and  keep a journal that was comprised of the pre-service teachers feelings about their accomplishments and failures during their instruction. as the methods course instructor, i associated the dialogue that occurred between the pre-service teachers and me with the arabic word haliqa. this word loosely means an “intellectually intimate chat.” such dialogue engenders learning and a mutual appreciation for the sharing of ideas, which can only occur when there is conversation based on intellectual intimacy linking teacher to student, student to student, and student to self. in general, the s-cpm methods course supported and encouraged the directional and bidirectional discourse model (yu & tawfeeq, 2011) for classroom tawfeeq & yu public stories journal of urban mathematics education vol. 5, no. 2 36 dialogue, both within the context of the methods course and within the context of the pre-service teachers future classrooms. before teaching the s-cpm methods course, 48 observations of pre-service teachers and their mentors in 12 high-needs middle and high schools in metro new york city were conducted. further observations of in-service mathematics teachers occurred as part of a larger professional development program funded by isa in several small learning high school communities 3 in atlanta, georgia, and east baton rogue, louisiana. 4 as the observations in these culturally diverse settings continued, a model was developed based on discourse observed between students of color and their teachers, where the questioning sequence provided a means for inquiry-motivated dialogue among participants. such inquiry-motivated discourse coupled with indepth content knowledge enabled the mathematics teacher to effectively guide their students’ efforts at learning through inquiry. figure 1 provides a visual framework or model that we identify as the directional and bidirectional discourse model which depicts a desired shift in dialogue in which the discourse between the pre-service teachers and students are fluid and shift based on the role of the students and teachers in the conversation. this model denotes not only the participants in the discourse but also the direction of the discourse. the person that initiates the discourse as listed first in the model determines its direction. more specifically, there are four levels of discourse: 1. teacher to student – teacher starting and carrying the conversation with the student, the lowest level of discourse. 2. teacher to student [responsible] – teacher starting the conversation with student; student responsible for maintaining the conversation. 3. student to student – students conversing about mathematical ideas. 4. student to teacher – student starting conversation with teacher, both equally responsible for keeping the conversation going; the highest level of discourse. additionally, we view the student’s role in the inner most circle (teacher to student) as the most passive role. as the circles move outward, the role of the student 3 small learning communities (slc) are small schools inside a traditionally large school building. slcs are often academically themed based (e.g., business, health, law/government, mathematics, or science) and serve a small population of students from the time they enter the high school until they graduate. 4 a hermeneutic dialectic process was used during the observation because it is interpretive and offers a comparison of contrasting and divergent ideas with the purpose of gaining a higher-level synthesis. the major purpose of this process is not to justify one’s own construction or to attack the weaknesses of the construction offered by others but rather to form a connection between them and allow for a mutual exploration of all ideas (guba & lincoln, 1989). tawfeeq & yu public stories journal of urban mathematics education vol. 5, no. 2 37 in discourse becomes more active. we want teachers to ask open-ended questions; however, asking open-ended questions is not enough. we advocate a more structured and rigorous approach to asking open-ended questions. for example, we would encourage teachers to require their students to compare and contrast other student responses. additionally, we would encourage students to verbally critique the questions of other students and the teacher, viewing such interaction as necessary and very important. figure 1. directional and bidirectional discourse. outcomes of s-cpm methods course in this section, we discuss outcomes of the s-cpm methods course, organized into two broad themes: (a) pre-service teachers’ confrontation of issues of race and culture in mathematics education and (b) pre-service teachers’ perception of the s-cpm methods course. confrontation of issues of race and culture one of the goals of the s-cpm methods course was to expose the preservice teachers to current research in culturally responsive teaching in mathematics education. while all the readings led to meaningful class discussions, the articles by martin (2003, 2009) were particularly challenging to the pre-service teachers’ notions of race in mathematics education. reading martin, for many, was the first time that they had heard someone take something which is generalize tawfeeq & yu public stories journal of urban mathematics education vol. 5, no. 2 38 in a social justice platform such as racism and turn it into something that is mathematical. when the learning of mathematics is racialized, it is clear that there are some powerful and oft-repeated assumptions about students’ abilities based on “race.” martin challenges not only the “mathematics for all” rhetoric but also what it means to learn and to assess learning within racialized learning environments. during one classroom episode, in an attempt to provoke discussion based on martin’s (2003, 2009) articles, i posed the following questions: “has mediocrity become the standard for black students? a black student getting a ‘c’ is the same as an asian student getting an ‘a?’” the students provided no direct response to the scenario. however, they redirect the conversation towards issues of academic resources of schools where black and brown students attend. generally speaking, the pre-service teachers were reluctant to engage the issue of racial perceptions in the learning of mathematics. rather, they deflected the issue to external circumstances such as lack of access to academic resources and other economic issues. the pre-service teachers seemed to be resolved to the idea that financial means primarily inhibited black and brown students’ positive performance on mathematical assessments—not racism. in short, the pre-service teachers’ position regarding financial means would appear to neutralize racialized learning experiences among diverse populations; it was easier for the pre-service teachers to engage issues of money (i.e., socioeconomic class) rather than issues of race and/or racism. nevertheless, in time, the pre-service teachers ability to confront racial issues in mathematics did begin to change. for example, roselyn initially interpreted martin’s (2003, 2009) position as blaming white people for the poor performance of black students on mathematics assessments. however, through class discussions and interactions with the high school students, roselyn began to see issues of racism not as an issue of blame but as a part of the reality of the learning experiences of the black and latina/o students she worked with throughout the semester. by the end of the semester, while the pre-service teachers may not have agreed with martin completely, they could not easily dismiss the possibilities of racialized mathematics learning experiences as they had in the beginning. perception of the s-cpm methods course we now consider the pre-service teachers’ reflections of the s-cpm methods course. the pre-service teachers were given a post-course, follow-up survey in the semester following their student teaching experience. sample questions included: during the course, in what ways did the issues learned in class seem relevant to your field-based experiences with the school aged students you worked with in the course? did this course influence your perception of teaching students who are african american or latina/o? if so, how? of the five students, only tawfeeq & yu public stories journal of urban mathematics education vol. 5, no. 2 39 morrison, ruth, and naomi responded. in general, all three respondents were satisfied with what they learned about teaching mathematics and what they learned from teaching students in a high-needs school. their responses showed that they were prepared with methods and experiences that would allow them to capitalize on time spent in schools during their field clinical and student teaching experiences. also, the pre-service teachers felt that the other mathematics methods courses should be restructured in a similar manner. in particular, the pre-service teachers indicated a variety of ways that the structure of the course provided a safe context that allowed them to make the issues in the course readings relevant in an actual teaching situation. naomi wrote: being able to apply the theories and issues that we had discussed to what we teach made it more practical. …knowledge acquired from readings is remembered for a semester, but practicing such theories and knowledge is something in which future teachers can apply to practice. similarly, morrison reflected: as teachers we understand and have an intended curriculum for our students. this includes lessons and unit plans produced by the teachers along with any projects or assessments. though when the teacher is in front of the class this curriculum often changes and adapts. …the teaching of these [high school] students without worrying about classroom management allowed for us to understand the different curriculum and how adjustments were always needed in the classroom. and ruth reflected: the students that came to our class benefited from our prep time and we became more in tune with their needs as we got to know them better. overall, i think we became better teachers and the students were able to obtain the help they needed. furthermore, morrison described how the s-cpm methods course provided opportunities to develop his mathematical knowledge necessary to teach (hill, rowan, & ball, 2005) as he reflected on ways to effectively present students with the mathematics content. for example, in a lesson on similarity in trigonometry he used manipulatives that modeled similar triangles. in preparing for and teaching this lesson to the high school students, morrison had to think through and reflect on conceptual issues related to similarity. during a debriefing and discussion session that took place right after the lesson on similarity was taught, he shared what he had learned about similarity through teaching the lesson during the field experience. morrison felt his strength in the lesson was knowing what to ask and how to transition from one topic to another with higher-level open-ended questions to students…understanding of student failure and student suc tawfeeq & yu public stories journal of urban mathematics education vol. 5, no. 2 40 cess. pacing of teaching and [knowing] struggles students have when math becomes open to more than one method or more than one answer. he continued: these students often came to us with less knowledge [regarding similar triangles] than needed for our intended material. as the teachers with our lessons we were forced to change our ways and adapt to the needs and understanding of the students. student questions and verbal and written assessment by the teacher allowed for us as teachers to veer off into different needed directions to fill in the gaps of all students. regarding the pre-service teachers’ perceptions of teaching black and latina/o high school students, the responses on the post-course survey were mixed. in response to the question, “did this course influence your perception of teaching students who are black or latina/o?” naomi reflected: this course had allowed me to see that urban students (such as the african american or latino students that we had worked with) are more appreciative of teachers who care and take the time out to put work into their lessons. in fact, they had seemed more appreciative of suburban students who have a tendency to take things for granted. furthermore, naomi commented on her professional willingness to someday teach in a similar setting: i originally was completely against teaching “at-risk” students. i thought i was not capable. but now i have become more open to the option after being exposed to a more urban setting. this statement is significant because it suggests that her exposure to an urban teaching environment, while different from her own pre-k–12 school experiences, provided her with a positive experience that she would otherwise not have considered or pursued. perhaps one way to aid in the recruitment of more qualified mathematics teachers to teach in urban schools is to provide pre-service teachers with authentic and meaningful teaching experiences with minority students in college courses with field-based teaching components. furthermore, we suggest that these courses explicitly address issues of race and culture as they pertain the teaching of mathematics. contrarily, on the same question regarding perceptions of teaching students who are black or latina/o, ruth responded: all the students seemed the same to me…students in need of extra help. there was really nothing different about these students from any other students that are slow in math. math can be a difficult subject and some students get it more readily than oth tawfeeq & yu public stories journal of urban mathematics education vol. 5, no. 2 41 ers. …this course taught me how to assist students in need whether they are latino, african american, or anything else does not matter. this response suggests that ruth did not seem to perceive a distinction between the culture of black and latina/o students and the culture of white students. instead, the black and latina/o students’ mathematics difficulties appear to be based on culturally neutral mathematical ability. this idea of neutrality is in contrast to the idea that mathematical difficulty may be related to a curriculum or pedagogy that lacks cultural relevance to students, as suggested in the course readings and class discussions. morrison, on the other hand, responded somewhat differently to the same question: more than influencing my perception of working with the african americans or latinos, i gained a higher respect of each of these students with their continued devotions each and every week. this statement may suggestion that morrison questioned black and latina/o students’ fidelity towards academics prior to this course. in the fall semester, after student teaching, morrison began teaching middle school mathematics in hawaii. in a telephone conversation, morrison expressed his amazement of the educational “plight” of the indigenous hawaiian students in comparison to their chinese, japanese, and white counterparts in the same school system and how the administration seemed to be apathetic about this situation. because morrison was “amazed” at the plight of native hawaiians, we assume that until this realization, morrison understood the concept of “at-risk” to be relative only to black and latino/a students. based on this assumption about morrison, we believe that other s-cpm methods courses should, through a broader list of required readings, extend the notion of educational disenfranchisement beyond black and latina/o students. prospective teachers of mathematics need to see the system of educational disenfranchisement in a world perspective. discussion an investigation of the use of a culturally pragmatic framework for mathematics education methods courses and field experiences is a multifaceted affair and has important implications regarding teaching mathematics at the middle and secondary levels. to maximize the learning experiences of pre-service teachers of mathematics in regards to sound mathematics content and culturally relevant pedagogy, the s-cpm methods course provided pre-service teachers with learning experiences that closely linked their course readings and discussions with actual teaching experiences. one limitation of this current discussion, however, is the tawfeeq & yu public stories journal of urban mathematics education vol. 5, no. 2 42 limited number of pre-service teachers; only three of the five completed the postcourse survey. of particular interest would have been the responses of roslyn and demi, both white females students in their early 20s, a common demographic of the pre-service teachers at the university. nevertheless, course readings with an explicit focus on issues pertaining to race and culture in mathematics education appeared to provided a perspective that allowed the pre-service teachers to critically consider issues of race and culture that exist outside of their own experiences. however, the manner in which the pre-service teachers applied this multi-cultural perspective in their field-based teaching experience was not as prominent as one would have hoped. as a point of improvement for a future s-cpm methods course, there needs to be a more explicit emphasis on the use of culturally relevant pedagogy. for example, one issue was that the pre-service teachers’ preoccupation with teaching mathematics lessons took their focus off considerations regarding a better understanding of their high school students’ cultures. one way to address this issue is to implement an activity in which the pre-service teachers interact with students in nonmathematical tasks. rather than mathematics, this interaction would be focused on understanding exactly who the students are. generally speaking, mathematics teacher educators must explicitly confront issues of race and culture in mathematic methods courses. regardless of their race and culture, or their own comfort level with such issues, they must recognize the need to expose pre-service teachers to critical literature to assists them in reflecting critically on their field experiences. not only should pre-service teachers be exposed to the literature regarding race and culture but also, when possible, they should be given opportunities to teach student populations reflected in the equity literature. given the increasing racial and cultural diversity in public schools, notions of mathematics pedagogical content knowledge as somehow culturally neutral knowledge should be confronted not only by expanding the literature base in mathematics methods courses but also by expanding the racial and cultural diversity of pre-service teachers’ field 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(2011). can a kite be a triangle? bidirectional discourse and student inquiry in a middle school interactive geometric lesson. new england journal of mathematics, xliii, 7–20. http://www.longislandindex.org/education.630.0.html http://www.longislandindex.org/education.630.0.html journal of urban mathematics education july 2015, vol. 8, no. 2, pp. 53–86 ©jume. http://education.gsu.edu/jume ksenija simic-muller is an assistant professor in the department of mathematics at pacific lutheran university, 1010 122nd st s., tacoma, wa, 98418; email: simicmka@plu.edu. her research interests include culturally responsive mathematics teaching and teaching mathematics for social justice. anthony fernandes is an associate professor in the department of mathematics and statistics at the university of north carolina at charlotte, 9201 university city blvd., charlotte, nc, 28223; email: anthony.fernandes@uncc.edu. his research interests include preparing mathematics teachers to work with english learners and understanding the use of multimodality in english learners’ mathematics communication. mathew d. felton-koestler is an assistant professor in the department of teacher education at ohio university, 79 s. court st. athens, oh, 45701; email: felton@ohio.edu. his research focuses primarily on prospective and practicing teachers’ beliefs about and knowledge for integrating social and political issues into their mathematics teaching. “i just wouldn’t want to get as deep into it”: preservice teachers’ beliefs about the role of controversial topics in mathematics education ksenija simic-muller pacific lutheran university anthony fernandes university of north carolina at charlotte mathew d. felton-koestler ohio university in this article, the authors report on the initial results of a mixed methods study that examined the beliefs that preservice teachers have about teaching real-world contexts, including those related to injustices, controversial issues, and children’s home and cultural backgrounds. data collection included a survey with 92 preservice pre-k–8 teachers and follow-up interviews with nine survey participants. analysis of the data suggested that preservice teachers were open to the idea of teaching mathematics through real-world contexts, but were ambivalent regarding the use of controversial issues, and often unable to provide concrete or non-trivial examples of what these different types of real-world contexts would look like in a mathematics classroom. based on the survey and interview findings, the authors make recommendations for future research and for the use of controversial issues in teacher education programs. keywords: preservice teacher beliefs, real-world contexts, social justice, teacher education range of arguments have been advanced for why it is important to connect mathematics to real-world contexts. one argument, frequently seen in policy documents, is that connecting mathematics to real-world contexts will better prepare students for their future careers and everyday lives (common core state standards initiative [ccssi], 2010; national council of teachers of mathematics, 2014). approaches based on this argument typically draw on examples that are likely to be viewed as largely uncontroversial or “neutral” in nature, such as shopa http://education.gsu.edu/jume mailto:simicmka@plu.edu mailto:anthony.fernandes@uncc.edu mailto:anthony.fernandes@uncc.edu mailto:felton@ohio.edu simic-muller et al. role of controversial topics journal of urban mathematics education vol. 8, no. 2 54 ping, counting or calculating with collections of objects, sharing food, or making connections to science and engineering. such “neutral” topics rarely result in or emphasize a critical perspective. a second argument is that connecting mathematics to students’ out-of-school practices—and in particular making connections to students’ homes, communities, and cultural backgrounds—can illustrate how all students’ backgrounds and knowledge are valued in the classroom (civil & andrade, 2002; de abreu & cline, 2007; gonzález, andrade, civil, & moll, 2001; mukhopadhyay, powell, & frankenstein, 2009; nasir, 2002; powell & frankenstein, 1997; tate, 1995). for example, educators have built on students’ backgrounds by creating projects about mathematics and gardening (civil & khan, 2001) or about mathematics practices of small businesses in the students’ neighborhoods (simic-muller, varley gutiérrez, & turner, 2009). finally, a number of scholars focus on the role of mathematics in preparing active and engaged citizens in a democratic society. this line of work is often referred to as teaching mathematics for social justice or critical mathematics, and while there are varying meanings for both of these terms (gates & jorgensen [zevenbergen], 2009; stinson & wager, 2012), for our purposes we highlight the common focus in this line of work on the importance of using mathematics to understand, question, and effect change in our world (frankenstein, 2009; gutstein, 2006, 2007; skovsmose, 1985, 1994; skovsmose & valero, 2001; tate, 1994, 1995; turner, varley gutiérrez, simic-muller, & diéz-palomar, 2009). some examples include using mathematics to analyze overcrowding in a school (turner, 2012) or to investigate the density of liquor stores in a neighborhood (tate, 1995). in both cases, students applied their mathematical knowledge to take action on an issue, but change can also be in the form of changed perceptions of the world. these topics can be engaging for students either because they are of immediate interest to them or because they have a broader importance that students can recognize. this engagement, in turn, often leads to students developing a greater sense of agency—a belief that they can take both an active role in shaping the mathematics they are learning and in improving the world around them (turner et al., 2009; gutstein, 2006, 2007; tate, 1995; turner, 2012). each of the three approaches to connecting mathematics to real-world contexts discussed—career and everyday preparedness, valuing students’ backgrounds, and engaging in social justice—involves different forms of real-world connections. we used these classifications of mathematical tasks, which were derived from existing literature and our own ongoing research (e.g., felton-koestler, 2015), as our starting point to design survey items that could be used to measure the preservice teachers’ (psts’) beliefs about using real-world connections in their future classrooms. in this article, we focus on psts’ beliefs about using controversial issues, which arise when teaching mathematics for social justice. simic-muller et al. role of controversial topics journal of urban mathematics education vol. 8, no. 2 55 literature review teaching mathematics with controversial issues there is a growing body of work focusing on teaching mathematics for social justice and on critical mathematics (see, e.g., gutstein & peterson, 2013; wager & stinson, 2012). however, mathematics educators have a range of perspectives on what these terms mean (bartell, 2012, 2013; gates & jorgensen [zevenbergen], 2009; stinson & wager, 2012). the vision of social justice mathematics that informs our work is one that involves using mathematics to support students in developing an awareness of injustices and ways in which they can critique and challenge injustices (frankenstein, 1995, 2009; gutstein, 2006; skovsmose, 1994). examples of this work include analyzing racial profiling and discrimination (gutstein, 2006); considering how government benefits should be distributed (skovsmose, 1994); and questioning the ways governments count and report data, such as unemployment rates (frankenstein, 2009). many of the examples of teaching social justice mathematics in the literature are with urban, minoritized, and/or underserved populations (brantlinger, 2013). in many cases, these issues can serve to make mathematics more engaging and accessible while also highlighting issues that are often of personal interest or concern to the students involved and supporting students to develop a sense of agency, especially when the students select the social justice issues to explore (gutstein, 2006, 2007; tate, 1995; turner, 2012; turner et al., 2009). however, in light of potential trade-offs between learning social justice mathematics and academic mathematics (brantlinger, 2013), we consider it critical that social justice mathematics is practiced with all students, not just underserved populations. based on our experiences and research (felton & koestler, 2012, 2015; koestler, 2010), most psts have little to no familiarity with “social justice” in the context of teaching mathematics. therefore, in our survey we do not explicitly mention social justice, but instead refer to “controversial issues,” which we define for the psts as follows: controversial issues are topics that will likely be viewed as contentious or debatable. not everyone agrees on what topics are controversial, but some examples might include the costs of the war on drugs, government spending, funding for schools, or climate change. we chose to focus on controversial issues because, in our experiences, most psts view social justice topics—such as racial profiling, analyzing government benefits, and questioning data reporting—as overtly political or controversial in nature. simic-muller et al. role of controversial topics journal of urban mathematics education vol. 8, no. 2 56 preservice teacher beliefs in recent years, psts’ beliefs have been the focus of an increasing number of research studies. there is no agreed-upon definition of beliefs in the field (philipp, 2007), but we found speer’s (2005) explanation most relevant for our purposes: “beliefs appear to be, in essence, factors shaping teachers’ decisions about what knowledge is relevant, what teaching routines are appropriate, what goals should be accomplished, and what the important features are of the social context of the classroom” (p. 365). this explanation is relevant as we consider the factors that shape teachers’ decisions about what mathematical knowledge related to real-world mathematics contexts is relevant and what goals it accomplishes, and whether creating a space for discussing controversial issues is an appropriate form of mathematics teaching. having spent years as students observing their own teachers before entering teacher education programs, psts already hold firm beliefs about what teaching should look like (ambrose, 2004; pajares, 1992). according to pajares (1992), beliefs that form early are more difficult to alter, and research often discusses how resistant psts’ beliefs are to change (cooney, 2001; grootenboer, 2008). pajares also claims that psts identify positively with the teaching profession, and are therefore interested in preserving the status quo. in particular, they do not see themselves as enacting societal change, because they support the system in its current form. for that reason, they may not see the need for engaging students in conversations about issues of social justice. there is still not enough research to support this claim, and we do not know much about whether teachers believe they “ought to bring about social justice… in their classrooms” (silverman, 2010, p. 294). according to ambrose (2004), one of the ways beliefs are formed is through cultural transmission. cultural transmission creates beliefs “that may be held at a subconscious level and can be thought of as resulting from the ‘hidden curricula’ of our everyday lives” (p. 93). culturally transmitted beliefs are often embodied in the form of stereotypes, for example about what mathematics is, how to teach it, or who is good at it. one of the most persistent beliefs that psts hold is that of teaching as caring, they have the tendency to shelter students rather than expose them to difficult experiences (ambrose, 2004). this belief can get in the way of difficult conversations and is prominent in psts’ stated reasons for avoiding difficult topics in the classroom. bartell (2011) proposes an alternative notion of caring, that of caring with awareness, which embraces difficult conversations, standing in solidarity with one’s students, and supporting students’ academic success by building on their cultural backgrounds and knowledge. while there is well-documented evidence of psts’ resistance to teaching focused on equity, diversity, or social justice in general, and in mathematics specifically (aguirre, 2009; castro, 2010; gay, 2009; rodriguez, 2005; villegas, 2007), simic-muller et al. role of controversial topics journal of urban mathematics education vol. 8, no. 2 57 there is also evidence that providing psts with opportunities to reflect, consider new perspectives, and engage in equity-oriented tasks can provide a powerful basis for shifting teachers’ and psts’ beliefs about integrating issues of equity, diversity, or social justice into mathematics teaching (bartell, 2012; ensign, 2005; felton, 2012; felton & koestler, 2012, 2015; felton, simic-muller, & menéndez, 2012; koestler, 2012; mistele & spielman, 2009; spielman, 2009). surveys about preservice teacher beliefs in our search of the literature, we found only a few surveys that related to our interest in psts’ beliefs about the use of real-world contexts. two surveys most relevant to us were concerned with issues of social justice, diversity, or equity in teacher education in general: enterline, cochran-smith, ludlow, and mitescu (2008) and silverman (2010), and one focused more specifically on mathematics: turner and colleagues (2012). at boston college, where strong emphasis is placed on preparing teachers to teach for diversity and social justice, enterline and colleagues (2008) designed the learning to teach for social justice-beliefs scale to measure teacher candidates’ and new teachers’ general beliefs about teaching for social justice. over one thousand participants completed five surveys from entrance to exit of boston college’s teacher education program, and for three additional years after exiting the program. this multi-year surveying allowed enterline and colleagues to follow the psts’ beliefs and self-reported practices over time. in their analyses, they found that the scores of existing teacher candidates were much higher (indicating beliefs more strongly aligned with teaching for social justice) than of those who were entering the program; these increases persisted after one year of teaching. perhaps most notably for our work, they argue that teaching for social justice is a measurable outcome that can be assessed using mixed methods. we have, therefore, created our survey with this idea of measurability in mind, but with a specific focus on teaching mathematics for social justice. silverman (2010) created the teachers’ sense of responsibility for multiculturalism and diversity scale to measure how psts understood commonly used terms like multiculturalism and diversity. she conducted her study as an outside researcher in an education psychology class (n = 88). silverman found that psts were more likely to endorse general statements about multiculturalism or diversity than terms such as gender or faith. the psts also believed that larger organizations, such as schools and communities, bear a greater responsibility for addressing multiculturalism and diversity than individual teachers. these findings suggest that psts may not be inclined to take action to address these issues. the items on our survey address the psts’ willingness to address similar issues in the context of mathematics teaching. simic-muller et al. role of controversial topics journal of urban mathematics education vol. 8, no. 2 58 turner and colleagues (2012) focused specifically on psts’ knowledge, dispositions, and practices around building on children’s multiple mathematical knowledge bases, including their cultural, home, and community-based knowledge. they conducted two surveys (one at the beginning and one at the end of the semester) with over 200 psts at six universities during their methods courses, which explicitly focused on the practices the surveys measured. the surveys consisted of items related to children’s mathematical thinking, students’ home language, and adapting curriculum materials to meet the needs of diverse students. they also conducted interviews with those psts who volunteered to be interviewed (n = 24). turner and colleagues used the survey, interviews, and samples of pst work to hypothesize a learning trajectory for how teachers learn to build on students’ multiple mathematical knowledge bases, including their family and community knowledge bases. our study also seeks to hypothesize trajectories, but, in our case, for preparing teachers to teach mathematics using real-world contexts, controversial issues, injustices, and family backgrounds and community practices. our survey was informed by the surveys and methods discussed as well as our own research and practice as mathematics teacher educators. while our study is similar to others in design and interest in issues of social justice, we note that no other survey specifically investigates psts’ beliefs about social justice in mathematics teaching. we provide the development of the survey, along with a description of the data collection and analysis in the next section. methods here, we used an explanatory design, a type of mixed study approach (creswell & plano clark, 2011). this method consists of two phases: a quantitative data collection phase, followed by qualitative data collection. our quantitative phase consisted of administering the survey, subsequently described in detail, to measure the psts’ beliefs about using real-world issues in their future teaching. the qualitative phase examined specific quantitative results that required further explanation. given the small number of studies in the area, using the explanatory design was particularly suited for our purposes of understanding and interpreting psts’ beliefs related to the use of real-world issues in the mathematics classroom. survey design and testing the survey design was primarily informed by existing research discussed previously and, as this literature thus far is limited, our combined 32 years of experience working with preand in-service teachers. in particular, our work was informed by felton-koestler’s (2015) framework for understanding psts’ beliefs about the sociopolitical nature of mathematics. one aspect of felton-koestler’s simic-muller et al. role of controversial topics journal of urban mathematics education vol. 8, no. 2 59 framework identifies a range of views psts may have about using mathematics to analyze the world, which include valuing connections between mathematics and real-world topics that are viewed or positioned as (a) neutral or apolitical in nature, (b) overtly political or controversial in nature, and (c) dealing explicitly with issues of perceived injustices. based on our experience, few psts have been exposed to issues of social justice, especially in mathematics (felton & koestler, 2012, 2015; koestler, 2010). therefore, to make the items comprehensible to psts, we relied on the research which claims that beliefs can be inferred from the potential actions someone would take (pajares, 1992), and tied these issues to the psts’ future teaching moves. for example, one survey item is “when i teach mathematics i will focus on mathematical concepts (for example, addition and subtraction, geometric shapes, etc.), and not worry about using controversial issues” (note that our definition of controversial issues was provided to the psts). this and other related questions aim to understand what psts think about controversial issues in relation to the usual mathematics that they might envision teaching. we initially designed a large pool of items and discussed these among the three of us. in these discussions, our goal was to ensure that the wording would be clear to the psts. through these discussions we created the four subscales of our survey (the definitions provided to the psts are in table 1). the first three of these subscales correspond to felton-koestler’s (2015) framework. in developing the survey we conceptualized the first three subscales—real-world situations, controversial issues, and issues of injustice—as interrelated, such that each new type of real-world context was a subset of the previous one. for instance, we would view an investigation of racial disparities in school funding as an issue of injustice and, likewise, as a controversial issue and a real-world situation. we chose the specific examples for the definitions of the four subscales provided to the psts (table 1) based on our experiences having used a variety of topics with psts. however, because not all psts take up all topics in the same way (felton-koestler, 2015), we were careful to emphasize that “not everyone agrees on what topics are controversial” and to ask the psts to “imagine topics that you consider controversial” when filling out the survey. in the next phase of our survey development, we sent the initial pool of items to seven experts in the area to ensure content validity. based on the feedback, we made further changes to the items and together we agreed upon a final list of 35 items that make up the connecting mathematics to the real world scale (cmrws); each item was scored a 5-point likert scale (1 = strongly disagree, 2 = disagree, 3 = neutral, 4 = agree, 5 = strongly agree). table 1 shows the directions that are given to the psts at the beginning of each section (subscale) of the survey; in each case, the subscale is designed to measure the degree of openness the pst simic-muller et al. role of controversial topics journal of urban mathematics education vol. 8, no. 2 60 display towards using or learning about these types of real-world contexts in their future teaching. the full text of the survey is provided in appendix a. in this reporting, we chose to focus primarily on psts’ responses to questions about controversial issues for two reasons: (a) the scores on this scale were lower than those on the other three and (b) the responses in this part of the survey were difficult to interpret. table 1 definitions of the four subscales subscalea definition real-world situations (6) real-world situations are everyday or career related topics. examples include choosing a cell phone plan, designing buildings, using mathematics for a job, connecting mathematics to artistic designs, or solving scientific problems. controversial issues (8) controversial issues are topics that will likely be viewed as contentious or debatable. not everyone agrees on what topics are controversial, but some examples might include the costs of the war on drugs, government spending, funding for schools, or climate change. when responding to questions on the survey about controversial issues, imagine topics that you consider controversial. issues of injustice (11) issues of injustice are topics that are likely to be seen as a form of injustice or wrongdoing. not everyone agrees on what topics represent injustices, but some examples might include animal cruelty, amount of living space in refugee camps, or deaths from preventable diseases. when responding to questions on the survey about issues of injustice, imagine topics that you consider injustices. family backgrounds or community practices (10) family backgrounds or community practices are things that people are involved in because of a connection to their family or community. for example, for some people activities like quilting, farming, cooking certain foods, or celebrating special holidays may be an important part of their family or community heritage. a the number of items for each subscale is given in parentheses. data collection we recruited our students and the students of colleagues around the united states to participate in cmrws. in the initial round we obtained a sample of 92 psts. approximately 40% of the survey participants were students in the first author’s mathematics content courses for preservice pre-k–8 teachers. the follow-up interviews were also conducted with psts enrolled in the first author’s courses, at the end of the spring semester. these psts were taking the first or second part of the university’s two-course sequence and had not yet been accepted to the educa simic-muller et al. role of controversial topics journal of urban mathematics education vol. 8, no. 2 61 tion program. the university is a middle-sized private liberal arts institution with a stated commitment to social justice, which is visible in the curriculum, though typically not in the sciences. the content courses, when taught by the first author, include some content related to social and political issues and to student backgrounds. in particular, when these psts took the survey, they had already had some exposure to the types of topics it addresses. the impact that these contexts had on the psts is difficult to gauge. while one pst (ann; all names are pseudonyms) shared that she thought social justice was a central tenet of the course she took, another (kayla) thought that social justice was mentioned “a little,” while a third pst (linda) thought that her exposure to these topics was so minimal that it made it difficult for her to respond to interview questions. based on these responses and the research that describes psts’ beliefs as resistant to change (e.g., ambrose, 2004), we believe that the course could not have had a large impact on the psts’ beliefs. furthermore, at this stage of survey design and administration, we were not as concerned with what could have caused psts beliefs as with what the cmrws was able to capture about them. data analysis survey. we collected demographic information from the 92 psts who participated in the survey and started the analysis for their responses by first reversecoding the responses to eight of the 35 items. for example in the case of the item “when teaching mathematics, real-world situations can distract students from learning the important mathematical concepts,” a value of 1 (strongly disagree) would represent more openness to incorporating real-world situations and hence we recoded the 1 to 5, 2 to 4 and 3 to 3. the recoding ensures that the higher values correspond to more openness to the issues under discussion. next, we worked with the missing values in the data. we did not want to force the psts to respond to every item and so their response was optional. in our data, we found that there were at most 2 missing values for any one of the 35 items. based on the recommendations of tabachnick and fidell (2012) for a small percentage of missing values (< 5%), we used the group mean to replace the missing data. using mahalanobis distance (p < 0.001) we did not find any outliers. cronbach’s alpha for each of the subscales and the entire scale of 35 items is shown in table 2. note that an exploratory factor analysis was not supported given that we only had 92 respondents. according to nunnally (1978), a sample of at least 350 would be required to reduce sample error. despite this constraint, we think that responses to the items within the scales, which demonstrate high reliabilities, along with qualitative interviews, can provide us with useful insight into psts’ beliefs about these issues. we describe our approach in the next section. simic-muller et al. role of controversial topics journal of urban mathematics education vol. 8, no. 2 62 table 2 cronbach’s alpha for the four subscales and the entire cmrws subscale cronbach’s alpha real world situations 0.79 controversial issues 0.86 issues of injustice 0.91 family backgrounds or community practices 0.90 cmrws 0.95 there were at most two psts who skipped any given demographic item and there were three psts who skipped the item that related to the educational level of their secondary parent or guardian. the majority of the survey participants were from four states—arizona (18%), north carolina (15%), texas (21%) and washington (41%). the psts mostly identified as white (64%), non-hispanic (80%), female (85%), and most were from 18–23 years old (76%). the vast majority of the psts (97%) had at least one parent with the equivalent of a high school degree or higher, and a majority (59%) had at least one parent with a bachelor’s degree or higher. this sample is fairly representative, in terms of gender and ethnicity, of the u.s. teaching population (e.g., zumwalt & craig, 2005). finally, 61% of the psts were interested in teaching grades pre-k–5, 28% in teaching grades 6–8, and 5% in teaching grades 9–12. interviews. based on the preliminary analysis of the survey results, we conducted semi-structured qualitative interviews with psts to help interpret our quantitative results. interviews were conducted with nine psts who had participated in cmrws and volunteered to be interviewed. all nine were women, eight of whom were identified as white and came from a wide range of socioeconomic backgrounds. two planned on obtaining a middle-level mathematics endorsement, while others represented a range of feelings towards mathematics, from frustration to acceptance. the interviewees also represented a wide range of interest in and openness to the ideas measured in the survey. only one pst had taken a prior course with the first author, and their exposure to the types of contexts investigated in the survey consisted of a single semester. the interview questions were tied to the constructs and items in cmrws as we sought to further understand the psts’ thinking about the use of real-world contexts in their future classrooms. the interviews were semi-structured, meaning that there was a fixed list of questions that all psts were asked (see table 3). depending on the responses, we asked some psts follow-up questions. all participants answered questions about their understanding of real-world issues, controversial issues, and injustices: what these terms mean, how important they are in mathematics teaching, and what real-world topics would fit into teach category. additional follow-up questions arose from the specific information the psts gave, and were usually based on the personal experiences they shared during the interviews. for simic-muller et al. role of controversial topics journal of urban mathematics education vol. 8, no. 2 63 example, one question asked how one pst’s educational experiences in both a lowincome and high-income school district would impact her teaching of real-world topics; another, posed to a pst with a stronger mathematical background, probed the types of mathematical knowledge needed to be successful in teaching real-world mathematical contexts. the interviews were 10–45 minutes each, depending on each pst’s interest, prior knowledge, and openness to the topic. if psts gave brief responses, 10 minutes were sufficient to respond to all the questions. if a pst was reluctant to talk or was unable to come up with examples for the contexts, we asked fewer follow-up questions. the two shortest interviews were with psts, who had difficulties creating examples of real-world issues, and who were not often vocal in the classroom; one of the two was also particularly ambivalent about using controversial issues in the classroom. table 3 interview prompts 1. some people think it is important to teach mathematics by making connections to real-world situations. what does that mean to you? [can you give me some specific examples? can you give me more specifics? how would you teach it? which mathematics would you use? what issues are relevant to you?] 2. some people think it is important to teach mathematics by making connections to controversial issues. what does that mean to you? [can you give me some specific examples? can you give me more specifics? how would you teach it? which mathematics would you use? what issues are relevant to you?] 3. some people think it is important to teach mathematics by making connections to injustice. what does that mean to you? [can you give me some specific examples? can you give me more specifics? how would you teach it? which mathematics would you use? what issues are relevant to you?] 4. how do you perceive the terms real-world situations, controversial issues, and injustice? are they separate, or is there an overlap? please give examples. 5. would these topics/activities be appropriate to use in pre-k–8 classrooms? please explain. give examples of topics that would and would not be appropriate to use. 6. if you had conditions in your school and districts that were ideal, would you use activities like these in the classroom? the analysis of the interviews was focused on developing a more robust understanding of the survey results. we independently examined summaries of the interviews to isolate common themes that related to the quantitative results. we paid special attention to how psts responded to items that had a high percentage of neutral responses. after the individual analysis, we discussed the themes and reached an agreement on significant beliefs that seemed to impact the psts re simic-muller et al. role of controversial topics journal of urban mathematics education vol. 8, no. 2 64 sponses on the survey, with additional multiple readings of the interviews in order to ensure that our claims about the beliefs were backed by pst responses. limitations. the sample size for the cmrws is not yet large enough to make generalizations about pst beliefs with confidence. we cannot perform a factor analysis based on a sample of 92. also, as may be the case with every survey, although it is anonymous, we are unable to ascertain if psts were responding truthfully to the questions or were agreeing with ideas that they knew their instructors or education programs would approve of, most notably the merit of teaching mathematics through real-world topics. even though the interviews were conducted at the end of the semester, and psts knew they would not be taking another course with the researcher, there is still a possibility that the inherently unequal teacher–student relationship impacted interview responses. also, because the interview was semi-structured, it means that not every pst answered exactly the same questions, which also may have had an impact on the results of the analysis. results and discussion survey the data from the 92 respondents were entered into ibm spss (ibm corp., 2012) for analysis. eight items were reverse-scored. we based the scoring on the degree of openness the psts displayed towards using real-world issues and grappling with the associated controversial aspects in their future teaching, and their interest in learning about these issues. thus a score of one would indicate a pst who was not at all open toward real-world issues and a five would indicate that the pst was quite open to these ideas. the means and standard deviations of the four subscales are listed in table 4. in addition, table 5 shows the percent of disagreement (d, combined disagree and strongly disagree), neutral (n), and agreement (a, combined agree and strongly agree), for the four items that asked about the psts’ willingness to use the four types of real-world connections emphasized in the four subscales. table 4 means and standard deviations for the four subscales real world controversial injustice backgrounds mean 4.29 (.53) 3.50 (.59) 3.62 (.64) 3.94 (.65) simic-muller et al. role of controversial topics journal of urban mathematics education vol. 8, no. 2 65 table 5 psts’ willingness to use different types of real-world connections item # item disagree neutral agree 1 when i teach mathematics i will make connections to real-world situations. 1% 2% 97% 7 when i teach mathematics i will make connections to controversial issues. 11% 61% 27% 16 when i teach mathematics i will make connections to issues of injustice. 13% 47% 40% 28 when i teach mathematics i will make connections to the family backgrounds or community practices of different peoples from all over the world. 4% 5% 90% note. percentages may not add to 100 due to rounding. tables 4 and 5 show that there is a much higher level of support for making real-world connections in the classroom than there is for making connections to controversial or social justice issues and to a lesser extent to family backgrounds or community practices. using a paired samples t-test, the mean differences between the scales show that there is a significant difference between real world and controversial (p < 0.05) and backgrounds and controversial (p < 0.05), and that the difference between injustice and controversial is not statistically significant. the high support for real-world connections is not particularly surprising, and it may be due to a variety of reasons: the psts may genuinely support the use of authentic real-world contexts in mathematics teaching; they may think they support the use of authentic real-world contexts without a deep understanding of what that means; they may have a broad understanding of what constitutes a real-world connection, thus including virtually any “contextualized” problem, such as simple word problems; or they may be repeating what they heard in their education programs. of all the subscales, the lowest scores were on the controversial issues (ci) subscale, which we chose to investigate for this reporting, not only because of the low scores but also because the responses in this section were difficult to interpret. we were also interested in the ci scale because of our own interests in developing future teachers who are committed to using mathematics as a tool for understanding our social and political world, which requires investigating controversial issues. the eight items of the ci subscale along with the percent of disagreement (d, combined disagree and strongly disagree), neutral (n), agreement (a, combined agree and strongly agree), and means are listed in table 6. note that the items are ordered by increasing mean scores and the “r” after the item number indicates that the scores were reverse-coded when calculating the means. simic-muller et al. role of controversial topics journal of urban mathematics education vol. 8, no. 2 66 table 6 items on the controversial issues subscale (n = 92) #a item disagree neutral agree mean 10r when i teach mathematics i will focus on mathematical concepts (for example, addition and subtraction, geometric shapes, etc.) and not worry about using controversial issues. 32% 47% 22% 3.10 11r when teaching mathematics, controversial issues can distract students from learning the important mathematical concepts. 42% 32% 26% 3.16 7 when i teach mathematics i will make connections to controversial issues. 11% 61% 28% 3.16 13 teaching mathematics with controversial issues helps students learn the mathematical concepts better. 10% 52% 38% 3.28 9 when i teach mathematics i will make sure my students have opportunities to take action to address controversial issues. for example, writing a letter to a government representative. 12% 30% 57% 3.45 8 i am interested in learning how to make connections to controversial issues when teaching mathematics. 12% 21% 67% 3.55 12 an advantage to teaching mathematics with controversial issues is that they help students learn about these issues in the world around them. 3% 15% 82% 3.78 14 an advantage to teaching mathematics with controversial issues is that some students identify these issues as important to them. 3% 13% 84% 3.80 note. percentages may not add to 100 due to rounding. there were two missing values in item 7 and one in item 9 since it was not possible to recode the serial mean that was used at the initial stages of the analysis. a items that were reverse scored are indicated by an “r.” looking across the items, we see that many of the psts are uncertain about whether they will use controversial issues or whether use of these issues supports student learning of mathematical concepts (see items 7 and 13, over half of the responses were neutral). in addition, responses are roughly split over whether integrating controversial issues can distract learners from learning mathematical concepts (item 11). it is interesting to note that, with the exception of item 13, the items below the mean (ci = 3.50), namely 10r, 11r, 7, 13, and 9, relate to the psts antici simic-muller et al. role of controversial topics journal of urban mathematics education vol. 8, no. 2 67 pated teaching practice and to potential disadvantages of integrating controversial issues. items 8, 12, and 14, which are above the mean, relate to the psts desire to learn more about controversial issues and their potential benefits in the classroom. the willingness to learn about controversial issues from others that is not necessarily accompanied with the desire to personally take action is consistent with silverman’s (2010) findings that psts often identify larger organizations as bearing the primary responsibility for addressing multiculturalism and therefore they may not be personally inclined to take action to address these issues in their own practice. we interpreted the high percentages of “neutral” responses in the “controversial issues” category, as well as similar percentages between the “agree” and “disagree” responses in many items, as pst ambivalence about these topics; and sought possible reasons for it. it is possible that, despite our definition, psts were still unsure about what counts as a controversial issue or could have been using a different definition; given that we allowed them to choose topics they viewed as controversial when responding to the questions. we also acknowledge that not all controversial issues are equally controversial to all people, which might also have an impact on survey responses. further, we also conjectured that the large number of neutral responses could be due to the fact that psts frequently have limited knowledge about how to integrate controversial issues into mathematics teaching (felton & koestler, 2012, 2015; koestler, 2010). this conjecture would seem to be supported by the results asserting that while only 27% of the psts indicated they would make connections to controversial issues (item 7), 67% were interested in learning how to do so. it may also be the case that, without clear evidence or experience to draw on, some psts could have been unsure whether implementing these issues would be worth the time and effort. finally, it was possible that the psts were not interested in implementing these issues, but did not want to show disagreement, even though the participants were told that the survey was anonymous. the fact that 67% of the psts indicated an interest in learning more about making connections to controversial issues is an encouraging result and can provide a foundation upon which teacher educators can build. interviews recall that to further investigate psts’ beliefs, the first author conducted interviews with nine of the psts enrolled in her mathematics course for preservice pre-k–8 teachers. in analyzing the interviews, two important themes were identified that shed light on the survey results. first, many of the psts considered a fairly narrow range of real-world contexts that could be used in the mathematics classroom, which might explain why psts so readily agreed with using real-world contexts in the survey. second, many of the psts thought that controversial issues were only appropriate to explore with some students, which might contribute to simic-muller et al. role of controversial topics journal of urban mathematics education vol. 8, no. 2 68 ambivalence toward controversial topics expressed in the survey. we explore these two themes below. a limited range of real-world situations. on cmrws, virtually all psts (97%) agreed with using real-world topics in their teaching, but, as discussed, a much lower percentage expressed interest in teaching using injustices or controversial topics. in the interviews, psts also emphasized the importance of learning mathematics through real-world situations. for example, monica, who strongly disliked mathematics, insisted on contexts that students could relate to, because “if they are not going to see how it’s relevant, then they are not going to care.” when asked for specific examples that they would consider using in their teaching, the psts provided contexts that are commonly seen in mathematics classrooms and curricula and that are not framed as controversial in nature. problems about food or money were mentioned in six of the nine interviews, though two of those psts noted that food was the easiest but not the ideal real-world context to use. other contexts included sports, telling time, changes in temperature, animals, and word problems about everyday objects. in response to the question about the kinds of context she would use with her students, kate said, “of course i am going to do the johnny has two apples kind [of problem], that’s technically a real world [problem],” implying that any story problem can be understood as a real-world context. food is one of the most common contexts for story problems, and the prevalence of food as a topic for psts to create story problems is noted in literature (e.g., lee, 2012). consider the following exchange from an interview: linda: you know, basically, if you are teaching fractions, i think it would be if you are talking to kids and they know what half of a pizza is, generally. you could cut a pizza in half and it’s like it’s a half of a pizza and that’s something they can relate to because they have seen that been cut up before and they know how that would work. so they are able to visualize it better than if you are maybe just took a circle and said this is like this. so just a little bit of context helps i think. and the addition you know there is always you give candy to your friends, how much candy do you have now? interviewer: do you consider those real, real-world contexts? linda: i think real world as to their world. hopefully they eat. it is true, as linda notes, that cutting pizza is an experience most students have had before. however, this example does not highlight how to use mathematics to solve or to investigate a genuine problem, but simply uses a familiar context to illustrate a mathematical concept (one half). when pressed, all but two psts (linda was one of the two) came up with examples of real-world contexts that were more overtly controversial in nature. it was simic-muller et al. role of controversial topics journal of urban mathematics education vol. 8, no. 2 69 a difficult question for them, and some needed additional prompts, but they eventually mentioned a range of topics, including environmental issues, homelessness and hunger, and racial and language diversity. apart from the two psts who were unable or unwilling to engage with controversial topics, psts were open to discussing these issues, and in some cases shared their personal interests, such as climate change or declining animal populations. they also readily agreed, consistent with survey results, that they would be interested in learning more about using controversial topics in their mathematics teaching. however, as is discussed below, the psts also frequently raised concerns about if or with whom these contexts could be used. the interviews suggest that the psts in our study genuinely value real-world contexts and believe that it is important for students to relate to the contexts used for mathematics problems. this conclusion is consistent with findings in research studies conducted with inservice teachers (chapman, 2009; gainsburg, 2008). however, psts’ current understanding of what kinds of real-world connections are possible in the mathematics classroom seems to have been shaped by their past educational experiences. the fact that two thirds of the interviewed psts gave examples related to food and money is not surprising, as it mirrors the extent of realworld mathematics that students often encounter in school. our conjecture that some psts equate real-world contexts with any contextualized or story problems was partially confirmed through kate’s observation that the “johnny has two apples” types of problems are a type of real-world problem, and the examples that other psts gave, which included sharing candy, counting bicycles around the school, or cutting pizza. there is certainly a place for these types of problems in the curriculum, but we would also like to see psts add mathematical complexity and meaning to them. for example, in addition to sharing candy, children could also discuss the fair sharing of resources; or in addition to counting bicycles around the school, children could engage in a neighborhood mapping project. controversial issues are only for certain populations. the psts generally believed that addressing controversial issues is only appropriate with certain groups of students. this occurred in three ways: (a) not exploring topics that directly affect students in negative ways because it might make them feel uncomfortable, (b) deciding whether or not to discuss topics based on how “diverse” the classroom is, and (c) avoiding controversial issues with young children. first, seven psts were concerned about discussing certain topics because they could “discourage students” or “offend people.” the other two were generally uninterested in talking about controversial topics. one of the seven, laura, whose responses were overall different than those of her peers, thought teachers should still address those topics, but with care; however, others were more willing to avoid them. for example, kate stated: simic-muller et al. role of controversial topics journal of urban mathematics education vol. 8, no. 2 70 i think maybe we could talk about poverty and homelessness, i just wouldn’t want to get as deep into it, … i feel like at that age … they haven’t realized yet oh, my family is poor when they are in kindergarten, because that’s not something you think about when you are that young, so i don’t want to be like, oh, this is how much your parents make, that means you are in poverty, you know. i guess that was my thought process, which is why i don’t want to teach about poverty in a low income school district, because i don’t want to be like oh, that sucks for you, you live in poverty, i don’t want to be the one to piece it together for them. this quote corresponds to research findings that many psts and teachers see teaching as caring and would rather shelter their students than expose them to difficult issues (ambrose, 2004; bartell, 2011). we find kate’s statement problematic because it assumes that children are not aware of the circumstances they live in. unlike kate, laura exhibited caring with awareness suggested by bartell (2011), in which she would have a conversation with this student, like, you know i really want to bring this up because i think it’s great for people to be aware and i don’t know if all students in our school are aware of all these things going on people’s lives and kind of explaining that. as we will discuss in more detail below, laura has had more opportunities to discuss controversial topics with others, and this experience translated into greater comfort with the idea of introducing them to young students. we are interested in developing similar tendencies in all psts. second, two of the psts directly stated and two others implied that the diversity of the students in their classroom would influence which controversial issues to explore. for example, monica talked about the need to talk about race, but only in a diverse classroom: it doesn’t apply to every classroom but i think that a really big deal is being different color than someone else. i think it would depend on the classroom because there aren’t always diverse classrooms. … but if you had a more diverse setting then you could talk about it, it’d be like you know this is okay we are not all the same, we are all different, and at the same time how many of use come from these backgrounds and data like that could be plotted or bar graphed. she also noted that she had gone to an all-white school where race was not discussed because “it was not an issue,” which is a perspective we have seen elsewhere (felton-koestler, 2015) that reflects the notion that “whiteness” is not a race (mcintosh, 1990). in contrast, ann talked about a classroom with only white students being a safer environment to investigate these issues: i feel like if you were in an all-white classroom, you could do stuff with injustice because you are not going to be likely risk, there is less controversy there, versus hurting simic-muller et al. role of controversial topics journal of urban mathematics education vol. 8, no. 2 71 someone’s feelings and in some ways it may be easier there than in a class with one kid that isn’t white. ann also talked about not wanting children to feel marginalized, thus again showing that caring can get in the way of critical conversations. both quotes highlight the perspective that whether or not a topic can be broached depends greatly on the makeup of the student body. while we believe it is important to be cognizant of and responsive to one’s students, we also see this perspective as problematic if it results in important topics, such as race, only being discussed with minoritized groups of students. in addition, psts mentioned concerns over parent and administrator disapproval of these topics. two stated that they did not want to offend anyone, and even laura said that she would check with an administrator before discussing a topic that could be deemed as controversial. if, as suggested by these data, psts are ready to discuss controversial topics only in certain contexts with certain students, then their ambivalence toward teaching controversial topics is understandable: they may be neutral because their final decision about whether to include controversial issues in the mathematics curriculum will depend on their school setting and the support they receive. this is an important notion for teacher educators to consider, as school administrators may not be aware of, informed about, or supportive of discussing controversial issues in mathematics. finally, the most prevalent version of the idea that controversial issues are only for certain groups centered on the age of the students. all psts interviewed were interested in teaching early elementary grades, with five of the nine choosing a grade level between kindergarten to third grade as their preferred grade to teach, and the other four undecided between early elementary and either middle school or special education. in their interviews, two thirds of the psts stated that controversial issues would be inappropriate for younger children (three explicitly mentioned firstand second-grade students as being too young) because they would have difficulty comprehending or caring about the issues that were being discussed or because they had not considered them before. kate’s previous quote about homelessness and poverty provides one example of this perspective. another can be seen in the following exchange: interviewer: can you give examples of making connections to real-world situations? how would you do it if you thought it was important? monica: that was hard for me to answer on the survey. i want to teach really young children and i feel like that’s going to be really hard. because a lot of the stuff i want to make connections to i feel like are more middle school high school level things like when i talked about the child soldiers [a project she did in high school]. kindergarteners are not even going to know what that means or will not care. so i feel like the simic-muller et al. role of controversial topics journal of urban mathematics education vol. 8, no. 2 72 things that i feel like are important are going to be out of the range of knowledge of the kids i want to teach. so like on the survey most of the questions i was like i would love to teach about them but i have no idea how i would do that. so i feel like it would have to be a lot lesser real-world situation like ice cream, which is still, like money. money is like for first graders especially i feel like it would be a really easy real-world connection to make but it’s not really an issue. in this quote, young children are viewed as incapable of understanding contexts beyond money or ice cream. though literature mostly documents work with slightly older students (e.g., turner et al., 2009), first-graders have also shown to be capable of discussing complex topics, such as power and inequality, through mathematics (murphy, 2009). interestingly, monica later stated that first-graders know a lot and have opinions about everything, which contradicted everything else she said in the interview, and which led us to conjecture that, at least in her case, the belief that children are not capable of understanding these issues was not firm. research points to the importance of field placements of psts in diverse schools and communities (e.g., villegas & lucas, 2002; zeichner, 2010), and we believe that it would be beneficial for the psts to interact with young students who live in marginalized or underprivileged communities and learn from them about their understanding of the issues they face. an illuminating case. amidst uncertainty about the meaning of and openness to teaching controversial topics, one pst’s interview responses were pointedly different from those of her peers. we include her case because we think that her different responses and background offer insight into possible paths for preparing psts to teach mathematics through real-world contexts, and, in particular, controversial issues. this pst, laura, had a strong mathematical background, grew up in the low-income area surrounding the university, and was a residence assistant (ra) on campus. we think that these three factors played an important part in shaping her interview responses, but we will only discuss the second and third here. unlike her peers, laura had ready examples for both real-world and controversial issues that could be used when teaching mathematics. for example, she talked about a problem that would involve finding the number of school lunches that a celebrity’s income could purchase, and noted that this problem would be fun while also relating to students’ lives. drawing on her experience volunteering in an elementary school in her community, she gave examples of story problems that students were likely to relate to about students who speak multiple languages, who have parents living out of state, or who live in very large households. in all three examples she emphasized that the goal would be for students to relate to the context and to see that they are not the only ones dealing with difficult situations such as living away from a parent. we think that her familiarity with the community and with the students’ circumstances helped her to see these circumstances as assets and simic-muller et al. role of controversial topics journal of urban mathematics education vol. 8, no. 2 73 potential contexts for mathematics problems rather than as issues to avoid because they are too controversial or too difficult for students to grasp. laura commented on this when she contrasted her own perspective with those of her peers, noting, “i feel like being from around here makes my perspective a little different.” also unlike her peers, laura did not think that controversial issues were only for some student populations. although she also thought that fifth grade was the right age for having in-depth conversations about real-world mathematical contexts, her concerns were more of a mathematical nature. for example, she worried about using school demographics with younger students because of percentages, but was still willing to create simpler problems for younger students (e.g., missing addend problems) using school data. laura was comfortable discussing social justice issues, due, it appears, to the social justice programming that she is responsible for as an ra. for instance, when initially asked about her interest in integrating realworld issues in the classroom she brought up social justice issues in connection with her work as an ra: just like with my ra position, i have learned about so much more about mostly social justice issues, how it’s more than just skin color obviously, your gender, so i think that i’ve learned a lot from that and i am excited to be able to incorporate what i have learned. she easily related some of her duties with teaching, in particular discussing the awareness of the lack of resources that many low-income students experience. when she later stated that discussing some issues, such as “abuse or jail rates… might hit [students] really hard if their family has been involved in that,” she again made connections back to her experience as an ra. she reflected on how she has to find ways to have conversations with residents who have been affected by particular issues, such as suicide, before making announcements or posting fliers for programming related to that topic. she maintained that, analogously, she would still incorporate controversial topics in the classroom, after checking in with students first. although she noted that checking in with students about discussing controversial issues that may affect them would be “something i would learn over time how to [do],” laura’s experiences as ra seem to have provided her with both a greater knowledge of social justice issues in general and a higher comfort level with discussing these issues. laura ended her interview with a question, “do people not want to include them in their teaching? i am thinking why would someone not want to include realworld issues in their teaching? maybe it’s just me being very naïve.” we found that almost all the survey respondents and all the interviewees said that they did want to include real-world issues in their teaching, and in the next section discuss our recommendations for helping psts learn how to do so effectively. simic-muller et al. role of controversial topics journal of urban mathematics education vol. 8, no. 2 74 recommendations and implications for research as mentioned, we are yet unable to make generalizations about psts’ beliefs based on our survey and interviews. however, based on our results we have several recommendations for research and practice in mathematics teacher education, with the goal of preparing psts to become more open, willing, and able to integrate controversial and social justice issues into their future mathematics teaching. in particular, we discuss the value of broadening psts’ understandings of what real-world contexts can be integrated with mathematics, challenging their assumptions about who can benefit from exploring controversial issues, thinking carefully about the sequencing of real-world topics, and exposing psts to social justice issues in a variety of contexts. we also identify directions for future work in each area. understanding of real-world issues. the psts we surveyed readily agreed with the use of real-world contexts in their future teaching, but further probing suggests that their understanding of what this looks like largely corresponds to familiar textbook scenarios and word problems as opposed to more authentic uses of mathematics outside the classroom in a variety of contexts. moreover, a significant number of the interviewed psts thought that many controversial issues were inappropriate for particular groups of students, and especially for young children. therefore, we suggest providing psts with a broad range of examples of realworld contexts and controversial issues that can be integrated with mathematics at a variety of grade levels. developing a rich set of examples of how to integrate mathematics and controversial issues, especially in the younger grades, remains an important task for educators committed to equity-oriented mathematics teaching. in addition, we suggest providing opportunities for psts to identify topics that they see as relevant or appropriate for students to explore and then support them in finding ways to explore these topics mathematically. such an approach would have the added benefit of supporting psts in developing their understandings of mathematical modeling by supporting them in seeing how to mathematize real-world phenomena (ccssi, 2010; felton, anhalt, & cortez, 2015; koestler, felton, bieda, & otten, 2013). challenging assumptions about who can benefit from exploring controversial issues. one of our possible explanations for why psts were ambivalent about controversial topics in the survey was that they did not consider them appropriate for everyone, but only some of the students, most notably older students, but also those belonging (or not belonging) to particular racial and/or ethnic groups. as discussed, we take the position that young children are capable of having difficult conversations about their lives and communities; researchers and educators have documented examples of rich and meaningful work with students of all ages, and in particular pre-k–8 students (e.g., turner et al., 2009; denny, 2013; gutstein, 2006; peterson, 2013; varley gutiérrez, 2013; turner, 2012), showing that children are not only simic-muller et al. role of controversial topics journal of urban mathematics education vol. 8, no. 2 75 aware of the issues that affect them and others, but are also capable of having complex conversations around these issues and engaging in complex mathematics to explore them in more depth. in response to the dismissal of young children’s ability to comprehend controversial issues, we have begun to respond to our own psts by arguing that children learn about many things in school that they will not fully comprehend, either immediately or over the course of their formal education. however, we start the conversation about these topics with students, and it grows over time. this growth in comprehension is true both with controversial real-world issues and with typical school content. for example, a young child will not “fully understand” everything about addition but can begin learning about it. we argue that, in principle, realworld, controversial, and social justice issues should be no different. however, as with other school content, we also acknowledge that topics must be introduced in ways that are attentive to children’s current understandings as well as their local contexts. therefore, as discussed, the development of a range of examples—not only of how to integrate mathematics and controversial issues but also of the kinds of conversations children are capable of having around these issues—can inform both the psts’ views about what is possible in the classroom and the field’s understanding of how to introduce these issues in age appropriate ways. it is also important to challenge the belief that social justice should only be discussed in classrooms with a particular racial and/or ethnic makeup. we find especially useful the notion that the (mathematics) curriculum should function both as a window into other perspectives or issues that students may not be familiar with and as a mirror reflecting back students’ interests and concerns (gutiérrez, 2007; tate, 1994). however, much of the work done focusing on social justice mathematics has been focused on “urban youth of color” or their teachers (brantlinger, 2013) and on the “mirror” portion of the above metaphor. thus, as has been emphasized in discussions of social justice more broadly (see, for example, swalwell, 2013), we again call for incorporating into mathematics teacher education programs a broader range of examples of the nuances of integrating social justice issues into mathematics across a diverse range of classrooms, in which mathematics serves both as a mirror and as a window into people’s lives. sequencing with care. according to enterline and colleagues (2008), psts differ in the degree “to which they understand, accept, believe and are prepared to teach in ways consistent with social justice principles” (p. 273). the goal of cmrws is to measure this degree of agreement, which should provide valuable information to teacher educators. in particular, the survey can help us identify social justice content that is easiest for psts to agree with and accept. because psts are especially resistant to ideas that clash with their beliefs (ambrose, 2004; pajares, 1992), one possible strategy is to introduce psts to ideas of social justice through simic-muller et al. role of controversial topics journal of urban mathematics education vol. 8, no. 2 76 contexts they find more acceptable. as cmrws results show, psts readily agree with the use of real-world contexts but have a more difficult time with controversial issues. while we did not focus on injustices here, we found that psts were somewhat more likely to agree with using injustices in their teaching than controversial issues (40.9% vs. 22.7%). we consequently conjecture that controversial issues that can be perceived as injustices (for example, past appropriation of indigenous lands, the gender wage gap, or unequal access to water around the world) may be easier for psts to engage with and can be more influential in changing their perceptions about controversial issues in general. more work is needed to examine (a) how psts respond to particular topics, (b) how the sequencing of topics affects psts reactions, and (c) how psts’ responses relate to their background and existing beliefs. it may be that there is large agreement among psts from a variety of backgrounds about which real-world topics they view as largely neutral and which they see as controversial or as dealing with injustice. or it may be that how they take up a topic depends greatly on their background and how the topic is framed. local context also plays a part in determining which topics are controversial. for example, while it is acceptable to discuss the cost of war in some parts of the country, in the area where the first author’s university is located (next to a military base), this topic is extremely controversial. one pst, briana, discussed this in her interview, noting, “you could talk about the war, but talk about supporting the troops” and “with all the military families, that’s where all the controversy comes in.” a better understanding of how psts interpret real-world contexts and which contexts they are initially most open to considering in their own teaching can inform mathematics teacher educators’ practice. exposing psts to social justice contexts in and out of the classroom. of the nine psts interviewed, linda stood out in her comfort with and willingness to discuss and explore issues of social justice. as discussed, this was due, in part, to her work as a residence assistant in the dormitories, which included training that required her to attend and organize social justice programming. the other psts interviewed did not have similar experiences; although through other courses they may have experienced similar topics, they had not participated in similar experiences outside of their academic coursework. we have to be realistic about what we as teacher educators can accomplish in a threeor four-credit semester-long course. for example, even though the interviewed students were exposed to some controversial issues in their mathematics methods course, they did not understand what these issues were or how to use them in mathematics teaching. despite evidence that mathematics content and methods courses can be powerful sites for beginning a conversation with psts about these issues (bartell, 2012; ensign, 2005; felton, 2012; felton & koestler, 2012, 2015; felton, simic-muller, & menéndez, 2012; koestler, 2012; mistele & spielman, 2009; spielman, 2009), there is little evidence that a single course will have a large or lasting effect on psts beliefs or their prac simic-muller et al. role of controversial topics journal of urban mathematics education vol. 8, no. 2 77 tices. teacher education programs committed to issues of equity and social justice must look for ways to provide opportunities for ongoing interaction with social justice issues both throughout their course work (mcdonald, 2005; nieto, 2000; villegas & lucas, 2002; zeichner, 2010) and outside of the classroom. we also suggest that teacher educators consider assignments within their own courses that will encourage psts to interact with or learn about social justice issues in their local context, such as observing a protest or interviewing students, parents, or community members about their concerns. conclusion in this article, we introduced the connecting mathematics to the real world scale (cmrws) for measuring psts’ beliefs about connecting mathematics to four kinds of real-world topics: (a) everyday or career related topics, (b) controversial issues, (c) issues of injustice, and (d) family backgrounds or community practices. the cmrws is unique in its focus on a range of types of real-world contexts and their use in the mathematics classroom. reflecting our interest in supporting psts in engaging in teaching mathematics for social justice, we focused our analysis on psts’ beliefs about using controversial issues in the mathematics classroom. psts’ showed the least willingness to use controversial issues. interviews with psts suggest that their concerns with controversial issues stem primarily from concerns about the types of students they believe are ready to have these conversations in terms of age and race and/or ethnicity. moreover, many of the psts seem to have limited experience with how to use mathematics to explore controversial issues. by developing a tool to triangulate psts’ beliefs with other qualitative forms of data we can better understand how teacher education programs affect psts’ beliefs and can ultimately lead to identifying learning trajectories based on different incoming past experiences. we believe this, coupled with a greater range of examples of teaching mathematics with controversial issues, may better prepare psts to engage in teaching mathematics for social justice. references aguirre, j. m. 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(2005). teachers’ characteristics: research on the demographic profile. in m. c. smith & k. m. zeichner (eds.), studying teacher education: the report of the aera panel on research and teacher education (pp. 111–156). mahwah, nj: erlbaum. simic-muller et al. role of controversial topics journal of urban mathematics education vol. 8, no. 2 82 appendix a connecting mathematics to the real world scale (cmrws)* we are interested in learning your views about connecting mathematics to real-world situations and to people’s lives. each section of the survey defines a specific kind of mathematical connection for you to consider. we appreciate your time in completing this questionnaire. using the scale below, please indicate how much you agree or disagree with each statement by circling a response. 1 = strongly disagree 2 = disagree 3 = neutral 4 = agree 5 = strongly agree real-world situations are everyday or career related topics. examples include choosing a cell phone plan, designing buildings, using math for a job, connecting math to artistic designs, or solving scientific problems. 1. when i teach mathematics, i will make connections to real-world situations. 1 2 3 4 5 2. i am interested in learning how to make connections to real-world situations when teaching mathematics. 1 2 3 4 5 3. when i teach mathematics, i will focus on mathematical concepts (e.g., addition and subtraction, geometric shapes, etc.), and not worry about using real-world situations. 1 2 3 4 5 4. when teaching mathematics, real-world situations can distract students from learning the important mathematical concepts. 1 2 3 4 5 5. an advantage to teaching mathematics with real-world situations is that they help students learn about the world around them. 1 2 3 4 5 6. teaching mathematics with real-world situations helps students learn the mathematical concepts better. 1 2 3 4 5 *not for use without permission of the authors simic-muller et al. role of controversial topics journal of urban mathematics education vol. 8, no. 2 83 using the scale below, please indicate how much you agree or disagree with each statement by circling a response. 1 = strongly disagree 2 = disagree 3 = neutral 4 = agree 5 = strongly agree controversial issues are topics that will likely be viewed as contentious or debatable. not everyone agrees on what topics are controversial, but some examples might include the costs of the war on drugs, government spending, funding for schools, or climate change. when responding to questions on the survey about controversial issues, imagine topics that you consider controversial. 7. when i teach mathematics, i will make connections to controversial issues. 1 2 3 4 5 8. i am interested in learning how to make connections to controversial issues when teaching mathematics. 1 2 3 4 5 9. when i teach mathematics, i will focus on mathematical concepts (e.g., add ition and subtraction, geometric shapes, etc.), and not worry about using controversial issues. 1 2 3 4 5 10. when teaching mathematics, controversial issues can distract students from learning the important mathematical concepts. 1 2 3 4 5 11. an advantage to teaching mathematics with controversial issues is that they help students learn about the world around them. 1 2 3 4 5 12. teaching mathematics with controversial issues helps students learn the mathematical concepts better. 1 2 3 4 5 13. an advantage to teaching mathematics with controversial issues is that some students identify these issues as important to them. *not for use without permission of the authors simic-muller et al. role of controversial topics journal of urban mathematics education vol. 8, no. 2 84 using the scale below, please indicate how much you agree or disagree with each statement by circling a response. 1 = strongly disagree 2 = disagree 3 = neutral 4 = agree 5 = strongly agree issues of injustice are topics that are likely to be seen as a form of injustice or wrongdoing. not everyone agrees on what topics represent injustices, but some examples might include animal cruelty, amount of living space in refugee camps, or deaths from preventable diseases. when responding to questions on the survey about issues of injustice, imagine topics that you consider injustices. 14. issues of injustice are also controversial issues. 1 2 3 4 5 15. when i teach mathematics i will make connections to issues of injustice. 1 2 3 4 5 16. i am interested in learning how to make connections to issues of injustice when teaching mathematics. 1 2 3 4 5 17. when i teach mathematics i will make connections to issues of injustice related to gender. for example, comparing women's and men's pay or looking at differences in the number of women and men in different professions. 1 2 3 4 5 18. when i teach mathematics i will make connections to issues of injustice related to people's income or wealth levels. for example, showing the difficulty of making ends meet on a minimum wage job. 1 2 3 4 5 19. when i teach mathematics i will make connections to issues of injustice related to people's race or ethnicity. for example, comparing differences in funding for schools in predominantly black, latino/a, and white neighborhoods. 1 2 3 4 5 20. when i teach mathematics i will make sure my students have opportunities to take action to address issues of injustice. for example, writing a letter to a government representative. 1 2 3 4 5 21. when i teach mathematics i will focus on mathematical concepts (e.g., add ition and subtraction, geometric shapes, etc.), and not worry about using issues of injustice. 1 2 3 4 5 22. when teaching mathematics, issues of injustice can distract students from learning the important mathematical concepts. 1 2 3 4 5 23. an advantage to teaching mathematics with issues of injustice is that they help students learn about these issues in the world around them. 1 2 3 4 5 24. teaching mathematics with issues of injustice helps students learn the mathematical concepts better. 1 2 3 4 5 25. an advantage to teaching mathematics with issues of injustice is that some students identify these issues as important to them. 1 2 3 4 5 *not for use without permission of the authors simic-muller et al. role of controversial topics journal of urban mathematics education vol. 8, no. 2 85 demographics gender 1 = male 2 = female 3 = other state race how do you identify your race? (choose all that apply) 1 = white or caucasian 2 = black or african american 3 = american indian or alaska native 4 = asian 5 = native hawaiian or other pacific islander 6 = other are you of hispanic, latino/a, or spanish origin? 1=no 2=yes what is your age? 1 = 18–23 2 = 23–30 3 = 30–40 4 = 40–50 5 = 50–60 6 = 60+ what is the highest level of school that your primary parent(s) or guardian(s) have completed? there is room to answer for up to two primary parents or guardians. parent or guardian 1 1 = less than a high school degree 2 = high school degree or equivalent (e.g., ged) 3 = some college but no degree 4 = associate degree 5 = bachelor’s degree 6 = master’s degree 7 = professional school degree (e.g., m.d., j.d.) 8 = doctorate degree parent or guardian 2 1 = less than a high school degree simic-muller et al. role of controversial topics journal of urban mathematics education vol. 8, no. 2 86 2 = high school degree or equivalent (e.g., ged) 3 = some college but no degree 4 = associate degree 5 = bachelor’s degree 6 = master’s degree 7 = professional school degree (e.g., m.d., j.d.) 8 = doctorate degree teaching interest after graduation (choose the one that best applies) 1 = preschool–2 2 = 3–5 3 = 6–8 4 = 9–12 5 = other classroom teaching experience (do not include experiences working in classrooms that were part of your teacher preparation program. include only the years you were a teacher.) 1 = none 2 = 0–4 years 3 = 5–10 years 4 = 11–20 years 5 = 20+ years journal of urban mathematics education december 2014, vol. 7, no. 2, pp. 1–5 ©jume. http://education.gsu.edu/jume david w. stinson is an associate professor of mathematics education in the department of middle and secondary education in the college of education, at georgia state university, p.o. box 3978, atlanta, ga, 30303; e-mail: dstinson@gsu.edu. his research interests include exploring socio-cultural, -historical, and -political aspects of mathematics and mathematics teaching and learning from a critical postmodern theoretical (and methodological) perspective. he is a cofounder and current editor-in-chief of the journal of urban mathematics education. editorial teaching mathematics for social justice: an ethical and moral imperative? david w. stinson georgia state university fear and stress in the police department (the new york times, march 5) – the acquittal last week of officer william l. walker by an all-white jury in brooklyn on charges that he murder a young black man named john brabham would be troubling in any event. unfortunately, in context the walker case is even more disturbing. over the past four years, three other blacks—clifford glover, claude reese jr. and randolph evans—have been shot and killed by white police officers in new york in circumstance that have frightened and enraged residents of black communities and have troubled thoughtful citizens everywhere. (editorial board, ¶ 1, emphasis added) rapes at hunter spark student protest (the new york times, september 30) – more than 100 student demonstrators, angered over the rape of three students at hunter college in the last two months, invaded the office of the school president yesterday to demand more guards and tighter security, which was cut over the summer as a result of the city’s fiscal crisis. … afterward, the dean announced several limited security measures. but she emphasized that students [i.e., the female students] should take greater personal precautions and noted that no funds were available to hire more guards to patrol the 16story main building, which has 15 exits, hundreds of classrooms, offices and laboratories and thousands of students. (mcfadden, ¶ 1 & 3; emphasis added) mexicans protest an intensification of inspections at border in el paso (the new york times, march 12) – mayor ray salazar castigated the immigration and naturalization service yesterday for having created “a potentially dangerous international situation” along the united states–mexico border here by stepping up its inspections. (crewdson, ¶ 1; emphasis added) he often-quoted epigram the more things change, the more they stay the same is attributed to the 19th century french journalist and satirist jeanbaptiste alphonse karr. those who often find themselves on the non-privileged side of discursive identity binaries (cf. derrida, 1974/1997) can certainly attest to the paradoxical truth found in the nearly two centuries old saying (e.g., white/non-white, man/non-man, wealthy/non-wealthy, able/non-able, christian/non-christian, citizen/non-citizen, english speaking/non-english speaking, t http://education.gsu.edu/jume mailto:dstinson@gsu.edu stinson editorial journal of urban mathematics education vol. 7, no. 2 2 heterosexual/non-heterosexual, etc.). the above headlines with accompanying introductory text, pulled from the new york times, clearly illustrate this change–same cycle, if you will. in that, these headlines, which can readily be mapped onto recent national events, are neither from last week, last month, or last year, nor even from the last decade, but rather from the mid-to-late 1970s (specifically, 1977, 1975, and 1979, respectively). why headlines from the mid-to-late 1970s? these years were my teenage years (i graduated from high school in 1979). i have been thinking a lot about my teenage years recently with the ongoing realization that many present injustices are, unfortunately and eerily, too often repeats of the past. nevertheless, in many ways, both as a child and as a teenager, i was oblivious to most national (and global) injustices that occurred during the 1960s and 70s. back then, it was as if children, even teenagers, were somehow protected or shielded from being aware of the injustices of the day; that is, unless the injustices were directed toward them and/or their community. or, more aptly, i should say, being shielded from the injustices of the day was true for most of the children in the racially (white) and religiously (protestant) segregated, blueand white-collar, lower middle class community in which i grew up. in making such a statement, i clearly recognize the danger in both romanticizing the past and generalizing my childhood. i wish to do neither. but, suffice it to say, most communities (those with privilege and those without) in the 1960s and 70s had some means of shaping messages about injustices for their children (even if that shaping meant not mentioning injustices at all). today, however, it is practically impossible for children and teenagers to escape from being aware (some more so than others) of present and past injustices. it matters not, for example, if the injustice happens in ferguson, missouri; charlottesville, virginia; or austin, texas; awareness of injustices is no longer isolated to particular individuals or groups and/or communities. with access to facebook; twitter; google; and tens of dozens of blogs, print and online magazines and newspapers, and radio and television stations (many specifically targeted to children and teenagers), children of all ages, from all communities, are aware (some more so than others) of local, national, and global injustices.1 and although children in the united 1 access to information is a change that will never be the same and will be forever changing. borgman (2000), however, provides some important caveats to this statement: in view of the undisputed magnitude of some of these developments [increased access to information through technology], it is reasonable to speak of a new world emerging. it is not reasonable, however, to conclude that these changes are absolute, that they will affect all people equally, or that no prior practices or institutions will carry over to a new world. nor is it reasonable to assume that any individual institutions, whether libraries, archives, museums, universities, schools, governments, or businesses, will survive unscathed and unchanged into the next millennium. strong claims in either direction are dangerous and misleading, as well as lacking in intellectual rigor. (p. 3) stinson editorial journal of urban mathematics education vol. 7, no. 2 3 states hail from literally tens of thousands of different communities, 50 million or so share a common experience: they attend one of the nearly 100,000 u.s. pre-k–12 public schools. furthermore, given the privileged status (justified or not) of the discipline of mathematics in u.s. public school curricula, these nearly 50 million children also share the common experience of mathematics instruction throughout the school year (if not every day, nearly every day). as i have been comparing my teenage years (or my childhood more broadly) with teenagers today, i have been reflecting on my current profession as a mathematics teacher educator as well as my previous profession as a public high school mathematics teacher. in doing so, i have been asking several questions in light of certain recent national events. given children and teenagers’ increased awareness of social injustices, what are the ethical and moral obligations of mathematics teacher educators and classroom teachers in using injustices as a catalyst for mathematics teaching and learning? does such an ethical and moral imperative exist? is a mathematics teacher educator or classroom teacher being ethical if she or he chooses to close the door (i.e., close off the world) to her or his mathematics methods course or algebra ii course to teach “best practices” or “families of function” without engaging in discussions about present (and past) injustices? as the most privilege discipline of study in schools, do mathematics teacher educators and classroom teachers have a unique civic responsibility in leading efforts of teaching and learning for social justice in our u.s. public schools? do mathematics teacher educators and classroom teachers have a unique pedagogical responsibility in demonstrating to stakeholders (i.e., students, teachers, administrators, school board members, communities members, etc.) that teaching for social justice is not either–or but rather both–and: both social justice pedagogical goals and mathematics (or any other specific discipline) pedagogical goals (see gutstein, 2006, p. 23). there appears to be an abundance of questions to ask around the increasingly unfiltered awareness about injustices that children wrestle with daily, and the ethical, moral, civic, and pedagogical responsibilities of teachers and those who teach teachers. additional questions include: how might a teacher assist a child in making sense of that which is senseless? how might a teacher assist a child in moving beyond awareness of injustices toward analyses of injustices? how might a teacher assist a child in moving beyond analyses of injustices toward selfempowering actions against injustices? as mathematics teacher educators and classroom teachers, we clearly understand that mere awareness is not enough in problem solving: awareness is a necessary condition but not a sufficient condition. problem solving requires doing science and taking action (and here, the phrase doing science is left open to its multiplicitous possibilities). furthermore, as mathematics teacher educators and classroom teachers, we clearly understand that within the context of schools there is no better place to do science on problem stinson editorial journal of urban mathematics education vol. 7, no. 2 4 solving than the mathematics classroom. it just seems natural, then, that the mathematics classroom would be one of the first places that the problem of injustice (in all its forms) would be used as a catalyst for teaching and learning rigorous science—in this case, the mathematical sciences (see, e.g., gutstein & peterson, 2013). given the profusion of injustices and children’s increasing awareness of those injustices, why has there not been a collective effort to integrate teaching mathematics for social justice throughout mathematics curricula (e.g., similar to integrating technology throughout mathematics curricula)? after more than three decades of research and scholarship on social justice (or critical) mathematics (see, e.g., d’ambrosio, 2012; frankenstein, 2012; gutstein, 2012; powell, 2012; skovsmose, 2012), is it not time for social justice mathematics to become not only an integral component of the “canon” of mathematics teacher education but also strategically integrated throughout the eight standards for mathematical practice? in the end, as mathematics teacher educators and classroom teachers, if we choose not to engage in the “empowering uncertainties” (stinson & wager, 2012, p. 3) of teaching and learning mathematics for social justice, are we failing to uphold our ethical, moral, civic, and pedagogical responsibilities? references borgman, c. l. (2000). the premise and the promise of global information infrastructure, in from gutenberg to the global information infrastructure: access to information in the networked world (1–32). cambridge, ma: mit press. crewdson, j. m. (1979, march 12). mexicans protest an intensification at border in el paso. the new york times, p. a14. d’ambrosio, u. (2012). a broad concept of social justice. in a. a. wager & d. w. stinson (eds.), teaching mathematics for social justice: conversations with educators (pp. 201– 213). reston, va: national council of teachers of mathematics. derrida, j. (1997). of grammatology (g. c. spivak, trans., corrected ed.). baltimore, md: johns hopkins university press. (original work published 1974) editorial board. (1977, march 5). fear and stress in the police department [editorial]. the new york times, p. a18. frankenstein, m. (2012). beyond math content and process: proposals for underlying aspects for social justice education. in a. a. wager & d. w. stinson (eds.), teaching mathematics for social justice: conversations with educators (pp. 49–62). reston, va: national council of teachers of mathematics. gutstein, e. (2006). reading and writing the world with mathematics: toward a pedagogy for social justice. new york, ny: routledge. gutstein, e. (2012). reflections on teaching and learning mathematics for social justice in urban schools. in a. a. wager & d. w. stinson (eds.), teaching mathematics for social justice: conversations with educators (pp. 63–78). reston, va: national council of teachers of mathematics. gutstein, e., & peterson, b. (eds.). (2013). rethinking mathematics: teaching social justice by the numbers (2nd ed.). milwaukee, wi: rethinking schools. stinson editorial journal of urban mathematics education vol. 7, no. 2 5 mcfadden, r. d. (1975, september 30). rapes at hunter spark student protest. the new york times, p. a34. powell, a. p. (2012). the historical development of criticalmathematics education. in a. a. wager & d. w. stinson (eds.), teaching mathematics for social justice: conversations with educators (pp. 21–34). reston, va: national council of teachers of mathematics. skovsmose, o. (2012). critical mathematics education: a dialogical journey. in a. a. wager & d. w. stinson (eds.), teaching mathematics for social justice: conversations with educators (pp. 35–47). reston, va: national council of teachers of mathematics. stinson, d., & wager, a. (2012). a sojourn into the empowering uncertainties of teaching and learning mathematics for social change. in a. a. wager & d. w. stinson (eds.), teaching mathematics for social justice: conversations with educators (pp. 3–18). reston, va: national council of teachers of mathematics. microsoft word final battista vol 3 no 2.doc journal of urban mathematics education december 2010, vol. 3, no. 2, pp. 34–46 ©jume. http://education.gsu.edu/jume michael t. battista is a professor of mathematics education in the school of teaching and learning, 341 arps hall, ohio state university, columbus, oh, 43210; email: battista.23@osu.edu. his research focuses on how students’ knowledge of and fluency with mathematics develops, and how teachers understand and use research-based learning progressions. response commentary engaging students in meaningful mathematics learning: different perspectives, complementary goals michael t. battista the ohio state university ecently, the question “where’s the mathematics in mathematics education research?” has been raised in kathleen heid’s (2010) editorial in the march 2010 issue of the journal for research in mathematics education (jrme) and in a research symposium at the 2010 national council of teachers (nctm) research presession (the symposium panelist were deborah ball, guershon harel, patrick thompson, and myself; jere confrey was the discussant). in this issue of jume, danny martin, maisie gholson, and jacqueline leonard (2010) provide a commentary on this question. as a panelists at the research presession, i respond to martin et al.’s commentary here. after a brief preface, i address disagreements that i have with martin et al.’s commentary, and then i address some of the issues within alternative perspectives. preface: there’s always an interpretative bias i accept that the critical theory approach of deconstructing accepted realities can lead to important insights. for instance, consider the september 29, 2004 news story in the boston globe: washington – the supreme court agreed tuesday to decide when governments may seize people’s homes and businesses for economic development projects, a key question as cash-strapped cities seek ways to generate tax revenue. at issue is the scope of the fifth amendment, which allows governments to take private property through eminent domain, provided the owner is given “just compensation” and the land is for “public use.” susette kelo and several other homeowners in a working-class neighborhood in new london, connecticut, filed a lawsuit after city officials announced plans to raze their homes to clear the way for a riverfront hotel, health club and offices. the residents refused to budge, arguing it was an unjustified taking of their property. (yen, 2004) r battista response commentary 
 journal of urban mathematics education vol. 3, no. 2 35 a deconstruction of this story that provides a different perspective than that of the journalist’s is as follows: “one group of rich and powerful people [the justices of the supreme court] will decide if smaller groups of locally powerful people [local governments] can force less powerful people [ordinary citizens] to sell their property.” the original perspective presented in the news story is based on cultural concepts that socially and mentally, but implicitly, take as given certain kinds of relationships between people, privileging some people over others. within the social institution we call “government,” referring to some groups of people as the “supreme court” or “local governments” gives these groups special powers and makes their actions and pronouncements seem unassailable. a similar theoretical lens can be used to view all of our social institutions. but in using critical theory, it is important to be mindful that maintaining any interpretive perspective on the world, including a critical-theory perspective, necessarily creates an interpretative bias. for instance, consider the following claim by martin et al. (2010): heid’s (2010) commentary and question, as well as the symposium summary and harel’s aforementioned statement, are not neutral (blair, 1998). they are political statements and represent particular stances and positions on the value and production of knowledge. they should be acknowledged, recognized, and deconstructed as such (2010, p. 13). consistent with a critical theory perspective, martin et al. (2010) seem to use the term “political” to connote some hidden, but explicit, agenda to disenfranchise certain groups of scholars. however, disregarding harel’s (2010) and heid’s (2010) intentions (which i do not know), i believe that a critical theory perspective is biased against accepting the notion that not every researcher statement of beliefs about the nature of mathematics education research should be construed as political in a manipulative sense.1 for instance, i believe that it is important to maintain a distinct identity for the field of mathematics education research, a field that struggled for identity at its inception, and is struggling again to find a role in the political battles for control of the education system in this country. but it would be a mistake to construe my statement as political in the sense that i wish to exclude certain kinds of scholarship from the field. i believe that it is legitimate and expected for researchers to debate what the identity should be for mathematics education research, and i believe that this identity will naturally and necessarily change, grow, and mature over time. furthermore, i contrast my “non























































 1 even if we expand the meaning of “political” to apply to the situation described by martin et al. (2010), as the oxford english dictionary definition below indicates, it is not clear that the term should have a negative connotation in the sense martin et al. are claiming: “1. of, belonging to, or concerned with the form, organization, and administration of a state, and with the regulation of its relations with other states” (oxford english dictionary, online at http://www.oed.com/). battista response commentary 
 journal of urban mathematics education vol. 3, no. 2 36 political” belief with intentionally political attempts to disenfranchise scholars in mathematics education as in the work of the national mathematics advisory panel (2008). despite my disagreement with several claims made by martin et al. (2010), as described below, i view their comments as important cautionary and evolutionary arguments about our field. while i may disagree with some of the tenets of critical theory, i agree that “this research can make, and is making, positive contributions to the identity of mathematics education research” (p. 16). disagreements and alternative perspectives before i provide alternative perspectives to some of the issues discussed in martin et al.’s (2010) commentary, i note several disagreements. first, i am not a researcher who has, as martin et al. argue, a “concern … regarding the lack of attention to mathematics” in mathematics education research. i actually do not know why i was asked to be a panelist at the research presession symposium. i suspect i was chosen because my research, which investigates cognition, nevertheless has a strong focus on mathematics. however, as i will outline below, i believe that many kinds of research have much to say about the overall task of educating children in mathematics. for instance, research on motivation, although not specific to mathematics education, is most certainly relevant for the big picture in mathematics education. thus, although i believe that the field of mathematics education research needs its own identity, i also believe that many other fields of research that do not focus specifically on mathematics are extremely valuable to mathematics education. furthermore, because these other fields of research are important to mathematics education, it is natural that mathematics education research expands to include sustained efforts of researchers to apply these fields to mathematics education. second, as martin et al. (2010) continue their discussion, they state: the implications for such exercises of power, under the auspices of an institutional and organizational entity such as nctm, are profound, as they have the potential to marginalize scholarship within particular areas of focus as well as marginalize scholars who devote themselves to this work. young scholars and graduate students are particularly vulnerable if the subtext of these statements is on pursuing what is valued in the field. (pp. 13–14) clearly the amount of marginalization of scholars’ work varies greatly with the local context. for instance, in my experiences with several universities’ colleges of education tenure and promotion deliberations, candidates’ whose scholarly work was broader than mathematics education, being for example well positioned in more general american educational research association contexts, actually battista response commentary 
 journal of urban mathematics education vol. 3, no. 2 37 had enhanced chances for tenure and promotion. and in my work on various nctm committees, members constantly and consciously sought inclusion of scholars whose areas of expertise focused on just the issues that martin et al. represent—actions oriented toward inclusion not marginalization. furthermore, special issues of journals, like the special issues of jrme on equity, are designed to highlight work, to bring its importance to the forefront. so publishing in special issues does not marginalize authors—it seems to enhance their reputations as specialists (and preeminence of expertise is what tenure and promotion committees look for during deliberations). finally, almost any scholar’s research can be marginalized in some contexts. for instance, recently a well-known senior scholar told me that he worried about receiving tenure in a department of educational psychology because he conducted his research in schools rather than university labs. and often, tenure and promotion committees in research i universities devalue scholarly articles written for nctm’s “practitioner journals,” seemingly disregarding the obligation that educational researchers should have to connect their research with instructional practice. third, consider the following statement by martin et al. (2010): mathematics, as a subject domain, is not acultural, without context or purpose, including the political… yet many students perceive school mathematics to be a narrow set of rules and algorithms that have little or no meaning to their lives. is this the mathematics to which heid, harel, and, perhaps, the other panelists might be referring? mathematics can also be a tool for understanding the world and, in the case of marginalized students, it can aid in understanding the social forces that contribute to their marginalization. (p. 14) even a cursory inspection of modern research in mathematics education would highlight that the kind of mathematics that most mathematics education researchers strive to promote in students is the opposite of “a narrow set of rules and algorithms that have little or no meaning to their lives” (p. 14). indeed, to use mathematics to understand the world requires that mathematics itself makes sense to students. so most researchers in mathematics education focus on promoting student understanding and sense making. unfortunately, some mathematics curricula that strongly emphasize applications do not attend carefully enough to supporting students’ mathematical sense making because they disregard research on students’ construction of specific mathematical concepts and ways of reasoning. i return to this point later. in general, implicit in much of the discussion of heid (2010), the research presession panel, and martin et al. (2010), is the question: what is mathematics education research? instead of me dancing around this question, it is more forthright for me to reply. one answer is to say that mathematics education research is research conducted by scholars with a ph.d. in mathematics education. but some battista response commentary 
 journal of urban mathematics education vol. 3, no. 2 38 people who do research in mathematics education have ph.d.’s in mathematics, educational psychology, or cognitive psychology—so a definition based on degree seems inappropriate. another answer is that mathematics education research is research conducted by scholars who know and build on the research in mathematics education as represented in research journals dedicated solely to research on mathematics learning and teaching.2 such research investigates a variety of important questions about teaching and learning mathematics, and it uses a variety of methods and theoretical perspectives. however, even this second answer leaves related questions unanswered. first, even though i do not consider general research on, for example, motivation, self-efficacy, and educational policy as mathematics education research, such research is often invaluable to understanding how students learn, and how we can teach, mathematics in schools. second, there is a whole body of valuable research conducted by cognitive psychologists that, in general, seems to intersect little with mathematics education research as i have defined it. although i do not consider this research “mathematics education research,” it is extremely unfortunate that most scholars in the two fields do not interact regularly and productively. in some sense, then, maybe we are asking the wrong question. perhaps a better question is: what kinds of research is needed for mathematics educators to understand and improve mathematics learning and teaching? omari and understanding students’ construction of mathematical knowledge martin et al.’s (2010) discussion of a black student, omari, seems like the typical exercise of setting up a caricature “straw man” that is easy to knock over. in this description, the caricature researcher “characterizes omari’s misconceptions as reflecting low cognitive ability” (p. 18). construing omari’s misconceptions as reflecting low cognitive ability seems to me to ignore all that researchers have discovered about mathematics learning in the last 3 decades. it is actually much more likely that his difficulties are due to an impoverished curriculum and a poor instructional environment than low cognitive ability. martin et al. go on to imply that clements and sarama’s statement, “although low-income children have pre-mathematical knowledge, they do lack important components of mathematical knowledge” (as cited in martin et al., p. 18), is of the same ilk as our strawresearcher’s statement. but in and of itself, what is wrong with their statement? given that research strongly supports the notion that instruction must build on 























































 2 although the journal for research in mathematics education, educational studies in mathematics, and the journal of mathematical behavior are classic examples of such journals, i also include journals such as the journal of urban mathematics education and the journal of mathematics teacher education. of course, mathematics education researchers also frequently publish in general education research journals. battista response commentary 
 journal of urban mathematics education vol. 3, no. 2 39 students’ current ways of reasoning, clements and sarama’s statement suggests that special care needs to be taken in thinking about the experiences instruction must provide for certain children. coupled with clements and sarama’s (2007) success in promoting learning among disenfranchised students, i have difficulty seeing the validity of this criticism. moreover, the inadequacies of omari’s teacher are not limited to urban school districts (although the prevalence of these inadequacies is higher in such districts). for instance, one of my son’s teachers felt compelled to insert a note into my son’s permanent school folder that he preferred to use non-standard computational algorithms, which the teacher did not understand and therefore labeled as strange and aberrant. i had helped my son discover these algorithms; he deeply understood them; and later he used his understanding of these alternate algorithms to fully understand traditional algorithms and related algebraic manipulations. what may differ between my son and omari is that i, well positioned in the community and education, could be a supportive advocate for him. unfortunately, such advocacy is often absent not only for students like omari but also many other students facing curricula that ignore their current ways of thinking about mathematics. having supportive and influential advocates can be especially important in obtaining the best educational opportunities for students who have been traditionally disenfranchised (berry, 2008). deficits versus cognitive plateaus martin et al. (2010) go on to say: content-focused studies that ignore or simplify the larger social context have often helped to normalize these [deficit-oriented] constructions by suggesting, for example, that poor and minority children enter school with only pre-mathematical knowledge and lack the ability to mathematize their experiences, engage in abstraction and elaboration, and use mathematical ideas and symbols to create models of their everyday lives. (p. 20) i will first deal with this statement directly, then put this and other relevant issues in a wider context. first, i believe that deficit-oriented perceptions of students’ mathematics learning still predominate the world of mathematics teaching (especially so for traditionally disenfranchised students). these perceptions exist despite mathematics education researchers’ total reconception of learning in terms of students’ construction of knowledge. it is unfortunate, and for many students tragic, that this view of mathematics learning still pervades the field of mathematics education. second, however, martin et al.’s (2010) objection to characterizing students as having “pre-mathematical” knowledge is inconsistent with modern, cognitionbased theories of mathematics learning. a common thread in these theories is that, battista response commentary 
 journal of urban mathematics education vol. 3, no. 2 40 before instruction, all students have pre-mathematical knowledge of mathematics topics that they are first learning (although the nature and amount of such experience varies). the point of modern learning theories is that effective instruction helps students build on and transform their pre-mathematical knowledge into more formal knowledge in personally meaningful ways. indeed, the whole notion of research-based learning progressions is founded on the idea that to effectively support students’ learning of mathematical concepts and reasoning, instruction must help students progress through a detailed cognitive terrain that consists of many plateaus of increasingly sophisticated (often pre-mathematical) knowledge and reasoning (battista, 2001, 2010). without knowledge of the cognitive steps that students can and must take in moving from their intuitive to formal ideas, students most often resort to rote memorization or withdrawal from learning. given the importance of this issue, it is worthwhile to examine relevant research in more detail. how do children learn mathematics? current major scientific theories describing learning agree that students must personally construct ideas as they intentionally try to make sense of situations (bransford, brown, & cocking, 1999; de corte, greer, & verschaffel, 1996; greeno, collins, & resnick, 1996; hiebert & carpenter, 1992; lester, 1994; national research council, 1989; prawat, 1999; romberg, 1992; schoenfeld, 1994; steffe & kieren, 1994). from a “constructivist” perspective, a student’s mathematical “reality” is determined by the set of mental structures that the student has constructed and is currently using to deal with mathematical problems and situations. it is through these established structures, sometimes called frames, that the student interprets and builds subsequent mathematical experiences. in fact, these structures determine the very nature of those experiences: “framing provides a means of ‘constructing’ a world, of characterizing its flow, of segmenting events within this world. … after becoming accustomed to a certain kind of framing, the strip of reality interpreted accordingly appears for the individual as natural, evident, and somehow logical” (krummheuer, 1995, p. 250; cf. bruner, 1990). in particular, research in mathematics education has demonstrated repeatedly that students build new mathematics understandings out of their current relevant mental structures (e.g., battista, 2008; battista & larson, 1994; bransford et al., 1999; cognition and technology group at vanderbilt, 1993; hiebert & carpenter, 1992; mack, 1990; mccombs, 1993). furthermore, students’ construction of mathematics is enabled and constrained not only by internal cognitive factors but by cultural artifacts such as language and symbol/representation systems; by the social norms, interaction patterns, and mathematical practices of the various communities in which students participate; by direct interactions with other peo battista response commentary 
 journal of urban mathematics education vol. 3, no. 2 41 ple (including teachers); and by cultural backgrounds and contexts (berry, 2008; bruner, 1990; cobb & yackel, 1995; de corte et al., 1996; tate & rousseau, 2007). also, “a learner’s motivation to learn and sense of self affects what is learned, how much is learned, and how much effort will be put into the learning process” (national research council, 2002, p. 126). in the constructivist paradigm, selection of instructional tasks must be based on knowledge of students’ mathematics (steffe & d’ambrosio, 1995); the choice of tasks should be “grounded in detailed analyses of children’s mathematical experiences and the processes by which they construct mathematical knowledge” (cobb, wood, & yackel, 1990, p. 130). and this finding is not restricted to mathematics learning: “there is a good deal of evidence that learning is enhanced when teachers pay attention to the knowledge and beliefs that learners bring to a learning task, use this knowledge as a starting point for new instruction, and monitor students’ changing conceptions as instruction proceeds” (bransford et al., 1999, p. 11). an abundance of research has shown that mathematics instruction that focuses and builds on students’ personal sense making produces powerful mathematical thinkers who not only can compute but also have strong conceptions of mathematics and problem-solving skills (ben-chaim, fey, & fitzgerald, 1998; boaler, 1998; carpenter, franke, jacobs, fennema, & empson, 1998; clements & sarama, 2007; cobb et al., 1991; cramer, post, & delmas, 2002; fennema et al., 1996; hiebert, 1999; muthukrishna & borkowski, 1996; silver & stein, 1996; villaseñor & kepner, 1993; wood & sellers, 1996, 1997). a framework of mathematics engagement in my work in one middle school that could easily have been omari’s, in an attempt to examine the relevance of different perspectives and research paradigms for the school system’s explicit goal of actively addressing the disparity between minority and white students’ mathematics achievement and course taking, i developed a simple model of levels of student engagement in schools (think of the levels as reference points on a continuum; see figure 1). i was actually prompted to develop this model based on a statement of a former teacher who was the grandmother of a minority student in the school. she said: the poorer students have been given permission to give up. we have lots of students who have given up. they really don’t care. … you can see that the children who are failing are not engaged in the intellectual life of the school. they just tolerate; they just sit. battista response commentary 
 journal of urban mathematics education vol. 3, no. 2 42 level 0: students disengage from the intellectual life of school probably because of a mismatch between the goals and culture of school with some students, some “drop out” of the intellectual activities in school and involve themselves only in its social aspects. level 1: students engage in the intellectual life of school, but disengage from mathematics learning students attempt to involve themselves in academic aspects of school. but, perhaps because of past failures, or because students see no relevance of mathematics to their lives, students decide that doing well in, or even enrolling in, mathematics courses is not important to their lives. level 2: students engage in learning mathematics as memorization and mimicry, but disengage from mathematical sense making students do not find intrinsic value in learning mathematics. but because they have embraced the overall academic values of school, they still try to get good grades and enroll in appropriate mathematics courses. however, because traditional instruction has made personal sense making inaccessible for most students, these students resort to memorization and mimicry as the primary focus of learning. level 3: students engage in learning mathematics as sense making students attempt to make personal sense of mathematics. they not only find extrinsic, career-oriented value in mathematics but also intrinsic value in learning mathematics. figure 1. levels of engagement of students in mathematics learning. comments on the model 1. many general efforts to improve schooling focus on moving students from level 0 to level 1, and many mathematics-specific efforts from level 1 to 2. successful programs for getting students to participate in school learning, and learning mathematics in particular, are extremely valuable. however, many of these programs inadvertently get students only to level 2. 2. level 2 engagement is extremely difficult to maintain over the long run because mathematics is too complex to be learned by rote memorization. also, level 2 engagement does not produce students competent in problem solving and prepared for future learning. almost all students involved at level 2 will drop out of mathematics as soon as they can because rote learning inevitably leads to failure. (but families and cultural contexts can affect how long students “put up with” school activities that make little inherent sense to students.) 3. students’ actual level of engagement and the level of engagement aimed at by instruction can be very different. some students are able to engage in mathematics at level 3 even if their instruction is focused on level 2. some students involved in instruction aimed at level 3 will engage in large parts of it at battista response commentary 
 journal of urban mathematics education vol. 3, no. 2 43 level 2. supporting level 3 engagement of students requires a deep knowledge of how students construct meaning for particular mathematical topics. 4. an important research question is whether there is a level between levels 2 and 3. in this level, call it temporarily level 2.5, students do not make full sense of mathematics, but they use mathematics to investigate things that interest them. this situation might occur in “applications” oriented curricula that are not firmly founded on research-based learning progressions. i hypothesize that students at this level make more than rote sense of the mathematics they learn, but they do not make full sense of the underlying ideas. the major advantage of applications focused curricula is that they may interest students more than traditional curricula (although applications that interest adults are often not interesting to students). the major drawback of this approach is that, because of the lack of learningprogression guidance, students often do not fully understand the mathematical ideas, so eventually they apply mathematical procedures in inappropriate situations (battista, 2001). indeed, the issue of whether mathematics is best learned in an applications context (as is often emphasized in some reform and social justice approaches), or in a carefully structured instructional context for gradually building on students’ cognitions (a learning progressions approach) has not been resolved. i am convinced by a significant amount of research that the latter approach is quite effective (see aforementioned references). furthermore, the work of marsh suggests that self-concept (which i believe is connected to personal sense making) is more important than interest in mathematics achievement (e.g., marsh, trautwein, ludtke, koller, & baumert, 2005). however, perhaps the best approach is a blend of these two approaches. what is important about the levels-of-engagement model is that it helps place different kinds of research and educational programs in perspective— different educators and researchers focus on different levels of engagement. they do this based on their beliefs, their research interests, and their understanding of student learning. by necessity, researchers often focus on small parts of the enormous problem of educating students in mathematics because of the detail and care needed to deeply investigate phenomena. in my research, i focus mostly on level 3. why? partly because of interest—i find all students’ mathematical thinking fascinating. but more fundamentally, i believe that the strongest and most robust research we have in mathematics education is that teaching that is based on research-based knowledge of the development of students’ reasoning about particular topics in mathematics produces better student achievement than teaching that does not. however, i am certainly not unaware of the broader picture. it’s just that i believe that even if we successfully engage non-engaging students in trying to learn mathematics, their continued engagement depends critically on their being able to make sense of battista response commentary 
 journal of urban mathematics education vol. 3, no. 2 44 mathematics, and their mathematical sense making depends critically on instruction that is founded on research-based learning progressions. the levels-of-engagement model also makes absolutely clear that research on student engagement, motivation, self-efficacy, and identity—examined in broader social contexts—is absolutely critical to research and practice in mathematics education. the picture of omari painted by martin et al. (2010) presents a critically important perspective that must be included in the overall research program in mathematics education. a closing thought determining how best to help all students learn mathematics is extremely complex. so researchers, out of necessity, each focus on small parts of the problem. that does not mean that they consider other parts unimportant. indeed, i believe that we are all working on the same problem, that our work is complementary, but because the problem is so large and complex, we are working on the problem from different perspectives, each doing our own part. there is no one “right” perspective on this work, just different perspectives, each adding its own set of insights. references battista, m. t. (2001). how do children learn mathematics? research and reform in mathematics education. in t. loveless (ed.), the great curriculum debate: how should we teach reading and math? (pp. 42–84). brookings press. battista, m. t. (2008). development of the shape makers geometry microworld: design principles and research. in g. blume & k. heid (eds.) research on technology in the learning and teaching of mathematics: cases and perspectives (vol. 2, pp. 131–156). charlotte, nc: national council of teachers of mathematics/information age. battista, m. t. (2010). representations of learning for teaching: learning progressions, learning trajectories, and levels of sophistication. in p. brosnan, d. erchick, & l. flevares (eds.), proceedings of the 32nd annual meeting of the north american chapter of the international group for the psychology of mathematics education, columbus, ohio. battista, m. t., & larson, c. n. (1994). the role of the journal for research in mathematics education in advancing the learning and teaching of elementary school mathematics. teaching children mathematics, 1(3), 178–182. ben-chaim, d., fey, j. t., & fitzgerald, w. m. (1998). proportional reasoning among 7th grade students with different curricular experiences. educational studies in mathematics, 36, 247–273. berry, r. q. (2008). access to upper-level mathematics: the stories of successful, african american middle school boys. journal for research in mathematics education, 39, 464–488. boaler, j. (1998). open and closed mathematics: student experiences and understandings. journal for research in mathematics education, 29, 41–62. bransford, j. d., brown, a. l., & cocking, r. r. (1999). how people learn: brain, mind, experience, and school. washington, dc: national research council. battista response commentary 
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(1993). learner-centered psychological principles for enhancing education: applications in school settings. in l. a. penner, g. m. batsche, h. m. knoff, & d. l. nelson (eds.), the challenge in mathematics and science education: psychology’s response (pp. 287–313). washington, dc: american psychological association. muthukrishna, n., & borkowski, j. g. (1996). constructivism and the motivated transfer of skills. in m. carr (ed.), motivation in mathematics (pp. 40–63). cresskill, nj: hampton press. national mathematics advisory panel (2008). foundations for success: the final report of the national mathematics advisory panel. washington, dc: u.s. department of education. national research council. (1989). everybody counts. a report to the nation of the future of mathematics education. washington, dc: national academy press. national research council. (2002). learning and understanding: improving advanced study of mathematics and science in u.s. high schools. committee on programs for advanced study of mathematics and science in american high schools. washington, dc: national academy press. prawat, r. s. (1999). dewey, peirce, and the learning paradox. american educational research journal, 36, 47–76. romberg, t. a. (1992). further thoughts on the standards: a reaction to apple. journal for research in mathematics education, 23, 432–437. schoenfeld, a. c. (1994). what do we know about mathematics curricula? journal of mathematical behavior, 13, 55–80. silver, e. a., & stein, m. k. (1996). the quasar project: the “revolution of the possible” in mathematics instructional reform in urban middle schools. urban education, 30, 476–521. steffe, l. p. & d’ambrosio, b. s. (1995). toward a working model of constructivist teaching: a reaction to simon. journal for research in mathematics education, 26, 146–159. steffe, l. p., & kieren, t. (1994). radical constructivism and mathematics education. journal for research in mathematics education, 25, 711–733. tate, w. f., & rousseau, c. (2007). engineering change in mathematics education: research, policy, and practice. in f. lester (ed.), second handbook of research on mathematics teaching and learning (pp. 1209–1246). charlotte, nc: information age. villaseñor, a. j., & kepner, h. s. j. (1993). arithmetic from a problem-solving perspective: an urban implementation. journal for research in mathematics education, 24, 62–69. wood, t., & sellers, p. (1996). assessment of a problem-centered mathematics program: third grade. journal for research in mathematics education, 27, 337–353. wood, t., & sellers, p. (1997). deepening the analysis: longitudinal assessment of a problemcentered mathematics program. journal for research in mathematics education, 28, 163– 186. yen, h. (2004, september 29). high court weighs eminent domain: conn. residents sue, questioning ‘public purpose.’ the boston globe. retrieved from http://www.boston.com/news/local/connecticut/articles/2004/09/29/high_court_weighs_eminent_domain/. goals , aimed at forging links between the educational and commercial sectors journal of urban mathematics education december 2011, vol. 4, no. 2, pp. 7–14 ©jume. http://education.gsu.edu/jume margaret walshaw is professor in the school of curriculum and pedagogy at massey university, centennial drive, palmerston north, new zealand; email: m.a.walshaw@massey.ac.nz. her research is focused on making connections between social theories of the postmodern and mathematics education. commentary positive possibilities of rethinking (urban) mathematics education within a postmodern frame margaret walshaw massey university ostmodernism and mathematics education are both crucial components of contemporary society, yet they have rarely addressed each other. coupling mathematics education with postmodernism allows us to explore what positive possibilities might ensue for the discipline in general and for urban schools in particular beyond the traditional contours of mathematics education. in discussing the postmodern potential, we first need to be clear about modernist thinking. that discussion takes us back to descartes’s search for certainty, order, and clarity—a search that was integral to the formulation of a modernist framework in the 17th century. from that time until recently, most western thinkers understood reality as characterized by an objective structure, accessed through reason by an autonomous subject. these characteristically modernist beliefs have tended to shape thinking about knowledge, representation, and subjectivity within the western intellectual tradition of which mathematics education is a part. during the 1960s a number of literary critics began writing about the limitations of modernist thinking. postmodern sensibilities then emerged and entered the full range of human sciences. this emergence was most keenly expressed through the publication of jean-françois lyotard’s (1984) the postmodern condition: a report on knowledge. in this work, lyotard argued that the “grand narratives” of western history and, in particular, enlightened modernity, had broken down. multiple factors have brought about postmodernism. they include political and social crises of legitimation, and the resulting changing nature of economies and social structures in western societies. these changes place complex and sometimes conflicting demands on people in ways that they are barely able to understand or predict. for example, increasingly, within mathematics education, we are becoming aware of the complex construction of our work emerging from, among other things, new forms of inclusive political tendencies, changing vocational needs, and advances in informatics and communication systems. the effects of these processes for mathematics education are unsettling. conceptual p walshaw commentary journal of urban mathematics education vol. 4, no. 2 8 tools and frameworks from postmodern thinking help us to develop an understanding of those effects. they help us to understand ideas that are central to mathematics education from beyond the standard categories of thought. in particular, they help us to understand cognition and subjectivity. cognition and subjectivity as explained within our traditions cognitive psychologists describe cognition as mental activity to do with interactions and reflections upon the environment. the tradition aimed to show that cognition can be structured and that it cannot but be inward directed. cognition is equated to intrapsychic activity in response to factors in the environment. in the internal information-processing model, for example, it is descartes’s individual, and more specifically, the individual’s developing internal representation within the mind that becomes the central unit of analysis. drawing on humanist sensibilities about the individual, constructivists’ accounts of cognition necessarily rely on the autonomous learner, understood as the stable, core, knowing agent. such notions underpin the well know, post-piagetian work of von glasersfeld (see steffe, von glasersfeld, richards, & cobb, 1983). within these accounts, the mind is privileged, while circumstances and conditions are minimized. sociocultural perspectives that draw their inspiration from vygotsky’s (e.g., 1978) work, developed largely independently of western cognitive psychology. in opposition to cognitive psychology’s privileging of interior mental processes, sociocultural theory highlights social contexts and experiences. in seeking to reverse the terms within the individual/social binary, sociocultural theory gives priority to shared consciousness, or intersubjectivity, arguing that conceptual ideas proceed from the intersubjective to the intrasubjective. cognition involves active construction by the individual and evolves through social interaction. in this formulation, given that social practice is a mechanism that informs thinking, the way in which mathematical truths are constituted interactively by the classroom community is integral to analyses of classroom life. in the route from cognitive psychology to sociocultural perspectives in the discipline, the “subject” (i.e., individual) has moved from the idea of self-centredness to one that is animated by negotiations of self with social structures and culture. contemporary interests and issues within mathematics education now concern the complex transactions that take place between the subject-in-process and the structures and processes of mathematics education but what is curious is that many aspects of the cartesian model continue to survive. postmodernism offers theoretical pathways that move beyond the cartesian self in order to account for the merging of the social, discursive, temporal, spatial, and the psychic. it achieves this move by explaining cognition in mathematics classrooms in relation walshaw commentary journal of urban mathematics education vol. 4, no. 2 9 to the dynamics of the spaces people share and within which they participate. in such an explanation, power and conflict and other important issues come to the fore. what does postmodernism offer? what exactly does postmodernism offer? broadly speaking, it offers a new attitude. it offers resources to help us understand an increasingly complex, plural, and uncertain world. its focus is not on foundations and efforts to establish authority. rather, its objective is to explore tentativeness and to develop scepticism of those principles and methods that highlight certainties. specificially, it problematizes impartial knowing, disinterested objectivity, and value neutrality. postmodern thinkers keep foremost in mind that reality does not have an objective structure, that research is fundamentally unrepresentable, and that representation is subjective and, hence, highly contested. truth, then, is multiple, historical and contextual, as well as contingent, and political. this multiplicity is not to suggest that all views are equal, but it does imply an ethically responsible engagement with specific complex problems that do not have generalizable solutions. knowledge, in postmodern thinking, is not neutral or politically innocent. cognitive products are merely that—products constructed by cognitive agents, enmeshed in a site of knowledge production that is unavoidably political. postmodern analyses might explore the contingency of power, privilege, and history on systems of knowledge, to reveal how knowledge implies forms of social organization and social practices that structure institutions and constitute individuals as thinking, feeling, and acting subjects. meaning construction becomes a form of critique that acknowledges its own complicity in the analysis. in this kind of thinking, mathematics education would be viewed, not in isolation, but head-on as a disciplinary endeavour situated at the interface of multiple and competing structures and processes. what would be emphasized are elements of practice characterized not only by regulatory practice but also the uncertainties of practice: both inside and beyond the classroom and school. for example, urban schooling would be interrogated as a construct, situated within institutions, historical moments, as well as social, cultural, and discursive spaces. importantly, in this formulation, identities, social conditions, and political dimensions all become highly significant. these kinds of priorities run up against portrayals within mass-mediated and ideological constructions of the roles and functions of urban schools that often assume an essentialist character. such portrayals have a tendency to offer a set of myths through which transmission strategies of teaching and high-stakes assessment come to the fore. in rendering this familiar story problematic, the postmodern approach views teaching, curriculum, and leadership within urban schools as walshaw commentary journal of urban mathematics education vol. 4, no. 2 10 constantly mobile, closely linked to interactions between people, past, present, and anticipated, situated in relation to one’s biography, current circumstances, investments, and commitments. what are magnified in the postmodern approach are complex practices, involving multiple dimensions and conflicting discourses, all of which prevent anyone from generalizing across settings and across teachers, learners, and schools. these ideas are helpful in understanding the current context in which many urban schools, searching for effectiveness and inclusiveness, struggle in their attempts to make a difference to all. on a day-to-day basis, they deal with diverse learner cohorts and are expected to minimize the effects of the differing behavioral and epistemic responses that go hand in hand with those cohorts. not only that: typically, teachers and institutional leaders confront heavy workloads, new technologies, and new curricular policy mandates, all of which operate to normalize and regulate their pedagogical practice, and, importantly, undermine their senseof-self within the schooling system. these difficulties can be viewed within a much larger complex social, cultural, and economic phenomenon. in a context in which mathematical proficiency is the cornerstone of a student’s self-empowerment, schools have become objects of scrutiny and critique. students’ lack of proficiency is, in the eyes of policy, to be blamed on schools, their infrastructure, their networks, and their teachers. increased surveillance has become the order of the day. demands for increased testing, scripted pedagogical interactions, and prescribed instructional leadership all operate within a context that privileges certain fundamentalist interests, values, and practices. trapped within a law of diminishing returns, schools struggle for expression against hierarchies of power and against their own marginalization. within our contemporary environment, postmodernism becomes a key resource for interrogating and understanding mathematics education. as a “system” of ideas, postmodernism allows us to build new knowledge about mathematics education within contemporary social and cultural phenomena. it enables us to chart urban school practice and the way in which identities evolve. it provides the tools for us to track reflections; investigate everyday classroom activities; analyze discussions with instructional leaders, mathematics teachers, students, and educators; map out the effects of policy; and so forth. such interrogations ask different kinds of questions. for example, we might ask: what power-knowledge lessons might be learnt for the discipline from the recent reconstruction of academic identities and new work environments, centred as they are on performativity and measurable research and publication outcomes? where does the postmodern collapse of the distinction between knowledge and commodity, with regard to technology, lead to in terms of the production of mathematical knowledge? in mathematics education research, postmodernism is associated with a range of different theoretical positions. each theory offers explanatory power in walshaw commentary journal of urban mathematics education vol. 4, no. 2 11 high-lighting and explaining particular aspects of mathematics education and each offers a new way of thinking. while different postmodern analyses have few concepts in common, all rely on the underlying assumption in the usefulness of new ideas for exposing aspects of practices previously situated beyond our vision. each is committed to approaches to mathematics education that question given understandings. when postmodern analyses explore lived experience, it is not with a view of capturing reality and proclaiming causes, but of understanding the complex and changing processes by which subjectivities are shaped. such analyses do not seek to legislate over the constitution and nature of reality. rather, they work at illuminating the dynamics of experience—how meanings are validated, and whose investments are privileged. in seeking to capture the fluidity and complexity of identity constitution, postmodern analyses reveal how different contexts carve out their own borders, and how each represents different and competing relations of power, knowledge, dependency, commitment, and negotiation. in doing so, they sensitize us to oppressive conditions, highlighting possibilities for where and in what ways practices, processes, and structures might be changed. irrespective of the standpoint of postmodern analyses—such as derrida’s (1978) work on deconstruction of taken-for-granted understandings; žižek’s (1998) explanation of how identities are constructed in relation to the other; bourdieu’s (1990) exposition of how everyday decisions are shaped by dispositions formed through prior events; fairclough’s (2003) insights about the way in which language produces meanings and positions people in power relations; foucault’s (1977) understanding of how practices are produced within discourses; lyotard’s (1984) explanations of language games as fundamental to the social bond; and gadamer’s (1989) insistence on interpretation as an ongoing process— the frameworks used and the questions prioritised are shaped in the belief that postmodernism offers a potential source of sophisticated analytical tools for understanding people and events in mathematics education. a few examples of how some of these frameworks might be put to use follow: walls (2010) used ideas drawn from psychosocial theory to explore the way in which teachers negotiate their way through contesting perceptions of effective teaching within a climate of compulsory standardized testing. in this work, identity is changeable and unpredictable, formed through a reconciliation of constructions of past, present, and future possible identity positions. walls revealed how systemic forces are lived as individual dilemmas, by demonstrating the ways in which teachers embody practices that they had wanted to change. with a focus on why and how teachers structure their teaching identities in the way they do, she highlighted the way in which walshaw commentary journal of urban mathematics education vol. 4, no. 2 12 teachers speak of their highly compromised (and limiting) practice within mathematics classrooms. in their investigation of the fragility of mathematical learning, stentoft and valero (2010) began with the notion that language constitutes social reality rather than reflects an already given reality. drawing attention to the interrelatedness as well as the fragility of classroom discourse, identity, and learning, they unpacked the ways in which students and teachers are involved with constructing multiple identities over the course of a mathematics lesson. they also showed how learning mathematics and constructing mathematical knowledge in the classroom is inextricably caught up in the discursive practices of the classroom. other approaches have used critical discourse analysis to study the classroom discourse and interaction. for example, de freitas (2010) grounded her work in the understanding that language not only produces meaning but also positions speakers in specific relations of power. discursive practices of mathematics education position people and contribute to the development of thinking in the classroom. they shape thinking by limiting the scope of what can be said and done. de freitas reported on what teachers chose to say and the way in which they said it, and the power relations that descended from those decisions. in particular, her research demonstrated the ways in which the discursive practices of teachers contributed to the kind of thinking that is possible within the classroom. nolan (2010) explored the development of an inquiry-based classroom in an undergraduate teacher education program. she showed how inquiryteaching approaches, that required a tolerance of ambiguity and uncertainty from students, met with resistance, challenge, and dissatisfaction from students. she drew on bourdieu’s conceptual framework to analyze the tensions between thought and action, knowledge and experience, and the technical and existential enacted in the pedagogical encounter. specifically, in providing an account of the dilemmas in trying to establish teacher authority in a context fuelled with contestation, she offered an explanation as to why reforms in teacher education do not always enjoy an enduring effect. postmodernism as a form of social critique a postmodern attitude demands a rethinking both of the question of research authority and of ways of representation. it offers a self-conscious consideration of the location of the researcher that can highlight the processes of meaning making walshaw commentary journal of urban mathematics education vol. 4, no. 2 13 and consciousness, and increase our curiosity about the activities of researchers and respondents in the field. as a form of social critique, postmodernism offers an understanding of how identities are produced through social interaction, daily negotiations, and within particular contexts and arrangements that are already heavily laden with the meanings of others. through such analyses, it is possible to develop insights about the struggle for self within wider meanings of and investments in schools and the way in which power insinuates itself into the discourses and practices of school and classroom life. in that sense, postmodernism offers a more expansive way of invoking ethical deliberation. it does not involve an outright dismissal of the ethical problems that guide modern thinkers. instead, it questions the specifically modern approach to confronting those problems. indeed, in postmodern thinking, ethical responsibility precedes all engagement with the other. crucially, such engagement is not dependent on the reciprocation of the other. educational transformation can be effected by making more visible the ways in which commonplace daily social relations are rearticulated. the process is important because it assists us in finding out where meanings and values are legitimated, whose investments are favored, and how those investments are sustained. such inquiry allows us to discover why our interests are sometimes silenced, how we are caught up in conditions of constraint, and where we might find weak points to imagine a space for creative change. by unpacking what seems “natural” and by locating the effects of constitutive power, we begin to think differently about constructing practices that are responsive and appropriate to specific sites of struggle. conclusion rereading the practices, processes, and structures within mathematics education through the understandings offered by postmodernism allows us to scrutinize the rules and practices of education. stinson and powell (2010) have shown that such understandings about mathematics practice—that neither stretch plausibility nor break with reality—emerge through practising teachers’ appropriation of postmodern ideas. importantly, they have shown how exposure to and engagement with postmodern ideas, leads to significant changes in teachers’ thinking about practice. standing up against discourses premised on remediation and salvation, an engagement with postmodernism reveals a commitment to engage in political struggle over the meaning of mathematics education itself, while simultaneously acknowledging that to speak of transformative change is to question the very meaning of empowerment. what it also means is that, with a postmodern sensibility, all of us involved with mathematics in schools can begin to reflect on what we are today, how we walshaw commentary journal of urban mathematics education vol. 4, no. 2 14 have come to be this way, and the consequences of our actions. it sensitizes us to our taken-for-granted assumptions and practices, and creates an opening in which knowledges, roles, and relationships are questioned and where new possibilities might be envisioned. choices become more apparent about how to speak, write, teach, and lead in ways that move toward the kind of arrangements in mathematics education that are more desirable, for the geographical settings and material conditions in which schools are located at this particular moment in time. such an opening is ripe for development within both the intellectual conditions and the material settings of our schools. references bourdieu, p. (1990). in other words: essays toward a reflexive sociology (m. adamson, trans.). cambridge, united kingdom: polity press. de freitas, e. (2010). regulating mathematics classroom discourse: text, context and intertextuality. in m. walshaw (ed.), unpacking pedagogy: new perspectives for mathematics education (pp. 129–151). charlotte, nc: information age. derrida, j. (1978). structure, sign and play in the discourse of the human sciences (a bass, trans.). london, united kingdom: routledge and kegan paul. fairclough, n. (2003). analyzing discourse: textual analysis for social research. new york: routledge. foucault, m. (1977). discipline and punish: the birth of the prison (a. sheridan, trans.). harmondsworth, united kingdom: penguin books. gadamer, h. g. (1989). truth and method (trans: j. weinsheimer & d. g. marshall). new york: crossroad press. lyotard, j-f. (1984). the postmodern condition: a report on knowledge (b. massumi, trans.) minneapolis, mn: university of minnesota press. nolan, k. (2010). playing the field(s) of mathematics education: a teacher educator’s journey into pedagogical and paradoxical possibilities. in m. walshaw (ed.), unpacking pedagogy: new perspectives for mathematics education (pp. 153–173). charlotte, nc: information age. steffe, l., von glasersfeld, e., richards, j., & cobb, p. (1983). children’s counting types: philosophy, theory, and application. new york: praeger. stentoft, d., & valero, p. (2010). fragile learning in the classroom: exploring mathematics lessons within a pre-service course. in m. walshaw (ed.), unpacking pedagogy: new perspectives for mathematics education (pp. 87–107). charlotte, nc: information age. stinson, d., & powell, g. (2010). deconstruting discourses in a mathematics education course: teachers reflecting differently. in m. walshaw (ed.), unpacking pedagogy: new perspectives for mathematics education (pp. 201–221). charlotte, nc: information age. vygotsky, l. (1978). mind in society: the development of higher psychological processes. cambridge, ma: harvard university press. walls, f. (2010). the good mathematics teacher: standardized mathematics tests, teacher identity and pedagogy. in m. walshaw (ed.), unpacking pedagogy: new perspectives for mathematics education (pp. 65–83). charlotte, nc: information age. žižek, s. (ed.) (1998). cogito and the unconscious. durham, nc: duke university press. journal of urban mathematics education july 2011, vol. 4, no. 1, pp. 98–119 ©jume. http://education.gsu.edu/jume roberta hunter is a senior lecturer in the college of education at massey university, building 94, oteha rohe, albany, auckland, nz; email: r.hunter@massey.ac.nz. her research focuses on the development of mathematical practices with pasifika and other diverse students in communities of mathematical inquiry. glenda anthony is a professor of mathematics education in the college of education at massey university, palmerston north, nz; email: g.j.anthony@massey.ac.nz. her research focuses on professional learning and development of effective pedagogical practices for the mathematics classroom. forging mathematical relationships in inquiry-based classrooms with pasifika students roberta hunter massey university glenda anthony massey university in this article, the authors report changes in mathematical disposition, participation, and competencies within a group of pasifika students as a teacher established the discourse of mathematical inquiry and argumentation. within a classroom-based design approach, the teacher used a communication and participation framework tool to support students to engage in a range of collective mathematical practices. drawing on analyses of student interviews conducted over one school year, the authors provide a narrative that illustrates how changes in agency and accountability accompanied shifts in the mathematical inquiry discourse. the results show positive learning outcomes for pasifika students when the general and mathematical obligations attend to the cultural, social, and mathematical well being of all students in mathematics classrooms. keywords: collaborative problem solving, culturally relevant pedagogy, educational change, inquiry communities, mathematics education, urban education nquiry-based classrooms position students as active participants in a community of learners. the belief is that active engagement will lead to the development of specific student dispositions and competencies that are presumed to make a positive difference in students’ life chances and their future civic participation (anthony & walshaw, 2007; goos, 2004). in the inquiry-based mathematics classroom students have a significant opportunity for engagement through activities involving mathematical discussion and argumentation. collectively, research based in western education systems (e.g., cobb, wood, yackel, & mcneal, 1992; forman & ansell, 2001; goos, 2004; walshaw & anthony, 2008), and in some asian systems (e.g., pang, 2009; sekiguchi, 2006), advocate that effective pedagogy affords students opportunities ―not only to share their ideas in a mathematical community but also to analyze and evaluate the thinking of other members‖ i hunter & anthony forging mathematical awareness journal of urban mathematics education vol. 4, no. 1 99 (mueller, 2009, p. 138). indeed, in a landmark study involving analysis of 42 mathematics lessons, wood, williams, and mcneal (2006) demonstrated that ―those interaction patterns that required greater involvement from the participants were related to higher levels of expressed mathematical thinking by children‖ (p. 249). there has been considerable attention within research literature to creating inquiry-based learning environments in which students not only express their own ideas but also use mathematical reasoning to challenge those of their peers and the teacher. there are a number of studies that provide exemplars of professional development programs to support reforms toward more inquiry-based practices (e.g., hunter, 2008; sherin, linsenmeier, & van es, 2009), and exemplars of inquiry-based classrooms in practice (e.g., kazemi & franke, 2004; staples & truxaw, 2010). this critical mass of studies has enabled detailed analysis of those teacher practices that facilitate effective mathematical discourse patterns—both in group work activities and whole-class discussions. for example, walshaw and anthony’s (2008) review of classroom discourse highlights the teacher’s role in establishing participation norms, in supporting and fine tuning mathematical thinking, and in shaping mathematical argumentation. similarly, stein, engle, smith, and hughes (2008) offer five key practices that teachers can use to orchestrate class discussions: anticipating, monitoring, selecting, sequencing, and making connections between student responses. however, these studies also highlight that incorporating practices of inquiry learning is challenging. in some cases, the intentions of equitable and inclusive participation are compromised. for example, planas and gorgorió’s (2004) classroom study illustrated the ways in which one teacher regulated participation by creating inconsistent rites across student groups. the researchers observed the teacher’s subtle systemic refusal of immigrants’ attempts to explain and justify their strategies for solving problems. other studies (e.g., ball, 1993; baxter, woodward, & olson, 2001) note the ways that highly articulate students can dominate discussions while ―low achievers‖ may choose to remain passive. in collaborative group work there is potential for unintended learning—students may learn incorrect mathematical strategies (good, mccaslin, & reys, 1992) or inappropriate social behaviors (hand, 2010). of more concern is that for some students the expected mathematical practices—such as constructing representations, making arguments, and explaining their thinking—may be difficult or make them feel uncomfortable. as esmonde (2009b) notes, this is particularly so ―if these practices are not common or they are interpreted differently in other community practices‖ (p. 1011). given these challenges, studies that provide insight into learning mathematics in inquiry-based classrooms from the perspective of the students, while less common, are particularly important. in new zealand, similar to many countries, we have an increasingly diverse student population. and similar to some other countries, it is clear that our educa hunter & anthony forging mathematical awareness journal of urban mathematics education vol. 4, no. 1 100 tion system has some way to go to meet the needs of all learners (bishop, berryman, cavanagh, & teddy, 2009). despite recent large-scale implementation of professional development opportunities in mathematics education (see higgins & parsons, 2009), we continue to record significant levels of underachievement for students who are from marginalized backgrounds. this underachievement includes a large percentage of pasifika students in new zealand schools (youngloveridge, 2009), students who are the focus of this study. in facing the challenge to address the low and inequitable mathematics performance of groups of students, boaler and staples (2008) urge researchers to gather more evidence on the ways that mathematics may be taught more effectively in different settings and circumstances. for our part, in looking to redress the inequitable opportunities afforded pasifika students in low decile 1 schools (ferguson, gorinski, wendtsamu, & mara, 2008), we collaborated with four teachers to explore how to enact inquiry teaching and learning practices that support students’ development of mathematical proficiency. in this article, rather than focusing on how the teachers (re)arranged their learning environment, we center our discussion on the students’ perceptions of ―being‖ mathematics learners in this changing environment. in examining closely how previously disaffected students come to develop productive relationships with both mathematics and with themselves as mathematical learners, we seek to understand more about the way learning—in all its forms—is occasioned within an urban mathematics classroom. forging relationships with and in mathematics the idea that teaching and learning are located within a complex social web draws its inspiration from vygotsky (1986) and the work of activity theorists. this body of work proposes a close relationship between social processes and conceptual development and is given a clear expression in lave and wenger’s (1991) well-known social practice theory, in which the notions of a community of practice and the connectedness of knowing are a central feature. as walshaw (2007) explains, social theories of learning suggest that thinking, meaning, and reasoning are constituted socially in a mutually relational manner—that is, ―the learner is inextricably connected to a dynamic social context‖ (p. 35). instruction that provides opportunities for students to engage in mathematical inquiry and in meaning making through discourse necessarily requires opportunities for learning in a social environment. through collaborative engagement in the context of shared activities and interests, students engage in discourse practices that involve the articulation and justification of their mathematical thinking. 1 schools in new zealand are ranked into deciles (low to high) as an indicator of the socioeconomic level of the school community. the lowest decile ranking is a decile 1; the highest is decile 10. students of pasifika ethnicity predominantly attend schools within decile ratings of 1–3. hunter & anthony forging mathematical awareness journal of urban mathematics education vol. 4, no. 1 101 to understand more about the interaction between students’ relationship with mathematics and the learning opportunities afforded the students in this study, we draw on cobb, gresalfi, and hodge’s (2009) analytic framework of the relation between the microculture established in a particular classroom and the students’ developing personal identities. in explaining ―how it is that students come to understand what it means to do mathematics as it is realized in their classroom and with whether and to what extent they come to identify with that activity‖ (p. 41) they highlight the role of classroom obligations—both general and mathematical—that are constituted in the course of the ongoing classroom interactions. obligations comprise both the ―general and the specifically mathematical obligations that delineate the role of an effective student in a particular classroom‖ (p. 43). general obligations concern the distribution of authority and the ways that students are able to exercise agency. mathematical obligations concern the ―norms or standards for mathematical argumentation and normative ways of reasoning with tools and written symbols‖ (p. 45). the complex web of relationships surrounding the organization and facilitation of knowledge production collide with students’ developing mathematical disposition, competence, and participation within the activity system of their classroom (esmonde, 2009a). here, we use the conceptual frameworks of disposition, competence, and participation to interpret students’ perceptions of their relationship with and in mathematics. disposition disposition, as we use the term, is set within the context of classrooms and constructed within a set of practices that are realized within the immediate classroom activity setting in which the mathematics teaching and learning takes place. students develop a mathematical disposition—that is a collection of notions about mathematics, the values of mathematics, and ways of participating in mathematics—in and through classroom activity (gresalfi & cobb, 2006). the idea that students learning ―to do mathematics‖ is inextricably linked with their learning ―to be mathematical‖ is clearly recognized in current mathematics education literature. studies (e.g., boaler, 2002; lampert, 2001; staples, 2008) illustrate how the role of authority, as exercised by teachers in classrooms, plays out in students’ ideas, values, and ways of participating in mathematics. boaler (2002), in her comprehensive study of mathematics classes in two different types of schools, illustrated that student opportunity to interact with, and in, mathematics was constrained in classrooms where teachers maintained a high degree of authority, particularly in relationship to determining preferred methods and correctness of procedures. in such settings, students constructed beliefs about their role as secondary to that of the teacher, and saw themselves as passive, non-questioning recipients of mathematical knowledge. the contrast is in classrooms where authority hunter & anthony forging mathematical awareness journal of urban mathematics education vol. 4, no. 1 102 is distributed more evenly among teachers and students. boaler illustrated how in settings where students were obliged to construct their own problem solving strategies, validate their reasoning, and regulate their behavior, they constructed different relationships with and in mathematics. they developed positive mathematical dispositions, which led to increased motivation to engage with mathematics at deeper levels. in this article, we examine how students’ dispositions changed as they aligned their ways more closely to the norms of practices of inquiry-based learning. competence rather than defining competence as an attribute of the individual—the ―what‖ a student needs to know or do in order to be considered successful—we draw on gresalfi, martin, hand, and greeno’s (2009) notion of ―systems of mathematical competence‖ (p. 49). constructed in classroom interactions, the ―opportunities for students to be understood as being competent depend on the tasks that they are assigned to work on, and on the agency and accountability with which they are positioned to do that work‖ (p. 67). for example, in one class, competence may be determined by using the right methods, and in another class, students may be constructed as competent when they participate in acts of sense making. importantly, within the social network of the classroom (and beyond) the teacher and the students are both central players in constructing competence. here, we illustrate how some of the students shifted in perceiving competence as a fixed attribute of themselves as individuals to view competence as related to their role and learning associated with specific activities as expressed by the perceived mathematical obligations (cobb et al., 2009). participation drawing on the perspective of lave and wenger (1991), learning can be defined as a change in participation in a set of collective practices. in mathematics, learning has come to be conceptualized as learning to participate in mathematical practices—the ―ways in which people approach, think about, and work with mathematical tools and ideas‖ (rand mathematics study panel, 2003, p. xvii). in inquiry-based classrooms, learning to participate in mathematical practices involves learning to construct representations, make arguments, reason about mathematical objects, and explain one’s thinking. rather than be on the periphery of participation, or constantly needing support, esmonde (2009b) argues that ―the goal is for learners to adopt central participation without the teacher’s direct help‖ (p. 1011). she notes, for example, that ―we need to look to see whether all forms of student participation allow students to move on a trajectory towards more central and competent participation in classroom practices‖ (p. 1011). however, in hunter & anthony forging mathematical awareness journal of urban mathematics education vol. 4, no. 1 103 moving toward those mathematical practices advocated in inquiry learning we need to be aware of the relational aspects involved in participation. for example, when working on problem solving tasks, forming and defending mathematical conjectures, and discussing with their peers, students need to be willing and able to engage in areas in which they are only partially competent (gresalfi et al., 2009). such engagement is likely to evoke a range of emotions such as, excitement, satisfaction, and frustration. bibby (2009) claims that relationships characterised by issues of emotion and trust actually frame what is knowable. for her, effective pedagogy needs to engage with the emotional labor and risk involved in trying to mutually understand something (and each other), and it must recognise the pain that constitutes not knowing. in this article, we examine how students took up or resisted opportunities to participate in ways advocated for by the teacher as he established inquiry-based learning. pasifika learners in the new zealand context as stated earlier, here we report on a group of pasifika students. the term pasifika, as we use it, refers to a multi-ethnic, heterogeneous group of people who originated from the island nations in the south pacific. in new zealand, pasifika students as a group exhibit significantly lower achievement results in mathematics than their european new zealand counterparts (crooks, smith, & flockton, 2010; ferguson et al., 2008). most of the pasifika population resides in an auckland region, which has the highest birthrate and holds the bottom ranking on all national indices including education (airini et al., 2007). although pasifika learners are a diverse group, anae, coxon, mara, wendtsamu, and finau (2001) draw attention to a set of cultural commonalities within pacific values, which include ―respect, reciprocity, communalism, and collective responsibility‖ (p. 14). these core values, whilst respected in culturally responsive pedagogical practices, may not initially be aligned with having students feel comfortable participating in problem-based mathematical activity and inquiry. for example, in considering the concept of respect, jones (1991) and clark (2001) describe how pasifika students in their studies identified listening to the teacher as an appropriate way to learn. the students considered the teacher to be their elder and therefore their knowledge unquestionable. likewise, the students viewed arguing with, or asking teachers questions, to be disrespectful because it was their responsibility to listen closely and learn from the teacher. awareness of these ―cultural-historical repertories‖ (guitiérrez, & rogoff, 2003) and encouragement for learners to ―develop dexterity in determining which approach from their repertoire is appropriate under which circumstances‖ (p. 22) are important equity con hunter & anthony forging mathematical awareness journal of urban mathematics education vol. 4, no. 1 104 siderations for socializing students into mathematical inquiry discourse (boaler, 2006; gutierrez, 2002; macfarlane, 2004). specific to the new zealand context, macfarlane (2004) advocates culturally responsive teaching which carefully structures student access to learning spaces that are ―culturally as well as academically and socially responsive‖ (p. 61). in this study, a key consideration was given to balancing the pasifika students’ beliefs with developing their participation and competence in inquiry discourse as a ―social language‖ (gee & clinton, 2000, p. 118). social language, as gee and clinton use the term, refers to ways of ―talking, listening…acting, interacting, believing, valuing and using tools and objects, in particular settings, at specific times, so as to display and recognize particular socially situated identities‖ (p. 118). other researchers (e.g., ferguson et al., 2008; hawk, tumama cowley, hill, & sutherland, 2005) highlight the problems caused by formality and competition in new zealand classrooms. they argue for classrooms to be more inclusive and to build on the cultural capital of pasifika students—and respect their concept of community and collectivism. in addition to the possibility that pasifika students are immersed in curriculum and pedagogical practices founded in euro-centric precepts (tate, 1994); they may also encounter hurdles related to low socioeconomic backgrounds. lerman (2009) uses bernstein’s theory to highlight how pedagogic discourse makes certain subjectivities available to middle-class students by establishing pedagogical relations that are highly ―visible‖—and similar to that which they have already experienced in the home. in such a classroom, ―being‖ mathematical is more difficult for students from working-class and culturally different backgrounds, who are less likely to recognise what mathematical knowledge should be engaged in within a given situation and are less likely to realize the required ―appropriate‖ behaviors. as a way forward, boaler and staples (2008) argue that pedagogical practices that ―evince social awareness and cultural sensitivity are critical if the desired outcome is student participation and academic success‖ (p. 612). in new zealand, hunter’s (2006, 2007, 2008) study illustrates the positive outcomes for pasifika students when teachers explicitly build on the pasifika values of reciprocity, communalism, and collectivity while sensitively observing the students’ notions of respect. in these classroom studies, pasifika students were positioned to engage in inquiry discourse and develop collective mathematical practices within a carefully crafted learning environment. responsive and caring relationships between the teachers and students in the classrooms were central in overcoming disparities and increasing inclusivity. as tate (1994) explains, ―connecting the pedagogy to the lived realities…of the students is essential for creating equitable conditions‖ (p. 478). the focus of hunter’s studies however were on the actions the teachers took to construct communities of mathematical inquiry with diverse hunter & anthony forging mathematical awareness journal of urban mathematics education vol. 4, no. 1 105 learners. our most recent study on inquiry classrooms provides an opportunity for us to hear the students’ voice and understand from their perspective how they view their relationship as users and doers of mathematics. the study in this article, we draw on data from one classroom within a classroombased design study (design-based research collective, 2003). the wider study was situated in two new zealand schools in an urban low socio-economic area (decile 1 and decile 3) where the large majority of students were of pasifika ethnicity. the aim of the study was to examine pedagogical practices that optimize equitable access to increasingly sophisticated forms of mathematical practices within inquiry-based classrooms. the trial of a communication and participation framework (cpf), designed to support teachers to scaffold student participation in a range of mathematical practices, was a key feature of the study. this tool (see appendix), adapted from the theoretical framework proposed by wood and mcneal (2003), details a set of collective reasoning practices related to the communicative and performative actions that support effective mathematical inquiry practices (for further details of the design of the cpf tool see hunter, 2007, 2008). over a period of one year, the teachers used the cpf as a tool to prompt and monitor student engagement in inquiry practices. as the study progressed, associated changes in social and sociomathematical norms supported students to make increasingly proficient mathematical explanations, representations, justifications, and generalizations. the case study over the course of the study, the teachers and their students adapted and adopted the practices detailed in the cpf in various ways. thus any analysis of student perceptions of their relationships with mathematics and with themselves as mathematical learners is bound by context. here, we have selected one case: a classroom of 20, elevento twelve-year-old students and their pasifika teacher. the students in this class were predominantly of pasifika ethnicity. the pedagogical changes in the selected case exemplified significant shifts toward inquirybased classroom practices as per the aims of the study. over the course of the school year, the teacher consistently pressed the students to develop the social and sociomathematical norms of mathematical inquiry. as part of his efforts, he drew on his own cultural knowledge to frame the social norms within social and cultural contexts that were familiar to the students. for example, the requirement that students work collaboratively was framed within an appropriate cultural setting (preparing an umukai [village feast] and the collaborative roles all participants hold). he guided student attention toward pasifika concepts of reciprocity, com hunter & anthony forging mathematical awareness journal of urban mathematics education vol. 4, no. 1 106 munalism, and collectivism as he had them develop mathematical explanations, representations, and justification within their groups. he drew on their concept of respect and reciprocity as part of their need to actively listen, question, offer help, check the understanding of all members of the group, and support each other when reporting back to a wider audience. data for the case study were collected from multiple sources over the timeframe of the school year; it included field notes, classroom artifacts, teacher interviews, and video and digital photo records of 10 mathematics lessons (with a focus on the teacher establishing small group and large group mathematical activities). immediately after each of the observed lessons, the four to six students who were the focus of the video capture were invited to participate in individual semi-structured interviews. the interviews provided opportunities for students to reflect on their role within the group and classroom activities experienced that day. additionally, in each of the interviews, students were asked to discuss how they were feeling about learning mathematics in this classroom. the data analysis we present here draws primarily on the audio-recorded interviews with the students. the first phase involved open coding (strauss & corbin, 1998) of the interview transcripts to look for emerging themes informed by our conceptual framework. a concurrent analysis of the video records and field notes of classroom episodes enabled us to compare and refine emerging patterns and themes in the interviews alongside shifts in the social and sociomathematical norms of the classroom. we have used this analysis to provide a collective narrative—presented as a trajectory—of student perceptions of their relationship with and in mathematics. student voices: relationships with and in mathematics at the beginning of the school year (the initial phase of the study) the pedagogical patterns used by the case teacher conformed to a more traditional pattern. the teacher taught content that consisted of preplanned numerical strategies in a procedural manner. any discussions about the strategy solutions typically followed an initiateresponse-evaluate (i-r-e) structure (mehan, 1979); that is, he initiated questions about the mathematical reasoning (mostly at the level of explanation of strategy steps), nominated who should respond, and then he took the responsibility for evaluating the responses. the students responded to his questions but they did not ask additional questions nor shift the discussion to other mathematical content or different ways of reasoning. initial relationships interviews with 12 students (assigned pseudonyms) conducted within a month of the project beginning (early in the school year) indicated that the students held a hunter & anthony forging mathematical awareness journal of urban mathematics education vol. 4, no. 1 107 range of views towards mathematics and how they saw themselves as mathematics learners. mele and tarai stated that they liked mathematics but they could not describe why they liked it other than they found it ―easy.‖ the remaining students either stated that they did not like mathematics or they expressed ambivalent views about their liking for, and relationship with it. for example mereana explained: ―i like it a little bit. it is all right. it helps me learn new strategies. but then i sort of understand and then i don’t, not all the time, just the hard work sometimes.‖ their explanations for not liking mathematics appeared to centre on their perceived inability to make sense of what the teacher explained. tama stated: i know that i am not that good because i know because i just get lost and confused. i get confused easily when i am doing maths. when i see too much numbers they just get all muddled up in my head and i cannot add them together or stuff. when describing how they could help themselves to learn mathematics, their responses suggested they held a view of themselves as passive participants in the learning process. they outlined how for them learning mathematics entailed listening to the teacher, working hard, and paying close attention to what the teacher said or did. similarly, they all reported that when they got stuck completing a mathematics task they asked the teacher for help. in contrast, they described their teacher’s role as an active one in helping them learn mathematics. they considered that it was his responsibility to tell them what to do, explain the mathematics, show them a range of different strategies, and question them. these views are well captured in mereana’s description of the day’s lesson where even when the teacher explicitly directed the students to actively participate and ask questions; but, by default, question asking still remained the responsibility of the teacher: we did fractions and he showed us strategies to do and we talked about them with the whole class and we sort of had to ask questions but the teacher asked the questions. beginning to change the initial focus of the inquiry intervention involved establishing new arrangements for learning. early in the school year, the teacher introduced a lesson format in which students first worked in small problem solving groups to construct mathematical explanations followed by presentation of their work to a larger group. group worthy contextualized problems 2 were used, and the students constructed ways to represent their reasoning. when interviewed about this new 2 for example, moana used 12 rolls of gift wrap to make 18 skirts for the trash to fashion show. how many rolls of gift wrap would she need to make 21? hunter & anthony forging mathematical awareness journal of urban mathematics education vol. 4, no. 1 108 arrangement, the students were unanimous in their preference for working together as a small group. they outlined how the smaller group provided opportunities to learn from each other, noting that they now regarded mathematics as more difficult to learn on their own. in this early stage of working collaboratively, students indicated a strong preference for working with the same people all the time (or with friends). while social arrangements were affirmed, views about the nature of involvement of expected mathematical practices were more tentative. confidence in one’s competence to provide a mathematical explanation, even in a small group situation, posed problems for many: ―it is quite difficult because i am not good at explaining things‖ (ana). while many reported uncertainty about their ability to speak and explain their mathematical reasoning in a small group, all of the students attributed a lack of confidence when required to speak to the larger group. commonly, a reticence to explain due to shyness was coupled with the students’ need to maintain ―face‖ both as individuals and as representatives of the group. for instance, the tensions experienced by sione were evident when he is asked to reflect on a photograph of himself providing an explanation. he recalls his feelings at that moment as being concerned that i better not muck this up. that is what is going through my head. i better not muck this up and if i do everyone is going to look at me; like, i have just got it wrong. upon viewing the photograph of this classroom episode, the other students who were listening to his explanation talked about the difficulties they had participating and actively listening and making sense of his reasoning. mahine stated that because sione’s reasoning was new, it was too difficult and confusing to understand. another student, dan, described his ―listening‖ as cuing his own thinking rather than active engagement in the reasoning being offered: ―i was trying to work out how we got our equation as they were explaining. i was listening but thinking about our work.‖ to increase student sense making of mathematical explanations, the teacher had introduced space during presentation of explanations to enable them to question parts they found confusing. however, all the students described feeling a lack of confidence to construct and ask questions during discussions: ―when we are talking in a big circle there are too many people and so i do not ask a question‖ (mahine). although they recognised the importance of asking questions for sense making their responses indicated that constructing appropriate questions posed difficulties. for example, when asked why they needed to ask questions, mereana explained: ―so i can understand better, but it is not really easy to ask questions. it makes it hard by figuring out what to ask.‖ hunter & anthony forging mathematical awareness journal of urban mathematics education vol. 4, no. 1 109 moving along the change trajectory as the learning environment moved more toward an inquiry-based classroom, so did the students’ relationships with and in mathematics change. in contrast to earlier accounts, all the students described their enjoyment in learning and being challenged in mathematics. reasons for liking mathematics varied but a common factor related to their feeling of mathematical capability associated with increased knowledge of strategies and ways of using them while problem solving. now, when the students were asked who helped them to learn mathematics during the lesson, they attributed their learning to the teacher, their classmates, and themselves. clearly, they felt part of a community where learning mathematics was an active process that involved them engaging with their own reasoning and the reasoning of others. for example, tere, in response to being questioned about whether she had previously learnt from other group members and the teacher, stated: ―i tried to but i didn’t know what they were saying. but now i ask myself some questions about what they are saying. like some things about the numbers.‖ the positioning of members of the classroom community had changed. they were taking increased responsibility for their own learning and the learning of other members of the group and they viewed the teacher’s role as one in which they worked with him as active partners: our teacher gives [a problem] to us and instead of just showing us a way to work it out, we have to get it in our group and then by writing it with him [reference to the teacher facilitating public sharing in plenary], and i get it now. (tama) a sense of their own, and their shared responsibility in the learning process, was evident when tama elaborated which of his group peers were involved in forming and writing the mathematical explanations: ―with my group pretty much, all of us because we are in the same group.‖ as noted by matiu, group work remained a positive feature of students’ learning experience: working in the group, i have never actually been good at it in previous years. but this year, it has been a lot better because i have people to help me and i learn different strategies from other people. working in a group this year has been important for my learning and that is what is helping me. however, in affirming the value of working in small groups, the following response by hemi exemplified how students now recognised that collaboration while an important way of helping also afforded them with opportunities to deepen their own understanding: hunter & anthony forging mathematical awareness journal of urban mathematics education vol. 4, no. 1 110 well, we had maurima in our group today and he was struggling with the maths and so he was going the hard yards and adding all the information in. and his questions, because he didn’t have the knowledge we had, made us understand better because it was like having somebody fresh in and he was keen and ready to go with lots of questions. now, students also indicated that they readily accepted working with a range of peers: ―i like working with lots of different people because it shows me how we can work with our friends‖ (koru). not only were students aware of the benefits of social and relational aspects of group work but also they were keenly aware that collaboration supported their use of effective mathematical practices. all the students used inclusive talk to describe their responsibility to construct a mathematical explanation that they could all understand and explain to the larger sharing group: ―yes, so the other groups understand what we are saying, what we mean from this‖ (mereana). a number of students described their gradual growth in confidence to speak and explain their mathematical reasoning. regarding the small problem solving groups as a safe setting for them to risk take—students were more comfortable in constructing and trialing the initial presentation of their explanations. their description of this change revealed an increased awareness of the value of being part of a community of learners. they expressed value in having their teacher hear what they knew, and value in sharing their reasoning, and value in teaching and helping other participants in the group to learn: ―it is just like if someone gets confused then you are there to help them where they are going and things like that‖ (ana). they indicated their acceptance that some listeners might be confused and that their explaining could help clarify their understandings, particularly if they asked questions: ―you can teach other people like if they do not know and when they ask questions you can teach them what they don’t know‖ (hemi). at the same time they outlined the mutual responsibility for their own learning and the learning of others’ as central to group activity: it is just like saying you don’t really get it and then others help you. your team helps you to explain it for the bigger group. you are learning by building your confidence. you are learning as well because you are working out a problem, you are working out a problem and you are speaking at the same time. before you speak, you have to think and work it out first. (ana) notably, students now viewed presentation of mathematical explanations to the large sharing group as an important part of their learning. the importance of asking questions in both the small and larger group was also readily acknowledged by the students. they stated an increase in confidence and competence to ask questions. however for some, like mele below, concerns hunter & anthony forging mathematical awareness journal of urban mathematics education vol. 4, no. 1 111 remained, particularly in regards to protecting the self-esteem of other members of the group: i find it a bit challenging as well because you are asking the questions but they might not have the answer. there are some bits that we do not understand and we have to ask, ―where did they get it from?‖ in the small group and in the bigger group. as well as asking questions, students were learning to engage in mathematical argumentation as they expressed agreement or disagreement with other students’ assertions. tarai noted: you could have others with different types of answers, some could disagree and some could agree. but then if you disagree you have to give an explanation on why you disagree. however, at this stage, agreeing or disagreeing with the reasoning of others was problematic for many students. matiu’s statement captures this tension: it is good to disagree if you strongly believe that it is not right, that the answer is not right. as long as you are not doing anything wrong it is good to friendly argue. it is like the teacher says friendly arguing in summary, within the mid-phase of the intervention, students showed an increased awareness and understanding of the role of mathematical argumentation. but it was still not part of their everyday repertoire of readily accepted actions, and held the potential to cause a loss of confidence. as mele stated: ―it made me think better but it is a bit scary.‖ continuing on the change trajectory at the conclusion of the school year, interview responses indicated that the students had made significant progress toward enacting the mathematical practices detailed on the cpf framework. the students outlined their confidence to explain, question, learn from mistakes, and use ―friendly arguing‖ to achieve group agreement or to provide mathematical justification. they talked about how they liked to have time to complete their mathematics independently as well as working within a group. group work they now described as important when the mathematical activity was difficult. they outlined how, in group interactions, the multiple sources deepened their learning: ―so you get different answers and different questions from your group members‖ (tarai). they also recognised the positive effects of listening to more proficient explanations and having their reasoning extended in the larger group setting: ―i didn’t really get it [what helped] hunter & anthony forging mathematical awareness journal of urban mathematics education vol. 4, no. 1 112 was working in our group and then the other group made us think even higher when they explained it‖ (teresa). they accepted initial confusion and being stuck as part of their learning. but at the same time, they recognised their emotional reaction when stuck, as hine described: i would be good with my basic facts and stuff but when it gets harder i just break down. i get stuck and i get shocked sort of. [then] i go back and think again. now, they had a range of ways to participate and meet mathematical challenges head on. these included discussion with other individuals, the teacher, their group, or with themselves. the teacher was positioned as only one source (among others) to draw on and they described how they requested his help only when they were really stuck. they stated that they only wanted clues from him to get them started but would not learn if he detracted from their struggle by giving too much information: i prefer to work with the group and struggle but get a little bit of help, just a little bit of help not the whole lot. [if the whole lot is given] i would just forget straight away. i would not remember because i have not struggled. (mele) for mele, the view of struggling as an essential component of learning (heibert & grouws, 2007) is captured as she continues and describes what mathematics means for her: ―it is about working out problems that are challenging and struggling, struggling well, it is to get somewhere further than you are. struggling is learning.‖ in summary, the roles of all participants in the classroom had been transformed. it was evidenced from the students’ responses that their active role in the classroom included taking ownership of their learning across a range of aspects of mathematical practices. their need for active involvement in learning is well captured in a tama’s comment: if you get involved, you will know lots about maths, and if you know lots, you will be successful. it’s like tell me i will forget, show me i might remember, but if you involve me, i will learn lots. discussion clearly, students can change their relationship with and in mathematics. in our case classroom, the students’ voices provide insight into the ways they changed how they came to understand what it means to both learn and do mathematics. to understand more about the students’ changing relationship with ma hunter & anthony forging mathematical awareness journal of urban mathematics education vol. 4, no. 1 113 thematics, we return to cobb et al.’s (2009) analytic framework based on the general obligations—concerning the distribution of authority and the ways that students are able to exercise agency and the mathematical obligations—concerning what counts as being mathematically competent within the classroom. in our case classroom, the teacher took great care to socially negotiate the obligations in a manner that was responsive to students’ cultural histories and valued practices. his careful and consistent shift in positioning from a central role of authority and agency to one which was shared, and indeed expected of all participants, was reflected in the students’ voices. as a result, the students developed increasingly more positive mathematical dispositions and their motivation to engage at progressively deeper levels increased. their change in disposition was embedded within and inextricably linked to their shift in authority and agency in the classroom. the classroom norms that held all students accountable for their own sense making and the sense making of others during mathematical activity were a pivotal factor in the strong sense of competence established by community members. as gresalfi and her colleagues (2009) suggest, opportunities for students to be seen as competent rely on assigned mathematical activity and how the students are held accountable for completion of it. in the case classroom, both the general and mathematical obligations associated with group mathematical activities shifted as the year progressed. as the teacher devolved responsibility for the solution strategies, it is evident that there were clear shifts in how the students viewed their competence to participate in the mathematics. initially, they considered the skill of listening and watching the teacher show them a solution strategy demonstrated competence (or incompetence) to learn mathematics. over time, however, they reconstructed this passive approach to learning to one in which competence was interpreted as active construction of mathematical meaning through participating in interactions with others using a range of mathematical practices. the students’ narratives of their learning experiences showed increased awareness of their role in the participatory practices of inquiry group work activities. their accounts reflected an increased confidence in their competence to be a useful and valued contributor in the learning community, alongside an increased awareness of and propensity to utilise productive mathematical practices as part of learning and doing mathematics. at the same time, their statements draw attention to the relational aspects that need to be considered when requiring students to participate in mathematical practices. the students initially voiced reluctance to participate in such mathematical actions as formulating and making conjectures, asking questions, or forming agreement or disagreement when not completely confident in their adequacy to do so. however, the perceived risks gradually diminished as the students (as individuals and as a collective) gained competence and were confident to participate in a variety of ways. nonetheless, for many stu hunter & anthony forging mathematical awareness journal of urban mathematics education vol. 4, no. 1 114 dents, even when they felt confident and competent to use a range of different mathematical practices, questioning and challenging others, and, in turn, being questioned or challenged by others, remained an emotionally charged activity. their descriptions of their reactions in the final interviews serve to remind us of the need for teachers to carefully consider what bibby (2009) describes as the emotional labor and risk involved for students in participating in development of mutual mathematical understandings. heeding esmonde’s (2009b) caution that different practices in different social and cultural groupings may be interpreted differently, one needs to be aware that many practices (here examples include questioning, agreeing, disagreeing, and challenging) are not common experiences for all students. here, there is evidence that the students acknowledged inquiry and argumentation as beneficial for their learning but they recognised their novice status in engaging in inquiry and argumentation—practices they were not familiar with or necessarily comfortable with using. they also held concerns over how others in the classroom community might interpret their intent when using these practices. whilst the main purpose of this article is to examine how students’ viewed the changes in their learning environment, as a way to understanding changes in their relationships with and in mathematics, we return at this point to the nature of the intervention. our assertion is that the case teacher (and students) successfully over time created an environment that supported inquiry learning and the risks inherent in being held accountable for one’s own learning and the learning of other members of the community. we make this assessment based on the observed pedagogic enactment of the communication and participation framework (cpf). through the use of this tool, the teacher attended to building the students’ use of inquiry discourse. as evidenced in student descriptions, this building resulted in them gradually developing what gee and clinton (2000) term a social language: ways of talking, acting, and doing mathematics within an inquiry environment. changes were reflected in the negotiated general and mathematical obligations that enabled these pasifika students to act as instructional agents. contributing pieces of mathematical knowledge and more advanced understandings held by different group members were combined to construct and progress a collective understanding. each member of the group was considered a knowledge component of the collective and accountable to the collective to participate and use the mathematical practices competently. in turn, the teachers’ responses tended to be directed more to a ―collective student.‖ in other words, for each contribution, the teacher sought to draw out implications for the learning of the whole class, rather than for each individual student. hunter & anthony forging mathematical awareness journal of urban mathematics education vol. 4, no. 1 115 conclusion enacting culturally responsive teaching to enable all students to participate and contribute competently in mathematics classrooms is a key equity issue (macfarlane, 2004). in the case classroom, the teacher drew on the cultural capital of the pasifika students and the core pasifika beliefs of reciprocity, collectivism, and communalism to frame both grouping arrangements and expected behaviors. in addressing the group norms for collaborative interactions, the teacher created a space for all participants to have opportunities to engage in equitable exchanges. within this environment, students constructed views of themselves as both competent individual learners and competent learners within collectives. for the most part, their accounts indicate awareness of changes in their relationship with and in mathematics. these findings have important messages for addressing the pressing concern of equitable participation of diverse learners in the mathematics classroom. they provide evidence that when general and mathematical obligations attend to the cultural, social, and mathematical well being of students, inquiry-based classrooms can be empowering and positive for students who have previously been marginalized. in this study, creating a mathematical community of inquiry that was culturally inclusive supported 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(2006). children’s mathematical thinking in different classroom cultures. journal for research in mathematics education, 37, 222–255. young-loveridge, j. (2009). patterns of performance and progress of ndp students in 2008, in findings for the new zealand numeracy development project 2008 (pp. 12–26). wellington, new zealand: ministry of education. http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/74/49 hunter & anthony forging mathematical awareness journal of urban mathematics education vol. 4, no. 1 119 appendix a communication and participation framework to engage students in mathematical practices within an inquiry classroom: develop conceptual explanations, including using the problem context to make explanations experientially real  provide a mathematical explanation. use the context of the problem not just the numbers. provide mathematical reasons (e.g. rather than ―tidying,‖ state 19+7=20+6 because 6+1=7 and 19+1=20).  develop two or more ways to explain a strategy solution.  analyse the explanation and construct ways to revise, extend, and elaborate on sections others might not understand.  predict questions that will be asked and prepare mathematical responses. active listening and questioning for sense making of an explanation  ask questions that clarify an explanation (e.g., what do you mean by? what did you do in that bit? can you show us what you mean by? could you draw a picture of what you are thinking?). collaborative support and responsibility for the reasoning of all group members  agree on the construction of one or more solution strategies that all members can explain.  work together to check, explain, and re-explain in different ways the group explanation. develop justification and mathematical argumentation  indicate agreement or disagreement (with mathematical reasons) for part of an explanation or a whole explanation.  justify an explanation using language (e.g., i know 3+4=7 because 3+3=6 and one more is 7).  use exploratory language (e.g., so, if, then, because, to justify and validate an explanation).  use questions that lead to justification (e.g., how do you know it works? can you convince us? why would that tell you to? why does that work like that? so what happens if you go like that? are you sure it’s? so what happens if? what about if you say…does that still work?). develop representations of the reasoning  represent reasoning as part of exploring and making connections (e.g., how can i/we make sense of this for my/ourselves?).  represent reasoning to explain and justify the explanation (e.g., how can i explain, show, convince other people?).  use a range of representations including acting it out, drawing a picture or diagram, visualising, making a model, using symbols, verbalising or putting into words, using materials. develop generalizations  extend the explanation and/or justification to a representation of the mathematical relationship in general terms.  identify the rules and relationships through making and extending the connections.  use questions that lead to generalisations (e.g., does it always work? can you make connections between? can you see any patterns? can you make connections between? how is this the same or different to what we did before? would that work with all numbers?). journal of urban mathematics education july 2011, vol. 4, no. 1, pp. 7–14 ©jume. http://education.gsu.edu/jume jo boaler is professor of mathematics education at stanford university, school of education, ceras, 520 galvez mall, stanford, ca 94305-3084; email joboaler@stanford.edu. her research focuses on the impact of different teaching approaches, grouping systems, and the promotion of equity. commentary changing students’ lives through the de-tracking of urban mathematics classrooms jo boaler stanford university hen schools and other education institutions move to de-track mathematics classrooms, opportunities for student learning increase. evidence for this fact comes from a number of different research studies that show that highachieving students achieve at the same levels in tracked and untracked groups but that middleand low-achieving students score at significantly higher levels when they are not working in tracks (boaler, 2002, 2008a; burris, heubert, & levin, 2006; nunes, bryant, sylva, & barros, 2009). but when schools de-track, mathematics classrooms changes are not only evident in test scores. teaching environments that encourage high achievement from all students provide a range of possibilities for student learning that go well beyond content knowledge. heterogeneous classrooms that are based upon co-operation among students change student perceptions of who they are and who they can be (boaler, 2005), they change perceptions of the nature of mathematics, they teach students about the different qualities and contributions of students who are different from themselves (boaler, 2008a, b) and they challenge the racial segregation that continues in schools. despite the role that de-tracking plays in promoting equitable and high achievement, schools across the united states continue to divide students by perceptions of ―ability‖ and communicate to students the idea that only some people— particularly white, middle class people—can be good at mathematics. the question of the ways students‘ learning opportunities vary in tracked and de-tracked mathematics classrooms is one that has been close to my heart for many years. my first teaching job was in an urban inner london comprehensive school, with extensive racial diversity—some 45 different languages were spoken there. i arrived in my first classroom, fresh and eager to implement ideas from my teacher education courses at london university, with a group of students who had recently been placed into the bottom track. their first words to me were, ―why should we bother?‖ i found it hard to answer that question, particularly as i knew that their placement into the bottom track meant that they had severely limited opportunity to achieve highly on the national examination in 2 years time. by my w boaler commentary journal of urban mathematics education vol. 4, no. 1 8 second year in the school, i had worked with the other teachers in the department to de-track all of the mathematics classes, and they remain de-tracked to this day. this meant that students who would previously have been placed in a low track with limited learning opportunities were able to achieve at the highest levels in the school in the following year, with focused encouragement and teaching. when i became a researcher, i was given the opportunity to investigate the impact of tracking and de-tracking more systematically and i have now conducted two different longitudinal research studies in which students in heterogeneous urban mathematics classrooms achieved at significantly higher levels than students who worked in tracked groups (boaler, 2002, 2008a; boaler & staples, 2008). in both cases, the teaching in the de-tracked groups was designed for heterogeneous classes; i review the essential features of such teaching environments later in this commentary. while much of my career has been spent studying and understanding the opportunities provided by de-tracked classrooms, i have also come to understand the challenges faced by teachers of such classes, particularly in urban schools, and realize that it is important to understand these challenges if we are to help more teachers teach heterogeneous classrooms effectively. in the first part of this commentary, i review some of the findings from research on de-tracked classrooms, highlighting the advantages and possibilities provided by de-tracking. in the second part, i outline some of the features of the teaching that is necessary to provide positive work experiences for all students in de-tracked groups. benefits of de-tracked mathematics classes the first reason to de-track mathematics classrooms is that they offer increased opportunities for student learning and for high achievement. in my own studies, i have followed hundreds of students through tracked and untracked groups, collecting quantitative data on student achievement as well as qualitative, focused data on the work of teachers and students. in england, i followed 300 students through two schools, one of which was untracked and the other tracked (boaler, 2002). in the united states, i followed hundreds of students through three schools, with one of them being untracked (boaler, 2008a; boaler & staples, 2008). in close, detailed studies of the teaching and learning in these schools, i was able to show the impact of tracking. in england, students from middle and low groups in the tracked school effectively gave up and became unmotivated when they were placed into their middle and low groups. at the end of 3 years, students in the untracked school scored at significantly higher levels on the national examination (boaler, 2002). this study confirmed others in showing that tracking significantly reduced the achievement of middleand low-achieving students. it also showed that students in the highest group, particularly girls, dis boaler commentary journal of urban mathematics education vol. 4, no. 1 9 liked their fast, high-pressure work environments and dis-identified with mathematics when they were placed into the highest group (boaler, 1997). in the untracked school, all students were given the opportunity to achieve, and this resulted in significantly higher achievement than students who worked in tracked groups. critics of studies of small numbers of schools argue that the schools could have achieved success for other reasons, such as exceptional teachers. this reoccurring critique makes a recent study all the more interesting as researchers at oxford university recently followed 16,000 students through schools considering (among other things) the impact of tracking for students in grades 4 and 6 (see nunes, bryant, sylva, & barros, 2009). their conclusions, even across many schools with different teaching styles, were clear: the achievement at the schools without tracking was significantly higher and the reason for this was that tracking diminished the achievement of students in middle and low groups. in the united states, there have been a number of studies examining achievement differences for students in tracked and untracked mathematics groups. for example, burris, heubert, and levin (2006) conducted an interesting study of a de-tracking innovation in mathematics that compared six annual cohorts of students in a diverse middle school in suburban new york. the student cohorts attending the school in 1995, 1996, and 1997 were taught in tracked classes with only high-track students being taught the advanced curriculum. but in 1998, 1999, and 2000 all students in grades 7 through 9 were taught advanced curriculum in mixed-ability classes and all of the ninth graders were taught an accelerated algebra course. burris and colleagues explored the impact of these different middle school experiences upon the students‘ completion of high school courses and their achievement, using four achievement measures, including scores on the advanced placement calculus examinations. they found that the students from de-tracked classes took more advanced classes, passed courses at significantly higher rates, and passed exams a year earlier than the average in the state of new york. the scores of the students were also significantly higher on various achievement tests, and the increased success from de-tracking applied to students across the achievement range, from the highest to the lowest achievers. in my own study of 800 students who went through three different u.s. high schools, i was able to observe an unusual and highly effective teaching approach, called ―complex instruction‖ (cohen & lotan, 1997). in the urban school, which i called ―railside,‖ there was a very wide spread of student achievement, and teachers spent a lot of time and attention teaching students to work in groups and to listen to and respect each other (boaler, 2008b; boaler & staples, 2008). the results were impressive, with students at the school achieving at significantly higher levels than students in tracked groups, as in other studies—but interestingly, the students who were most advantaged in the de-tracked school were the boaler commentary journal of urban mathematics education vol. 4, no. 1 10 highest achieving students. it was these students whose achievement increased the most over 3 years and who achieved significantly higher than those in high-track groups. the reason for their high achievement was that they spent time explaining work, which strengthened their own understanding, and they were encouraged to broaden their mathematical ways of working. whereas the high-achieving students came into the school able to execute procedures quickly, the teachers encouraged the students to be broader: to consider different ways to solve problems, to view different mathematical perspectives, and to reason and interpret situations. at the beginning of this study, some of the high achievers complained about always having to help others, but by the second year they had changed their minds as they realized that the explanations and the depth of their work was helping their own achievement. in the following excerpt, a senior reflects upon working in a group with lower achieving students who might need her help: i think people look at it as a responsibility, i think it‘s something they‘ve grown to do like since we‘ve taken so many math classes. so maybe in ninth grade it‘s like oh my god i don‘t feel like helping them, i just wanna get my work done, why do we have to take a group test? but once you get to ap calc you‘re like oh i need a group test before i take a test. so like the more math you take and the more you learn you grow to appreciate, like oh thank god i‘m in a group! (imelda, railside, year 4) when considering the achievement opportunities provided to students in tracked and de-tracked groups the evidence is clear: students who work in detracked groups are given opportunities to learn (porter, 1994), opportunities that are not always afforded to students in low groups. but recently, i was given the opportunity to consider the longer-term impact of ability grouping on students‘ achievement in the years after they attended school. i managed to track down the students who had progressed through the two different schools i had researched in england, some 8 years after they had left the schools. by that time, they were adults of about 24 years of age. i administered surveys to determine the jobs of these young adults, which allowed me to categorize their social class, revealing something very interesting. it revealed that students who had worked in untracked groups had significantly moved up the social-class scale, compared to their parents; whereas, those who had worked in tracked groups had stayed at the same social-class levels. in interviews, the students gave clear reasons for this lack of social mobility, telling me that the tracking they had experienced in school had constrained their achievement and made them feel that they were in ―psychological prisons,‖ which impacted their lives well beyond school (boaler, 2005). the benefits of good, heterogeneous grouping, enacted by skilled teachers, are many, ranging from high achievement, enhanced respect for other students (boaler, 2008b), and even social mobility. but research also tells us that teaching boaler commentary journal of urban mathematics education vol. 4, no. 1 11 de-tracked mathematics classes is challenging—specifically, in a culture such as the united states where parents (and other stakeholders) hold strong beliefs about the need for tracking in schools. inside the classroom, further challenges exist for teachers—in particular, providing materials that are appropriate for all of the different achievement levels, and encouraging students to work well together, especially when they have developed deficit ideas about students with lower achievement. these challenges are considered in the following section. successful teaching methods in de-tracked groups i have been fortunate to conduct two different longitudinal research studies in which i was able to observe highly successful teachers working with heterogeneous groups. in a third study, my graduate students and i taught four mixedability classes of sixth and seventh graders in order to put into practice some of the teaching practices that we had studied (boaler, sengupta-irving, dieckmann, & fiori, in preparation). in all three cases, the teachers used different teaching methods—one being to give open-ended projects, one to give less open problems but more careful teaching on students working together, and one that combined the other two approaches. but in all cases, teachers used a number of particular practices that i have come to regard as critical in the teaching of de-tracked groups: 1. students worked on mathematics tasks that were appropriate for many different achievement levels. in england, the teachers achieved this by giving open, exploratory tasks that students could take in different directions. different students used different mathematics, depending on where they chose to take the tasks. in the united states, the students worked on the same problems and worked together to agree upon answers, but the problems were chosen to be ―multi-dimensional‖— requiring different ways of being mathematical, such as asking questions, seeing problems in different ways, and drawing and representing. by broadening the mathematics that students worked on, teachers found that all students could offer important contributions. the mantra of the u.s. approach was, ―nobody is good at all of these ways of working but everyone is good at some of them.‖ whether teachers differentiate by task or by outcome, it is critical that students gain opportunities to work at the right level for them, that all students can contribute their thinking, and that all are challenged appropriately. these opportunities cannot occur when students are given the same narrow procedural questions. boaler commentary journal of urban mathematics education vol. 4, no. 1 12 2. students were taught to respect each other and to work well together. in the english study, students were allowed to choose whether they worked alone or in pairs or groups and they were left to choose their work partners. teachers always gave messages about all students being capable and i did not ever witness students being disrespectful or putting each other down during the 3 years i observed lessons. in the u.s. study, the grouping was much more deliberate and teachers arranged mixedachievement groups, which changed every few weeks. the teachers also spent a lot of time teaching students how to listen to each other, work together, and be respectful, which included spending the first 6 weeks of freshman year teaching students how to work well together. the teachers also enacted the methods of complex instruction, a pedagogical approach that has been designed to help make group work equal (cohen & lotan, 1997). 3. teachers gave messages about learning and smartness that are consistent with what dweck (2006) has termed a ―growth mindset.‖ this means that teachers always emphasized learning as a process, and stressed that high achievement was a reflection of effort not of innate ―ability‖ and that all students could reach the highest levels. teachers also found value in all students‘ thinking. in the complex instruction approach, they also used specific pedagogical practices to raise the status of ―low status‖ students, such as publicly praising their contributions. the different teaching methods that are used in successful de-tracked classes are methods that are completely in line with research on effective mathematics teaching and learning more generally. unfortunately, few teachers in the united states have received careful, sustained opportunities to learn these teaching methods (lampert & ball, 1998), which is why many teachers are daunted by the idea of working with de-tracked groups. in my current work, i am working with teachers in england who have de-tracked their urban classrooms for the first time. they are finding the teaching a challenge but they have also been encouraged by the responses of students, particularly those who would have been placed in a low track. as one of the teachers reported to me, ―it was my low ability children who had the greatest ideas!‖ despite the use of ―fixed mindset‖—labeling suggesting that students have a particular ―ability‖—the teachers are clearly changing their perceptions of what students can do, increasing their expectations for previously low achieving students, which is absolutely critical. boaler commentary journal of urban mathematics education vol. 4, no. 1 13 concluding thoughts in other countries such as finland and japan—countries that top the world in achievement and where all classes are untracked—teachers expect students of different achievement levels to work together and help each other, and they view different achievement levels as a resource rather than a challenge. as one teacher from japan reflected: japanese education emphasizes group education, not individual education. because we want everyone to improve, promote and achieve goals together, rather than individually. that‘s why we want students to help each other, to learn from each other…to get along and grow together—mentally, physically and intellectually. (as cited in boaler, 2008a, p. 108) when heterogeneous teaching is done well, students also come to appreciate working with students from different levels, as one of the students at railside reflected: everybody in there is at a different level. but what makes the class good is that everybody‘s at different levels so everybody‘s constantly teaching each other and helping each other out. (zane, railside, year 2) as we move into a new era in which teachers are using more reasonable content standards that broaden the definition of mathematics, and give more students opportunities to contribute positively, it is my greatest hope that many more teachers will learn the value of working with de-tracked groups and of giving students from across the achievement spectrum the opportunity to work at their own highest potential. such changes are central to a united states in which racial segregation, low and inequitable achievement, and widespread fear and hatred of mathematics are a thing of the past. references boaler, j. (1997) when even the winners are losers: evaluating the experience of ‗top set‘ students. journal of curriculum studies, 29, 165–182. boaler, j. (2002) experiencing school mathematics: traditional and reform approaches to teaching and their impact on student learning. mahwah, nj: erlbaum. boaler, j. (2005). the ‗psychological prison‘ from which they never escaped: the role of ability grouping in reproducing social class inequalities. forum, 47(2), 135–144. boaler, j (2008a). helping children learn to love their least favorite subject—and why it’s important for america. new york: penguin. boaler, j. (2008b). promoting ‗relational equity‘ and high mathematics achievement through an innovative mixed ability approach. british educational research journal, 34, 167–194. boaler, j., sengupta-irving, t., dieckmann, j., & fiori, n. (in preparation). the many colors of algebra: engaging disaffected students through collaboration and agency. boaler commentary journal of urban mathematics education vol. 4, no. 1 14 boaler, j., & staples, m. (2008). creating mathematical futures through an equitable teaching approach: the case of railside school. teachers college record, 110, 608–645. burris, c., heubert, j., & levin, h. (2006). accelerating mathematics achievement using heterogeneous grouping. american educational research journal, 43, 103–134. cohen, e., & lotan, r. (eds.). (1997). working for equity in heterogeneous classrooms: sociological theory in action. new york: teachers college press. dweck, c. s. (2006). mindset: the new psychology of success. new york: ballantine books. lampert, m., & ball, d. (1998). teaching, multimedia, and mathematics: investigations of real practice. new york: teachers college press. nunes, t., bryant, p., sylva, k., & barros, r. (2009). development of maths capabilities and confidence in primary school (research report no. dcsf-rp118). retrieved from department of children, schools, and families https://www.education.gov.uk/publications/eorderingdownload/dcsf-rr118.pdf. porter, a. c., (with associates). (1994). reform of high school mathematics and science and opportunity to learn. new brunswick, nj: consortium for policy research in education. retrieved from http://www.cpre.org/images/stories/cpre_pdfs/rb13.pdf. https://www.education.gov.uk/publications/eorderingdownload/dcsf-rr118.pdf http://www.cpre.org/images/stories/cpre_pdfs/rb13.pdf microsoft word final stinson vol 3 no 1.doc journal of urban mathematics education july 2010, vol. 3, no. 1, pp. 1–8 ©jume. http://education.gsu.edu/jume david w. stinson is an associate professor of mathematics education in the department of middlesecondary education and instructional technology in the college of education, at georgia state university, p.o. box 3978, atlanta, ga, 30303; e-mail: dstinson@gsu.edu. his research interests include exploring socio-cultural, -historical, and -political aspects of mathematics and mathematics teaching and learning from a critical postmodern theoretical (and methodological) perspective. he is a co-founder and current editor-inchief of the journal of urban mathematics education.

 editorial the sixth international mathematics education and society conference: finding freedom in a mathematics education ghetto david w. stinson georgia state university n march 2010, i had the opportunity to attend and present at the sixth international mathematics education and society conference (mes 6)1 in berlin, germany. this opportunity provided a unique and rare experience for me as a relatively new mathematics education scholar, social scientist, and teacher educator, an experience that i characterize as finding freedom in a mathematics education ghetto. i use the term ghetto to juxtapose my professional experience of finding freedom as a mathematics educator to think (and act) differently found at mes 6 with my personal experience of finding freedom as a gay/queer2 man to think (and act) differently found when i discovered, in the mid-1980s, the “gay ghetto” in my hometown atlanta, georgia, usa. that is to say, both experiences provided a sense of self-empowerment and liberation—albeit, one professional and the other personal. here, it is important to note that i do not intend to romanticize the oftenharmful consequences of the ghetto.3 but like many members of historically mar























































 1mes 6 was held in johannesstift, a planned charitable community located approximately 10 miles northwest of berlin, germany, march 20–25, 2010. for complete information about mes 6, including plenaries, project presentations, research papers, and symposia, see http://www.ewi-psy.fu-berlin.de/en/v/mes6/. for complete information regarding previous mathematics education and society conferences, see the following websites: • mes 5 – 2008 albufeira, portugal: http://www.mes5.learning.aau.dk/ • mes 4 – 2004 golden coast, australia: (website not available) • mes 3 – 2002 helsingør, denmark: http://www.mes3.learning.aau.dk/ • mes 2 – 2000 montechoro, portugal: http://nonio.fc.ul.pt/mes2/ • mes 1 – 1998 nottingham, united kingdom: http://www.nottingham.ac.uk/csme/meas/conf.html 2 “‘queer’ can function as a noun, an adjective, or a verb, but in each case is defined against the ‘normal’ or normalizing” (spargo, 1999, pp. 8–9). 3 sears (1991), in his book growing up gay in the south: race, gender, and journeys of the spirit, cautions that although gaining access to gay communities (i.e., gay ghettos) is often liberating that i stinson editorial 
 journal of urban mathematics education vol. 3, no. 1 2 ginalized racial, ethnic, religious, cultural, gendered, sexual, intellectual, and so on communities, the ghetto has a different meaning to those who find themselves as members of such ghettos, as opposed to those from dominant groups. historically, the term ghetto has its origin in the jewish ghetto of medieval cities of europe (wirth, 1927), and, of course, the world became all too familiar with the concept with the infamous jewish ghetto of warsaw, poland during world war ii. the united states has its infamous, or famous—depending on perspective— ghettos as well. the most well known of these ghettos has been harlem new york city, usa, the center of black american intelligentsia and artistry since the early 1900s and, in many ways, the center of american intelligentsia and artistry of the twentieth century in general (gates & west, 2000). louis wirth (1927), a jewish immigrant and an american sociologist of the early chicago school, in providing a natural history of the ghetto, noted a unique, and i might add promising, characteristic of the urban ghetto. wirth claimed that it is within the ghetto “where one finds freedom from hostile criticism and the backing of a group of kindred spirits” (p. 60). langston hughes (1945/1994), one of america’s literary giants, an african american and member of the harlem ghetto, wrote the poem the heart of harlem4 that captures this sense of freedom and backing from kindred spirits. i provide hughes’s poem in its entirety, because it is an aspect of his poetic description of harlem that i use to frame my discussion of the concept ghetto, and the rare experience of finding freedom as a mathematics education scholar, social scientist, and teacher educator in the mathematics education ghetto of mes 6. the heart of harlem by langston hughes the buildings in harlem are brick and stone and the streets are long and wide, but harlem’s much more than these alone, harlem is what’s inside— 






















































































































































 a gay or lesbian identity can be as oppressive as it is liberating, as reactionary as it is revolutionary. it promises possibilities; it poses problems. identifying oneself as a lesbian or gay man enhances self-understanding and raises social consciousness; it also limits potential sexual experiences, reinforces the norm of heterosexuality, reifies the “homosexual,” and lessens opportunities for growth of the spirit. becoming a homosexual invites further sexual categorization…and social segregation…. within a society that too readily sorts, categorizes, and segregates people. (pp. 407–408) unfortunately, i fear, similar such statements could be made about other ghettos, including those constructed around professional identities. 4 langston hughes’s (1945/1994) the heart of harlem was written in conjunction with a musical score composed by duke ellington, a giant of american classical music (i.e., jazz). stinson editorial 
 journal of urban mathematics education vol. 3, no. 1 3 it’s a song with a minor refrain, it’s a dream you keep dreaming again. it’s a tear you turn into a smile. it’s the sunrise you know is coming after a while. it’s the shoe that you get half-soled twice. it’s the kid you hope will grow up nice. it’s the hand that’s working all day long. it’s prayer that keeps you going along— that’s the heart of harlem! it’s joe louis and dr. w.e.b., a stevedore, a porter, marian anderson, and me. it’s father divine and the music of earl hines, adam powell in congress, our drives on bus lines. it’s dorothy maynor and it’s billie holiday, the lectures at the schomburg and apollo down the way. it’s father shelton bishop and shouting mother horne. it’s the rennie and the savoy where new dances are born. it’s canada lee’s penthouse at five-fifty-five. it’s small’s paradise and jimmy’s little dive. it’s 409 edgecombe or a cold-water walk-up flat— but it’s where i live and it’s where my love is at deep in the heart of harlem! it’s the pride all americans know. it’s the faith god gave us long ago. it’s the strength to make our dreams come true. it’s a felling warm and friendly given to you. it’s that girl with the rhythmical walk. it’s my boy with the jive in his talk. it’s the man with muscles of steel. it’s the right to be free a people never will yield. a dream…a song…half-soled shoes…dancing shoes a tear…a smile…the blues…sometimes the blues mixed with the memory…and forgiveness…of our wrong. but more than that, it’s freedom— guarded for the kids who came along— folks, that’s the heart of harlem! “folks, that’s the heart,” is a refrain that is easily remembered, but often forgotten, even when in the company of kinder spirits. but in the heart of harlem, hughes’s (1945/1994) chief purpose, i believe, is to remind us to move beyond focusing on the structures to celebrating the people—the heart—without romanticizing the inequities and injustices of the ghetto’s structures (physical and otherwise). in other words, hughes limits his focus on the structures. i must admit, however, that after the first agora (i.e., business meeting) of mes 6, i began stinson editorial 
 journal of urban mathematics education vol. 3, no. 1 4 to focus on the “structure” of mes 6 rather than its people.5 in so doing, i became somewhat disenchanted with the conference, given that i perceived some aspects of the structure of the agora to be too similar to the structures found in education conferences in the united states; structures that are designed (most often?) to maintain rather than transform the status quo. (i found similar such structures within the gay ghetto of atlanta in the mid-1980s.) unfortunately, and in too many ways, i believe that even for members of ghettos it is difficult to think the unthought (cf. foucault, 1969/1972) in our individual and collective attempts to construct spaces that might be more ethical and just. in that, members of ghettos, like members of dominant groups, have been so discursively constituted within the multiplicities of unethical and unjust sociocultural and sociohistorical structures and discourses (cf. foucault, 1969/1972) that we often—unintentionally, i suppose—duplicate the very structures and discourses that positioned us as members of ghettos in the first place. i include this brief, but important, critique of mes 6 to make clear that it was not without its flaws. but, on the other hand, and more importantly, when i redirect my focus from the structure of mes 6 back onto its people, its folks, my participation at mes 6 was invaluable. that is to say, as a mathematics education social scientist who works to deconstruct (cf. derrida & montefiore, 2001)6 “the fictions, fantasies and plays of power inherent in mathematics education” (walkerdine, 2004, p. viii), the folks of mes 6 provided me an invaluable experience of selfempowerment and liberation. at mes 6, i found a critical mass of kinder spirits who understand and acknowledge, either explicitly or implicitly, the discipline mathematics education as a discursive formation, limited by sociocultural and sociohistorical assumptions, conditions, and power relations (cf. foucault, 1969/1972). in other words, a critical mass of scholars who acknowledge the discipline mathematics education as a system of unjust and unethical capital-t truths “linked in a circular relation with systems of power which produce and sustain it, and to effects of power which it induces and which extend it” (foucault, 1977/1980, p. 133). in short, the discipline mathematics education, i fear, has become a “‘régime’ of truth” (p. 133). present at mes 6, however, was an extraordinary group of scholars—from the novice to the accomplished—who use a multiplicity of philosophical, theo























































 5 i thank paola valero, a leading scholar of the mathematics education and society conferences, for reminding me, during a private conversation held after the first agora, to redirect my focus on the people, a conversation that in part motivated this editorial. 6 st. pierre (2000) argues that derrida’s concept deconstruction “is not about tearing down but about rebuilding…looking at how a structure has been constructed, what holds it together, and what it produces” (p. 482). she further claims that because deconstruction acknowledges that the world has been constructed through language and cultural practices, it can be deconstructed and reconstructed again and again (st. pierre). stinson editorial 
 journal of urban mathematics education vol. 3, no. 1 5 retical, and scientific concepts as tools in their attempts to smash this regime of truth, this system of power.7 i found it to be self-empowering and liberating—in a word, freeing—to be in the company of this group of scholars who had moved beyond “traditional” mathematics education research and were asking different questions and framing, theoretically, those questions differently. this group, most fortunately, did not represent the “growing concern among many mathematics education scholars regarding the lack of attention to mathematics in much of the current work in mathematics education” (ball, battista, harel, thompson, confrey, 2010, p. 60).8 in that, the question “where’s the math?” (heid, 2010, p. 102) was not asked.9 which is not to say that mathematics was absent in these scholars’ arguments. indeed, mathematics as the discursive formation mathematics education was forever present, everywhere! how does mathematics education sustain itself as an institutional space of whiteness (martin, 2010)? how does mathematics education induce the continued marginalization of bilingual and multilingual learners (chronaki, planas, setati, & civil, 2010)? how does mathematics education produce a specific “regime of rationality” (kanes, morgan, & tsatsaroni, 2010)? the aforementioned are a mere sampling of questions explored, scientifically, at mes 6. broadly speaking, questions explored at mes ask: how does mathematics education function as a discursive formation? how is it produced? how is it regulated? how does it exist (cf. bové, 1995)? but even as these (and other) “how” questions are explored from a multiplicity of socio-cultural and 























































 7 similar to foucault (1975/1996), i like to think of philosophical, theoretical, and scientific concepts as tools to smash or short-circuit systems of power: all my books…are like, if you like, little tool boxes. if people want to open them, use a particular sentence, idea, or analysis like a screwdriver or wrench in order to short-circuit, disqualify or break up the systems of power, including eventually the very one from which my books have issued…well, all the better! (p. 149) 8 i thank danny martin and maisie gholson for bringing my attention to the 2010 national council of teacher of mathematics’ research presession symposium titled “keeping the mathematics in mathematics education research”; the quote is extracted from the symposium’s description found in the 2010 program for the research presession (p. 60). as listed in the program, deborah ball, michael battista, guershon harel, and patrick thompson were the symposium presenters; jere confrey was the symposium discussant. 9 i hope that the statement and question by ball et al. (2010) and heid (2010) are not marking the being of a new “war” within mathematics education (schoenfeld, 2004). history demonstrates that, in the end, such wars are unproductive. given the extreme focus on mathematics education within the current political environment, as a community of mathematics education social scientists, we should be encouraging the expansion of the science of mathematics education, not attempting to contract it. in short, as the disciplined science mathematics education continues to expand beyond cognitive psychology (kilpatrick, 1992), we must allow each generation to “address anew what doing research in mathematics education is all about” (sierpinska & kilpatrick, 1998, p. 527). the chief purpose of the journal of urban mathematics education has been, and continues to be, to not only encourage expanding the disciplined science mathematics education but also to provide an intellectual and accessible outlet for the dissemination of such expanding disciplined science, specifically, within urban contexts. stinson editorial 
 journal of urban mathematics education vol. 3, no. 1 6 political theoretical paradigms, pais, stentoft, and valero (2010) caution that we should not forget the “why” questions. they argue that placing too much emphasis on how questions takes mathematics education for granted and limits radical alternatives. to engage in why questions can ultimately lead to the question why mathematics education, which implies a critical exploration of its very existence (pais et al.). exploring how and why questions from a multiplicity of socio-cultural and political theoretical paradigms, however, might be too much discomfort for some (many?) mathematics education social scientists. but like lather (2006) who suggests that a proliferation of theoretical paradigms is a good thing to think with when conducting education research, i believe that mathematics education social scientists need a proliferation of theoretical paradigms to smash the systems of power inherent in mathematics education. as i and others have argued elsewhere (stinson, 2006; weissglass, 2002), for those social scientists who are focused on issues of equity and justice within mathematics education, the critiques and explorations of mathematics education must become much broader than those found in the confines of the instructional triangle (cohen & ball, 1999; also see national research council, 2001, p. 314). as a community of mathematics education social scientists, if we wish to take an ethical stance, adopting a degree of social consciousness and responsibility in seeing the wider social and political picture of mathematics education (gates &vistro-yu, 2003), we must continue to take the social turn (lerman, 2000) or, better yet, the sociopolitical turn (gutiérrez, in press) in our research, exploring not only questions of how but also questions of why (pais et al., 2010). if taking up and supporting such an ethical stance relegates me (and others) to a mathematics education ghetto because such a stance resides outside the simplicity of the instructional triangle—so be it. i am happy that i discovered a mathematics education ghetto at mes 6. after all, my personal experience as a member of a gay ghetto, as well as the history of ghettos generally, has demonstrated that kinder spirits relegated to ghettos can (do) build empowering and liberating communities, and, in the end, transform society at large. that is, if in the process of building such communities, the folks—the heart—are maintained as the focus. i encourage you, as a reader of the journal of urban mathematics education, to plan to attend mes 7 in 2012,10 so that you too might have a similar such experience as i did in finding freedom in a mathematics education ghetto (just remember, keep your focus on the folks). and in the meantime, i suggest that you explore past mes proceedings,11 so that you might become familiar with 























































 10 the location of mes 7 in 2012 has yet to be determined. 11 my first introduction to the mathematics education and society conferences was in 2002, during graduate school in the department of mathematics education at the university of georgia, when two of my then fel stinson editorial 
 journal of urban mathematics education vol. 3, no. 1 7 a critical mass of kinder spirits who are conducting “good” education research (hostetler, 2005), producing different knowledge and producing knowledge differently (st. pierre, 1997). references ball, d. l., battista, m., harel, g., thompson, p. w., & confrey, j. e. (2010, april). keeping the mathematics in mathematics education research. research symposium at the national council of teachers of mathematics research presession, san diego, ca. bové, p. a. (1995). discourse. in f. lentricchia & t. mclaughlin (eds.), critical terms for literary study (2nd ed., pp. 55–65). chicago: university of chicago press. chronaki, a., planas, n., setati, m., & civil, m. (2010). same questions different countries: use of multiple languages in mathematics learning and teaching. in u. gellert, e. jablonka, & c. morgan (eds.), proceedings of the sixth international mathematics education and society conference (vol. 1, pp. 72–79). berlin, germany: freie universität berlin. cohen, d. k., & ball, d. l. (1999). instruction, capacity, and improvement. consortium for policy research in education. retrieved from http://www.cpre.org/images/stories/cpre_pdfs/rr43.pdf.
 derrida, j., & montefiore, a. (2001). “talking liberties”: jacques derrida’s interview with alan montefiore. in g. biesta & d. egéa-kuehne (eds.), derrida & education (pp. 176–185). new york: routledge. foucault, m. (1972). the archaeology of knowledge (a. m. sheridan smith, trans.). new york: pantheon books. (original work published 1969) foucault, m. (1980). truth and power (c. gordon, l. marshall, j. mepham, & k. soper, trans.). in c. gordon (ed.), power/knowledge: selected interviews and other writings, 1972–1977 by michel foucault (pp. 109–133). new york: pantheon books. (original work published 1977) foucault, m. (1996). from torture to cellblock (j. johnston, trans.). in s. lotringer (ed.), foucault live: interviews, 1961–1984 (pp. 146–149). new york: semiotext(e). (original work published 1975) gates, h. l., & west, c. (2000). the african-american century: how black americans have shaped our country. new york: free press. gates, p., & cotton, t. (eds.). (1998). proceedings of the first international mathematics education and society conference. nottingham, united kingdom: the center for the study of mathematics education, nottingham university. gates, p., & vistro-yu, c. p. (2003). is mathematics for all? in a. j. bishop, m. a. clements, c. keitel, j. kilpatrick, & f. k. s. leung (eds.), second international handbook of mathematics education (vol. 1, pp. 31–73). dordrecht, the netherlands: kluwer. gutiérrez, r. (in press). the sociopolitical turn. journal for research in mathematics education (special equity issue). heid, m. k. (2010). where’s the math (in mathematics education research)? journal for research in mathematics education, 41, 102–103. hostetler, k. (2005). what is “good” education research? educational researcher, 34(6), 16–21. 






















































































































































 low doctoral students, amy hackenberg and brian lawler, attended and presented at mes 3; upon their return, they gave me a copy of the proceedings of the first international mathematics education and society conference, edited by peter gates and tony cotton (1998) (the copy was inscribed by ole skovsmose). this edited volume has been an important resource in both my research and teaching. i thank my current institution, the college of education at georgia state university, for providing the travel funds to experience the mathematics education and society conference firsthand. stinson editorial 
 journal of urban mathematics education vol. 3, no. 1 8 hughes, l. (1994). the heart of harlem. in a. rampersad & d. roessel (eds.), the collected poems of langston hughes (pp. 311–312). new york: knopf. (original work published 1945) kanes, c., morgan, c., & tsatsaroni, a. (2010). analysing pisa’s regime of rationality. in u. gellert, e. jablonka, & c. morgan (eds.), proceedings of the sixth international mathematics education and society conference (vol. 2, pp. 272–282). berlin, germany: freie universität berlin. kilpatrick, j. (1992). a history of research in mathematics education. in d. a. grouws (ed.), handbook of research on mathematics teaching and learning (pp. 3–38). new york: macmillan. lather, p. (2006). paradigm proliferation as a good thing to think with: teaching research in education as a wild profusion. international journal of qualitative studies in education, 19, 35–57. lerman, s. (2000).the social turn in mathematics education research. in j. boaler (ed.), multiple perspectives on mathematics teaching and learning (pp. 19–44). westport, ct: ablex. martin, d. b. (2010). not-so-strange bedfellows: racial projects and the mathematics education enterprise. in u. gellert, e. jablonka, & c. morgan (eds.), proceedings of the sixth international mathematics education and society conference (vol. 1, pp. 42–64). berlin, germany: freie universität berlin. national research council. (2001). adding it up: helping children learn mathematics. j. kilpatrick, j. swafford, & b. findell (eds.), mathematics learning study committee, center for education, division of behavioral and social sciences and education. washington, dc: national academy press. pais, a., stentoft, d., & valero, p. (2010). from questions of how to questions of why in mathematics education research. in u. gellert, e. jablonka, & c. morgan (eds.), proceedings of the sixth international mathematics education and society conference (vol. 2, pp. 369– 378). berlin, germany: freie universität berlin. schoenfeld, a. h. (2004). the math wars. educational policy, 18, 253–286. sears, j. t. (1991). growing up gay in the south: race, gender, and journeys of the spirit. new york: harrington park press. sierpinska, a., & kilpatrick, j. (1998). continuing the search. in a. sierpinska & j. kilpatrick (eds.), mathematics education as a research domain: a search for identity: an icmi study (vol. 2, pp. 527–548). dordrecht, the netherlands: kluwer. spargo, t. (1999). foucault and queer theory. new york: totem books. st. pierre, e. a. (1997). circling the text: normadic writing practices. qualitative inquiry, 3, 403– 417. st. pierre, e. a. (2000). poststructural feminism in education: an overview. international journal of qualitative studies in education, 13, 467–515. stinson, d. w. (2006). african american male adolescents, schooling (and mathematics): deficiency, rejection, and achievement. review of educational research, 76, 477–506. walkerdine, v. (2004). preface. in m. walshaw (ed.), mathematics education within the postmodern (pp. vii–viii). greenwich, ct: information age. weissglass, j. (2002). inequity in mathematics education: questions for educators. the mathematics educator, 12(2), 34–39. wirth, l. (1927). the ghetto. the american journal of sociology, 33, 57–71. polish school report journal of urban mathematics education december 2012, vol. 5, no. 2, pp. 133–156 ©jume. http://education.gsu.edu/jume stephen o’brien is a college lecturer in the school of education, university college cork, ireland; e-mail: s.obrien@ucc.ie. fiachra long is a senior lecturer in the school of education, university college cork, ireland; e-mail: f.long@ucc.ie. mathematics as (multi)cultural practice: irish lessons from the polish weekend school stephen o’brien university college cork, ireland fiachra long university college cork, ireland in this article, the authors challenge the erroneous assumption that mathematics is universal, and thus culturally neutral, by critically investigating diverse cultural meanings and “ways of knowing” that influence individual/social (affective) forms of identity. the authors begin by briefly detailing the structural features of a polish weekend school and providing an overview profile of the polish community living in ireland. the rationale for the “weekend” school is then discussed from both polish and irish perspectives. empirical data suggest a greater need for “parallel integration,” whereby two divergent education systems attempt to culturally coalesce at some level of school policy and/or mathematics classroom practice. keywords: irish cultural lessons, multicultural mathematics, parallel integration, polish weekend school polish education in ireland background he polish “weekend” school in cork (szkolny punkt konsultacyjny w cork) was first established in 2007, an organisation that shares the same physical site as its “host” irish primary school and is so-named because it operates on saturdays and sundays. students are taught three curricular subjects: polish history, literature, and mathematics. each subject is assessed yearly and student records are kept in accordance with polish school standards. it is possible for polish children living in ireland to participate in the irish state examination system and, as such, undertake a subject test in the polish language. it is also possible for polish children living in ireland to participate in the polish state examination system, though doing so requires travel to warsaw. in effect, polish children are preparing for the possibility of both options by virtue of their dual membership of host and weekend schools. in the irish school, they meet irish students and irish teachers, engage an irish curriculum and irish textbooks, and speak english in class (some study irish as well). in the polish weekend school, students and teachers are exclusively polish; all work with polish textbooks and a polish curriculum, and speak polish in and out of class. students in the school range in years: pre-primary (przedszkole, 3–6 t o’brien & long mathematics as (multi)cultural practice journal of urban mathematics education vol. 5, no. 2 134 years), primary (szkola podstawowa, 6–13 years), and lower secondary general (gimnazjum, 13–16 years). the ten teachers in the school are all polish qualified (to master’s degree level) and are called by their first name. the school day itself reflects polish operational norms (e.g., parents may visit classrooms without prior appointments, there are five to ten minute breaks after each lesson, and a school council operates in accordance with polish legislation). the weekend school, in effect, is a polish enclave—an ambassadorial centre for the 40 million poles still living in poland, a social place where new polish friends can be made, a home away from home, a place where the polish school system can continue its work. the students, teachers, and parents of this weekend school constitute a small share of the polish population living in ireland, estimated to be between 140,000– 160,000 persons. the last population census published in 2006 reveals that there were more polish persons (1.5% of the population) than there were black (of african or irish identity) persons (1%), asian or asian irish (1.3%), and irish travellers (0.5%). polish persons then accounted for 26% of all immigrants to ireland. the polish language unofficially became the second language of the republic of ireland given that “more people spoke it on a daily basis than spoke irish” (hegarty, 2009, p. 199). at the time of the 2006 census, the vast majority (over 90%) were catholic; 20% of the polish working community were in construction (29% of all polish males were in this category); 20% were in the manufacturing industry; 15% worked in the wholesale and retail trade; and 15% were employed in hotels and restaurants (28% of all polish females were represented here). a mere 117 polish teachers were in “official” teaching posts, only 2 persons were members of the gardaí (police force) and 4 were in the irish army. in 2006, 38.3% of non-irish nationals leaving full-time education had a third level (i.e., university) qualification, compared with 28.2% of irish nationals. in relation to the polish community: 55% were educated up to upper secondary level; 3% at non-degree third level (i.e., diploma or an equivalent university qualification); and 23% at degree or higher level (there were almost twice as many polish women as men educated at this latter level). in terms of the overall non-irish national population, 6% declared that they had some form of disability (with almost equal division between males and females). just 2% of the polish community made such a declaration, which included 191 persons who claimed to have “difficulty in learning, remembering or concentrating,” 120 declaring a “learning or intellectual disability,” and 183 persons having a “psychological or emotional condition.” no details were given regarding the measure of language difficulties and/or a broader appraisal of “cultural adaptation.” 1 1 at the time of writing, analysis of the latest 2011 census was being officially conducted (the full report is due to be released late 2012). for the first time, questions relating to individuals’ native language (other than english or irish) were asked, including their levels of proficiency in english. beyond some basic linguistic measure, however, a more thorough enquiry into levels of cultural adaptation was still lacking. o’brien & long mathematics as (multi)cultural practice journal of urban mathematics education vol. 5, no. 2 135 rationale for the weekend school the vast majority of the polish population in ireland has extensive experience of the polish education system—these are mostly adults. by contrast, many children have had to adjust to student life in the irish national system. indeed, some of the youngest children cite the irish system as their sole educational experience. the weekend school serves to connect a migrant community with the polish education system, drawing on individuals’ past experiences and offering opportunities for first-hand or repeat experience. cultural identification is strengthened by a common educational purpose, marked in particular by the study of polish literature and history. further, the study of polish mathematics “officially” connects with a strong sense of national economic purpose. the weekend school, then, presents as a key means through which polish immigrants can (re)connect with their cultural identity. consequently, it remains in high demand from parents of diverse social strata. enrolment has increased from an initial figure of 250 pupils in 2007 to just over 600 in the 2010–2011 school year. this increase in participation represents no mean feat, particularly in the context of a deep economic recession. net emigration figures in ireland now outstrip those of net immigration. in recessionary times, immigrant communities are oft hard-pressed, even to the point of being forced to re-emigrate. 2 yet, the weekend school continues to benefit from financial (albeit reduced) support from the polish government. looking back at an interview on 7 december 2007 with a senior polish embassy official in dublin, we were reminded of the reasons for such sponsorship. the weekend schools (across 5 locations in ireland) were presented as: proffering real cultural connections between the polish diaspora and the “mother nation,” enabling young polish nationals to “keep alive” their language and sense of history, and preparing polish families for possible re-entry into polish society and its employment market. the senior embassy official believed that, for polish parents living in ireland, their “rational” motivation for school participation was grounded in “not wishing to complicate their children’s life” under returned migration circumstances. of course, polish children’s lives can also be complicated under migration circumstances. there was some concern expressed on the part of the polish official about how polish children (and their parents) 2 to illustrate, hegarty (2009) notes that as the recession took hold in 2008 the numbers of nonirish nationals seeking unemployment assistance rose a full 100% in just one week—from 16,000 to 32,000 persons. recent cutbacks to unemployment assistance, educational supports (e.g., english second language teachers [e2l] and special needs assistants [snas]), and premium wage rate entitlements are likely to create more conditions of adversity for the non-irish national population. evidence suggests that both irish and non-irish nationals are exiting in significant numbers— figures for the 12 months to april 2010 show about 65,300 people leaving ireland, with both groups being almost equally represented (hennessy, 2010). the emigration total in 2011 is estimated to have increased to 100,000 persons. o’brien & long mathematics as (multi)cultural practice journal of urban mathematics education vol. 5, no. 2 136 “become integrated” into the irish school system and irish society. however, this concern lay outside the immediate realm of weekend schools given that “they are not dedicated to integration…a connection that is missing.” thus, as we have highlighted elsewhere (see long & o’brien, in press), a situation of “parallel accommodation” appears to exist, where two education systems (weekend and host schools) operate alongside one another without official mutual judgement or comparison. from an irish perspective, the principal of the host primary school spoke to us in april 2008 about the establishment of the weekend school. he had learned of the unique plan from his polish special needs assistant (sna) who was undertaking valuable work helping non-irish national children integrate into the irish system. the plans for the weekend school received full backing from the host school principal and it was arranged, for a nominal rental fee, that the host school’s physical site be utilised. interestingly, much of the sna’s official work in the host school focused on english language instruction. this focus may have influenced initial irish reactions to the polish school’s existence, with teachers in the host school questioning whether polish students would be exposed to english language lessons on the weekends. such a cultural deficit assumption—namely, a principal concern for perceived linguistic shortfall (despite the fact that many polish children speak a number of languages)—soon yielded to another culturally accepted/acceptable attitude. this assumption was centred on the analysis that the polish weekend school would serve to “re-assimilate” children upon their return to the polish education system. in the words of the host school principal: the polish government obviously needs to keep in contact with their own people— they need to do something to attract them back to their country and they also need to do something so that when they do come back, they can assimilate easily back into the system. both weekend and host schools’ support for re-assimilation appears perfectly rational. however, such support is likely to reinforce, what maceinri (2010) calls, “culturally defined and racially bounded notions of ethno-nationalism.” thus, the poles are viewed as having their own people, their own country, and their own education system, and the irish as having theirs. this reinforcement serves to further legitimate a parallel accommodation of separate education systems. of course, there are some benefits to the establishment of parallel institutions, not least the preservation and maintenance of so-called native and migrant cultures (e.g., gibson, 1984). respected on both sides is the intractable purpose of both education systems to further the socialisation of children and the replication of a society’s values (long & o’brien, in press). however, such separation can be stark. parallel institutions cannot presuppose inclusionary practices, particularly o’brien & long mathematics as (multi)cultural practice journal of urban mathematics education vol. 5, no. 2 137 of a multicultural quality. furthermore, separation often fails to purposely speak to the lived reality of migrant pupils who regularly inhabit diverse cultural worlds. aims of the study the weekend school is thus accommodated—literally by the host school and methodologically by both irish and polish systems. its spatial temporality mirrors a transitory identification with which polish (economic) migrants and their children are perceived by both native and migrant cultures. 3 yet, we found some evidence of “temporary accommodation” being disturbed, or at least questioned. this temporary accommodation appeared in an interview conversation with the principal of the weekend school following her return from a professional development conference in warsaw. the conference in question was organised by officials from the ministry of national education and the ministry of science and higher education for the diaspora of polish teachers living and working in europe. as well as providing information about recent changes to the polish education system, the conference stressed the importance of maintaining and improving educational outcomes. this emphasis in itself is not surprising, especially in the context of europe’s preoccupation with standardised forms of educational commodification (see, e.g., o’brien & brancaleone, 2011). however, the conference’s interest in educational outcomes took a specific shape, centring on a perceived dissonance between polish children’s school achievements in poland and polish children’s school achievements abroad. here, the weekend school principal impressed upon us the widespread view that returning migrant children “were experiencing big problems coming back to poland.” questions were being asked about how weekend schools were preserving standards so cherished by the polish education system. how, the principal was frequently asked, could weekend schools (in accordance with their official function) further help polish children upon their return? what were weekend schools doing about returning children’s “language and maths problems”? while some questions were also asked about “what polish children are doing in irish schools,” this particular interest in irish school practices lacked rigour—due in no small part to a seemingly powerless position to influence “a different system, an irish one.” thus, from a polish perspective, the primary emphases on return and “ease of transition” rested with the weekend school. 3 contrary to transitory images, many polish migrants stay longer in host countries. a survey of polish migrants in the united kingdom (centre for research on nationalismm, ethnicity and multiculturalism, 2004), for example, revealed that approximately 30% of the population adopted a “wait and see” approach to the duration of their stay, 15% indicated that they wanted to stay in the united kingdom permanently, and 30% intended to bring their families and children over or that their families were already in residence. o’brien & long mathematics as (multi)cultural practice journal of urban mathematics education vol. 5, no. 2 138 in reality, ease of transition is of analogous importance for the irish schooling system. for teachers and pupils of irish and non-irish backgrounds, many challenges persist in pursuit of more integrated classroom practices. research seminar discussions and interviews with irish student teachers clearly reveal these challenges. 4 in particular, student teachers made known to us how emotionally invested they had become in their attempts to more fully integrate non-irish national children in their classrooms. strong feelings of empathy, support, and personal/professional responsibility appeared alongside those of frustration, powerlessness, and unfamiliarity. many described integration as personally and professionally demanding. to illustrate, one respondent spoke of irish pupils’ “poor attitudes” towards non-irish national groups and his personal/professional reaction: some of my students said, “sir, i don’t like poles.” “what do you mean”? i replied. they said, “sir they’re taking all the jobs”…and i finally brought all the kids around and said: “if you go to university or become a plumber or a carpenter, you have a right to go abroad and work in the eu.” and part of me felt, as middle class, we are not competing for jobs with polish people whereas children from this socioeconomic background, their dads and mums might be. 5 the above quote forefronts the weight of teachers’ own multicultural beliefs and predicaments. it also highlights the potential impact of hidden curriculum messages on individual/social forms of identity. further, it demonstrates genuine opportunity, arising from real-life learning moments, for multicultural commitment in the classroom. in subsequent seminar discussions, student teachers also indicated a number of demanding formal curriculum challenges. in particular, they noted the predominance of irish forms of cultural knowledge: “they [nonirish national pupils] study irish mountains most of the time, irish historical figures, irish literature.” with reference to mathematics instruction, one respondent noted: i teach maths. we have lots of examples like “sean went to the shop,” “mary bought an ice cream.” what’s the most popular name in poland, lithuania or algeria? if 4 between us, we have some 31 years of experience working with student teachers and we both teach on the inclusive multicultural education module on the professional diploma in education (de) programme in university college cork. the series of research seminars we refer to were conducted in the academic year 2008–2009 as part of the ubuntu research project (sponsored by irish aid). there were 3 seminars in total with an average of 10 student teachers in attendance on each occasion. 5 significantly, there was no non-irish national pupil present in the class—had there been, the student teacher admitted that his own actions “may have been different.” this suggests a sense of reluctance and uncertainty on the part of student teachers to engage controversial (seen as possibly divisive) class discussions, particularly in relation to matters of ethnicity and race. more critically, such a position may be viewed as an avoidance of multicultural commitment in the classroom. o’brien & long mathematics as (multi)cultural practice journal of urban mathematics education vol. 5, no. 2 139 everyone around you is irish and you’re not, they [irish pupils] think you’re different. while this perspective presents as some challenge to monocultural approaches to mathematics teaching and learning, it appears fragile in effecting substantial methodological change. deeper cognitive and affective forms of learner engagement were manifest in one student teacher’s english lessons. specifically, this student teacher utilised portfolio work to elicit pupils’ (cognitive) understanding of a lesson on biography, thus engaging the theme multiculturally. moreover, much (affective) importance was placed on personal experiences and feelings, in the interest of advancing pupils’ knowledge of self. thus, one such portfolio entry began: my name is agnieska. i am 13 years old. i was born on the 8 th february 1994. i am from poland but three months ago i moved to ireland. i am going to irish school and am living here with my family. my father is economist, but for one year he works as carpenter. my mother is teacher but she does not work now. i am a happy cheerful teenager. i am often smiling. sometimes i feel very badly, but very fast i forget about the sadness. from such insights, we became curious as to why some subjects (mostly associated with the humanities) appeared more culturally relevant than others (ladson-billings, 1995c). specifically, we questioned why mathematics might be commonly viewed as more culturally “neutral.” we set out, therefore, to explore if polish mathematics—measured against the weekend school’s teaching and learning arrangements—disclosed a particular cultural character. we wished to particularly explore some key cognitive and affective learning effects to establish any such cultural significance. thus, we were interested in the cultural dynamics of “coming to know” mathematics and sought to examine how diverse cultural meanings and “ways of knowing” might influence individual/social (affective) forms of identity. furthermore, we were interested in the potential impact of our study’s findings. if polish mathematics did exhibit a discrete cultural way of knowing and being, what lessons could be learned by irish teachers and irish schools? moreover, we asked: can two divergent education systems culturally coalesce at some level of school policy and/or classroom practice? these key questions inform the research direction of this article. mathematics as (multi)cultural practice culturally relevant pedagogy (e.g., gay, 1995, 2000; grant, 1978; ladsonbillings, 1995a, 1995c) is a well-established discipline in educational literature. emerging from a diverse interest and scholarly base (from anthropology and cul o’brien & long mathematics as (multi)cultural practice journal of urban mathematics education vol. 5, no. 2 140 tural studies to critical race theory and inclusive multicultural education approaches), culturally relevant pedagogy signifies a collective worldview that affirms difference amongst students, teachers, and their school communities. discrete cultural knowledge (or ways of knowing) is recognized, alongside important influences on affective forms of identity. schools are a key (though not exclusive) means wherein certain teaching and learning practices are culturally formed and beliefs about one’s culture and others are developed. traditional practices and beliefs are recognized as being in a state of constant flux, even if this may not be immediately obvious. advocates of culturally relevant pedagogy seek to challenge school traditionalism through critical reform of such areas as: home-school partnerships, use of pupils’ prior experiences, inclusive methodologies, formal and hidden curriculum messages, assessment policy and practice, and teachers’ personal/professional development. there is paramount concern in this empowering educational approach for positive cognitive and affective outcomes. in the case of the former, culturally relevant pedagogy is epistemologically concerned with “how to become a better learner” (e.g., using cultural referents to encourage meaningful problem solving and support academic success). in the case of the latter, culturally relevant pedagogy is ontologically concerned with “how to relate to oneself and others” (e.g., using cultural referents to explore one’s emotional attitudes to cultural identity and participate in an inclusively diverse collective). in both cases, cognitive and affective outcomes rely heavily on the relative strength of culturally relevant teaching and learning practices in a school. these practices are aimed at all students, not just those who are culturally diverse. mathematics education is not exempt and, necessarily, engages the multicultural dimensions of “content integration, knowledge construction, prejudice reduction, equity pedagogy and an empowering school structure and social structure” (ladson-billings, 1995b, p. 144). this position belies the common image of mathematics as a universal, culture-free discipline. from its history, mathematics remains the product of many cultures and is, in essence, “pan-cultural” (greer, mukhopadhyay, powell, & nelson-barger, 2009, p. 2). in addition, swetz (2009) notes that mathematics both reflects the culture it serves and is, in turn, shaped by that culture. thus, even as a pure intellectual activity abstractly manipulated, mathematics must adhere to formal systems of analysis and expression established by the mathematics community (p. 12, emphasis added). discrete cultural groups (e.g., a polish school community) thus make sense of mathematical experiences together and classify, codify, and communicate these experiences symbolically. this intimate relationship between mathematics and culture generates critical cognitive and affective consequences. o’brien & long mathematics as (multi)cultural practice journal of urban mathematics education vol. 5, no. 2 141 cognitive outcomes cultural references abound in mathematics textbooks. many textbook examples speak to deeply shared cultural values, such as attitudes towards family, kinship, occupational roles, the nature of lifestyles or friendship, preferences for competition or cooperation, and so on. these values run deeper than surface cultural references to food, religious festivals, celebrations, and the arts. in effect, they often speak to unconscious norms and to profound levels of emotion attaching to cultural membership. 6 such cultural content influences the choice of mathematics materials and instructional methods used in classrooms. thus, ways of knowing mathematics are culturally imbued, supported mostly by learning activity that remains strongly rooted in local community practices. the community in which the student is “situated in” and learns his or her mathematics, then, significantly affects his or her understanding and application (boaler, 2000). this situated perspective restructures mathematics from an image of smart people and individualistic endeavour to one of smart contexts and co-constructive activity (barab & plucker, 2002). 7 in concert with culturally relevant approaches, teachers utilise meaningful cultural referents to support their students’ deeper understanding of mathematics. teachers also strengthen their students’ academic success in reasoning, analysing, justifying, inferring, and deducing. this strengthening is authenticated by the exchange and co-construction of community beliefs, social practices, codes, classifications, and symbols. while ensuing cognitive outcomes may be more difficult to quantify than test scores, they can, nonetheless, be most affirmative. it is claimed that culturally relevant approaches frequently produce higher levels of student engagement and attendance, parental involvement, and general learning fulfilment (ladson-billings, 1994; mukhopadhyay, powell, & frankenstein, 2009). if ways of knowing and learning mathematics are culturally situated (lave & wenger, 1991), then teachers will need to have a deeper understanding of their students’ cultural worlds. in the case of non-irish national students, irish teachers will need to appreciate and value their cultural practices, especially those relating 6 we are grateful to our colleague karl kitching for his conversational insights into these deeper forms of cultural membership. 7 contemporary political/ideological moves towards individual testing and content or productdriven approaches to numeracy (e.g., department of education and skills, 2010) strongly endorse an image of individualistic endeavour and “smart people.” this approach has the effect of disconnecting individuals from their collective cultural identity and disembodying their relative position within formative power social structures. culturally relevant pedagogues challenge this apolitical image of mathematics and argue that the underscoring of cultural connectivity and relational embodiment is ultimately aimed at advancing more equality in schools and society (e.g., nieto, 1992; ladson billings, 1995a). o’brien & long mathematics as (multi)cultural practice journal of urban mathematics education vol. 5, no. 2 142 to communication, peer/adult interaction, and knowledge construction (moschkovich & nelson-barber, 2009). beyond typifying cognitive behaviour, 8 teachers will need to understand how individuals culturally filter their learning to include “preferences for thinking, observing and (inter)acting…, as well as how they approach schooling” (p. 117). such understanding allows for flexible learner management, including a respect for students’ opportunity to “code switch” in mathematics education. code switching here refers to students’ concurrent use of two or more language and/or symbolic forms. in acknowledging code switching, teachers can affirm the variety of students’ linguistic/symbolic, cognitive and cultural forms of communication. this affirmation is especially important for bilingual students who read, listen, write, speak, and think in two or more languages for different purposes, in different contexts, with different people (moschkovich & nelson-barber). code switching between languages does not appear to affect the quality of conceptual thinking (a. cumming, as referenced in moschkovich & nelson-barber). it may be beneficial, allowing students to problem pose and solve in different ways and in different contexts (a type of additive bilingualism?). or code switching may be a hindrance, such as when a student does not fully understand a mathematics word problem (a type of restrictive bilingualism?). simple conclusions, then, cannot be drawn about a student’s mathematical proficiency on the basis of his or her code switching (moschkovich & nelson-barber). nevertheless, for studies such as ours on bilingual learners in bicultural school settings, code switching presents as an absorbing area of cognitive concern. of course students also communicate mathematically and this shapes, and is mirrored by, culture. beyond technical language, discourse practices include modes of argument, ways of abstracting and generalising, as well as mathematical claims, representations, and imagining (moschkovich & nelson-barber, 2009). everyday, as well as academic, mathematical discourse is communicated (moschkovich & nelson-barber). for the bilingual learner, this presents both linguistic and conceptual challenges in understanding: everyday words (in both polish and english), formal mathematical words and symbols (in polish and irish settings), as well as diverse mathematical problems and their inherent logic. 9 teachers of multicultural student groups need to be particularly mindful of how mathematics 8 gutiérrez and rogoff (2003) caution against viewing variations in cultural practices, including cognitive behaviour, in terms of traits of individuals or collections of individuals. rather, beyond essentialist assumptions, variations exist as “proclivities of people with certain histories of engagement with specific cultural activities” (p. 19). the focus of explanation, therefore, ought to be on individuals’ and groups’ experience in activities, not their traits. 9 we draw heavily here on miller-jones and greer’s (2009) discussions on students’ (whose birth language was spanish) understanding of english written mathematics tests. vygotsky’s relationship between language and concept formation likewise informs the challenges outlined here for bilingual learners (e.g., vygotsky, 1978). o’brien & long mathematics as (multi)cultural practice journal of urban mathematics education vol. 5, no. 2 143 is communicated via assessment procedures. an assessment for learning approach (national council of curriculum and assessment, 2010) challenges traditional modes (e.g., standardised testing) and emphasises the need for a greater qualitative understanding of students’ experiences, interests, and abilities over time. this approach engages students in their own learning development and facilitates a sharing of learning focus, tasks, and assessment criteria. while there can be no built-in relationship between learning style and minority group membership, culturally responsive teachers understand and value the “proclivity” of students to cognitively act in certain ways (gutiérrez & rogoff, 2003). they employ assessment processes that are more flexible, on-going, personalised, and developmentled. in this way, they come to learn more about their students (“get inside their cultural heads”) and themselves (e.g., their own cultural norms and practices). this cultural interaction (of self and other) displaces the so-called objective image of mathematics (i.e., facts speak for themselves) with a more humanised representation (gay, 2009). in turn, mathematics reveals the sharing of cultural knowledge as essential to human development (gardner, 2004). from a cognitive perspective, polish students’ experiences of the weekend school’s cultural activities are worthy of investigation. if certain cultural ways of knowing are made more visible, then irish teachers can learn more about their students and themselves. specifically, a culturally responsive approach to mathematics proffers teachers the opportunity to, inter alia: affirm their students’ cultural identity and knowledge; challenge the expert metaphor of mathematics as a topdown activity; and provide greater access to different ways of seeing, understanding, and doing mathematics. affective outcomes the polish weekend school embodies a specific cultural membership, with its own use of language, codes, social relations, customs, and ways of communicating and reasoning. for its members, the migrant experience endures as a strongly shared attachment. while migrant children are affected by the same issues as all children, they are especially required to develop complex strategies for coping with migration (ni laoire, bushin, carpena-mendez, & white, 2009). these strategies are profoundly influenced by peer culture and parents’ experiences and material circumstances. such strategies are similarly shaped by the efficacy, or otherwise, of host migration policies and practices, including educational arrangements (ni laoire et al.). thus, community, parental and institutional milieus strongly sculpt migrant children’s emotional responses and strategies to schooling. multicultural education aims to support children in their understanding of home culture, while simultaneously freeing them from cultural boundaries (banks, 1992). at its heart lies an “ethics of diversity” informing “respect, soli o’brien & long mathematics as (multi)cultural practice journal of urban mathematics education vol. 5, no. 2 144 darity and cooperation with difference” (d’ambrosio, 2009, p viii). mathematics and mathematics education that “have everything to do with this ethical necessity” (d’ambrosio, 2009, p viii) seek a transformed image of mathematics and its representation in people’s lives. thus, the familiar image of mathematics as universal, and its narrow association with nationalistic and competitive interests, is challenged by a human relations vision of mathematics, with its stress on individual differences and collective affinity (gutstein, 2009). this transformation enables us to acknowledge that, in different cultural contexts, children and adults may participate differently in conversation, discussion, debate, and explanation. thus, diverse mathematical discourses are shown to co-exist due to the high dependence of mathematical practices “on natural language as well as other semiotic systems” (moschkovich & nelson-barber, 2009, p. 112). bilingual students in bicultural/multicultural classrooms, in particular, use different languages and symbolic forms to express mathematics differently (moschkovich & nelson-barber). moreover, teacher-student expectations may differ across diverse cultural contexts and these are likely to impact on students’ knowledge of subject material, as well as their motivations to learn. this possible situation highlights the importance of emotional interactions between teachers and students (hargreaves, 2000). in particular, teachers’ knowledge of the everyday experiences of their students is essential in informing inclusive classroom methodologies (gay, 2009). these inclusionary methodologies reveal an enduring relation between the cognitive and affective dimensions of learning. diverse cultural meanings and ways of knowing mathematics, then, directly relate to individual/social forms of identity. these affective outcomes are observable in students’ feelings about mathematics, including the value they place on the discipline. in bicultural settings (e.g., polish and irish schools), differences and affinities naturally co-exist. in negotiating these, polish children code switch their feelings about mathematics, alongside modes of thinking and practice. concomitantly, questions of self-identity and group membership emerge, such as: how do i feel about the weekend school? how do i feel about the irish school? what is my relationship to language—polish, english, and irish? how do i view, and adjust to, both the polish and irish communities? do i internalise any sense of difference or similarity in relation to educational provision and/or social mobility opportunities? some questions are more explicit than others. all are not exclusively cross-cultural, because diversity exists within, as well as between, cultures. also, identities are never static, as multiple affiliations become realigned over time in different contexts (gardner, 2004, p. 38). such insights highlight the developing role of education in nurturing key affective competences. of these, “multicultural capital” has a critical and contemporary value: a person who has “multicultural capital” is someone who is able to decode and understand the diverse signs of their social and cultural world, someone who is com o’brien & long mathematics as (multi)cultural practice journal of urban mathematics education vol. 5, no. 2 145 fortable residing in such a world and who is able to relate to that world with confidence. such a person would neither feel that their own identity is threatened by difference, nor would they feel superior to those who appear different. (gardner, 2004, p. 29) in this article, we argue for the educational development of this affective competence, via the promotion of mathematics as (multi)cultural practice. methods collectively, we have some 31 years of experience working with student teachers and we both teach on the inclusive, multicultural education module on the professional diploma in education (pde) programme. we approach this research project as “interested scholars,” a phrase that denotes how we come to this research project by now engaged with various multicultural theories, concepts, and/or ideas. thus, we are interested by virtue of our extant research interests, regular presentations and scholarly writings in the field. from a contemporary perspective, we are interested in ireland’s newfound multicultural context and by emergent challenges to cultural integration. this article’s particular interest rests with a concern for inclusive classroom practices in mathematics. why mathematics? mathematics is frequently associated with a universal knowledge base and is associatively presented as culturally neutral. here, we are interested in the cultural dynamics of coming to know mathematics and are seeking to examine how diverse cultural meanings and ways of knowing might impact on individual/social (affective) forms of identity. secondly, we approach this study as investigative researchers. this description reflects the fact that, hitherto, not much was known about so-called polish weekend schools. thus, by utilising a broad-based qualitative approach to research, we set out to critically investigate school and classroom practices, with a view to investigating what meaningful cultural lessons, if any, could be gleaned for irish education. it was envisaged that our, as yet, informed position could have important parallel integration opportunities, whereby two divergent education systems attempt to culturally coalesce at some level of school policy and/or mathematics classroom practice. we chose a broad-based qualitative research approach for this investigation. the previous literature review makes the case for mathematics as (multi)cultural practice, providing important thematic focus to both its cognitive and affective outcomes. we were mindful of these conceptual insights in seeking further explanation from empirical findings and elaborating upon their understanding. the empirical work sought to critically describe, organise, and analyse lived accounts of our numerous visits to the school. a collection of qualitative research methods engaging key actors were utilised for this purpose, including: semi-structured in o’brien & long mathematics as (multi)cultural practice journal of urban mathematics education vol. 5, no. 2 146 terviews with the principal and head maths teacher of the weekend school, observations of a sixth class primary mathematics class (12 year olds), and individual conversations and focus group interviews with pupils of this class. in addition, analyses of polish and irish mathematics textbooks were undertaken. since 2007, we have visited the polish weekend school on numerous occasions. the following empirical insights emerge from our latest lived accounts of these visits (four visits in 2010–2011). specifically, classroom observations, semistructured interviews, and textbook analysis combine in an attempt to unearth participants’ real cultural engagement with mathematics. particular research focus is on sixth class primary mathematics (12 year olds) and the ways in which students think about, and identify with, polish mathematics. insights into the polish weekend school cultural ways of thinking in mathematics from early on, many students told us that polish mathematics was “harder” than its irish counterpart. it was difficult to argue with this assertion. in classes that we observed, some challenging mathematics problems habitually emerged. thus, on the topic of fractions, students in sixth class primary (12 year olds) were asked to solve: 4 1/3 – 7/9. the majority of students successfully attempted this problem and the teacher quickly reviewed the solution on the board. interestingly, analysis centred on the advanced use of equivalent and improper fractions—not the lcm (lowest common multiple), as we (from an irish perspective) might have expected. 10 thus, the solution read: 4 1/3 – 7/9 = 4 3/9 – 7/9 = 3 12/9 – 7/9 = 3 5/9 another challenging problem presented itself in a geometry lesson: here, students were asked to find the area of the trapezium abcd, but with a difference. moving beyond relatively undemanding solutions (e.g., composite shape 10 one might have also expected (from an irish perspective) to read the solution in a scrolled (i.e., vertical step-by-step) format, as opposed to its linear (i.e., horizontal step) appearance. o’brien & long mathematics as (multi)cultural practice journal of urban mathematics education vol. 5, no. 2 147 or formula approaches), students were tasked with dividing the shape into two equal areas in order to prove that the areas of these two shapes equalled the entire area of the trapezium abcd. thus, the solution read: the area of the left hand side shape (square adef) = length x width = 2 x 2 = 4 cms² the area of the right hand side shape (small trapezium bcef) = [(top + bottom) x height] ÷ 2 i.e. [(1 + 3) x 2] ÷ 2 = 4 cm² the area of the big trapezium abcd = [(3 + 5) x 2] ÷ 2 = 8 cms² which is equal to the two smaller areas added together. the above examples demonstrate variances in levels of conceptual difficulty experienced by students in polish and irish schools. the weekend school appeared to demand greater investigative problem solving. students were encouraged to use critical thinking and were not easily coached or supported in finding “easy” routes to the solution. polish mathematics, it appeared, did not stress rule-based or formula approaches as much as (our knowledge of) irish classroom practice. for example, the polish mathematics teacher noted the reliance of students on the bimdas rule (brackets before indices, multiplication, division, addition and subtraction) learned in irish school as limiting a more “natural” approach to solving problems. moreover, students appeared over-eager to apply formulae to speed, distance, and time problems, instead of “working out the meaning behind them.” alongside more abstract expectations, polish students were required to manually “work out” arithmetic problems, as calculators were disallowed for both school and homework use. additionally, the full range of real numbers was normalised, as (what might be commonly termed as) irregular solutions frequently appeared, for example 7/125 or 1.33, and so on. similarly, it was not unusual for polish students to regularly negotiate square roots and powers and engage with substantially small and big numbers. to illustrate, even as early as fourth class primary (10 year olds), students were tasked with notating large numbers, moving beyond thousands, hundreds, tens and units (as required in irish school). thus, one ex o’brien & long mathematics as (multi)cultural practice journal of urban mathematics education vol. 5, no. 2 148 ample read: convert 25150040500 into polish. 11 answer (translated from the polish language): twenty five billion, one hundred and fifty million, forty thousand and five hundred. this last example illustrates not just variances in conceptual difficulty but also in language proficiency demands. as highlighted earlier, socio-cultural learning theory forges a direct connection between language and conceptual development (e.g., vygotsky, 1978). it is contended that meaningful access to words, terms, and/or symbols has the power to liberate more developed forms of thinking. conversely, developed thinking is restricted by a limited access to discursive meaning. in our classroom observations, we had become aware that some polish children had become so “used to using english,” as one student put it, that polish terms (including those mathematical) posed “a real problem.” 12 polish textbooks, like irish ones, appeared very “wordy” and students had to constantly decode print, as well as mathematical literacy structures. some were more successful than others in this regard. thus, students’ ability to negotiate within and across discourses was key to their effective communication with mathematics. of course, this communication also involves code switching between different symbolic systems. to illustrate the necessity for such translation: the decimal point in polish mathematics denotes multiplication, commas represent decimal points not thousands (or three-place positioning), and the division sign in polish equates to the ratio sign in irish mathematics (see table 1). table 1 different symbolic notation polish symbolic system irish symbolic system 68·3 68 x 3 79,3 79·3 42 : 15 42 ÷ 15 moreover, the ways of “working out” problems differed. thus, one can appreciate the challenge for bilingual/bicultural students when faced with this division problem: find 30,24 : 1,5 (i.e., find 30·24 ÷ 1·5) (see figure 1). in attempting this question in the polish maths class, we recall our own difficulties in symbolic 11 polish mathematics does not use commas to denote thousands (or three-place positioning); indeed, as we illustrate later, commas are only used to denote decimal points. 12 this demonstrates that the polish school is just as concerned with promoting its own language in a multicultural/multilingual social context. irish schools’ concern for language integration (including gaeilge) is equally valid, though it frequently receives higher attention. indeed, it may be argued that both perspectives ought to be critically engaged, especially in light of globalising influences on language and culture (see, e.g., long & o’brien, in press). o’brien & long mathematics as (multi)cultural practice journal of urban mathematics education vol. 5, no. 2 149 code switching, including working from right to left. a number of students told us that “this was so confusing” and that most were “not allowed to do it the other division way.” hence, both polish and irish schools frequently endorsed, as one student put it, their “own way of doing things.” figure 1 polish vs. irish division. that said, the two mathematical systems did exhibit similarities. these appeared most evident in a general emphasis on more traditional forms of teaching (e.g., demonstration approaches) and an associated reliance on textbook work: the teacher everyday corrects the homework and talks to us about new things and she gets us doing lots of work in the textbook. we do this in irish school and at the weekend school. (student 1) my teachers show us lots of examples on the board, we listen to them and then we do out some in the textbook. (student 2) both places are sort of like each other because we have to do writing out of examples and we get homework and we correct it right or wrong and we take notes in our copies. (student 3) textbook problems in both systems were positioned in terms of the perceived difficulty that they posed. in the irish texts, part c of a problem indicated greater challenge than parts a and b, whilst polish texts displayed a cactus symbol to ex o’brien & long mathematics as (multi)cultural practice journal of urban mathematics education vol. 5, no. 2 150 emplify a higher degree of complexity. polish students indicated to us that irish text problems were “much easier” and that they regularly received top marks in exams. their confidence in irish mathematics was generally high and they associated this with their exam success, less arduous homework and, as one student put it, the fact that “you don’t have to think in irish maths.” polish textbooks went to some lengths in explaining solutions, offering sequential steps to logically work out problems. there were plenty of investigations presented that sought to “tease out” an appropriate investigatory stance from the reader. this appoarch encouraged a particular (cultural) way of working, invoking skills of reasoning, analysing, justifying, inferring, and deducing. in addition, texts were very colourful and were replete with cultural references specific to polish topography, nature, economy (e.g., agricultural practices), food, currency, and measurements (e.g., the use of decagrams). much of the homework was based on such textbook problems. students indicated to us that homework demands were more challenging in polish school, as it took longer to do and demanded a greater degree of “working out.” 13 because of the perceived gap in conceptual difficulty and workload demands, polish parents indicated to us that it was “tricky,” as one put it, “to keep the kids interested in irish school.” beyond such difference, however, many parents believed that “maths is the same.” furthermore, as the principal of the polish school indicated, “our cultures are not that different.” while this may, to some degree, ring true, discussions here and in previous sections indicate that mathematical practices do exhibit discrete cultural ways of thinking and doing. perhaps polish parents and teachers are not fully cognisant of this difference. one might assume that they are not alone—that irish parents and teachers have a similar (mis)understanding. certainly, it appears that the children are more fully aware of the cultural similarities and differences that co-exist in mathematics learning. while they may not always be attentive to such cultural distinction, and/or capable of its elucidation, it is they, first and foremost, who are faced with the (necessary) task of cultural negotiation. cultural identification with mathematics throughout our visits to the polish school we were always struck by the presence of a real sense of community. after dropping off their children, parents would gather outside the school, exchange pleasantries, share experiences and sometimes, as the principal noted, form “support networks.” the school operates a news bulletin on the polish community in ireland, hosts holiday festival events, and makes available polish newspapers for purchase in the staffroom. a number 13 this “higher” demand on time and critical reflection needs to be considered in the context of intense periods of polish instruction and the heavy requirement to “cover” the polish syllabus at weekends. o’brien & long mathematics as (multi)cultural practice journal of urban mathematics education vol. 5, no. 2 151 of parents work on a voluntary basis, supervising corridors, ringing the school bell, helping with lunches, and assisting with the school’s attendance register. students refer to all teachers by their first name and the staff attempts to create a “more relaxed” environment, very conscious of the fact that children are attending weekend school. the principal was keen to inform us that, given that more and more families “had joined their fathers” in migrating to ireland, the polish community had become more “solid.” the school both mirrors and shapes this newfound identity. in many ways, it has helped to create a unique identity, serving parents and children that, as the principal put it, “are different from those in poland” and “are different within the [polish-irish] community.” the majority of students, we spoke to, had an attractive blend of cork and polish accents. they told us the names of their irish schools, some of which in gaeilge (the irish language), they pronounce expertly. we meet mateusz who refers to himself as matthew, for our (and one suspects) his irish peers’ benefit. this “name switching” is not unusual as children engage dual community membership that reveals interesting facets of their multicultural identity. to further illustrate, we asked sixth class students if they considered themselves to be polish or irish. at the outset, a slight majority of the 16 class members said they were polish, the rest claimed they were irish. the latter felt that it was “cool” that their irish friends thought of them as irish. interestingly, one boy in a half joking response to this “irish group” exclaimed “traitors” (in polish), followed by “shame on you” (in english). when asked specifically if class members might consider themselves to be both (polish and irish), an overwhelming majority concurred. notably, the dissenting boy did not. one girl in the class, sporting a scouts scarf inscribed “cork polish scouts,” was very positive about her dual identity. another boy proudly declared his knowledge of the polish and irish languages and stated that he now wanted to learn french. many appeared to welcome dual community membership, telling their irish friends about the polish school and teaching them some polish phrases. others appeared guarded, fearing it would look “uncool” (as one put it) to declare school attendance at the weekends. all were in agreement that it was, as one put it, “hard to go to school all week.” many of the students were divided about their favoured way of “working out” mathematical problems—more tending to opt for “irish ways” of doing maths over “polish ways.” the polish mathematics teacher informed us that a greater exposure to irish schooling meant that “this is always the way it goes.” interestingly, the vast majority of students counted in polish in their heads, only some did so in english. these insights indicate to us that students’ shared migrant experience forms a core element of their identity formation. individually, students are likely to negotiate their identity over time and in concert with personal contexts and experiences. their feelings towards polish and irish schools were important in this regard. many commented on the “friendliness” of irish teachers, their “close” irish o’brien & long mathematics as (multi)cultural practice journal of urban mathematics education vol. 5, no. 2 152 friends and their “belonging” to the irish school and local area. many felt that these feelings applied to the polish school also. accordingly, students viewed having two groups of friends and communities as being advantageous. at the same time, students appreciated that both sets of friends and communities were distinct. this appreciation might appear more obvious with respect to: the life experiences of polish students and parents vis-à-vis those of their irish peers; differences in food, cultural festivals, and patterns of family relations; the separate establishment of polish and irish schools; differences in conceptual difficulty between irish and polish mathematics; and the pressures to perform well in either or both, depending on future familial expectations, and so on. undoubtedly, these apparent features of division affectively shape polish children’s self-identity. it was difficult for us to access students’ appreciation of more covert differences affecting self-identity. an indication of more in-depth (and covert) feelings around difference emerged in a number of students’ conversations on mathematical practice. some had noted their feelings of frustration in polish school at, as one student put it, “having to think and explain my answers.” for these students, feelings of frustration were conflated with having to adopt an investi-gatory learner stance. comparisons with irish school were constantly made by all students with some feeling more comfortable with irish ways of doing maths. this comfort level may be linked to their relative success in the irish system and an associated confidence in the subject. certainly, many students appeared to accept difference between polish and irish mathematics. interestingly, this accept-ance was guided by the formal and hidden curriculum messages they received in the mathematics classroom. to illustrate, a number of students told us that they were not allowed to “use polish maths in irish school.” when asked why, one girl replied: “they [irish teachers] don’t understand it.” it is interesting to note that it is polish children, first and foremost, who are charged with affectively negotiating the acceptance of difference. this acceptance is not really considered by teachers and/or parents in both school contexts, particularly in relation to mathematical practice. despite this, teachers and parents do influence polish students’ attitudes to diverse mathematical practices. thus, parents who wish to return to poland, the principal told us, tend to place more emphasis on the weekend school; those intent on staying in ireland “focus more on irish school.” many parents value both sites of education and children are no doubt influenced by this assessment. students indicated that they received cultural support at home, particularly with symbolic translation, polish expressions and mathematical techniques. indeed, many students culturally supported their parents in decoding irish schooling demands and clarifying letters of communication. some, noted the principal, help their parents in real-life mathematics by frequently “communicating with banks and post offices.” teachers in both settings tended to value their own work with the children, charged as they were with covering o’brien & long mathematics as (multi)cultural practice journal of urban mathematics education vol. 5, no. 2 153 separate curricula. this, as highlighted above, was explicitly and implicitly communicated to the students who internalised this separation. perhaps the greatest source of cultural support emerged through peer networks. in our observations of mathematics classes, we witnessed students helping one another translate from irish ways of doing maths to polish ways, using both polish and english instruction. through this decoding, students scaf-folded their understanding and developed their confidence in mathematics. as noted previously, some were able to use their knowledge of both systems to advance their mathematical competence (a type of additive bilingualism /biculturalism). others were more comfortable within one system. we spoke to one boy who had little trouble with polish mathematics but, due to his limited access to english (including academic english), sought to “have help with irish maths” (a type of restrictive bilingualism/biculturalism). these insights indicated to us the following: firstly, students’ perseverance and ability to access and utilise cultural supports in mathematics was key to both their academic success and positive self-identity; and secondly, their capacity to draw on multicultural capital (gardner, 2004) and relationally “fit in” was central to their fluid cultural identifi-cation with mathematics. parallel integration: irish lessons from the polish school here, we have demonstrated that it is polish students who are, first and foremost, faced with the (necessary) task of cultural negotiation. this negotiation shapes new learner identities and engages significant cognitive and affective adjustments. polish children therefore exemplify cultural literacy (i.e., act as “cultural readers”) in response to malleable social and schooling experiences. given that neither parents nor embassy officials nor teachers meaningfully attempt this mediation, the child is left to make his or her own way. specifically, the teacher is inevitably absent from this area of cultural exchange in the classroom and from many minor actions that matter most to the child. the polish teachers we met had never taught in an irish school, nor did they know much about the irish curriculum and schooling experience; similarly, from our knowledge of working with irish teachers, they know little or nothing of the polish curriculum and schooling experience. the teachers, parents, community workers, embassy officials we spoke to were all in the dark about the “other” system. in sum, the integration of two cultures occurs principally in the cultural identity of the child. the case for the existence of parallel institutions that preserve and maintain discrete cultural values may be compelling in any multicultural society. mutually incompatible values may never be reducible to a common agenda, even if this is worthy of pursuit. whilst it is important to acknowledge and affirm such difference, a multicultural society that is interested in greater social cohesion will o’brien & long mathematics as (multi)cultural practice journal of urban mathematics education vol. 5, no. 2 154 seek to systematise “unity of diversity” conditions. at the heart of such efforts lies critique of the (oft implicit) acceptance of dominant cultural norms. it is likely that host teachers may be ill-prepared—apart from obvious language issues—to deal with different cultural constructions of knowledge and wide-ranging identities. weaker multicultural policies and practices in schools—for example, a predominant focus on english language lessons for migrant children and the provision of exceptional curriculum pockets and learning supports—frequently distract from the lived experiences of the migrant child. these policies and practices are habitually informed by a need for “multicultural management” and the systemic priority of assuaging “challenges.” deeper multicultural policies and practices reject problem-focused interventions, seeking instead to affirm and respond to new social and schooling realities. these realities are not easily captured however, as they take time and effort to understand and require continual and personalised levels of response. school structures and supports remain commonly centred on mainstream provision and teachers are frequently engaged in normalising individuals’ efforts and exercising standardised forms of curriculum and assessment. accordingly, the qualitative substance of children’s lives, including their learning development, is too often sidelined. this sideling is particularly relevant to migrant children. the chairperson of the polish school’s parent association, seemed to understand this point well; it was he who prompted the view that the polish school appears “at times [as] something separate.” moreover, he suggested to us the idea of a state-sponsored “multicultural centre,” one that provides for all children— “polish, irish and others.” for us, this indicates the need for an educational partnership that addresses the lived multicultural experiences of all children. further, it recognises that identities are circumscribed by material and structural opportunities; that they are never “fixed” or “simply there” but, instead, emerge as a social process, made and re-made in context and in relation to others (reay, 2010). in irish schools, irish and non-irish national children share the same physical space. any expected benefits of multicultural exchange will depend on how seriously (and deeply) the respective school engages with lived cultural experiences. this level of engagement will require moving beyond the foci on official school policy and the establishment of set curriculum programmes. whilst welcome, their efficacy rests on dynamism, particularly practice-based engage-ment and the enactment of culturally relevant teaching. this is as relevant to mathematics teaching and learning as any other knowledge area. as we have demonstrated, there are many cultural lessons to be learned from the polish school. fundamentally, these lessons will emerge via the re-imagination of mathematics and its enactment as a living multicultural discipline. o’brien & long mathematics as (multi)cultural practice journal of urban mathematics education vol. 5, no. 2 155 acknowledgments we are grateful to colleagues in the school of education, university college cork, and the anonymous reviewers who provided critical feedback on the substance of this article. and special thanks to cillian long for the graphical presentation of the mathematics problems presented in the article. references banks, j. 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(1978). mind in society: the development of higher psychological processes (m. cole, v. john-steiner, s. scribner, & e. souberman eds.). cambridge, ma: harvard university press. http://www.irishexaminer.com/home/youth-exodus-138359.html microsoft word final gutstein vol 3 no 1.doc journal of urban mathematics education july 2010, vol. 3, no. 1, pp. 9–18 ©jume. http://education.gsu.edu/jume eric (rico) gutstein is a professor in the department of curriculum and instruction in the college of education, at the university of illinois at chicago, 1040 west harrison street, m/c 147, chicago, il 60607; email: gutstein@uic.edu. his research interests include mathematics education; teaching for social justice and critical literacies in urban, multicultural contexts; freirian approaches to teaching and learning; and chicago school policy. commentary the common core state standards initiative: a critical response eric (rico) gutstein university of illinois at chicago so make no mistake: our future is on the line. the nation that out-educates us today is going to out-compete us tomorrow. to continue to cede our leadership in education is to cede our position in the world. – president barack obama, the white house, 2010 billions of new competitors are challenging america’s economic leadership. – u.s. department of education, 2006 our nation is at risk. our once unchallenged preeminence in commerce, industry, science, and technological innovation is being overtaken by competitors throughout the world.… what was unimaginable a generation ago has begun to occur—others are matching and surpassing our educational attainments. – national commission on excellence in education, 1983 ome things change slowly. i start my response to the common core state standards initiative (ccssi) with the preceding quotes to suggest that the u.s. government’s proclamations today on maintaining and extending its “once unchallenged preeminence” are consistent with its past positions. i contend that the ccssi is part of a larger agenda shaping u. s. education, economy, international relations, and domestic policy whose purpose is to serve u.s. supremacy. my intent in this commentary is to critique the ccssi from this perspective and examine its effects on the mathematics education of urban u.s. students— primarily, low-income and working-class african american and latina/o youth. because i live and work in chicago and am involved in grassroots movements to strengthen democratic public education, and because u.s. department of education secretary arne duncan came directly from chicago to washington, dc, i have much on-the-ground experience with his policies.1 some refer to the 20th century as the “american century” and desire that the 21st century be a repeat. this desire is captured in the words of an influential, conservative think tank: “at present the united states faces no global [military] 1 although i use chicago as an example throughout, i acknowledge the unique differences that exist within and across urban environments in the united states. s gutstein commentary journal of urban mathematics education vol. 3, no. 1 10 rival. america’s grand strategy should aim to preserve and extend this advantageous position as far into the future as possible” (project for a new american century, 2000, p. i). this drive for u.s. dominance—military, economic, scientific, and political—directly relates to education, including the ccssi. other things, however, have changed—a lot. although the year 1983 was a period of severe economic dislocation—the year that a nation at risk (national commission on excellence in education, 1983) was published—today’s crisis is dissimilar on several levels and also deeper (foster & magdoff, 2009). in the education arena, the current era is marked by privatization, exemplified by huge increases in charter schools; mayoral takeovers of school systems; gutting of democratic school practices; attacks on teacher and school-employee unions; performance-based measures (i.e., standardized test scores) to evaluate schools, administrators, teachers, and students; massive cuts in public education funding at the pre-k–20 level; and choice and voucher programs (brown, gutstein, & lipman, 2009; compton & weiner, 2007; lipman & haines, 2007; pedroni, 2007). these changes strongly influence the education and lives of urban youth. the common core state standards initiative the ccssi is an initiative of the national governors association center for best practices (ngacbp) and the council of chief state school officers (ccsso). the standards include those in college readiness, english, and mathematics. the goal is for all 50 states, the district of columbia, and u.s. territories to adopt the common standards so that students will “graduate high school able to succeed in entry-level, credit-bearing academic college courses and in workforce training programs” (ngacbp/ccsso, 2010). as of may 2010, two territories, the district of columbia, and all but two states, texas and alaska, have joined. the standards are supposed to be coherent, clear, consistent, rigorous, aligned with college expectations, research based, and informed by standards in highperforming states and countries “so that all students are prepared to succeed in our global economy and society” (ngacbp/ccsso, 2010). there have been criticisms on specific aspects of the ccssi’s mathematics standards. one is a plea for critical revisions to the common core state standards for mathematics,2 and another is from the national council of teachers of mathematics (nctm, 2010). both concur on certain points (e.g., that the ccssi’s view of kindergarten children’s place value knowledge is inappropriate for their development); but i do not analyze the ccssi content standards here. there may be good or bad aspects of specific content standards, but, as michael 2 a plea for critical revisions is a blogspot.com petition circulating on the internet; see http://commoncorematheducatorsrespond.blogspot.com/2010/03/test-number-3.html. gutstein commentary journal of urban mathematics education vol. 3, no. 1 11 apple (1992) critically advised the mathematics education research community years ago regarding the nctm’s 1989 standards (nctm, 1989), there are bigger and more important questions that should be asked: whom do standards benefit? what is the purpose of standards, of mathematics education? what should be the purposes? what is our role? without putting standards in broader sociopolitical contexts, especially in our current volatile world, we can neither understand them nor know how to respond to them. the larger sociopolitical context the ccssi is central to the president obama–secretary duncan education agenda, which includes, among other initiatives, the race to the top (rttp) grant program. in rttp, states compete for grants, and one selection criterion is whether they adopt common standards (e.g., the ccssi). to be awarded grants, states need to track students’ test scores, focus on “producing” excellent teachers and principals (i.e., those whose students score high on tests), tie teacher evaluations to students’ scores, expand charter schools, improve the most lowperforming schools, and promote mayoral takeover of school districts. states that focused on science, technology, engineering, and mathematics (stem) education are awarded extra points in the grant competition. by may 2010, 40 states had applied for rttp funding, 16 states were finalists, and 2 states, delaware and tennessee, were first-round winners. beyond rttp, mathematics education figures prominently elsewhere in the obama–duncan education agenda, including in the educate to innovate campaign, intended to complement rttp. the following is a statement from the obama white house announcing the campaign: president obama has launched an “educate to innovate” campaign to improve the participation and performance of america’s students in science, technology, engineering, and mathematics (stem). this campaign will include efforts not only from the federal government but also from leading companies, foundations, nonprofits, and science and engineering societies to work with young people across america to excel in science and math. (the white house, 2009) the educate to innovate press release contained the usual statements about u.s. students doing poorly on international assessments and stated the campaign’s goals: • increase stem literacy so that all students can learn deeply and think critically in science, math, engineering, and technology. • move american students from the middle of the pack to top in the next decade. • expand stem education and career opportunities for underrepresented groups, including women and girls. (the white house, 2009) gutstein commentary journal of urban mathematics education vol. 3, no. 1 12 nothing is new here, including the paltry stand on equity. in fact, a nation at risk begins: “all, regardless of race or class or economic status, are entitled to a fair chance and to the tools for developing their individual powers of mind and spirit to the utmost” (national commission on excellence in education, 1983, p. 1). of course, the experiences of most urban youth are nothing like either pronouncement, and weak or superficial calls for expanding access are a far cry from the fundamental transformation of society that a social justice perspective demands. the george w. bush and obama education agenda i argue that educate to innovate, rttp, and the ccssi are all straightforward continuations of the georgia w. bush administration’s education agenda, and that little difference exists between the collective goals of the obama–duncan education initiatives and the bush administration’s american competitiveness initiative (aci). this is particularly true for stem education (e.g., the national mathematics advisory panel [nmap, 2008)] was part of the aci and its recommendations surely influenced the ccssi). like the current administration’s initiatives, the aci emphasized that u.s. students should help reclaim the nation’s leading global position. the domestic policy council (2006) under then-president bush declared: in the years to come, the united states will face increased economic competition from a number of countries around the world. we will have to work harder to maintain our competitive edge. by laying the foundation today for expanded scientific and technological excellence, we will continue to lead the world tomorrow in inquiry, invention, and innovation. (p. 23) president obama’s launch of educate to innovate is similar: whether it’s improving our health or harnessing clean energy, protecting our security or succeeding in the global economy, our future depends on reaffirming america’s role as the world’s engine of scientific discovery and technological innovation. and that leadership tomorrow depends on how we educate our students today, especially in math, science, technology, and engineering.... and that’s why my administration has set a clear goal: to move from the middle to the top of the pack in science and math education over the next decade. (the white house, 2009) the aci’s premise was that innovating technologically, strengthening stem education, and upgrading the skills and knowledge of u.s. students and workforce would heal the ailing, second-rate u.s. economy. the framing of the aci was that u.s. economic problems affected us all, and therefore, all would benefit from the cure. president obama’s dire warning that “our future is on the gutstein commentary journal of urban mathematics education vol. 3, no. 1 13 line” (emphasis added) suggests that all will suffer if we do not educate to innovate and “mak[e] sure our students are prepared for success in a competitive 21st century economy and workplace” (the white house, 2010, ¶ 4). education for productivity—for whose benefit? the political and neoliberal ideological assumptions embedded in the economic policies of the obama administration and previous ones dating back at least to reagan’s are that the wealth that goes to the wealthiest, whether through tax cuts for the rich (the reagan and w. bush administrations) or through bailing out major u.s. banks, will eventually “trickle down” and benefit the rest of us. these economic suppositions influence u.s. education policy, which has been strongly influenced by several recent reports, such as tough choices or tough times (national center on education and the economy, 2007). like the aci, this report suggested that the u.s. people as a whole will suffer unless we transform our educational system; it also stated that capital has no allegiance to the united states and will flow anywhere to maximize profit: if we continue on our current course, and the number of nations outpacing us in the education race continues to grow at its current rate, the american standard of living will steadily fall relative to those nations, rich and poor, that are doing a better job. if the gap gets to a certain—but unknowable—point, the world’s investors will conclude that they can get a greater return on their funds elsewhere, and it will be almost impossible to reverse course. although it is possible to construct a scenario for improving our standard of living, the clear and present danger is that it will fall for most americans. (p. 8) however, this argument misleadingly suggests that the u.s. standard of living will improve—for “most”—if we fix what is broken. in fact, u.s. productivity overall has increased over the past 45 years, but simultaneously, income and wealth polarization within the u.s. population has grown. dew-becker and gordon (2005) documented that from 1966 to 2001, “nobody below the 90th [income] percentile received the average rate of productivity growth” (p. 58; emphasis in original). where did the gains go? they answer: “only the top 10 percent of the income distribution enjoyed a growth rate of real wage and salary income equal to or above the average rate of economy-wide productivity growth” (abstract; emphasis in original). since 2001, the gap between rich and poor has widened. edward wolff (2010), an economist well known for studying both income and wealth inequality, wrote, “all in all, the 2000s witnessed a moderate increase in income inequality, a small rise in wealth inequality, and a significant jump in non-home wealth inequality” (pp. 12–13). gutstein commentary journal of urban mathematics education vol. 3, no. 1 14 exacerbating inequality for urban students—issues of race and class thus, educating to innovate by focusing on stem education may improve u.s. productivity, but if the past is prelude, the gains will benefit only the wealthiest and not the majority of urban youth of color in public schools. in fact, current education policies can further marginalize these youth in other ways. concretely, this agenda already has done so for these students in chicago under duncan’s leadership (although we in chicago recognize that chicago public schools policy was much bigger than duncan alone). first, de-democratizing public education and increasing educational privatization in rttp (charter schools, mayoral takeover, school choice, evaluations tied to test scores, school turnarounds) are tied to neoliberal ideologies that promote the primacy of the market in all spheres, bottom-line “performance indicators” as the only viable metric of all activities, parents as educational consumers in a “free” educational market, erosion of the public good including resources and funding, and deregulation. our experience in chicago is with mayor dailey’s renaissance 2010 plan to close neighborhood public schools and open charter and contract ones (lipman & haines, 2007)—but these policies echo from los angeles, to philadelphia, to st. louis, and elsewhere. in chicago, as the board of education closes schools, low-income students and communities of color experience dislocation (from school to school), displacement (massive gentrification and destruction of public housing), spikes in violence (as students cross neighborhood and gang lines), and profound disrespect for community wisdom and democracy (brown, gutstein, & lipman, 2009). the board proposes school closings, and community members and school personnel learn about them through the media. the board then holds hearings that board members rarely attend, while community members, school alumni, parents, teachers, and students pour out their hearts for increased resources and for their schools, often longtime anchors in economically devastated, deindustrialized, and disinvested neighborhoods. a common perspective on school closings in low-income african american and latina/o communities is: “it’s a done deal.” evidence? at a 2009 board meeting, i heard a teacher ask if any board member (all mayoral appointees) had read the testimony gathered from hundreds of people at public hearings on the 16 schools to be closed. not one member had read the testimony, yet all voted unanimously to close/turn-around all 16 schools. also, one contract-school operator posted on its website teaching positions for a school eventually turned over to it (i saved the postings), but the application closing date was before the board even voted! this shredding of democracy does—and will—affect those with least power in society. what people experience in chicago public schools bodes badly for the nation. additionally, the u.s. economy does need a stratum of stem professionals, gutstein commentary journal of urban mathematics education vol. 3, no. 1 15 but a stratified labor force reflects inequity onto education. chicago has elite public high schools, which less than 10% of chicago’s students attend. these are disproportionately white, asian, and middle and upper income. the vast majority of students attend under-resourced, heavily segregated, and disproportionately lowincome neighborhood high schools. though these have a smattering of advanced courses, they are for the most part not college-preparatory schools, and experience dropout rates of 50% or more and mean act scores around 15 (illinois state board of education, 2009–2010; swanson, 2008). no neighborhood high school has a majority of students at or above state standards, and the school-to-prison pipeline is enacted through discipline policies and disproportionate suspensions of african american male students (karp, 2009). the “mis-education” of students of color, as woodson (1933/1990) named it, is obvious in chicago. when resources—new buildings, the most academically qualified teachers, smaller classes, scholarships, enrichment programs, extra-curricular research opportunities, world-class technology and equipment, university partnerships, and more— are concentrated on high-performing students being groomed to boost the flagging u.s. economy, we not only have the shredding of democracy but also of equity and social justice. despite weak rhetoric to the contrary, nothing in the ccssi, rttp, or educate to innovate turns this situation on its head and transfers the necessary massive resources to benefit the majority in urban schools. in fact, given that the u.s. department of labor projects that in 2018 more than half of all jobs will require no more than “short” or “moderate” on-the-job training (e.g., truck drivers, secretaries, salespeople, wait staff)—and not college (lacey & wright, 2009)—capital and government have little incentive to demand that neighborhood public schools have college-preparatory stem programs. despite this stratification, urban schools have some advanced offerings, albeit few. for one, there are certainly government officials who care about access and opportunity. but beyond this, to legitimate selective, exclusionary programs, one has to create the illusion of equal opportunity. furthermore, brilliance emerges despite adverse conditions, and few in positions of power are so ideological that they would reject someone who can fight through the barriers and potentially help alleviate the crisis. but even this can subvert others who work as hard but, for whatever reasons, do not make it through the sieves. as conforti (1992) pointed out, “each instance of success by individual black people undermines racial discrimination as an explanation of the lack of success on the part of other black individuals” (p. 235). some closing remarks there is a profound disconnect between the economic and education crises facing urban u.s. communities and the ccssi. as martin (2008) pointed out in gutstein commentary journal of urban mathematics education vol. 3, no. 1 16 critiquing the nmap, “race is conspicuously absent in the national mathematics advisory panel’s final report despite a review of 16,000 research publications and policy reports and testimony from 110 individuals” (p. 389; emphasis in original); the same is true for the ccssi mathematics standards. in them, like bush’s national mathematics advisory panel, race, racialization, racism, equity, african americans, and latinas/os do not exist. the lives and voices of people and scholars of color are “conspicuously absent.” there is also no mention of class or gender. it is as if one could develop a common core of standards and ignore these issues. yet institutional and structural racism and political economy loom large in the experiences of urban youth, both within and outside the mathematics classroom. so what is our role? for guidance, i turn to dr. martin luther king, jr.’s famous 1967 riverside church speech against the vietnam war: i am convinced that if we are to get on the right side of the world revolution, we as a nation must undergo a radical revolution of values. we must rapidly begin the shift from a thing-oriented society to a person-oriented society. when machines and computers, profit motives and property rights, are considered more important than people, the giant triplets of racism, extreme materialism, and militarism are incapable of being conquered…. a true revolution of values will soon look uneasily on the glaring contrast of poverty and wealth. with righteous indignation, it will look across the seas and see individual capitalists of the west investing huge sums of money in asia, africa, and south america, only to take the profits out with no concern for the social betterment of the countries, and say, “this is not just.”…. a true revolution of values will lay hand on the world order and say of war, “this way of settling differences is not just.” this business of burning human beings with napalm, of filling our nation’s homes with orphans and widows, of injecting poisonous drugs of hate into the veins of peoples normally humane, of sending men home from dark and bloody battlefields physically handicapped and psychologically deranged, cannot be reconciled with wisdom, justice, and love. a nation that continues year after year to spend more money on military defense than on programs of social uplift is approaching spiritual death. (as cited in washington, 1986, pp. 240–241) dr. king’s radical words call us to certain tasks. i believe that readers of the journal of urban mathematics education are committed to racial justice, like dr. king, yet he spoke plainly that opposing racism was necessary but insufficient— one should also stand against “extreme materialism,” the “glaring contrast of poverty and wealth,” and against war and a military budget of nearly $1 trillion versus $50-plus billion for education. we have to go beyond mathematics education for urban students because the “giant triplets” are inextricably related. mathematics educators and teachers can raise and link these issues in all our work; push the boundaries on the accepted, narrow discourses of “standards,” curriculum, and pedagogy; and find a place in struggles against education privatization, urban displacement, climate destruction, and whatever else in concert with students, par gutstein commentary journal of urban mathematics education vol. 3, no. 1 17 ents, community members, and citizens of the world. finally, dr. king did not specifically advocate ending capitalism—but had he lived longer perhaps he would have because the seeds of its rejection were in his words and trajectory. a revolution in values, to reclaim spiritual life, demands a deeper understanding of the roots and systematic nature of our present crisis, and a vision of society that, as dr. king said, puts people and nature before profit. call it socialism, like millions across the planet, or call it what you will—our present order is fundamentally flawed, and our responsibility is to create a new one. i close with words from amilcar cabral (1973), leader of the independence struggle of guinea-bissau against almost 500 years of vicious colonialism: “[join] the difficult but inspiring struggle for the liberation of peoples and humankind and against oppression of all kinds in the interest of a better life in a world of peace, security, and progress” (p. 15). acknowledgements i would like to acknowledge ongoing conversations with danny martin about race, racism, and mathematics education and pauline lipman about political economy—both have helped and pushed my thinking in these areas. references apple, m. w. (1992). do the standards go far enough? power, policy, and practice in mathematics education. journal for research in mathematics education, 23, 412–431. brown, j., gutstein, e., & lipman, p. (2009). arne duncan and the chicago success story: myth or reality? rethinking schools, 23(3), 10–14. cabral, a. (1973). return to the source: selected speeches. new york: monthly review press. compton, m., & weiner, l. (eds.). (2007). the global assault on teaching, teachers, and their unions. new york: palgrave. conforti, j. m. (1992). the legitimation of inequality in american education. the urban review, 24, 227–238. dew-becker, i., & gordon, r. j. (2005, september). where did the productivity growth go? inflation dynamics and the distribution of income. paper presented at the 81st meeting of the brookings panel on economic activity, washington, dc. domestic policy council. (2006). american competitiveness initiative: leading the world in innovation. washington, dc: u.s. government office of science and technology policy. foster, j. b., & magdoff, f. (2009). the great financial crisis: causes and consequences. new york: monthly review press. karp, s. (2009). black male conundrum. catalyst in depth, xx(5), 4–7. illinois state board of education (2009–2010). ereport card public site. retrieved from http://webprod.isbe.net/ereportcard/publicsite/getsearchcriteria.aspx. lacey, t. a., & wright, b. (2009). occupational employment projections to 2018. monthly labor review, 132(11), 82–123. retrieved from http://www.bls.gov/opub/mlr/2009/11/art5full.pdf. lipman, p., & haines, n. (2007). from accountability to privatization and african american exclusion: chicago’s “renaissance 2010.” educational policy, 21, 471–502. gutstein commentary journal of urban mathematics education vol. 3, no. 1 18 martin, d. b. 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(1989). curriculum and evaluation standards for school mathematics. reston, va: national council of teachers of mathematics. national council of teachers of mathematics (2010). nctm public comments on the common core standards for mathematics. retrieved from http://www.nctm.org/about/content.aspx?id=25186. national governors association center for best practices, & council of chief state school officers. (2010). common core state standards initiative: about the standards. retrieved from http://www.corestandards.org/about-the-standards. national mathematics advisory panel. (2008). foundations for success: the final report of the national mathematics advisory panel. washington, dc: u.s. department of education. pedroni, t. c. (2007). market movements: african american involvement in school voucher reform. new york: routledge. project for a new american century (2000). rebuilding america’s defenses: strategy, forces and resources for a new century. washington, dc: project for a new american century. swanson, c. b. (2008). cities in crisis: a special analytic report on high school graduation. bethesda, md: editorial projects in education. u. s. department of education. (2006). answering the challenge to a changing world: strengthening education for the 21st century. washington dc: u.s. department of education. washington, j. m. (1986). a last testament of hope: the essential writings and speeches of martin luther king, jr. new york: harpercollins. white house. (2009). educate to innovate. retrieved from http://www.whitehouse.gov/issues/education/educate-innovate white house. 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(original work published 1933) microsoft word 6 final xenofontos vol 9 no 1.doc journal of urban mathematics education july 2016, vol. 9, no. 1, pp. 94–116 ©jume. http://education.gsu.edu/jume constantinos xenofontos is a lecturer in the department of education at the university of nicosia, 46 makedonitissas avenue, engomi, 1700 nicosia, cyprus; email: xenofontos.c@unic.ac.cy. his research interests include mathematics teacher education, pro-blem solving, comparative studies in mathematics education, and cultural issues in math-ematics education. teaching mathematics in culturally and linguistically diverse classrooms: greek-cypriot elementary teachers’ reported practices and professional needs constantinos xenofontos university of nicosia, cyprus in this article, the author presents and discusses findings from a small-scale qualitative project that included greek-cypriot elementary mathematics teachers working in schools with high percentages of immigrant pupils. working in three neighbouring schools in the same underprivileged urban area, 16 teachers were individually interviewed regarding their experiences of teaching mathematics in those settings. the author discusses the teachers’ reported instructional practices in facilitating their pupils’ mathematical understanding, as well as their professional needs for doing so. the author’s analysis suggests that teachers focused solely on providing linguistic support for their pupils, without any attempt to incorporate their pupils’ diverse cultural backgrounds in classrooms. also, the participants reported a number of professional needs not currently addressed by the ministry of education and culture of the republic of cyprus. some implications for policy and practice are discussed. keywords: cyprus, diversity, elementary teachers, mathematics education, professional needs, reported practices eaching mathematics in diverse classrooms is a complex and challenging endeavour (anhalt & rodríguez-pérez, 2013; clarkson, 2004), not least because in such settings (as in all classrooms) mathematical content intertwines with issues of language and culture. researchers have shown that in multilingual mathematics classrooms several types of discourse—ranging from the everyday to the formal mathematical register—interchange and coexist (clarkson, 2009; slavit & ernst-slavit, 2007). at the same time, pupils who are learning the language of instruction alongside mathematics may have difficulty accessing mathematical content presented in verbal forms (elbers & de haan, 2005). furthermore, it has been well documented that the content of mathematics curricula as well as the teaching and learning of the subject vary significantly across countries (andrews, 2007, 2014; bishop, 1994; campbell & kyriakides, 2000). immigrant and minority pupils bring their ethnic cultures’ mathematical values and aspirations to classt xenofontos greek-cypriot elementary teachers journal of urban mathematics education vol. 9, no. 1 95 rooms (gorgorio, 2006; stathopoulou & kalabasis, 2007), which are not, however, always welcomed by teachers. in this article, i draw on data from a small-scale project examining greekcypriot elementary teachers’ beliefs about mathematics teaching and learning in diverse settings. the following two research questions are addressed: 1. what are teachers’ reported practices in facilitating immigrant pupils’ learning of mathematics? 2. what professional needs do teachers have in assisting their immigrant pupils’ learning of mathematics? a small mediterranean island and a member of the european union (eu) since 2004, the republic of cyprus has had a diverse demographic character even before its declaration as a republic in 1960 (hajisoteriou, neophytou, & angelides, 2012). over the last few years, however, a great number of immigrants and labour workers from east asia, eastern europe, the former soviet union, and the middle east has arrived to the island, along with some internal movement of turkish cypriots from the north part of the island—occupied by turkish military forces since 1974—to the south part, controlled by the internationally recognized republic of cyprus (zembylas & lesta, 2011). according to the latest census for the government-controlled part of the island (statistical service of the republic of cyprus, 2011), 667,398 inhabitants have cypriot citizenship (including people with dual citizenship); 106,270 have eu citizenship; and 64,113 a non-eu citizenship. about half of the people who migrated to the cyprus republic in the last decade are female domestic workers, while a high percentage of the rest is employed in agriculture, the tourism industry, and in construction (zembylas & lesta, 2011). the changing demographics of the population have certainly altered the pupil population of the republic (angelides, stylianou, & leigh, 2003; zembylas & lesta, 2011). the ministry of education and culture (moec) first expressed an explicit interest in diversity issues in 2001 (hajisoteriou et al., 2012). in 2003, the program for zones of educational priority (zep), a unesco strategy for positive discrimination (or what in the u.s. context is called affirmative action), was introduced as a way to promote “tolerance and dialogue in order to eliminate stereotypes through education” (moec, 2008, p. 21). zep schools are typically located in areas of low socioeconomic status, and their pupil population typically comprises high proportions of immigrant or other-language children. the vast majority of these pupils do not speak greek (the language of instruction), and so they have to take reception classes a few hours a week to learn the language (papamichael, 2008). at the same time, they have to attend “regular” classes with their local peers (hadjitheodoulou-loizidou & symeou, 2007) for all school subjects. the same teachers who teach in the so-called regular class typically xenofontos greek-cypriot elementary teachers journal of urban mathematics education vol. 9, no. 1 96 teach reception classes. these teachers do not need to have any further qualifications (i.e., teaching greek as a second, additional, and/or foreign language) to teach in the reception classes. this decision is simply a matter of how the school management distributes the teaching periods to its employees. the moec has provided most zep schools with a particular book series, originally developed in greece, as part of a research project for teaching the language to children of the greek diaspora. however, anecdotal evidence indicates that most zep teachers in the cyprus republic do not use the series, because, as they claim, it was developed for different purposes and does not take the particularities of the greek-cypriot educational system into consideration. instead, they would rather prepare their own teaching materials (i.e., worksheets) or treat reception classes as tutorials to what has been taught in the regular class. in a previous article (xenofontos, 2015), i discuss findings from the project reported here regarding particular zep schoolteachers’ views of immigrant and other-language pupils as learners of mathematics. as concluded there, the participants made direct references to language as the main (in most cases, the only) factor that prohibits pupils’ mathematical learning. it was observed that if pupils managed to overcome the linguistic barriers by learning basic greek, then learning the mathematical content would be easy given that mathematics was perceived as a universal language. nevertheless, some of the teachers made peripheral references to pupils’ ethnic cultures as another factor that might impact the latter’s mathematical ability. more specifically, teachers commented on how pupils from certain ethnic groups appeared to perform better than those of other ethnic backgrounds (i.e., russian speakers performing better than arabic speakers); they also noted that the parents of those who performed well seemed to value school and mathematics more than the parents of those who underperformed. the teachers’ comments, however, were based solely on anecdotal observations, making it apparent that they lacked the necessary vocabulary and scientific knowledge to justify such claims. based on the same dataset, here i discuss teachers’ reported teaching strategies in facilitating immigrant pupils’ mathematical understanding, as well as their professional needs. because research on greek-cypriot mathematics teachers’ practices and needs for teaching in diverse classrooms is largely overlooked (xenofontos, 2015), i turn my focus to international literature in this area, which has been helpful in framing and locating my work. while i am aware that diverse educational policies have a local character and are perceived differently in dissimilar educational systems (gorski, 2006; gropas & triandafyllidou, 2011), i am confident, despite the cultural specificities of my work, that the research reported here will enable dialogue among colleagues with similar concerns around the globe. xenofontos greek-cypriot elementary teachers journal of urban mathematics education vol. 9, no. 1 97 teachers’ concerns and practices in diverse mathematics classrooms in this section, i summarize the relevant international literature on mathematics teachers’ concerns and instructional practices in diverse classrooms. from my readings, it could be inferred that research in this field typically falls into two clusters. the first cluster comprises research focusing on the issue of language. language plays a vital role in developing a deep conceptual understanding of mathematical ideas (durkin, 1991); therefore, pupils learning mathematics in a language other than their home language need explicit and deliberate linguistic support (anhalt & rodríguez-pérez, 2013). in view of this, clarkson (2004, 2009) highlights several practices employed by teachers in such settings, which have been found to be effective for pupils’ learning. for example, teachers may encourage different types of language practice; that is, informal talk in the pupils’ home language can lead to more formal mathematical talk in the language of instruction (salehmohamed & rowland, 2014). such encouragement has proven to have a positive impact on pupils’ conceptual understanding of mathematics (moschkovich, 2007; setati & adler, 2000; webb & webb, 2008). furthermore, teachers could adopt academic, mathematical language in verbal discourses and endorse the anticipation that pupils will come to practice such language. the use of a simplified form of the official language by teachers does not guarantee that pupils have better access to the mathematical content. on the contrary, it may constitute an additional barrier because it prohibits learners’ acquisition of rich mathematical concepts by obscuring them (adler, 1997; gorgorio & planas, 2001). recently, moschkovich (2012) proposed a more comprehensive set of instructional techniques for teachers dealing with language issues in mathematics classrooms. these include focusing on pupils’ mathematical reasoning, not accuracy in adopting a language; shifting to a focus on mathematical discourse practices and moving away from simplified views of language; recognizing and supporting pupils to engage with the complexity of language in mathematics classrooms; treating everyday language and experiences as resources, not as obstacles; and finally, uncovering the mathematics in what pupils say and do. studies deriving from the so-called developing world conclude that teachers seem to have concerns about the socio-political status of language in mathematics classrooms. for instance, in many african countries, a colonial language (i.e., english, french, portuguese) is the official language of instruction, while, at the same time, it is neither the teachers’ nor pupils’ home language (chitera, 2011). for example, in a study in botswana (kasule & mapolelo, 2005), primary teachers appeared to hold strong views about having to adhere to the official language policy because it is believed that children learn better mathematics when they are taught in english than in any other language. guided by such beliefs, teachers of botswana xenofontos greek-cypriot elementary teachers journal of urban mathematics education vol. 9, no. 1 98 employed a number of language games and organizational strategies in an attempt to help their pupils learn mathematics, as taught in english. similarly, in the context of post-apartheid south africa, setati (2005) reports the case of a primary teacher, who taught in a school where the majority of the pupils spoke setswana as their first language, even though the official language of instruction was english. the teacher was confronted with two conflicting and competing identities when teaching mathematics: the identity of a mathematics teacher in an english-language dominated world, who believed that pupils should be able to communicate mathematics in english; and the identity of a south african who had experienced language discrimination during apartheid, and who now wanted to promote the use of the pupils’ home language to enhance its standing as a legitimate language of interaction in a mathematics classroom, as well as to empower the learners’ identities. in the second cluster, there are studies focusing on how pupils’ diverse cultural backgrounds could be leveraged in mathematics classrooms. such studies typically examine the effects of intervention programs on pupils and teachers (i.e., attainment, conceptions, etc.). for programs of this kind to create environments where pupils are able to demonstrate significant progress, the philosophy of their design and the teachers’ instructional philosophy need to be compatible (civil & wiles, 2005). examples of such programs are reported in esmonde and caswell (2010) and gutstein (2003), who locate their work in equity and social justice, and are concerned with how inequitable mathematics classrooms can be transformed into equitable ones. in turn, gay (2002) talks about culturally responsive mathematics teaching. such a notion, she claims, implies, among other things, teachers’ general and specific knowledge about the cultural values, traditions, and learning styles of different ethnic groups; knowledge in how to determine the multicultural strengths and weaknesses of curriculum designs and instructional materials and how to make necessary changes to improve their overall quality; and thorough and critical analyses of how ethnic groups and experiences are depicted by mass media and popular culture. in a similar vein, averil and clark (2013) conclude that certain practices employed by mathematics teachers are seen as respectful towards pupils’ cultural backgrounds, both by the teachers themselves and the pupils. these include: being well prepared for mathematical challenge and listening; enabling pupils’ mathematical decision-making; providing mathematical assistance and feedback; and differentiating teaching while, at the same time, holding high academic standards for all learners. much research on social justice and culturally responsive mathematics teaching has taken place in australia, new zealand, and the united states (see, e.g., averill et al., 2009; bartell, 2013; boaler, 2008; zevenbergen, niesche, grootenboer, & boaler, 2008). common issues that arose from these studies regard xenofontos greek-cypriot elementary teachers journal of urban mathematics education vol. 9, no. 1 99 the tension experienced by teachers in accommodating pedagogies of this kind, as well as the necessity for more professional development programs so that the opportunities such teaching raises are better embraced by the teachers. mathematics teacher education and diversity “mathematics education in practice is, and always should be, mediated by human teachers” (bishop, 1988, p. 189). as bishop states, teachers must be able to identify the values inherent in the subject they are responsible for; they must be informed about the cultural history of their subject and reflect on their relationship to those values; and they must be aware of how their teaching contributes not only to the mathematical development of their pupils but also to the development of mathematics in their culture. nonetheless, bartell (2011) points out the lack of published work on how to support in-service mathematics teachers in developing pedagogies that address diversity issues. changing the mind-set of experienced (mathematics) teachers in order to accommodate culturally and linguistically responsive pedagogies is a hard process (gorgorio & planas, 2001) and not always a successful one. for instance, in their work with experienced in-service teachers in three european countries (italy, spain, and portugal), favilli, oliveras, and césar, (2003) discuss how participants highlighted the need for support in their attempt to teach mathematics to minority and immigrant pupils. the teachers, however, appeared to be reluctant in shifting their practices for accommodating changes even after a series of professional development seminars and the provision of specially prepared digital supportive materials (césar & favilli, 2005). yet, designing and implementing successful professional development programs is possible, as planas and civil’s (2009) study with secondary mathematics teachers in barcelona informs us. participants, who were working in schools located in impoverished areas with high percentages of immigrant pupils, were involved as co-researchers in a program investigating their local contexts and practices. by the end of the program, the teachers developed awareness of their local situation, and questioned and reformed their own practices to encourage pupils’ active participation in class. from the perspective of pre-service mathematics teacher education, the lack of research related to cultural and linguistic diversity may imply that these issues are missing from many preparatory programs around the world. for instance, chitera (2011) underlines the need for inclusion of courses in language and mathematics and in diversity and mathematics in the teacher preparation programs of many african countries. because most programs were developed before these countries gained independence, pre-service teachers are currently being trained to teach all subjects, including mathematics, in their former colonial languages. in the context of new zealand, patadia and thomas (2002) examined seven teacher education centres and xenofontos greek-cypriot elementary teachers journal of urban mathematics education vol. 9, no. 1 100 assessed whether multicultural perspectives on mathematics teaching had been introduced into these programs. in their research evaluations, they have found that there are no formal or specific guidelines available for a particular model or approach to mathematics teacher education programs, as far as diversity is concerned. similar observations have been made by xenofontos (2014b) with respect to the reception programs in the republic of cyprus. various successful efforts to develop pre-service teachers’ awareness and competence for mathematics teaching in diverse settings have been reported. many of these efforts have taken place in the united states. for instance, aguirre, zavala, and katanyoutanant (2012), who see the development of culturally responsive mathematics teachers as the intersection of two sets—culturally responsive teaching and pedagogical content knowledge—discuss their work with pre-service teachers to develop the latter’s knowledge in mathematics teaching in diverse settings. aguirre and colleagues assigned student teachers the task of analysing their own lessons from their school placements by giving them a rubric, as well as the task of writing reflections. other similar examples in a u.s. context finding a positive impact on participants’ beliefs are those of white, murray, and brunaud-vega (2012) and turner and colleagues (2012). eliciting and making sense of pupils’ cultural, home, and community-based knowledge, and its relevance to mathematics instruction is a process that begins much earlier, during pre-service teachers’ preparations, and continues developing as teachers enter the field (anhalt & rodríguez-pérez, 2013; civil, 2007). if pre-service teachers “carry this awareness into their future careers as mathematics teachers, this is a start on negotiating mathematics classrooms in which cultural diversity is affirmed and valued” (presmeg, 1998, p. 336). in general, it appears that in different parts of the world, mathematics teachers have context-specific concerns with respect to teaching the subject to pupils from diverse backgrounds. regardless of the particularities of each setting, however, these concerns are related to language and culture. it is important for teachers to learn about the links between these two factors and mathematics, as a school subject, during their teacher education studies and as part of their in-service professional development. to improve teachers’ pedagogical repertoire in the republic of cyprus, i conducted this study, aiming to map the local landscape and set the ground for professional development programs to be organized. methods participants this project employs a collective case study methodology (goddard & foster, 2002; yin, 2009). as goddard (2010) explains, “collective case study involves more than one case, which may or may not be physically collocated with other cases” (p. xenofontos greek-cypriot elementary teachers journal of urban mathematics education vol. 9, no. 1 101 164). in such an approach, “cases are chosen because it is believed that understanding them will lead to better understanding, perhaps better theorizing about a still larger collection of cases” (stake, 2005, p. 446). the participants here were 16 greekcypriot teachers from three neighbouring urban elementary schools belonging to the same zone for educational priority (zep), while the pupil population of each school includes more than 90% of immigrant pupils. all 16 teachers comprise the collective case of this article. the graduates of the three schools attend the same lower-secondary school, which is also in the same area. initially, the three head-teachers were contacted and asked whether they would like their schools to participate in this study. all responded positively and informed their colleagues about the project. sixteen volunteers expressed an interest to get involved. table 1 presents some demographic information about the teachers, their gender, and years of teaching experience in general, and specifically, with immigrant pupils. due to the fact that teachers in cypriot state schools are obliged to be fluent in greek to be employed, the majority of the teacher population comprises native greek-cypriots (who speak the variety of standard modern greek, the cypriot dialect, and in most cases, english), with a small percentage coming from greece. typically, teachers belong to the dominant greek-orthodox ethnic culture, while, in sociolinguist terms (see yiakoumetti & esch, 2010), most greek-cypriot teachers could be labelled as bilingual-bidialectal, speaking english and two varieties of greek. for ethical reasons, and to maintain participants’ anonymity, each is given a pseudonym. the teachers are presented according to the alphabetical order of their pseudonym. table 1 the participants pseudonym gender years of teaching experience years of experience w/ immigrant pupils andreas male 15 6 athena female 20 12 elenora female 9 4 elina female 7 4 ioanna female 9 4 leonidas male 7 1 martha female 6 2 mary female 24 16 nikos male 23 13 panayiota female 25 19 petros male 20 4 phaedra female 7 3 sonia female 15 14 sophia female 7 1 stephanie female 15 14 theodoros male 8 3 xenofontos greek-cypriot elementary teachers journal of urban mathematics education vol. 9, no. 1 102 data collection and analysis each participant was invited to an individual semi-structured interview. elements of narrative research were involved in the interview questions for the purpose of establishing honesty and trust between the researcher and participants (connelly & clandinin, 1990; lichtman, 2013). more specifically, the participants were encouraged to share stories of their lives as mathematics teachers of immigrant pupils. in a previous article, i discuss findings from the same project addressing how teachers view their immigrant pupils as learners of mathematics (xenofontos, 2015). here, as noted previously, i address two questions (a) what are teachers’ reported practices in facilitating immigrant pupils’ learning of mathematics? (b) what professional needs do teachers have in assisting their immigrant pupils’ learning of mathematics? each interview lasted 30–40 minutes, and was audio-recoded and transcribed soon after. table 2 includes examples of questions related to each of the two research questions. table 2 examples of questions included in the interview protocol and their relationship to a specific research question research question examples of questions included what are teachers’ reported practices in facilitating their immigrant pupils’ learning of mathematics? 1. what specific practices, if any, have you adopted that you consider effective for immigrant pupils’ learning? 2. how do you manage pupils’ difficulties, when they arise? what do you do to help them overcome them? 3. to what extent do you differentiate your teaching to compensate their needs? 4. how can teachers support immigrant pupils so they can develop a better understanding of mathematics? what professional needs do teachers have in assisting their immigrant pupils’ in learning mathematics? 1. to what extent do you consider yourself prepared to manage a mathematics class with immigrant pupils? 2. what kind of support do teachers have for their efforts? 3. what kind of reinforcement do you wish you had in order to assist immigrant pupils in mathematics? no predetermined, specific coding scheme was utilized for data analysis. by employing both the ideas of coding and categorization (kvale & brinkmann, 2009; miles & huberman, 1994) and the constant comparison process (strauss & corbin, 1998), several categories were identified and clustered under general themes, which corresponded to each of the initial research questions. xenofontos greek-cypriot elementary teachers journal of urban mathematics education vol. 9, no. 1 103 participants’ reported instructional practices data analyses identified four reported instructional practices employed by the teachers to facilitate immigrant pupils’ learning of mathematics. briefly, these are: language support during reception classes, minimizing verbal language use and visualizing mathematical concepts, lowering expectations, and getting help from “translators.” each reported practice is discussed below. language support during reception classes ten of the teachers argued that during reception classes the main goal for their immigrant pupils was to learn greek. acquiring basic communicative skills, they claimed, were most important, for the specific pupils. during these classes, no support in mathematics was given because learning the subject was expected to come naturally once pupils grasped the greek language. similar practices are reported by gorgorio and planas (2001) in catalonia, during reception classes, whereby “from the point of view of the educational administration, not knowing the language is the only problem immigrant students face” (p. 8, emphasis in original). here, mary, an experienced teacher and head of her school, commented: “we need to focus on language and pupils’ communicative skills. learning mathematics will come naturally later because mathematical skills are based on linguistic skills.” in a similar vein, panayiota, also head of her school, explained that during reception classes, “emphasis is given to language. for these classes, pupils are grouped according to their language competence, beginners, intermediate, and advanced, so we basically work on language.” only one teacher, sophia, questioned this practice, saying: maybe our school system has not realized that mathematics learning is based on language and that in mathematics classes, there are too many instructions expressed in verbal form. in many teachers’ and school inspectors’ minds, mathematics is just about symbols and numbers, so they think that there is no need for language support. andreas was the only teacher who claimed to split his reception class time equally between language and mathematics: during these classes, i explain what has been learnt in the regular class during that day or week, by emphasizing the understanding of keywords so that the kids can understand what the lesson was about. for example, if we’re learning about multiplication, they need to understand the concepts of “factors” and “product.” … we have a regular blank notebook that we named “mathematical dictionary,” in which we write the name of each concept and then use symbols so that the kids will match the word with a mathematical symbol. for example, we’d write the word “addition” and then the symbol “+” next to it. xenofontos greek-cypriot elementary teachers journal of urban mathematics education vol. 9, no. 1 104 minimizing verbal language and visualizing mathematical concepts for the vast majority of the participants, the use of language and verbal explanations, instructions, and examples in mathematics lessons need to be minimized. in particular, 14 of the teachers claimed that they tried to reduce language activity in their mathematics teaching for immigrant pupils. for example, elina said, “it’s better to avoid having many language elements in mathematics lessons.” her response demonstrates what has also been expressed by other participants; for example, stephanie complained, “verbal tasks are difficult for them [the pupils], so i focus on simple things, like place value and sums.” she added: honestly, i believe that they need to learn the language first and then enter the classroom. … yes, mathematics is an international language, numbers and stuff, but it includes verbal problems and instructions as well. how can you explain these when you can’t communicate with the pupil? in a similar spirit, panayiota commented on how in her teaching she often uses “tasks, which are more general”: take tasks with numbers and sums, with little circles and arrows that show which arithmetic operation needs to be done, for example. with these, you can differentiate your goals without changing their structure, but by changing the numbers. if, for example, you have a pupil who has many difficulties, and not only with language, you can give her/him exercises of addition, subtraction, multiplication, and division with numbers below 20. the same task can be given to a stronger pupil, but with bigger numbers. … tasks of this kind do not involve language. [immigrant pupils] can solve them without feeling they are different. all teachers here argued that to minimize language interaction in mathematics lessons, pupils need to visualize mathematical concepts by means of pictures and mathematical manipulatives, both physical and virtual. according to leonidas: visualization and the use of materials are very important since the pupils do not understand language. when they don’t understand a written text, then you have to present it in many different ways. materials do not involve language. with these, they can understand mathematics more easily. technology assists teachers in this attempt, the teachers claimed. “there is mathematical software, and these media help other-language pupils,” stated nikos. athena, in turn, offered examples of using online games in her teaching. she reported: i use various games on the internet all the time, and i’ve come to realize that they are very helpful. there are games like football, basketball, rugby, monopoly, stuff that xenofontos greek-cypriot elementary teachers journal of urban mathematics education vol. 9, no. 1 105 kids like and find helpful. for example, when i was teaching how to round numbers up, i used many online games, which had a timer and were really challenging for the pupils. later, when they had a test, they all did well! they were rounding up to the nearest unit, tenth, hundredth, thousandth, or tens, hundreds, thousands, without any difficulties. indeed, visualizing mathematical concepts has been found to be effective for other-language pupils’ learning mathematics by a number of related studies (anastasiadou, 2008; césar & teles, 2005). however, the teachers here do not seem to recognize the important role language plays in mathematics lessons (anhalt & rodríguez-pérez, 2013; durkin, 1991). approaches like simplifying or even minimizing language use in mathematics classrooms have also been reported in other countries (e.g., gorgorio & planas, 2001); however, these may constitute an additional obstacle for other-language pupils by actually preventing the development of a deep conceptual understanding of mathematics (adler, 1997; moschkovich, 2012). lowering expectations in regular class with respect to the regular class, lowering their expectations to what immigrant pupils can achieve in mathematics is an issue discussed by 10 of the participants. these teachers argue that classroom realities, a heavy curriculum, and tight timeframes do not leave space for differentiation in instruction, methods or materials; on the contrary, lowering their “expectations regarding their pupils’ learning capacities, skills, and outcomes is all teachers can do,” claimed martha. likewise, phaedra argued that differentiation of instruction is impossible! if you had only 2 or 3 kids on different levels, then you could work something out. but if you see my class, oh, there are so many kids on different levels. i wish i could clone myself endless times to satisfy everyone’s needs. but i can’t please everyone at once. from a similar perspective, panayiota added: let’s be honest with ourselves. our school programme and curriculum do not leave time for this. when you have a class of 25 pupils, you can’t just differentiate your teaching for immigrant pupils. besides, everyone has different needs. there will always be someone whose needs are not met. theodoros’s opinion below corresponds to those of his colleagues: differentiating instruction in such settings is a myth. this whole idea is really nice and works in theory, but in reality it’s not always easy. school inspectors share this rhetoric as well. ok, i’m not saying it’s always impossible; there are some cases where xenofontos greek-cypriot elementary teachers journal of urban mathematics education vol. 9, no. 1 106 this can happen, but you can’t do it throughout the school year because differentiation is not just for mathematics. … all you can do is lower your expectations and simplify things for immigrant pupils. two of the teachers claimed that differentiation was necessary and they applied it consistently in their instruction. however, when they were asked to give examples of such practices, they referred to giving immigrant pupils “worksheets with simpler tasks” (ioanna) or having them practice with “computer software and applets for younger kids” (petros). in general, the participants in this study seem to have very specific views on what is possible to be differentiated in a mathematics classroom. by lowering their expectations for immigrant pupils, they somehow shift the responsibility away from themselves. such a perception is far from the views of scholars, like adler (2001), leonard (2008), zaslavsky (1996), and tucker, singleton, and weaver (2006), who propose various strategies for adapting mathematics lessons so that they meet diverse learning needs. others, like civil and wiles (2005), esmonde and caswell (2010), and gutstein (2003), who position their work within a perspective that fosters social justice for all, argue that mathematics teaching needs to be reconceptualized and tailored to the needs and interests of immigrant pupils so that they can be part of a more inclusive educational system. in countries like the united states, where research on the issues of race has taken place for sometime, the lowering of teachers’ expectations to compensate for the differences in minority pupils is perceived as a form of racism (davis, 2003; solórzano, 1998). however, it might not be the case that the cypriot teachers in this study lower their expectations because of racist dispositions but due to a lack of knowledge and informed insight on related pedagogical approaches, the poor support zep schools receive, as well as the overall bad organization of the moec. these issues are discussed subsequently. getting help from translators including pupils as assistants in mathematics lessons, who are competent both in greek and another language, was a practice reported by four of the teachers. sophia, for instance, commented: “when i know the language competence level of my pupils, i make sure i get help from other pupils. for example, if there were two bulgarian kids, and one of them spoke greek while the other didn’t, i would ask the former to explain the instructions in the common language.” sophia, elina, and martha talked about getting help from other pupils who could translate mathematical instructions in another language. panayiota, however, who agreed with them, talked about an extension of this practice outside school and the involvement of people from the community, like parents. she said: xenofontos greek-cypriot elementary teachers journal of urban mathematics education vol. 9, no. 1 107 a few years ago when i was at a school where i had to teach many immigrant pupils who had just arrived to cyprus, we established the practice of incorporating people who served as “translators.” they were parents-volunteers who would translate worksheets for us, and/or would come to class to sit with kids and translate mathematical verbal problems and instructions. this way, you expand the school’s role in the community, and you bring the community in, since parents get involved and help children other than their own. gorgorio and planas (2001) reported similar practices in barcelona. in that context, as they explain, adult translators were incorporated only when immigrant pupils would first arrive to the country and would have to take entry examinations so that the school would allocate them to a particular group. in such cases, adults from the community outside school would sit with them during the examination and help the pupils with language issues. participants’ professional needs the participants indicated a number of professional needs in their effort to support immigrant pupils in mathematics. briefly, the teachers talked about difficulties because of the ministry of education and culture’s (moec) “bad” organization; the need for appropriate preand in-service teacher education; the need for a specialized curriculum; and the need for a teaching assistant. presented below is a discussion of each need. difficulties because of the moec bad organization all 16 teachers emphasized how the bad organization of the moec, as far as zep schools are concerned, has created many difficulties to them as professionals. theodoros, for instance, pointed out: “the ministry needs to decide what it wants. we have no specific guidelines on how we should work with our immigrant pupils, and this, to me, is unprofessional. we don’t know what we should expect them to learn.” sadly, the participants’ views here are in line with the conclusions of other national studies (e.g., hajisoteriou, 2009, 2010) that indicate how, despite the republic’s official intention of adherence to the educational goals of the eu, a coherent state-derived multicultural policy does not exist. in addition to the lack of clear guidelines, sophia summarized the views of her colleagues by talking about other prohibiting factors that derive from the moec’s bad organization. she said: “due to the financial turbulence cyprus is experiencing in recent years, the ministry has decreased the teaching periods of reception classes for other-language pupils. this way, instead of dealing with the problems, we create more.” sophia went on by stating that moec’s policies did not fully take research evidence into consideration: xenofontos greek-cypriot elementary teachers journal of urban mathematics education vol. 9, no. 1 108 take you, for example. you’re a researcher working in this area in order to help these pupils learn mathematics, and i really appreciate your efforts. i don’t want to sound too pessimistic but i don’t really think they [the moec] will listen to what you have to say. they don’t utilize the results of academic research. the ministry promotes people and ideas based on other criteria, like politics, and not according to one’s qualifications and the importance of what they have to say. the need for appropriate preand in-service teacher education addressing the issue of teachers’ needs for appropriate preand in-service education regarding diversity, and the teaching and learning of mathematics, emerged from all participants’ responses. angelides and colleges (2003) made similar observations, more than a decade ago, and argued that teacher education from the perspective of cultural diversity in the cyprus republic was practically non-existent. as mary, one of the teachers here, argued, “we are basically ‘experimenting’ because we are not properly prepared for teaching mathematics in diverse classrooms,” an opinion shared by many of her colleagues. martha responded: “it’s up to each teacher to discover strategies that work in a classroom. from my experience, no one has ever told me how to approach mathematics teaching for pupils with diverse backgrounds.” nine of the teachers in this study were below the age of thirty and with less than nine years of work experience. they all drew upon their pre-service teacher education experiences, arguing that their undergraduate programs did not include modules on diverse mathematics education. studies examining the content of teacher preparation programs in other countries have reached similar conclusions (see, e.g., chitera, 2011; patadia & thomas, 2002). during her first years as a teacher, ioanna argued that she felt hopeless! assigning young teachers with no experience to zep schools is tragic. we were not properly prepared during our undergraduate studies. i felt that i wasn’t qualified to teach greek or mathematics to non-native speakers. we were not prepared at all. our studies did not include modules on teaching language, mathematics or any other subject to non-indigenous learners. in a similar vein, leonidas stated that teacher preparation programs should include diverse mathematics education classes because the demographics of cypriot society have changed. more specifically, he said: i guess the reason we didn’t have such classes during our undergraduate studies was because back then our schools and classrooms were not so diverse. but in recent years this has changed. it is necessary that prospective teachers are appropriately prepared. additionally, all 16 participants stressed the lack of diverse mathematics education seminars for in-service teachers. as theodoros said: “i don’t think i’ve xenofontos greek-cypriot elementary teachers journal of urban mathematics education vol. 9, no. 1 109 ever heard of seminars in diverse mathematics education. there are a few about language and education in general, but nothing related to mathematics.” for participants, such seminars were important because as stephanie said, “[seminars] would be practical because most of the times we are lectured about teaching practices which are far from classroom realities.” nikos claimed that seminars of this kind would need to take place during school time, and not in the afternoons: on the one hand, we all need to be properly trained. on the other, teachers need motivation. what do i mean by this? maybe they should give us less teaching hours and instead encourage us to participate in training programs in diverse mathematics education. maybe it’s wrong, i don’t know, but teachers need motives. if such seminars take place in the afternoons, then not many teachers would express interest. this is why the seminars need to be in the morning. unfortunately, most teachers want rewards. if this could be done by exempting them from a few teaching hours, then let it be. the need for a specialized curriculum twelve participants commented on the inappropriateness of the mandatory national curriculum and textbooks in relation to immigrant pupils’ mathematical learning. in particular, teachers who raised this issue argued that the current mathematical materials, prepared by the moec, mainly refer to typical indigenous pupils and do not take into consideration the needs of culturally diverse learners. in a characteristic response, sonia said: let’s take the mathematics textbooks of grade 4 that i’m teaching this year. i’m not saying that the problems included are hard for a typical 10-year-old. but they definitely don’t refer to other-language pupils. they refer to children who speak greek as a native language. the fact that there are no official materials (i.e., books) for other-language learners of mathematics was time consuming for teachers, who have to dedicate extra time and effort to prepare something else, said elina. as she claimed: “it would be helpful if we were given some teaching materials specifically designed for diverse pupils. this would save us a lot of time. something like an online platform with worksheets and teaching ideas.” other teachers argued, because mathematics textbooks rely a lot on language, it would be helpful if they were translated into other languages as well. for instance, according to mary: it’s well known that in cyprus most immigrant pupils are arabic speakers, russian speakers, and bulgarian speakers. these are the main languages of immigrants in our schools. i think it would be helpful if our mathematics textbooks were translated to these three languages. xenofontos greek-cypriot elementary teachers journal of urban mathematics education vol. 9, no. 1 110 martha was the only teacher who talked about mathematics learning outside the classroom and did not rely on textbooks. in her view, zep schools should have participated in other activities in the real world: there should have been other programs for zep schools. we shouldn’t follow the same curriculum and textbooks. for example, our pupils could visit a real work environment, like a market, and participate in activities there, related to mathematics. or we could organize a mathematics week, during which our pupils could learn mathematics through hands-on activities, like cooking or through drama. this would be different, more interesting, and, i believe, effective for diverse learners. the fact that only one teacher commented on mathematics learning through activities that do not involve the use of the national textbooks is not surprising. in the cyprus republic, the vast majority of mathematics teachers rely on the textbooks provided by the moec and do not easily deviate from them (mullis, martin, & foy, 2008; xenofontos, 2014a). nevertheless, similar concerns about the need of specialized teaching materials for immigrant pupils were expressed in the past by teachers in italy, spain, and portugal (favilli et al., 2003). furthermore, those teachers hesitated to modify their instructional practices so to incorporate digitally supportive materials developed by a team of researchers (césar & favilli, 2005). culturally responsive mathematics teachers, argues gay (2002), need to determine the multicultural strengths and weaknesses of curriculum designs and instructional materials and learn how to make necessary changes to improve the overall quality of such materials. the teachers in this study recognized weaknesses in the current curriculum, but did not seem to be able to adapt parts of it to fit their needs, as well as their pupils’ needs, in a more direct way. quite the contrary, they waited for top-down changes initiated by the moec. the need for a teaching assistant four of the teachers talked about the need of a teaching assistant in mathematics classrooms with diverse pupils. such a practice is quite common for the general classrooms of state schools in countries like the united kingdom. in the cyprus republic, however, teaching assistants are provided only in cases where there are pupils with special learning needs, (angelides, constantinou, & leigh, 2009), that is, pupils who have mental and/or kinaesthetic disorders. culturally and linguistically diverse pupils are “really wronged. they are not allowed to have support as learners with special needs,” said sophia. “having a teaching assistant in mathematics classrooms would be ideal,” said nikos. “that person could help immigrant pupils with both the mathematical content and the issue of language in the tasks. ok, teachers do help them, but when you have 25 kids in a class, your time is limited.” if immigrant pupils had support from an assistant, “if they had someone especially for them, they would learn more naturally,” claimed ioanna. xenofontos greek-cypriot elementary teachers journal of urban mathematics education vol. 9, no. 1 111 concluding remarks overall, the findings from this study suggest participants’ instructional practices for supporting the mathematical learning of their pupils focus mainly on the issue of applying language as a means of basic oral communication. this finding is not surprising, considering, as argued elsewhere (xenofontos, 2015), the same teachers hold a strong belief that claims linguistic barriers are the main source for the many difficulties that immigrant pupils encounter in mathematics classrooms. as a result, the teachers here do not seem to employ any strategies that take pupils’ cultural background into consideration (e.g., civil & wiles, 2005) or to cultivate attitudes towards social justice through mathematics (e.g., esmonde & caswell, 2010; gutstein, 2003). quite the contrary, some of their reported practices, for example, minimizing verbal language use and lowering expectations, are not considered to have a positive impact on diverse pupils’ learning of mathematics by most scholars (e.g., adler, 1997; moschkovich, 2012). the implementation of priority policies to support disadvantaged pupils, like immigrant children, is typical of many european countries (muskens, 2011). however, although the “classic” positive discrimination programs, as, for example, the republic’s zep policy, differ across countries in administrative approach and intensity, “they have proved to be remarkably similar in one respect, that is, the limited results they have achieved” (karsten, 2006, p. 277). in the case of the republic of cyprus, research evidence (e.g., hajisoteriou, 2009, 2010) indicates that there are no clear local guidelines regarding the implementation of eu diversity policies in education, a fact reflected in the mathematics teachers’ comments reported here. this is, in fact, in opposition to the philosophy of the new curriculum, launched in 2010, which makes explicit references to the culturally diverse character of contemporary cypriot society, and aims at creating democratic and humane schools (moec, 2010). in spite of the presence of this reference in the official documents, no specific guidelines are provided for zep schools, and teachers working in diverse environments are required to follow the same curriculum and national textbooks as do schools with a mainly native population (xenofontos, 2015). as far as teacher education is concerned, the participants’ views, here, echo my previous observations that none of the undergraduate preparation programs in the republic of cyprus include modules on diverse mathematics education (xenofontos, 2014b). in addition, this fact does not correspond, unfortunately, to the research of scholars, like presmeg (1998), which propels the condition that mathematics teachers’ cultural awareness be developed as early as possible, preferably, during their teacher preparation studies. every study carries a set of limitations. in the work presented here, the findings are based solely on interview data, and, unfortunately, no other method was utilized for triangulation. for this reason, i have chosen to talk about teachers’ xenofontos greek-cypriot elementary teachers journal of urban mathematics education vol. 9, no. 1 112 reported practices, given that, at this stage, they could not be compared to observational data on what actually happens in classrooms. nevertheless, due to the lack of other related studies in the context of 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(2008). creating equitable practice in diverse classrooms: developing a tool to evaluate pedagogy. in m. goos, r. brown, & k. makar (eds.), proceedings of the 31st annual conference of the mathematics education research group of australasia (pp. 637–643). adelaide, s.a., australia: merga. journal of urban mathematics education july 2011, vol. 4, no. 1, pp. 75–97 ©jume. http://education.gsu.edu/jume mindy kalchman is an assistant professor of mathematics education in the college of education, depaul university, 2320 n. kenmore avenue, chicago, il, 60614; email: mkalchma@depaul.edu. in her research, she focuses on the interaction between the demands of classroom pedagogy and problem solving outside of school, with implications for preservice teachers and school-age children. preservice teachers’ changing conceptions about teaching mathematics in urban elementary classrooms mindy kalchman depaul university in this article, the author reports on a project intended to gain insight into the effect a specific constructivist learning opportunity might have on preservice teachers’ beliefs and attitudes about the value of conceptual-based instructional methods for urban children. the project context was an elementary mathematics methods course; the weekly learning opportunity asked students to write about an authentic mathematical experience that they had had during the week. students were required not only to summarize the experience but also to explain how they solved the problem in ways that did not involve a school-taught algorithm or a calculator. the author argues that completing this assignment resulted in more than building preservice teachers’ mathematical knowledge and skills; it also provided them with an opportunity to learn within a constructivist framework, and to see that learning is about the relevance of curriculum and the meaning individuals make of it rather than the demographics of learners. keywords: elementary teacher education, constructivist teaching, mathematics teacher education, teacher beliefs, urban education romoting mathematics learning environments that privilege students’ conceptual development over the ―traditional‖ rule-based, procedural methods of instruction has been a primary focus for the national council of teachers of mathematics (nctm) for over two decades (nctm, 1989, 2000). the enriched learning opportunities for students who experience a conceptual-based learning environment are well documented in the mathematics education literature (fitzgerald & bouck, 1993; hiebert, 2003; spillane & zeuli, 1999; sutton & krueger, 2002). students in such environments often excel with respect to greater flexibility, sophistication, confidence, and competence with both routine and non-routine problems and computations (donovan & bransford, 2005; national research council [nrc], 2001). the pedagogical methods that foster such conceptual learning, however, are not routinely reaching and/or being implemented in urban classrooms (berry, 2003; mckinney, chappell, berry, & hickman, 2009). too often in urban classrooms an ―initiation-response-evaluation‖ (ire) pattern (hiep kalchman changing conceptions journal of urban mathematics education vol. 4, no. 1 76 bert & stigler, 2000) remains the dominant instructional practice (lubienski, 2002; strutchens & silver, 2000). here, students listen to what their teachers say and do, try to remember it, and then attempt to parrot it back on homework assignments and tests (heuser, 2000). within this method of instruction, there is little emphasis on developing conceptual understandings by having students explain their thinking, make conjectures, or discuss ideas and strategies (franke, kazemi, & battey, 2007). thus, many teachers in urban classrooms, as well as others who employ such instructional methods, should rethink their pedagogical practices to keep their students competitive when it comes to mathematics (ladson-billings, 1997; nrc, 2001). putting the onus on teachers is not meant to be punitive. rather, it is meant to self-empower teachers by acknowledging that within the context of schools, teacher quality is the most direct measure of students’ academic achievement and success (brown, 2002; haberman, 2005; ladson-billings, 1994; steinberg & kincheloe, 2004). in other words, effective teaching matters! bringing about change in urban teachers’ pedagogical methods and practices, however, must begin by addressing their beliefs about what constitutes effective mathematics instruction for their students (klein, 1998; steele & widman, 1997). here, beliefs are defined as convictions that resist change and are not necessarily contingent upon either reason or evidence (watters & ginns, 1997). such beliefs have been found to be far more influential than knowledge in determining how individuals organize and define tasks and problems and are stronger predictors of behavior (pajares, 1992). moreover, it has been argued that beliefs about ―good‖ teaching are well established by the time students get to college (grossman, 1990; pajares, 1992). therefore, it is important for teacher educators to ensure that preservice teachers’ incoming beliefs about teaching are explored, discussed, and revised (if necessary). this recommendation is especially relevant for prospective teachers’ initial beliefs about teaching in urban classrooms because research has shown that preservice teachers believe that urban students require mathematics instruction that focuses on basic skills (gilbert, 1997; walker, 2007), rote teaching and learning (anyon, 1997; breitborde, 2002), and repetition (walker, 2007). addressing this cultural bias among preservice teachers is critical in light of the previously mentioned literature on the persistence of ineffective pedagogical methods and practices in urban mathematics classrooms and the resilient nature of established beliefs. moreover, interventions aimed at preservice teachers’ beliefs about teaching mathematics in urban schools must result in authentic conceptual change by first confronting their original perspectives (harrington & enochs, 2009; klein, 1998; steele & widman, 1997). without such an authentic struggle, many preservice teachers may revert to teaching in the traditional ways they experienced in their own schooling rather than implementing the sort of conceptual-based in kalchman changing conceptions journal of urban mathematics education vol. 4, no. 1 77 structional approaches they are taught and exposed to in their teacher education programs (ebby, 2000; merseth, 1993). accordingly, the purpose of the project reported here was to gain insight and perspective into the effect constructivist learning opportunities might have on preservice teachers’ beliefs and attitudes about the value of conceptual-based instructional methods for urban elementary mathematics students. the context for the project was an undergraduate elementary mathematics methods course at an urban university where the college of education has an explicit mission of preparing teachers for urban, multicultural settings. specifically, preservice teachers were given a weekly assignment for which they had to write about an authentic mathematical experience they had during the week between classes. they had to write a summary of the situation as well as explain how they solved the problem in ways that did not involve a school-taught algorithm or a calculator. completing this assignment resulted in more than building preservice teachers’ mathematical knowledge and skills; it also provided them with an opportunity to learn within a framework consistent with the nctm, and to see that learning is about the relevance of curriculum and the meaning individuals make of it rather than the demographics (e.g., race, gender, class, etc.) of learners. in this article, i first outline the theoretical perspective for the methods course i teach, followed by a description of how i address constructivism as a pedagogical framework. i then share the structure of my methods course before introducing the participating preservice teachers, data sources, and methods for analysis. i then present the results of my findings and provide examples of preservice teachers’ work. finally, i elaborate on the results in the context of preservice teachers’ changing perceptions about the value of conceptual-based pedagogical methods and practices in urban mathematics classrooms. my mathematics methods course theoretical perspective of the methods course the previously noted literature supports a vital and practical purpose for this project. nonetheless, the original motivation for it came from the in-class experiences i was repeatedly having with preservice teachers surrounding their attitudes and beliefs about the value of constructivism for teaching mathematics in urban classrooms. i decided to examine these experiences systematically and to study the effects that the design of my methods course might have on preservice teachers’ attitudes and perceptions about teaching mathematics in urban settings. constructivism (defined later) is the central theoretical and organizing perspective for my methods course; i introduce it introduced at the beginning of the quarter as a general theory for how people learn, and one that informs myriad in kalchman changing conceptions journal of urban mathematics education vol. 4, no. 1 78 structional practices in mathematics teaching and learning and other disciplines. however, despite a general affinity for constructivism as a framework for practice, preservice teachers often challenge the value of it for urban children’s mathematics learning experiences. specifically, preservice teachers too often contend that urban children need to be ―kept on task,‖ motivated, and taught in ways that constructivist-influenced classrooms cannot and do not support (jepsen, 2009). contrary to this popular belief, research shows that urban children and youth do indeed thrive in constructivist learning environments because of the theoretical and practical foundations such environments provide (griffin, case, & siegler, 1994; mcnair, 2000). therefore, one of the most important components of my methods course is not only to expose preservice teachers to literature that supports constructivist learning environments but also to ensure that the preservice teachers themselves have mathematical learning experiences that substantiate the general value of constructivism in teaching and learning. this means that it is not enough for me to talk to preservice teachers about the value of constructivism for pedagogical decision-making; that would merely be modeling and sanctioning an ire strategy of instruction. rather, i must ensure opportunities for authentic constructivist learning opportunities as part of preservice teachers’ overall course learning experiences. discussing constructivism as a framework for pedagogical practice early in the quarter, i introduce the origins of constructivism as a theoretical construct most often attributed to piaget’s theory of child development (piaget, 1952). we discuss the evolution of constructivism in the context of criticisms of piaget’s theory, including challenges to the universality of his stages and the undefined role for a teacher in children’s learning (e.g., laurenco & machado, 1996). we then look at the ways these challenges to piaget’s theory have led to diverse interpretations and adaptations of constructivism for mathematics education, most specifically by those situated in sociocultural frameworks for learning (e.g., cobb, 2006; cognition and technology group at vanderbilt, 1997; lampert, 2003; vygotsky, 1962). a particularly robust theoretical contribution of piaget’s that we revisit throughout the course relates to a basic definition of constructivism. in this definition, constructivism is described as the process of cognitive structures changing, and thus individuals learning, as they are exposed to external, authentic environments and integrate information by either assimilating or accommodating it (bollinger, 2006). admittedly, this is an extremely oversimplified and abbreviated definition. however, two important aspects of it are presently relevant. first, the reference to individuals implies that the theory does not discriminate based on age, race, gender, domicile, or any other demographic data. rather, research shows consistent developmental progressions within children across cultures (okamoto, brenner, & curtis, 2002), social class (grif kalchman changing conceptions journal of urban mathematics education vol. 4, no. 1 79 fin, case & siegler, 1994), and so on. these progressions are not necessarily seen uniformly with respect to the timing and rate of achievement, but the progressions do appear to occur in a predictable and structured fashion. second, interactions with external, authentic environments are critical to a learner’s development (bollinger, 2006; webb, 1980). in typical school settings, students of all ages and demographics engage with mathematics in ways that mimic, model, or even demonstrate the mathematical demands of the ―real world‖ (ninnes, 2000; sundberg & goodman, 2005). however, situations that reflect the ―real worlds‖ of students are not the same as authentic situations that are in real-time, and are unique to and current in their lives (gravemeijer, 1997). these latter, real-time situations are found in children’s mathematical activities such as shopping, playing sports, and cooking, and require spontaneous and functional applications of mathematical knowledge and skills. having a personal connection with curriculum in order for effective learning to take place is neither a new proposition, nor attributable to piaget. indeed, dewey wrote about it over a century ago (dewey, 1902). dewey’s ideas along with those that fall within the framework of situated learning and cognition are also discussed extensively in my methods course. these perspectives are elaborated on in class as we extend the general argument to include the need for mathematics curriculum to be personally meaningful in order to motivate children—in particular, urban children ―whose life experiences often are farthest from the traditional school curriculum experience‖ (mcnair, 2000, p. 552). structure of the methods course the relevant methods course met once per week for 10 weeks. each class session was 3 hours, and had time allocated for a variety of learning opportunities, including the discussion of field experiences, assigned readings, and doing mathematicsrelated activities including the math in the everyday life assignment (discussed later). also, independent of class time, individual students (i.e., preservice teachers) attended an assigned field placement once per week for 10 weeks. each visit lasted for approximately 90 minutes. during these visits, students were required to go beyond observing, and had to plan for and teach mathematics lessons in collaboration with and independent of their cooperating teachers. the research project participants twenty-three undergraduate elementary education students participated in the project (20 women; 3 men). eighteen of them were seniors and five were juniors. two of the men identified as latino, and one as african american. eight of the women kalchman changing conceptions journal of urban mathematics education vol. 4, no. 1 80 identified as latina, four as african american, and eight as caucasian. the mean age was 21, and 15 of the students indentified as urban by birth and 3 by current domicile. as mentioned, our college of education has an explicit mission for preparing teachers to teach in urban, multicultural settings, with an emphasis on serving the poor and disenfranchised. thus, all students must engage with field experiences in schools that reflect this demographic. however, most students also spend time in contrasting environments including suburban schools, independent secular schools, public schools in middle-class neighborhoods, and parochial schools. the methods course described here was among the last of the courses preservice teachers take before student teaching. thus, most had been in at least one school with a population of low-income families, an independent parochial school, and a public school in a middle-class municipal or suburban neighborhood. for this particular course, students were individually placed in schools with a population where they either needed to accumulate experience and time, or were most interested in ultimately teaching. accordingly, 13 of the preservice teachers were placed in urban, ―high-needs‖ schools with respect to serving low-income, historically marginalized and underserved students. data sources and analyses as ebby (2000) points out, ―a constructivist perspective focuses on the process of coming to know rather than on only the outcomes‖ (p. 75). accordingly, i used ethnographic methods of data collection and analysis in order to gain a deeper understanding of participants’ perceptions, attitudes, and beliefs that drive their decisions (cohen, manion, & morrison, 2000). the study involved three main data sources collected throughout the quarter. two sources used for analysis were instructor-initiated whole class discussions, and a review of preservice teachers’ written work. the math in everyday life (miel) assignment described next was used to contextualize and analyze students’ learning and comments. details of the analytic methods are elaborated within descriptions of each data source. the miel assignment. the purpose of the miel assignment was to provide each student with the opportunity to identify and then to do mathematics in authentic and personally relevant ways that reflect the values and premises embedded in constructivism. each week, students submitted an miel, which was an account of having done some mathematics that was real and necessary in their non-teaching lives (kalchman, 2009). typical contexts included tipping, exercising, dieting, shopping, cooking, and paying bills. preservice teachers were required to communicate the relevant situations and to show how they did the requisite mathematics, and encouraged not to use calculators and formal, school-taught algorithms. instead, they were encouraged to apply strategies that were contextdriven and situation-dependent and responsive to the circumstances at hand. for kalchman changing conceptions journal of urban mathematics education vol. 4, no. 1 81 example, it is not typically convenient to pull out a calculator or pencil and paper when sitting in the back of a taxi trying to calculate a tip. thus, students had to describe and explain their thinking, and communicate the steps they took to solve their problems. finally, they were encouraged to ask other people how they would solve the same problems. then, each week students either voluntarily or at my request shared their miel with the class. i typically asked those whose problems presented unique challenges and required some alternative thinking. furthermore, if possible, i tried to choose submissions that would have some relevance to an urban child’s lived experiences, not only to support our college’s mission but also to address the educational needs of urban children. groups of preservice teachers then engaged with the mathematics of the selected miel and shared their solutions and strategies. then, as a class, we discussed the constructivist processes and implications of each group’s problem solving strategies. for example, we explored particular features of the process such as how difficult it was to think like and interpret problems like someone else and how surprising it was that there were so many ways to solve a seemingly simple and routine problem. instructor-initiated whole class discussions. twice in the quarter, i initiated a whole class discussion specifically about the relevance of constructivism for discrete populations such as urban students. significant to the resulting conversations was the fact that in my methods course we do not delve into the unique complexities of teaching and learning in urban classrooms as a course topic per se. these issues are confronted in other courses specific to the foundations of education, and i expect students to come to my classes with some experience with and knowledge of the pertinent issues. rather, in our discussions, we focused on the implications of different learning cultures and environments, rather than on the environmental and cultural circumstances that necessitate the conversation. the first discussion was initiated in the second week of the course following a lecture-style overview of constructivism as per the theoretical perspective previously described. i opened this conversation by asking students if they believed that putting constructivist theory into practice with all children in all mathematics classes is both plausible and desirable, and to elaborate on and justify their beliefs. the ensuing conversation was completely organic to their ideas. the second discussion was in week 10 of the quarter after we discussed literature related to changes preservice teachers make in their beliefs about teaching and learning mathematics as a result of their experiences in a methods course with highly integrated field experiences (ebby, 2000). i opened this conversation by asking them to consider and share any changes they experienced over the course of the quarter as they relate to teaching and learning mathematics for themselves and for children. i asked them to refer specifically to and comment on different student populations and on different instructional styles they had observed in their field expe kalchman changing conceptions journal of urban mathematics education vol. 4, no. 1 82 rience for the mathematics methods course and in their teacher preparation program in general. these conversations were audiotaped. transcriptions were coded to reflect categories of initial beliefs about the pedagogical needs of urban children: initial beliefs of not believing in constructivism (in), initial beliefs of being undecided about the value of constructivism (iu), and initial beliefs of believing in constructivism (iy). codes for the second conversation involved two levels of categorization. first, transcripts were coded to reflect students’ beliefs at the end of the course: outgoing beliefs not favoring constructivism (on), undecided outgoing beliefs (ou), and outgoing views of believing in the value of constructivism (oy). transcripts then were coded to reflect conceptual shifts attributed to a particular aspect of the course:  miel,  fe (field experiences),   cd (class discussions), and combinations of the three. to analyze the data, i first looked quantitatively at the number of relevant comments the preservice teachers made in each of the categories during both structured discussions. then, i attributed each comment to individuals in order to account for each student’s preand post-quarter positions. i then coded the reasons they gave for maintaining or changing their perspectives and recorded those to reflect each contribution. review of preservice teachers’ written work. students were not required to write about their perspectives on urban mathematics education per se. but because of the explicit mission of the college, many of them were completing their field experiences in urban classrooms; therefore, the topic of urban classrooms appeared with some frequency in their written work, especially toward the end of the quarter. in addition to the miel, preservice teachers had two other written assignments. the first was their ―weekly contribution.‖ for the first 10 to 15 minutes of each class session, students wrote short pieces about a topic, or topics, they hoped to discuss in the forthcoming class session. the topics they wrote about were at their discretion and ranged from those related to the readings and/or their field experiences to relevant current events. these pieces were coded using the same codes previously described if content warranted it. however, each code was preceded with a ―wc‖ to indicate that the comment was made in the context of the weekly contribution. to illustrate, if a student mentioned an experience in a classroom that affected or contributed to a change in his or her perspective on pedagogy for urban children, the code would read ―wc  fe.‖ the second written assignment was a final essay describing and reflecting on a mathematics lesson the preservice teachers taught in their field-placement experience. of particular relevance to this project were sections on ―pedagogical choices‖ and ―what you learned about teaching and learning elementary mathematics.‖ in the case of the former, they were required to explain and justify the kalchman changing conceptions journal of urban mathematics education vol. 4, no. 1 83 sort of pedagogical approach they intended to use in the pre-planning of their upcoming lessons (artzt & armour-thomas, 2002). this assignment was due in the final week of the quarter; thus, the pedagogical choices they made were significant. if the pedagogical choice reflected a constructivist perspective, then the entry was coded as pcc; it was coded as pct for a choice of traditional pedagogy. a ―u‖ was added to each code if the lesson was specific to an urban population. the reflection section of the final papers were also coded for relevance to preservice teachers’ beliefs and perceptions about the pedagogical needs of urban children. these codes were consistent with those reflecting change previously described but were prefaced with ―fp‖ for final paper. i coded and counted the number of relevant written remarks made in preservice teachers’ weekly contributions and again in their final assignments. after establishing a quantitative basis for believing that the miel assignments were influential in the changes preservice teachers made in their perspectives on teaching mathematics in urban settings, i began reviewing their comments and written remarks qualitatively. for example, i looked for comparative statements, oral or written, that included before and after remarks and the context and timing for any conceptual epiphanies. i looked for comments they made regarding their observations of instruction in urban mathematics classrooms and how those observations conflicted with or supported our class discussions and their own constructivist experiences. findings sample miel assignments the following miel samples provide context, and are representative of the sort of miel assignments i selected to use as in-class activities. the mathematics activities were challenging enough for elementary-level children, involved a variety of conceptual strands (nctm, 2000), and may have resonated more with preservice teachers as urban experiences than submissions that focused on more generic tasks such as tipping, cooking, or banking. figures 1, 2, and 3 are examples of miels. in figure 1, a student shared her process for finding a primary care physician within a certain radius of her home. she used chicago’s block system to calculate distances. indeed, the process is a bit dizzying for someone unfamiliar with the city. however, calculating distances and giving and receiving directions using the block system is standard for chicagoans and an essential code for all who live there. the mathematics involved was diverse and ranged from standard computation to two-dimensional algebraic thinking as the student considered traveling both south and west on the grid. kalchman changing conceptions journal of urban mathematics education vol. 4, no. 1 84 math everywhere this week my husband and i searched online for a primary care physician for myself. in the selection process, the website instructed us to indicate how far we would be willing to travel to visit the doctor’s office. the question was written in miles (5,10,15,20,25), so i had to convert to city blocks (since we don’t “talk” miles in the city). first, i had to remember that 1mile equals 8 city blocks. therefore, i had to multiply 8 by 5 which is 50. next, since i live on the 60th block south, i had to add forty to sixty. i knew that 4+6 = 10, add 0+0 = 0, which is 100. so since i was willing to travel to the 100th block in chicago, i continued on to 10 miles. first, i multiplied 8 by 10. since i know that any number times ten is that number with a zero, then i knew it equaled 80. next, i added 80 to sixty. first, i added forty to sixty to get one hundred, and then i added the remaining forty to get one hundred forty. well, the 140th block is a bit far, so i wondered if i would go to the 100th block and 40 blocks west (that is not even past cicero since i know that cicero is 52 blocks west), so i would be willing to go to cicero & 100th. so i continued on to 15 miles. since i already figured out that ten miles equals 80 blocks, i calculated five more miles and i know that 8 times 5 equals 40. so 80+40, i know that 8+4= 12 and 0+0=0, so 12 and 0 is 120. would i be willing to travel 120 blocks? let’s see. i live on the 60th block, so i added 60 to 120. i know that 12+6 is 18 and attaching the zero brings us to the 180th block. i would be willing to travel to 115th block south. the remaining blocks i would want to travel west. ten more to 115 would be 125, so 20 more would be 135, 30 more would be 145, 40 more would be 155. fifty more would be 165, sixty more would be 175, and so five more would bring us to 180. sixty and five, or 65 tells me how many blocks i would travel west. as i mentioned before, cicero is 52 blocks west, and 65 blocks is only a few more west. 115th & beyond cicero seems far enough from my house. i would not want tot travel any farther, so i selected 15 miles. after i finally selected a primary care physician, i noticed that their office was on the 94th block and 20th block (western). i wondered how many miles? first, i subtracted sixty from 94, 9–6 is 3, 4–0=4 so 34. i added 34 and 20. i know timetables up to 12, i know that 8 times 7 is 56, which is too much. so 8 times 6 is 48. it takes two more to get to 50, and 54 is 4 more that fifty. so, 4 and 2 is 6. the office is 7 miles and 6 blocks away from my house. hmmm, it doesn’t seem that far! figure 1. finding a primary care physician using chicago’s city block system. in figure 2, a student discussed the cost of zoned parking meters in downtown chicago. the zoned meter system is not made explicit on the machines themselves, but it is essential that visitors and natives alike negotiate it to avoid parking tickets. the mathematics involved included addition, subtraction, multip kalchman changing conceptions journal of urban mathematics education vol. 4, no. 1 85 lication, and division with whole numbers as well as multiplication with decimals. working with money, specifically, is also part of the measurement strand of the nctm principles and standards for school mathematics (nctm, 2000). math everywhere this week math was involved a lot in my life. most of the time i didn’t realize i was using math at the moment but quickly recognized that math was being used in my daily routines. a friend and i were walking down kedzie street and there were meters that read “zone 6” and i asked her what “zone 6” meant and we both didn’t know the answer. however, we figured that it had something to do with number of hours or minutes one quarter gives. we thought that the zoning had to do with time and money because there are certain places in the city that give you more or less amount of time for one quarter. so we quickly changed the subject up to how much money we thought could be spent on one meter daily. we knew that on kedzie, one quarter gave you an hour and there are obviously twenty-four hours in a day. so to figure out how much money could be deposited into the meter on any given day, we multiplied twenty four times 25 cents. i couldn’t think of the product of twenty four times 25, but twenty times 25 gave me 500. i then multiplied 4 times 25 which gave me a hundred and i added the two which gave me 600. finally, i moved the decimal over two places because the twentyfive was in cents not dollars. so after moving the decimal over two places to the left, we got $6. that particular “zone 6” meter on any given day could have up to $6 in quarters deposited into it. we then talked about how for different meter zonings more money could be deposited since less time is given for one quarter. the methods i took to solve this problem were convenient because a lot of mental math was done. however, we made an error because there are certain times in the day when money does not need to be deposited. so we would have to subtract those amounts from the original total of $6. figure 2. feeding parking meters in a downtown center. finally, in figure 3, a student wanted to determine how much topsoil to buy to cover a 1.5' border surrounding her 8' x 8' yard. this problem involved using algebraic and geometric thinking along with computing with whole numbers, fractions, and decimals. this problem is relevant to an urban lifestyle because of the ways city-dwellers must often harvest their space if they have a yard and would like to enjoy any part of it as a garden. kalchman changing conceptions journal of urban mathematics education vol. 4, no. 1 86 figure 3. buying topsoil for a city garden. structured discussions discussion 1. the first structured discussion (week 2) lasted for 30 minutes and all 23 students contributed to the conversation. a total of 36 comments were coded and distributed to reflect the findings in table 1: 61% of the preservice teachers expressed initial beliefs that constructivism was not an appropriate instructional approach for urban elementary mathematics students, 26% were undecided, and 13% believed it was appropriate. reasons they gave to support their negative reactions fell primarily into three categories. the first was urban students’ apathy toward learning: ―those students don’t want to learn, and so trying kalchman changing conceptions journal of urban mathematics education vol. 4, no. 1 87 to engage them in discussions about their own thinking in order to reflect on it is not realistic. the conversations would never get started.‖ the second and most popular reason for why urban students are not suitable candidates for constructivist learning environments was illustrated by the following comment made by a mature, latina: urban kids need more structure and discipline than that. there’s no way a group of kids at my school would stay on task if the teacher put them in groups and told them to solve a problem without step-by-step instructions for how to do it. they’d start throwing things or pulling out their cell phones. the final reason was that ―students don’t have the background knowledge they need to build new knowledge.‖ those who were undecided about the value of constructivism for urban children fell into two categories. some admitted that they did not really understand the scope and application of constructivism prior to my lecture and were reconsidering their understanding of it before deciding on their position. while others admitted that they had never thought about constructivism in the context of mathematics education before because all of their experience with the term had been with literacy or science education. furthermore, their incoming model for teaching mathematics was one that focused on a textbook that essentially stipulated what and how they would teach. finally, preservice teachers who believed from the beginning that constructivism was the best approach to teaching mathematics in urban schools supported it as a framework for instruction in general, and it was already a part of their evolving philosophy on teaching. table 1 number of preservice teachers’ with each incoming and outgoing perspective per structured discussion discussion initial perspectives initial no initial undecided initial yes discussion 1 14 (61%) 6 (26%) 3 (13%) outgoing perspectives outgoing no outgoing undecided outgoing yes discussion 2 0 7 (30%) 16 (70%) kalchman changing conceptions journal of urban mathematics education vol. 4, no. 1 88 discussion 2. the second discussion (week 10) also lasted 30 minutes and all 23 students contributed. a total of 32 remarks were coded and attributed to individual students. as recorded in table 2, at the end of the quarter, zero students disagreed with the idea of constructivism as a guide for instructional practice for urban children. thirty percent were undecided and 70% agreed with it. ultimately, 74% of students changed their perspectives. the three students who began the quarter supporting constructivism for urban children remained committed to it. of the original six who were undecided, three remained so. four students moved from not originally believing in the value of constructivism for urban children to being undecided, three went from being undecided to agreeing with the notion, and 10 students went from not agreeing with it to agreeing with it. table 2 number of preservice teachers per reasons for changing perspectives reason for change discussion miel alone miel + fe* miel + cd** miel + cd + fe discussion 2 3 5 2 7 note: fe indicates ―field experience‖; cd indicates ―class discussion.‖ all preservice teachers who reported a change in attitude included the miel assignment as a factor in their final perspective. table 2 itemizes the number of preservice teachers who reported a change in perspective per reason or reasons for change. three cited only their experiences with the miel as homework and as an in-class activity as the main reason for changing their perspective. for instance, a white woman who came into the class believing she knew everything she needed to know mathematically for teaching and only needed instructional strategies said: every week when we did the [miel] in class i would look around and always be sure that my way of solving the problem was the right way and that my classmates would see why. then, every week that just wasn’t true. other people would share their work and have answers that made sense and that other people even understood better than how i did it. i learned so much from that about teaching. i can’t always believe that my way is the right way or the only way. i have to be able to listen to others and be open to how they solve problems otherwise my students won’t relate to me and won’t learn from me. i never thought before about needing to figure out how other people do math, especially kids. i figured if it wasn’t familiar it wasn’t right. two preservice teachers attributed their changes to a combination of the class discussions and the miel. representing this change is the following from a kalchman changing conceptions journal of urban mathematics education vol. 4, no. 1 89 white female student who went from not seeing the value of constructivism for all learners to appreciating it: in the beginning i really tried to think about the types of students that [constructivism] would and would not be suitable for. the more we talked about it though and the more we did each other’s and our own everyday math assignments, i started to think that as long as something is approached in an appropriate manner, it can be suitable for any class, anywhere, any age. five attributed their change to a combination of the miel and their field experiences. a representative remark from these students was the following, spoken by a white man: i am doing my clinical in a 2nd grade classroom at [an urban school]. in the beginning i couldn’t imagine how i would use any of the constructivist stuff we were learning about. students seemed so out of control. then, my cooperating teacher gave me a small group of students struggling with multiplication. i decided to try some of what we talk about in class. i asked them if they ever had to multiply in real life. they said just for homework. then i told them how i had to multiply that morning to know how many counters i needed to bring for them. after that, the students wanted to find times in their lives that they used multiplication. seven preservice teachers attributed their change in perspective to a combination of the miel, their field experiences, and class discussions. for example, the following quote came from a latina student who began the quarter not supporting the idea of constructivism for urban children because she believed they needed much more structure and discipline than she perceived a constructivistinfluenced classroom could provide: when you first asked us what we thought about constructivism for urban students i thought no way. even listening to my classmates who thought it was a good idea didn’t convince me. i’ve done a lot of my hours in urban classrooms and all of my teachers just give worksheets and tell kids to answer the problems how they were just shown. then, i got really interested in [my classmate’s] garden problem. i love gardening but found it harder than i expected to explain my solution even though i have to solve these sorts of problems a lot. i also know the kids just did area and perimeter in math. for my lesson i brought in soil and seeds and asked students if they could figure out how big of a box we need to make an indoor garden on the science table. i had them sketch different possibilities asking them to maximize and minimize the perimeter and the area. they were so into it and didn’t finish the lesson. they asked if they could finish the next day. i’m not sure if they thought they were doing math because they weren’t doing a worksheet, but they were doing math in a big way. kalchman changing conceptions journal of urban mathematics education vol. 4, no. 1 90 results from written work weekly contributions. in the first half of the quarter, seven students brought up the issue of constructivism for urban settings in their weekly contributions—all in the form of a question and coded as undecided. for example, one student who was placed in an inner city school wrote the following: ―should i be trying to use constructivism with the students in my placement? they just stare at me if i ask them what they think rather than giving them an answer or referring them to an example in their textbook like their teacher does. i don’t think they’re ready for it.‖ in the second half of the quarter, however, 17 preservice teachers brought up the topic in their weekly contributions, and 10 of those referred specifically to the miel in some way. these contributions were all coded as ―oy‖ or leaving the quarter with a positive view of constructivism for urban students. eight of these weekly contributions were about planning for an upcoming lesson and wanting to talk about developing ―a constructivist lesson to give kids something different and to challenge them in new ways.‖ seven of those eight referred specifically to somehow wanting to incorporate a ―real-world situation, like we do for our everyday math assignment.‖ the nine remaining weekly contributions were from students who wanted to share a teaching experience in which they tried to use constructivist principles to plan for and implement the lesson. even students whose lessons ―failed‖ in their eyes were mindful of how difficult a new teaching style can be for students and teachers alike. and how it is important to remember that this is a process. i can see that i didn’t like doing math like this in the beginning and it took time to really get why it was helpful and important. i think instruction like this has to be the norm from the very beginning of the year. i don’t think it is at [my urban] school. justifying pedagogical choices. all 23 preservice teachers had to plan a lesson and justify their choice of pedagogy. fifteen of them planned for a lesson based in constructivist theory. the remaining eight did not either because their cooperating teachers wanted them to teach something directly from the class textbook, or because they were afraid to deviate from the teacher’s methods for fear of losing students’ attention and respect. thirteen of the 23 preservice teachers were placed in schools with predominately low-income, african american or latina/o populations. of those 13, eight planned for constructivist-based lessons and five did not for the same reasons just mentioned. the comments that follow are exclusively from those students who were in urban schools. essentially, the preservice teachers who were in urban settings and planned for constructivistinfluenced lessons said they did so because they wanted to experience it as a teacher and not just as a learner, and/or because they wanted to try to motivate kalchman changing conceptions journal of urban mathematics education vol. 4, no. 1 91 what were in their eyes apathetic or disinterested students. for instance, an african american woman shared the following: for my lesson i am going to use constructivism. i am going to ask kids if they’ve ever heard of the word probability and what they think it means and when they think it’s used. then i’m going to give them a real situation where probability is important and see if they can solve the problem. i want to try not to just give them a definition of it and then give them problems to do. i’m not sure how this will go over because i have never heard the teacher ask their opinions about math or to tell her something she hasn’t told them first. i’m nervous but i’m also excited to try this and to give the kids a different experience. as noted, preservice teachers who did not plan for a constructivist-oriented lesson were either given the pages from a textbook to cover, or were anxious about students’ and their cooperating teachers’ reactions to an unfamiliar instructional style. a mature, white woman, wrote: i have been taught throughout my studies that often times when schools and students are not meeting state standards, their teachers feel pressure to teach their students through textbooks and worksheets and drills. with such teaching tactics employed, students are more apt to memorize a topic temporarily than they are to gain any deeper understanding of the subject matter. this is not how i want to end up teaching, but this is how my teacher teaches and what the kids are used to. i am going to do the lesson my teacher gave me and teach how she usually does but also think about how i would change it if i were the actual teacher. final reflective essay. in the final, reflective essay assignment, 13 preservice teachers who admitted to initially believing that urban children were illsuited for a constructivist-influenced pedagogy reconsidered this position after having a placement in an urban classroom and planning for and teaching a lesson in such a setting. this change occurred not only for preservice teachers who were able to teach using constructivist principles but also for those who did not: i wish i had been able to teach about equalities in a way other than what was in the book. the kids weren’t engaged and i had to keep reminding them of which way to put the ―mouth of the alligator.‖ i think that if they had been able to come up with their own ways of remembering how to put the signs and why, the whole lesson would have had a different feel and a different outcome. i wish i had taken more risks with the teacher and the students. twelve of these 13 students also referred to the miel as a significant influence in their changing perspective. for example, early in the quarter many skeptical students cited chronic off-task and recalcitrant behavior in urban children as an impediment to a successful constructivist classroom. four students wrote specifically about this belief and replaced it with the perspective that such traits like kalchman changing conceptions journal of urban mathematics education vol. 4, no. 1 92 ly stem from a disconnect between child and curriculum rather than a demographic identifier. for example, a latina preservice teacher who was undecided in her initial remarks in discussion 1 because she did not believe the students in her urban placement could stay interested in a problem for any prolonged period of time, wrote the following: i learned a lot about the importance of children connecting with what they’re learning. i am at [an inner city] school, and i wish my students could do our weekly math assignment. they just don’t see why they have to learn math. when i saw my classmate’s example of figuring out how far she needed to walk from the [subway] station to the concert in [the] park, i thought that would be a great problem for my students….the school is right there! i see now that teaching from a textbook is boring for these students because it has no purpose. it scares me to think about teaching in the city without a textbook, but now i see how important it is to have the students learn in constructivist ways. a different student, whose initial beliefs were also undecided because of his desire for a textbook that would structure and guide his teaching, wrote the following summarizing what he had learned about teaching and learning elementary mathematics: i never thought that ―real-world‖ problems needed to be changed for different students. the ―real world‖ is the real world, isn’t it? ...it never occurred to me that word problems used in textbooks could be so far removed from so many students’ lives. for the first time, i see why constructivism, and having students use their own situations, strategies and problems, would actually be better than using a textbook. everybody has a different reality. discussion the data reported here describe the constructivist teaching and learning experiences one group of preservice teachers had with the miel assignments and how they interacted with them and other aspects of the methods course to change their attitudes and perceptions about the pedagogical needs of urban elementary mathematics students. however, i see the present findings as reflecting the sort of learning paths preservice teachers consistently report and demonstrate from quarter to quarter. this finding is not surprising given that the biases preservice teachers brought to the methods course were consistent with what literature tells us more generally about the beliefs of teachers in urban classrooms. preservice teachers who were initially either undecided or skeptical about the value of constructivism for urban classrooms held many of the same conceptions research has found practicing teachers to have. for example, some preservice teachers had the incoming belief that low-income, urban children do not want to learn. this attitude reflects the literature showing that teachers have low expec kalchman changing conceptions journal of urban mathematics education vol. 4, no. 1 93 tations of urban students, which typically results in a transmission model of pedagogy that emphasizes basic skills (zeichner, 1996). furthermore, many of the preservice teachers’ incoming perspectives focused on their beliefs that urban children are difficult to control, present intimidating behavioral challenges, and require more structure than a reform-based classroom can provide. this preconception aligns with the finding that some teachers expect african american students to be harder to control and in need of more restraint in the classroom (ladson-billings, 1994). it seems likely that without some sort of intervention in their teacher education program, these preservice teachers could be bound for perpetuating a ―pedagogy of poverty‖ (haberman, 1991). most preservice teachers in this project acknowledged that doing the miel weekly led to a reassessment of their perspectives on urban mathematics education. some spoke about the miel as being the only truly constructivist-learning opportunity they had that was not a contrived classroom experience meant to model the sort of reform pedagogy they are encouraged to use in their own teaching. they shared their initial frustrations and ultimate appreciation for the struggles they had in finding their own learning paths and constructing their own understandings not only about mathematics but also about pedagogy. these authentic experiences seemed to highlight and instantiate many of the pedagogical impediments urban teachers and students routinely face. for example, one paradigm for change was reconsidering the role of a textbook for urban students’ mathematics education. many of the preservice teachers came in assuming that a textbook would provide the scope and sequence for their curriculum and that their role in planning for content and pedagogy would be minimal. however, for many of these students, doing the miels and developing confidence with doing, owning, and explaining mathematics was transformative. thus, they came to appreciate the importance of all children connecting with curriculum in personal and authentic ways. this epiphany was relevant to preservice teachers’ attitudes toward teaching children of all demographics. however, it was especially true for their attitudes toward teaching urban children, who most often do not relate to the traditional examples, analogies, and artifacts found in mainstream textbooks (ninnes, 2000). this revelation about the need for children to experience a curriculum in authentic ways also had an effect on preservice teachers’ initial belief that urban children are far too recalcitrant and difficult to thrive in a constructivist learning environment. this seemed to come about because many of the preservice teachers shared that they initially felt ―marginalized‖ from the methods course curriculum, including the miel assignment, because it was not what they expected and they were frustrated by not being told how and what to do. however, as time went on they began to take ownership of and responsibility for their learning and their teaching by doing their own and their classmates’ miels from week to week. kalchman changing conceptions journal of urban mathematics education vol. 4, no. 1 94 consequently, they came to understand why there would be such apathy and resistance among disenfranchised youth who generally do not connect with a textbook curriculum. that feeling of frustration and resistance would never go away. thus, constructivism became, as one student put it, ―not only the preferred pedagogy, but the essential one‖ for ―at-risk‖ populations. concluding thoughts in light of research literature pointing to the contrasting quality of education accessible to students from a range of social classes, communities, and cultures, misconceptions about reform-minded teaching and its suitability for urban mathematics classrooms need to be addressed in teacher preparation programs. although literature, in-class learning opportunities, and field experiences are critical to this education, a more concrete and personal relationship with it in general and urban children’s authentic mathematical environments in particular need to be facilitated. here, a weekly assignment asking students to share, discuss, and explain their day-to-day encounters with mathematics appeared to be influential in fostering at least a rudimentary change in preservice teachers’ beliefs about the value of constructivism as a framework for reform in urban elementary mathematics classrooms. most of the preservice teachers discussed here made a conceptual shift and came to appreciate that a student population does not frame or limit the efficacy of or need for reform pedagogy. rather, it is a teacher’s commitment to providing an environment that supports rich, conceptual learning opportunities and a direct connection to the mathematics explored that determines a successful school mathematics program. consequently, they ultimately expressed informed opinions about coming to see constructivism as a model for how people learn and develop regardless of race, class, gender, domicile, or any other demographic descriptor. in effect, the experience of being a constructivist learner in the context of the miels was significant for recognizing the value and impact of constructivism for all: ―using the weekly math discussions to examine how i looked at math was very helpful. 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(1996). educating teachers to close the achievement gap: issues of pedagogy, knowledge, and teacher preparation. in b. williams (ed.). closing the achievement gap: a vision for changing beliefs and practices (pp, 56–76). alexandria, va: association for supervision, curriculum and development. journal of urban mathematics education december 2012, vol. 5, no. 2, pp.55–86 ©jume. http://education.gsu.edu/jume dionne i. cross is an assistant professor in the department of curriculum and instruction, indiana university, 201 n rose avenue, bloomington in; email: dicross@indiana.edu. her research interests include examining the psychological constructs that impact teachers’ classroom practices. corresponding autor rick a. hudson is an assistant professor at the university of southern indiana, evansville, in; email: rahudson@usi.edu. his research interests include the teaching and learning of data analysis and statistics and issues in mathematics teacher education. olufunke adefope is an assistant professor at georgia southern university, statesboro, ga; email: oadefope@georgiasouthern.edu. her research interests include equity in mathematical education and students’ geometric reasoning. success made probable: creating equitable mathematical experiences through project-based learning dionne i. cross indiana university bloomington rick a. hudson university of southern indiana olufunke adefope georgia southern university mi yeon lee indiana university bloomington lauren rapacki indiana university bloomington arnulfo perez indiana university bloomington in this article, the authors describe a 16-hour project-based learning statistics unit designed for and implemented with elementary-aged, african american children. the unit was designed to provide the children with mathematical learning experiences that allowed them to make personal sense of mathematics or to use mathematics to critique and analyze issues within their communities or in the wider society. the authors worked with thirteen, elementary-aged, african american girls to address an authentic, school-based problem; four dimensions of equity—power, access, identity, and achievement—were used as a lens to examine the quality of the project and the impact of the mathematical experiences on the students. keywords: equitable mathematics teaching and learning, project-based learning, statistics education, urban education espite the high percentage of african american and latina/o students in the u.s. education system, too many african american students struggle to meet grade-level competence in core academic subjects (e.g., mathematics, science, reading, and social studies) (howard, 2003). identifying and understanding the factors that contribute to low educational achievement for students of color has been the focus for a significant portion of educational research in the past d cross et al. success made probable journal of urban mathematics education vol. 5, no. 2 56 decade (e.g., johnson, 1984; ladson-billings, 1997; lee, 2002; national center for educational statistics, 2009; secada, 1992). the high rates of underachievement for students of color in general have pushed many educators to rethink their approaches to schooling to achieve the goal of academic success for all students. with the publication of the national council of teachers of mathematics’ (nctm) principles and standards for school mathematics (2000), where the organization advocated quality mathematics education for all, attention to equity and diversity issues became more prominent. equity in this regard means that every child, irrespective of race, socio-economic status, or personal characteristics, should be afforded worthwhile and meaningful mathematical experiences. embodied in this definition is the view that if students are treated competently, they will achieve high levels of competence (ladson-billings, 1994). with this view in mind, we designed and engaged a group of elementaryaged, african american girls in a 16-hour, project-based learning (pbl) statistics unit. we were confident the children would be able to learn complex statistical concepts if the unit built on the students’ cultural knowledge and lived experiences (civil, 2007; ladson-billings, 1994, 1997), and presented opportunities to use these ideas to reason about and solve problems in their world (gutstein, 2007). here, we describe how we designed and implemented the unit; the following questions guided our efforts: 1. what happens to students’ “mathematics learning” when taught in “mathematically meaningful” ways? 2. how can students develop “mathematical power,” and simultaneously, use mathematics as an analytical tool with which to investigate problems that are personally meaningful to them? 3. what difference might such efforts make in the lives of students and also in the larger society, in both the shortand long-term? in the following sections, we explain the theoretical grounding of our work and describe our data sources and analytical approach. we organize the discussion around mi yeon lee is a graduate student in mathematics education at indiana university bloomington; email: miyelee@indiana.edu. her research interests include k–8 students’ geometric and informal algebraic thinking and pre-service teachers’ understanding of students’ reasoning. lauren rapacki is a graduate student in mathematics education at indiana university bloomington; email: lrapacki@indiana.edu. her research interests include the development of elementary mathematics specialists/coaches, the use of technology in mathematics education, and pre-service and in-service teacher efficacy and pedagogical content knowledge. arnulfo perez is a graduate student in mathematics education at indiana university bloomington; email: perez4@indiana.edu. his research interests include the analysis of national math achievement scores and factors that support successful student transitions from middle school to high school. https://www.exchange.iu.edu/owa/redir.aspx?c=865174ca62e147228cf8f939fd22bb68&url=mailto%3amiyelee%40indiana.edu mailto:lrapacki@indiana.edu mailto:perez4@indiana.edu cross et al. success made probable journal of urban mathematics education vol. 5, no. 2 57 gutiérrez’s (2009) four dimensions of equity—power, access, identity, and achievement—and conclude with a discussion addressing the questions that guided our work. framing our approach equity embodies the notion that all students irrespective of gender, race, class, or socio-economic background can learn and should have opportunities for a high-quality education. across proposed frameworks that incorporate this construct, a common thread of meaning is that equity involves fair distribution and access to the physical, intellectual, and technological resources that contribute to learning. both fennema (1993) and allexsaht-snider and hart (2001) defined equity in terms of (a) distribution of resources to schools, students, and teachers, (b) quality of instruction, and (c) outcomes for students. in addition to curriculum, teaching, and assessment, other researchers have argued that in establishing goals for equity in mathematics education, we must attend to the social, economic, and political issues that impact what happens in schools and classrooms (e.g., apple, 1992; tate, 1997). nctm’s (2000) principles and standards for school mathematics addresses many of these criticisms, yet there are still calls for specific guidelines regarding how these ideas can be best translated into district and school policies (allexsaht-snider & hart, 2001). although these equity perspectives attend to some of the concerns posed by apple (1992) and tate and rousseau (2002), they mainly attend to the physical and quantifiable aspects—specifically, external resources and achievement outcomes. examining equity issues through an achievement lens tends to ignore the interpersonal and societal factors pivotal in discussions about the education of children. primarily it minimizes the importance of power and identity, overlooks deficiencies in measurement instruments, and provides a limited perspective of the student as an individual and solely in comparison to the dominant group (gutiérrez, 2009). we therefore broadened our lens to include issues that were central to the individual. as such, our work reflects an approach that addresses both the internal (psychological) and external (physical, technical, quantitative) aspects of equity. we use gutiérrez’s (2009) four dimensions of equity—power, access, identity, and achievement—to frame our discussion about the unit we designed and how we evaluated its ability to support the development of the statistical literacy of elementary-aged, african american children. power the power dimension reflects the degree to which the curriculum and classroom instruction enable students to use mathematics to consider, analyze, and cri cross et al. success made probable journal of urban mathematics education vol. 5, no. 2 58 tique societal structures and the injustices embedded within these structures. it encompasses how the classroom is organized, who gets to speak, who gets heard, and how students are provided with opportunities to examine and critique their world. gutiérrez (2002) situates the discussion of power within the tensions of a dominant vs. critical mathematics. the former relates to mathematics aligned with the status quo in society, reflecting a western colonial perspective. in contrast, critical mathematics encompasses mathematics that attends to the idea that students are members of a society organized around power structures and systems of domination. it acknowledges the importance of students’ cultural identities and “builds a mathematics around them in such ways that doing mathematics necessarily takes up social and political issues in society, especially highlighting the perspectives of marginalized groups” (p. 151, emphasis added). therefore, adopting a critical mathematics perspective would mean challenging society’s established power structures and using mathematics to critique and transform oppressive structures (gutstein, 2006). power also encompasses learning mathematics in ways that are culturally relevant. ladson-billings (1994, 2001) proposed a culturally relevant pedagogy that advocates producing academically successful students who are both sociopolitically and -culturally competent. essential to teaching in this way is acknowledging that students’ identities are shaped by sociocultural and sociohistorical factors and that the cultural knowledge brought to the classroom can be leveraged to enrich their learning experiences. enrichment in the mathematics context goes beyond developing deep conceptual understandings of mathematical ideas; enriching experiences enable the student to be critical of the content they are learning and challenge them to use this content to transform the world they live in (ladson-billings, 1994, 2001; gutstein, 2007). gutstein’s (2007) pedagogy of questioning also embodies the tenets of this dimension as this instructional approach empowers students. they make mathematics relevant, interesting, and meaningful to them as the questions they pose drive the instruction and the learning (boaler, 2008). the issues they investigate using mathematics as an analytical tool allow students to better understand their own life experiences within the broader sociopolitical context (dime, 2007). this dimension involves allowing students to set personal goals with regard to mathematics and providing the knowledge, context, and support needed to actualize these goals. allowing students to have a voice and to be involved in making decisions that will impact the mathematics they learn and their environment captures the social transformation described in the power dimension. access in addition to developing strong reasoning and problem solving skills, students should have mathematical experiences where they can see the beauty of cross et al. success made probable journal of urban mathematics education vol. 5, no. 2 59 mathematics and appreciate its complexity. these goals align with the vision detailed in the principles and standards for school mathematics (nctm, 2000). an integral part of enacting this vision is the six principles; one of which is equity, stating: all students, regardless of their personal characteristics, backgrounds, or physical challenges, must have opportunities to study—and support to learn—mathematics. this does not mean that every student should be treated the same. but all students need access each year they are in school to a coherent, challenging mathematics curriculum that is taught by competent and well-supported mathematics teachers. (p. 2) access means that students should have the necessary resources to develop a broad mathematical knowledge base and the reasoning skills to apply that knowledge appropriately. these essential resources include a rigorous curriculum, high-quality teachers that can implement the curriculum well, a learning environment that invites and sustains engagement and a school infrastructure that supports learning outside of class and beyond school hours (gutiérrez, 2009). although, we contend that all these resources must be made available to all students, we appreciate that equal access may not mean the “same” access, so the type and quantity of resources may vary given the population and the individual student. access to the same resources would only be equitable if in the past, universally and historically, all students had the similar opportunities. in particular, african american and latina/o students tend to reside in communities that are segregated and have experienced long-term educational injustices. in this regard, given the limited educational opportunities these students may have experienced in the past, having the same resources as other students with greater access to educational opportunities would in itself continue to perpetuate academic disparities. therefore, access as it relates to equity must attend to the specific needs of the community and make available the types and quantity of resources necessary for those members of the community to achieve success (boaler, 2008; tate, 1997). identity students’ abilities to negotiate between who they perceive themselves to be, how they are perceived by others, and who they want to become tend to affect their participation and engagement in educational activities broadly (gutiérrez, 2009; cobb & hodge, 2002), and more specifically within the mathematics classroom (martin, 2006). therefore, identity becomes an important construct to consider in discussions about the participation and achievement of african american students in mathematics. identity has both an individual and social dimension; it is shaped by cultural factors and social processes (martin, 2006) but also encom cross et al. success made probable journal of urban mathematics education vol. 5, no. 2 60 passes the individual’s perception of him or herself and how the individual evaluates him or herself in relation to others (roeser, peck & nasir, 2006). examining the relationship between identity and students’ connections and associations with school has been quite revealing about the academic achievement of students, particularly minority students (nasir, 2002; cobb, gresalfi & hodge, 2009; martin, 2007). research has shown that students who identify strongly with school tend to have higher academic achievement and remain in school longer, often pursuing post-secondary studies (dolby & dimitriadis, 2004). for many african american students, participating meaningfully within the educational context involves connecting their cultural identities with their classroom experiences (nasir, 2002). this connecting is often difficult because of the sociocultural and sociohistorical factors that impact their lives (boaler & greeno, 2000; gutstein & rogoff, 2003; stinson, 2009). these factors tend to shape how students see themselves as participants in mathematics, influence the extent to which they have developed a commitment to, and have come to see value in mathematics as it is presented in the classroom (stinson, 2009). based on “master-narratives” of what it means to be and the expectations of black students in the classroom, some african american students often see themselves as inferior to whites and asian americans, and see failure to attain mathematical (and more broadly academic) success as the norm and what society expects of them (martin, 2007). given these prevailing negative stereotypes of african american youth and the likely impact on their academic identity, to successfully teach this group of students, teachers must draw on their cultural knowledge (dominguez, 2011; matthews, 2003), providing a bridge between their cultural identity and the normative classroom identity, thereby increasing both students’ learning and engagement (ladson-billings, 1994; nasir, 2002). achievement in addition to attending to the aforementioned components of equitable mathematics, it is necessary to give an account of student outcomes. we define achievement broadly, to include scores on standardized tests and other types of assessments (e.g., non-traditional, performance-based) that measure conceptual growth, critical thinking, and reasoning. moreover, and particularly important in defining achievement through an equity lens, is extending the notion of achievement to include an analysis of the relation between students’ ways of participating in mathematics and the norms and practices of the mathematics classroom (cobb & hodge, 2002; gutiérrez & rogoff, 2003). in acknowledging that students’ ways of participating may be influenced by norms of engagement in non-school activities (civil, 2007; lubienski, 2002), we were pushed to find ways of examining students’ talk, reasoning, and justifications in ways that value their contributions whether or not they aligned with the normative practices of the classroom. cross et al. success made probable journal of urban mathematics education vol. 5, no. 2 61 we also incorporated into our definition of achievement students’ competencies in mathecracy, a term introduced by d’ambrosio (1999) and elaborated by gutiérrez (2002), which refers to the ability to read data, draw conclusions from data and calculations, and propose hypotheses. we found it necessary to include these set of competencies within our criteria for assessing achievement for two reasons: (a) it captured the objectives of the instructional unit we designed, and (b) it targeted the knowledge students need to be functional citizens. methodology participants the participants were 13 african american girls in grades 4–6 from a gendered elementary school called brayton elementary (a pseudonym, as are all proper names) who were enrolled in a school-based summer camp. brayton is situated in the metropolitan area of a large midwestern city. similar to other schools located in urban areas, brayton is a school community surrounded with high levels of poverty and crime. the school has a disproportionate number of students who identify as african american (99% african american and 1% multi-racial) and qualify for free or reduced lunch (88 % of the student population qualify for free or reduced lunch). over the past decade students in grades 4-6 in the wider school district consistently scored below the state average according to the state’s standardized test for mathematics. as a part of the district’s response to restructure and reform public schools, brayton had recently transitioned to a singlegender school for girls. brayton’s test scores had been on the rise for the past few years having made adequate yearly progress (ayp) in the year prior to the project. the summer camp was held at the school as an academic alternative for students required to attend summer school. sessions were held for four hours over the course of one week with the content focus being mathematics. although some teachers at the school stopped by and observed the students periodically, none were integrally involved in facilitating sessions. the research team included six researchers (five graduate students and a teacher educator). during the academic year preceding the summer camp, we conducted professional development at the school; therefore, we were aware of the kinds of educational opportunities available to the students. the objectives of the summer camp were to engage available students in authentic statistical inquiry and to help students develop 21st century skills (described below). members of the research team also served as the instructors for the summer camp, which was held at the school. the project: food for thought according to the buck institute for education (www.bie.org), project-based http://www.bie.org/ cross et al. success made probable journal of urban mathematics education vol. 5, no. 2 62 learning (pbl) affords students a lengthened and supported process of inquiry and construction, which occurs in response to a complex question, problem, or challenge posed or encountered. the summer camp focused on the students completing a project-based statistics unit that was aimed at developing students’ knowledge of sampling and surveying, measures of central tendency, dispersion, and use of data representations to make statistical arguments. in a conversation about how students can best prepare for learning, students made suggestions that included doing their homework, eating a filling breakfast, and being prepared with necessary materials. following the discussion, students read a newspaper article about a school that had improved test scores after ensuring that all students ate breakfast prior to taking a standardized test. drawing on this article, the students suggested that more students might eat breakfast at the school if they could select the breakfast the school would serve on the mornings prior to taking the standardized tests. given this information, the students investigated the following authentic student-generated question: assuming students who eat breakfast perform better on standardized tests, what should our school serve prior to standardized tests to ensure that the most students will eat breakfast? to investigate this question, the students brainstormed possible breakfast options and narrowed the list to four choices. the students designed surveys to measure other students’ opinions about the four options and used these surveys to collect data from their peers. using tinkerplots 1 software, they analyzed the data and created representations to support their recommendations. finally, the students presented their findings to representatives of the school administration. throughout the investigation, we planned instructional activities to foster understanding of particular concepts. consistent with other project-based units (e.g., barron et al., 1998), these activities were integrated into the unit as the need arose. for example, prior to analyzing the data, we introduced the concept of mean, ensuring that students had both a procedural and conceptual basis prior to engaging in analyses. we used activities designed to elicit a conception that the mean could be found by “leveling” data values (cai, 1998). projects such as ours allow for students’ autonomy and agency and are not solely about helping students learn key academic content. they also have to learn to work as a team and contribute to a group effort. they must listen to others, make their own ideas clear when speaking, be able to read a variety of material, write or otherwise express themselves in various modes, and make effective presentations. these skills, competencies, and habits of mind are often referred to as 21st century skills. in addition to expanding students’ knowledge of statistics, developing 21st century skills was an instructional goal. 1 tinkerplots is a dynamic, data analysis software program primarily designed for students in grades 4–8 ; see http://www.keypress.com/x5715.xml for more information. http://www.keypress.com/x5715.xml cross et al. success made probable journal of urban mathematics education vol. 5, no. 2 63 data collection mathematics interviews. all participants were interviewed. initial interviews were conducted on the first day of the camp and focused on students’ views and knowledge of elementary mathematics broadly and statistics specifically. the second interview was conducted on the last day of the camp and focused on the students’ learning experiences and their statistical knowledge. elementary level students often have difficulty expressing their thoughts and reasoning in writing, so these interviews were integral for the following reasons: (a) to determine students’ level of reasoning prior to the implementation of the project, and (b) to identify the ways their reasoning developed throughout the project. interviews were conducted in groups of two or three and lasted about 15 minutes on average. all interviews were video recorded. video-recordings. all classroom activities included in the statistics unit were videotaped. three cameras were used; each focused on one group of students for the entirety of each activity. during whole-class conversations, one of the group’s cameras was refocused on the facilitator to capture teaching behavior that engaged the students and supported their reasoning. students’ work from all the activities and photographs of the students engaged in activity were collected. analysis given our theoretical framework, we examined the two primary data sources (i.e., interviews and the video-recordings) to locate student statements and classroom episodes that provided evidence that the design and implementation of the unit aligned with gutiérrez’s (2009) four dimensions of equity—power, access, identity, and achievement. our first round of analyses occurred immediately after the camp. the research team assembled and discussed our initial ideas about the level of success of the pbl unit. we identified specific aspects of the unit’s design and implementation that aligned with our equity framework and what elements may have thwarted our goal. we documented these ideas in short descriptive narratives and organized them into four groups with respect to how well they aligned with our descriptions of the four dimensions of equity. each of the graduate students (co-authors) selected one dimension and examined the videos (approximately 48 hours of video) and other relevant data (e.g., project documents, classroom artifacts) to identify episodes that supported or refuted our initial ideas and thoughts about that dimension. the teacher educator (the first author) independently engaged in a similar process but with all four dimensions in mind. for each dimension, video episodes and other relevant data were discussed between the two team members assigned to that dimension to determine the degree of alignment of their interpretations. where there was disagreement, the episodes were discussed until we achieved consensus. cross et al. success made probable journal of urban mathematics education vol. 5, no. 2 64 evaluating success across the four dimensions applying the equity framework previously described, we examined the results of our pbl intervention through gutiérrez’s (2009) four dimensions of equity: access, power, identity, and achievement. access to ensure that all students received equitable mathematics education, we considered the qualitative aspect of access by examining the quality of the learning opportunities afforded to students. in light of this, we analyzed access from two perspectives: the quantity of resources available and the quality of opportunities provided. quantity of resources. “good” mathematics instruction requires rigorous curricula, highly qualified instructors, and the physical (cognitive tools) and technological resources to enhance learning. keeping in mind that equal opportunities to learn do not necessarily mean the same opportunities for all students (gutiérrez, 2002), we considered the broader sociohistorical context in which the students were situated. based on our pre-implementation project observations, students were primarily taught with a focus on memorization and repetition, rather than problem solving and reasoning. given the type of educational experiences the students had prior to the implementation, we exaggerated the quantity of resources available to the students. first, there were five instructors for thirteen students so the student-teacher ratio was very low. second, all five instructors were mathematics educators. in addition, prior to starting this project, the instructors specifically focused on developing expertise in common statistical knowledge and specialized knowledge for teaching statistics. we examined and discussed in depth the fundamental ideas in elementary statistics and the literature on statistics teaching and learning to ensure that the students had meaningful mathematical experiences. third, all students were taught to use technology in ways that would support their learning. to facilitate sustained engagement, students were paired during computer use. working collaboratively during computer use provides a better learning situation for girls; they tend to be more social so there is increased enjoyment and benefit because of the collaboration (doerr & zangor, 2000; underwood, 1994). quality of opportunities. our view of quality includes access to resources and experiences that we consider academically beneficial for the students and that serve the interests of individuals and their immediate community (martin, 2011). as such, we designed the curriculum based on research findings on the positive impact of pbl (krajcik & blumenfeld, 2006) and inquiry-based activities (boaler, 2008) on students’ learning. with regard to quality, we designed the project and sequenced the activities so students’ statistical experiences would be meaningful. cross et al. success made probable journal of urban mathematics education vol. 5, no. 2 65 we wanted the project and related activities to have four main components. it should: (a) be designed to address an authentic problem, (b) be embedded within a culturally relevant scenario, (c) include technology to enhance learning, and (d) engage students in the statistical investigation cycle. for the first component, the context of the pbl unit was a real situation that was relevant to the students’ lives—improving students’ test scores. the students decided on a reasonable approach to solving the problem, which was to ensure that all students ate well before the test. because the problem was relevant to them, they were motivated to solve it, thereby increasing engagement. through whole-class discussion, students completed the first two stages of the statistical investigation cycle—defining the problem and creating a plan. initial suggestions were revised based on students’ realizations that other students’ opinions must be considered. in the process of refining the ideas, students recognized the importance of selecting a sample and administering a survey to determine breakfast options that were optimal for everyone. designing activities where students were invested in the outcomes motivated students to participate (gal, 1998; groth, 2006; nicol & crespo, 2005). the following excerpt provides an example: instructor: we need to narrow down what options people really don’t like. what do you think people really don’t want to eat for breakfast? student: a peanut butter sandwich. instructor: you don’t think peanut butter will get many votes. do you agree or disagree? students: (almost all students said aloud) i agree. students: (simultaneously) i don’t want toast. students: (simultaneously) i hate bagels. students: (other students said loudly) oatmeal instructor: why do you think we should erase oatmeal? students: because i hate it. instructor: do you think many people agree with you or disagree with you? students: agree. students: disagree. later in the discussion… instructor: what do we need to do in order to convince her [the principal] that option one or option three is the best? how would you advise her? student: we should figure out which one of the options is the best. student: collect the data. instructor: what do you mean by that? student: how many people like this and how many people like that? which one is the one that most people like? cross et al. success made probable journal of urban mathematics education vol. 5, no. 2 66 for the second component, the activities embedded in the project targeted important statistical ideas such as sampling, surveying, distribution, data representations, and measures of center. 2 several of these activities were based on scenarios that addressed social and political issues relevant to the african american community and to which the students could connect. the activities included people, such as michael jackson and barack obama, who is the first black president of the united states about whom the students were quite knowledgeable (see figure 1). one activity was embedded within current events of the time related to the bp oil spill. students were able to realize the importance of reasonable sampling by connecting the context with experiences within their own family. figure 1 sampling task. for the third component, students used tinkerplots to expand their understandings of statistics. the goal of including the technology was to provide students with opportunities to analyze data and reason with the statistical tools without the tedium of paper-and-pencil calculations, hopefully increasing the strength of their arguments. figure 2 provides a screenshot from tinkerplots. the figure shows how students used the tool to create multiple representations of the data, 2 also referred to as measures of central tendency. over half of 45,000 michael jackson fans who have voted in a music-focused website’s world-exclusive “death hoax poll” say the king of pop did, in fact, fake his death of a heart attack and is still alive today. visitors to the website were asked to sign up for an account (at a charge of $2.50) to give their opinion about whether or not michael jackson was really dead. it turned out that 56% of the callers felt that michael jackson was alive. 1. identify the population of interest and the sample actually used to study that population in this example. a. population: b. sample: 2. do you think that 56% is an accurate reflection of beliefs of all americans on this issue? if not, identify some of the flaws in the sampling method and suggest how it could have been improved. cross et al. success made probable journal of urban mathematics education vol. 5, no. 2 67 including interpreting the data as individual case cards (upper left), within tables (upper right), and as graphs (below). tinkerplots provided an opportunity for students to organize and compare datasets efficiently. for example, they used the software to visually inspect the data, calculate means, compute percentages, and graph statistical results. through using the tool, students were given more access to instruction that focused on conceptual development, rather than on computation or tedious tasks, such as creating graphs by hand. figure 2 sample student exploration of data using tinkerplots. in the following dialogue, a pair of students analyzed data collected from the survey using the tinkerplots program. to enter the data in tinkerplots, they had to discuss appropriate ways to code categorical data. additionally, the program provided percentages and values for mean, mode and median (see figure 2), so instead of spending time manually calculating these values, students discussed what these values told them about the data. specifically, they used percentages cross et al. success made probable journal of urban mathematics education vol. 5, no. 2 68 and the means to decide which options to recommend for breakfast on the day of the test. in this way, access to tinkerplots was useful for students to efficiently analyze the data. in the excerpt below, two students used tinkerplots to organize, analyze (using percentages and means), and interpret the breakfast data (making decisions about which breakfast option is best). during their discussion, one of students, moriah commented on the average of the menu item grilled cheese sandwich, 2.493, as shown in figure 2. kaycia: what are you supposed to be doing? moriah: figuring out which one had the most appeal? kaycia: let me do something. [kaycia plotted the stacks of preferences according to the options available by moving plots and adding attributes to the two axes of the tinkerplots graph.] moriah: (trying to figure out how to code the data) which one is preferred; which one is not preferred? kaycia: um…oh, wait. sort them in order of greatest preference; we use 1, 2, 3, 4, 5. where 1 is the most preferred, 2 is simply preferred, 3 is unsure, 4 is disliked, 5 is strongly disliked. moriah: this is the preference for waffles. 87 people really liked it. instructor: [teacher reconvened students to explain how to use “average” icon in tinkerplots] okay. stop everybody. what tool can be used to help analyze the data? kaycia: mean, average, mode, and range instructor: right. sometimes statisticians calculate the mean by computer. at the top, can you see a bluish, purplish triangle? if you click on that triangle, it tells you the mean. kaycia: (dragging various attributes into the horizontal axis of tinkerplots graph) the average of “waffles and sausage” is 4.036, and this (average) is 3.645. moriah: let’s go to french toast. it is 3.987. let’s try grilled cheese. it’s 2.493. which one is the most do you think? kaycia: actually “waffles and sausage” is the most. it’s 64% and its mean is 4.036. moriah: right. the highest one will be first choice. for the fourth component, we engaged students in the practices of statisticians by designing activities that corresponded to the statistical investigation cycle. these steps included identifying the problem and the research questions (which breakfast foods would appeal to the most students), planning the procedures to collect data (deciding on a suitable sample and an appropriate survey), engaging in the data collection process (administering surveys to fellow students), analyzing the data (using statistical tools within tinkerplots) and drawing conclusions (determining which breakfast food most students would eat). they were not restricted to data analysis, which is usually foregrounded in a traditional elementary cross et al. success made probable journal of urban mathematics education vol. 5, no. 2 69 statistics unit. engaging in a curriculum designed around the statistical investigation cycle allowed students to enact the practices of statisticians and provided mathematical experiences that were practical and relevant (to the problem being addressed). for example, after defining the problem, in thinking about data collection instruments and procedures, students were not just told to use a survey with a likert scale; instead, the activity was designed to enable students to think about the advantages and disadvantages of various survey types (see appendix a). below is an excerpt from the discussion about which survey would be best to use: instructor: why do you say survey b? student: because you can know how they really feel about it. instructor: what’s our goal again? what do we want to know? student: what people really like. instructor: we want to know which the best breakfast is. what survey do you like best? student: i like the second one best…because it tells us what is strongly disliked, disliked, undecided, liked, or really liked. with scaffolding by the instructor, students reasoned about and discussed the different surveys to decide why using a likert scale was most appropriate given the data they wanted to collect. power gutiérrez (2009) states, “equity is ultimately about the distribution of power—power in the classroom, power in future schooling, power in one’s everyday life, and power in a global society” (p. 5). below we describe how we tried to position and self-empower the students to be change-makers. establishing voice. classroom power relations can be characterized in many ways, such as who controls the conversation, who determines the mathematical accuracy of statements, and who determines how the lesson unfolds. we deliberately planned the activities and orchestrated classroom conversations so students would feel a sense of ownership over the project from inception. for example, on day one of the summer camp we engaged the students in a conversation about what ideal conditions would be for taking a test in order to maximize students’ scores. suggestions were elicited from the students, one of which was that eating a hearty breakfast on the morning of the exam would increase the students’ scores. the students determined that if others liked the breakfast then most would eat it. in the excerpt below, we see the students making decisions about how to maximize scores on the test and selecting breakfast options. instructor: so what are some things you would do on the day of the test? student: think hard. cross et al. success made probable journal of urban mathematics education vol. 5, no. 2 70 instructor: what are some of the things that make you think hard? if i say think hard, is it like a magic button? student: eating breakfast. instructor: so to think hard you would need to eat breakfast. [students provided additional responses including go to bed early, get enough sleep, relax, concentrate, etc.] [5 minutes] instructor: so what are some of the foods you would like to eat that you think would help you to think hard on the test? … and is really good and a lot of people eat it? ... [students’ thinking] students: waffles…eggs…bacon…sausage…orange juice…grits…oatmeal…biscuits…grilled cheese…french toast…soup…pancakes later in the discussion the students narrowed these options down to four 3 based on what they thought were the most popular breakfast items among their peers. from this initial conversation the students knew that their ideas and thoughts were heard and that they had a powerful voice in this classroom. these early conversations established the students’ voices as primary in decisionmaking, allowed students to take ownership of the task, and set the tone for the rest of the project. later, with scaffolding, the students identified their target population, all the students at brayton elementary school taking the state exam, and their sample, all summer school students at brayton. in order to collect the data needed to accurately represent their population, the students developed a survey with the food items discussed in the group discussions. we also organized the classroom to facilitate personal interactions and communication among students. over the duration of the summer camp we saw the girls gain confidence in stating their ideas and independently providing justifications for their responses supported by data. we attributed these changes to deliberate efforts on our part to ensure that the students had a clear and powerful voice within the classroom. we ensured that all major decision-making aspects of the project were driven by the students and incorporated the choices they made. controlling decision-making. statistics educators have developed theories about statistical thinking and the cycle of investigation in statistics (wild & pfannkuch, 1999). these frameworks suggest that statisticians make many choices in the process of conducting a statistical investigation, such as how populations 3 the fours options were: (a) waffles and sausage, (b) french toast and sausage, (c) grilled cheese and soup, and (d) grits, eggs, biscuits, and bacon (the options will be referenced by the first food item from here on). cross et al. success made probable journal of urban mathematics education vol. 5, no. 2 71 will be sampled, what instruments will be used to collect data, what measures and representations will be used to analyze the data, and how the results will be disseminated. throughout the project, we provided opportunities for the students to make these choices themselves. in particular, students made suggestions about possible breakfast items, and with scaffolding, students determined the appropriate survey to use to gather data. one of the primary goals of the unit was to have the students identify a single breakfast option to recommend to the principal. after conducting the analyses, the students determined that the waffles and french toast options had the highest mean. further analyses showed that a higher percentage of students chose “like” or “really like” for waffles (78%) than french toast (76%). thus, the students and instructors initially determined to make waffles the breakfast of choice. this decision was reconsidered following kaycia’s objection. below she describes why she thought another recommendation was viable: kaycia: well, we realized that the most choices were the waffles and the french toast, and you know, not a lot of people like waffles. i know i don’t eat waffles…so we were thinking. why don’t they have both waffles and french toast out there…that way, whoever wants which could just grab them and go. instructor: well, that’s a recommendation that you could have, right? kaycia: those are the highest [percentages of the four options], and they’re almost the same. instructor: then, it’s hard to distinguish between… kaycia: [nodding her head in agreement] yeah, so we should give what the students really, really like. that way, half the kids won’t be eating cause they didn’t like the food. in this dialogue, we see that kaycia was likely motivated by her personal opinion about waffles; however, she also acknowledged that both options had similar data. rather than suggest that only one option was viable, the instructor suggested to kaycia that she could include both options as part of the students’ recommendation, which she supported with data showing that the percentage of students that liked both breakfast items was very close. in such cases, the students were empowered to make their own recommendations and adapt these recommendations when necessary, granted they could support their decisions with data. of significance here is not that the students were allowed to make the decisions but they had acquired the necessary statistical knowledge and skills to consider the situation and make informed decisions that would impact their school community. identity developing identity in this framework relates to optimizing students’ cultural and linguistic experiences when partaking in mathematics teaching and learning cross et al. success made probable journal of urban mathematics education vol. 5, no. 2 72 activities (gutiérrez, 2002). in particular, we wanted to situate the mathematics within contexts that the students would connect with and consider relevant, allowing the students to see mathematics and statistics as meaningful and providing them an opportunity to legitimately participate in disciplinary practices. in this section, we describe what we observed with regard to identity development as we engaged students in activities to help them develop strong mathematical identity, specifically that of a statistician. embracing the name. we ensured that from the first day of the project students made connections between what they were doing in the activities with the work of statisticians. deliberate efforts included discussing what statisticians did and the importance of their work to society, emphasizing that the work they would be engaging in was important and the ways it would impact their school community. also, to strengthen this identity we thought it important that in addition to the students seeing themselves as statisticians, the larger school community also needed to identify them as such. to support this goal, we enlisted the help of the principal and the other instructors at the school to acknowledge the students as statisticians and inquire about the daily activities in which they were engaged. the pivotal moment was on the morning of data collection when the principal announced to the entire school that: “statisticians are coming to collect some important information for the school.” this act on the second day of the project was sufficiently powerful that afterwards several students began to refer to themselves as statisticians. the following statements from the students reflect this emerging identity: paula: (while smiling and covering her mouth) they’re talking about us. eugena: we are s-t-atisticians. engaging in the practices of statisticians. consistent with frameworks on statistical thinking (see, e.g., wild & pfannkuch, 1999) and research findings (see shaughnessy [2007] for a review) that recommend engaging students in all aspects of statistical inquiry, we designed the project so the students progressed through the various stages of the statistical investigation cycle (previously described). we considered this important because we wanted the students to complete the project successfully so they would be able to see themselves as problem solvers and statisticians. in our interviews with the students at the end of the unit, we tried to elicit how students defined their roles in the project and how they described the work of statisticians. the excerpt below provides an example of how students viewed their role on the project: interviewer: what else did you learn this week? kiley: ahhh…i learned some new, funny words. cross et al. success made probable journal of urban mathematics education vol. 5, no. 2 73 interviewer: like what? kiley: statisticians interviewer: so who are statisticians? kiley: for the past week, we’ve been statisticians. interviewer: so what do statisticians do? kiley: we had to give out surveys and use the data to figure out what breakfast to serve. interviewer: what do other statisticians do? kiley: they do the same kinds of things but for different reasons…different purposes. other students had difficulty saying the word “statistician” and articulating what statisticians did in statistical terms. instead of using words like “data” and “analyze,” these students described some of the activities they had completed during the week. when asked, “can you tell me what statisticians do?,” some simply replied, “what we did.” achievement equity concerns include the conditions under which learning occurs as well as the outcomes (gutstein et al., 2005). it is a foremost concern because students’ level of mathematical achievement tends to have greater impact on their longterm economic and professional goals. we do not limit our measures of achievement to test scores, but we sought to identify growth in the set of knowledge and skills that would create opportunities for the students to take advanced mathematics courses and to consider math-related careers; that is, knowledge that will help students progress through the mathematics pipeline (gutiérrez, 2009). in examining the data for evidence of this growth, we attempted to answer the question, “what knowledge did the students have at the end that was not present at the start of our work with them?” an analysis of the pre-interviews showed that the students varied widely in their knowledge of statistical concepts—ranging from no knowledge of statistics 4 to being able to state definitions for the measures of center. specifically, 85% of the students stated that they did not know what statistics or data meant. interestingly, three students were able to describe the “add and divide” formula for mean and state that median meant “middle,” but either had not heard the word statistics before or did not know what it meant. sixty-nine percent did not know what average, mean, median, or mode meant or gave unclear responses. for example, when asked what came to mind when they heard the word mean (in relation to mathematics), responses included: “answer to a problem,” “the total,” “like when you have a c average or a b average.” thirty-one percent were able to state the defini 4 no knowledge refers to not having the terms (such as statistics, data, median, mode) in their vocabulary and little to no understanding of meanings of the terms. cross et al. success made probable journal of urban mathematics education vol. 5, no. 2 74 tions of the mean, mode, and median (although some confused them) but were not able to use the definition in their reasoning. none of the students were able to use the measures of center to make data-based arguments; for example, solve and provide justification for problem in figure 3. more that 50% of the students were able to state that the diagram in figure 3 was a graph, but only two students stated it was a bar graph. below, we describe growth in two main areas: statistical knowledge and generalized knowledge. figure 3 task adapted from released naep item 2007-4m11 #4 (national center for educational statistics, 2011). statistical knowledge. in this category we included growth in students’ understanding of statistical ideas. alternative conceptions of mean. results from studies on the learning of statistics state that students tend to hold narrow conceptions of mean and do not simultaneously conceptualize mean as typical value, fair share, data reducer, and as a signal amid noise (cai, 1998; konold & pollatsek, 2002; mokros & russell, 1995; watson & moritz, 2000). one of our goals was to help students develop a 1. the students in a class each counted the number of letters in their first names. the class made the graph below of the results. a. a new student, victor, joined the class. draw on the graph to include the data for victor. b. what effect will victor have on the mean of the number of letters in the first name of the class—will the mean increase, decrease, or stay the same? cross et al. success made probable journal of urban mathematics education vol. 5, no. 2 75 broad conception of mean to encompass the above mentioned. students who knew the term came in with one way of thinking about mean, specifically, applying the procedure of adding the numbers and dividing by the number in the set of data. throughout the sessions we engaged students in activities that targeted mean as “fair shares,” “leveling out,” and “balance point.” we observed students who already had knowledge of the formula (add all data values and divide by the number of data values) grow in their understanding to include mean as leveling out, providing meaning to their procedures (see figure 4). figure 4 example of student work from the “leveling out” task. in the post interviews when asked about their understanding of the mean, some students, in addition to talking about the procedure, they included the idea of leveling out in their descriptions. when asked what she understood by the term mean, kaycia responded: well it’s what you get when you add up the numbers and divide by the numbers— the amount of numbers…you can level them out; like with the cubes and get the mean. you can have different sets of numbers, like a lot of different numbers [that have the same mean]. in addition to conceptualizing mean as leveling out, some students considered the notion of mean as balance point. after calculating the mean grade of a particular dataset, students were asked to examine the numbers in the dataset and state their observations about the numbers in relation to the mean. students observed that some of the numbers had to be above the mean and some had to be below the mean. the excerpt below shows the classroom conversation that supported students’ understanding of this idea: instructor: if you were a teacher, so you taught your class about fractions. you gave them a test with ten questions. after you grade everyone’s pa cross et al. success made probable journal of urban mathematics education vol. 5, no. 2 76 per and the class average was five. would you be happy or would you be sad? students: sad. instructor: why? students: because that is just half of the test instructor: what percent is that? students: fifty percent? instructor: is that good or bad? students: bad…it’s alright… instructor: so if the average is 50, does it mean that everyone got a 50? does it mean that tana got a 50, and jada got a 50? students: not really. instructor: does it mean that paula had to get a 50? or heather? students: no. instructor: so what kind of scores would everybody have to get to get an average of 50? dara: ten. instructor: if everyone got ten, would it average out to 50? let’s see… [instructor lists ten repeatedly on the board to represent ten students’ scores.] instructor: would that average out to 50? ashley: no, that would be ten. instructor: how do you know? ashley: because if i added then up and divide by ten, i get ten dara: because they are already level out instructor: well, can you give me some other numbers that would give me an average of ten? [students silent for about 90 seconds] instructor: so what if this one was 15? [teacher writes five on the board]. what is another number that could be included? alyssa: five [teacher places 5 next to 15] instructor: what’s another one? ellie: eight. instructor: so do the numbers have to be close to 10 to average out to 10? students: no. [instructor directs the students to look at a problem completed earlier in the lesson where they had to find the mean.] instructor: what was the average of these numbers? students: five. instructor: were they all close to five? students: no…yes…no. instructor: some of them are what? students: above… instructor: some are above 5 and the others are? students: below five. cross et al. success made probable journal of urban mathematics education vol. 5, no. 2 77 instructor: ok, so some are above and below the average. so if we wanted to create a set of numbers that have an average of 10, what do we need to do? alyssa: have some numbers above 5 and some below 5. [students made suggestions for other numbers that were above and below ten to create a dataset with an average of ten.] although initially students did not appear to see the numerical relationship between the mean and data values on each side of the mean they were able to apply this notion of “balancing” to finding a new dataset with a mean of 10 reasonably well. additionally, they used compensation to justify their increasing of one number with a corresponding decrease of the same value to another number. we must note that the students who were able to make these connections initially were the students who had prior exposure to these ideas. as the activity progressed, more students were able to recognize that you did not have to always give and take away from the same two numbers but that the simultaneous increasing and decreasing could be spread across the individual numbers in the dataset. mean not necessarily a part of the dataset. one major misconception that students hold is that the mean must be a value in the dataset. we directly targeted this misconception by giving the students a task where they were required to create several datasets that had a given mean. applying the notion of mean as leveling out, students were able to create several datasets for a given mean. in particular, we asked students to create a dataset of five numbers that had a mean of five. students worked in groups and started by creating five towers, each having five blocks. then they redistributed the blocks to make five towers of different heights. students repeated this process several times and recorded each of the new sets of heights as a new dataset with a mean of five; some datasets did not include five as a data value (see figure 5). figure 5 student-constructed datasets with a mean of 5. from preto post-interview, we progressed from 85% of the students not being able to define “statistics,” “average,” or “mean” to 90% of the students describing data, statistics, the work of statisticians, and how statisticians use mean data and statistics in their work. this reversal shows that by the post-interview, paula: 10, 5, 4, 5, 1 kaycia: 3, 4, 5, 6, 7 and 2, 3, 4, 7, 9 tyisha: 8, 5, 5, 4, 3 and 9, 6, 5, 3, 2 eugena: 17, 2, 2, 2, 2 cross et al. success made probable journal of urban mathematics education vol. 5, no. 2 78 students were better able to talk about and explain relevant statistical concepts in context. for those students who had knowledge of mean, mode, and median in the pre-interview but could not use the measures to reason about data within a context, by the post-interview, they understood that the mean did not need to be a part of the dataset and could conceive of mean as fair shares. general knowledge. in this section, we highlight the skills students acquired that were not specifically related to statistics. self-identification of concepts they still misunderstood. often when we refer to knowledge and understanding we only include in this description ideas students can clearly articulate. the ability to reflect on and think about one’s own thinking, referred to as metacognition, is essential for developing mathematical expertise (schoenfeld, 2006; schunk & zimmerman, 2006). an awareness of what one does not know and needs to learn shows metacognitive awareness. eugena’s statement provides an example. she stated, “what i think about median…i’m not so sure; i think i need to talk more about that.” her statement shows that she knew there was another statistical tool called the median but she was aware that currently she did not know how to define this tool or use it to summarize data. in our pre-interviews, most students showed no awareness of these tools (69% stated they did not know what average, mean, median, or mode meant), and those who did struggled to differentiate between them. being able to distinguish between the three measures of center and recognize deficiencies in their understanding, we considered notable improvements. 21st century skills. our project focused on statistics but also intentionally tried to help students develop communication, critical thinking, collaboration, and oratory skills. although most students would readily talk in small groups, they were initially very hesitant to interact with the instructors or explain their thoughts and solutions in whole-class discussions. students demonstrated growth in these areas during the final presentation of the findings. several of the students were able to confidently explain their part of the presentation, making significant eye contact with the audience well-composed and speaking knowledgeably. equitable mathematics education or not? in 2005 the journal for research in mathematics education published an article written by the nctm research committee, the goal of which was “to raise the awareness about equity and issues surrounding equity from a research perspective as well as to support the nctm’s commitment to the equity principle” (gutstein et al., 2005, p. 92). the authors strongly supported equity research in mathematics education and encouraged researchers to use a critical equity lens to examine their work with the goal of better understanding the complexities of teaching and learning mathematics. the authors also included important equity cross et al. success made probable journal of urban mathematics education vol. 5, no. 2 79 related research questions that remain unanswered within the field and implored researchers to use them to guide their advancing research agendas. three sets of questions directly targeted the dimensions of equity relevant to our project; here, we use modified versions of those sets to frame the discussion of our findings. how can students develop mathematical power, and at the same time, use mathematics as an analytical tool with which to investigate problems that are personally meaningful to them? can they also then begin to see themselves as mathematicians capable of shaping their communities? how might this occur, and under what conditions? we designed a unit we considered to be mathematically rich and personally meaningful so the students would not only learn important mathematical ideas but also learn how to use mathematics as a critical, analytical tool. the statistics concepts were embedded in a context that was authentic and that the students saw as relevant. students, with support, were able to describe and suggest possible solutions to the problem of low test scores; determine that a survey would be an appropriate data collection instrument and select the items for the survey; describe what an appropriate sample would be so we could make inferences about the school population; collect and analyze the data, and use the data to make recommendations to the principal with regard to the most appealing breakfast. community (within and outside of the school) support is essential. specifically, enlisting the principal and other members of the school community to support the students as they went through the data collection and analysis phases motivated and increased the level of confidence the students had in their ability to solve the problem. in addition to school staff, we also had support of parents; one parent provided feedback on the students’ presentations. additionally, the camp was funded by a national organization that promotes educational equity for girls. members of the local chapter provided and still provide support to the school with members serving as “big sisters” to some of the girls at brayton. along with feeling empowered to have a voice in making meaningful decisions, students also felt like they had a right to critique other issues within the school environment that troubled them. in particular, kaycia thought that in addition to selecting breakfast, they could also engage in similar statistical processes to determine whether or not the breakfast should be catered or made at the school. this suggestion was based on her observations from her years as a student that irrespective of what was provided for breakfast, only a limited number of students would partake. with learning how to use statistics to identify and solve problems, as statisticians do, students were now informed about how to use mathematical tools to examine and critique issues in their environment. we purposely sought to cultivate a positive identity towards mathematics as research studies have identified cross et al. success made probable journal of urban mathematics education vol. 5, no. 2 80 this as a key contributing factor towards mathematical success for african american students (martin, 2009; stinson, 2010; berry, 2008). through engaging the students in the practices of statisticians, they were able to positively identify with the role. what happens to students’ mathematics learning when taught in mathematical meaningful ways? the activities within the pbl unit addressed core statistical ideas that are not only included in the standards for elementary school but are tools used by a range of professionals in their jobs to make sense of the world and construct databased arguments. given the variation in statistical knowledge the students possessed at the start of the project, reporting aggregate achievement scores would not accurately capture the students’ learning over the course of the project. so instead we measured achievement based on analyses of the students’ responses focused on answering the question: “what do the students know about statistics now that they did not know at the start of the project?” for example, students who began the project not knowing how to describe statistics or data were able to provide sensible descriptions and relate them to the work of statisticians, with whom they identified. others who had only procedural conceptions of mean, mode, and median broadened their conceptions to include mean as leveling out, mean as balance point, and mean as typical value. we acknowledged this broadening as a notable achievement as the literature in statistics education states that these ideas are particularly difficult for students to grasp and for which students hold major misconceptions (konold & pollatsek, 2002; mokros & russell, 1995). although some students still struggled to provide clear explanations of what these measures tell us about datasets, they could articulate that they are important tools that statisticians (and others) use to make decisions and state clearly which ideas needed further refinement. having this exposure will not only position students to be critical of the statistical ideas they are taught in the future but also to be critical of the basis on which decisions are made that impact them. additionally, pbl is often touted as instructional reform that will motivate students because its context-rich nature tends to readily engage students securing their investment in solving the posed problem. our findings suggest that not only is pbl implementation motivational but also it supports equitable teaching practices (boaler, 2008). in this case, the approach allowed students to build on their own cultural experiences, develop expertise relevant to solving the problems and foster 21st century skills. twenty-first century skills are often not associated with or addressed in mathematics classes; however, we cannot overlook the importance of preparing our students for the future. as it pertains to collaboration, thinking, and communication, there were visible improvements among the students. with scaffolding by the instructors, students were able to describe the problem they ad cross et al. success made probable journal of urban mathematics education vol. 5, no. 2 81 dressed, the phases of the inquiry process, and explain how they arrived at their conclusions in presenting their recommendations to the principal. what difference might such efforts make in the lives of students and also in the larger society, in both the shortand long-term? it is widely accepted within the educational community that attaining equity in education is worthwhile and essentially “the right thing to do” (gutstein, 2005). nevertheless, teaching for equity is extremely challenging even with abundant resources. the data supports our conclusion that the intervention provided access to the resources necessary for quality mathematical learning. students saw mathematics as an activity in which they could be full participants and felt empowered to use statistics to enact change while embracing the role of statistician. therefore, in the immediate short-term we are confident that the tasks and activities the students engaged in during the summer camp left important mathematical residue; specifically, students had more productive dispositions towards mathematics, gained insights into the nature of statistical inquiry, and developed useful problem solving strategies (hiebert et al., 1996, 1997). one major drawback is that the project design did not allow us to assess the long-term impact on the students’ subsequent learning, views of mathematics, and mathematical identity. although we know, based on the data, students saw statistics as an analytical tool that could be used to help make important decisions and to make changes within their school community, we are cautious here with our statements. we restrict them to the school environment and to statistics because we are not confident that their identity as statisticians developed to be a part of their core mathematical identity (gee, 2001). specifically, we are not certain that the students would see regular school mathematics as a tool to enact change or if they would embrace their role as change-makers outside of the school context. we can only hope that the residue from our intervention helped students to position themselves differently with regard to mathematics, more as constructors and not just consumers of mathematical ideas, so they can successfully navigate through the mathematics pipeline. facing the realities of “equity for all” we were able to provide access to the physical and intellectual resources necessary to engage the students in mathematically meaningful activities, yet we faced significant challenges. many of these challenges we were able to overcome but question how realistic it is to hold similar expectations for individual classroom teachers. although the students who were enrolled in the camp were not considered struggling students, many of them had weak conceptions of number cross et al. success made probable journal of urban mathematics education vol. 5, no. 2 82 sense. we were able to address these deficiencies readily because of the approximately 2:1 ratio of students to teacher. given the number of instructors, we were able to attend to all students even if it meant rearranging the responsibilities of the instructors in a particular session. it was possible to pinpoint the students who were struggling and provide individual support. providing this level of scaffolding is extremely difficult if not impossible in regular classrooms. in addition to the scaffolding, all the instructors were particularly sensitive to equity issues, so we were consistent as it relates to holding the students to high standards. therefore, in situations where we felt students were not grasping the concepts, we more readily questioned our practices than the intelligence of the students and modified our approach when necessary. we spent hours collectively reflecting on the day’s activities, identifying instances of student learning and evidence of struggles, and using this information to develop student-centered (some student-specific) instructional goals for upcoming sessions. given the realities of schooling, this kind of teacher collaboration and support is often not feasible. the main challenge to our success was the mathematical deficiencies of the students. these deficiencies are reflective of the harsh social realities these students face in their daily lives. our intervention could not erase the larger societal and fiscal barriers that have impacted and will continue to influence the mathematical development of these students. however, our goal was to provide worthwhile mathematical experiences for african american students so that they could envision the possibilities—as mathematics learners and as citizens with resources to enact change. we also hoped to be able to tell a tale of success adding to the other counter-narratives about african american students (e.g., see, martin, 2007; stinson, 2010). we can only speculate about the long-term impact that our intervention may have had on the students. as stated by apple (1992) and tate (1997), teaching for equity cannot solely reside on the shoulders of teachers; without a complete reshaping of the political and social structures that impact these students’ lives, the long-term impact will be minimal, at best. cross et al. success made probable journal of urban mathematics education vol. 5, no. 2 83 appendix a sample surveys survey a survey b survey c circle which breakfast food you think is the best: i. waffles and sausage ii. grits, eggs, biscuits, and bacon iii. french toast and sausage iv. grilled cheese sandwich with soup rate each of the following breakfast foods by responding: strongly dislike, dislike, undecided, like, or strongly like. circle your response. waffles and sausage strongly dislike dislike undecided like really like grits, eggs, biscuits, and bacon strongly dislike dislike undecided like really like french toast and sausage strongly dislike dislike undecided like really like grilled cheese sandwich with soup strongly dislike dislike undecided like really like rank the following breakfast foods, with #1 being your favorite breakfast, #2 being your second favorite breakfast, and #3 being your least favorite breakfast. waffles and sausage grits, eggs, biscuits, and bacon french toast and sausage grilled cheese sandwich with soup 1. _________________________ 2. _________________________ 3. _________________________ 4. _________________________ cross et al. success made probable journal of urban mathematics education vol. 5, no. 2 84 references allexsaht-snider, m., & hart, l. 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(1999). statistical thinking in empirical enquiry. international statistical review, 67, 223–248. http://www.nctm.org/publications/article.aspx?id=26303 journal of urban mathematics education december 2011, vol. 4, no. 2, pp. 44–66 ©jume. http://education.gsu.edu/jume allison w. mcculloch is assistant professor of mathematics education at north carolina state university, department of science, technology, engineering, and mathematics education; 2310 stinson drive; 502l poe hall; raleigh, nc 27695-7801; email allison_mcculloch@ncsu.edu. her research focuses on the role of affect and culture in the mathematics teaching and learning process. patricia l. marshall is professor of multicultural studies at north carolina state university, department of curriculum, instruction and counselor education; 2310 stinson drive; 602h poe hall; raleigh, nc 27695-7801; email patricia_marshall@ncsu.edu. her research focuses on how intersections between elements of teachers’ personal identities and their professional concerns about working with ethnoracially diverse students influence understanding and adoption of critical equity pedagogies. k–2 teachers’ attempts to connect out-of-school experiences to in-school mathematics learning allison w. mcculloch north carolina state university patricia l. marshall north carolina state university in this article, the authors report on a 3-year professional development research project. the project focused in general on early mathematics teaching and learning in urban schools and in particular on promoting teachers’ awareness of the importance of making connections between students’ out-of-school experiences to promote deep understanding of k–2 school mathematics. children cross into school spaces bringing with them a wide variety of out-of-school experiences; this is especially true in the early elementary grades when they have spent more of their lives out of school than in school. effective teaching at this level requires that teachers put forth concerted efforts to make connections between these outof-school experiences and formal curricular content. the authors present the strategies that participating teachers (n = 49) employed in their attempts to make such connections as well as implications for future professional development research. keywords: cultural connecting, culturally relevant pedagogy, elementary teachers, mathematics education, professional development, urban education ultural diversity in k–12 classrooms in the united states is increasingly becoming a prominent topic of discussion in education policy and research, and is even often heard on popular media outlets. these discussions often highlight that the cultural backgrounds of too many teachers today differ considerably from a growing number of children in u.s. schools (zumwalt & craig, 2008). the reality of these divergent cultural backgrounds has special implications for teacher education, specifically in regards to the structure of professional development as children today cross into school spaces bringing with them a wider variety of outc mcculloch & marshall out-of-school experiences journal of urban mathematics education vol. 4, no. 2 45 of-school experiences than in previous eras. this “bringing with” is especially true in the early elementary grades where children have spent more of their lives out of school than in school. “effective” teaching at this level therefore requires that teachers put forth concerted efforts to make connections between children’s out-of-school experiences and formal curriculum content. but too often mathematics (and the formal mathematics curriculum) is seen by many to be “culture free,” lacking in any truly useful out-of-school experiences that might be drawn upon. nevertheless, young children do engage in a variety of wide-ranging experiences with number and operations (albeit informally) before formal schooling begins. reaching across borders—metaphorically, the out-of-school and in-school wall—is especially necessary for effective teaching of early elementary mathematics. therefore, in this article, we report on a 3-year professional development research project focused in general on early mathematics teaching and learning in urban schools and in particular on promoting teachers’ awareness of the importance of making connections between children’s out-of-school experiences to enhance deep understanding of the content of k–2 school mathematics. we present here some of the strategies teachers who participated in our study employed in their attempts to connect out-of-school experiences to in-school mathematics learning, as well as outline implications for future professional development research. importance of making connections in recent years, research has highlighted the underachievement of u.s. students in mathematics compared to many of their international peers (gonzales et al., 2009). there exists a well-documented “gap” between present levels of achievement in mathematics and the levels of excellence set as a goal for all students, particularly for african american students (hilliard, 2003). to address these major concerns, many scholars and researchers have begun to examine the connection between the development of mathematical knowledge and culture (e.g., boykin, coleman, lilja, & tyler, 2004; civil, 2002; cobb & nasir, 2002; gutstein, 2003; leonard, napp, & adeleke, 2009; lipka, 2005; martin, 2000; moschkovich, 2002; moses & cobb, 2001). research on the connection between knowledge and culture is grounded in the perspective that students come to understand phenomena as they are related to what they already know (foster, lewis, & onafowora, 2003; ladson-billings, 1994). studies have shown that drawing on informal mathematics knowledge leads to increased understanding and attitudes towards the relevance that school mathematics has in students’ lives (e.g., carpenter, fennema, peterson, chiang, & loef, 1989; gutstein, 2003; gutstein, lipman, hernandez, & de los reyes, 1997; mcculloch & marshall out-of-school experiences journal of urban mathematics education vol. 4, no. 2 46 hiebert & grouws, 2007; smith, 2000; wearne & hiebert, 1989). as such, it is imperative that teachers provide opportunities for students to develop links between new mathematical concepts and extant understandings. this imperative is especially true in the early grades where “formal” mathematical ideas are limited for most children. an emerging area of research is focused specifically on teachers making connections to children’s informal mathematics knowledge in the context of their cultural knowledge. this work has produced some promising results. for example, boykin and colleagues (2004) studied the effect of a communal learning environment (one that is designed to be consistent with the home social culture of low-income african american children) and found that african american children in a high communal context (i.e., wherein the success of the group as a whole is valued in conjunction with the success of individuals) performed significantly better on a mathematics post-test than did their peers in a low communal context. another project that attempted to link school mathematics to out-of school experiences was the yup’ik project which culminated in a curriculum based on the culture of the yup’ik people, an indigenous group in alaska. the researchers documented those children whose teachers used the yup’ik curriculum (after receiving professional development) showed higher achievement than comparable groups of children (lipka, 2005). though research suggests that making connections to both cultural knowledge and informal mathematical knowledge is important for the development of formal mathematical knowledge, there is little discussion in the literature on how teachers actually might do so, especially at the early elementary level. one exception is civil and andrade (2002) who have been working toward making explicit connections between the “authentic” mathematics in which children participate at home, and school mathematics. working with elementary teachers in workingclass mexican-american communities, teachers and researchers made household visits to acquire understanding of the mathematics in which the children participated at home. utilizing this knowledge, they designed thematic mathematics units intended to capitalize on the children’s informal mathematics experiences and understandings. for example, visits to the homes of second-grade children revealed families exposed children to a wealth of knowledge about construction (e.g., materials and shapes used in building homes, painting, and tiling) and the inherent mathematics therein. making use of this information, the researchers developed a 2-month-long curriculum project based on construction that integrated mathematics, writing, reading, social studies, and science. the mathematical goals included: measurement (including perimeter and area), identifying and classifying shapes, estimation, and patterns. while there were many very positive outcomes from this project, the researchers also noted difficulties they encountered with this model. when working toward merging required curriculum with children’s eve mcculloch & marshall out-of-school experiences journal of urban mathematics education vol. 4, no. 2 47 ryday lives, it was often difficult to extract school mathematics from the everyday (civil & andrade, 2002). they noted: “the transformation of these everyday experiences into pedagogical knowledge for the classroom involves a balancing act between the real world situations and the teacher’s mathematical agenda” (p. 158). nurturing mathematics dreamkeepers: the project nurturing mathematics dreamkeepers (dreamkeepers) was a quasiexperimental, longitudinal professional development intervention study designed to explore how k–2 teachers in urban schools understand and adopt standardsbased teaching practices (national council of teachers of mathematics [nctm], 2000) that have potential to promote young children’s deep understanding of early mathematics concepts. two ideas formed the dreamkeepers conceptual framework and served as the focus of the professional development intervention. these ideas were standards-based mathematics instruction (nctm, 2000) and culturally relevant pedagogy (ladson-billings, 1994). standards-based teaching refers to practice consistent with goals set by nctm in the principles and standards for school mathematics (nctm, 2000). the standards state that teaching “requires understanding what students know and need to learn and then challenging them and supporting them to learn it well” (p. 16). the standards were a call for teaching mathematics with a goal of promoting deep understanding for all students. standards-based teaching characterizes teachers who operate from the assumption that a student’s mathematical reality is not independent of that student’s ways of knowing and acting. this characterization implies that what a student “sees, understands, and learns is constrained and afforded by what the student already knows,” and that “mathematical learning is a process of transformation of one’s knowing and ways of acting” (simon, tzur, heinz, kinzel, & smith, 2000, p. 584). that is, the teacher must let go of the notion that “we understand what we see” and recognize that “we see what we understand” (p. 585). moreover, what we “understand” is greatly informed by the cultural lens through which we view the teaching-learning process (foster, lewis, & onafowora, 2003; perry, steele, & hilliard, 2003). for the dreamkeepers project, the concept culture was defined as “the consistent ways in which people experience, interpret, and respond to the world around them” (marshall, 2002, p. 8), including the tendencies, styles, and/or orientations commonly exhibited in academic contexts by persons from the same cultural/ethnoracial community or background. thus, culture was understood as a critical element in the teaching learning process because learning itself is “fundamentally contextual” (perry, 2003, p. 4). mcculloch & marshall out-of-school experiences journal of urban mathematics education vol. 4, no. 2 48 similarly, the concept race was presented as a central element of culture. we explored unique issues surrounding race as a critical social construct that permeates all facets of life in u.s. society, including the experiences of teachers and students in schools. foster, lewis, and onafowora (2003) characterize culture as “anthropology’s essential concept” (p. 261) that has particular implications for the teaching-learning process. yet they stopped short of equating culture with other related social constructs such as race, noting, “whereas culture, race, ethnicity, and nationality are intertwined in complex ways, culture is not coterminous with any one of these constructions” (p. 262). even so, in the united states, race represents a peculiarly complex construct that historically has been deliberately conflated with group culture. in part, this is because of the longstanding practice in the united states of classifying people by “race” (marshall, 2002). correspondingly, most individuals in the united states acquire group and personal identities that are aligned with the particular worldviews and ways of being (i.e., culture) that are directly and commonly associated with the racial group within which they are classified (helms, 1993). thus, although we made distinctions between the two concepts—culture and race—we highlighted that the concepts are often used synonymously in discussions of schooling. in so doing, we took into account some of the professional literature related to behavioral styles. hilliard’s (1992, 2003) focus on the behavioral styles of african american students as a cultural/ethnoracial group and the implications for teaching and learning was instructive. in conjunction with this broad interpretation of culture, the dreamkeepers project participants were introduced to the notion of cultural relevance as a professional pedagogical orientation. according to ladson-billings (1994) there are three primary tenets of cultural relevance: high academic achievement, cultural competency, and sociopolitical awareness. teachers who effectively promote high academic achievement among students use an array of resources and presentation styles that align with their students’ unique learning tendencies. they incorporate students’ cultural realities into instruction and, in turn, create a learning atmosphere that is unique to the children in the class, and thereby enhances their interest in academic learning. in a similar vein, cultural competency refers to teachers’ abilities to “capitalize on the cultural practices and sensibilities of their students” (nasir, hand, & taylor, 2008, p. 219). it also addresses teachers’ abilities to acknowledge themselves as cultural beings and recognize that their worldviews and frames of reference are likely to differ from (and perhaps be in conflict with) at least some students in their classrooms. lastly, sociopolitical awareness speaks to teachers’ knowledge of structural inequities (i.e., status, resource, and power differentials among diverse groups) in the larger society, and how these come to bare on schooling. teachers who adopt a culturally relevant pedagogical orienta mcculloch & marshall out-of-school experiences journal of urban mathematics education vol. 4, no. 2 49 tion work with colleagues, parents, communities, and the students themselves to neutralize or counteract these effects. our goal in the dreamkeepers project was to capture and analyze the extent, if any, to which teachers incorporated tenets of cultural relevance into their pedagogy in general and their mathematics instruction in particular. that is, as the teachers acquired more critical understandings of culture and race through the project intervention we sought to facilitate an epistemological shift in the pedagogical orientations of most (if not all) of the teachers (gay, 2002; haberman, 1995; hooks, 1994; ladson-billings, 1994; moses & cobb, 2001; simon, tzur, heinz, & kinzel, 2004). the key objective throughout the project was to promote (provoke) changes in their classroom interactions and professional worldviews. thus, the idea of pedagogical orientation was interpreted as the actual techniques and strategies used to deliver mathematics lessons. it also referred to the substance and focus of the teacher’s articulation of pre-teaching intents and postteaching reflections. ultimately, the shift we sought involved the teachers themselves perceiving a need to alter, or change outright, their pedagogy in response to being challenged to consider and capitalize on children’s culture and out-of-school experiences in their planning and implementation of in-school mathematics lessons. we felt confident in our assumption that few (if any) of the participants were currently enacting this orientation to their mathematics pedagogy. this assumption was based on participants’ input during early retreat activities (including participation in the cross cultural simulation bafá bafá [shirts, 1977]), responses on a survey of multicultural course/professional development experiences, and early classroom observations of mathematics lessons. professional development for primary teachers: the nurturing mathematics dreamkeepers project intervention all experimental group teachers in our project were required to participate in an extensive professional development intervention that combined standardsbased mathematics and culturally relevant pedagogy. the professional development was organized into four, 2-day sessions (“retreats”) spaced throughout each academic year of the project. each retreat addressed both foci and activities included exploration of the teachers’ mathematics content knowledge. also through the retreats, teachers developed critical awareness of how young children understand key early elementary mathematics concepts, and explored the influence of cultural factors on the teaching-learning process. the overall themes for one year of retreats appear in table 1. retreat activities were interactive in that teachers were doing, analyzing, and reflecting with respect to both mathematics and culturally relevant pedagogy (e.g., banilower & shimkus, 2004; loucks-horsley, hewson, love, & stiles, 1998). they also included mcculloch & marshall out-of-school experiences journal of urban mathematics education vol. 4, no. 2 50 examination of student work samples (e.g., kazemi & franke, 2004) and classroom videos (e.g., van es & sherin, 2008). particular care was taken to design the dreamkeepers project intervention to be in alignment with characteristics of high quality professional development as noted in the literature. for example, darling-hammond, wei, andree, richardson, and orphanos (2009) summarized the research on effective professional development, indicating that it needs to be intensive and ongoing. moreover, it should connect to practice, address the teaching of specific curricular content, focus on student learning, build strong relationships among teachers, and align with school improvement goals. table 2 outlines the manners in which the dreamkeepers project intervention was consistent with these critical components of professional development. table 1 nurturing mathematics dreamkeepers retreat intervention topics culturally relevant pedagogy standards-based mathematics retreat i  understanding culture  nctm standards  constructing counting systems  base 10 and base 4 number systems  counting strategies retreat ii  professionalism: the dreamkeepers identity  defining crp  high academic achievement  cultural competence  teaching mathematics for understanding  mathematical models and tools  learning trajectories for addition and subtraction retreat iii  promoting academic achievement  language and cultural identity  early algebraic thinking  patterning retreat iv  cultural competence  enthoracial identity  sociopolitical consciousness  early rational number sense  fair shares table 2 critical components of the nurturing mathematics dreamkeepers professional development design critical component professional development design intensive and ongoing  approximately 90 hours per year  four intensive, 2-day retreats each year  one week intensive summer institute each project year mcculloch & marshall out-of-school experiences journal of urban mathematics education vol. 4, no. 2 51 connected to practice  reflection on own teaching of specific lessons  analysis of peers lessons  focus on cultural relevance in mathematics teaching  focus on student understanding addresses the teaching of specific curricular content  emphasis on teaching and learning of k–2 mathematics  highlighting understanding of number and operation focused on student learning  analysis of student work samples  reflection on student understanding in the context of specific lessons facilitates relationships among teachers  observed same grade peer’s mathematics lessons  analysis of peers teaching  engaged in cross cohort/grade/school collaborations  worked together to develop workshops for conference presentations aligned with school improvement goals  schools were invited to participate based on their student population and school improvement goals  participants were leaders within school-based professional learning communities  participants drew upon nurturing mathematics dreamkeepers project goals in formulation of school improvement plans in addition to attending the project retreats, participants were required to observe each other teaching two consecutive mathematics lessons four times each year. to facilitate these observations, they were organized into same-grade “buddies” (pairs or triads) with a teaching peer from their same school. following the observations (at a mutually convenient time before or after school), the teachers engaged in a structured reflective session with a trained research assistant in which they discussed the children’s understandings of mathematics concepts along with the role of culture in the planning and implementation of the lesson. according to garet, porter, desimone, birman, and yoon (2009) these types of activities (i.e., observation, reflection, feedback) lead to not only enhanced knowledge and skills but also change in teacher practice. methods participants the participants in the dreamkeepers project were practicing kindergarten, first, and second grade teachers (n = 49), organized into three intervention groups or “cohorts” based on the year they joined the study. they were drawn from six elementary schools in a large, urban school district in the southeastern part of the united states. some participating schools had multiple cohorts of teachers in the project. the teachers’ classroom experience ranged mcculloch & marshall out-of-school experiences journal of urban mathematics education vol. 4, no. 2 52 from 2 to 20+ years, with the mean years of teaching experience being nearly 10 years. all participants were women, with 37 whites, 11 blacks, and 1 asian. table 3 presents the cohort breakdown by grade-level. the dreamkeepers project teachers as a whole did not exhibit deep understanding of many of the mathematics concepts typically taught in early elementary grades (as determined by project mathematics assessments), though their understanding did improve over the course of the project. as noted earlier, we cannot be certain individual teachers were not practicing culturally relevant pedagogy intermittently in their teaching prior to their participation in the dreamkeepers project. however, we are confident that none were enacting this orientation to their pedagogy on an ongoing basis, based on our early observations and interviews of the teachers. furthermore, our inter-actions in retreat sessions revealed that the knowledge of nearly all of the teachers in relation to features of culture and its effect on teaching and learning was slight at best. table 3 participants by cohort and grade level kindergarten first grade second grade total cohort i (3 schools) 2 4 2 8 cohort ii (3 schools) 5 4 1 10 cohort iii (6 schools) 13 7 11 31 data collection and analysis the dreamkeepers project was a mixed-methods investigation in that we collected and analyzed data drawing on both qualitative and quantitative methods (tashakkori & creswell, 2007). data reported here are drawn from video recorded mathematics lessons and post-teaching reflective sessions that were collected at four intervals during each academic year. video recorded mathematics lessons were analyzed using a multi-step process involving the collection of both qualitative and quantitative data. each video recording was first viewed in its entirety. then each lesson was mapped according to its structure (e.g. lesson opening /closing, whole group instruction, small group activity) (stigler, gozales, kawanaka, knoll, serrano, 1999). next, each lesson was coded using a rubric developed specifically for the project and consistent with the goals of standards-based mathematics instruction and culturally relevant pedagogy (see decuir-gunby, marshall, & mcculloch, 2011). the rubric categories focused on teacher communication and actions that mcculloch & marshall out-of-school experiences journal of urban mathematics education vol. 4, no. 2 53 were consistent with the guiding theories and goals of the project. categories included: learning connecting, illuminating thinking, affirming multiple representations, extensions of tasks, language matching, relevance making, cultural connecting, and communalizing. finally, critical events identified by the lesson rubric were transcribed verbatim and those transcripts were open coded for further interpretation (corbin & strauss, 2008). four-hundred-eighty-two (482) k–2 mathematics lessons (each ranging in length from 20 to 40 minutes) were coded using the lesson rubric. for the purposes of this article, we were most interested in teacher actions that were attempts to connect children’s out-of-school experiences to school mathematics. as such, we focused on a particular set of codes that were created to capture these actions: cultural connecting, language matching, and relevance making. the definition and an example of each of these codes are provided in table 4. table 4 code, definition, and example code definition example cultural connecting teacher makes reference to child(ren)’s home/out-of-school experiences with purpose of affirming and illuminating connections to particular mathematics topic, skill, concept, being explored. does not include attempt to highlight how knowing skills will be useful; intention is to connect and affirm child(ren)’s out-of-school experience with in-school mathematics. “this jar is filled with different colors of beads. beads like these are good for putting on braids, just like the one’s jade is wearing in her hair. how many beads do you estimate are in this jar?” language matching teacher makes use of child(ren)’s own language (as opposed to standardized mathematical language) when giving an instructional directive, explanation, or following up on child’s comments. purpose is to affirm child(ren)’s own ways of articulating their understanding of the mathematics, by using child’s own language. “okay carante, “we gon’ put them two numbers together” and then, what we ‘gon do next?” relevance making teacher makes deliberate effort to help children make association between mathematics task, concept, skill, etc. (or asks child to), and then illuminates for children or asks child(ren) to demonstrate how knowing the task, concept, skill, etc. will be helpful in nonschool situations. “how many of you buy things at the “cruizers” store just down the street? if you went to that store and wanted to buy some candy and gum, how could you use what we learned today about adding numbers?” mcculloch & marshall out-of-school experiences journal of urban mathematics education vol. 4, no. 2 54 reflective sessions were included as a data source for this article to add to the reliability of the findings. the purpose of the reflective session was to explore the nature of teachers’ thinking about mathematics instruction. this exploration included being asked to reflect on the execution of lessons, articulation of the nature and observed evidence of students’ understanding of mathematics concepts, and the role of culture in the planning and implementation of the lesson. these sessions provided another perspective (the teachers’) to the analysis of the lessons. the duration of the reflective sessions ranged from 1 to 2 hours, depending on the verbosity of the participants. all sessions were analyzed using thematic content analysis (coffey & atkinson, 1996). a codebook was developed through theory and data driven coding that resulted in 27 codes (see decuir-gunby, marshall, & mcculloch, 2011). once coding was completed, the codes were organized into larger themes, guided by either theory or patterns that emerged from the data. finally, findings from the video recorded lessons and the reflective sessions were compared and interpreted. all analyses were completed by the research team, which consisted of three principal investigators and five research assistants. with such a large research team, we found it necessary to extend miles and huberman’s (1994) approach for establishing inter-rater reliability; therefore, we created a process focused on group “consensus” (harry, sturges, & klinger, 2005). all members of the research team coded practice artifacts until interpretations of all codes reached 100%. reliability was checked throughout the data analysis process by periodically choosing a data piece to be coded by multiple coders in order to ensure that coding remained consistent (see decuir-gunby, marshall, & mcculloch [in press] for further detail). results connecting out-of-school experiences to in-school mathematics learning during instruction the analysis of the video recorded mathematics lessons revealed that instances in which teachers explicitly made connections between children’s out-ofschool experiences and the curricular content were not commonplace. moreover, those that were made were superficial in nature. in the 482 lessons coded, there were only 149 instances in which one of the codes associated with such actions was used. all but 12 instances of out-of-school/in-school mathematics connections occurred after the teachers had participated in at least two retreats. we believe the relatively late introduction of these actions suggests that the project intervention prompted the attention towards such connections for these teachers. it is important to note that even though there were 149 instances, the instances oc mcculloch & marshall out-of-school experiences journal of urban mathematics education vol. 4, no. 2 55 curred in 102 different lessons, taught by 42 different teachers, spread relatively evenly over the three cohorts and years of the project. this distribution suggests that although the vast majority of the teachers engaged in out-of-school/in-school mathematics connecting, they did so infrequently. nevertheless, the fact that the majority of them did engage in these actions suggests that the culture/mathematics focus of the intervention was having an impact, if only minimal. the cohort structure of the dreamkeepers project allowed us to gain some insight into the cumulative effect of the professional development on the dreamkeepers project teachers. for example, 100% of the teachers in cohort 1 had at least one instance of an out-of-school/in-school mathematics connection, whereas 90% of the cohort 2 teachers, and 83% of the cohort 3 teachers did. at the lesson level we found that 31% of the cohort 1 teachers’ lessons had at least one instance, compared to 19% of cohort 2 lessons, and 17% of cohort 3 lessons. taken together these statistics suggest that the length (number of retreats per year) and depth (years in project) of teachers’ exposure and engagement with intersections between the concepts culture and mathematics did impact their incorporation of features of a culturally relevant orientation in their mathematics lessons. our analysis of the data that fell into the three out-of-school/in-school connections codes (i.e., cultural connecting, language matching, and relevance making) revealed that in their efforts to connect mathematics lessons to children’s out-ofschool lives the teachers’ actions fell into four categories: clarifying mathematical context, making mathematics relevant, introduction of formal mathematics, and validating students’ mathematical contributions. in the next sections, we describe the dreamkeepers project teachers’ actions in each of these categories as they relate to connecting out-of-school experiences to specific mathematics lessons. examples that typify such actions are also presented. clarifying mathematical context. one of the ways that the dreamkeepers project teachers connected to children’s out-of-school experiences during mathematics lessons was through use of contextual problems. when using such problems teachers were often very careful about making sure that all of the students understood the situation and/or object being discussed. for example, teacher 3-f27 1 presented an informal measurement task to her students that referred to a bathtub. in the task, students were asked to determine the length of a bathtub with respect to a soap bar, a toothbrush, a washcloth, and a towel (teaching integrated mathematics and science, 2004). during an initial discussion of the task she paused and stated: okay, think about your bathtub at home. everyone close your eyes and think about your bathtub at home. okay, some of you might say i know i grew up, i didn’t have 1 teacher id codes were generated for project use to maintain confidentiality. the middle symbol reflects the grade taught (k = kindergarten, f = first grade, s = second grade). mcculloch & marshall out-of-school experiences journal of urban mathematics education vol. 4, no. 2 56 a bathtub at my dad’s house; i had a stand-up shower. so i would be sitting here going mrs. smith i have no idea what you are talking about ‘cause i got a stand up shower, i don’t have a bathtub. so if you don’t have a bathtub then it’s kind of hard for you to think about how long it is. she went on to explain that a bathtub is typically as long as the table that was placed at the front of the classroom. in the context of this lesson, it served as a referent for the children. in some instances, the teacher did not immediately clarify the context as above, but instead surveyed the children to check for their understanding of an out-of-school context she planned to draw upon. for example, in preparation for an activity in which they would be asked to create different coin combinations at an imaginary store, rather than assuming the context was familiar to all of the children, teacher 3-s-16 said, “raise your hand if you go grocery shopping or clothing shopping with your mom or dad.” such a check for understanding was typically followed either by a clarification or by continuing forward, depending on the children’s responses. making mathematics relevant. the dreamkeepers project teachers also tried to connect in-school mathematics learning to children’s out-of-school lives by pointing out why knowing particular mathematical ideas will be important in their futures. summarizing a lesson on doubling, teacher 1-f-8 asked: “why do we need to learn doubles? what would be the point? how will it help me out in the world? how will doubling help one in life?” at first the children responded that they would need to know it to “do math homework.” the teacher probed further: “ok, but that’s just work that i give you. what about outside of school work?” at this point, the children offered ideas such as “it will help me count my money” and “counting pop tarts, because they come in twos.” this conversation continued with the teacher posing situations. placing a hand on a girl’s shoulder, the teacher posited: “suppose you’re going on a date and it costs $5 for a ticket. how will you figure out how much for you both?” to which the children shouted, “double it!” in some instances, teachers’ attempts to make mathematics relevant included highlighting examples of mathematical properties. when teacher 1-f-8 discussed symmetry, for example, she asked children to brainstorm objects that were symmetrical. remarking on one child’s braided hairstyle, she queried: “does your hair have symmetry? if i put a line down the middle i have the same amount of braids on one side and the same amount on the other side. so, hair designers need to know symmetry.” continuing the hair reference to illustrate the importance of symmetry she asked: “what if i was a hair designer and you wanted me to do your hair for a ball you were going to. ladies, would you want one side curled and the other side punk rock?” similarly, during a lesson on grouping by 2’s teacher 2-f-1 pointed out objects that came in groups of 2 to illustrate the mathe mcculloch & marshall out-of-school experiences journal of urban mathematics education vol. 4, no. 2 57 matical concept: “what comes in groups of 2? earrings. some people have more than 2 earrings, but [teacher’s name] only has 2.” this listing of objects that the children were familiar with in their every day lives was her way of trying to connect their experiences to the mathematics being learned. introduction of formal mathematics. young students often rely on terms they use regularly out-of-school to describe new phenomena and objects introduced in school. using children’s out-of-school language in the classroom was another way that teachers made connections to out-of-school experiences. such actions demonstrated to the children that their informal mathematical ideas and ways of expressing those ideas were valued in the classroom. when this occurred, the dreamkeepers project teachers often used the opportunity to introduce formal mathematical language. for example, when introducing an activity in which children were to sort objects based on weight, teacher 3-k-13 brought out a balance scale and asked if anyone recognized the tool. one child shared that it is a “weigher” and went on to note that she has one at home. the teacher responded, “it’s a weigher, good. we can also call it a balance.” this response let the student know that what she called it at home was perfectly acceptable, while also introducing her to the term that would be used for the tool in school. validation of students’ mathematical contributions. the act of validating a mathematical contribution refers to situations in which children offered a contribution to the mathematical conversation that drew on their out-of-school lives, and the teacher acknowledged the contribution with respect to the lesson. for example, as teacher 3-k-29 began her mathematics lesson she asked the children if they remembered what they had been learning in math. one child offered that during dinner with her dad she had eaten “4 peas and 2 corn.” the teacher responded: “you had 4 peas and 2 corn, so you asked the question how many do i have in all? how many did you have in all?” this response acknowledged the child’s contribution and tied it to the mathematics they had been learning. at times children’s contributions were not in the form of sharing a specific experience, but instead sharing their out-of-school (informal) language. in these instances validation occurred through the teacher’s restatement of the child’s contribution to the mathematical conversation using the same language as the child; no corrections were made to the student’s word choice. in a whole class discussion at the end of a lesson on estimation, teacher 3-s-25 asked the whole class “what does a good estimator do?” to which one child replied, “he wonders.” the teacher validated this statement by repeating, in an upbeat voice, “okay, an estimator wonders.” in another example, during an introductory lesson on 3dimensional figures, teacher 1-k-6 asked children to explain the difference between a circle and a sphere. many of the children noted that the sphere “sticks up.” the teacher then used that language to introduce a cube, wherein she noted that it (a cube) is different from a square because, “as y’all say, it is sticking up.” mcculloch & marshall out-of-school experiences journal of urban mathematics education vol. 4, no. 2 58 drawing on the lesson rubric findings, we were quite unsettled by the scarcity and the overall quality of the out-of-school connections demonstrated by the teachers. this unsettled response was because based on informal conversations with the teachers during retreat sessions, it was clear to us that they were thinking about, and in some cases struggling with, how to make these connections. we wondered whether there might have been other instances (besides actual lessons) where the teachers demonstrated connections through their other pedagogical activity. the rubric created to code the lessons focused on the teachers’ verbal and physical actions during a lesson. this restriction meant that it could not capture decisions made prior to teaching the lesson. we speculated about the possibility that such connections could have been considered in the context of post teaching activity. thus, to gain a different perspective on the teachers’ considerations with respect to making out-of-school/in-school connections we turned to analyses of their reflective sessions. reflecting on lessons and out-of-school connections analysis of the teachers’ reflective sessions revealed that most of them did in fact consider their students’ out-of-school experiences in the planning of their lessons. in many cases, the enacted lessons reflected decisions they had made regarding how to best set up a task and the structure of the lesson as a whole with students’ out-of-school experiences in mind. for example, when reflecting on a task referring to a yak teacher 3-f-1 noted that she knew many of her students had not ever heard of a yak: “so you know, i went on the internet real quick and showed them what a yak was.” though she was aware that many children were not familiar with yaks, the term itself was left in the task intentionally. she explained, “i know they’re not going to know; but i leave it in there so that they can be exposed to different words.” in contrast, some teachers very carefully chose the context of problems to carefully match children’s out-of-school experiences. teacher 3-s-26 intentionally chose to include buttons for a task, noting: “well, with the buttons, some of the students could relate. you know, they had said, ‘oh, my grandmother has a box of buttons.’ and they were really interested in the buttons.” while this teacher was relatively certain most of the children were familiar with buttons, her assumption about the extent of their familiarity was obviously measured: we talked about how buttons are sewn on. and we even looked at how people used buttons—you know, in the past. we had some pictures from a book that we had read. so [i’m] just kind of building on some of their background experiences. there were some situations in which teachers engaged children in lengthy discussion at the beginning of a mathematics lesson about the context of the les mcculloch & marshall out-of-school experiences journal of urban mathematics education vol. 4, no. 2 59 son (not the mathematics) to set the stage appropriately for all students. for example, when reflecting on one of her first lessons involving money, teacher 2-k-2 explained why she spent so much time at the beginning of the lesson talking with the children about money, and how they spend it: different cultures interpret money differently. some children don’t touch money, don’t own any money. but some other cultures may give it to them as a reward and different celebration types. it just depends on how they use it at home. she felt this time was well spent as they moved forward with the money unit. according to the teacher, it provided an opportunity for all of the children to share experiences with money, and it informed her decisions about future lessons in the unit. notably, this teacher explicitly mentioned culture in her explanation, suggesting that the expected interconnection between culture and mathematics promoted in the dreamkeepers project had influenced her pedagogy. while we saw evidence of teachers’ using children’s informal language in the context of mathematical discussions, one of the main concerns and considerations of the dreamkeepers project teachers was how to work with children for which english is a second language (esl). connecting the out-of-school language experiences with in-school mathematics language and curriculum proved to be challenging in such cases. analyses of reflection sessions revealed that when discussing the context of a problem that students might be unfamiliar with, teachers asked children with non-english home languages to share the names/words used in their homes for particular objects. for instance, discussing a lesson on number modeling, teacher 3-k-5 explained that she tried to make sure to include number words in all the various languages represented in her class: “they were brainstorming all the ways to model that number. we did the spanish word. and then one of the kids said, ‘we need to do french’ and the students all turned to charlotte and asked her. then we also did arabic.” in addition to drawing on home languages when setting up problems, teachers also focused on classroom interactions that would be most helpful in providing esl students the structure needed to engage in mathematical activities with english-dominant peers. some teachers made deliberate decisions about seating arrangements with esl students in mind. teacher 3-k-12, who discussed small group seating and interaction strategies for esl students during a lesson on sorting, provides an example of this decision-making process: just that they’re able to watch what somebody else is doing. and see, and pick up, and—you know, for them i think sometimes it’s better for them to sit back and listen and watch and observe, and absorb things. because they’re picking up on terms that the kids are using when they’re talking to each other. and you know, they’re learning well, what is shiny. because they—you know, in their language shiny is some other word. …in that sense i think it was good for them to do it with their group, ra mcculloch & marshall out-of-school experiences journal of urban mathematics education vol. 4, no. 2 60 ther than trying to do it by themselves. because if they were doing it by themselves, they would have no one to feed off of. they wouldn’t have anybody to prompt them, or—you know, to physically show them what they’re supposed to be doing. many of the teachers shared such strategies for their esl students. in addition, they also noted intentional use of pictures and objects, “so if their english wasn’t that good, they could see the visual” and the adaptation of curriculum materials. teacher 2-f-1 noted: “on that homework sheet, i did see it was a little too wordy. i mean, it was describing the hand span and the cubit and the directions were a little bit wordy too.” she went on to explain in detail the words she cut out and the illustrations she added to facilitate the children’s ability to complete their homework assignment. discussion analysis of video recorded lessons revealed that the number of explicit outof-school connections made during mathematics lessons was low. furthermore, many of the connections were quite simplistic and were clearly not effectively capitalizing on the children’s out-of-school prior knowledge and experiences in mathematically meaningful ways. however, we see in the teachers’ reflections that indeed they were seriously considering, and often struggling with, how to make connections between the school mathematics curriculum and out-of-school experiences. we recognized that our teachers generated few out-of-school connections in their lessons that were mathematically meaningful. so the question we were faced with was: what might be some reasons so few of the teachers demonstrated facility with making connections? this question surfaced because there are outstanding examples in the literature of teachers who are successful in this venture. descriptions of these successful teachers however, reveal that they often engage in curriculum development activity drawing on their students’ out-of-school realities and experiences (e.g., boykin et al, 2004; civil, 2002; civil & andrade, 2002; gutstein, 2003; gutstein, et al., 1997; lipka, 2005; moses & cobb, 2001). for our teachers, we believe at least one factor precluded their willingness to modify their curriculum in any manner, including engaging in substantive out-ofschool/in-school connecting. this factor was the district-wide “pacing guide” that the teachers perceived they were required to implement. our teachers were acutely aware that they were expected to teach particular topics on particular days, and to cover a certain amount of material by a certain point in the academic year. for them, the power of the orthodoxy surrounding the pacing guide was reinforced by district-wide standardized testing that occurred during and at the conclusion of each academic year. we theorize that another explanation for teachers’ minimal acknowledgement of students’ out-of-school experiences in the enactment of their lessons may mcculloch & marshall out-of-school experiences journal of urban mathematics education vol. 4, no. 2 61 have been their lack of deep understanding and/or comfort with mathematics. opening up to students’ informal understanding is risky for teachers who do not have deep understanding of the mathematics themselves. all of this, coupled with knowing that their lessons were being video recorded, might have resulted in less receptivity to encourage children to share their informal mathematical understandings. scholars report that teachers whose mathematical knowledge background is slight are far less likely to engage in mathematical conversations or unconventional mathematical teaching, that have the potential to reveal and affirm the knowledge capital that students bring with them to school (hilliard, 2003; leonard, brooks, barnes-johnson, & berry, 2010). nevertheless, those dreamkeepers project teachers who were successful in drawing on children’s experiences to make mathematics more meaningful were successful in making the thick borders (metaphorical walls) between school and home more permeable. whereas those who were not might have inadvertently erected an additional border that served to even further separate school mathematics from out-of-school experiences. in hindsight, we have reason to believe that the creation of additional borders as noted above would have been less likely to occur in the context of literacy instruction. this is because for most of the dreamkeepers project teachers, literacy was an area with which they felt comfortable and knowledgeable. similarly, many held the common (mis)perception that literacy is necessarily a more “natural” content area fit with which to make out-of-school cultural connections. 2 it is true that literacy often provides obvious out-of-school connections, yet some of the teachers made effective connections to mathematics topics that are commonly recognized as directly applicable to real life. among these were counting money, telling time, measurement, and fair shares. not only did these types of concepts seem easier for the teachers to engage children in conversations about mathematics with respect to their previous experiences but also the teachers were quite familiar with stories (literature) in which characters were in situations that required use of the mathematics. these stories were often used to begin such conversations. on the other hand, teachers seemed to have more difficulty finding the outof-school connections with concepts they viewed as “school mathematics” (e.g., addition and subtraction algorithms) (civil & andrade, 2002). an unexpected issue we encountered during data collection was that some teachers tended to maintain a somewhat narrow understanding of the concept of culture despite the broader definition used in the project. during the reflective sessions, teachers were asked to discuss the role of culture in their lessons. in part, this question was intended to elicit their perspectives on how they attempted to draw upon the children’s out-of-school realities in mathematics lessons. when 2 such misconceptions may be owing to the fact that early content integration as a dimension of multicultural education (banks, 2002) often focused on literacy/literature and social studies curriculum (marshall, 2002). mcculloch & marshall out-of-school experiences journal of urban mathematics education vol. 4, no. 2 62 responding to the question however, some teachers tended to highlight and isolate race without considering its complex relationship to culture. typically they would indicate that they did not consider race (for them a proxy for culture), in their mathematics teaching. thus, we believe one critical block that prevented teachers from making connections to children’s culture was the simplistic connections they drew between race and culture. indeed discomfort with the issue of race manifested in some teachers embracing a “colorblind perspective.” doing so meant that they would ignore and/or avoid reflection session questions where they were being asked to describe the influence of children’s backgrounds on the teaching and learning process. this avoidance coupled with their deeply entrenched beliefs that mathematics is culture free, and in turn, translated to mathematics is race free. for teachers who understand culture as a euphemism for race, a child’s cultural (racial) background is irrelevant to the teaching-learning process. likewise, for such teachers making cultural (racial) connections to early mathematics is an unnecessary, if not impossible, undertaking. in hindsight, we recognize that there were a few limitations to aspects of the design of our study. for example, the wording of a question intended to prompt conversation about the ways the teachers’ drew upon children’s out-of-school experiences (i.e., explicit use of the term “culture”) may have inadvertently resulted in a conflict for many teachers. also, the substance of the data resulting from the reflection sessions relied heavily on the ability of the research assistants to ask meaningful follow-up questions to teachers’ initial responses. as is the case with most research teams involving graduate students as emerging researchers, some of them completed this task better than others. another limitation was the nature of the video data collected. each lesson was recorded using one stationary camera aimed at the teacher. as a result, we did not capture individual or small group discussions that took place more than a few feet from the camera. hence, it is possible that our teachers’ did engage in one-on-one or small group conversations (in addition to the whole group discussions analyzed here) in which connections were made, but not captured by the camera. these limitations notwithstanding, the data (both video recorded lessons and reflection sessions) reveal that the dreamkeepers project teachers were indeed attempting to reach across the borders of school to capitalize on children’s culture-based outside-of-school experiences with mathematics. furthermore, the increased frequency with which this occurred as the project progressed suggests that our project was the impetus for making out-ofschool/in-school connections. conclusion the nurturing mathematics dreamkeepers project sought to facilitate teachers’ ability to make connections between in-school mathematical content and stu mcculloch & marshall out-of-school experiences journal of urban mathematics education vol. 4, no. 2 63 dents’ out-of-school mathematical experiences. in short, we sought to promote cultural relevance in early mathematics teaching and learning. our findings suggest that adopting a culturally relevant orientation towards teaching early elementary mathematics does not come easily for many teachers. still, the promising results of other scholars (e.g., boykin, et al., 2004; civil, 2002; cobb & nasir, 2002; gutstein, 2003; lipka, 2005; moschkovich, 2002; moses & cobb, 2001) along with our findings suggest that more professional development focused specifically on making out-of-school connections to in-school mathematics is needed. similarly, the cohort structure of the nurturing mathematics dreamkeepers project suggests that such professional development needs to be intensive and long term. we found that the longer the teachers participated in our project, the more likely they were to make out-of-school/in-school connections in mathematics lessons. the struggles the dreamkeepers project teachers experienced with the term culture and its relation to mathematics teaching and learning suggests that connections between culture and mathematics may need to be more explicitly grounded in the realities of practice. while we do not believe that critical orientations towards teaching, such as cultural relevance, can be (or should be) fully prescriptive, our results suggest that some “real-world evidences” may be warranted. 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(2008). who is teaching?: does it matter? in m. cochran-smith, s. feiman-nemser, d. j. mcintyre, & k. e. demers (eds.), handbook of research on teacher education: enduring questions in changing contexts (3rd ed., pp. 134–156). new york: routledge/taylor & francis group & association of teacher educators. latina/o student beliefs about mathematics, language, and agency journal of urban mathematics education december 2011, vol. 4, no. 2, pp. 26–43 ©jume. http://education.gsu.edu/jume maura varley gutiérrez is the director of teaching and learning at elsie whitlow stokes community freedom public charter school, 3700 oakview terrace, ne washington, dc 20017. email: mauravarley@gmail.com. her work now and as an educator/researcher at the university of arizona focuses on critical mathematics education and the intersections of gender and race in elementary schools. craig willey is assistant professor of mathematics education at indiana university school of education at indianapolis, 902 w. new york street–es 3156, indianapolis, in 46236; email: cjwilley@iupui.edu. his research interests include the preparation and development of mathematics teachers of latinas/os and other bilingual student populations, as well as the improvement of mathematics curriculum to provide more access and increase the engagement of emerging bilingual students. lena licón khisty is professor emerita in the department of curriculum and instruction at the university of illinois at chicago; email: llkhisty@uic.edu. her expertise integrates the two areas of bilingual/bicultural and mathematics education. her research interests include creating equitable learning environments for chicanas/os in mathematics, the preparation of teachers for multilingual mathematics classrooms, and the dialogic ecologies of mathematics classrooms for equity and social justice. (in)equitable schooling and mathematics of marginalized students: through the voices of urban latinas/os maura varley gutiérrez elsie whitlow stokes community freedom public charter school craig willey indiana university purdue university indianapolis lena l. khisty university of illinois at chicago in this article, the authors present the mathematics counterstories of a marginalized, non-dominant group of students: urban latinas/os. the presentation rests on a key tenet of critical race theory: that the experiential knowledge of nondominant people is legitimate and critical for understanding and remedying the factors and processes that subordinate groups, in this case, urban latinas/os in mathematics. the authors use data from research on afterschool mathematics projects to provide latina/o students’ perspectives, or counterstories, on their experiences with learning mathematics. throughout their counterstorytelling, themes are uncovered that relate to latina/o students’ perspectives on their mathematics learning experiences and ways in which they sometimes resist these experiences. these counterstories, in turn, offer insights that shift assumptions about marginalized students and mathematics instruction. keywords: critical race theory, latina/o education, mathematics education, urban education hile there has been (some) progress in reforming mathematics teaching and learning, the same cannot be said for turning schools into equitable environments with concomitant outcomes for significant populations of students, especially students who are linguistically, ethnically, and racially diverse and who w varley gutiérrez et al. voices of urban latinas/os journal of urban mathematics education vol. 4, no. 2 27 live in urban areas. this lack of equitable learning environments is particularly true for latinas/os. too often, latinas/os hold minority status in terms of equity and sociopolitical power, even though they may outnumber other student populations, and are educationally marginalized because of their status (lópez leiva & khisty, 2010; solórzano & villalpando, 1998). most unfortunate, the schooling practices and processes for latinas/os continue to be linked to discrimination, segregation, and failure (moll & ruiz, 2002). latinas/os tend to be over-represented in classrooms where reductionist pedagogies are emphasized (lipman, 2004), they have the lowest achievement levels in mathematics on standardized tests (darling-hammond, 2010), and they often suffer discriminatory microaggressions that marginalize or exclude them from critical participation in mathematics (lópez leiva & khisty, 2010). in this article, however, counter to these often-repeated circumstances, we are concerned with understanding the experiences in mathematics of latinas/os in urban schools and the factors and practices that hinder or nurture their learning. we do so by listening to and foregrounding the students’ voices—a term critical race theorists use to mean people of color naming their own realities (ladson billings, 1998). while explanations for the persistent and pervasive underachievement of latinas/os in school have been put forth elsewhere—namely, deficit-based explanations that place the blame within the students, their parents, or their culture (valencia & black, 2002)—it most often has been from the perspective of the majority with its inherent roots in the status quo. these explanations can be considered majority stories. on the other hand, latina/o students have seldom been asked for their perspectives on their classroom mathematics experiences or what insights they might provide about the possibilities of enhanced mathematics learning opportunities. the latina/o students’ perspectives and insights are what critical race scholars call counterstories (e.g., parker & lynn, 2002; solórzano & yosso, 2001; yosso, 2006); in this case, stories that illuminate the conditions of schooling and perhaps challenge accepted assumptions about schooling practices, particularly those related to mathematics and latinas/os. we begin our discussion with setting forth our assumptions related to counterstories. we then describe our methodological approach and discuss our findings related to students’ counterstories. these counterstories tell of student resistance to their exclusionary experiences of mathematics and shed light on potential alternative learning arrangements. importantly, we consider the interaction of students’ home language and mathematics—as told through the voices of the latina/o students—because of the critical role that language plays in their sense of self, or identities (gonzález, 2001). we close with some concluding thoughts and implications of our work. varley gutiérrez et al. voices of urban latinas/os journal of urban mathematics education vol. 4, no. 2 28 background and the use of counterstories latinas/os now comprise the largest “minority” student population in the united states, and the majority of students in many metropolitan school districts across the country (moll & ruiz, 2002). the label “latino” represents a wide variety of groups from south and central america and the caribbean. however, the majority of latinas/os are puerto rican (9%) and mexican or mexican descent (mexican american) (65%) (u.s. census bureau, 2010). in this discussion, we focus on the latter group. this distinction is significant because while other latina/o groups may think of themselves as immigrants and carry with them the willingness and disposition to assimilate into the u.s. society, mexican americans— and puerto ricans—generally do not. these two groups have deep roots in the geographical united states and histories marked by colonization (donato, 1997; suárez-orozco & páez, 2002). the history of mexican americans is one that includes a struggle for civil rights during the 1960s (donato, 1997) and selfdetermination since the mid-1800s (valencia & black, 2002); their schooling has been characterized by segregation and control, and coercion through the content (i.e., a curriculum that does not reflect their histories, promotes assimilationism, and has a colonizing effect) (moll & ruiz, 2002). consequently, to analyze and understand the current state of latinas/os’ education (i.e., mexican americans in particular), requires a perspective that takes into account the social, political, and historical patterns of exclusion, degradation, and racism that permeate latinas/os’ education. for this reason, we base our conceptual approach and analysis in the work of critical race theory, and specifically, in the use of counterstorytelling (e.g., parker & lynn, 2002; solórzano & yosso, 2001; yosso, 2006). critical race theory was developed by legal scholars in response to the need to acknowledge and account for the systemic unequal and biased societal conditions for people of color when considering legal arguments (delgado & stefancic, 2001). applied to the field of education, this perspective rests on the premise that systemic unequal and biased schooling conditions, access, and opportunities exist (delgado & stefancic, 2001). because these circumstances negatively impact later life outcomes (yosso, 2006), improved schooling for non-dominant students cannot occur without addressing these conditions. a counterstory serves to illuminate these biases and the processes by which they operate and influence educational acts. furthermore, a key tenet of critical race theory is that the experiential knowledge of people of color is “legitimate, appropriate, and critical for understanding, analyzing, and eradicating racial subordination. critical race research in education draws explicitly on the lived experiences of students of color” (yosso, 2006, p. 7). counterstories seek to document the processes of discrimination, bias, and racism from the perspective of those injured or affected by these factors (ladson varley gutiérrez et al. voices of urban latinas/os journal of urban mathematics education vol. 4, no. 2 29 billings, 1998; yosso, 2006). they also offer the unique opportunity to locate patterns of resistance to these processes that are often embedded in systems of schooling. solórzano and delgado bernal (2001) describe these patterns of resistance as ranging from subtle to outwardly activist, yet all contributing to transformation of oppressive conditions. for the purposes of this article, we combine counterstory with critical ethnography, ethnography that attempts to not only describe but also challenge the lived conditions of the marginalized (foley, 2002). we weave together the stories of individual students to present a collective voice of latinas/os’ lived experiences in mathematics education. this data representation allows us to understand their schooling in the context of such processes of discrimination, bias, and racism and allows us to counter the deficit-based perspectives of latina/o students that pervade educational discussions today. however, we also recognize the complexities evident in their stories. while the stories sometimes reflect a narrative of exclusionary schooling, the counterstories also offer insights into ways these students resist such schooling processes and into possibilities for disrupting these processes. setting and methods we draw on data gathered as part of a larger ethnographic study of thirdto sixth-grade urban latinas/os in after-school mathematics settings (e.g., khisty & willey, in press; varley, willey, & khisty, 2009). the after-school contexts are relevant to the present discussion because of their role as a methodological tool. the after-schools were designed specifically to offer an alternative mathematics learning environment, one that explicitly defined and utilized students’ home language, communities, and knowledge as learning resources; they also offered more advanced mathematics opportunities and practices than what the students typically received in their classes. the after-schools, thus, were intellectual spaces where students did academic work, but not like school; in a physical school-like environment, but not of school; employing learning skills and resources for thinking, but not as they did in school. consequently, the after-schools became counter spaces (gutiérrez, rymes, & larson, 1995) that naturally provoked students to compare, analyze, question, and comment on their mathematics experiences in both the after-school and their regular classrooms. our work was conducted in two geographical sites (the midwest and southwest united states) in urban schools that are predominantly latina/o and with students who run the range of proficiencies in spanish and english. with respect to language use in schools, the two contexts are very different. in the case of the midwestern site, the students attended a school that strives for biliteracy and which has content instruction in spanish and in english up to 6th grade; it is a dual language school in most respects. at the southwestern site, there are institu varley gutiérrez et al. voices of urban latinas/os journal of urban mathematics education vol. 4, no. 2 30 tional sanctions against using spanish in schools for instruction. consequently, our participants have different experiences related to spanish and english in mathematics classrooms. yet, in spite of these differences, as we will see in a later section, bias based on language ideologies affects both contexts equally. critical ethnographies of the after-schools were carried out for approximately three years. during this time, the after-schools generally took place for 8 to 12 weeks each school semester, meeting for approximately two hours once or twice a week. students volunteered to participate in the mathematics afterschool “clubs,” and a majority of them chose to participate throughout the running of the afterschool projects. between the two sites, there were approximately 34 students (distributed nearly equally across both contexts) many of which participated in the after-schools all 3 years. about two-thirds of the participants were young females. the after-schools emphasized collaborative, project-based activities that involved cross-generational participants (i.e., students, preservice teachers, and researchers). in this way, we were active participant observers with a relatively constant group of students across multiple years. in this general context, students frequently offered natural, unsolicited comments and questions—which became an essential component of their counterstories. all activities in the after-schools were videotaped, audiotaped, and/or described in extensive field notes. our discussion draws from these data. the themes that form the counterstories emerged from students’ spontaneous and more structured comments. as they emerged, data sources were revisited and analyzed to confirm and elaborate the more salient of these themes. in addition, semistructured and informal interviews with students were conducted as the occasion warranted, to further elaborate the themes. regular focus group discussions were also held to garner students’ feedback and insights regarding the after-school or any other topic they chose; these, again, served to confirm the relevance of the themes. in this way, the stories that emerged include the students’ own perspective of their schooling experiences, which we then situate within a larger societal context, following tenets of both critical ethnography and critical race theory (foley, 2002; yosso, 2006). the themes we discuss represent the composite voice of the schooling experiences of a particular group of students across settings, presented as counterstories. in some cases, we use the voice of one student to represent the experiences of others, and in other cases, one student’s “story” offers the full counterstory. the nature of counterstories allows for the voices of students to form a “living theory” in itself. the themes from the students’ counterstories represent their situated perspectives on mathematics education and the tensions they perceive in their collective mathematics learning. because these students share similar systemic forms of exclusion, we present their voices as a collective reflection of this exclusion. varley gutiérrez et al. voices of urban latinas/os journal of urban mathematics education vol. 4, no. 2 31 in the following section, we discuss four of the most salient themes to emerge from student’s comments over the years of participating with them in mathematics in the after-schools. we present their own perceptions of their mathematics education, as well as their stories of resistance to school mathematics, examples of resistance to current arrangements concerning language and mathematics, and illustrations of resistance revealed through the alternative arrangement of the after-school learning experience. the counterstories white, middle-class students often experience mathematics as challenging and collaborative problem solving (gandara & contreras, 2008), and therefore personally beneficial. consequently, they co-construct shared perspectives about mathematics consistent with their experiences. latinas/os and other political minorities, on the other hand, too often have very different experiences with mathematics, experiences often characterized by rote learning of rules and steps to solve problems (khisty & willey, 2008; martin, 2000); likewise, they develop a shared perspective consistent with those experiences. because the experiences are different, latinas/os’ attitudes and perspectives about mathematics differ accordingly. this reality begs the questions of what are latino students’ perspectives of their regular school experiences, and what are their perspectives given significantly different experiences with mathematics. in addition, in what ways do latina/o students resist these schooling experiences? moll and diaz (1987) raised the following concern: although student characteristics certainly matter, when the same children are shown to succeed under modified instructional arrangements it becomes clear that the problem these working class children face in school must be viewed primarily as a consequence of institutional arrangements that constrain children and teachers by not capitalizing fully on their talents, resources, and skills. (p. 302) the issue, consequently, becomes understanding (from the perspective of the students) the “social arrangements” of classroom mathematics juxtaposed against a different environment (i.e., after-school) that is aimed at capitalizing on students’ linguistic and cultural resources, and their lived experiences. how do their perspectives of mathematics in the different learning environments compare? we begin with the students’ expressions of their perceptions of mathematics learning. latina/o students’ perceptions of mathematics learning if we are to understand students’ affiliation to or defection from learning a specific content or persisting in schooling, then it is relevant to understand how varley gutiérrez et al. voices of urban latinas/os journal of urban mathematics education vol. 4, no. 2 32 students think about or define those experiences: do these experiences draw them in or push them to the periphery of the learning community? do they align with their own sense of what they should be learning? our analysis of students’ perceptions of mathematics and mathematics learning did indeed reveal patterns reflective of constricting institutional arrangements. although the bulk of our argument lies within the students’ counterstories to this kind of dominant, exclusionary narrative of schooling, which is presented in the subsequent section, we begin with a brief description of ways in which students’ stories sometimes reflected these dominant narratives. this portraiture is important because it presents a picture of mathematics education as experienced by the students. representative of the majority of after-school participants’ perceptions, mariana 1 described mathematics as follows: i think of numbers, like 2 times 20 or 2 times 100. …when i think of math, i just see a whole bunch of numbers, and problems go around me, and i get dizzy, and then i fall because it’s so much work. even when pushed to stretch their definition, students rarely broke out of the “mathematics-is-numbers” box. when students were asked to consider what it takes to be “good at mathematics,” they largely gravitated toward external measures of mathematical success, such as grades or test scores. this perception was closely linked to comments that good mathematics students “do homework,” “practice,” “pay attention to the teacher,” and “show up for class on time.” these notions of what it means to do mathematics have little to do with engaging in mathematical practices as a learner, and more importantly, they disenfranchise students who do not readily conform to these behavioral norms. despite these “dominant” stories, students told stories that countered this narrative either in ways in which they resisted the narrative or in ways in which the after-school mathematics setting offered an alternative or contrasting experience. the following counterstories shed light on an alternative narrative of mathematics education, one that reveals patterns of resistance to the dominant narrative and the potential of an empowering mathematics education. latina/o students’ counterstories of resistance to school mathematics: defining mathematics differently even though students expressed limited conceptions of mathematics learning, they still told counterstories of resistance, or what we deem stories that challenge the school system’s narrative of what the students can or should do in math 1 all student names are pseudonyms. varley gutiérrez et al. voices of urban latinas/os journal of urban mathematics education vol. 4, no. 2 33 ematics. several students expressed resistance to being defined as “good at mathematics” through external evaluations such as by the teacher or from test scores. margarita related: i like to be able to think a certain way, like not somebody telling you, you have to think this way in order to do that. you just think your own way to find out the answer your way. there [are] different strategies, and you can find your own strategy. in another instance, margarita said: some kids hate math, but at this point, i’m acing math because math is really easy for me. i like math. it’s one of my favorite subjects. it’s just something that i click right away. …probably ‘cause i pay attention so i understand. i find all the details that help you find it. my teacher, he finds a way of doing it like finding division, but i have a different way of finding it so for me it’d be easier. these comments reflect margarita’s confidence in her mathematical abilities and a problem-solving orientation to what it means to do mathematics (e.g., “find your own strategy”). although margarita did describe these problem-solvingbased mathematical practices, she also acknowledges an existing tension as she speaks about more discrete activities and an emphasis on speed. margarita explained: i like to work alone in math because usually when i have other kids with me i have to help them and i really want to get my work done and to me if i don’t finish my work then i feel like i’m gonna get in trouble. at the same time, margarita described the importance of taking “your time”: “it’s like when you think of something and you take your time and that’s when you pass something. but you have to take your time to get it right.” although she says that mathematics “clicks right away” for her, and that it is important for her to finish her work or she might “get in trouble,” she also describes the need to “take your time” in order to find the strategies that make the most sense and to really “understand.” we see these somewhat contradictory ideas (e.g., emphasis on outcome vs. process) as evidence that students believe they have strengths that are not fully recognized or appreciated in classrooms. margarita was both playing the game of school, while resisting the form of learning that was highly valued in her classroom because she felt it was not necessarily the best way to “understand.” these same ideas may be reflections of the dual presence in many u.s. schools of both an emphasis on a reform curriculum intended to develop student understanding and the pressures from a timed, standardized testing and tracking-based school culture. despite an emphasis on numbers and operations, and being good at math varley gutiérrez et al. voices of urban latinas/os journal of urban mathematics education vol. 4, no. 2 34 ematics as measured by external factors such as test scores or grades, some students, like margarita, resisted these notions and articulated more productive perspectives of themselves as mathematics learners; that is, she acknowledges a sense of agency, one that reflects her confidence and capacity to access tools and approaches to solving mathematical tasks beyond those typically promoted by the teacher. in addition, we were surprised by a final, subtle pattern related to critique and resistance that emerged from some students: a sense that they were being misjudged and abandoned mathematically—or “short-changed”—in their classrooms. yesenia conveyed how she would like “a lot of math things like the charts [referring to mathematical graphs],” higher-level content she knew existed but did not have sufficient lessons on. similarly, another fifth grade latina, katrina, wanted to know “how shapes and polygons are,” a clear reference to wanting more advanced mathematics, but fall outside the frequent emphasis on numbers and operations. finally, a third latina, maribel, offered the following assessment of what was missing from her fifth grade class: “que nos enseñaran come hacer pi…como hacerlo porque nos enseñaron un poco pero pararon porque nos tuvieron que enseñar mas de ciencia. [that they would teach us how to make pi…how to make it because they taught us a little, but they stopped, because they had to teach us more science.]” juliana noted that she felt that her participation in her mathematics classroom was “to make me feel like crap ‘cause i don’t know how to do the work.” this feeling was in direct contrast to her description of her participation in the after-school that was “actually there to help me,” and “i’m not being graded on it.” other students similarly noted they felt that good grades in math came from behavior that reflects the ability to play the game of school, such as, “doing your work,” “turning in your homework,” and “paying attention.” these comments are significant indicators that these students did not trust the system of schooling, in part, because they know they were not getting all they should (e.g., “advanced” or more challenging mathematics), and that they recognized there is more to mathematics than the basic operations that tend to dominate their curriculum. moreover, as maribel implied, she was being abandoned, relegated to know less than she could have known—all because there is not enough time. these comments are real indictments of these students’ classroom learning arrangements, some of which are dictated by ill-conceived policy or district mandates. latino students’ counterstories of language and mathematics a frequent theme to emerge among students’ comments related to the role spanish and english played in the context of mathematics. given the essential role that language plays in learning and its connections to identity, this is an important element in understanding latinas/os’ perceptions about mathematics edu varley gutiérrez et al. voices of urban latinas/os journal of urban mathematics education vol. 4, no. 2 35 cation and their sense of affiliation or alienation toward it. moreover, we recognize that language bias and language ideologies represent perhaps the most overt form of bias, discrimination, and racism that latinas/os experience; they are precisely the processes that need to be exposed through the students’ counterstories. when we asked students for some background related to spanish and english use in their homes, we found a diversity of experiences consistent with what is generally known about latino families and language use (zentella, 2005). most of the latina/o students in our after-schools came from homes where one or both parents speak only spanish, but where many family activities are conducted in both spanish and english. for example, elisa described her mother as “speaking english so, so…[and] my dad knows a lot [english]…and i always talk to him [in english], well i talk to him in spanish, too.” students noted that their siblings tended to speak english amongst each other because they are english-dominant. given the various proficiencies, students made many language choices. in other words, they moved between the two languages depending upon with whom they spoke, and as in the example above, depending simply on “how they feel.” furthermore, many students described themselves, their family, and their home as “bilingüe” (bilingual). the self-identification as “bilingüe” seemed important to students, because as katrina noted, “i like spanish because there’s a good connection with my mom and my parents,” and as alonzo explained, “when i speak spanish, it feels like i am in mexico.” however, some students clearly gave english status: “because i am attracted to english” (alonzo), and because “solo aprendemos palabras mas grandes [in english] [we only learn bigger words {in english}]” (maribel). in addition, the majority of the english-dominant or english monolingual latina/o students at the southwestern site—where spanish was all but outlawed in schools—spoke about a desire to speak spanish or to speak more spanish. for example, vanessa (vane) described herself as of mexican american and native american descent and her family as spanish-speaking. vane explained that she spoke some words in spanish and understood a bit more, although she said she would like to learn more spanish and have her classes in spanish because “i’m mexican, and i don’t know that much spanish.” although spanish was not her first language, because she is latina, she explained, she identified with spanish. she went on to say, “my whole family is mexican and most of my family speaks spanish.” like vane, the majority of the students in the after-schools referred to their affiliation with spanish because they were latinas/os, whether they were english or spanish-dominant. interestingly, in the mathematics context, students tended to view different roles for the two languages: one language was used to read or comprehend a problem and another was used to express their ideas—and maybe something different yet for working with peers. in the following statement from katia, a fifth grade varley gutiérrez et al. voices of urban latinas/os journal of urban mathematics education vol. 4, no. 2 36 latina, we can see how she uses her bilingualism as a tool. (note that “i” refers to the interviewer and “k” refers to katia.) i: ok, ¿cuando trabajas en matemáticas qué idioma usas el inglés o el español? ok, when you work in mathematics, do you use english or spanish? k: los dos. the two. i: …¿cuál idioma sientes que es mas fácil para ti para usar para que puedas comprender el problema, cuando está en inglés o en español? …do you feel is easier to understand the math problems in english or spanish? k: en español. in spanish. i: ¿…. y para explicar tus ideas en el problema? and which one is easier to express your ideas? k: ingles. english i: ¿inglés, te explicas mejor en inglés? pero te gusta mas cuando está el problema en español. ¿y para hablar en general? you express yourself better in english? but you like it better when the problem is in spanish. what about when you are speaking? k: spanglish i: uh? k: spanglish while this student demonstrated a keen sense of meta-linguistic awareness (i.e., recognizes the need to use one language for one thing and another for another objective), some students were just as aware of the fact that their skills in the two languages were not equal, and because of this imbalance they had to rely on one language or the other for a given linguistic task. students noted that they “don’t read english well enough” or “speak spanish well enough for expressing ideas” to really be able to use the two languages as tools for learning. in the following interview response, we see an older student who exemplifies an awareness that something is missing from her repertoire of resources for learning mathematics. margarita speaks spanish at home with her parents and grandmother, though she clearly describes her struggle with spanish as an academic language: (note that “i” refers to the interviewer and “m” refers to margarita.) m: if i’m doing multiplication or take away, then i would probably speak in spanish or english, but if i was doing division, like the harder ones, division or fractions, i would have to speak english ‘cause fractions right now, doing the minus and stuff like that and times and stuff like that with fractions gets to your head and stuff like that…so i would have to probably speak in, in english for that. …because like sometimes in spanish i, like menos is take away so if i were to talk, i would know how to say take away in spanish so if i were to explain it to somebody i would tell them, “this menos this equals what?” and if i were to say division, i have no idea how to say division in spanish, so i wouldn’t know how to explain to somebody who just knew how to speak spanish. i: and is explaining part of the math class, learning math? varley gutiérrez et al. voices of urban latinas/os journal of urban mathematics education vol. 4, no. 2 37 m: yeah, it’s also if my teacher asks me questions and i can only speak spanish, then that would be, that would be hard for me, because i don’t know how to say fractions or division in spanish. i know how to say math and all that stuff, but i just have no idea how to say division and fractions. although margarita focused explicitly on vocabulary, she referred to specific instances of communicating mathematical ideas in spanish and the struggle she anticipated if she did not have the spanish vocabulary to communicate her thinking effectively. what we hear from the students is their pride in their bilingualism or its potential, and the recognition that the two languages are learning resources— not only for literacy development but also for mathematics. current research supports this idea that two languages can serve as powerful tools in students’ mathematical development (e.g., planas & setati, 2009). unfortunately, the students also remind us that their bilingualism is an untapped and underutilized resource, especially in mathematics. from the students’ comments, we can conclude that at least two phenomena are occurring here: (a) the students are accessing and utilizing two languages for various sub-tasks (within a larger mathematical task) to communicate and make meaning of mathematical ideas, and/or (b) students regret their lack of proficiency in academic spanish, effectively deeming it a lost learning resource. in either case, the importance of maximizing spanish as a resource becomes clear, whether it be to facilitate cognitively challenging tasks or to help verify the notion that spanish is a tool worthy of development and use in academic settings, a reality that has major implications for latinas/os’ mathematical identities (gonzález, 2001; khisty, 2006). regardless of the language context, their stories point to a need for a re-orientation to the role of language in mathematics. the dominant narrative tells us that english is important because of its role in mainstream instruction, tests, and general school and social advancement. while this may be true in some respects, it is also true that spanish has critical functions as a cognitive tool to help students make sense of complex mathematical ideas, as a means of maintaining family and community connections, and as a marker of historical and present identity. as the students’ counterstories imply, bilingualism is a strong alternative to monolingual schools and mathematics classrooms. latino students’ stories of the after-school setting students’ comments offered evidence that the after-schools began to challenge and alter some of the socialization patterns of the regular classroom that we described above. as noted earlier, the afterschool environments clearly provided students with alternative experiences with mathematics, ones that both supported their mathematical understanding, and also connected to who they are by valuing varley gutiérrez et al. voices of urban latinas/os journal of urban mathematics education vol. 4, no. 2 38 and integrating their language (i.e., adults often spoke in spanish and tasks were written in two languages), experiences, and communities. when we asked students to compare the after-schools to their mathematics classrooms, students frequently noted how their regular classroom experiences were limited in the diversity and quality of activities. maribel, noted: “we don’t use those games [in class] to make it more enjoyable; we only use them for fractions.” katrina pointed to the fact that “in class, we don’t make our own math problems” (a very common activity in the after-school). several students spoke about the benefits of problem solving in the after-school, a reference to working on more challenging, open-ended problems frequently embedded in games or projects. vane described: you don’t know, first of all with the investigation, 2 you don’t know what type of math you’re doing. that’s what you have to find out: ‘oh, i have to do multiplication for this, or division, or addition, or subtraction. often, the mathematical investigations in the after-school developed from the interests and experiences of the students and were grounded in their communities, rather than being based on a content area, as is often the case in classrooms. moreover, other students noted the different patterns of interaction in the after-schools, for example: “if i don’t understand something in the classroom, i ask the teacher, and he explains it more, and in los rayos [the name of one of the after-schools], i can ask a lot of different people” (yesenia). another student expressed that in the after-school, the facilitators ask questions rather than offering explanations; or, in other words, they engaged students in dialogue. still a different student, marlena, described the learning environment in the following way: instead of—you can’t get up on your feet in the normal class, like you have to stay with the person you are working with. you can’t go around and check what they’re doing to see if you or your answer…to see if whoever you are working with, to see if you got the answer right with another pair. but when you’re at the after-school, you can move around and ask them, “oh, what did you get? because i got this…” and then we look at each other’s work, and we see if one of us got it wrong. and it’s kind of better than in class. marlena spoke about seeking out peers as resources in finding valid strategies for solving problems, and how the after-school environment facilitated this freedom, even if only in the sense of physically being able to move around the room. these responses in particular are indicative of students’ recognition of the normative 2 investigation refers to a mathematical investigation that took as its starting point the concerns of students; in this case, the potential closing of their school. varley gutiérrez et al. voices of urban latinas/os journal of urban mathematics education vol. 4, no. 2 39 behaviors required in the classroom as compared to the different ways of interacting and communicating in the after-schools. it should be noted that such comments or counterstories came from students from both project sites. these comments also reinforce the idea that latina/o students recognize that there are other possibilities for learning. finally, students’ experiences with community-based mathematical investigations opened a window into the role of mathematics as a tool for more outward forms of resistance, resistance to the current structures of society. some of the mathematical investigations that we conducted with the students involved projects that specifically linked social issues in their community to mathematics. these instances gave us an opportunity to explore students’ ideas about the role or usefulness of mathematics for social change. we explored their ideas about resisting conditions in their lives and connections to mathematics. drawing heavily upon their experiences with community-based mathematics projects aimed at creating change, students shared stories about making change in their lives and the role of mathematics in making that change. vane described how she would like to make her nana’s (grandmother’s) neighborhood safer, referring to specific actions undertaken during a project to save her school from being closed. when asked if she felt that she could do anything about the crime in her nana’s neighborhood, she replied: “not unless the whole community comes together and talks about it. …because we might hear other people’s thoughts about their feelings about getting robbed; having their whole neighborhood robbed.” she linked these ideas to what she had learned through her involvement in the community movement to save her school, explaining, “’cause our community kinda, sorta got together on that thing we had, the rally we had on that board meeting.” interestingly, students detailed specific ways in which they could leverage mathematical investigations to make social change. more specifically, they expressed confidence in their ability to collect, organize, and represent data in ways that would inform and convince others that there is a problem that needs to be addressed. for example, vane felt that her grandmother’s community could use mathematics, as well as community members in positions of power, to improve the safety of that neighborhood: “[we could use math] finding out how many people feel about the robberies and how many houses have already been robbed.” vane went on to explain her belief in the power of organizing by saying, “when we did this [community-based project], i felt like we could do whatever we wanted and that the board members are listening to us.” in this case, the board members were the community members that needed to be convinced to not close her school. she referred to specific, concrete actions done in the community-based project to describe what her nana’s neighborhood could do to change the situation. varley gutiérrez et al. voices of urban latinas/os journal of urban mathematics education vol. 4, no. 2 40 in reflecting on the community-based projects, students spoke about their belief in the power of using mathematics to make their arguments and how this belief contributed to their sense of voice. adán, who participated in a project to document and share issues about bullying with the teachers at his school, remarked, “i like seeing all the statistics in action.” given that many students find little utility in mathematics beyond consumer practices, it is striking that this student saw how statistics could be utilized in an authentic context. other students remarked on the ability to show their thinking and “show drastic differences” using mathematics, and how this could lead to social change. in spite of all the emphasis given in their classrooms to rote manipulation of numbers, students exhibited a flexible understanding of how the mathematical tools they used and developed in the after-school investigations could be applied to other situations. their comments about their after-school experiences suggest the value of directly connecting mathematics to their lived realities and providing latinas/os with the requisite tools to imagine not just ways to participate in mathematics, but ways to participate in making social change. based on these counterstories, it is clear that for students who are marginalized, not just in schools but in many facets of society, the development of tools to engage in social change could be an important component to reversing this trend of marginalization. concluding thoughts and implications the latinas/os’ words we bring forth in this discussion tell important counterstories. at a time when there is extensive concern about latinas/os’ progress in mathematics, their stories confirm that, in fact, instruction appears to be constraining with an over emphasis on lower-level content. in light of this context, it is little wonder that many latinas/os become distanced from mathematics and that few embrace mathematics and identify with it. however, our findings also point to latinas/os’ unwillingness to accept this condition. their counterstories tell of resistance to schooling practices related to what it means to do and learn mathematics. for example, they tell stories of resistance to the kind of learning environment where the teacher is the sole holder of knowledge and instead declare their own ability, as noted earlier: “i like to be able to think a certain way, like not somebody telling you.” in addition, the students clearly point to their recognition that there is learning power in two languages, power not being utilized. the students’ recognitions in this regard are fractures in the façade of the dominant narrative that learning mathematics can or should be done in only english. in addition, based on the students’ experiences using mathematics in community-based projects, the students told stories of using the mathematics content for a more overt form of transformative resistance (solórzano & delgado bernal, 2001) to oppres varley gutiérrez et al. voices of urban latinas/os journal of urban mathematics education vol. 4, no. 2 41 sive social structures. we can use these stories as insights of what and how to take steps towards better mathematical experiences for latinas/os. these findings raise questions about schools’ effects on latinas/os’ mathematical agency, in this case, referring to their overall sense of power, selfdetermination, and positive identity. are these students given sufficient and appropriate intellectual space in classrooms to genuinely engage in developing higher-level mathematical understanding—or to become empowered? are there spaces for their voices of resistance to current schooling conditions? if yes, are we positioned to faithfully listen and act on their insights? unfortunately, latina/o students do not control the curriculum, which raises further questions about the preparation of teachers and other educators to move beyond simply mastering teaching practices related to content and pedagogical approaches. how do we develop educators to move toward understanding student histories, listening to students’ stories, building on students’ unique knowledge (e.g., home language), and granting abundant and explicit opportunities for students to be inducted into authentic mathematical practices through communities (gee, 2005)? ultimately, there are questions about how to challenge educators’ own stereotypes and biases concerning latinas/os and other non-dominant students. moreover, the students’ counterstories raise questions about ideologies— especially those related to language—and how they constrain both students and teachers, causing many learning tools and resources latinas/os have—including spanish—to be grossly neglected. is mathematics education complicit in holding latina/o students back by not paying any or enough attention to the role of students’ home language (spanish) in instruction and in curriculum? despite the differences in contexts, the prevalence of english-only or assimilationist ideologies becomes clear through the students’ voices. students are bombarded daily with schooling practices (i.e., instruction, course requirements, standardized tests, textbooks, etc.) and tacit or explicit messages that convey a primary and urgent goal of abandoning spanish and developing proficiency in english. these students’ counterstories convey the damage that is caused by the overwhelming quantity and abrasiveness of language ideologies that denigrate spanish, a critical component of their being or identity, but one that is systematically extracted from their repertoire of learning and communicative resources. because critical race theory tells us, “that these traditional paradigms act as a camouflage for the self-interest, power, and privilege of dominant groups in u.s. society,” (solórzano & delgado bernal, 2001, p. 313), the counterstories we have presented are all the more crucial to transforming educational experiences. the students’ counterstories suggest that unless fundamental changes are made, oppressive schooling conditions and schooling failure will persist. varley gutiérrez et al. voices of urban latinas/os journal of urban mathematics education vol. 4, no. 2 42 acknowledgments this research was funded in part by a national science foundation center for learning and teaching grant to cemela (the center for the mathematics education of latino/as), grant no. esi-0424983. any opinions, findings, and conclusions or recommendations expressed are those of the authors and do not necessarily reflect the views of the national science foundation. references darling-hammond, l. 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(2005). building on strengths: language and literacy in latino families and communities. new york: teachers college press. http://factfinder2.census.gov/faces/nav/jsf/pages/index.xhtml journal of urban mathematics education july 2015, vol. 8, no. 1, pp. 83–118 ©jume. http://education.gsu.edu/jume lorraine m. baron is an assistant professor in the institute for teacher education at the college of education – university of hawaiʻi at mānoa, 1776 university avenue, honolulu, hawaiʻi, usa, 96822; email: baronl@hawaii.edu. her research interests include curriculum implementation implications for the classroom teacher, community mathematical literacy, and philosophy of mathematics education. financial literacy with families: opportunity and hope lorraine m. baron university of hawaiʻi at mānoa in this article, the author explores the link between citizens’ quantitative literacy abilities and their financial prosperity. the author applies a robust social justice research vision and a freirean approach to describe personal flourishing within the context of numerical, mathematical, and financial literacy (nmfl) education. four families participated in a weekly evening community program that was designed to inform them about nmfls. analysis of the interview data showed that participants described a sense of personal flourishing, gained confidence and skills, and felt financially empowered enough to teach/transfer that knowledge to their children. the author proposes a conceptual framework linking personal flourishing with nmfls and suggests the framework be used to investigate and describe quantitative literacy and financial literacy in future empowering pedagogies research. keywords: community research, financial literacy, flourishing, quantitative literacy, social justice research umerical, mathematical and financial literacies (nmfls) are pressing economic, social, and cultural challenges. for example, as lusardi and mitchell described (2011), international research shows that financial illiteracy is widespread. this lack of financial literacy is true in various countries, even “when financial markets are well developed as in germany, the netherlands, sweden, japan, italy, new zealand, and the united states, or when they are changing rapidly as in russia” (p. 497). moreover, a strong link has been shown between citizens’ basic numerical or mathematical abilities and their financial prosperity and civic engagement (human resources and skills development canada [hrsdb], 2012). steen (2001) argued that quantitative literacy (ql), which includes personal finance and other mathematical expressions, is increasingly necessary for fully contributing citizens of tomorrow. the problem of financial illiteracy can be particularly limiting for families who live in communities where there is a low median income, where unemployment is common, and where financial capability is low. for “members of marginalized and low-income groups, it is basic financial knowledge and skills—in addition to a meager subsistence level income—that prohibits their amassing the necessary savings to even consider investment options” (ek-udofia & spotton visano, 2012, n http://education.gsu.edu/jume mailto:baronl@hawaii.edu baron financial literacy with families journal of urban mathematics education vol. 8, no. 1 84 p. 5). literacy (reading and writing) is most commonly targeted as an approach to provide spaces for empowerment for marginalized communities. when we can read and write, we can advocate for ourselves and participate more broadly in society. the work and scholarship of the brazilian educationalist, theorist, and activist paulo freire has become the (international) model for self-empowering and transformative pedagogies through reading and writing (see, e.g., 1970/2000). in this study, these same principles of self-empowering and transformation are applied for mathematical literacy. the count on yourself (coy) project applies ideas of self-empowerment through numerical, mathematical, and financial literacies. count on yourself provided a parallel program for adult and child numerical, mathematical, and financial literacies: while parents were involved in a financial literacy course, their children participated in a math camp led by preservice teachers from a local university. the goal of the project was to provide the space for both adult and child participants to become more self-empowered with respect to mathematical and financial literacies. for the adults, this meant that they learned more about aspects of financial literacy that they could apply to their lives, and for the school-aged children, this involved participating in math camp activities that emphasized conceptual understanding of numbers, place value, fractions, decimals, money, and financial concepts. this initiative aimed to integrate research and practice and to address practical needs of an urban community. based on the coy project, a research project was designed to analyze how participants might express what they learned about financial literacy and what they felt were the greatest benefits from the program. one of the purposes of the study was to develop a conceptual framework linking personal flourishing (subsequently described) with numerical, mathematical, and financial literacies (nmfls) and the participants’ “voices” and to suggest that this framework could be used in future studies to investigate and describe work with communities with respect to nmfls. using an in vivo approach (see andersen & kragh, 2010), i applied a robust social justice research vision (grant, 2012) to develop a framework for describing how participating families benefited from this financial and mathematical literacy project. three primary questions guided the project: 1. how did the participating adults describe what they learned about financial literacy from the coy program? 2. how did this knowledge impact (or not) them and their families? a. what did adult participants notice, understand, or come to believe? b. how, if at all, did this impact their motivation and ability to act to improve their financial circumstances? 3. what were the features of the coy program that adult participants felt were most beneficial to them and to their families, and why? baron financial literacy with families journal of urban mathematics education vol. 8, no. 1 85 numerical, mathematical, and financial literacies and community needs why nmfls? a review of the literature identified various definitions and meanings for mathematical literacy. some used the idea of numeracy (using mathematics in the world) as parallel to that of literacy (using reading and writing in the world). steen’s (2001) definition of quantitative literacy became the most useful overarching definition of what it meant to be able to use number and mathematics in the world, and of its importance for fully participating in the democratic process. financial literacy is considered, within steen’s definition of ql, one of the expressions of ql. steen explained that we express our quantitative literacy when we enact our financial capabilities. with respect to the word numeracy, i have also found that, in practical settings such as schools, numeracy is often used to mean “basic facts”; hence, for example, a numeracy course might be assigned to students who were deemed not to be able to do “real” or “important” or “serious” mathematics. the course would consist of an overly simplified curriculum. the idea of numeracy or quantitative literacy too often involved an oversimplification of the four operations of addition, subtraction, multiplication, and addition, and a focus on speed drills, rather than encompassing the broad and applicable notions as described by steen (2001). for this reason, the word numeracy often fails to capture the essence of ql. though the adult course in this project involved financial literacy, the children’s program included concepts typically taught in the k–6 mathematics classrooms as well as ideas about money and finances. for this reason, i included the ideas of ql and renamed the focus of my project as numeracy, mathematics, and financial literacies. these competencies are linked to each other, as illustrated below. numeracy and its link to financial literacy as defined by the hrsdb (2012), quantitative literacy or numeracy is “the knowledge and skills required to effectively manage mathematical demands” (p. 22). the hrsdb study addressed the objective of facilitating “the creation of opportunities for canadians to acquire the learning, literacy and essential skills they need to participate in a knowledge-based economy and society” (p. iii). hrsdb found that approximately half of respondents (49%) scored only level 1 or 2 [of 5 levels] on the numeracy proficiency scale. these low results are significant because numeracy and problem solving have been linked to financial prosperity and civic engagement (behrman, mitchell, soo, & bravo, 2010; hrsdb, 2012; smith, mcardle, & willis, 2010). measures of financial capability, which included making ends meet, planning ahead, managing financial products, and financial knowledge baron financial literacy with families journal of urban mathematics education vol. 8, no. 1 86 and decision-making, are disquieting. for example, more than half of the u.s. population has no rainy day fund, only one-third consistently pay the full monthly owing amount on their credit cards, and fewer than 40% could answer 4 or 5 [out of 5] fundamental financial literacy questions correctly (finra investor education foundation, 2013). similarly, a canadian report (statistics canada, 2008) found that 49% of all adults say they do not engage in basic household budgeting (57% between the ages of 18 and 29 do not have a budget, while 47% between the ages of 60 and 69 do not maintain one). financial literacy and civic engagement innumeracy, or “an inability to deal comfortably with the fundamental notions of number and chance, plagues far too many otherwise knowledgeable citizens” (paulos, 2001, pp. 3–4). numeracy was determined by smith and colleagues (2010) to be “by far the most predictive of wealth among all cognitive variables” (p. 18). in short, mathematical literacy has an impact on an individual’s financial health and prosperity. behrman and colleagues (2010) worked to isolate the causal effect of financial literacy on wealth accumulation and found that, compared to other variables, wealth accumulation was dependent on financial understanding and skills more than any other variable. in these studies, an understanding of mathematics and finances early in life was found to have a significant impact on a person’s wealth. there is substantial evidence that financial literacy programs “can make an important contribution to the well-being of vulnerable groups” (mcfayden, 2012, p. 1). gutstein (2006) wrote, “reading the mathematical word is equivalent to developing mathematical power” (p. 29); he claimed that opportunities to learn, access, and equity all demand that historical marginalized groups get the chance to develop the tools for mathematical empowerment (p. 30). this study engaged grassroots community resources to support nmfls and applied a qualitative research approach for the purpose of improving awareness, access, and the quality of programs available to the participating school community. the community and its needs the elementary school in which this project took place is one of three elementary schools in a smaller, urban community (population 28,000) within a larger city (population 115,000). the larger city’s main activities include tourism, agriculture, light industry, forestry, manufacturing, and the high tech industry. the school is considered a community school, which means that it serves as a space for local community events on weekends and evenings as agreed to by the city council and the school board. this smaller urban center is generally perceived as a lowersocioeconomic region and a “less desirable” location. compared to the larger surrounding city, this small urban center has the lowest median income, the highest baron financial literacy with families journal of urban mathematics education vol. 8, no. 1 87 unemployment and underemployment rates, the highest number of single parent families, the highest number of rental properties, the highest number of social assistance recipients, the highest number of drug-related arrests, and the highest number of visible minorities. despite these statistics, this community also has high family values and a proud community history. though others may perceive this smaller urban community as less desirable, its residents exhibit a sense of pride and cohesiveness, as has been demonstrated regularly through community events and publications. leonard (1984) characterizes marginalized people as remaining “outside the major arena of capitalist productive and reproductive activity” (p. 181). because of the indicators of low socio-economics and because of its location outside the newer town center, the citizens of this smaller urban center have less access and information about free financial or educational programs, or other forms of social assistance that might be available in the newer and larger town center. the needs of this urban community determined the setting for this study. at the time of the study, the school principal had been an administrator in the community for over 20 years and was well trusted by the community and parent council. the schools in which this principal worked had historically poor academic performance, a prevalence of large numbers of both lowand high-incidence special needs students, and a high rating on an index measuring early childhood vulnerability. data indicated that students from these schools had poor achievement in intermediate grades, and were less likely to graduate than their peers (british columbia ministry of education, 2006). the principal was a colleague and our professional relationship was based on the fact that i had supported k–12 teachers in a district role as a mathematics consultant for over a decade. knowing this particular principal’s history of working with the most marginalized communities in our school district, i consulted with her regarding which school community might benefit from a financial literacy and math camp project such as coy. because she was highly trusted and a consistent and dedicated member of the community, and i had been a support for the school and district, the parent advisory council of the school was quick to approve the program and the research. the project was announced at a community forum, and four families chose to participate in the coy quantitative and financial literacy program. though all members of the school community were invited to participate, it was fortunate that a fairly small group signed up for the first iteration of this program. a trusting environment could be more readily built with a small group. during the progression of the program, more families and other school communities became interested in coy. it had been expected that the program would be repeated, either in the same location, or at another school in the community. a nearby school had requested the program be provided for them in the following school year. baron financial literacy with families journal of urban mathematics education vol. 8, no. 1 88 the participants determined the definition of “family” in this study. though some families may have been more traditional (e.g., mother, father, two children, etc.), the participants defined others. this distinction is important because a mother had asked for her fiancé to participate even though he did not live in the community. this “family” was formed from blending the mother’s two children from previous relationships with the father’s two children from a previous marriage and a shared child on the way. furthermore, in other cases, the fathers did not participate in the project, and in one case, neither did the children. it was important to be inclusive of all who wished to participate. the composition of the families and some demographics are provided in table 1. table 1 participating families family description mother father mother and father (family of four); two participating children in grades 4 and 5 valerie: 35-year-old mother of two; works at a financial institution brent: 43-year-old with teacher training/unemployed father of two; husband of valerie mother and fiancé (family of six), four children plus one on the way, (combined family: two from ema, two from phil); one of phil’s children participating in grade 2, the other three not yet in elementary school ema: 29-year-old mother; ema works as an office assistant; during the course, she was pregnant with a fifth child on the way from this combined family. phil: 42-year-old father occasionally employed as a heavy equipment operator; father of same children and fiancé of ema mother and three children (family of five), grade 1 and grade 6 child participating, (third 8th grade child served as a babysitter) anna: 41-year-old married mother of 3 children; home business owner; married to tradesman husband not participating in coy mother (family of four); husband and children not participating in coy) carolyn: 42-year-old married mother of two boys; works outside of home for family business; married to business owner husband not participating in coy theoretical framework this study utilized the in vivo approach to build theory (as described in andersen & kragh, 2010). according to andersen and kragh, the theoretical framework of a project is evolving and acts as a mechanism for understanding the empirical data. as dubois and gadde (2002, see also 2014) described, this approach involves systematic combining and the creation of a tight and emerging framework where an overriding theoretical stance is initially claimed, and subse baron financial literacy with families journal of urban mathematics education vol. 8, no. 1 89 quently used to frame and reframe the findings throughout data collection, analysis, and then re-combining of theoretical ideas. dubois and gadde explained that newly developed research frameworks are formed by critically evaluating emerging constructs against ongoing observations. the theoretical framework subsequently evolves during the study. as dubois and gadde explained, the “framework is successively modified, partly as a result of unanticipated empirical findings, but also of theoretical insights gained during the process” (p. 559). the goal is to develop a theory that demonstrates both novelty and continuity; it must be connected to existing theory and literature and seek to resolve paradoxes, but differ from it as the researcher’s attention turns to unexpected data. critical theory as burrell and morgan (1979) explained, social sciences researchers “cannot operate in more than one paradigm at any given point in time” (p. 25). i designed my study with a critical theory perspective as a foundation with the goal of providing the space for a marginalized group to “become critically aware of their ‘true’ situation, intervene in its reality, and take charge of their destiny” (stinson & bullock, 2012, p. 1166). freire (see, e.g., 1970/2000) wrote about the fallacy of teaching the technical skills of reading as disconnected from social and political contexts. freire immersed himself in the community, spending time with them, “learning himself the words that are important to the people” (crotty, 1998, p. 148). he also believed in dialogue, explaining that the “oppressed cannot be liberated without their reflective participation in the act of liberation” (p. 155). similarly, mclaren (2007) claimed, “any genuine pedagogical practice demands a commitment to social transformation in solidarity with subordinated and marginalized groups” (p. 189). with these ideas in mind, i immersed myself as a participant in the adult’s financial literacy course, inviting the discourse and active participation of the adults in the program. leonardo (2004) argued that critical theorists must confront inequality and come “to terms with social arrangements that create social disparities” (p. 13). critical mathematics education enables learners “to use mathematics as a tool for social critique and personal empowerment… for the purpose of transforming one’s place within [the] world” (appelbaum, 2009, p. 194). according to grant (2012), a number of core practices are essential to achieve a robust social justice vision of education. here, i apply those practices to educational research. to begin, the researcher’s beliefs and values with respect to research must be critically examined. kemmis and mctaggart (2005) contended that it would be naive to argue that the facilitator is completely equal as a coparticipant in the group; however, the facilitator can engage as a coparticipant with “some special expertise that may be helpful to the group in its endeavors” (p. 594). they maintained that the facilitator should not be seen as overriding, but rather “as someone aiming to estab baron financial literacy with families journal of urban mathematics education vol. 8, no. 1 90 lish or support a collaborative enterprise in which people can engage in exploratory action” (p. 595). in this study, i kept my role and my voice in balance (or, at least made great efforts to do so) and critically, intentionally, and regularly reflected on my own influence upon the group. i participated in the adult financial literacy course, sitting among the participants, and sometimes (but rarely) asking questions of the facilitator. though i had clearly been the organizer of the project, i was also a learner. i was careful not to participate in a way that would limit the other voices. i consistently reflected on my involvement as i worked to facilitate the overall program. in qualitative studies, research validity is often associated with the development of ethical relationships between participants and the researcher. the reason for this focus on ethics is that more credible data can be collected if the study has been designed to establish a trusting climate. guba and lincoln (2005) deepened this idea by asserting that “the way in which we know is most assuredly tied up with both what we know and our relationships with our research participants” (p. 209). the researcher’s persona and experiences also have an effect on trust and authority; in this case, participants may have been more accepting of me because of my previous work in the community, or because the school’s principal was trusted and also trusted me to take on this work. the study reflected a significant amount of catalytic validity: “validity [that] points to the degree to which research moves those it studies to understand the world and the way it is shaped in order for them to transform it” (attributed to p. lather in kincheloe & mclaren, 2005, p. 324). from this perspective of praxis (i.e., a continuous cycle of refection and action; see, e.g., freire, 1970/2000), this study sought to deepen participants’ “understanding of society [leading] to engagement in social movements, at whatever level people are capable of participating given the daily struggles for survival” (gutstein, 2006, p. 25). gutstein (2006) viewed “writing the world with mathematics as a developmental process, of beginning to see oneself capable of making change, and…developing a sense of social agency (gradual growth)” (p. 27). the coy project sought to identify how participants described their gradual growth of understanding and engagement with mathematical and financial literacies. theme and theory development as described in the in vivo approach to theory building, one immerses oneself into the data; in this case, the voices of the participants, and their collective ideas, and as data themes emerge and questions arise, other relevant theoretical frameworks may be called upon. though i remained open to multiple themes “emerging” from the data, two research perspectives seemed appropriate as tools to make sense of the data. the two perspectives, grant’s (2012) personal flourishing and steen’s (2001) quantitative literacy, assisted in adjusting and refining the data themes. baron financial literacy with families journal of urban mathematics education vol. 8, no. 1 91 personal flourishing. grant (2012) proposed how to practice and cultivate a more robust social justice vision of education and argued for a democratic education for what he described as a flourishing and whole life. brighouse (2006) and grant (2012) described how, in order to make meaning and sense of important aspects of their lives, individuals should experience personal flourishing and personal autonomy. this meant that they must develop the confidence and skills to contribute to society and to the (broadly defined) economy, democratic competency, and the facility and desire for cooperation (grant, 2012). according to grant, personal flourishing also includes living and doing well, having a positive identity, having family and friends as support mechanisms, financial stability, education, and a commitment to children’s flourishing minds. people who flourish experience the opposite of languishing, yearning for more, or feelings of being stuck in a rut (keyes & lopez, 2002). quantitative literacy. steen (2001) studied various international documents describing what it meant to be “numerate” and found some commonalities he called elements. he also described that one could look at quantitative literacy in terms of actions and how one expresses ql. he called these expressions of ql. steen’s elements and expressions of ql are listed in table 2. table 2 elements and expressions of quantitative literacy elements and expressions steen’s (2001) list elements of ql confidence with mathematics, cultural appreciation, interpreting data, logical thinking, making decisions, mathematics in context, number sense, practical skills, prerequisite knowledge, and symbol sense expressions of ql citizenship, culture, education, professions, personal finance, personal health, management, and work steen (2001) described elements and expressions of ql that included confidence with mathematics, being able to use mathematics in context, expressing quantitative literacy through citizenship (e.g., understanding data, projections, inferences, etc.), the application of mathematics in one’s educational trajectory, the application of quantitative literacy in one’s personal finance and management, and the ability to make quantitative decisions with respect to one’s personal health (e.g., options, dosages, risks, nutrition and exercise data, etc.). steen described personal finance as an expression of quantitative literacy that is “probably the most common context in which ordinary people are faced with sophisticated quantitative issues” (p. 13). he also argued that it is “an area greatly neglected in the traditional academic track of the mathematics curriculum” (p. 13). quantitative literacy “empow baron financial literacy with families journal of urban mathematics education vol. 8, no. 1 92 ers people by giving them tools to think for themselves, to ask intelligent questions of experts, and to confront authority confidently [;]… skills required to thrive in the modern world” (p. 2). steen’s description of ql is broad, powerful, and matched some of the other literature on teaching for social justice (e.g., grant, 2012). methods the adult’s financial literacy curriculum i designed and led this research project; however, it resulted from the collaborative efforts of several groups including the faculty members from the local university and employees and volunteers of four non-profit community partners. the adults’ financial literacy course was facilitated free of charge by an accredited chief executive officer of a local financial advising and management company who also participated in non-profit work and whose personal mission was to offer free advice and programs for the citizens of the city, in particular, marginalized groups and individuals. the course was based on momentum (see table 3), a curriculum that was developed in calgary, canada (momentun community economic development society, 2010). though a curriculum plan structure was in place, the financial literacy facilitator organized the adult learning according to feedback received from participants during every session. during the first session of the course, the facilitator provided each participant with a binder of the course content (provided at no cost) and described the overall structure (see table 3). he asked participants which areas were of most interest to them, and which were of least interest. he explained that he would attend to all the areas generally, but was happy to address those areas of greatest interest. adjustments were made over the following eight sessions depending on the financial interests of the participants. in the first session of the course, participants were engaged in describing their assets. this exercise is an important starting point because one’s assets not only include financial ones, but also human assets, such as one’s skills and experience, physical assets, such as housing, food, clothing and transportation, personal assets such as motivation, self-confidence, or sense of humor, and social assets such as the support one has from family and friends. this broad perspective of assets is an example of how the course materials and the facilitator showed participants to value what they have—a competency model rather than a deficit model. if participants originally felt they did not have assets, this part of the course dispelled those beliefs and created a strong base and place of trust for future discussions about credit, budgeting, consumerism, and banking. baron financial literacy with families journal of urban mathematics education vol. 8, no. 1 93 table 3 momentum (2010): financial literacy curriculum topic content assets participants set shortand long-term goals for five types of assets (human, personal, social, physical, and financial). credit participants learned to take charge of their credit. they developed an awareness and understanding of the advantages and disadvantages of credit, using it wisely, and about consumer credit tools available to them. budgeting participants learned the difference between needs and wants, which is a critical step in setting a realistic budget and being able to follow it. they identified and tracked their spending habits over a one-week period and they used this information to build a budget that was reasonable for them. then, they set shortand long-term goals for saving or paying off debt. consumerism participants uncovered the effects of consumerism on their lives and aspects of advertising that threatened their financial stability. they discussed and worked on their ability to control their consumeristic tendencies, and developed strategies to live more simply. banking participants learned about banking account options and the benefits of each. they discussed how to use banking services and how to access banking tools and resources. the children’s math camp the coy project provided access to one-on-one, needs-based teaching during the math camp for children involved in the project. the four math camp leaders had been students in my elementary mathematics methods course for one semester. they were at the end of a 2-year program for licensure and on practicum in the field full-time when they volunteered to participate as math camp leaders to build on their “capstone project” (required for graduation) through service-learning activities. it was not necessary for them to engage in such a complex and timeconsuming plan for the completion of their capstone project; however, as i had introduced them to my research idea during our coursework, they became committed to completing the coy project with me. the design of the course was left up to them; however, i assisted by providing some extra curricula and activities materials. the math camp leaders assigned themselves to individual children, prior to the sessions beginning, according to their grade-level teaching interests (each participating child was in a different grade in school), and planned each session according to the child’s needs, desires, and interests. early in the program, the math camp leaders planned “icebreakers” and “climate builders.” they surveyed the children to find out what they enjoyed, how they felt about math, what they liked to eat, what they were allergic to, and more. this knowledge of the child helped them baron financial literacy with families journal of urban mathematics education vol. 8, no. 1 94 tailor the session to each child’s preferences. the evening sessions included nutrition and physical activity breaks, with some time together, but with most of their time dedicated to the children with whom they had planned to work. most of the time early on in the math camp was spent on mathematical content chosen for the child’s grade level. however, because the children asked about what their parents were learning and showed an authentic interest in why their parents were participating in the coy project, the math camp leaders decided to introduce financial literacy at the child’s level—linking the content of the children’s learning with that of their parents in the parallel sessions. for the last half of the 8 weeks, financial literacy for children became a large part of the focus during math camp. overall, the content of the math camp included games, lessons, and activities based on numbers, place value, fractions, decimals, money, and financial concepts. a series of children’s books on financial literacy were provided (e.g., phillips, 2010) as well as adult and teen financial literacy books (e.g., vaz-oxlade, 2010). they closed the program on the last day by spending their budgeted play money at a “store” set up by the math camp leaders. this activity was a highlight for the children in the program. the program consisted of eight meetings on wednesday evenings in the school’s library. if a family chose to participate, it was important that they be able to attend all sessions, and so, the principal and i worked with the participants to make sure the wednesdays scheduled for the financial literacy course were feasible for all who intended on participating. a meal and refreshments was available for participating adults and children. funds for purchasing and preparing meals were raised by a group of college students from a local institution. the families met in the school library, and the school-aged children were taken to the multi-purpose room. pre-school-aged children attended a child-care service provided free of charge in a small classroom in the school. data collection and procedures the proceedings of the adult financial literacy course were digitally recorded, as were individual preand post-interviews. adults were individually asked three questions at the beginning of the program in an informal interview setting that was selected by each participant (either at their homes, in the school, or at a coffee shop). they were asked six questions at the end of the program. these were digitally audio-recorded. each participating adult was interviewed for approximately 35 minutes at the beginning and at the end of the program. adults were also asked to complete a financial literacy program exit survey (canadian centre for financial literacy, 2013) near the end of the program (see appendix a1). 1 see prosper canada at http://prospercanada.org. http://prospercanada.org/ baron financial literacy with families journal of urban mathematics education vol. 8, no. 1 95 the semi-structured interview questions asked early in the program (presurvey) dealt with the perceived quantitative needs for the families involved and the immediate community, the kind of mathematical learning and support the participants would find useful, and the predicted benefits of the program. the participants, in their pre-survey interviews, clarified the definition of the community; they referred to the small urban center as their community and described the needs of that community with respect to mathematical and financial needs. near the end of the program, questions were asked (post-survey) about what kind of support was most helpful; what were the benefits and disadvantages of the coy project; and which specific teaching strategies, advice, structures, or “technologies” were believed to be of most benefit (see appendix b). though the interviews were semi-structured, i often allowed the participants to lead the conversation towards whatever topics on which they felt they wanted to comment. similarly, the facilitator of the fl course also allowed for divergent conversations. this flexibility in data collection resulted in data that reflected more closely the participants’ perspectives on the coy project. all of these data were used, along with the literature, to generate themes for the study, and to answer the research questions. data coding and analysis i attended to the transcription of the audio-recorded data within a week of each session to best reflect the conversations. as i coded the data, i selected five themes (or categories) to organize the data. i gradually deepened my knowledge of the key concepts that arose, and continually related them back to both grant’s (2012) and steen’s (2001) theories to refine the categories, keeping in mind that some data may not fit. initially, i merged grant’s (2012) and steen’s (2001) work to create a proposed conceptual framework that linked personal flourishing with quantitative literacy. this proposed theoretical framing acted as a lens through which i re-analyzed the data. acknowledging the reciprocity of theory informing data analysis and data analysis informing theory, i engaged in a back-and-forth process that allowed me to initially analyze the data using this initial framework. after transcription and multiple readings of the text, i coded and classified the data into initial themes that addressed the research questions and also corresponded well with grant’s (2012) ideas of flourishing lives and with some of steen’s (2001) elements or expressions of quantitative literacy. using the initial framework, i concluded that participants made sense of financial literacy within the context of their lives, and they gained personal financial knowledge. participants also seemed to describe themselves as more confident and able to engage in financial matters. they produced budgets and attended to them, and they engaged more in financial matters through their banks and other financial institutions. they not only described themselves as learners but also as impacting their children and families. baron financial literacy with families journal of urban mathematics education vol. 8, no. 1 96 nonetheless, using this original framework, i found that some themes did not accurately describe what it seemed grant (2012) and steen (2001) had originally intended. i also found that some data did not exactly fit. as a result, some responses were difficult to code, and some could have been included in more than one category. at this stage of the coding, i tightened the themes to better reflect the data. as described in dubois and gadde (2002), this is the stage when the analysis finally turned into a product, and there seemed to be “no confusing pieces left” (p. 558). in many ways, the existing literature supported the five themes derived from the participants’ voices and collective ideas, providing a framework for nmfl educational research (see table 4). the first theme financial knowledge described what kinds of financial knowledge the participants felt they had learned during the coy program. when participants described how they made sense of or understood financial matters as they applied to their lives, their comments were coded under this theme. the second theme re-imagining self and possibilities involved the participants challenging their previously held beliefs and assumptions that they may have perceived held them back. are they stuck in poverty? are they alone in their financial struggle? is there hope for the future? in this data theme, participants noticed the barriers that might have previously been invisible to them. it represents shifts in beliefs. this theme can also bethought of as connecting grant’s (2012) ideas of positive identity with steen’s (2001) description of confidence with mathematics. the third theme taking action was used to code participants’ responses when they described specific actions they took to better their financial situations. in theme two, they may have expressed agency, but in theme three, those beliefs about agency have now been realized through action. the participants had done something for themselves that represented the actualization of those beliefs they may have described. when the participants acted, they engaged in society, and this action, though not completely equivalent, can be related to grant’s (2012) ideas of contributing to society and steen’s (2001) comments on citizenship. the fourth theme, which i called impact on family, is perhaps the most significant. this theme was used to code instances when the adult participants described changes through the empowerment of their children and the agency they felt in how they could impact their children’s financial futures. this newfound sense of self-empowerment created a shift in possibilities for future generations, and provided hope for the future of their children’s financial lives. finally, i created a fifth theme features of the coy program to describe the technical and structural features of the project that participants described as helpful or productive which did not necessarily fit in the four previous themes. by describing the structures used while implementing coy, this theme calls upon the critical theorist’s perspective supported by the literature on re-imagining schooling (see, e.g., mclaren, 2007; stinson & bullock, 2012). the findings section discusses the baron financial literacy with families journal of urban mathematics education vol. 8, no. 1 97 five resulting themes that were supported by the initial theoretical perspective and best organized the data collected. table 4 framework for nmfl educational research theme description 1. financial knowledge data were coded this way when participants indicated a shift in financial knowledge or skills or described their financial knowledge. 2. re-imagining self and possibilities data were coded this way when participants examined their assumptions and either realized that things should change, that things could change, or that they were not alone in the struggle for change. this theme represented a shift in beliefs about what was possible and about their agency. 3. taking action data were coded in this category when participants felt so strongly about the possibilities for change that they took action. these shifting practices were realizations of their beliefs about their agency. 4. impact on family data were coded in this theme when participants described the impact of the coy project on their relationships with other members of their family, including their ability to communicate with their children and impact their children’s financial futures. 5. features of the coy program this final theme included data that described how and why the structures of the coy program were beneficial. findings this study sought to discover what financial knowledge was gained through the count on yourself program and how this understanding impacted (or not) adult participants and their families. did new understandings lead them to take action with respect to their financial lives? how did coy impact their families, and what did they feel were its most beneficial features? in the following data sections, the findings are described to express the voices of the participants directly as quotes from the data recordings. some of the responses from exit survey questions are also provided to support the themes when appropriate. only survey questions where the participants’ responses were consistent were reported here (e.g., they all agreed or disagreed) because this showed a strong commitment by the participants to the ide baron financial literacy with families journal of urban mathematics education vol. 8, no. 1 98 as presented in the survey. (a summary of the exit survey results can be found in appendix c.) theme 1 – financial knowledge for this theme, participants made meaning and sense of financial aspects of their lives in context. this theme describes adults who have the practical skills, knowledge, and ability to solve authentic problems in real contexts. they can appreciate, understand, and interpret financial information in their worlds. participants described their financial knowledge and, therefore, the potential for better financial stability. this theme describes adults who have the personal financial skills, understanding, and knowledge to manage their world. to confirm the notion that participants felt they had the skills to manage their worlds, three exit survey questions are described here. survey question 8 “what do you currently do to manage your money?” had a number of parts that included, for example: “(a) pay my bills on time (b) make sure that my spending isn’t more than my income each month, and (f) pay my debt, when i owe money” (exit survey questions). the manner in which the question was asked made it difficult to tell if the participants were reporting on what they learned from the course or simply what they knew and had practiced all along. nevertheless, these data indicate a level of confidence with respect to healthy financial knowledge and habits. for these survey questions, all participants agreed that they usually or always paid their bills on time (3 of 6 marked “usually,” while 3 of 6 selected “always”), that they made sure that their spending isn’t more that their income each month (all six marked “usually”), and that they paid their debt when they owed money (5 of 6 marked “usually,” while 1 of 6 selected “always”). survey question 7 also dealt with confidence. similarly, it is difficult to tell whether the participants’ knowledge was a result of the course or whether the course improved their confidence. however, because the question represents knowledge and understanding, it is included here. the survey item asked: “tell us how you feel now about managing your money”; part (e) stated, “i feel that i will improve my financial situation” (exit survey questions). for this survey item, all but one participant reported that they always (3 of 6) or usually (2 of 6) felt that they would improve their financial situation. only one participant reported that they sometimes felt that way. brent described his place in the larger picture of consumerism (one of the topics in the momentum curriculum). he understood and interpreted consumerism in his world: consumerism, everything that you see out there is an ad. you are bombarded on television, media, print, advertising: “you deserve it, it’s about you! you’re entitled to this. you’ve worked hard. enjoy it! spend it! here’s a way that we can make it more af baron financial literacy with families journal of urban mathematics education vol. 8, no. 1 99 fordable to you.” people are sucked into this, and they are really not aware of it until it’s too late, and that’s for me, what i think this community, city, province, country, nation, i’m not going to say the world, it’s north america that’s got the problem. (brent’s post-interview) an open-ended survey question brought richer data for this theme in that it delved more deeply into the practical and meaningful knowledge and information the participants took from the financial literacy course, including momentum’s budgeting, credit, banking, and assets course topics. survey question 1 asked: “what are the most important and useful things that you learned from this program?” valerie listed: “tools for handling/managing our finances—budgeting and online resources, insight to ‘good’ credit and how the ratings work, and that there is more than one way to do things well with budgeting.” brent listed: “the mystery of a credit score, how to find a credit score, understand it, and improve it, and how to fight consumerism!!” ema described: “the importance of being in control of your finances and being aware and responsible about where it is going and what it is doing.” phil added: “financial awareness, appreciating and seeking [out] community resources for information, and money management strategies.” anna included: “i really liked learning about all the different existing accounts/services in our banking and credit system.” carolyn listed: “budgeting, and to use many accounts.” my researcher’s notes also revealed that, as an active learner in the financial literacy course, i had participated in learning and had made some personal gains with respect to financial knowledge. i made notes to discuss these questions and ideas further with my family. through rich discussions during the sessions, we described various financial strategies, and we discussed and deepened our personal financial skills, understanding, and knowledge. theme 2 – re-imagining self and possibilities for this theme, participants described beliefs of agency and confidence. this theme depicts adults who noticed some of the assumptions they may have made about what was possible for them. the data described beliefs, values, or perspectives that described how participants felt empowered enough to make changes in their lives with respect to their finances. realization 1: i am not alone in my financial struggles. during anna’s postinterview, she referred to the financial literacy course sessions, everyone’s ability to give examples of their own financial experiences, and for others to empathize with them. the participants and facilitators were honest in their descriptions of their real life experiences: i think that everybody was comfortable and confidential—it’s nice to hear other examples as well—you know you’re not by yourself. you learn from other people. the other members talk and you learn more. stories that somebody else tried. it was nice. baron financial literacy with families journal of urban mathematics education vol. 8, no. 1 100 anna was expressing not only that she was making sense of her own financial contexts but also that the conversations during the coy project allowed her to understand and make sense of others’ financial situations. their successes and challenges were not different from hers. realization 2: hope for the future. valerie made a significant declaration during her post-interview as she considered the benefits of her participation. she articulated: “it’s part of taking control, and giving yourself a future as opposed to just letting your future take control of you.” this response demonstrated a healthy mindset and indicated a good level of confidence with respect to making a difference for herself in her financial life. during valerie’s post-interview, she described how she now felt that she had the ability to accomplish something of value: i finally felt that i had hope for the future and less of a defeatist attitude, because of what we had learned, we can make as small or as big of a change as you want, and any change would be better than where we were at. realization 3: general beliefs about their empowerment. for valerie, her child’s success in mathematics in school was evidence of empowerment: “my child is doing very well in math and i think her opinion of math being more fun is what helped that. i truly believe that came from the time spent in the coy program.” when asked how the program impacted them, both phil and brent individually described feeling empowered: “i felt enthusiasm, empowering….” brent stated, “being more empowered.” important information was revealed in one of the exit survey questions. survey question 7 asked what participants felt comfortable doing: “tell us how you feel now about managing your money: (c) i feel comfortable getting help with my money (examples: finding resources online, seeing a credit counselor, help with my taxes or talking to someone at the bank)” (exit survey question). for this survey question, all participants reported that they always (1 of 6) or usually (5 of 6) felt comfortable seeking help for their financial situation. theme 3 – taking action for this theme, participants engaged in, participated in, and contributed through personal agency in society. this theme describes adults who take action on their beliefs. during anna’s post-interview, she described that participating in the coy program gave her the confidence to know that she should take action and what action to take, which, in her case, was to visit her credit union. she stated, “now i have to do something and the program helped me and gave me confidence and gave me the idea to ask and to go to the credit union.” during carolyn’s postinterview, she described how the coy program gave her the skills and understand baron financial literacy with families journal of urban mathematics education vol. 8, no. 1 101 ing to run her household budget as well as the family’s business budget. she explained: i got the financial support that i needed to start and maintain a budget. my household budget—i run my business budget as well. it will have an impact on my husband because i do run the household budget. it will have accountability impact on him. instead of going with the flow, there will be more proactive planning involved. both phil and anna separately described situations where they took action for themselves: “there are different avenues that i am exploring. i have been to an investment group, and we are going to a mortgage broker and to a bank” (phil’s postinterview); “[i went to] find out about the bank and credit union. i went to a credit union workshop and i also participated in a financial group” (anna’s postinterview). one of the exit survey questions addressed participants’ actions. when asked: “what do you currently do to manage your money?: (h) get help with my money (examples: filing taxes, financial advisor, credit help, credit counseling or using online tools and resources)” (exit survey question), all participants (6 of 6) reported that they usually sought help. a second open-ended survey question brought richer data for this theme in that it delved more deeply into the specific actions the participants took as a result of their involvement in the financial literacy course. survey question 16 asked: “is there anything else that you have started doing to manage your money during this program?” all six of the participants added some of the steps they had taken as a result of participating in coy. valerie listed: “opened an online budgeting tool, contacted credit card to lower interest rate, opened second account to keep bills separate, always making sure money is set aside, and opened savings account for invariable expenses as added to the budget.” brent added, “started to use mint.com.” ema stated, “consolidate my debt (asked my family to help me pay my credit card debt so i am not accruing more interest)”; and phil added, “being more aware and talking to the right resources.” anna said that she “called a credit card financial department, and tried to get a loan, but it didn’t work out,” and carolyn listed: “started proper monthly budget, wanting to do full financial overview!” all of the adults in the study showed that they had a newfound sense of agency to take whatever financial actions they felt were important. theme 4 – impact on family in this theme, participants saw the coy program as impacting their families. this theme describes adults who take responsibility for children’s flourishing minds. they are committed to developing children’s habits of mind to help prepare them to become autonomous financial thinkers. baron financial literacy with families journal of urban mathematics education vol. 8, no. 1 102 during brent’s post-interview, he worked to make sense of the impact of consumerism on his and his children’s lives: so, the coy program is chock full of ideas for us to apply with our kids; we talk to them about consumerism, and we’ll use those moments in time as a teachable moment. what does this “buy now, pay later” for 12 months really mean? they say “you deserve this” but what’s wrong with the widget we already have? it works—why do you need a new one? and you’ll be happy for a little bit of time but then, you won’t be happy any more, because you’ll have to worry about paying it off! brent’s expression was twofold. he not only expressed that he knew how to use his financial knowledge in real-life situations but also that he knew how to converse about it with his children. the first setting was as a consumer, and the second setting was as a parent. anna described how she gained the confidence to speak to her children about financial matters that were age-appropriate: i was talking to the kids about the finances. now i can ask the kids “can we afford” to have that. before i was like ok…“no…we can’t buy it” i always understood the concepts but it gives me a little bit more confidence in talking to the kids. during brent’s post-interview, he described his parenting role as also the role of being a teacher for his children. he saw himself as a model of financial literacy for his children and described how the coy program equipped him with the necessary skills to make a difference with his children: i’ve always believed that i’m not just raising my kids but i’m teaching my kids to be a good parent, so there’s a responsibility and there are tools in the program that now i can use to explain finances to them. during ema’s post-interview, she described the importance for her daughter to make the connection between education, schooling, mathematics in school, and financial skills: my child was disappointed that it [coy] ended, which is huge for her because she hates school and anything to do with school. math can be fun and can be incorporated into numeracy. [my daughter] now sees the connections to numbers in real life, not just about numbers. brent also described how the coy program and its facilitators helped them to become responsible for themselves as adults, and also for their children: “you’ve empowered us to become teachers, whether it is with our kids or with other people who are struggling and need some advice” (brent’s post-interview). baron financial literacy with families journal of urban mathematics education vol. 8, no. 1 103 two open-ended survey questions brought further data in support of this theme. survey question 16 asked: “is there anything else that you have started doing to manage your money during this program?” valerie pointed to the fact that she and brent had started including their kids in more financial conversations. survey question 6 asked: “would you recommend this program to someone else?” all respondents answered yes to this question. one respondent added: “i already have!” these responses indicate a group of adults who believe they, with the help of the coy program, can and have positively influenced their children and others in their community with respect to numerical, mathematical, and financial literacies (nmfls). theme 5 – features of the coy program momentum curriculum. one exit survey question supported this theme of making sense of nmfls in context. survey question 2 asked: “feedback on what i learned: (b) i understand the information that we discussed.” here, all participants agreed that they understood the information discussed in the financial literacy course. they either “agreed a lot” (5 of 6) or “agreed” (1 of 6). the point that all participants understood the level of materials in the financial literacy supports the use and appropriateness of the momentum (2010) curriculum chosen for this coy research project for this particular group. survey question 2 stated: “feedback on what i learned: (c) i feel this program will change how i manage my money” (exit survey questions). for this survey question, all participants agreed that they would change how they managed their money. they either “agreed a lot” (2 of 6) or “agreed” (4 of 6). from this question, it appears that all participants felt to some degree that they would change how they managed their money. these changes can be seen as indicators of newfound confidence with nmfls. safe environment. this theme also shows evidence of the comfort level that was developed over the 8-week implementation period of the program. during informal conversations, other participants repeated anna’s comment that everyone felt more comfortable as time went on. trust was built over the course of the sessions in the group through the telling of personal stories, some of financial success, and others not. the focus group sessions (the adult financial literacy course sessions) became the space for building trust within this community group. the community aspect of the program, including eating meals together, learning together, parallel programs for children and adults, and the giving and philanthropic nature of the facilitators and contributors created a trusting space, and also seemed to assist in nurturing a sense of inter-responsibility between members of the group. during ema’s post-interview, she described a comfortable and healthy atmosphere during the program sessions due to everyone’s excitement at participating baron financial literacy with families journal of urban mathematics education vol. 8, no. 1 104 and sharing their time. she described how that positive emotion translated into conversations she then had with her family as a result: the researcher and facilitators were willing to be there, to be involved and share in personal experiences. it was obvious to everyone that you were excited about what you were doing and, because of your enthusiasm it’s easier for us to be excited about it when we see your enthusiasm because we can see that it matters to us, and your ability to educate us, and to see the children all excited about learning? that’s priceless! it looked awesome to my family, so i showed my daughter off to my mom. though the following quotes do not describe specific actions taken by the participants, they are the participants’ comments about how they noted that others had taken social action with them in mind to deliver the coy program. when asked what was important about the coy program for the community, carolyn noted that she noticed a form of social action by offering the program: “to make a program like this and integrate it into the community is fantastic” (carolyn’s postinterview). when asked what the community needed, valerie added: “learning to fight to learn; we need more fight in us to learn to equip ourselves, empowering ourselves to take charge” (valerie’s post-interview). as i reviewed my notes about what participants had described as important features of the coy program, they said that they appreciated that the whole family had been taken care of, including their pre-school aged children. moreover, they did not have to concern themselves with preparing dinner or such inconveniences as finding parking because their program had been brought directly to them in their community and at their school. they described a feeling of safety and familiarity in that environment. i could see, in part, what the coy program had achieved: to question who are the learners in schools, to re-imagine the role of the school, and to re-examine the role of curriculum. discussion and implications grant (2012) called for “tools that assess how well we are cultivating a flourishing life … and the extent to which we are creating a caring, democratic community” (p. 927). moreover, the financial literacy and education commission (flec, 2011) stated that it was “important that all individuals and families, including those of diverse and underserved populations, [be] aware of and have access to reliable, clear, timely, relevant and effective financial information and educational resources” (p. 8). the coy project is an example of a freirean approach to numerical, mathematical, and financial literacies that attends to families and to a robust social justice vision. the five themes describe a new theoretical framework for describing nmfl research. baron financial literacy with families journal of urban mathematics education vol. 8, no. 1 105 urban families need catalysts that provide them opportunities to breach the barriers that stand in the way of them becoming more critically aware, developing agency to take action, and making shifts to impact their and their children’s futures. if financial literacy programs are such vehicles, then it follows that continued work on projects such as coy are needed. the framework for nmfl educational research suggested in table 4 provides answers to the three research questions. the first theme addresses how the participating adults described what they learned about financial literacy. themes 3, 4, and 5 describe how that knowledge impacted them and their families in terms of beliefs, actions, and the impact on their families; and the fifth theme addresses the last question that asked what were the features of the coy program that participants felt were most beneficial. i suggest that the framework for nmfl educational research can be used in future studies to investigate and describe participants’ learning and beliefs about their transformation and their actions, as well as the important features of the nmfl program they experienced. the participants in this study described the learning that resulted from their participation in the count on yourself project, and their responses could be framed with respect to these five themes. it is yet to be determined, through future studies, how well this framework can be used to fully describe participants’ learning. further themes may need to be added or the themes clarified. the momentum (2010) curriculum was originally designed to develop knowledge and skills for people who are living on a low income or who are experiencing significant barriers in their lives. this curriculum, which includes knowledge of assets, taking charge of credit, budgeting strategies, effects of consumerism, and concepts of banking was designed to be very accessible, particularly for marginalized communities, and so that participants could learn to manage their finances more effectively. the flec (2011) indicated that unbiased and understandable financial education resources should be provided that included core financial competencies in plain-language and user-friendly information. all of the participants agreed that they understood the information discussed in the financial literacy course. the flec (2011) also argued the importance of “increasing rigorous research and evaluation on financial literacy” (p. 11) for the purpose of better understanding how to establish effective programs and practices. in this article, i suggested a framework that can be used in community empowerment and financial literacy studies to investigate nmfl programs. the framework for nmfl educational research (see table 4) could also be used as a framework to elicit and track the important qualities, as described by the participants, that emerge from community programs that address these needs. the flec also sought to “identify customized programs that effectively address local needs, such as … underserved communities” (p. 12). there are many factors that have impacted the results of this study, baron financial literacy with families journal of urban mathematics education vol. 8, no. 1 106 including the user-friendly and relevant characteristics of the momentum (2010) curriculum, the overall design of the program, the facilitation skills of those who worked with the participants, and the level of trust developed in the community. these factors address a possible re-imagined vision of the role that community schools can play. in this case, the school provided a space for a non-traditional curriculum and addressed complete families. furthermore, a key feature of the coy project was its inclusion of entire families that served to encourage family discussions around financial literacy. recent literature (see, e.g., research and evalution committee: flec, 2015) on this matter suggests: parents and caregivers play an extremely influential role in the development of the financial capability of young people. this is done through imparting financial skills, knowledge, habits, and attitudes about what is normal and expected with regard to financial behavior. … it is important to support parents’ acquisition of financial knowledge, build their understanding of and confidence in their role as a financial mentor for their children, and provide strategies and guidance for supporting their children in developing effective financial skills and habits. (p. 300) the coy program strove to meet this call. participants demonstrated their financial literacy by indicating that they were more successfully managing their money by paying their bills on time, making sure they did not spend more than they earned each month, and paying their debts when they owed money. they became more knowledgeable about credit scores, being more in control of finances, and budgeting. they considered the impact of consumerism, and learned about the various services that banks could provide. participants also displayed more agency and self-confidence with finances. being heard was essential. the practice of “letting research participants speak for themselves” (guba & lincoln, 2005, p. 209) allowed them the voice to tell me they felt empowered themselves, in their ability to speak to their children, and that their children also felt more numerically, mathematically, and financially empowered. all of the participants felt that the coy program would change how they managed their money. guba and lincoln (2005) discussed the validity of a study as catalytic authenticity when the researcher creates capacity in participants for positive social change and forms of emancipatory community action. grant (2012) also spoke of the researcher’s ability to encourage social action. in this study, there is evidence that the participants appreciated the community aspect of the project, and took action by visiting banks and credit unions, investment groups, and mortgage brokers. they began budgeting processes online and using software, contacted credit companies to ask for a reduction in their interest rates, opened separate bank accounts to better keep track of their expenses, consolidated their debt, and applied for loans. the exit survey supported the findings and indicated that all of the participants not only felt baron financial literacy with families journal of urban mathematics education vol. 8, no. 1 107 comfortable getting help, but also did get help with their finances. taking action is powerful evidence of empowerment, and, there is evidence that the participants took action to improve their financial circumstances. perhaps the most significant result of this project was the belief from participants that they had influence upon their children. though not all families participated as complete units, those who had children in the coy program felt more able to discuss their financial circumstances with their children when appropriate. all participants agreed that they found value in the program and would recommend it to someone else. the implications for this result is that programs such as count on yourself, that involve the whole family, might offer steps towards breaking cycles of poverty in communities in which they are implemented. as i continue to consider grant’s (2012) vision, i am struck by his statement: you learn about how the practice of democracy can be made to work for you or against you and that it is important that you understand the differences as well as you know what you can do to influence an outcome that befits those who are marginalized. (p. 925) grant’s intent was to clarify what it means to practice democracy. i find that this statement is reflected both in my practice as a researcher and in the evidence that is produced from this study through the voices of the participants. i entered this work to make a difference and to provide a learning space for families who could participate more democratically in society because they chose to learn to be more numerically and financially literate. i have learned that i can make a difference, and they have learned that they can make a difference in their own life situations. this most certainly reflects grant’s “practice of democracy” (p. 925). approximately two years after the completion of the coy project i received the following email from valerie: both the girls have two bank accounts opened and an investment account too. they use the saving concepts they learned at coy with the cereal box piggy bank they made and are applying it to their bank accounts now. they each saved $500 to start their investments and have watched it fluctuate with the markets. they love seeing how it has grown just by changing the type of account. they have applied their savings and spending accordingly. the funny thing is when they ask us why we are using our credit cards for purchasing!! so funny. they don't bring up coy all that often but it does creep back into our conversations at times. brent and i have fixed our bad credit and are now ‘a’ rated. such a huge blessing. we haven’t maintained all the principles we learned, just baron financial literacy with families journal of urban mathematics education vol. 8, no. 1 108 the ones that work for us. the biggest change is that we do it all together and are on the same page. our financial outlook is worlds better than when you met us. i feel like we had the best of both worlds. we had a great experience with coy partnered with the exposure and help at [the bank where valerie works]. i definitely think the program would be beneficial to families. it’s a huge key missing from people’s awareness with finances and we were offered practical tools. the best part was the lack of judgment associated with it. i valued that the most. another aspect was the networking. i have run into [name of course facilitators] a few times. in fact i interviewed with him for a job but had to decline as it was right when brent had lost his job. we needed the stability that [my job] offered. however, i felt honored he thought of me for the position. so i’ll just wrap up by saying we are grateful for the experience. we would love to continue staying in touch and appreciate all the heart you pour into this project and the people. you are a gem! there is no better overall description of the benefits of this coy project. through the research and its proposed research framework, valerie’s and others’ voices can been heard. references andersen, p. h., & kragh, h. 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(2010). easy money. edmonton, ab, canada: grass roots press. http://www23.statcan.gc.ca/ baron financial literacy with families journal of urban mathematics education vol. 8, no. 1 111 appendix a financial literacy program exit survey* (*for downloadable copy see http://prospercanada.org/getattachment/1697a108-196d-4487-95b9e79cd0c236e9/exit-and-post-assessment.aspx.) 1. what are the most important and useful things that you learned from this program? 2. feedback on what i learned: circle a number to show how much you agree or disagree … i d is a g re e a l o t i so m e w h a t d is a g re e i a m n o t su re i a g re e i a g re e a l o t most of the information i heard was new to me 1 2 3 4 5 i understand the information that we discussed 1 2 3 4 5 i feel this program/activity will change how i manage my money 1 2 3 4 5 3. what other things would you like to learn about managing your money? 4. what additional financial literacy supports and services did you receive from staff? (please check all that apply)  they referred me somewhere else for help  they helped me file my taxes  they helped me fill out government applications and forms (e.g. a social insurance number application)  the made phone calls for me  they connected me to a bank  they advocated for me and/or helped me advocate for myself  they helped me to get a loan  other please specify __________________ http://prospercanada.org/ http://prospercanada.org/getattachment/1697a108-196d-4487-95b9-e79cd0c236e9/exit-and-post-assessment.aspx http://prospercanada.org/getattachment/1697a108-196d-4487-95b9-e79cd0c236e9/exit-and-post-assessment.aspx baron financial literacy with families journal of urban mathematics education vol. 8, no. 1 112 5. how can we make this program better? 6. would you recommend this program to someone else?  yes  no  unsure thinking about money … 7. tell us how you feel now about managing your money… circle a number to show how often you agree with the following: n e v e r r a re ly s o m e ti m e s u su a ll y a lw a y s i feel confident managing my money 1 2 3 4 5 i worry about how much debt i have 1 2 3 4 5 i feel comfortable getting help with my money (examples: finding resources online, seeing a credit counselor, help with my taxes or talking to someone at the bank) 1 2 3 4 5 i worry about being able to pay my bills each month 1 2 3 4 5 i feel that i will improve my financial situation 1 2 3 4 5 8. what do you currently do to manage your money? please circle the number that best explains how often you do the following: n e v e r r a re ly s o m e ti m e s u su a ll y a lw a y s pay my bills on time 1 2 3 4 5 make sure that my spending isn’t more than my income each month 1 2 3 4 5 keep track of my spending and income 1 2 3 4 5 save money 1 2 3 4 5 baron financial literacy with families journal of urban mathematics education vol. 8, no. 1 113 please circle the number that best explains how often you do the following: n e v e r r a re ly s o m e ti m e s u su a ll y a lw a y s compare prices when shopping 1 2 3 4 5 pay my debt, when i owe money 1 2 3 4 5 learn about money topics that might affect me 1 2 3 4 5 get help with my money (examples: filing taxes, financial advisor, credit help, credit counseling or using online tools and resources) 1 2 3 4 5 9. do you budget your money?  yes (go to question 10)  no (go to question 11) 10a. if yes, how do you budget your money? (please check one)  i write out a budget  i keep a budget in my head  other: please tell us ___________________________________________________________ 10b. if yes…how often do you follow your budget? please circle the number that best explains how often you do the following: n e v e r r a re ly s o m e ti m e s u su a ll y a lw a y s i follow my budget 1 2 3 4 5 11. if no… why don’t you budget your money? (please check the one that best applies)  i don’t know how  i don’t believe in budgeting  i did it before and it didn’t work  it is just not that important to me right now  other: please tell us ___________________________________________________________ 12. do you have a goal for saving money?  yes  no baron financial literacy with families journal of urban mathematics education vol. 8, no. 1 114 13. if yes, what are you planning to save for? (please check all that apply)  my education  my child’s education  first and last month’s rent  paying back money i owe  something big – like a car or appliance  paying back taxes owed  a trip  an emergency  home ownership  retirement  to finance a business  other please tell us ______________________ 14. do you have any savings set aside?  yes  no (go to question 16) 15. did you save and put aside any of your money in the past month? (please check one)  yes  no 16. is there anything else that you have started doing to manage your money during this program? 17. have you filed your income tax forms with help from this program?  yes  no we're interested in knowing what you think! please sign up below if you are interested in filling out a short survey in a few months from now. your ideas will help us continue to improve the way we deliver financial education. may we contact you in six months to a year to find out what you think about this program?  yes signature: __________________________________  no thank you! baron financial literacy with families journal of urban mathematics education vol. 8, no. 1 115 appendix b preand postsemi-structured interview questions adults early in the program 1. what do you believe are the needs of your community as far as mathematics, numerical, or financial learning? 2. what kind of support would you like for yourself? for your spouse/other? for your children? 3. can you imagine what would be the good features of a program like count on yourself? what would be the potential benefits for you? for your spouse/other? for your children? adults end of the program 1. what do you believe were the most important learning needs of your community as far as mathematics, numerical, or financial learning? 2. what support did you get with the count on yourself program that you liked for yourself? for your spouse/other? for your children? 3. what were the best features of count on yourself? what were the benefits for you? for your spouse/other? for your children? 4. were there any disadvantages of participating in the count on yourself program? if so, what were they? 5. do you believe there will be any long-term benefits of this program? a. what were some of the best teaching strategies? b. what was some of the best advice from the facilitator? c. what was the best part of how it was organized (timing, treats, location, etc.)? d. which types of technologies helped you the most during your learning (games, videos, social media, websites, etc.)? 6. talk about what you appreciated about the facilitators, researchers, school staff, college students, or other community members who made your experience positive? baron financial literacy with families journal of urban mathematics education vol. 8, no. 1 116 appendix c complete exit survey results 1. what are the most important and useful things that you learned from this program? participant a (female)  tools for handling/managing our finances budgeting and online resources  insight to “good” credit and how the ratings work  more than one way to do things well with budgeting participant b (male)  the mystery of a credit score  how to find a credit score, understand it, and improve it  how to fight consumerism!! participant c (female)  the importance of being in control of your finances and being aware and responsible about where it is going and what it is doing participant d (male)  financial awareness  appreciating and seeking community resources for information  money management strategies participant e (female)  i really liked learning about all the different existing accounts/services in our banking and credit system participant f (female)  budgeting  to use many accounts 2. feedback on what i learned [disagree a lot 1, i am not sure 3, agree a lot 5] a. most of the information i heard was new to me (4, 2, 2, 2, 2, 2) b. i understand the information that we discussed (5, 5, 5, 4, 5, 5) c. i feel this program will change how i manage my money (5, 5, 4, 4, 4, 4) 3. what other things would you like to learn about managing your money? participant a  ways for money to grow, outside of a simple day-to-day savings account  how to balance savings with debt reduction participant b  how to become tax literate  ways to increase your tax return participant d  discuss present/future investing options and strategies participant e  i believe i’ve learned a lot, and since i am [just starting to be ]the bookkeeper for my house[hold finances], i should start doing [the] budgeting as well.  i might need someone to [watch over me] though, just [in the] beginning participant f baron financial literacy with families journal of urban mathematics education vol. 8, no. 1 117  investing 4. what additional financial literacy supports and services did you receive from this program? they referred me somewhere else for help, they referred me somewhere else for help, they referred me somewhere else for help, they referred me somewhere else for help, they referred me somewhere else for help, they helped me advocate for myself; n/a 5. how can we make this program better? participant a  show how the dinners provided were broken down in cost. families that don’t pay attention can see how little it can cost to feed the family (a “per person” cost factor) participant b  incorporate the use of media more often in the presentations (video, powerpoint, internet, iphone, etc.) participant d  consistent feeding time or suggest an alternate time  investment segment 6. would you recommend this program to someone else? yes, yes, yes, yes, yes, yes – already have 7. tell us how your feel now about managing your money? [never -1, sometimes 3, always 5] a. i feel confident managing my money (3, 5, 4, 3, 4, 3) b. i worry about how much debt i have (4, 3, 3, 1, 3, 3) c. i feel comfortable getting help with my money (examples: finding resources online, seeing a credit counselor, help with my taxes or talking to someone at the bank) (4, 5, 4, 4, 4, 4) d. i worry about being able to pay my bills each month (2, 2, 3, 2, 3, 2) e. i feel that i will improve my financial situation (5, 5, 4, 3, 4, 5) 8. what do you currently do to manage your money? [never 1, sometimes 3, always 5] a. pay my bills on time (5, 5, 4, 4, 4, 5) b. make sure that my spending isn’t more than my income each month (4, 4, 4, 4, 4, 4) c. keep track of my spending and income (3, 4, 2, 4, 5, 3) d. save money (3, 3, 3, 5, n/a, 4) e. compare prices when shopping (4, 4, 5, 4, 4, 3) f. pay my debt, when i owe money (4, 4, 4, 5, 4, 4) g. learn about money topics that might affect me (2, 4, 4, 5, 3, 3) h. get help with my money (examples: filing taxes, financial advisor, credit help, credit counseling or using online tools and resources) (4, 4, 4, 4, 4, 4) 9. do you budget your money? yes, yes, yes, no, no, yes 10. a. if yes, how do you budget your money? i keep a budget in my head, i keep a budget in my head (just starting to use mint.com), i keep a budget in my head, i keep a budget in my head (i have learned to [make a] budget better!) b. if yes, how often do you follow your budget? [never 1, sometimes 3, always 5] 3, 3, 4, 1 baron financial literacy with families journal of urban mathematics education vol. 8, no. 1 118 11. if no, why don’t you budget your money? it is just not that important to me right now (it’s a time/effort management thing), not enough time in my day 12. do you have a goal for saving money? yes, no, no, yes, yes, yes 13. if yes, what are you planning to save for? my child’s education, paying back money i owe, retirement; a trip, an emergency, home ownership, retirement; paying back money i owe, something big like a car or appliance, a trip, an emergency; my child’s education, a trip, an emergency, retirement) 14. do you have any savings set aside? yes, yes, yes, yes, yes 15. did you save and put aside any of your money in the past month? no (paid down debt instead), yes, no, yes, no, yes 16. is there anything else that you have started doing to manage your money during this program? participant a  opened an online budgeting tool  contacted credit card to lower interest rate  opened second account to keep bills separate  always making sure money is set aside  opened savings account for invariable expenses as added to the budget  started including our kids in more conversations participant b  started to use mint.com participant c  consolidate my debt (asked my family to help me pay my credit card debt so i am not accruing more interest) participant d  being more aware and talking to the right resources participant e  called a credit card financial department  tried to get a loan, but it didn’t work out participant f  started proper monthly budget  wanting to do full financial overview! 17. have you filed your income tax forms with help from this program? no, no, no, no, no, no (this wasn’t a part of the curriculum) 18. may we contact you in six months to a year to find out what you think about this program? yes, yes, yes, yes, yes, yes journal of urban mathematics education july 2015, vol. 8, no. 1, pp. 62–82 ©jume. http://education.gsu.edu/jume teresa k. dunleavy is an assistant professor of the practice of mathematics education in the department of teaching and learning in peabody college at vanderbilt university, pmb 230 gpc 230 appleton place, nashville, tn, 37203; email: teresa.dunleavy@vanderbilt.edu. her research interests include teaching and learning practices that strive toward equity, that improve students’ competence and their perceptions of competence, that support the development of student discourse, and that attend to status issues in the classroom. (the study reported here was conducted while at the university of washington.) delegating mathematical authority as a means to strive toward equity teresa k. dunleavy vanderbilt university in this article, the author provides insight into the pedagogical processes for delegating mathematical authority to students, through the use of specific classroom structures, as a means to strive toward equity. employing qualitative methods, the author analyzes transcripts of classroom video, along with field notes and teacher and student interviews, collected during one semester of the participating teacher’s algebra i course. the author addresses how the teacher’s practice was striving toward equity through the use of classroom structures that delegated mathematical authority to students. analyses revealed that the teacher delegated mathematical authority through the use of student presentations, shuffle quizzes, and participation quizzes. each instance featured was chosen to highlight a different facet of the ways in which delegating authority repositioned students as competent sensemakers. keywords: delegating mathematical authority, equitable teaching and learning practices, mathematics education ttention to complex practices involved in striving toward equity has become a growing priority in mathematics education (see, e.g., bartell, 2013; cohen & lotan, 1997; esmonde, 2009a; gutiérrez, 2007; martin, 2003; nasir & cobb, 2007; national council of teachers of mathematics [nctm], 2000; zavala, 2014). with a few exceptions (e.g., boaler & staples, 2008; jilk, 2007; staples, 2008), the field does not yet have significant qualitative empirical evidence of equitable teaching and learning practices in secondary mathematics classrooms. in this article, i adopt esmonde’s (2009b) definition of equity as “a fair distribution of opportunities to learn” for all students (p. 1008). teachers who strive toward equity intentionally pursue practices that help students to view everyone as capable of learning highlevel content (cohen & lotan, 1997; esmonde, 2009b; nctm, 2000). this definition of equity, which assumes all students are competent to do high-level mathematics, also implies that equitable classrooms open up interactional space for a broad range of competent ideas. attention to equity suggests the need to evaluate enacted classroom structures and practices (boaler, 2002). previous research on classroom structures has attended to: teacher discourse through revoicing (o’connor & michaels, 1993); students’ a http://education.gsu.edu/jume mailto:teresa.dunleavy@vanderbilt.edu dunleavy delegating mathematical authority journal of urban mathematics education vol. 8, no. 1 63 use of explanations as tools (esmonde, 2009a); and teachers’ discursive positioning moves facilitating english learners’ opportunities to take on agentive problemsolving roles (turner, dominguez, maldonado, & empson, 2013). because mathematics educators do not yet deeply understand pedagogies that move toward equitable learning opportunities (esmonde, 2009b), this study is motivated by the need for qualitative documentation of equitable teaching and learning practices. given that cooperative learning practices such as complex instruction (e.g., boaler & staples, 2008; cohen & lotan, 1997, 2014; featherstone et al., 2011) have been linked to equitable teaching and learning opportunities, here i share findings from a single case study on one teacher whose pedagogical practices centered on the use of complex instruction. my analysis of the collected data (transcripts of classroom video, classroom field notes, and teacher and student interviews) attends to the elements of the teacher’s practice that strive toward equity. the purpose of the study was to contribute to filling the gap in the empirical research on equity to “uncover a range of solutions focused on what works, where, when, and why” (martin, 2003, p. 18). analysis of the teacher’s pedagogical practices reveals empirical evidence of this teacher’s process for striving toward equity by the use of specific classroom structures that served to delegate mathematical authority to her students. gresalfi and cobb (2006) found that positioning students to have mathematical authority made students responsible for a higher level of mathematical argumentation. they described mathematical authority as the degree to which students are given opportunities to be involved in decision making and whether they have a say in establishing priorities in task completion, method, or pace of learning. thus authority is not about “who’s in charge” in terms of classroom management but “who’s in charge” in terms of making mathematical contributions. (p. 51) gresalfi and cobb asserted that who has mathematical authority in a particular classroom depends on who has been given the opportunity to verify that a given mathematics contribution is reasonable. researchers have studied the delegation of authority as a way that teachers can make students responsible for their own and their classmates’ learning (bianchini, 1999; lotan, 1997); support students to use one another as resources (cohen, 1997a); increase student-student interdependence and shift checking for understanding to students (ehrlich & zack, 1997); and make students’ life experiences, opinions, and points of view legitimate components of what is learned (hand, 2003; lotan, 2003). teachers delegate mathematical authority to students when they require students to convince their peers that their solutions make mathematical sense (gresalfi, martin, hand, & greeno, 2009) and when they require both individual and group products (lotan, 1997). delegating authority empowers students to argue, evaluate, dunleavy delegating mathematical authority journal of urban mathematics education vol. 8, no. 1 64 and confirm the validity of their mathematical ideas. students take on responsibility to explain mathematical concepts, answer questions, demonstrate multiple solution methods, and co-construct their overall conceptual understanding (bianchini, 1999; boaler & staples, 2008; gresalfi, 2009). when the teacher analytically scaffolds opportunities for students to participate in sustainable mathematical discourse, the delegation of authority is visible (nathan & knuth, 2003). delegating authority can also manifest through (a) substantive student discourse, (b) task cards, (c) cooperative norms, and (d) use of procedural roles that set clear expectations for group and individual products (cohen, 1994). the reporting of this study shows delegating mathematical authority as a significant pedagogical practice that offers equitable learning opportunities to students. specifically, findings suggest that delegating mathematical authority to students can be (is) a significant aspect of striving toward equity in mathematics classrooms. striving toward equity in mathematics education invokes constant, purposeful work from the teacher as she or he seeks to diminish differences in access, opportunities, and outcomes for students. this study builds on and contributes to current understandings of striving toward equity by investigating the pedagogical processes in which mathematical authority might be delegated from teacher to students (esmonde, 2009b; zahner, 2011). two research questions guided the study: 1. what structures and practices did the teacher engage to delegate mathematical authority to students? 2. how was the teacher’s process of delegating mathematical authority to students linked to striving toward equity? framing learning a sociocultural perspective on learning informs this study. this perspective allows the researcher to observe learners through their social and historical contexts as they develop and change in the classroom (rogoff, 2003). in this study, students engaged in small group and whole class interactions. a sociocultural framing suggests students co-constructed their learning experiences, positioning themselves and their classmates at various levels of competence to complete mathematical tasks. i draw on the sociocultural perspective in order to attend to learners’ participation and interactions during group work; i simultaneously use status and positioning theories to explain the dynamic nature of learners’ participation and interactions. status theory explains which attributes have been valued in the classroom and offers an explanation for the ways students generalize their performance expectations for another student’s contributions (cohen & lotan, 1997, 2014; kalkhoff & thye, 2006). positioning theory attends to the dynamic process of positioning, or the ways the teacher and students actively position students to be mathematically competent dunleavy delegating mathematical authority journal of urban mathematics education vol. 8, no. 1 65 (or not), and also offers the opportunity to understand discourse moves that established students’ positions during social episodes in the activity of the classroom (harré & van langenhove, 1999). i couple these two theories to investigate how the teacher and students positioned one another to generalize expectations for mathematical competence. methods site and participant selection as public school classrooms in the united states continue to increase in racial, ethnic, linguistic, and socioeconomic diversity, teachers are challenged to use pedagogies that are successful with heterogeneous student populations. martin (2009) emphasized that examining race and racism, as well as other social and cultural markers of historical marginalized groups, are imperative considerations in research that strives toward equity. as such, i situated this study in an urban school with a racially, ethnically, linguistically, and socioeconomically diverse student population. i sought to understand how historically marginalized students (e.g., african americans, latinas/os, students impacted by poverty, immigrants, and girls in general) made sense of their mathematical learning within their classroom context. because i was interested in learning primarily from a teacher whose students had not previously been successful in mathematics, i elected to work with a teacher and her students in a ninth-grade algebra i class, knowing that many of the other ninth-graders were entering high school already prepared to take geometry or algebra ii. (as i discuss later, it is significant that the students who were taking algebra i in high school were less advanced in mathematics than some of their peers.) i invited one teacher, ms. martin (all proper names are pseudonyms), to participate in this study because i believed that she intentionally made pedagogical choices to counter status issues in the classroom (cohen, 1997b). she had been teaching algebra i for five years when the study started; i came to know ms. martin about two years prior. through these prior professional interactions, i learned that she cared about and intentionally developed sociomathematical norms (kazemi & stipek, 2001; yackel & cobb, 1996) which fostered interdependence during group work (lotan, 2003); that she regularly engaged students in mathematics discourse (cazden, 2001); and that she organized her classroom practices around student-centered learning. i also learned that she and i shared a common interest in supporting students who had previously been unsuccessful in mathematics. some mathematics educators and researchers have argued that effectively implemented group work is an important pedagogical tool that teachers (and students) can use to strive toward equity. for this reason, group work has become a consistent focus of research in the teaching and learning of mathematics (e.g., boaler & dunleavy delegating mathematical authority journal of urban mathematics education vol. 8, no. 1 66 staples, 2008; cohen & lotan, 1997, 2014; esmonde, 2009b; webb, 1991). “effectively” implemented group work has been shown to increase student participation, engage students more deeply in their learning, develop their academic thinking (boaler & staples, 2008; herrenkohl & guerra, 1998), promote positive mathematics identities (hand, 2006; jilk, 2007; nasir, hand, & taylor, 2008), and foster classroom relational equity (boaler, 2006). ms. martin’s classroom practices feature group work facilitated by the principles of complex instruction (ci), a form of cooperative learning developed by elizabeth cohen and rachel lotan. ci is a form of ambitious teaching (lampert, 1990; lampert & graziani, 2009), because the effective implementation of ci offers opportunities for every student to construct understandings of and reason about authentic problems. much of ms. martin’s learning and interpretation of ci was based on work that came out of railside high school (boaler & staples, 2008), which had been presented to her during her teacher education program as a project that sought to understand equitable teaching and learning practices in mathematics classrooms. the presumption that every student is capable and competent in learning high-level mathematics is the key to a faithful implementation of ci. this presumption of competence drives ci teachers to the practice of regularly randomly assigning students to groups, which ms. martin did every 2 weeks during the study. ci teachers also aim to disrupt typical hierarchies that develop expectations for competence by using status theory and paying attention to the role of status as students work together (cohen & lotan, 1997, 2014). ci teachers facilitate learning in small groups by delegating mathematical authority to students and by assigning tasks that require multiple abilities and multiple entry points. here, student time spent in class (i.e., classwork) was used to create opportunities for equal-status interactions in the classroom (cohen & lotan, 1997, 2014), thereby striving toward equity. data collection i used a qualitative case study approach to offer a thick and rich description (corbin & strauss, 2008) of the mathematical learning experience of one group of students. the data collection period took place over 50 classroom visits during one period in the fall semester of ms. martin’s algebra i class during the 2011–12 school year. this algebra i class had 28 students. twenty-four of the 28 students formally agreed to participate in the study. seventeen students were african american or african immigrants,1 five were asian american, four were european american, and two were latino/a. twenty-two students were girls, six were boys. when 1 because student demographics were not made available to me, these calculations are on the basis of my own observation, so any error or misrepresentation of how students perceived their identities is my error. dunleavy delegating mathematical authority journal of urban mathematics education vol. 8, no. 1 67 compared with the school demographics, this algebra i class had an overrepresentation of african americans, latinas, and girls in general, and an underrepresentation of asian americans, european americans, and boys. i outline the racial and ethnic demographics here because it signifies the larger systemic inequities present in the opportunities these students had access to prior to arriving in this algebra i class. in fact, the overrepresentation of african americans, latinas, and girls in general also mirrors a national trend in which, prior to entering high school, disproportionately more european americans, asian americans, and boys successfully completed algebra i, while disproportionately fewer african americans, latinos/as, and girls in general successfully completed algebra i (allexsaht-snider & hart, 2001; oakes, 1990; spielhagen, 2006; tate, 1995). above all, the data collection site for this study was motivated by the opportunity to feature successful mathematics learning opportunities for historically marginalized students. i focused the data collection period around two units taught in ms. martin’s algebra i class during the fall semester. the first focal unit (unit 1: linears) was taught during 4 weeks at the beginning of the school year in the fall semester. the second focal unit (unit 4: lab gear2 and solving linear equations) was taught for 2 weeks at the end of fall semester. the purpose of selecting one unit at the beginning and one unit at the end of the semester was to understand the nature of the students’ experiences over time. during the focal units, i attended class every day. during the middle of the semester, i attended class about twice per week to collect field notes on student learning and to continue to build relationships with students. data collected for this study included: (a) field notes taken during 50 classroom observations; (b) qualitative records in which field notes were typed and fleshed out shortly after each classroom observation; (c) audio and video recordings of 26 classroom sessions focused on small-group participants during units 1 and 4, placing one camera near the five small groups with study participants; (d) student and teacher artifacts, including but not limited to classwork, homework, projects, quizzes, and tests, gathered from observation days; (e) audio and video recordings of 22 semi-structured interviews with 12 of the 24 students who volunteered for interviews outside of class time; and (f) four semi-structured individual interviews and more than 100 informal conversations conducted with the teacher before, during, and after the study over a 4-year period between 2009 and 2013. during the semester of the study, i became a participant observer in the class. field notes attended to students’ capital-d discourse (see gee, 1996), positioning moves, storylines, displays of status characteristics, the teacher’s pedagogical choices and positioning moves, evidence of status generalization, and equity. at times, i sat near groups to hear their conversations with one another; at times, i re 2 lab gear was created by henri picciotto, published by wright group/mcgraw hill – creative publications; for more information on using lab gear in a mathematics classroom, see http://www.mathedpage.org/manipulatives/labgear.html. http://www.mathedpage.org/manipulatives/lab-gear.html http://www.mathedpage.org/manipulatives/lab-gear.html dunleavy delegating mathematical authority journal of urban mathematics education vol. 8, no. 1 68 sponded to questions about their work, and, as often as possible and per ms. martin’s expectation, i redirected them to work with one another when questions arose. as i started building relationships, some students called me over to ask questions more than others. when this occurred, i redirected them to use one another as mathematical resources. positioning students to use one another as mathematical resources was critical; one of ms. martin’s focal goals was to support students in finding themselves and one another as mathematically competent, often for the first time in their mathematics career. positioning students to use one another as mathematical resources also became a way that ms. martin delegated mathematical authority to her students. in all cases, to understand the teacher’s classroom structures and the process for delegating authority, i maintained a focus on the activity of learning mathematics in groups. data analysis because mathematical learning and delegating authority were co-constructed between ms. martin and her students, my unit of analysis is the activity of delegating authority as students worked together in their groups. this group focus does not take away the importance of the community and personal planes of analysis (rogoff, 2003). rather, the examination of perceptions of learning from the mathematics community in this classroom and from individual learners provides a frame for studying student interactions in groups. data collected on the nature of group interactions contributes to the empirical evidence of status, positioning, and equitable secondary mathematical learning practices. initial analyses of classroom observations and interview data took place by creating qualitative records. the qualitative records contained typed and tagged versions of field notes and coded video sessions. the field notes and coded video sessions reflected ms. martin’s interpretation and use of ci. the video sessions were coded for teacher moves such as “assigning competence,” positioning moves such as “positive self-positioning,” and discursive interactional moments such as “on-task verbal group interaction” and “heads in.” i wrote memos related to equitable pedagogical practices, students’ perceptions of competence, and discursive positioning moves. i took multiple passes through the data, reading field notes, qualitative records, coding schemas of classroom sessions and interviews, coding for common themes and interesting moments. i used open coding (emerson, fretz, & shaw, 1995; merriam 2009) to allow codes such as “pointing and explaining” to develop. i found the teacher frequently reinforced the kind of group interaction she was hoping to see. i used linguistic microanalysis (corbin & straus, 2008) during a second phase of analysis; i studied the instances in which pointing and explaining developed, and eventually found its relationship to the delegation of mathematical authority from ms. martin to the small groups. dunleavy delegating mathematical authority journal of urban mathematics education vol. 8, no. 1 69 once preliminary findings were determined, i questioned the data, so to speak, at multiple subsequent stages of analysis to better understand the nature of the delegation of mathematical authority in this classroom (corbin & straus, 2008). questioning the data allowed me to probe more deeply into a particular interaction in class that i was aiming to understand and helped me when i was stuck at a certain stage of analysis. for example, i may have suspected, preliminarily, that when students made eye contact with one another, it meant they were interested in what the other person was saying. this assumption would have caused premature assignment of positioning codes. questioning the data through multiple stages of analysis allowed me to determine that when eye contact was coupled with other discursive acts, it could take on different meanings between individuals, including interest, positioning, and even dominance. i used the constant comparison method (corbin & strauss, 2008; merriam, 2009) to analyze segments of data alongside each other to seek confirming or disconfirming evidence. i used constant comparison to analyze classroom observation and video data for evidence of assigning competence and positioning students as competent. i also used constant comparison to examine interviews for themes and trends. i searched for similarities and differences in how students talked about mathematical competence, and i triangulated their perceptions with specific moments from the data. i used constant comparison to focus on more than a single case and to examine my findings in greater detail. delegating mathematical authority – three cases the teacher’s practices for delegating mathematical authority included positioning all students to offer genuine mathematical contributions and cultivating a classroom community in which all students were given opportunities to display competence. in the following sections, i address how ms. martin framed her work with students and i offer examples of delegating authority through student presentations, shuffle quizzes, and participation quizzes. teacher positioning of students as competent sense-makers during one interview, ms. martin explained to me that she knew that all of her students were competent to contribute meaningful mathematics ideas: zoom out completely and know that i come into this whole-heartedly knowing that all of my kids have something genuine to contribute. all my kids are capable of learning math. i hope that exudes from me, as a teacher, everywhere i go in that classroom. and over time, i’ve gotten other kids convinced that yes, they’re equal contributors to this class. ... i believe that big teacher belief that all kids can learn, and learn in equally valuable ways. and i’m going to do what i can to get them there. dunleavy delegating mathematical authority journal of urban mathematics education vol. 8, no. 1 70 ms. martin here illustrated that she knew all students were able to make authentic mathematical contributions, and that part of her job was to help students discover ways they were capable of contributing to the class. ms. martin’s perspective, that all students “have something genuine to contribute” and are “capable of learning math,” aligns with cohen’s (1997a) assertion that equitable classrooms start with teachers who position all students as “capable of learning both basic skills and highlevel concepts” (p. 4) and with nctm’s (2000) expectation that all students be given opportunities to learn mathematics. by saying she convinced students they were “equal contributors” to the class, she indicated a vision for students to hold mathematical authority in a supportive classroom environment, in line with gresalfi and cobb’s (2006) suggestion that mathematical authority is visible by who is “in charge” of the mathematical contributions. ms. martin also regularly cited status theory (cohen & lotan, 1997, 2014) for positioning students as competent to offer authentic mathematical contributions. she described responding to status issues by publicly assigning competence to students whenever she could: in order to combat status, i try to assign competence to kids. and that only works if it’s genuine and totally absolute. so there may be a day when i know a student has low status in his group, and i try really hard to get him back in the conversation, or get kids to turn him. but [maybe that day] it doesn’t happen genuinely to actually bring him in. the cool part is that i haven’t seen a kid where i’ve never been able to assign him competence. i’ve always been able to figure out how a kid is smart, or [find they had] some secret thing written down on their paper they didn’t share with their group and [i’ll be able to] make a big deal about it. like maybe they had a perfect graph and i can use it as the example for the whole class. in this excerpt, ms. martin discussed combatting status issues by positioning students as competent and assigning competence to low-status students for ways they authentically contributed to the mathematics of their group. she noted that assigning competence “only works if it’s genuine and totally absolute.” in this way, ms. martin stated that this strategy was contingent on assigning competence for genuine mathematical contributions. she indicated that it was always possible to find a way that a student was competent in mathematics. evidence that ms. martin’s classroom structures were shaping students’ productive dispositions (national research council [nrc], 2001) surfaced when students described their role in working with others. in one interview, a student shared: you do need to be able to like, explain thoroughly and like, several different ways. and just, so then, everybody can get it. or at least like, if both of you have the right answer, you both have to explain how you got it, to each other. dunleavy delegating mathematical authority journal of urban mathematics education vol. 8, no. 1 71 this student was developing accountability to themselves and others, acknowledging that mathematics learning means explaining methods and understanding group members’ methods. other students discussed expectations to justify mathematics, saying for example, “c’mon you guys, we have to explain to each other. she’s gonna ask us to explain how we know!” these excerpts illustrate how these students were seeing the utility of explaining and understanding mathematics in multiple ways. case 1: delegating mathematical authority through student presentations ms. martin began to delegate mathematical authority during the first student presentation of the year. in her classroom, student presentations were an opportunity for students to listen to a peer’s justification of his or her mathematical thinking. most often two or more students would present in a row, and each student would have different ways of justifying the presented problem. student presentations were a classroom structure that allowed ms. martin to delegate mathematical authority to students; and, in turn, asking students to justify the mathematics through student presentations reinforced the delegation. in another example, students were working on a problem where they were meant to assume a fictitious student had made mistakes on a graphing problem. the task directions were “circle the mistake(s) and tell what the student did wrong” (see figure 1). figure 1. i love m&ms. during this student presentation, ms. martin asked for a volunteer to go up to the front of the room to lead the discussion on the “i love m&ms” problem featured in figure 1. when ms. martin asked a volunteer to be the first student to pre dunleavy delegating mathematical authority journal of urban mathematics education vol. 8, no. 1 72 sent that year, naima raised her hand. naima’s willingness to volunteer is salient because in spite of failing algebra i as a freshman in another class, on the eighth day of this algebra i class, she was willing to take a role where she was justifying mathematics ideas in front her peers. naima started: 1 naima: are they s’posed to tell me? 2 ms. martin: sure, you can do whatever you want. someone tell her. or, you run the 3 class. 4 naima: you guys, raise your hands to me. a few students laughed when naima said line 4, possibly because naima had taken, for the first time, what is often seen as a teacher role. ms. martin’s encouragement in lines 2–3 indicated a delegation of responsibility from teacher to student. in this way she delegated authority to naima to decide how to proceed. in line 4, naima asked students to raise their hands, and when the first student did, naima called on him and he talked about missing subtitles. naima circled the empty spaces along the x and y axes, and wrote “no labels (subtitles)” next to the graph. the class conversation continued, and when the class again became quiet, ms. martin added: 5 ms. martin: what else? (pause.) there’s more. there’s other big ideas too. 6 (naima quietly wrote in the left-hand corner: “needs to start @ 10.”) 7 ms. martin: why does it need to start @ 10? tells us about that. 8 naima: because you can start this one and go from 0 to 20. (pointing to the y 9 axis.) this one, you go from 0 to 1. 10 student: it’s going up by 20. (naima looked back at the graph, shrugged, and 11 erased “needs to start @ 10”.) 12 ms. martin: i was actually, i was thinking it was a good idea, “needs to start @ 10.” or 13 some other number? 14 student: it’s going up by 20! 15 student: …doesn’t start at 0… 16 ms. martin: yeah, okay, let me back that up. it’s going by 20s, right? so the spacing is 17 okay. 18 naima: but there’s stuff in between it (pointing.). it would be easier to graph if 19 you did it by 10s, because there’s still stuff in the 10s spot. 20 ms. martin: yeah. look how big that grid is. and look how much space the line takes 21 up. (naima rewrote “needs to start @ 10.”) so if we went by 10s, or 22 even by 5s, we could space that 60 out, we could use the whole y-axis. so 23 that’s actually really important. i’ll ask you to make a full-page graph. 24 you want to use the whole graph to show the line, to show the whole 25 curve. so, spacing by another number would actually help us to see the 26 mistake up there. what else? … in line 7, ms. martin fostered the immediate expectation for naima to provide mathematical justification for “needs to start @ 10.” by using non-evaluative language, “why does it need to start at 10? tell us about that,” ms. martin positioned naima as genuinely able to justify her thinking while simultaneously conveying the dunleavy delegating mathematical authority journal of urban mathematics education vol. 8, no. 1 73 expectation that students contribute mathematical ideas. starting the origin at 10 with no other axes changes would be mathematically problematic because the spacing would be 10, 20, 40, 60. ms. martin, however, took naima’s assertion “needs to start @ 10” in line 6 as an opportunity for her to explain her thinking, which naima did in lines 8–9 and 18–19. ms. martin’s non-evaluative question in line 7 and revoicing in lines 20–24 positioned naima as a competent sense-maker. this excerpt illustrates how naima received the mathematical authority to highlight spacing as a common graphing mistake. the “needs to start @ 10” example addresses one way ms. martin used student presentations to delegate mathematical authority to students. she maintained a specific role of keeping the cognitive demand of the talk rigorous while orienting students to one another’s mathematical ideas. ms. martin also reframed naima’s contribution as valid, by offering the opportunity to justify mathematical thinking. in this way, “needs to start @ 10” brought authentic mathematical competence to the conversation. during the 26 class days captured on video for this study, 15 different students were delegated mathematical authority to lead whole-class conversations through student presentations. student presentations, then, were one of the classroom structures that delegated mathematical authority to students by positioning students to learn from one another’s mathematical ideas. case 2: delegating mathematical authority through shuffle quizzes ms. martin also used a second classroom structure, shuffle quizzes, to delegate mathematical authority and build equitable learning opportunities for students. a shuffle quiz is a strategy that ms. martin learned through her training in complex instruction. she described a shuffle quiz as a small-group structure in which one member of a small group would be chosen randomly and asked to explain a mathematics concept or problem their group was working on. prior to ms. martin approaching a group for the quiz, the groups were expected to work together, reach mathematical consensus, and to practice justifying their understanding of the mathematics in question. when she approached, she would shuffle their papers, and the student whose paper landed on top would be responsible for explaining the given work. the reason for randomly calling on students was two-fold. first, she told me that she wanted them to work together to explain the mathematics to one another. second, she was positioning them all as capable to represent the mathematics ideas of their group. when a group’s response was deemed mathematically valid, the group was allowed to move on to the next part of the assignment, thus “passing” the shuffle quiz. the interaction between ms. martin and one group during the first unit demonstrated how she used shuffle quizzes to delegated mathematical authority to students. ms. martin started the semester by developing students’ expectations for justifying mathematics during shuffle quizzes. part of developing the expectations dunleavy delegating mathematical authority journal of urban mathematics education vol. 8, no. 1 74 for justification meant leaving groups when their justifications were not yet mathematically sound, setting the expectation that they needed to continue working to understand the mathematics in a particular way, and then returning to groups for another opportunity to explain the mathematics and pass the shuffle quiz. after previously setting this group to a more in-depth discussion about explaining the meaning of variables, ms. martin re-approached the group for their second attempt to pass this shuffle quiz: 27 ms. martin: okay, did we continue that variable conversation? 28 students: sort of. 29 ms. martin: sort of? okay. i feel like there’s a lot of places where you, like, you sort 30 of have the right idea, but you’re not a hundred percent on, so i’m going to 31 bug you guys to be really detailed about it. (jaelyn’s paper lands on top, 32 indicating she begins the shuffle quiz.) so, jaelyn, part a, what do t and 33 d mean in this situation? 34 jaelyn: t means total amount of money. d means number of days. 35 ms. martin: okay, so when i was here last, we had a discrepancy about what, what do 36 you mean about total amount of money? can you put more words around 37 that? 38 jaelyn: no… it means total amount of money! like, in this case, it means, would 39 equal, total amount of money. correct? 40 ms. martin: yeah, well, i’m not sure that you guys actually had this conversation. did 41 you guys have this conversation? no, no, keep going. try again. total 42 amount of money, say more about that. you’re right, but it needs more 43 detail. 44 tamira: okay. she wants us to give more detail. like, stretch it out! 45 jaelyn: okay. i’m just putting words to it. total amount of money he has, each 46 week. 47 (pause.) 48 jaelyn: okay. i give up. 49 ms. martin: okay, so i think this is a misconception, i’m not sure. but, t and d are 50 variables, right? that means that the numbers they represent vary? they 51 change? so what does t represent? 52 tamira: the varying, the change. 53 ms. martin: every, each day. it’s 80 on day 0, so total amount of money left on each 54 day. 55 jaelyn: what did we start with? 80 it would be at the beginning. 56 ms. martin: 10 is happening each day. where is 10 in the table? 57 (continued shuffle quiz with other students.) 58 ms. martin: okay, so there were a couple hiccups here, where you had about the same 59 thing, but not exactly the same thing. so stuff like your tables are 60 important to check, and stuff like talk about a and b, where it asks you 61 in a and b to explain. i’m going to ask you not just to read, but to say 62 more about it. this excerpt illustrates one of the times ms. martin asked students to work together to achieve mathematical consensus. by telling them to further discuss the dunleavy delegating mathematical authority journal of urban mathematics education vol. 8, no. 1 75 significance of variables when she left, and then by walking away, monitoring their conversation from afar, and returning when she thought they might be ready, she held the group accountable for justifying mathematical understanding to one another before justifying it to her, as their teacher. when ms. martin returned, she first asked, in line 27, whether they were accountable to explaining mathematics to one another, “ok, did we continue the variable conversation?” she then made it clear that students had the responsibility to justify the mathematics, saying, in lines 30– 31, “i’m going to bug you to be really detailed about it.” when asked to describe what t and d represented, jaelyn repeatedly replied, “total amount of money” and “number of days” (lines 34, 38–39, and 45–46). ms. martin continued to press jaelyn to explain more, saying, in lines 35–37, “what do you mean about total amount of money? can you put more words around that?” and then in lines 42–43, “say more about that. you’re right, but it needs more detail.” after jaelyn said, “i give up” in line 48, ms. martin explained to the group in lines 49–51 that she was pressing jaelyn to describe t and d as variables. because this was the first shuffle quiz of the year, ms. martin explained, in lines 58–62, her expectations that she was going to ask students to work together to achieve mathematical consensus, “there were a couple hiccups here, where you had about the same thing, but not exactly the same thing.” this excerpt illustrates how ms. martin emphasized justifying mathematics and working toward mathematical consensus as ways to delegate mathematical authority to students in groups. the shuffle quiz excerpt displays the ways ms. martin used shuffle quizzes to delegate authority to students. shuffle quizzes became opportunities for ms. martin to regularly hold students accountable for individual and group sensemaking. holding students accountable for being on task, providing mathematical contributions, and justifying mathematics are all examples of the ways that delegating mathematical authority was realized in her classroom. shuffle quizzes, then, became a second classroom structure that ms. martin used to position students to learn from one another’s mathematical ideas and therefore delegate mathematical authority. case 3: delegating mathematical authority through participation quizzes ms. martin’s use of participation quizzes represented a third classroom structure that she used to delegate mathematical authority to students. the participation quiz was a public record, displayed on a smartboard at the front of the classroom, recording students’ group interactions while they worked together on mathematics. see figure 2 for an example of one participation quiz from early in the school year. dunleavy delegating mathematical authority journal of urban mathematics education vol. 8, no. 1 76 figure 2. using participation quizzes to delegate mathematical authority. before a participation quiz started, ms. martin prepared the purple and red phrases shown in figure 2 at the bottom of an otherwise blank smartboard screen. when students interacted in ways ms. martin found appropriate, she copied the prepared purple phrases into the space for their group. when students interacted in undesirable ways, she copied the prepared red phrases into the space for their group. when something happened that was desirable but not part of her prepared list, she wrote it in by hand, in blue. desirable ways of interacting from this participation quiz included, for example, “quick start” and “reading directions out loud.” undesirable ways of interaction during this participation quiz included, for example, “too quiet,” and “talking outside group.” students’ group grade was meant to represent how well they worked together as a group. a group’s significant number of dunleavy delegating mathematical authority journal of urban mathematics education vol. 8, no. 1 77 blue and purple ways of interacting, along with no red, was worth full credit on a participation quiz. the participation quiz was a classroom structure that ms. martin used to reinforce students working together in particular ways. she described her intent with participation quizzes in one interview, “i visibly grade students and show them what they’re doing well. making it public is really important, as is giving immediate feedback on the behaviors that contribute to good group work and ultimately good learning.” this quote indicates that ms. martin used participation quizzes to create and publicly reinforce “good group work.” for example, under “group 1” in this participation quiz, she wrote (a) “quick start,” (b) “talking [about] rate,” (c) “reading directions out loud,” (d) “that’s not making sense,” (e) “showing on calc[ulator],” and (f) “does it matter…[?]” the purple and blue phrases emphasize students’ ways of interacting, including (a) getting started quickly on mathematics, (c) reading directions out loud, and (e) showing what is displayed on a calculator to group members. the phrases also highlight students’ ways of speaking, including (b) talking about the mathematics topic, such as rate, (d) asking whether something matters, and (f) talking about whether something makes sense. the participation quiz was a classroom structure that ms. martin used to introduce and reinforce group work interactions that positioned students to use one another as mathematics resources. making the participation quiz a public representation was also one of the ways that students had the authority to monitor the expectations for mathematical justification. ms. martin used the participation quiz to delegate authority to students by reinforcing specific kinds of high-quality group interactions, including “get off to a quick start,” “point and explain,” and using “why?!” and “because!” sentences. the participation quizzes displayed the ways in which students were enacting the expected ways of interacting while working together, and they reinforced the type of group interactions ms. martin expected to see, thus serving to reinforce the ways ms. martin expected individuals to engage as students and as mathematicians. summary of ms. martin’s processes for delegating mathematical authority the cases presented in this section illustrate how ms. martin used three classroom structures—student presentations, shuffle quizzes, and participation quizzes—to position students as competent, to attend to status issues, to require students to justify mathematics, and to position students to work with one another. student presentations were used as opportunities for students to stand in front of the room and take intellectual and mathematical risks. ms. martin often used opportunities like “needs to start @ 10” to position students’ ideas as authentically mathematically valid. shuffle quizzes became opportunities for students to rehearse justifying mathematics with one another before being required to do so by ms. martin. participation quizzes were opportunities for students to receive public acknowledgement dunleavy delegating mathematical authority journal of urban mathematics education vol. 8, no. 1 78 and course credit for interacting in the ways that ms. martin expected. cases 1, 2, and 3 offer evidence of these classroom structures as processes ms. martin used to delegate mathematical authority to her students. delegating mathematical authority – striving toward equity this study was purposefully situated in an algebra i class where underrepresented and historically marginalized students were delegated mathematical authority to engage in high-level algebraic thinking and learning. in the context of a nation struggling to reposition students who have been historically perceived as having less competence than their peers, ms. martin’s classroom structures offered her students opportunities to engage in high-level mathematics sense-making. these classroom practices aligned with the nctm (2000) equity principle that all students must have opportunities to learn rich mathematics. furthermore, while this study focuses on three specific classroom structures, each was chosen to highlight a different facet of the ways in which delegating authority can position and reposition students as competent sense-makers. in case 1, ms. martin used non-evaluative language in a student presentation by inviting naima to explain her initially unclear but valid mathematical thinking about “needs to start @ 10.” in cases 2 and 3, ms. martin held students accountable to justify mathematics to one another through shuffle quizzes and participation quizzes. ms. martin used these three particular classroom structures to position students to do the intellectual heavy lifting when it came to sharing mathematical ideas, facilitating equitable opportunities to students who had not previously been successful in mathematics. she delegated authority by featuring classroom structures that positioned students as competent sense-makers, positioned students to use one another as mathematical resources, and required students to use valid mathematical justification. one student, helen, was particularly eloquent in summarizing ms. martin’s enactment of delegating authority to students: at first i was like, ms. martin is secretly teaching us! at first i had to try to get used to the fact that it was about working with groups. but it’s helped me a lot to improve my math skills and stuff. all of my other teachers will show you the formulas and show you the little tricks about how they will do the problem. but ms. martin will make us try and figure it out. ms. martin’s “secret teaching” offered students the mathematical authority to share their own methods for understanding mathematics problems. she created a learning environment in which students were comfortable sharing their mathematics thinking during whole-class and small-group interactions. through repeatedly drawing attention to many students’ different competencies, she increased the number of smart contributions available in the room. once the number of smart contri dunleavy delegating mathematical authority journal of urban mathematics education vol. 8, no. 1 79 butions increased, the amount of smartness students had access to, while learning high-level algebra i content, increased. closing thoughts although clearly illustrating how a mathematics classroom teacher might create processes that delegate mathematical authority to students, the study reported here is limited by this explicit focus on ms. martin’s classroom practices. while i collected data on students’ practices and perspectives, those data are not included here. nevertheless, the nature of the questions i asked during students’ interviews often uncovered students’ self-reflections of what was needed to improve their own mathematics experiences. for example, one two-part question i asked was, “think of a moment when you learned something really well. how did you know that you learned it well?” the process of interviewing students in this way may have been positively correlated with their changed perceptions of their experiences in ms. martin’s mathematics classroom. another limitation of the study is that data were collected during lessons in one class period and from one teacher who had training in ci. the study, therefore, does not attend to the scope or complexity that would be offered by simultaneously studying the delegation of authority in multiple pedagogies, classrooms, teachers, schools, and/or districts. in the end, the analysis reporting here contributes to the ongoing conversation about the importance of striving for equitable learning opportunities, while simultaneously revealing classroom structures that delegate mathematical authority. while previous research has analyzed mathematics classroom structures and 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(2014). latino/a youth’s perspectives on race, language, and learning mathematics. journal of urban mathematics education, 7(1), 55–87. retrieved from http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/188/152 http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/188/152 microsoft word 7 final ragland et vol 7 no 2.doc journal of urban mathematics education december 2014, vol. 7, no. 2, pp. 76–96 ©jume. http://education.gsu.edu/jume   tamra c. ragland is an education consultant at the hamilton county educational service center, 11083 hamilton avenue, cincinatti, oh 45231; e-mail: ragland.tamra@wintonwoods.org. her research interests include the intersectionality of self-perceptions, beliefs, goals, motivation, and self-efficacy and their impact on mathematics achievement. shelly sheats harkness is an associate professor in the college of education, criminal justice, and human services, at the university of cincinnati, p.o. box 210002, teachers/dyer hall, cincinnati, oh 45221; e-mail: shelly.harkness@uc.edu. her research interests include ethnomathematics; mathematics and social justice; mathematics and art connections; and teachers’ use of elbow’s “believing game” as they attempt to understand, honor, and respect students’ answers to problems. recruiting secondary mathematics teachers: characteristics that add up for african american students tamra c. ragland hamilton county educational service center shelly sheats harkness university of cincinnati in this article, the authors provide portraits of three mathematics teachers: one european american man, one african american man, and one middle eastern woman. all three taught in secondary schools with predominantly african american student populations. semi-structured interviews and observations were conducted to create a comparative case study that analyzed the teachers’ “star” background experiences and skills (haberman, 1995). the analysis also focused on whether or not there was a connection between their star background experiences and skills and their use of relational and field-dependent methods of teaching. data suggest that there was a connection between haberman’s star framework and teachers who taught relationally and used field-dependent methods. based on these results, the authors argue that alternative certification programs should broaden their criteria and recruit preservice teachers with star background experiences and skills. keywords: mathematics education, teacher recruitment, urban education there comes a moment…where we have to discuss “the black issue” and what’s appropriate education for black children. –lisa delpit, 1995 t some point we have to address what is best for african american students. while what (content) we teach and how (pedagogy) we teach mathematics are definitely important, in this article, we seek to continue the conversation about who should teach mathematics to african american students. martin (2007) raised this question and called for more research about mathematics teachers that a ragland & harkness recruiting secondary teachers journal of urban mathematics education vol. 7, no. 2 77 emanated from the experience lens rather than the achievement lens. the experience lens gives a “face” to the teacher and allows for the teachers’ beliefs, experiences, knowledge, and dispositions to rise to the surface; while the achievement lens places an emphasis on the teacher’s credentials (martin, 2007). the achievement lens may address the question of who is “qualified” to teach mathematics to african american students by mainstream standards, but it is the experience lens that addresses what qualifies them to teach mathematics to african american students. the experience lens informs the teacher’s instructional decisions and practices and how these decisions impact the teaching and learning of students. researcher positionality as researchers who explore critical issues in mathematics teaching and learning, we believe that it is important to the research process to reveal (although briefly) our perspectives and experiences that inform, inspire, and motivate our work. tamra i am a “biracial” woman who identifies as african american. i earned a bachelor’s degree in mathematics from a historically black college and university (hbcu), gained licensure through a master’s degree program at a predominately white institution (pwi), and achieved national board certification in adolescence and young adulthood mathematics. i taught high school mathematics for 13 years in the same urban district where i was a former student. as a teacher and instructional coach, i witnessed other teachers having difficulty teaching mathematics to african american students. i see the importance of credentials that deems one “highly qualified”; however, i also recognize that those credentials alone do not qualify a teacher to teach mathematics to african american students. i too often witnessed teachers, both black and white, with strong content knowledge, little or no classroom management skills, and sometimes low expectations who could not develop professional-personal relationships with their students to make the classroom conducive to learning. as a former mathematics teacher educator at a hbcu who prepared primarily african american students to teach secondary mathematics, i saw another problem. too often students with high gpas in mathematics content graduate without taking and/or passing the state licensure mathematics content examination (there is no requirement to take and/or pass the examination before graduation). while these students have the skills and disposition, they do not receive an initial teaching license, and have difficulty acquiring permanent teaching positions. because there is no required test preparation course at the university, they too often ragland & harkness recruiting secondary teachers journal of urban mathematics education vol. 7, no. 2 78 procrastinate and delay taking the examination until after they graduate. to address this void, a new required class was created to assist students with passing licensure requirements. given the diverse student population at the university, there is a great opportunity to impact the diversity of the teaching profession in secondary mathematics if we can overcome this hurdle of too many students either not taking or not passing the state licensure examination. shelly i am a white woman who taught middle and secondary mathematics in urban settings for 12 years prior to becoming a mathematics teacher educator. my research interest focuses on mathematics and social justice. for the past three years, i have participated in the revival of a grant program for college students of color who expressed an interest in teaching either mathematics or science. before i began working with the program, over 20 students participated and only one of them graduated with a teaching license. several of the students earned bachelor’s degrees in education but did not pass the licensing examinations. for the past 3 years, i have worked with five students of color, four of whom have graduated, passed the licensing examinations, and secured teaching positions in urban school districts. unfortunately, a disproportionate amount of mentoring time was spent on studying for the licensing examinations despite the fact that all of these students’ gpas in mathematics content courses and teacher education courses were above 3.25. sharing similar research interests, i served on tamra’s doctoral dissertation committee. conceptual framework star background experiences and skills haberman (1995) argued that “star teachers” (p. 1) focus less on teaching the content standards and objectives and more on turning students onto learning. in fact, haberman noted that only about 5 to 8% of teachers in schools entrenched with poverty are stars. nearly 20 years ago, haberman espoused the belief that teacher education programs should recruit candidates with star attributes and create immersion programs in urban school settings. according to haberman, the recruitment and selection is more important than the training. star teachers have background experiences that include many of the following: they are more than 30 years old; they are of color; they are parents or guardians; they have had military experience but not as officers; they live or have lived in urban areas; they majored in a field other than education as an undergraduate; and they have experienced living in poverty or can empathize with the challenges of living in poverty (haberman, 2005). ragland & harkness recruiting secondary teachers journal of urban mathematics education vol. 7, no. 2 79 haberman and post (1998) argued that too often urban children and youth of color “control the urban school’s agenda by making educators spend most of their time and energy reacting to street values rather than proactively implementing the stated curriculum” (p. 96). star teachers, however, do not let “street values” dominate in their classrooms. they promote their students’ successes by creating communities of learners, not by threats of force or coercion, but rather by teaching students how to become self-empowered and to take control of not only their classroom behavior but also their learning (haberman, 2008). according to haberman (as cited in ladson-billings, 2001, p. 80), the best teachers for poor children use these specific skills: • protect learners and learning, • put ideas into classroom practice, • challenge external labels given to students (e.g., “at-risk”), • develop a professional-personal orientation toward students, • satisfy school bureaucracies without comprising teaching quality, • recognize their own fallibility, • have emotional and physical stamina, • have good organizational ability, • focus on student effort rather than a vague notion of ability, • focus on teaching students rather than sorting them, • convince students that they are needed in the classroom, and • serve as allies with students against challenging material. relational and field-dependent teaching methods hale (1982) asserted that social class and ethnicity shape cognitive style, and that children learn their cognitive style through family socialization. african american children tend to employ a relational cognitive style and schools tend to value an analytical cognitive style (bonner, 2000; hale, 1982). african american students, therefore, tend to be more dependent on their relationships with teachers than other races (irvine, 2009); that is to say, they more often than not need to establish positive affirming and caring relationships with teachers. furthermore, african american students’ learning styles are typically field-dependent while learning styles for white students are typically field-independent (shade, 1994). people who reason analytically and sequentially are field-independent thinkers whereas people who reason holistically, simultaneously, or relationally are fielddependent thinkers (malloy, 1997; shade, 1994). it is important to note that fielddependence and field-independence characterize learning styles and behaviors, but they are not mutually exclusive to race or ethnicity (shealey, lue, brooks, & mccray, 2005). additionally, learning style is “not a psychological trait but a dy ragland & harkness recruiting secondary teachers journal of urban mathematics education vol. 7, no. 2 80 namic state resulting from synergistic transactions between the person and the environment” (joy & kolb, 2009, p. 71). stiff and harvey (1988) claimed that “field-independent teaching and learning styles value analytical thinking and systematic approaches to problematic situations…what one finds is that school mathematics and the manner in which it is delivered are in opposition to the communication and learning styles of black americans” (pp. 196–197). effective culturally responsive teachers of black children, therefore, not only acknowledge the cultures, backgrounds, and strengths of their students but also they utilize students’ learning styles (gollnick & chinn, 2009). to recap, african american children most often are socialized in their families to be relational in cognitive style and field-dependent learners. therefore, mathematics instructional practices for field-dependent learners should include holistic approaches, cooperative environments, intuitive thinking opportunities, and informal discussions (malloy & malloy, 1998; morgan, 2009; sadler-smith, 1999; saracho, 1991; slavin & oickle, 1981; stiff, 1990). teachers who provide positive feedback to students in secondary mathematics courses may encourage field-dependent students’ achievement (adegoke, 2011). in general, research suggests that teachers who employ the above methods, which value relationships and field-dependent learning, validate the actions and cognitive styles of african american students. the study purpose this study is based on interviews and observations with three secondary mathematics teachers who taught in urban secondary schools. the purpose of this study was to analyze the teachers’ star background experiences and skills (haberman, 1995). additionally, the study focused on whether or not there was a connection between the teachers’ star background experiences and skills with regards to their use of relational and field-dependent methods of teaching. the following three research questions guided the study: 1. what star background experiences have the teachers had? 2. what star teaching skills do the teachers use? 3. what connections emerge between star background experiences and teaching skills and instructional methods focused on relational learning styles and field dependency? ragland & harkness recruiting secondary teachers journal of urban mathematics education vol. 7, no. 2 81 research design according to patton (2002), “the purpose of the case study approach is to gather comprehensive, systematic, and in-depth information about each case of interest” (p. 447). thus, we used a case study approach to gather information about the three teachers. we first examined each teacher as an individual case. then, we conducted a collective case analysis (creswell, 2012) to provide insight into teachers’ star background experiences and skills with regards to relational and field-dependent teaching methods. participants and sampling procedures the school district in which the participating teachers were employed was located in a mid-size metropolitan city in the mid-western region of the united states. a purposeful sampling method was used to select participants. the intent was to select a diverse but small group of mathematics teachers for maximum variation (creswell, 2012). to achieve this variation, the participants needed to meet the following criteria: (a) possess unique race, ethnic, and/or gender identities; (b) teach mathematics at different grades levels; and (c) had earned their teaching licensure through an alternative route. the first author identified three participants whom she knew personally and who met all the criteria, invited them to participate, and they, in turn, accepted. the only consistency among the participants was that they all took a nontraditional or alternative route to licensure and taught in the same urban school district. after achieving minimal consistency, we looked for variability. we wanted to see, given the variability, how each teacher’s case study uniquely aligned to the literature. our three teachers were: grant, hayden, and wisnewski (pseudonyms). data collection teacher interviews. the first author interviewed each teacher twice during the study. the first semi-structured interview was designed to learn about the teachers’ background experiences and their beliefs and philosophies about education and teaching. all interviews were audiotaped and transcribed. each interview lasted 45 to 60 minutes. the second interview was conducted after the classroom observations (described below) were completed. this second interview was openended and its purpose was to have the teachers clarify instructional strategies observed and to give each teacher the chance to describe his or her teaching style, strategies, and teaching behavior. field notes from the observations drove the development of questions to be asked of each teacher for the second interview. ragland & harkness recruiting secondary teachers journal of urban mathematics education vol. 7, no. 2 82 classroom observations. in addition to the two interviews of each teacher, the first author conducted five observations of each teacher interacting with his or her students. the first author entered the room before the start of each class period to be observed to minimize altering or interrupting the classroom routine. field notes were taken to captured student–teacher conversations and interactions. data analysis we immersed ourselves in the data by reading and re-reading interview transcripts and field notes from the observations. we continually referred back to the star background experiences and teaching skills as a priori categories to document evidence from the data that might align with these categories. table 1 shows an example of one of the tables we created for one of the teachers, grant, in our study. in table 1, we listed the star teacher skills, the date the data were collected, the source (observation or interview) of the data, and the page number(s) of the source. the tables for each teacher assisted in determining how the data supported (or not) our findings. table 1 grant’s evidence of star teaching skills teaching skill date source page protect learners and learning 3-12-08 observation 4 interview 2 p. 2 p. 5–6 (i try to hinder…) put ideas into practice interview 2 p. 1 (pd) challenge external labels interview 2 p. 3 (not out of same mold) develop professional-personal relationships interview 2 p. 3 (tie the mainstream with a little ghetto) satisfy school bureaucracies interview 2 p. 3 (follow the rules but….) recognize own fallibility interview 2 p. 3 (be more creative) have emotional and physical stamina * possess good organizational skills * focus on student effort 3-11-08 observation 3 interview 1 interview 2 p. 2 (all can learn …at different pace) p. 5 (everybody has unique attributes) teach students rather than sort convince students they are needed observation 3 interview 2 p. 3 p. 7 (some things i can’t get across to them, but their friends can) serve as allies with students 3-5-08 observation 1 interview 2 p. 1–2 p. 8–9 (word walls … language) * evidence from professional experiences with grant. ragland & harkness recruiting secondary teachers journal of urban mathematics education vol. 7, no. 2 83 next, we totaled the number of experiences and skills for each participant. then we compared the participants’ background experiences and skills to the extant literature related to relational or analytical cognitive styles as well as fielddependent and field-independent learning styles. validation of the findings and delimitations we had numerous researcher meetings to discuss our interpretations. additionally, we shared drafts of our analysis with colleagues in teacher education programs at two other universities. these colleagues provided feedback and furthered our discussion about our interpretations to validate our findings. we do not claim that the findings are generalizable because the data are from only three teachers. furthermore, the teachers were selected from one school district, and the first author knew all three personally prior to the study. the teachers were chosen in a type of purposeful sampling method (creswell, 2012); each had diverse background experiences before and after he or she became mathematics teachers. findings teachers’ background data in table 2, we summarize the individual teachers’ demographic data collected during the interviews. what follows are individual descriptions of the teachers’ backgrounds and two additional tables in which we summarize their star background experiences (haberman, 2005) and star skills (haberman, 1995) in table 3 and table 4, respectively. table 2 teachers’ demographic data demographic grant hayden wisnewski gender male female male race and/or ethnicity black iranian white years teaching 17 years (urban) 20 years (urban) 29 years (urban) school type middle school high school high school subject/grade mathematics 8th grade algebra ii 11th grade precalculus 12th grade university training public, hbcu public, pwi private, catholic undergraduate degree business education engineering/ mathematics mathematics ragland & harkness recruiting secondary teachers journal of urban mathematics education vol. 7, no. 2 84 grant. grant was a 51-year-old, male african american who had been teaching mathematics for 17 years at the time of the study: 14 years at the high school level and 3 years at great river middle school. he came from a family of educators who also had worked within the same school district. grant grew up in the city where he taught and also had graduated from a local high school. his ties to the district were strong because of his family, and he was still living in the community. grant attributed his father with introducing him to teaching at an early age because his father allowed him to coach 10-year-old children in a tennis program for inner-city youth. although he admitted to being a “horrible student,” grant attended a hbcu and earned a business education degree. after graduation, he became a long-term substitute teacher for 2 years at a high school and then decided to join the army. while in the army, he became an ammunitions specialist. he trained military personnel to use firearms and ammunition using “a little math.” after military service, he took courses at a local university and received a mathematics certification and his master’s degree in education. within 2 years, he completed a second master’s degree in education administration from the same university. hayden. hayden had been a high school mathematics teacher for 20 years. she was originally from iran and left there when she was 16 years old due to the political and religious unrest at the time of the khomeini regime. she was very close to her sister who also taught mathematics. when she started college, she majored in engineering with a geology option. her intent was to return to iran and “make lots of money,” but she married an american. her engineering profession required traveling, which her husband did not like, so she went back to school to pursue a degree in mathematics. while pursuing this degree, she tutored mathematics majors, and her tutees suggested that she become a mathematics teacher because she was “really good at helping them understand it.” over the years, hayden developed a reputation for helping her students pass the ap calculus exam. she reported that she had seven ap calculus students at the time of the study. wisnewski. wisnewski had been a high school mathematics teacher for 29 years and worked the last 23 years in the same building while serving as the department chair. he admitted that he majored in mathematics because he liked it in high school and did well. instead of asking the teacher for help, several of his classmates often asked him for help. while in college, his roommate and other friends asked him for help with calculus. although he graduated with a mathematics degree, wisnewski did not consider the teaching profession. after graduation, he worked in a furniture warehouse for five years. he decided to become a teacher because he wanted to do “something worthwhile” and he attended the university where he earned his undergraduate degree to pursue a master of science in teaching degree. he started teaching after completing his student-teaching assignment within this same district. ragland & harkness recruiting secondary teachers journal of urban mathematics education vol. 7, no. 2 85 teachers’ star background experiences table 3 illustrates the star background experiences of all three teachers: grant, hayden, and wisnewski. based on the interview data, we summarize the attributes that haberman (2005) noted star teachers typically share and predict their effectiveness and retention in schools serving poor minority students. table 3 star background experiences experience g h w 1. more than 30 years of age (when started teaching) no no no 2. parent or has extensive relationships with children no yes no 3. had military experience but not as an officer yes no no 4. lived in a metropolitan area yes yes yes 5. majored in a field other than education as an undergraduate yes yes yes 6. experienced living in poverty or can empathize with the challenges of poverty yes yes no 7. attended schools in a metropolitan area yes yes no 8. had out-of-school experiences with diverse children yes yes no 9. earned a bachelor’s degree from a non-highly selective or non-elitist university yes yes yes 10. had extensive work experience before teaching no yes yes 11. engaged in paid or volunteer activities with diverse children yes yes no 12. multitasks quickly or for extended periods (e.g., parenting and working part time) yes yes no 13. a member of a minority group or working class family yes yes no 14. lived in a city or would move to the city for residency requirement yes no no 15. part of family/church/ethnic community in which teaching is still regarding as a high status career yes yes no key: g = grant; h = hayden; w = wisnewski teachers’ star skills and teaching methods grant. grant typically started class with a bell ringer (a released item from the state achievement test), which students were expected to complete within the first 10 to 15 minutes of class. he then explained the expectations for the day and mentioned the mathematics standard and benchmark that was written on the whiteboard. afterwards, he moved into whole group instruction for about 20 minutes and then gave similar exercises for the students to solve in their groups. grant wanted his students to respect each other enough to listen to what each had to say. when students asked him a question, he responded with, “what did your ragland & harkness recruiting secondary teachers journal of urban mathematics education vol. 7, no. 2 86 teammate say?” or, if a group member made a plausible argument, he encouraged the other group members to take heed. he knew that his students needed to become better problem solvers and less dependent on him for answers. thus, he wanted them to believe in themselves enough to find an answer and convince the other group members: (one student asks mr. g a question.) mr. g: what did your group members say? what do you think? well why aren’t you sticking up for your own thoughts? (mr. g moves to next group. there’s a lot of loud conversation between group members.) mr. g: y’all need to listen to him (points to one student). he explained what y’all was tryin’ to find. students: aren’t we going to find area? mr. g: did they say area? what did they say? (mr. g continues to make his way around the room to check on other groups.) mr. g: (to group in back) y’all straight? (he turns to other group in front with same problem.) i don’t know why they got it and y’all didn’t. they ain’t no smarter than you are. some of y’all clown too much but you can solve it. grant allowed students to work in groups even though he did not like the loud noise that usually occurred. he admitted, however, that he used peer groups because the students learned more from each other than they did from him. he believed all of his students were capable of learning mathematics and could be successful but the key, for him, was getting the students to believe in themselves: i believe that all kids can learn. they might learn at a different pace. some may not really try. they don’t see the benefits education can afford in life and so i try to be an agent of change in their attitude and get them to understand why education and learning is so important. one way he accomplished this change in attitude was by using out-of-school experiences and connections to the students’ lives to support the mathematics content. he artfully told stories and admitted that he sometimes told “white lies” if he could somehow relate a real-world experience to what was studied in class. for example, during one lesson he used going to the store as the context of a problem. he frequently used the students’ language, black english vernacular, but not in a condescending way. for example, when a student did not explain the process of finding a solution using “standard” english he said, “i know what you are trying to say, but those test graders in “north cackalacky” [his reference to north carolina] don’t know so we need to use proper english,” or when he checked for understanding he would say, “y’all wit me? are we here?” moving his finger back and forth between himself and the students. he acknowledged that ragland & harkness recruiting secondary teachers journal of urban mathematics education vol. 7, no. 2 87 his two all-male classes and one mixed-gender class were comprised of african americans and a few latinas/os who were into sports so he made sports references when they were relevant. he also took students to an nba basketball game every year: i take them out on the floor. they can see the scorekeeper. …these guys make over $100,000 for sitting, checking the period of averages, who has the most rebounds. all that’s math. the ticket takers…scanners for people who enter the game. there are numbers involved in that. they have to post attendance of every game for the newspapers. …to express the importance…what type of education you need for that and that sort of thing. the camera men…because most of our kids see, all they see is the basketball players. grant was known to be a disciplinarian and whom students perceived as “mean.” his classes were “well-behaved.” he grew up attending segregated schools—with black children and black teachers—and when he went through school it was a family-like atmosphere, with direct instruction, and a lot of discipline. he described himself as a horrible student. when younger, he saw how his father exhibited understanding and compassion, along with a healthy dose of tough love. grant said he adopted the same strategies. during a parent-teacher conference, one student said, “mr. grant, he too mean. he so mean he make me do my work.” he noted how he and his students worked out behavior and academic problems among themselves. he may have given students a piece of his mind, but the students knew grant had the students’ backs and would be there for them. most discipline he handled in house rather than referring students “to the system” (e.g., sending them to the principal’s office). he admitted that he did not always articulate his discipline lectures with political correctness, but the students appreciated his care in not writing them up for discipline referrals. grant allowed for plenty of discussion and conversation about topics that were school related but was willing to entertain non-school related topics as well. he understood that some of his students came to school with more “street knowledge” than “school knowledge”; thus, he used their knowledge to bridge and share knowledge between himself and his students. he acknowledged differences and diversity among his students. he said, “everybody brings unique attributes [to the classroom].” he thought students learned in different ways and at different paces. he had a different definition of successful teaching of african american students, and believed that if he could get a student to try a math problem, he had succeeded. he said: i don’t look at success as a, b, c, d, or f. if i could just get my students to try to learn something, even if they are not successful the first time, they could learn from their mistakes. as long as i get them to try, i am having some type of success. ragland & harkness recruiting secondary teachers journal of urban mathematics education vol. 7, no. 2 88 he thought that his students needed as many experiences as possible and looked for any possible means for success. as illustrated in table 4, grant displayed 10 star skills during his classroom instruction and interviews. grant’s teaching focused on building relationships and encouraging his students to think holistically by using field-dependent teaching strategies. he did this by using cooperative groups (slavin & oickle, 1981) for students to engage in discussion (shade, 1994). he valued the benefit of the group and encouraged cooperation among all parties in order to complete the tasks. he respected and had knowledge of students’ lives, culture, and experiences and used this knowledge in instruction (malloy & malloy, 1998). he validated his students by using their language and their experiences as the social context to teach mathematics (berry, 2008; stiff & harvey, 1988; sheppard, 2011). hayden. hayden taught in a school where the “arts came first and the academics came second.” some of her students were extremely talented in art, dance, or music, but she encouraged her students to pursue careers in mathematics rather than the arts. she told them with a career in mathematics they “may not make a million dollars, but they won’t sleep in the streets.” she typically presented new material at the beginning of class and if there was time she walked through each row talking to each student as she passed. some of her comments were related to the work they were doing; some were not. if a student had a question, she answered it and then moved to the next student. sometimes she just stopped and looked at their papers, and if she noticed a mistake, she would point it out to them: ms. h: (to a girl) i know this is your favorite (pointing to a polynomial long division paper) but you have to do this too (moves polynomial long division paper and puts another paper on top). after a while this become your favorite too. student: i know. (student smiles) ms. h: (to a boy) where your graphing calculator? student: somebody stole mine. ms. h: hun, what about you, you don’t ask me any question. student: i don’t have any. ms. h: let me see—come on, let’s sit together. she turned and moved toward her desk and the student followed her. individual students or students in small groups approached her desk and sometimes she asked students to assist other students at her desk: you should not be so egoistic and sometimes understand that kids understand from other kids better. sometimes when i see a kid who explain really well, i say, “wow, she explains really good.” so i encourage the kid to explain to other kids…whatever works for them. ragland & harkness recruiting secondary teachers journal of urban mathematics education vol. 7, no. 2 89 if students were late to class she reprimanded them. however, there seemed to be an understanding that if they had to get a drink or use the restroom, they could do these things without permission. she rarely had discipline problems. she believed all of her students could succeed and that it was her job as a teacher to “turn the light on” for her students. she gave them encouragement and a healthy dose of tough love: “just keep teaching them harder and be hard on them and make them be the best that they can possibly be. and don’t let them say they can’t learn this, or they can’t do this.” she viewed their academic achievement as an opportunity to overcome their circumstances: we shouldn’t make them feel that they are different than the white or the yellow or the japanese. they come to our classroom and they should be learning and you have every chance to succeed. you have every chance to become whoever you want to be. we owe them that much. we have to make them feel that way, because they can go home and feel miserable. i always tell them that a mind is a beautiful thing. it really is. there is no price you can put on when you have a good thinking process. you can do such a beautiful problem and later on you can get a good job. you can do a lot of things. hayden saw herself as a part of her students’ families. she thought all teachers were extensions of students’ families because teachers touched students’ lives and served as role models: my life is not a secret. they ask me a lot, i share a lot of things about myself, because somebody who teaches them, somebody who’s with the kids, they think you become family, because whether we like it or not we are touching their lives. like maybe they say that in 10, 15 years, i want to be like mrs. h. i want to have two kids. i want to be like her. hayden respected the knowledge her students brought to school. she learned from them, and they learned from her. she admitted she was not aware of black– white racial issues when she arrived in this country, but her students had shared their experiences with her and she could relate. in her native country of iran, there were not racial issues per se, but there were issues regarding gender and social class. she said, “if you go to middle east, rich people have better lives. they don’t have black and white issues, they have, where is the money…our job is to teach them to rise above that.” she acknowledged that she had an accent that became thicker when she was excited and talked fast. she knew her students did not always use standard english, but admitted she did not use it either. however, she wanted to know what her students were saying, and how they were saying it. therefore, she asked them to explain their language to her. it was evident that she was passionate about mathematics and teaching. when students told her that they wanted to major in math ragland & harkness recruiting secondary teachers journal of urban mathematics education vol. 7, no. 2 90 ematics when they went to college, she became excited. when she explained how to use the equation to find h, k and the foci so the students could graph an ellipse, she asked them, “you see the beauty in this?” as illustrated in table 4, hayden displayed 11 star skills during her classroom instruction and interviews. hayden’s teaching focused on building relationships with her students where she specifically thought of herself as a member of their family. she supported field-dependence by giving students straightforward problems and provided immediate individual feedback (stiff & harvey, 1988) for the entire class by walking through the classroom and examining students’ work after she explained the exercises. she assisted students with questions, and when those students used correct procedures, she asked them to help others. by encouraging this collaborative effort, she promoted a sense of security and interdependency within the classroom (stiff & harvey, 1988). she respected and had knowledge of students’ lives, culture, and experiences (malloy & malloy, 1998) and encouraged students to overcome negative experiences by presenting mathematics concepts as both powerful and beautiful. wisnewski. the first day in wisnewski’s classroom, a student, bianca, introduced herself as the next political star. the class had 23 students and they were noisy. after the bell rang, there were students out in the hallway knocking on the door. as the late seniors walked in, wisnewski said, “minus 2, minus 4, minus 6,” marking students’ tardiness. he shut the door and soon after another late student knocked on the door. he told her to come in and he admonished her for being late. he started his lecture by talking about transformations of periodic functions and over the next 5 days he taught and re-taught how to identify a dilation, horizontal shift, vertical shift, and period of a function using equations and graphs. he worked through each exercise step-by-step because, it seemed, he assumed his students did not know what to do to solve the exercises on their own. wisnewski wrote h(x) = 6 sin(10x – 2)” on the board and asked: mr. w: what is the amplitude? students: six. mr. w: remember amplitude is the same as dilation. remember to factor out 10. student: n/5. mr. w: right, n/5. (he wrote h(x) = -6 sin10 (x – n/5) on the board, which was incorrect.) mr. w: i know it’s not the pre-calculus stuff that causes you problems; it’s the fractions stuff. the fifth grade stuff. a formal teacher–student relationship existed in his room. he was the authority and he had the knowledge to be dispensed through lecture and no one challenged him. after the lecture, students on the left side of the room worked quietly on problems individually while students on the right side of the room formed ragland & harkness recruiting secondary teachers journal of urban mathematics education vol. 7, no. 2 91 pairs or groups. one group on the right side of the room, comprised of six girls, lined their desks up together in the front of the classroom. they loudly debated back and forth about how to draw the graph, shared their notes, and looked at each other’s solutions. when the girls could not agree on a solution they asked wisnewski to settle the disagreement about the problem. however, he spent more time with the quiet students working independently than he did with the students who worked in small groups. he seemed to connect with certain students while he walked around the classroom answering questions and collected homework at the beginning of class. the same students often asked him questions and interrupted him during his lectures. however, he did not get upset when students interrupted him even though, at times, he could not finish a sentence. wisnewski said that he “believes that all students can learn,” but admitted: an awful lot of kids nowadays with a background that is unbelievably bad…it seems to me that if they are weak in understanding such basic things that it is really somewhat unrealistic to get them to think at a higher level, if they have not mastered the basic skills. he did not think that it was realistic for students with a “low-level” mathematics background to expect to pass the graduation test in high school if they did not pass previous tests in elementary grades. it bothered him that high school teachers were blamed when this happened. he saw his students as having “life and death experiences that suburban kids haven’t had” and a lot more going on outside of school than suburban kids such as taking care of younger sisters and brothers, and jobs. he noted, “a teacher should not assign much homework because if the student does not complete it in school, it will not get done.” he did not attempt to make connections with mathematics to the students’ lives or acknowledge attributes of african american culture except for imitating their speech. he said, “you know like they may say, i don’t understand, and i’ll come back with, i know why you not be understanding ‘cause you not be doing homework, and they’ll laugh at that and say, yeah, yeah, i know i need to be doin’ my homework.” he said that his students were: too social for cooperative learning groups and are highly likely to get off task. i don’t see enough kids who are really good enough maybe to help the other kids. …i just see them getting off task too easily and again the goal seems to be the social interaction and not the concept i am trying to teach. wisnewski did not seem to believe his students to be capable and knowledgeable about mathematics. he did not appear to understand how they learned or what knowledge they brought with them to the classroom. his conception of knowledge was very analytical and linear. he noted: ragland & harkness recruiting secondary teachers journal of urban mathematics education vol. 7, no. 2 92 i think that these kids are more dependent on the teacher and each other. simply for one reason, they don’t have the general background because they don’t do much outside of school. the white suburban kids, they are more likely to do their homework outside of school, they are more likely to get parental support, go to the library, and get the research done. this needs to play a much bigger role with inner city or african american students. as illustrated in table 4, wisnewski used or described his use of four star skills in interviews. he used three additional star skills but only with some students and not with all students. wisnewski’s teaching did not support his students to be field-dependent thinkers. he did not support students working in collaborative groups, even those who grouped themselves. he saw their dependence on each other and on him as a problem that came as a result of their limited background. he saw their culture and life experiences as having little value. for the most part, winsnewki taught his class as if they were field-independent learners by encouraging students to “focus on detail and use sequential/structured thinking…focus on the task, learn from formal lecture, achieve individually, and emphasize facts and principles” (malloy & malloy, 1998, p. 251). he was frustrated by his students because he taught the same skills repeatedly. perhaps his frustration came as a result of repeated attempts to teach field-dependent learners with field-independent strategies. summarizing teachers’ star skills and teaching methods all three teachers readily admitted that they were not “prepared” specifically to teach african american students in urban settings during their teacher preparation programs. wisnewski, however, had an african american cooperating teacher and student-taught in this same urban district. but both wisnewski’s and hayden’s introduction to african american culture was on-the-job training. it was while working with other african american teachers and principals that they learned more about their students. wisnewski admittedly did not use african american culture in his class. he clearly saw deficits, not differences, and thought that his students could not learn what he was teaching until they learned the “lower-level” mathematics first. to connect to her students, hayden used examples of african american mathematicians in her class, among other things. and grant used real-world examples to teach mathematical concepts, among other strategies, to connect to his african american students. as illustrated in table 3 grant and hayden met 12 out of 15 of haberman’s (1995) star background experiences while wisnewski met only 4. when we combined the results of haberman’s star background experiences (before they started teaching) and star teaching skills (after they started teaching), we had a clearer picture of which of these teachers’ skills supported relational and field-dependent ragland & harkness recruiting secondary teachers journal of urban mathematics education vol. 7, no. 2 93 methods in teaching mathematics to african american students. in fact, as illustrated in table 4 haberman’s background characteristics aligned to the teachers’ star skills and teaching methods prior to their teaching. table 4 summary of star teaching skills teaching skill g h w protect learners and learning yes no yes put ideas into practice no yes no challenge external labels yes yes no develop professional-personal relationships yes yes * satisfy school bureaucracies yes yes yes recognize their own fallibility yes yes no have emotional and physical stamina yes yes yes possess good organizational ability no yes yes focus on student effort yes yes no teach students rather than sort yes yes * convince students that they are needed yes yes no serve as allies with students yes yes * key: g = grant; h = hayden; w = wisnewski * yes and no, because yes with only some students implications for alternative licensure programs perhaps, teacher recruitment for alternative programs should go beyond analysis of gpas and the passing scores on teacher licensure exams; this is especially true when disproportionate numbers of minority students fail content licensure exams in mathematics (goldfaber & hansen, 2010; tyler, et al., 2011). furthermore, the study conducted by goldfaber and hansen (2010) claimed that “replacing the failing black teacher with a passing white teacher considerably decreases student outcomes” (p. 245). what are the implications for alternative teacher licensure programs such as teach for america (tfa, 2013), math for america (mfa), and the woodrow wilson teaching fellowship (ww)? these programs seek to take bachelor’s degree holders and turn them into teachers. mfa (2013) and ww (2013) place their emphases on gpas and praxis ii test scores for entry into their respective programs; such an emphasis is an achievement lens focus on teaching (martin, 2007). tfa, however, places less emphasis on the gpas, the minimum being a 2.5, but more emphasis on skills or characteristics that cannot be measured by gpa or praxis ii (such as skills outlined in table 1). these characteristics provide a holistic look at the teacher candidate, just like haberman’s (2005) background experiences criteria. applying haberman’s ragland & harkness recruiting secondary teachers journal of urban mathematics education vol. 7, no. 2 94 fifteen attributes to each of these programs’ self-reported selection criteria, only tfa comes close to using what could be called an experience lens focus (martin, 2007) approach to prospective teacher selection. while haberman’s star background experiences alone may not guarantee success, teachers with these experiences along with star skills may likely be more effective in teaching african american students in mathematics. conclusion there is an additional richness and depth to teacher qualifications and characteristics researched by haberman (1995, 2005, 2008), ladson-billings (1990, 1994, 1995a, 1995b, 1997, 2001), and others. this past research should not be overlooked today as old or outdated because it is crucial for the education of all students, particularly african american students, that teachers understand their students and are prepared to teach them. with the student population becoming increasingly more diverse, especially in urban areas (snyder, 2008), it is critical that teachers be prepared to teach differences without labeling the differences as deficits. based on our findings, more research is needed that focuses on the recruitment of teacher education. perhaps alternative licensure programs should attempt to recruit mathematics teachers with an equal focus on both the experience lens and the achievement lens (martin, 2007) to teach african american students. additionally, we wonder what impact and power recruiting preservice teachers with star background experiences (haberman, 1995, 2005) into traditional teacher education programs might have on preservice teachers without these experiences. references adegoke, b. a. 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(2013). re: what we look for in a fellow [website]. retrieved from http://www.wwteachingfellowship.org/apply/what-we-look-for.php caring teachers build on the cultural knowledge and resources students bring to the classroom: journal of urban mathematics education july 2011, vol. 4, no. 1, pp. 50–74 ©jume. http://education.gsu.edu/jume tonya gau bartell is an assistant professor of mathematics education at the university of delaware, school of education, 16 w. main street, newark, de, 19716; email tbartell@udel.edu. her research focuses on aspects of preparing teachers to teach for social justice and equity. caring, race, culture, and power: a research synthesis toward supporting mathematics teachers in caring with awareness tonya gau bartell university of delaware in this article, the author draws on theories of care to lay out a theoretical map of sorts on what an effective, caring teacher–student relationship that supports student learning might “look like.” in so doing, theories of culturally relevant pedagogy are considered, as these not only illustrate effective practices caring teachers employ but also because such theories provide models of classroom practices that consider explicitly issues of race, culture, and power. the author aims to illuminate the complex, nuanced, and, at times, overwhelming descriptions of what it means to be a caring teacher in the service of student learning. the author concludes by considering models of professional development that hold potential for supporting mathematics teachers in developing teacher–student relationships reflective of “caring with awareness.” keywords: care theory, culturally relevant pedagogy, mathematics education, teacher professional development ddressing equity or, more appropriately, inequity, remains at the forefront of current reform efforts in mathematics education (national council of teachers of mathematics [nctm], 2000, 2008). persistent gaps in opportunities to learn mathematics between historically underrepresented students and their middle-class white counterparts remain (flores, 2008). urban schools, populated largely by minority students who too often live in disadvantageous economic circumstances, are under-resourced and underachieving (darling-hammond, 2007; ladson-billings, 2006). low-income and minority students are less likely to have qualified teachers and well-resourced schools (hill & lubienski, 2007; oakes, 2005) and are more likely to experience school mathematics as disconnected from their out-of-school experiences (civil, 2007; nasir, hand, & taylor, 2008). addressing these gaps in opportunities to learn requires teachers to see mathematics as not only relevant to but also part of students‘ lives and communities. it requires teachers to move beyond a narrow focus on measurable performance as dictated by the pressures of standardization and mathematics testing to attend to students‘ a bartell caring with awareness journal of urban mathematics education vol. 4, no. 1 51 interests, cultural backgrounds, and concerns; it requires getting to know students well enough to engage them in learning and relating to students across cultural, racial, and socioeconomic lines. looking to theories of care in education can provide insight. given society‘s pervasive deficit orientation toward urban students and communities—too often reflected in preservice teachers‘ (and, likely, inservice teachers‘) stereotypical views of urban education environments (aaronsohn, carter, & howell, 1995; gomez, 1996)—in this article, to counter this deficit orientation, i make a case for mathematics teachers to be provided learning opportunities to understand the need for and importance of developing caring relationships with students. one long-accepted characteristic that is often repeated in the literature of an ―effective‖ teacher is the ability to cultivate and maintain strong interpersonal relationships with students (good & brophy, 2000; ladson-billings, 1994; noddings, 1992). this literature suggests that certain relationships, such as those promoting an ―ethic of care‖ between teachers and students (noddings, 1984, 1992) lead to higher levels of student engagement and achievement (pianta, 1999). furthermore, research documenting effective practices for traditionally marginalized students, such as culturally relevant pedagogy, suggests that care is an integral component of these practices (irvine, 2002; ladson-billings, 1995). these relationships are built on teachers‘ understanding of each student ―in nonstereotypical ways while acknowledging and comprehending the ways in which culture and content influence their lives and learning‖ (darling-hammond, 2002, p. 209), are necessarily political (gutstein, 2006), and allow teachers to utilize the cultural and linguistic resources students bring to the classroom to further their learning of content (cochran-smith, 1999). establishing productive teacher–student relationships in the mathematics classroom has direct implications for equity. teachers consider a number of factors when determining whether and how to engage in relationships with their students (davis, 2001). these factors include a teacher‘s sense of a student‘s likelihood for success (muller, katz, & dance, 1999), how teachers understand their role as teachers (woolfolk hoy, davis, & pape, 2006), and their beliefs about students‘ abilities and motivations (stipek, givvin, salmon, & macgyvers, 2001). in the teacher–student relationship, these varied beliefs can take the form of greater attention for more highly regarded students, valuing their responses and evaluating their performance more positively. in contrast, teachers are more likely to accept poor performance from students for whom they hold low expectations (brophy & good, 1970) and students who are perceived to be low in ability may be given fewer opportunities to learn new material, asked less stimulating questions, or given briefer and less formative feedback (cotton, 1989). in the context of mathematics education, the nature of a teacher‘s relationship with her or his students impacts whether and how the teacher views a child as mathematically bartell caring with awareness journal of urban mathematics education vol. 4, no. 1 52 competent; this view, in turn, impacts the subsequent mathematical situations posed to a child to further her or his mathematical understanding (hackenberg, 2010a). teachers also tend to have preferences for students whom they perceive to be most like themselves (spindler & spindler, 1982). given that the mathematics teacher population, reflecting the teacher population generally, consists primarily of white, middle-class, females teaching a student population that is increasingly racially, ethnically, linguistically, and socioeconomically diverse (howard, 1999), what teachers perceive to be ―most like themselves‖ necessarily falls along lines of race, culture, and class. often teachers, particularly white teachers, have more negative attitudes and beliefs about minority children than about white children (irvine, 1985). these attitudes and beliefs, coupled with teachers‘ stereotypical views of urban schools and communities and the fact that cultural distances between teachers and students are likely greater in urban areas than in smaller communities, suggest that attending to teacher–student relationships can support student success in urban areas. moreover, for african american, latina/o, immigrant, and other students traditionally marginalized in mathematics (who often make up the majority in urban schools) to achieve well, an understanding of effective student–teacher relationships is imperative given that research documents that the presence of positive teacher–student relationships—driven by teachers‘ robust knowledge of their students—is an important factor for their mathematics success (berry, 2008; borman & overman, 2004; gutstein, 2006; ladson-billings, 1994). here, i draw on theories of care (beauboeuf-lafontant, 2002, noddings, 1984, 1992; rolón-dow, 2005; thompson, 1998) to begin to lay out a theoretical map of sorts on what an effective, caring teacher–student relationship that supports student learning might ―look like.‖ next, i turn to theories of culturally relevant pedagogy (gay, 2000; ladson-billings, 1994), as these not only illustrate effective practices caring teachers employ but also because this work provides models of classroom practices that consider explicitly issues of race, culture, and power and have had demonstrative effects on the academic achievement of traditionally marginalized students. (for a detailed and explicit example of culturally relevant and critical pedagogy enacted in the mathematics classroom see gutstein, 2006.) given that the development and influence of caring teacher–student relationships has been largely understudied in mathematics education (vithal, 2003) and that there is still relatively little published empirical research examining culturally relevant mathematics pedagogy, the majority of the research discussed here draws on work outside of mathematics education. 1 whenever possible, how 1 two fairly recent published books in mathematics education have drawn explicit connections to mathematics education and culturally relevant (responsive or specific) pedagogy: culturally responsive mathematics education, edited by brian greer, swapna mukhoopadhyay, arthur powell, and sharon nelson-barber (2009), and culturally specific pedagogy in the mathematics classroom: strategies for teachers and students, written by jacqueline leonard (2008). (for reviews of these two books, see jume vol. 3, no. 2, and vol. 2, no. 2, respectively.) bartell caring with awareness journal of urban mathematics education vol. 4, no. 1 53 ever, i connect the research on caring relationships to the context of mathematics education. the synthesis provided here serves to illuminate the complex, nuanced, and, at times, overwhelming descriptions of what it means to be a caring teacher in the service of student learning and brings to the surface a general description of a teacher that cares with awareness. given that caring with awareness might be overwhelming for teachers—due to the knowledge and dedication required to promote effective learning, including effective mathematics learning—professional development models are needed that can support teachers in these endeavors. thus, in conclusion, i consider models of professional development that hold potential for supporting mathematics teachers in developing teacher–student relationships reflective of caring with awareness; models that can support mathematics teachers in particular in building caring relationships with students to insure their success in mathematics. to explore the intersections between theories of care and culturally relevant pedagogy in conceptualizing the nature of effective teacher–student relationships, i begin by providing an overview of care theory, connecting it explicitly to the theory of culturally relevant pedagogy. in connecting the two theories, i then describe four key components of the nature of effective teacher–student relationships: racial, cultural, political, and academic. care theory: an overview building on the work of carol gilligan (1982), a pioneer of care theory, noddings (1984) modified and expanded gilligan‘s work, considering its application to education (1992). care theory posits, in part, that the development of caring teacher–student relationships is central to supporting students‘ academic achievement (noddings, 1984, 1992). caring is necessarily relational, requiring both the teacher and the student to contribute to the formation of a caring relationship (noddings, 1992). such relationships involve the caregiver (e.g., the teacher) understanding the cared for (e.g., the student) from the perspective of the cared for (mayerhoff, 1971), which noddings (1984, 1992) calls ―engrossment‖ and ―motivational displacement.‖ engrossment, or ―feeling with‖ another person, is different from imagining what one would feel in someone else‘s situation (noddings, n.d.). engrossment is not about one‘s own feelings. rather, it occurs when the teacher is completely taken up with what the student is feeling—when a teacher accepts students‘ feelings and acknowledges the relevance of students‘ experiences. for noddings (1984, 1992), this acceptance suggests the other critical component of a caring relationship, motivational displacement, where ―when i care…my motive energy flows toward the other and perhaps, although not necessarily, toward his ends…i allow my motive energy to be shared; i put it at the bartell caring with awareness journal of urban mathematics education vol. 4, no. 1 54 service of the other‖ (1984, p. 33). during motivational displacement, the caregiver puts aside her or his needs in order to care for the other individual. noddings (1992) also argues that caring relationships are incomplete unless the student actively receives the care: ―no matter how hard teachers try to care, if the caring is not received by students, the claim ‗they don‘t care‘ has some validity‖ (p. 115). dialogue, then, is an important factor in contributing to the development and maintenance of caring relations, because it allows students to connect to the teacher through language and shared experience. through dialogue, teachers seek to understand students‘ relationships to the subject matter, including what the students‘ goals are and ways that the subject matter may connect with students‘ lives (noddings). another component of a caring relationship is that such relationships develop inter-subjectivity, where teachers and students develop shared interests and common understandings of each other (tharpe, estranda, dalton, & yamauchi, 2000). goldstein (1999) refers to the attainment of inter-subjectivity as a teacher‘s creation of a ―shared intellectual space‖ with their students. in the process of achieving inter-subjectivity, teachers work to share with students their understanding of a concept while simultaneously working to understand students‘ understanding(s) of a concept. goldstein and tharpe and colleagues contend that through the process of jointly negotiating the meaning of concepts and activity, teachers demonstrate care for individual students and for the subject matter itself. finally, caring relationships require confirmation. noddings (1992) claims that for caring teachers, confirmation is an act of affirming or encouraging the best in others, and confirmation of students comes through establishing trust. caring teachers accomplish confirmation by developing relationships with students and knowing their students well. to know students well in this context means to realize what they are trying to become or to see what they are really striving for. it is important to note—if not obvious by the complexity implied in the aforementioned descriptions—that caring is a process; it is something teachers do rather than something teachers feel. to care is to take an ethical stance. goldstein (1998) expresses care as a moral stance that leads to ethical action: an action rather than an attribute, a deliberate moral and intellectual stance rather than simply a feeling—offers a powerful alternative to the conceptions of caring currently shaping our thinking about the term. (p. 18) moreover, caring requires personal contact and varies according to individuals and situations: ―two students in the same class are roughly in the same situation, but they may need very different forms of care from their teacher‖ (noddings, 2002, p. 20). noddings (1992) also notes the ―difficulties of knowing another‘s nature, needs, and desires when one party holds power over the other or is a member of a group that has historically dominated another‖ (p. 3). in this bartell caring with awareness journal of urban mathematics education vol. 4, no. 1 55 way, culturally relevant pedagogy provides a way to think about teacher care with respect to supporting teachers in successfully addressing the different educational and cultural needs of students from various ethnic and racial backgrounds. care theory and mathematics classrooms some research on caring teacher–student relationships has also been situated in the context of mathematics education, documenting that caring teacher–student relationships can support students‘ learning of mathematics and engagement with mathematics. for example, when secondary school students perceived that their mathematics teachers cared for them, they reported increased effort (muller, 2001; stipek, 2006). additionally, for students deemed by their teachers to be atrisk of dropping out of high school, when they perceived their teachers cared for them they performed better in mathematics compared to ―at-risk students‖ who did not perceive that their teacher cared for them (muller, 2001). in other words, the perception that their teacher cared, listened to them, and expected them to succeed mitigated the negative effects of having been deemed at-risk in the first place. yet, little is known about how the nurturance of caring student–teacher relationships might be involved in the process of mathematics learning. recent work by hackenberg (2010a, 2010b) stands alone in its examination of caring in relation to mathematics teaching and learning. in her work, hackenberg moves beyond descriptions of mathematics teachers as caring in the general sense to develop a model of mathematical caring relations. she conceives of mathematical caring relationships as a teacher engaged in a dynamic process with students ―harmonizing with students‘ schemes and energetic responses to mathematical activity…making interpretations of students‘ current schemes and operations and basing interaction on those interpretations so that tasks posed to students are sensible to them (2010a, p. 59). if a teacher struggles with harmonizing with a student‘s schemes, she likely experiences difficulties in posing appropriate mathematical challenges that will further the student‘s learning (hackenberg, 2010b). what makes these caring relationships mathematical, then, is the cognitive decentering required of a teacher; ―the construction of new mathematical ways of operating that fit with the teacher‘s experience of the students‖ (p. 266). nonetheless, hackenberg‘s work does not explicitly consider issues of race, culture, and power. care theory and race descriptions of care theory, outlined by gilligan (1982), noddings (1984, 1992) and others, has been criticized as being ―colorblind‖ (beauboeuf-lafontant, 2002; thompson, 1998, 2003) in that care theorists ―fail to acknowledge and address the whiteness of their political and cultural assumptions‖ (thompson, 1998, p. 524). colorblindness is the inability (or unwillingness) to acknowledge that bartell caring with awareness journal of urban mathematics education vol. 4, no. 1 56 race matters, that racism exists, and that race and racism have a significant influence on whether and how students are successful in school (bonilla-silva & forman, 2000). a colorblind approach is problematic because it ignores the fact that inequity and discrimination are current issues and are not easily remedied by simply ignoring race (decuir & dixson, 2004; ladson-billings & tate, 1995). students‘ learning opportunities may be hindered when teachers fail to consider their own and their students‘ racial background and instead adopt colorand culture-blind beliefs and practices (milner, 2007). thompson (1998) draws on ethical positions grounded in black women‘s lives and black feminist ethics to reexamine and reinterpret early work on the ethics of care. her work, in part, suggests that caring is not limited to the private sphere as is often implied with early work on care theory. rather, caring in the black community is as much public as it is private. caring for a student is viewed as both a collective and individual responsibility. the emphasis of care is on cultural, communal, and political solidarity; an emphasis shared by not only the teacher but also by the community, extended family, and/or the local black church. furthermore, ―caring means bringing about justice for the next generation and justice means creating the kinds of conditions under which all people can flourish‖ (p. 529). thompson (2004) argues that caring teachers implement antiracist curriculum, reject a colorblind approach, and instead embrace ―colortalk‖: ―acknowledging racial identity, culture, racism, and racial privilege as factors that shape and color experience, colortalk recognizes that a person‘s color is a significant dimension of her or his experience‖ (p. 26). similarly, beauboeuf-lafontant (2002, 2005) provides a racial critique of care theory by describing characteristics exhibited in the pedagogy of exemplary african american, female teachers as a means to expand what it means to ―care‖ for students and in turn illuminate the colorblind descriptions of previous work. she describes ―womanist caring‖ as consisting of three characteristics: an embrace of the maternal, political clarity, and an ethic of risk. embrace of the maternal requires teachers to treat all children as if they are her or his own and to meet children‘s particular needs by whatever means necessary. this embrace involves sharing responsibility with families and communities to ensure that all children succeed. womanist caregivers also demonstrate political clarity, or the awareness that society and schools are structured to ensure differential success for different groups of students. in other words, oppression and inequity are systematic rather than individually motivated. thus, caring teachers are not simply promoting an agenda that seeks to reward everyone. rather, because teachers are affiliated with schools, they acknowledge they are products of (and culprits within) an inequitable system and that caring must involve action and commitment to fight injustice. this action and commitment is the ethic of risk—caring teachers‘ commitment to understand, confront, and transgress oppressive structures. bartell caring with awareness journal of urban mathematics education vol. 4, no. 1 57 care theory and culturally relevant pedagogy education theories aimed to support work toward equity in mathematics education need to attend to how issues of race, culture, and power intersect with and inform our understanding of effective mathematics education (dime, 2007). theories of culturally relevant pedagogy serve this purpose, providing models of classroom practices that consider explicitly issues of race, culture, and power. furthermore, these theories have documented the nature of teacher–student relationships that support students‘ mathematics learning in ways that intersect with and expand descriptions within theories of educational caring. culturally relevant pedagogy is fundamentally about the academic success of students of color. it is a ―pedagogy that empowers students intellectually, socially, emotionally, and politically by using cultural referents to impart knowledge, skills, and attitudes‖ (ladson-billings, 1994, pp. 17–18). as a bridge between students‘ home and school cultures, culturally relevant pedagogy facilitates teachers incorporation of students‘ cultural values, experiences and perspectives into the curriculum (gay, 2002). a necessary requirement of effective culturally relevant pedagogy is students‘ academic success. culturally relevant pedagogy builds on students‘ home cultures as a means to foster success in school. moreover, it ―uses student culture in order to maintain it and to transcend the negative effects of the dominant culture‖ (ladson-billings, 1994, p. 17). culturally relevant teachers exhibit an ethic of care, enable their students to think critically about their world and its injustices, and equip students with the skills to change it (ladson-billings). care, race, culture, politics, and learning mathematics in this section, i look across the literature on care theory and culturally relevant pedagogy that portrays teachers exhibiting an ethic of care to illuminate complex and nuanced descriptions of what it might mean to be a caring teacher for all students in as specific a way as possible. the sub-sections that follow serve to illuminate the specific practices caring teachers engage in and to highlight four key components (or categories) of the nature of effective teacher–student relationships: racial, cultural, political, and, inherently, academic. these categories are not mutually exclusive nor are they separated in an effort to be prescriptive. rather, these categories serve to organize the literature under consideration. it is also important to note that throughout the discussion, i draw primarily on literature that describes pedagogical practices and caring relationships as reflected (or not) in the schooling experiences of black and brown children. in doing so, i do not intend to essentialize the lived experiences of black or brown children or to suggest that the discussion pertains only to black or brown children. but rather to bartell caring with awareness journal of urban mathematics education vol. 4, no. 1 58 draw on this literature to make a strong case for caring with awareness for mathematics teachers (and teacher educators)—a caring with awareness that i believe benefits all children in learning and doing mathematics. caring teacher–student relationships and race as noted previously, caring relationships reject a colorblind approach and instead embrace colortalk, ―acknowledging racial identity, culture, racism, and racial privilege as factors that shape and color experience‖ (thompson, 2004, p. 26). in mathematics education, martin‘s (2006) work illustrates one way to conceptualize caring teachers. caring teachers are identified according to the degree to which they value, devalue, or challenge african american students‘ status, identity, and prior knowledge. how white teachers in particular interact with african american learners, in the name of ―engrossment‖ or ―motivational displacement‖ (noddings, 1984, 1992), often from colorblind frames (thompson, 1998), places limits on teachers‘ expectations of who their students are and who they can become (martin, 2000), a key component of a caring teacher-student relationship. outside of mathematics education, siddle walker‘s (1993) work on the construct of interpersonal caring garnered from her examination of how caring functions successfully for african american students suggests how caring relationships attend explicitly to issues of race. in this work, caring teachers were explicit with their students about what was expected of them as black children, exposing racist structures in society by revealing and examining white privilege in a myriad of ways. explicit discussions of how race functioned in school and in society facilitated students‘ academic success. beck and newman‘s (1996) work examining caring at a high school also shows that teacher caring was made evident to students when teachers acknowledged racial differences, confronted actual and potential tensions, and involved students in developing solutions and strategies. drawing on her work with puerto rican girls in middle school, rolón-dow (2005) proposes ―critical care praxis‖ as a conceptual framework to examine teachers‘ relationships with students. critical care asks teachers to center issues of race and ethnicity in their relationships with students. rolón-dow found that ―deficit-based, racialized caring narratives were often articulated when teachers used their own experiences as well as the historical experiences of white immigrant groups as ideological foundations‖ (p. 104). thus, her work asks teachers to get to know students well—to gain a historical understanding of students‘ lives—in order to improve teaching and learning. the teachers described in rolón-dow‘s work suggest that caring teachers examine and confront their raceand ethnicitybased assumptions about the students‘ family lives, as one participant notes: ―[to] understand from where they come…you need to watch very closely. you need to listen very closely before you attack. and it‘s so easy for all of us to attack. but bartell caring with awareness journal of urban mathematics education vol. 4, no. 1 59 you have to understand from where they come‖ (p. 103). this understanding and listening facilitates teachers in seeing that separating the homes of students from the school can serve as a barrier to the development of caring relationships. further, caring teacher–student relationships require teachers to elicit and respond to counter-stories about race and ethnicity present in students‘ communities. for example, for the puerto rican middle school girls in rolón-dow‘s research, counter-stories existed that pointed to how structural and institutional factors influenced the care given to particular students and to how important racial and ethnic factors are in caring for students not only personally but also as members of multiple communities. caring teachers seek out, listen for, and attend to such counternarratives. the research of rolón-dow (2005) also serves to caution teachers and teacher educators that ―dominant stock stories‖ of care can serve to normalize racism. for example, building on a belief that students‘ homes are uncaring places, teachers ―care‖ by saving students from those homes rather than joining parents in collaborative efforts to care. this misguided care is similar to the ―white missionary paternalism‖ that martin (2007) describes where teachers conceptualize their work as saving african american children from themselves and their culture (p. 13). instead, caring teachers work to understand african american students‘ experiences with mathematics as african americans (martin, 2000, 2006, 2007), serving to support the development of positive mathematical and racial identities. caring teacher–student relationships and culture a caring relationship requires that the teacher understand the student from the perspective of the student; that she or he becomes ―engrossed‖ in the student and experiences ―motivational displacement,‖ putting her or himself at the service of the student (noddings, 1992; mayerhoff, 1971). further, caring relationships require confirmation (noddings, 1992), where the teacher identifies a student‘s potential and encourages its development. without knowing students well, one cannot see what the student is really striving for, or what their true potential may be. thompson (1998), in critiquing the colorblindness of care theory, suggests that caring relationships are about knowing the whole child, which includes knowing her or his situation. knowing a student‘s situation requires, but is not limited to, knowing something about the student‘s home life, cultural history, and the political situations that she or he confronts outside of the classroom. thompson argues, ―teachers cannot tap into and develop the possibilities latent in students‘ cultural knowledge if they do not understand students‘ cultural situations‖ (p. 536). in a study examining the role of peer influences in urban high school students‘ academic success in mathematics (primarily, african american and latina/o students), walker (2006) linked students‘ academic behaviors and success to bartell caring with awareness journal of urban mathematics education vol. 4, no. 1 60 a historical tradition of intellectual networks in their communities. these findings point to another way to think about the notion of understanding students‘ situations. walker argues that to fully value the cultural contributions that students bring with them to school, teachers must understand ―the depth of students‘ academic communities and the ways in which students and their peers foster intellectual communities among themselves‖ (p. 68). thus, knowing students well and explicitly attending to issues of culture means recognizing, valuing, and drawing on students‘ historical and current traditions of community support and engagement. of course knowing students well and understanding their situations is not a simple endeavor, as ―two students in the same class are roughly in the same situation, but they may need very different forms of care from their teacher‖ (noddings, 2002)—determining the ―different type‖ of care students might need can be facilitated by recognizing students‘ definitions of care. suarez-orozco, rhodes, and milburn (2009) examined the academic achievement of immigrant youth and found that while chinese and mexican students reported the lowest levels of relational engagement at school, these levels were not similarly predictive of academic performance. suarez-orozco and colleagues surmised that students from different cultural backgrounds might have different cultural expectations of schoolbased supportive relationships, or different expectations and conceptions of caring relationships. similarly, owen and ennis (2005) argue that understanding and articulating the cultural meaning of caring for different members of a school is one essential component in supporting disenfranchised students. knowing students well by explicitly attending to issues of culture, then, includes understanding students‘ cultural conceptions of caring and their expectations for caring relationships. yet another way teachers might work to know students well that requires explicit attention to issues of culture is through building commonalities with students. christensen (2000) suggests forging a ―curriculum of empathy‖ with students where the curriculum itself provides opportunities for students (and, i would argue, teachers) to learn about one another and to develop empathy. sharing stories and engaging in a curriculum of empathy requires students and teachers to look beyond their own world and share the lives of others. as a result, both teachers and students learn how to go beyond stereotypes to look for and reflect on common feelings, ideas, and facts. ladson-billings (1994) describes a similar characteristic of culturally relevant teachers where small acts of civility and kindness (e.g., giving students a birthday card) reflect teachers ―consciously working to develop commonalities with all the students,‖ working not just to make ―idiosyncratic and individualistic connections with certain students, [but working] to assure each student of his or her individual importance‖ (p. 66). bartell caring with awareness journal of urban mathematics education vol. 4, no. 1 61 developing commonalities with students is a complex endeavor, as illustrated by the work of cooper (2002), who examined effective white teachers of black children juxtaposed with literature on effective black teachers. cooper notes that the white and black teachers were similar in their emphasis on respect for students‘ culture and community. a key difference, however, was that black teachers evaluated their behaviors by what the community wanted whereas white teachers evaluated their behavior by personal views and experiences. cooper warns that white teachers‘ seeming inability to ―view the schools as places that reflect the greater communities‘ ideals‖ (p. 159) could encourage black students in such classrooms to receive messages that inadvertently contrasted with their community norms, placing the students‘ success in jeopardy. cooper‘s work highlights the importance for teachers to build connections with students across cultural lines; constant dialogue about how students receive that care is critical. working to know students well and make connections with students requires teachers to extend themselves beyond the classroom walls to build caring relationships with students (howard, 2003; valenzuela, 1999). in rolón-dow‘s (2005) research exploring the schooling experiences of puerto rican girls, one frustration of the girls was that many of their teachers did not know them beyond interactions within the classroom, often leading to teachers‘ misconceptions about students and making it difficult for students to feel cared for in significant ways. rolón-dow argues, ―building relationships of authentic care must move beyond making assumptions about who students are… [to] create sustained interactions that allow students to share their perspectives of how ethnicity, social class, and gender dynamics affect their daily lives‖ (p. 106). an important aspect required of teachers reaching beyond the classroom walls to build caring relationships with students is the explicit rejection of deficit perspectives of students and their communities. deficit perspectives place the blame of school failure upon individual students and families based on beliefs about certain cultural, racial, or economic characteristics of a group (valencia, 1997). a caring teacher, however, does not attach failure to the student, but to themselves, searching within to find a more effective way to reach students (collier, 2005). caring teachers explicitly reject deficit-based thinking and embrace the belief that students from culturally diverse backgrounds are capable learners (ladson-billings, 1994). confronting deficit perspectives is not an easy task. in my examination of teachers learning to teach mathematics for social justice, i found that though the teachers worked hard to create mathematics lessons that confronted deficit ideologies about academic achievement, their conversations about their own students could often be interpreted as manifestations of a deficit perspective (bartell, 2011; gau, 2005). howard (2003) contends that to become culturally relevant (and thus to reject deficit-based thinking about students) teachers need to engage in honest, critical reflection that includes ―an examination of bartell caring with awareness journal of urban mathematics education vol. 4, no. 1 62 how race, culture, and social class shape student‘s thinking, learning, and various understandings of the world‖ (p. 197). a teacher‘s care manifests in a willingness to reflect on her or his attitudes toward diverse students; this reflection communicates a sincere commitment that the teacher has toward students‘ success. caring teacher–student relationships and politics (and power) gutstein (2006), in describing his efforts to read and write the world with mathematics with his students, notes that the quality relationships he built with his students were explicitly political; relationships where teachers take up ―active political stands in solidarity with students and their communities about issues that matter‖ (pp. 132–133). caring relationships keep issues of injustice, and efforts to fight for justice, at the forefront. in caring for students, teachers provide students opportunities to conduct political analyses of the world, share their own political understandings with students, and support students in their struggles to change unjust conditions in their lives. in turn, as gutstein demonstrates, students learn mathematics with understanding, develop mathematical power, and grow in their ability to understand complex aspects of society, where mathematics becomes a necessary and powerful analytical tool that students use to study their sociopolitical existence. similarly, ladson-billings (1994) describes culturally relevant teachers as those who are not only concerned with ensuring that their curriculum reflects the lived experiences of their students but also recognize the importance of taking a critical look at curriculum content. this critique includes teachers seeing themselves as vital to the process of helping students identify oppressive elements in society and in schools and arming students with the knowledge, skills, and critical attitudes necessary to struggle against those oppressive elements (ladsonbillings). in the context of mathematics education, ladson-billings provides the example of ms. rossi, a sixth-grade teacher who found an old set of algebra books so that she could engage her students in algebra, even though it was beyond what the district‘s curriculum required. ms. rossi recognized the gate-keeping role mathematics, and in particular algebra, plays with respect to students‘ opportunities to learn and she was explicit with her students about why she was making the choice to teach them algebra. beauboeuf-lafontant (2002, 2005), in similar ways to gutstein (2006) and ladson-billings (1994), argues that caring teachers have ―political clarity,‖ or recognize the existence of oppression in their students‘ lives and seek to use their own status to encourage students to understand and undermine those oppressive conditions. beauboeuf-lafontant, as previously noted, describes womanist caring as consisting of three characteristics: an embrace of the maternal, political clarity, and an ethic of risk. of relevance here are the latter two, political clarity and an ethic of risk. political clarity is ―the recognition by teachers that there are relation bartell caring with awareness journal of urban mathematics education vol. 4, no. 1 63 ships between school and society that differentially structure the successes and failures of groups of children‖ (beauboeuf-lafontant, 2002, p. 77). teachers that care do so with the recognition and explicit discussion with students of the fact that society marginalizes some children and not others. furthermore, caring teachers demonstrating political clarity not only recognize and discuss these structures but also actively work against them. the latter of the two is the ethic of risk—teachers‘ commitments to understand, confront, and transgress oppressive structures. caring relationships are explicitly political in that they are, in the work of the aforementioned scholars, acts of solidarity with students emerging from shared commitments to equity and justice. caring teachers must also work to confront unequal power relations within their classrooms, working to neutralize status differences so that all students can achieve. as they build caring relationships with students, caring teachers work to understand whether and how each student has status with respect to engagement with the mathematics so that the teacher might support students with lower status in gaining access. boaler‘s (2006, n.d.) work with mathematics teachers at railside using complex instruction (cohen, lotan, scarloss, & arellano, 1999) speaks to this idea. a key approach of mathematics teachers using complex instruction is that of assigning competence. here teachers work explicitly to raise the status of students who may be of lower status by bringing to the class‘s attention—through statements, questioning, or asking students to share—something of intellectual value students have said or done. if a student is not expected to be competent, she or he may not be asked for an opinion or asked about her or his thoughts related to the mathematics. railside students in classrooms where teachers enacted complex instruction started at significantly lower levels of achievement when compared to students at two schools where more traditional mathematics teaching, without the use of complex instruction, took place, but within 2 years they were achieving at significantly higher levels and demonstrated more positive views about mathematics. caring teacher–student relationships and academic achievement a primary objective of caring teacher–student relationships is students‘ academic achievement. noddings (1984) notes that caring teachers have two responsibilities: ―to stretch the student‘s world by presenting an effective selection of that world with which she is in contact and to work cooperatively with the student in his struggle toward competence in that world‖ (p. 178). this cooperation includes understanding students‘ relationships to the subject matter in order to build and extend their knowledge (noddings, 1992). goldstein (1999) argues that caring relationships are a central part of a student‘s intellectual growth and development. she contends that a teacher‘s use of scaffolding to match the demands of a task to students‘ abilities in an effort to maximize their chances of success is a key bartell caring with awareness journal of urban mathematics education vol. 4, no. 1 64 aspect of a caring relationship. in short, learning and caring are inextricably entwined. in the context of mathematics education, hackenberg‘s work (2010a, 2010b) demonstrates that caring mathematical relationships support a teacher‘s ability to choose appropriate problems to pose for students based on students‘ previously demonstrated mathematical reasoning, in turn supporting students‘ mathematical learning. theories of culturally relevant pedagogy are helpful in showing that supporting students academically includes ―learning how and where to help students connect what they know to what they do not know and how to use prior skills to learn new ones‖ (cochran-smith, 1999). this support entails teachers learning to meet students where they are in order to help students participate meaningfully in the knowledge development process (ladson-billings, 1994). caring teachers strive to discover the knowledge students bring to the classroom, and explore and utilize that knowledge to support student achievement (ladson-billings). caring relationships are also academic in that teachers expect and demand academic excellence from all of their students. vasquez‘s (1988) characterization of a teacher as a ―warm demander‖ is illustrative of this notion. he conceptualized a warm demander as a teacher who will not lower standards for students and who will reach out to students and provide needed assistance to help them reach high standards. similarly, kleinfeld‘s (1972) work with teachers of eskimo and native american students, recently relocated to urban settings, suggests caring teachers demand academic excellence from their students. kleinfeld summarizes: the essence of the instructional style which elicits a high level of intellectual performance from village indian and eskimo students is to create an extremely warm and personal relationship and to actively demand a level of academic work which the student does not suspect he can attain. village students thus interpret the teacher‘s demandingness not as bossiness or hostility, but rather as another expression of his personal concern, and meeting the teacher‘s academic standards becomes their reciprocal obligation in an intensely personal relationship. (p. 34) furthermore, ladson-billings (1994) describes what i would call a caring classroom culture as one not focused on individual achievement, but rather focused on collective growth and liberation. students care about one another‘s achievement, teach each other, and take responsibility for one another‘s learning: ―psychological safety is a hallmark of each of these classrooms… [students] realize that the biggest infraction they can commit is to work against the unity and cohesiveness of the group‖ (p. 73). students in such classrooms, then, need an opportunity to collaboratively practice care (noddings, 1992). students learn what it means to support and teach one another, to care, and they experience the contribution to community that occurs when ―all are in this together‖ (ladson-billings, 1994). this classroom cul bartell caring with awareness journal of urban mathematics education vol. 4, no. 1 65 ture resonates with what boaler (2006, n.d.) terms relational equity, or classroom practices that facilitate students treating each other with respect and responsibility. in the successful story of railside high school, students learned to respect students from different cultural, social, gender, and ability groups. boaler argues that the respectful relationships students developed with one another were made possible by the particular mathematics approach used by teachers—an approach that valued students‘ many different perspectives, strategies, and contributions as they collectively solved mathematics problems. the teachers at railside worked hard to create classrooms that approached learning as a collective, rather than an individual, endeavor. they reminded students to work together as a group, modeled respectful behavior for students, taught students to take responsibility for one another‘s learning, and reinforced this message with their grading practices. as noted previously, this resulted in significant mathematics learning for students. summarizing the four components in the discussion of the four components—racial, cultural, political, and academic—i illuminated that when looking to understand more clearly and specifically what it means to build a caring relationship with one‘s students, in particular in an effort to promote equity in mathematics education, a complex, nuanced, and seemingly overwhelming description emerges. teachers that care with awareness know their students well mathematically, racially, culturally, and politically. they work to understand and make connections with students‘ cultures and communities; help students develop positive racial, cultural, and political identities; reflect critically on their own assumptions and practices about students‘ cultures and communities, including rejecting and confronting deficit and colorblind perspectives; and labor to neutralize status differences within and beyond the classroom walls. teachers that care with awareness engage in discussions of race and racism with their students, listen for counter-narratives that might help shape a more caring bond between them and their students, and stand in solidarity with students. they use the knowledge gained from all of these avenues to support students‘ academic success, accessing students‘ existing mathematical and cultural knowledge and scaffolding tasks based on where students are to engage them meaningfully in building new mathematics knowledge. similar to freire‘s (1998) conception of ―armed love,‖ care should be authentic, based on respect, considerate of issues of race, culture, and power, and focused on providing students with both an academically rigorous and liberatory, self-empowering education. toward a model of professional development i could stop here, having laid out a theoretical map of sorts on what it means to build caring relationships with awareness for all students, within and across bartell caring with awareness journal of urban mathematics education vol. 4, no. 1 66 lines of race, culture, and power, to support equity in mathematics education. but as mathematics teacher educators, we must also consider how teachers might begin to operationalize these conceptualizations of care in the daily practice of teaching mathematics and, more specifically, consider professional development models that can support teachers in translating theory into practice. one would be hard-pressed to find a teacher that would say they do not care for their students. this is likely due to the fact that teachers often consider care to be a personality trait or a warm, fuzzy feeling that one has toward others, rather than an ethical stance (goldstein, 1998, 1999). the work of care theorists synthesized in this article, however, serves to illustrate that caring is more than a warm feeling; it indeed is a complex, nuanced concept. teacher education and professional development therefore need to support teachers in developing this ―deliberate moral and intellectual stance‖ (goldstein, 1998, p. 18). some work in teacher education suggests that narrative methods are an important way to teach teachers about care (see, e.g., rosiek, 1994; young, 1998). additionally, supporting teachers in attending to individual students with awareness is important in the development of an ethic of care (rabin, 2008). in an examination of preservice teacher education courses with a commitment to care ethics, rabin found that teacher educators‘ use of questions to direct novice teachers‘ attention to their learners‘ needs and course activities that focused novice teachers‘ attention on listening to students‘ stories with awareness were instrumental in developing their ethic of care and informing related changes in their practice. furthermore, explicit discussion about how the ―larger structural restraints—which these novice teachers may meet in the often-overburdened urban schools where they teach—may thwart their efforts at constructing a caring practice‖ (p. 11) supported teachers in navigating these constraints and prioritizing a focus on what a student‘s story suggests about what is required for her or his care. moreover, rabin notes that one particular assignment that engaged novice teachers in articulating and reflecting on a dilemma of practice with respect to one student ―stood out as an opportunity for the novice teachers … to construct a relational moral stance‖ (p. 6). with respect to research in mathematics education, little published work documents professional development models effective in supporting teachers in developing caring relationships with their students in the ways described here. one particular model that i believe holds promise focuses teachers on dilemmas of practice involving an individual child (see, e.g., rabin, 2008), serving as an orienting experience not only with respect to that child but also with respect to the teachers‘ mathematics pedagogy. foote‘s (2008, 2009, 2010) research on designing, facilitating, and examining a study group that engaged teachers in exploring the mathematical thinking and inand out-of school experiences of an individual child is one such model bartell caring with awareness journal of urban mathematics education vol. 4, no. 1 67 that holds promise and can inform future research. each of the six teachers in the study group—all white, female teachers, working at the same elementary school—were asked to choose an african american child they believed was struggling in mathematics as their ―target‖ student. participants were specifically asked to choose an african american learner ―to minimize issues of essentializing based on comparisons across cultural groups of students‖ (foote, 2010, p. 44). participation in the study group consisted of teachers taking an in-depth look at their target child, gathering specific information to share with the group for feedback. teachers shadowed their target child in a school setting, conversed informally with the target child to learn more about the activities she or he enjoyed outside of school, and collected samples of the child‘s work in mathematics. at the same time, the children‘s parents took photographs of their child demonstrating competence, participating in household routines (i.e., cooking or grocery shopping), engaging in activities the child found interesting, and engaging in activities that involved mathematics or attention to number. between teachers‘ first and second study group presentations, they met with their target child‘s parent to learn about the child‘s out-of-school experiences and to learn more from the parent about the child‘s strengths, competencies, and interests. results of foote‘s (2008, 2009, 2010) work demonstrate that this model supported teachers in reaching beyond the mathematics classroom into children‘s home and community and in moving them from seeing their children‘s home environments as problematic in some way to instead viewing children‘s home environments as supportive. for one teacher in particular, this experience and the knowledge gained about her target child supported her in changing her classroom practice to build on the child‘s interests and support his learning of mathematics. additionally, these experiences supported the teachers in developing strong, or stronger, relationships with their students. three of the teachers in particular ―talked passionately about the strong relationships they had forged with their target students‖ and the deep commitment that they felt for them in ways that ―positioned them to be more effective teachers of those children‖ (foote, 2010, p. 55). furthermore, the teachers not only connected with their target child but also developed open channels of communication with the parents of all of their children in an effort to learn more from children‘s parents about how best to support their learning. the results of such professional development opportunities suggest that teachers with these experiences begin to develop caring teacher–student relationships, particularly with respect to caring relationships that center on issues of culture. this work extended beyond classroom walls, supporting teachers in confronting deficit views they had about children‘s home environments and facilitating the adaptation of one teacher‘s mathematics classroom practices to better service a child‘s learning needs. bartell caring with awareness journal of urban mathematics education vol. 4, no. 1 68 in spite of the connections teachers forged with these target children, explicit discussion of issues of race, which one might expect to surface given that these were white teachers working with african american children, was less salient. while the researcher attended to issues of race in that the white teachers were asked to select african american students as ―target‖ students, explicit discussion around this idea did not take place. perhaps the fact that issues of race did not surface is reflective of our national tendency to be colorblind or colormute (pollack, 2004). in any case, professional development models that aim to support teachers in caring with awareness for all students to insure their learning of mathematics must directly address and attend to issues of race. i now turn to another model of professional development that supported teachers in reflecting on a dilemma of practice with an individual child as an orienting experience, but also brought to the forefront discussions of race and its intersection with mathematics teaching and learning. in their work supporting mathematics teachers in using children‘s thinking to guide instruction, battey and chan (2010) draw on the work of franke and chan (2009) to describe a multi-year professional development program that moved from a focus solely on students‘ mathematics thinking to one that also explicitly grappled with issues of race in mathematics classrooms. this progression took place over the course of 3 years where teachers engaged in not only more ―traditional‖ professional development meetings but also where battey, chan, and other project researchers were in teacher‘s classrooms and schools on a weekly basis engaging in informal conversations around mathematics teaching and learning. these project researchers labored to not only develop individual teacher‘s practice but also to use what they learned to be individual teachers‘ strengths to foster collaboration among teachers. this history is important as the teachers and battey and chan built a community, forming trusting relationships with one another that allowed for teachers to present dilemmas of practice in a safe environment. battey and chan (2010) posit that one outcome of their ongoing work with teachers is that teachers began to pay ―attention to individual students, created environments that met individual students‘ specific needs, and supported different kinds of participation in relation to mathematics‖ (p. 142). this attention to individual students, coupled with the creation of a teacher community over time, laid a foundation for battey and chan, as professional developers and education researchers, to begin to challenge deficit discourses arising for individual teachers as they paid attention to the achievement of students of color in different ways. in these individual conversations, teachers often began by describing the ways in which african american students were lacking, rather than focusing on evidence of their mathematical understanding. at these moments, project researchers interjected, asking questions about what these african american students did know. bartell caring with awareness journal of urban mathematics education vol. 4, no. 1 69 when framed in this way, teachers recognized the skills their african american students had, considered ways to build on these skills, and in turn confronted their deficit assumptions. specifically, in describing one teacher, they note: ―as the year went on, the teacher created more opportunities for the three african american students to share, showing her and the other students that these children were mathematically capable‖ (p. 145). in this context, a focus on an individual child in the classroom served to alter teachers‘ stance in a way more reflective of an ethic of care, which in turn informed their mathematics teaching practice. what is also important to note about this professional development program is that these kinds of interactions with individual mathematics teachers supported battey and chan (2010) in addressing deficit notions within the entire community of teachers; engaging mathematics teachers in discussions of ―labels‖ placed on students and exposing damaging metanarratives that often frame african american children in specific ways with respect to being doers (or not) of mathematics. situated within this broader discussion, they engaged teachers in similar discussions as those that occurred in individual interactions, reframing the issue to one of focusing on what students can and are doing. this shifting attention to what students can do ―allows for some of our relationships to begin to address issues of student participation, success, and classroom practices that support the achievement of students of color‖ (p. 149). concluding thoughts although additional research is needed about professional development that can support teachers in operationalizing caring with awareness, particularly with students across racial and cultural lines, this article serves as a first step in wading through the complex, nuanced, and seemingly overwhelming theoretical knowledge base that can be useful in informing mathematics teaching. a model of professional development that begins to emerge from these examples is one that focuses on individual student‘s mathematical thinking and, in particular, on what mathematics students do know and what their specific competencies are. these models leverage teachers‘ dilemmas of practice with individual children to foster the development of caring teacher–student relationships that explicitly attend to issues of race, culture, and power. this focus on an individual student also serves as an orienting experience for teachers, not only with respect to that one child but also with respect to their practice, supporting teachers in rejecting deficit assumptions about students and instead working as caring teachers to learn about and address students‘ specific learning needs. bartell caring with awareness journal of urban mathematics education vol. 4, no. 1 70 references aaronsohn, e., carter, c. j., & howell, m. 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(2006). urban high school students‘ academic communities and their effects on mathematics success. american educational research journal, 43, 41–71. http://legacy.lclark.edu/faculty/gradfac/objects/ctp.pdf bartell caring with awareness journal of urban mathematics education vol. 4, no. 1 74 woolfolk hoy, a., davis, h., & pape, s. (2006). teachers‘ knowledge, beliefs, and thinking. in p. a. alexander & p. h. winne (eds.), handbook of educational psychology (2nd ed., pp. 715–737). mahwah, nj: erlbaum. young, l. (1998). care, community, and context in a teacher education classroom. theory into practice, 37, 105–113. in this paper we examine how the identities of two alternatively certified (ac) mathematics teachers are impacted by conflicts they encounter over the course of their first year of teaching in an urban environment journal of urban mathematics education december 2011, vol. 4, no. 2, pp. 67–95 ©jume. http://education.gsu.edu/jume mary q. foote is an associate professor in the department of elementary and early childhood education at queens college, cuny, 6530 kissena boulevard, flushing, ny 11367; email: mary.foote@qc.cuny.edu. her research attends to equity issues in mathematics education and broadly stated examines issues in mathematics teacher education. beverly s. smith is an associate professor in the department of secondary education at the city college of new york, cuny, 160 convent avenue, new york, ny 10031; email: besmith@ccny.cuny.edu. her research focuses on the development and support of mathematics teachers in urban schools. laura m. gellert is an assistant professor in the department of teaching, learning, and culture at the city college of new york, cuny, 160 convent avenue, new york, ny 10031; email: lgellert@ccny.cuny.edu. her research interests include the support and education of elementary school teachers in mathematics, the teaching of mathematics in collaborative team teaching environments, and professional learning communities. evolution of (urban) mathematics teachers’ identity mary q. foote queens college city university of new york beverly s. smith laura m. gillert the city college of new york city university of new york in this article, the authors examine teacher identity development to understand more completely the intellectual and pedagogical demands of the profession. drawing on interviews, observations, and reflections of two alternatively certified mathematics teachers, and contextualizing these data with surveys from 157 teachers in the same certification program cohort, the authors examine how identities evolved over the course of the teachers’ first year of teaching. these identities are explored in relation to the teachers’ knowledge of mathematics, their mathematics teaching, and their relationships to urban youth. keywords: alternative certification; mathematics teacher identity; mathematics teacher education he research reported in this article is part of a large-scale study on alternatively certified mathematics teachers in new york city conducted by metromath – the center for mathematics in america’s cities. 1 here we examine how the identities of two alternatively certified mathematics teachers are influenced by conflicts they encounter over the course of their first year of teaching in an urban environment. the teachers in this study—mathematics teaching fellows (mtf)— entered the teaching profession through the new york city teaching fellows program (nyctfp), the largest alternative certification program in the united states. 1 for complete details of metromath see http://www.metromath.org/. t http://www.metromath.org/ foote et al. mathematics teachers’ identity journal of urban mathematics education vol. 4, no. 2 68 established in 2000, the nyctfp was designed to provide certified teachers for “hard-to-staff” new york city schools. hard-to-staff schools are defined as schools that have high teacher turnover due to job dissatisfaction and school staffing actions (ingersoll, 2000). the nyctfp has provided teachers for new york city public schools in the high-need areas of mathematics, science, bilingual education, and special education. at the time of this study, over 25% of all new york city mathematics teachers entered the profession through this program (new york city teaching fellows, 2010). undoubtedly, mtf are having a significant influence on the education of new york city students. understanding more clearly how these teachers might begin to develop their pedagogical and mathematical knowledge provides useful information for both preparation and mentoring programs and professional development designed to support these teachers. theoretical perspective examining identity development among teachers is one way in which aspects of the knowledge base for teaching mathematics might be understood in a more nuanced manner. because identity is not only shaped by the knowledge and skills one acquires but also shapes the very knowledge and skills one seeks to develop (battey & franke, 2008), we frame our study by first examining teacher identity development more broadly. we then turn to considerations of mathematics teacher identity development and the development of teacher identity as it pertains specifically to teaching urban youth. teacher identity development gee (2000) provides an analysis of identity development that allows us to see the possibility of various aspects of identity operating concurrently. important to our study, these include: (a) institution-identity (a position) authorized by authorities within institutions, (b) discourse-identity (an individual trait) recognized in the discourse/dialogue of/with “rational” individuals, and (c) affinity-identity (experiences) shared in the practice of “affinity groups” (p. 100). gee views discourse-identity as being the most salient, while affinity-identity as increasingly being an important perspective in western society. gee contends, however, that all views of identity can operate concurrently as a person (such as a teacher) lives within a particular context. gee’s (2000) views of identity assisted us in exploring what might happen as the identities of mtf develop within the classroom. the mtf are supplied with institutional-identities as teachers. nonetheless, if students do not recognize as “valid” the discourse-identities that the teacher develops within the classroom foote et al. mathematics teachers’ identity journal of urban mathematics education vol. 4, no. 2 69 the institutional-identities bestowed by the school system most likely are undermined. a lack of affinity-identity may in turn also undermine the institutionalidentities bestowed by the school system. kohl (1991) discusses this situation in his book i won’t learn from you: the role of assent in learning. we believe that undermined identities of sorts might be a result of limited shared experiences between students and mtf. this limited shared experiences often makes it difficult for the fellow to establish an affinity group that is inclusive of herself (or himself) and her students, thus inhibiting the development of an affinity-identity within the context of the classroom. this lack of affinity-identity, which in part is a result of varying discourse-identities between teacher and at least some students, might then undermine the institutional-identity of the mtf. in this situation, issues of power most likely come to the fore. can the power of the institutionally sanctioned teacher identity prevail against a discourse-identity that is not sanctioned, or possibly even rejected, by the students in the middle school classroom? complementing the work of gee (2000) is the work of holland, lachiotte, skinner, and cain (1998). they argue that identity develops within the figured worlds in which one participates. identity therefore is situated (or contextual) and socially dependent (or relational) and cannot be appreciated apart from its social context. the notion of figured worlds and the movement of the mtf among the various worlds they inhabit (e.g., classroom, school, university, home, community, etc.) is a useful one for considering the development of their identities as urban mathematics teachers. the figured world that most centrally interests us is the mathematics classrooms of the mtf and their identities as teachers within that particular context. the mtf participate in other figured worlds, which relate to and inform the identity/identities they develop as classroom teachers of mathematics. they have identities within the nyctfp itself as well as concurrent identities as graduate students in the figured world of universities, where they aim to connect the theory and practice of teaching as they work as teachers of record in new york city schools. building on the work of gee (2000) and holland and colleagues (1998), we look at identity as a relational construct. in our study, more individualistic notions of self have been replaced by an understanding that identity is situated and relational. a person has multiple concurrent identities, for example mother, teacher, researcher, wife (or partner), and so forth. a particular identity in this sense (such as that of a teacher) develops within a particular situational context (such as that of a classroom) and in relation to others (such as students) operating within that context. furthermore, the “stocks of knowledge at hand” (schutz, 1967) upon which teachers draw include their own educational experiences and the way in which they learned mathematics in those educational settings. these various life foote et al. mathematics teachers’ identity journal of urban mathematics education vol. 4, no. 2 70 experiences of an individual are available to her as she moves from context to context. here in our attempts to more deeply understand the mtf identity development, we turn to mtf narratives. the role of narrative as a vehicle for understanding identity has an important lineage that can be traced through the work of bruner (1987), who claims that an individual’s identity is exposed in the course of storytelling, and, more recently, through the work of sfard and prusak (2005), who claim that narrative is identity. we draw on mtf narratives by using written reflections and interviews to trace the development of their identities as mathematics teachers in the urban context of new york city. furthermore, through observations, we examine how these identities play out in the context of the classroom. mathematics teacher identity development what knowledge is necessary to support the development of a situated classroom identity that might be leveraged to support successful achievement in mathematics for urban students of mathematics? mathematics content knowledge is vital. ball and colleagues (e.g., ball & bass, 2000) have identified a subset of mathematics content knowledge that they term mathematical knowledge for teaching (mkt). they argue that mkt is crucial in becoming a successful teacher of mathematics. we believe that the fellows’ development of mkt is one that supports (or whose lack of it undermines) the development of an identity as a mathematics teacher (meagher & brantlinger, 2011). furthermore, pedagogical content knowledge is equally important. one might define mathematics pedagogy as teachers possessing the ability to impart knowledge while becoming facilitators of students’ learning (nctm, 1991). teachers must select appropriate mathematical tasks to support learning, promote classroom discourse to deepen understanding, monitor student thinking and adjust instruction, and help students connect new concepts to prior knowledge. scholarship by van zoest and bohl (2005) proves useful in thinking about the various components that may be at play in mathematics teacher identity development. they developed a framework that takes into account “the knowledge, skills, and understanding that teachers carry with them from one context to the next” (p. 316). van zoest and bohl refer to this knowledge as “cache of capacity,” and for our purposes the framework provides the important feature of connecting content as well as pedagogical knowledge for teaching. but we must also consider how teachers draw on their own personal experiences with learning mathematics—stocks of knowledge at hand (schutz, 1967)—as they negotiate this work. foote et al. mathematics teachers’ identity journal of urban mathematics education vol. 4, no. 2 71 identity development as teachers of urban youth ladson-billings (1995) has suggested that in addition to being mathematically substantive, teaching must be culturally competent. accordingly, one of the necessary characteristics for teaching is the ability to support cultural competence in students. when the teacher comes from a community that is different from the community of the students, there is work to be done. teachers need to not only understand the lived experience of the students but also build on the strengths students bring to the classroom. important work has been done in looking at identity development that attends to issues of students as well as issues of content both in science and mathematics (e.g., enyedy, goldberg, & welsh, 2006; van zoest & bohl, 2005). this research suggests that classrooms and schools must be contextualized within the communities they serve if teachers are to understand the complexities involved in teaching students who come from communities in which they are under (or miss) informed. in sum, the literature leads us to conclude that to be successful mathematics teachers of urban youth, teachers must develop multiple expertise in knowledge of mathematical content, pedagogy for teaching mathematics, and knowledge of students. therefore, we ask: how and to what extent teachers might develop their mathematics teacher identities by concurrently developing mathematical knowledge for teaching, knowledge of mathematics pedagogy, and knowledge of students and the communities in which they teach? we also ask: how teachers might draw on their experiences (preservice training, in-service support, previous schooling) in forming their identities, and in what ways they resolve (or not) conflicts among past learning and present experiences? methods as we analyze the development and evolution of the identities of two alternatively certified mathematics teachers during their first year of teaching, we draw on several sources of data gathered as part of the larger metromath study previously mentioned. first we contextualize the development of our participants by drawing a large-grain portrait of teacher identity through considering surveys administered to all teachers in their mtf cohort at the beginning and end of the first year of teaching. then, we use extensive case study data to provide a nuanced view of the developing identities of our two participants during their first year of teaching. the case study data sources include lengthy preand post-first year interviews, videotaped classroom observations, field notes, post-observation interviews, and written lesson reflections. foote et al. mathematics teachers’ identity journal of urban mathematics education vol. 4, no. 2 72 participants our two participants, kate and kelly (pseudonyms), were both white women in their first year of teaching mathematics in middle schools in new york city. they each concurrently attended similar graduate programs (master of arts for teaching) at different universities. these programs were specifically designed in a collaborative effort among the new york city department of education, the new york state education department, and the local universities to educate and support mtf during their first 2 years of teaching. kate was 30 years of age at the start of her first year of teaching. she had attended a magnet high school in a large eastern city. in this respect, kate is like 36% of the 106 people in her cohort who responded to this question saying they had gone to an urban high school (50% had attended suburban high schools, 8% rural high schools, and 7% attended high schools abroad). after high school, she had attended an elite, eastern liberal-arts college where she majored in international relations. her college mathematics background consisted of one year of calculus. after completing college, kate lived in africa while working for an international service organization. following this service, she worked for several years for an educational publisher editing mathematics textbooks. kelly was 25 years of age when she started teaching. she had attended a rural high school in the midwest. after high school, kelly had attended a large, topranked, eastern university where she majored in international politics and religion. kelly had taken and passed ap calculus in high school; her college level mathematics consisted of one statistics course. kelly had a master of arts degree in islamic studies from a prominent canadian university. she had lived abroad before becoming an mtf. school contexts kate worked in a small middle school of approximately 300 students (middle schools in new york can be as large as 1,000 or more students). the student body was 37% white, 33% latino/a, 23% african american, 6% asian, and 1% native american; 42% of the students were eligible for free or reduced-priced school lunch and 4% of the students were considered to have limited english proficiency (these figures are for the 2006-07 school year, kate’s first year of teaching, and vary little from the previous 2 years). the school did not receive title i funding and had met adequate yearly progress (ayp) for all students and for all targeted sub-groups in both english language arts and mathematics in 2007-08. the new york state education department rated the school in “good standing,” the highest rating a school can receive from the state, which rates all foote et al. mathematics teachers’ identity journal of urban mathematics education vol. 4, no. 2 73 public schools not receiving title i funding 2 (new york state education department, n.d.). kelly worked in a middle school of approximately 800 students, and one with a higher level of student need. the student body was 56% latino/a, 42% african american, 1% asian, and 1% native american; 93% of students were eligible for free or reduced-priced school lunch and 16% of the students were considered to have limited english proficiency (again, these figures are for the 2006-07 school year, and vary little from the previous 2 years). the school did not meet all its goals for ayp in 2007-08. although all students met ayp in mathematics, they did not in english language arts. the federal government has rated the school as “restructuring year 2,” the lowest rating a school can receive from the federal government, which rates all schools receiving title i funding (new york state education department, n.d.). survey data and analysis informing our work is survey data that includes preand post-responses from approximately 157 of the first year mtf 3 that made up kate’s and kelly’s cohort. these surveys consisted of both likert-like and open-ended questions related to mathematics teaching and learning, urban teaching, self-efficacy as a mathematics teacher, views on mathematics procedures and concepts, and mathematics learning and student readiness. here we focus on the survey questions that related to identity. responses to the survey were analyzed using a mixed method approach. for the likert-like items, descriptive statistics were used to explain the responses. open-ended items were coded using the framework from van zoest and bohl (2005) described below. case study data and analysis nine mtf agreed to be participants for in-depth case studies as a part of the larger metromath study. as noted, we present results from kate and kelly, two teachers who were in their first year of teaching and whose data corpus was suffi 2 this school does not have the “typical” characteristics of student underachievement too often found in hard to staff schools. although, at the onset of the nyctfp, fellows were placed in hard to staff schools, as the program has evolved fellows now find their own jobs; many do not begin their teaching careers in hard to staff schools. 3 not all respondents who completed the survey answered each question. the number of respondents per likert-like question varied from a low of 151 to a high of 163. for some of the other questions, responses were even fewer. in those cases, we will state the number of responses on which the presentation of the data is based. foote et al. mathematics teachers’ identity journal of urban mathematics education vol. 4, no. 2 74 ciently complete. case study data for kate and kelly, including videotaped lessons and field notes, lesson reflections, and teacher interviews, were initially analyzed using a closed-coding scheme framed by van zoest and bohl’s (2005) mathematics teacher identity framework. data from each source were coded for instances that fit into the framework domains: (a) content/curriculum domain, (b) pedagogy domain, or (c) professional participation domain. at the same time, data were coded as belonging either to one of what is designated in the framework as aspects of self-in-mind ([a] knowledge or [b] beliefs, commitments, intentions) or aspects of self-in-community. the latter includes three dimensions of competence: (a) mutuality of engagement, (b) accountability to an enterprise, or (c) negotiability of a repertoire as well as perceptions of self and others. close examination of the coded data indicated that much of the identity development of kate and kelly during their first year of teaching involved the negotiation of conflicts. in the discussion below, we identify the salient conflicts that shaped their identity development and provide narratives for kate and kelly. we begin the discussion by drawing on survey data to provide an overview of some of the characteristics as well as beliefs and attitudes of the cohort of mtf examined. we do so to demonstrate the extent to which kate and kelly are typical members of their cohort. we then present in more detail the particular ways in which the identities of kate and kelly developed during their first year of teaching. survey results an analysis of the data shows that many mtf are from middle-class backgrounds, suburban schools, and are graduates of elite colleges (prestigious private colleges or large, top-ranked research universities). of the 105 members of the cohort who indicated their undergraduate institution, 64% graduated from elite colleges or universities. in this respect, kate and kelly are typical of their cohort, both having graduated from these types of schools. in addition, the majority of mtf did not major in mathematics or a mathematics related field. of the 80 members of the cohort who indicated their major, just 10 (or 13%) majored in mathematics; another 22 (or 28%) majored in a field related to mathematics such as accounting, finance, or engineering. the majority, 60%, majored in a liberal arts or science field. again, kate and kelly are typical of this majority in that they are among this group with limited learning experiences in college-level mathematics. in reflecting on their first year of teaching mathematics, the majority of mtf were confident in their ability to teach mathematics (79% agreed or strongly agreed with the statement: i currently feel confident in my ability to teach mathematics effectively; 89% agreed or strongly agreed with the statement: i think i eventually can become an exemplary urban mathematics teacher; 98% agreed or foote et al. mathematics teachers’ identity journal of urban mathematics education vol. 4, no. 2 75 strongly agreed with the statement: i will be able to improve my mathematics teaching ability). both kate and kelly are among these respondents. looking ahead to future plans, the majority of mtf were positive about teaching (82% agreed or strongly agreed with the statement: i think a teaching career will continue to engage and challenge me; 84% agreed or strongly agreed with the statement: if i had to do it all over, in view of my present knowledge, i would decide to become a teacher again). despite these positive attitudes toward teaching, both kate and kelly agreed with the majority of their cohort (80% of respondents) saying that they found teaching stressful. although both agree that they would become teachers, if they had it to do over again, kate agreed that teaching would continue to engage and challenge her, while kelly strongly disagreed that it would. when asked about their long term career plans 59% of the mtf indicated that they would continue to teach (41% in new york city and 18% outside of new york city); 13% responded that they would work in a job related to education; 10% said they would attend graduate school; and 18% said they had no plans as yet. again, kate and kelly were fairly representative of their cohort. kate planned to continue teaching in new york city, while kelly expected to move out of the city after 2 years with an ultimate goal of teaching at the college level. and yet again, kate and kelly were fairly representative of their cohort. both planned to continue teaching in new york city or elsewhere. case study results not unexpectedly, kate and kelly negotiated several conflicts during their first year of teaching. here we focus on three conflicts that emerged from our data analysis. the first involved the tension between understanding mathematics and the implications that this understanding had on their mathematical knowledge for teaching. the second revolved around mathematics instruction including which forms of instruction best suited the developmental needs of their students. and the third centered on how kate and kelly related to their students and the families of their students. we take each conflict in turn and examine the development of kate and kelly as they negotiated these conflicts. conflict #1: knowledge of mathematics and of mathematics for teaching there was a difference between the ways in which kate and kelly described learning mathematics and the ways in which they were expected to teach mathematics by the new york city department of education (nycdoe). from their descriptions of their own schooling, they learned mathematics as a discrete set of procedures and skills presented by teachers and memorized and practiced by stu foote et al. mathematics teachers’ identity journal of urban mathematics education vol. 4, no. 2 76 dents. they were asked, however, to teach in an environment that looked at mathematics as a connected set of concepts, skills, and problem solving (america’s choice™, n.d.), and to use a “comprehensive approach to balanced mathematics” (cabm), a curriculum model adopted by the nycdoe in 2003. at the heart of cabm was a model of instruction called the workshop model (new york city department of education, 2003). 4 exacerbating this conflict was the limited depth of their knowledge of mathematics. as we will see, kate and kelly negotiated this conflict in significantly different ways. attitude toward mathematics. in kelly and kate’s mtf cohort, the range of completed mathematics coursework varied from no college-level coursework in mathematics to master’s degrees in the discipline. while kelly’s formal background in mathematics was on the lower end of the spectrum, she entered the program professing that she not only liked mathematics but also that she was a strong student of mathematics. kelly had been accelerated in elementary school mathematics; in sixth grade she was placed with a small group of students who taught themselves sixthand seventh-grade mathematics with little teacher support. when kelly reached seventh grade, she studied algebra allowing her to complete ap calculus in high school. upon entering college, kelly received credit for ap calculus and used a course in statistics to complete her undergraduate mathematics requirements. kelly had the opportunity to study mathematics further during her first year of teaching. yet she found the mathematics course of little value and had difficulty understanding the material: every single class period [the professor] would start out with the basics. …and then he would go to some complex, weird extrapolation of what he’d started on and for the next hour and a half no one had a clue what he was doing. (post-first year interview, 2007) this lack of understanding led kelly to revise her ideas about mathematics: “i don’t really like math. never have really liked it” (post-first year interview, 2007). her survey responses echoed these comments. she strongly disagreed that teaching mathematics would be a lifelong career and disagreed with the statement: “i plan to continue to study math.” 4 quoting from the new york city doe cabm for grade 7: “the curriculum balances structured learning, direct instruction, and creative problem-solving. student discovery plays as significant a role in the learning process as does teacher-directed instruction. …there is a balance of basic skills and conceptual understanding; students build new mathematical ideas and at the same time practice needed procedures” (p. 3). the workshop model used in cabm outlines an instructional agenda that included a 7to 10-minute mini-lesson of direct instruction, 20 minutes of group work, and 10 minutes of sharing results. foote et al. mathematics teachers’ identity journal of urban mathematics education vol. 4, no. 2 77 kate had a similar background in school mathematics, having taken only calculus i as an undergraduate. nonetheless, she chose to study more mathematics and completed two additional college-level courses before she entered the nyctf summer pre-service program. at the end of her first year of teaching, kate was still eager to learn more mathematics. in discussing her graduate program and areas in which she still wanted support, kate responded, “i pretty much want advanced math classes” (post-first year interview, 2007). so here we see two first year teachers with similar school-based formations in mathematics but who at the end of their first year of teaching have quite different orientations toward and confidence in their knowledge of mathematics. moreover, kate and kelly have devoted differing energy to the improvement of that knowledge and have differing views of the knowledge they would seek to develop. understanding the mathematics they teach. both kate and kelly had been successful (i.e., received good grades) in school mathematics. yet even though they were teaching at the middle school level, they did not have a deep understanding of the concepts they were teaching. kate was nonetheless enthusiastic about the prospect of developing this conceptual knowledge base for teaching: this time we literally made a paper plate as a unit circle and drew everything on there and figured out why. really the content just reinforced all the different ways to see every kind of mathematical concepts, stuff i’d never thought about before. (prefirst year interview, 2006) additionally, kate clearly saw the link between mathematical knowledge and mathematical knowledge for teaching. she saw that understanding student thinking was intimately linked with knowing mathematics. when asked what she felt were characteristics of effective mathematics teachers in urban settings, kate mentioned the need to know mathematics: i think you really have to get the math. they want so many math teachers. they get all these people who don’t really have much of a math background, and you can’t teach math if you don’t know what’s going on, because the kids come up with the most bizarre ways to explain the way they think about different problems, and nine times out of ten it makes complete sense. (post-first year interview, 2007) kelly, on the other hand, did not share the desire to learn more mathematics, even though she recognized that her traditional background could limit her ability to use different methods to solve problems: the way that my teachers taught me these things is really different than how we learned them in these [university methods courses], so it was really interesting to see the different [methods]. i just memorized things. our [methods] professors are telling foote et al. mathematics teachers’ identity journal of urban mathematics education vol. 4, no. 2 78 us it’s not really about the formula, give them a calculator or both, or give them the formula. and i’m—wait a minute, that was all i learned, how to memorize the formula and plug it in. (pre-first year interview, 2006) on one particular occasion, kelly expressed her concerns about these differences by focusing on the mathematics at the superficial level of terminology. when asked if she knew mathematics well enough to teach it, kelly somewhat hesitantly responded: i think i know it. i know it if i can review it right before i teach it. but, as i’m learning, a lot of the technical terms for things were not the same for what they call it here and what they called it in [my home state]. so when [my professor] said we were going to add and subtract mixed numbers with grouping. now i had never in my life heard of grouping. i didn’t know what grouping meant and i assumed i had done it somewhere along the line but i had no idea. (pre-first year interview, 2006) at the end of the year, when asked the same question as kate about the effective characteristics of urban mathematics teachers, unlike kate, kelly did not mention that knowledge of mathematics was important. conflict #2: mathematics instruction both kate and kelly worked in middle schools that, in line with directives from the nycdoe, used the cabm and workshop model (previously described) for teaching mathematics. given their background in learning mathematics via traditional mathematics pedagogy, it is not surprising that both encountered some conflict around the idea of teaching mathematics using cabm. it was a different way to look at the teaching and learning of mathematics from what kate and kelly had experienced as learners, and at times they questioned whether these new methods they were encountering best served their students. and yet, they both found something enticing and intriguing about the idea of studentcentered learning and discovery pedagogy. initially, kelly expressed reservations about implementing the workshop model: so i have to present [the material] first, then they’re supposed to work alone for five or ten minutes and then do something in a group. i don’t understand how i can go through a whole workshop model in 43 minutes—instruction, independent work, and group work. (pre-first year interview, 2006) although kelly expressed concerns about implementing the workshop model on a daily basis, by the end of the year she could to see that a routine such as the workshop model provides might have promoted better classroom behavior: foote et al. mathematics teachers’ identity journal of urban mathematics education vol. 4, no. 2 79 i get bored very easily with routines, like doing the same thing every day. but i found that if i changed things up, then the students’ behavior went nuts because they weren’t used to that. …coming in with the same thing every day, [the students] know what’s going to happen and it does happen. i had less issues with their behavior. …and i liked [the workshop model] a lot because it doesn’t put a lot of emphasis and time—a lot of time to the teacher driven part because i’m not going to hold their attention with examples for more than a few minutes. they would rather talk it out with their neighbor than listen to me. (post-first year interview, 2007) but even after a year of teaching, kelly could not articulate how she believed mathematics should be taught in this urban environment: “i don’t really know because i haven’t seen—i mean i haven’t watched any other teachers in my building. i guess it’s just what i think would work. …we haven’t taken a class on it” (post first-year interview, 2007). one area where kelly felt she had improved over the course of the year was in the delivery of her lessons. planning had been a major focus of her preservice program. after the first few months of teaching, however, during which she had devoted several hours each week outside of school to planning, she decided to set aside only one hour in the morning before school to plan. “i thought we [teachers] would spend a lot more time doing things like lesson planning when in fact i did not. that was more of a thing that i did in the morning when i got there” (postfirst year interview, 2007). by thanksgiving, kelly had decided that if she was going to survive the year, she needed more control over her schedule outside of school: so, i did all of my prep in the morning because by the end of the day, i didn’t care. …i didn’t stay ever afterwards. …this is my revelation. i don’t know if it’s good or bad. …[initially] i was planning for the whole week on sunday for a couple of hours, but none of my classes ever got done what i wanted. (post-first year interview, 2007) by the end of the year kelly found herself not only to be more confident but also better able to read how the lesson was progressing and adapt if necessary: i didn’t feel like i needed it written down word for word, but i was also more responsive of how it was going in class. i think at the beginning of the year i chugged along on my lesson plan regardless. and after a while, i could read their expressions more and they were more willing to say, “i don’t get this.” (post-first year interview, 2007) while making connections between the mathematics taught and real-world applications is one of the many goals of cabm, it seemed to be the only one that kelly focused on. before she began her classroom teaching, she voiced concerns as she realized that her limited and largely procedural knowledge of mathematics was interfering with her ability to access real world contexts: foote et al. mathematics teachers’ identity journal of urban mathematics education vol. 4, no. 2 80 so when i’m trying to think…how am i supposed to make them want to…learn imaginary numbers? …well, i’ve got [the other fellows] raising their hands about codes with the cia. i have no idea because i learned the formula. (pre-first year interview, 2006) kelly repeatedly voiced similar concerns about what it would take to engage her students in doing mathematics problems that were related to their lives and experiences. at the end of her first year of teaching, kelly underscored the importance from a motivational perspective of including material drawn from the students’ lives: the lessons that i had that were most successful and my students paid attention the most when i could somehow manage to relate what we were doing to their life. when i did something with money and sales and going to the store and you had a certain amount of money in your checking account and you wanted to buy this and that. …so, i think effective teachers will find ways to make the material applicable and resonating in their students lives so that they actually see a reason to learn it. (post-first year interview, 2007) yet during this first year of teaching, kelly rarely drew on real-world contexts and there was no evidence that she developed investigations when teaching mathematics. instead, she too often resorted to teaching lessons that were more procedural in nature. as we saw previously, kate was favorably disposed to using methods consistent with cabm and focused on engaging with discovery pedagogy in her classroom. yet, throughout the year we see the tension between teaching mathematics in an engaging way and reverting to the more familiar traditional pedagogy she had experienced as a student. toward the end of a lesson early in the year, for example, when a student verbally expressed a rule for  a c b c stating that the equivalent expression was  (ab) c , kate responded, “that is the rule!” after a short pause, kate asked, “why does it work? how do we write it out in variables?” although this question could easily have supported some investigation and discovery by the students, kate (evidently, prompted by time constraints) waited only a few seconds and receiving no immediate response said, “i’ll just tell you because we’re running out of time” (observation, 9/06). in a reflection later in the fall, kate defended her decision to focus at least periodically on repeated drill and practice of a skill: almost everyone can factor and expand a bit, which is very reassuring [because] i have spent so much time going over it. they are sick to death of it i’m sure, but i don’t care. i’m still going to keep doing the distributive property problem of the day for as long as it takes. (reflection, 12/06) foote et al. mathematics teachers’ identity journal of urban mathematics education vol. 4, no. 2 81 nonetheless, in that same reflection she looked ahead with excitement (although also with a certain sense of anxiety due to classroom management issues) to a unit that she felt would be able to involve more discovery learning: we are all so excited to be done with algebra for a bit and i think they’ll really like the geometry unit. it’s so much more crafty though, and i’m a bit worried about classroom management when most of every period is them making stuff and doing the whole discovery exploratory thing. we’ll see. maybe they’ll surprise me. (reflection 12/06) some of her concerns about managing the discovery classroom were apparently born out, as just a month later kate expressed distaste for discovery learning: i hate doing discovery lessons because you notice my room is a mess…discovery mathematics and this whole like new way. …i would love to put a formula on the board and give them 25 problems to do—that’s the way i learned it. (postobservation interview 1/07) in the same interview, however, kate expressed amazement at how well a portion of the lesson had played out: “that whole making the net thing was actually amazing—i can’t believe it worked out so well today. they’ve never been able to do that before” (post observation interview 1/05/07). shortly after this interview, and after a lesson linked to valentine’s day that had developed particularly well (on estimating areas and finding areas of irregular shapes), kate gave a strong endorsement of discovery learning saying: i want to spend more time doing problems like the heart problem, ones that get the kids really involved in loving a problem that is complex but accessible, where they have to use all different kinds of math (like with the heart problem—geometry, algebra, measurement, proportions) to find a solution. these can take a lot of class time, but i think the students learn more math than sitting there, listening to me and then doing practice problems about something they hardly understand. it is easier if they are given ways to make connections to what they already know and they sort of need bigger problems, not just one-step drills. (reflection, 2/07) in spite of successful lessons such as the heart problem that used discovery pedagogy, kate expressed in a reflection in june some frustration with the insistence by the school administration that she adhere to the workshop model. she said that she liked to have more control in the classroom and so she often limited the amount of classroom discussion. in an interview after the completion of her first year of teaching, it is apparent that kate still felt that a certain amount of drill and practice served her students well. when asked how her image of teaching had foote et al. mathematics teachers’ identity journal of urban mathematics education vol. 4, no. 2 82 changed since the beginning of the year (when she was enthusiastic about discovery learning), kate responded: i think at the beginning of the year i felt like the whole just like practice problems rote learning thing was really out of fashion and not that valuable now. i mean i know that it worked for me. …and i think that i felt guilty about doing that this year, like “open to page 328 and do 1 through 30,” but i think it’s actually very valuable for them and i don’t feel ashamed of it anymore. …i think you just have to balance it. i mean some things work better for different topics and you just have to do all of it all the time kind of, instead of picking one thing. (post-first year interview, 2007) so we see that although kelly was able to verbalize the need for relating mathematics lessons to students’ lives, she was unable to follow through in her lessons. kate continued to be much more responsive to using the cabm model, although she too felt she needed at times in her classroom to use a pedagogy that focused more on the learning of procedures. conflict #3: teaching urban youth along with the content issues involved in teaching mathematics, there are also relational issues that come into play in classrooms that are equally important to achieving success in teaching (grossman & mcdonald, 2008). while knowing mathematics content and pedagogy is crucial for being an effective teacher of mathematics, knowing the students whom one teaches is also crucial to being a successful teacher of mathematics (ball, lubienski, & mewborn, 2001; national research council, 2001). there are two aspects of teaching mathematics to urban youth in which kate and kelly exhibit conflicts during their first year of teaching: are the urban youth they are teaching capable of engaging in conceptually challenging mathematics? and how do teachers relate to the students (and their families) as people who bring rich experiences and resources to the classroom? beliefs in students as capable learners. the conflict kate and kelly exhibited around their beliefs in their students as capable learners was two pronged. on the one hand, they held some deficit views of students and/or their families (for a more complete discussion of the deficit views held by the kate’s and kelly’s mft cohort, see brantlinger, cooley, & brantlinger, 2010). but on the other hand, they recognized that, at times, their students’ exhibited clear understanding of grade-level content knowledge and the ability to engage in problem solving. having a belief and recognition of students as capable learners however, did not lead either teacher to incorporate student-based knowledge into mathematics lessons. initially, kelly talked about motivating students by connecting what they were learning to real-life applications and worried that her mathematics background was not strong enough to incorporate applications into her lessons. how foote et al. mathematics teachers’ identity journal of urban mathematics education vol. 4, no. 2 83 ever, when the opportunity arose (a basic statistics lesson reviewing mean, median, and mode), kelly neglected to add even one example that might relate to the lives of the students in her class. she reflected on her lesson and justified her lack of planning for student engagement: i was kind of bored with the material so i can only imagine what the students thought. i think i achieved my objective; however, i would like to make it more interesting as time goes by. …i will focus on more creative lessons when i have the basics [classroom procedures] under control. …i have been getting advice from colleagues that these students are really motivated so i need to make sure they are challenged. yet, i just found that while [reviewing] the homework they [easily gave] up when the questions were not that difficult, they gave up without trying. (reflection, 9/06) as the year proceeded, kelly continued to use classroom management issues as a justification for limiting her lessons to a short teacher directed mini-lesson with the remainder of the class period used for students working together on skillbuilding problem sets. while she considered offering more inquiry-based lessons or providing a focus on applications, kelly did not take opportunities to incorporate any of these strategies into lessons. in her post-first year interview, kelly indicated that her students were reluctant to explore and problem solve independently: if they could not get it, they wanted me to be over there to help them, show them how to do it “right now.” [they] can’t make mistakes. they don’t like to get anything wrong in class. struggling to figure it out on their own is not something that they like to do. (post-first year interview, 2007) from the start of the school year, kate had a more positive view than kelly of her students’ capabilities. she was also dedicated to supporting them to develop confidence in doing mathematics as she saw a lack of confidence, rather than a lack of ability, as hampering her students’ achievement. when, during an interview following a classroom observation early in the fall semester, the researcher commented, “you keep telling the class they can do the given assignments.” kate responded by stating: a lot of my students get stuck on not knowing how to do something immediately. i’m trying to help them understand that they can figure stuff out; give them the confidence they need to figure out the tough problems. …i remind them constantly that they do know and i know they know. (post observation interview 10/06) nonetheless, kate appeared to lack confidence in her students’ abilities as capable learners in other respects, “they don’t have the vocabulary or the ability to think about it well enough to discuss it afterwards” (reflection, 1/07). she be foote et al. mathematics teachers’ identity journal of urban mathematics education vol. 4, no. 2 84 lieved that following a structured/procedural approach to mathematics might address these deficits and support them in becoming confident learners of mathematics: “they can solve a word problem by writing an equation and then following the set of rules. as long as they follow these rules, they can do anything they want, and they can solve any problem” (reflection, 12/06). yet in spite of her repeated advocacy for the need to use this procedural approach, she was also aware of the benefits that could be had by students engaging in inquiry activities in the classroom. on one occasion when the students successfully engaged in discovery learning, kate was impressed and surprised by what they accomplished. and at the end of the year, kate spoke positively about her students saying again that it was a lack of academic confidence rather than a lack of competence that negatively influenced her students’ performance: i’d really like it if my students were able to look at a math problem, decide what they need to do to solve it, and figure out how to use what they know to find the answer. they don’t seem to have a lot of faith in what they already know, which is a lot. total lack of confidence. (post-first year interview, 2007) relating to students. both teachers demonstrated the ability to relate to each student as an individual to support student learning. but because of her middleclass upbringing, and the more modest socio-economic status of many of her students, a social distance (ladson-billings, 1997) existed between kelly and her students. social distancing was not the case for kate, however. her conflict with relating to students was more centrally concerned with a personal distance she sought between herself as a teacher and her students. here we elaborate on the two different ways that kate and kelly were conflicted in their relationships to their students. additionally, we will see that while both kate and kelly may have understood that incorporating home and community knowledge in the classroom could be motivating for students, they demonstrated little ability or effort to do so. even when lessons appeared to be ripe for making connections from students’ own lives to mathematics (such as in the statistics problem previously discussed) it was rarely done. the views that kate and kelly had of students and families as lacking a rich and valuable knowledge base did influence their thinking about what was possible for their students to accomplish mathematically. negotiating social or personal distance. kelly had been told in her interview with the nyctfp that being raised in upper-middle-class, rural america would influence her ability to connect with her students, but she did not find that to be the case. at the end of her first year, she said: like when you go to these interviews with the teaching fellows, they kept pointing that out [the social distance], so i didn’t know how it would manifest itself, but somehow [they seemed to be saying that it] was going to be difficult for me to co nnect with my students and maybe they wouldn’t like me because of this or that. …i foote et al. mathematics teachers’ identity journal of urban mathematics education vol. 4, no. 2 85 wouldn’t understand where they’re coming from. but they’re just kids. it really didn’t actually—i guess as long as you’re an empathetic person and you listen to their problems, they’re just kids. and it didn’t really matter that much because i was capable of listening and understanding about what was going on in their lives. and i don’t think they really cared where i came from. (post-first year interview, 2007) on an individual basis kelly had a respectable rapport with students. this respectable rapport was evident in a lesson observation where an african american student sitting in the back of the room did not move to participate in group work after the mini lesson. he was quiet, but visibly uninterested in the class and looked somewhat upset. rather than reprimand the student in front of his classmates, kelly went over and spoke quietly with the young man, making him laugh. he eventually moved and started to participate with his group (observation, 12/07). additionally, kelly had experiences she brought with her to teaching that may have supported her in bridging any predicted social distance between herself and her students. she had been an exchange student in the same south american country that many of her students’ families came from. because of this exchange program she spoke some spanish. she also had lived in the middle east and spoke arabic. her ability to speak arabic provided an opportunity to talk about cultural differences and, along with her ability in spanish, might have helped to position her as someone with multicultural understandings: it was interesting and the only time we had like a real serious discussion was one of my students in my homeroom speaks arabic. he’s muslim. and i heard him—i asked him in arabic something because i saw his name. and then a couple of the students started laughing and were like, “oh, so you’re a terrorist too?” so, i was— so, we talked a little bit about arabic and islam, which [i studied in college], so we talked about that for a while. (post-first year survey, summer 2007) but kelly also expressed opinions about the nationalities of her latino/a students indicating that perhaps she did not have a fine-tuned sense of the importance of students’ cultural identities: the biggest deal was between students, a student making stupid jokes and stupid comments to one another about their own cultures, which is something we dealt with all year and never really got dealt with, never got handled. …they had names for everybody depending on where they came from. and, of course, that upset about half the people in the classroom. and they would yell that out in the middle of class. … there were differences between the dominican republic and haiti and i’m dominican and you’re a mexican. i am guatemalan and you’re mexican. and they would just make fun of each other. no real point to it and they’re similar anyways. (postfirst year interview, summer 2007) foote et al. mathematics teachers’ identity journal of urban mathematics education vol. 4, no. 2 86 clearly, for the students, there were differences in identity between being mexican and guatemalan, for example. although to kelly the cultures and languages may have seemed too similar to be the basis for ridicule and arguments, this perspective of similarity was not the case for the students, however. kelly’s lack of planning and limited mathematics background also may have influenced her ability to connect with her students. in a lesson on the laws of exponents, which she assumed would be a review, kelly stumbled in her response to a problem on order of operations, -2 2 , indicating that one would evaluate it as the expression -2 *-2 resulting in 4 (instead of the correct evaluation of [-1][2 2 ] = -4). this inaccurate result was questioned by the students and led to a series of vague, inaccurate, or, at best, imprecise responses on kelly’s part through the remainder of the lesson. it was clear by the end that kelly was frustrated and the students were confused. but rather than recognizing her mistake, she reflected on her lesson in the following way: their teacher from last year told me they had already mastered all the laws of exponents and i think she may have been exaggerating or they all forgot, because it took much longer than i imagined. also i have a hard time understanding their questions because i think most of the time they want to trick the teacher and i rarely understand what they are getting at. …i am trying to teach them that they may not understand the minute it comes out of my mouth but through practice and examples it will come to them. this is a constant struggle. (reflection, 10/06) thus, although kelly never saw the social distance as an issue, she failed to see that her latino/a students saw themselves as different from each other, in this way exhibiting that there did exist some social distance between her and her students. furthermore, she did not trust the students enough in their abilities to question her content knowledge or to allow them to engage in inquiry-based learning. kate’s conflict around her students takes a different form from that of kelly. as we will see, kate does not believe that social distance between her and her students is an issue. her identity as a teacher, however, compels her to try to maintain somewhat of a personal distance from them in the interest of being “professional.” she is concerned about being perceived by her students as not sufficiently strict. although kate is conflicted about the nature of her relationships with students, she does not experience the same enacted social distance between herself and her students as kelly appears to. in fact, kate sees more similarities than differences between herself and her students: i think we’re really similar. i mean the school that i went to was a public school. it was a magnet school. it was really, i mean it was tracked and sort of segregated, was sort of dangerous, but it still had a really good reputation. i mean i can like see in my students like which one i would be and who my friends would be. i don’t think foote et al. mathematics teachers’ identity journal of urban mathematics education vol. 4, no. 2 87 they’re that different at all. they’re pretty much the same. some of them seem more mature than anybody i knew at that age. (post-first year interview, 2007) kate’s desire was to have a somewhat impersonal relationship with her students. she wanted to remain somewhat of an unknown entity to her students so that she could maintain a personal or professional distance that she perceived was difficult for her to do. this desire led her purposefully to withhold personal information from her students: i mean i feel like they were all so different, like they’re all from so many different places…and they didn’t really know where i was coming from because i didn’t want to share it with them. (post-first year interview, 2007) kate, however, wanted to employ a pedagogy of caring for her students (noddings, 1984). she saw herself as someone who both related well to students and let them know she cared: i think i get how to work with kids and how to like really listen to them. …i was able to really connect with them and i think some teachers can’t. i mean they know that i care. they knew that i really, really wanted them to do well, they knew that i loved math and they knew that i expected them to work their hardest. and what else can you really want, you know? (post-first year interview, 2007) kate demonstrated this pedagogy of caring and camaraderie to her students in many ways. she was very expressive in her class discussions with students using such words as “great” and “fantastic.” she smiled often. after the heart problem (previously discussed), kate congratulated her students on a great job: “so, first thing i want to do is talk for a moment about the heart problem we did on wednesday. that was fantastic” (observation, 2/07). while working on perimeter and area problems, she demonstrated not only enthusiasm for the students’ work but also made a connection among the work the students were doing, saying: “ooh – okay – nice. so you’re pretty much doing the same thing but different order as to how you are thinking about the problem. fantastic.” (observation, 3/07) nonetheless, she did not want to be perceived as too caring. at the beginning of her teaching, she had acknowledged that she had this persona with students: i seem to convey this, “i care about you. i’m supportive,” like you-can-come-to-me openness, which is really, really useful in a lot of ways in life. it’s great as a camp counselor, but it’s not great as a teacher all the time. i really have to figure out how to not project that all the time. i’m also a little too expressive. i need to have a little bit more of a mask. (pre-first year interview, 2006) when asked at the end of her first year, how her students perceived her, kate replied: foote et al. mathematics teachers’ identity journal of urban mathematics education vol. 4, no. 2 88 they loved me. they thought i was really nice. i mean not the kinds of things you want your students to think of you. i want them to be scared of me, like they are of ms. christianson [pseudonym]. you know, i want them to be happy to see me but be kind of like mentally checking to make sure they’re like not chewing gum when i see them. and it’s not like that. i feel like i’m more like mom. so i’m going to work on being scarier this year. (post-first year interview, 2007) negotiating personal distance, as kate perceives it, continued to be a work in progress. she began and ended the year wanting to project an image that one can imagine might call commanding respect. while she cared for her students and was enthusiastic and effusive in the classroom setting, she was uncomfortable that at times she passed over a boundary of professionalism that she wished to establish and maintain. are parents the problem? both kate and kelly saw a difference between the parents of their students and the families in which they had grown up with regards to the reaction of parents to failure in mathematics. in kelly’s case, not only did her parents have high expectations of her achievement in mathematics but also they encouraged her to accelerate in mathematics. she found her students’ parents did not necessarily hold these same expectations. as she put it: actually, i found it interesting that at the parent-teacher conferences, quite often parents would say things [about] their kids like, “oh, it’s all right they’re not doing good at math. i was never good at math.” …and if you’re getting messages at home that it’s okay to not be good at math or it’s okay because it’s hard, then it’s easy for them to accept so-so grades when they could be doing better. (post-first year interview, 2007) i think the biggest problem for my students is their parents, because probably at least half of the parents who came to talk to me on parent-teacher night tell me that they hate math. they don’t understand math. they can’t help their kids with math because they don’t get it. (post-first year interview, 2007) yet kelly found herself holding mixed feeling about parent involvement in their child’s education. she tried to call parents when students were in danger of failing or were causing trouble in her classroom; however, she rarely actually talked with parents when she called—most calls resulted in messages left on answering machines. while it was easy to blame parents for lack of interest in their children’s education, she also recognized that many were doing the best they could: i mean we didn’t see very many of the families [during parent conferences]. the support and involvement by the families wasn’t that much. …but that’s not fair to say that they weren’t supportive because a lot of them just had so many jobs that they were working all the time and couldn’t make it up to school. so, i want to think foote et al. mathematics teachers’ identity journal of urban mathematics education vol. 4, no. 2 89 that they were just as supportive of education, but i don’t know. we just never really interacted that much with the families. (post-first year interview, 2007) kate had sounded a similar strain, also indicating that the onus for success lay with the parents, that there were instances where the teacher did not have the tools to influence learning given the conditions at home: “the kids who get no support [at home] struggle in school. their parents don’t help them and the kids are a mess and are too disorganized to get stuff done on their own. and their teachers can’t help them all” (post-observation interview, 2/07). however, in spite of this view of parents, kate reflected at the end of her first year of teaching that she had been able to develop relationships with some of her students’ families and an understanding of their lives. the development of these relationships, kate credited to her involvement with the parents even when that involvement began due to behavior problems in the classroom: so most of the kids whose parents i talk to a lot were kids who were more of a pain in the butt unfortunately, but whenever i had to make those phone calls i also made myself make a couple of happy phone calls, just to call and tell them [their child] was doing well. so i tried to keep in touch.” (post-first year interview) so we see that both kelly and kate experienced conflicting views regarding the relationships that they had with their students and their students’ families as well as the influence that these relationships (or lack thereof) have on the students learning. kelly was the most conflicted about family support, identifying it as limited and problematic, but also understanding the dilemma low-income parents faced as they struggled to provide food and shelter for their families. kate was less sympathetic toward the parents of her students, but recognized the importance of sharing positive as well as negative feedback with parents in an attempt to build relationships. clearly, there were relational issues that influenced kate’s and kelly’s effectiveness as first year teachers. while both felt there was little social distance between them and their students, kelly made limited use of her diverse experiences and knowledge of latino/a students to make her class more engaging or to help address ethnic rivalries. kate, on the other hand, although identifying with her students, sought to maintain a professional distance from them, believing this would improve her ability to manage her classroom. discussion we know that identity is not only shaped by the knowledge and skills we acquire but also shapes the very knowledge and skills we seek to develop (battey & franke, 2008). educational policymakers, mathematicians, and mathematics foote et al. mathematics teachers’ identity journal of urban mathematics education vol. 4, no. 2 90 teacher educators agree that teachers need sufficiently deep content and pedagogical knowledge to teach mathematics for understanding (ball et al., 2001; hiebert & carpenter, 1992; ma, 1999; national council of teachers of mathematics, 2000; new york state department of education, 2010). there is little agreement, however, on what knowledge, skills, and dispositions make up this deep content and pedagogical knowledge. kate and kelly clearly did well in the mathematics courses they completed prior to entering the nyctfp and successfully passed state content knowledge requirements to teach secondary mathematics. nonetheless, both teachers experienced difficulty in resolving conflicts between the mathematical knowledge needed for teaching and their past learning and present experiences in mathematics. both kate and kelly are in many ways typical members of their mtf cohort and thus provide pertinent examples for thinking about teacher preparation for not only the nyctfs but also for alternatively certified teachers of mathematics more broadly. although they are similar to many other members of their cohort, and to each other, the ways in which they developed throughout their first year of teaching are different, leading to different trajectories of identity as teachers of mathematics in an urban context. the orientations that kate and kelly had toward mathematics as a discipline and toward the teaching of mathematics significantly affected the development of their identities as mathematics teachers. in the discussion that follows, we first discuss the orientation of kate and kelly toward mathematics, then toward the teaching of mathematics, and finally consider issues they encountered as teachers of urban youth. we conclude with implications for policy, practice, and research. the orientation of kate and kelly toward mathematics influenced their identity development as teachers of mathematics. although they began the nyctfp with similar school-based formations in mathematics and both initially said they “liked” mathematics, their relationship to the discipline of mathematics diverged during the first year. kate recognized before entering the program that she needed to learn more mathematics, perhaps because she lacked confidence in her ability or perhaps because she had an interest in mathematics. at the end of her first year, kate remained enthusiastic about the subject looking forward to studying further advanced mathematics. it may be that kate’s belief that she would continue to be challenged and engaged by being a mathematics teacher is tied to her enthusiasm for mathematics. kelly also began the nyctfp with a positive attitude toward her abilities in mathematics. but by the end of her first year of teaching, she was not positive about mathematics as a discipline, saying simply that she did not like mathematics. kelly had experiences in the university mathematics courses she was taking in her graduate program that might be construed as discouraging. she was not easily able to grasp the content of her mathematics courses and did not find it particular foote et al. mathematics teachers’ identity journal of urban mathematics education vol. 4, no. 2 91 ly useful. she ended the year not mentioning that knowledge of mathematics would be a characteristic of an effective urban mathematics teacher. to have a positive identity as a teacher of mathematics it would seem clear that an affinity for or interest in the subject would be necessary. we see here that kate alone remains positive about her orientation toward mathematics during her first year of teaching and with her plans to study further mathematics is on a trajectory toward having a more fully developed identity as a mathematics teacher. the orientation of our participants toward teaching mathematics is also consequential for their identity development as teachers of mathematics. given that both kate and kelly had been schooled by teachers using a more traditional pedagogy, they both turned away (kelly) or longed at times to turn away (kate) from the cabm classroom and toward a more procedurally oriented one. complicating their points of view on the teaching of mathematics was the issue of classroom control. while seeing some of the positive benefits of discovery/inquiry-based learning, connections to real-world applications, and a student-centered classroom both kate and kelly also found the environment in which this played out difficult to manage, both in terms of student behavior and in terms of time management. from the beginning of the school year, kate was enthusiastic about what she called the “new way” of teaching mathematics. and although she experienced a conflict between the cabm philosophy and traditional pedagogy she remained aware of some of the positive aspects, especially in the area of discovery learning. however, she was also conflicted about the cabm because she wanted to have more control over the classroom conversation and because she wanted time used more efficiently. while kate perceived she would have more control with a traditional pedagogy, she did not always resort to it, but rather continued to struggle at resolving the conflict between discovery and traditional pedagogy. kelly on the other hand, began the year not understanding how she would be able to implement the cabm and its workshop model. while she recognized conceptually the importance of relating mathematics to students’ lives, she rarely drew on contexts familiar to students in her teaching. in kelly’s case, her lack of confidence in using cabm meant that she turned to a more procedural model of teaching. another reason that kelly may have turned to this teaching model is that she spent so little time on planning, abandoning after teaching for only a few months any out-of-school planning. this limited planning minimized the amount of time she devoted to thinking about how the strategies advocated in cabm and the structure of the workshop model might be incorporated into her lessons. in essence, kelly did not confront the conflict between the cabm philosophy and traditional teaching. classroom control was difficult for her and a focus on classroom control rather than on instruction pervaded her thinking. kate’s and kelly’s identities as teachers of urban youth also developed in different ways as the year progressed. while kate struggled with how to teach her foote et al. mathematics teachers’ identity journal of urban mathematics education vol. 4, no. 2 92 students and felt that their parents did not provide enough support, she believed they could learn mathematics. she longed for more balance to teach in a manner more consistent with her philosophy, believing that practice would help her students develop the confidence they needed to be successful in school. kelly, on the other hand, who was able to establish a respectable rapport with her students on an individual basis, had trouble connecting with the class as a whole. she never mastered the ability to engage the class in whole-group discussions about mathematics, but could have rich discussions about islam and arabic when the opportunity presented itself. when a mathematics lesson did promote good student questions, kelly’s limited mathematics background and/or planning led her to believe that students were trying “trick” her by asking questions she did not understand and could not answer. unlike kate, kelly did not place blame entirely on the parents for lack of support of their children’s education. she recognized that these parents were often working hard to provide basic needs for their children and may not have had the time, or felt they did not have the language skills to return a phone call or attend a parents night. to use gee’s (2000) terms, although both kate and kelly have had institutional-identities as mathematics teachers bestowed on them by the nyctfp and the nycdoe, only kate has adopted a discourse-identity aligning herself with other competent doers and teachers of mathematics. this adoption has put kate on a trajectory toward an affinity-identity as a member of a practicing group of mathematics teachers. kelly, to some extent, at least due to her rejection of mathematics as interesting and accessible to her personally, continues to walk around the edges of the profession, not really moving in the same way as kate toward becoming a full member of the mathematics teaching profession. implications for policy, practice, and research alternative certification policymakers and teacher developers need to understand how the limited backgrounds alternatively certified teachers can influence their identity development. while insisting that alternatively certified mathematics teachers hold a major in the discipline is unreasonable, simply requiring them to complete equivalent mathematics major courses as part of their preparation may not be in the best interest of the teacher or the children whom they teach. alternatively certified teachers should be provided the opportunity to study mathematics that is directly connected to the content they teach. these connections must be explicit and not left to the teacher to build. therefore, instructors of mathematics courses must have knowledge of secondary school curricula and an understanding of how adolescents learn. they should model discovery pedagogy as they deepen the teachers’ content knowledge and ability to teach for under foote et al. mathematics teachers’ identity journal of urban mathematics education vol. 4, no. 2 93 standing, giving teachers the opportunity to experience both discovery and inquiry learning. along with enacting an unfamiliar pedagogy, alternatively certified teachers also are asked to teach in schools different from those they attended. this difference often results in a lack of understanding how student-centered models of instruction can be implemented in the urban classroom and how nontraditional pedagogy such as discovery learning can actually enhance student learning. because alternatively certified teachers have little opportunity to practice their craft with accomplished master teachers prior to becoming a teacher of record, they need substantial support in their classrooms to develop the skills not only to manage discovery learning classrooms but also to plan and implement standards-based instructional strategies (foote, brantlinger, hadar, smith, & gonzalez, 2011). early in their first year of teaching, alternatively certified teachers should engage in professional development activities in their school environments that focus explicitly on developing the knowledge, skills, and dispositions to implement instructional strategies like those identified by cabm. alternatively certified teachers of urban youth, especially those whose own backgrounds do not provide them with a basis for understanding the culture of urban schools, must learn to foster student persistence and confidence in their ability to do mathematics. they can do so by carefully planning classroom activities that give their students opportunities for success while providing challenging and meaningful work. they can also help their students and their students’ parents to understand the importance of mathematics in preparing for future employment. to do so, the alternatively certified teacher must have a positive disposition toward the discipline and an understanding of how mathematics is applied. further research is needed on what types of preservice experiences and firstyear professional development can support new teachers in improving their mathematics content knowledge and implementing instructional models such as cabm (models with which they too often have had little experience as learners). a teacher’s first year has a significant impact on the trajectory of her identity. it is particularly important that alternatively certified teachers have positive, educative experiences to support growth in mathematics content knowledge, mathematics knowledge for teaching, and knowledge of urban youth. not only are these experiences beneficial for the teacher but also it has been found that retention of teachers increases with more positive mentoring and positive educational experiences (national commission on teaching and america’s future, 2005). furthermore, a positive disposition or orientation to mathematics as a discipline figures prominently in our discussion of why one new teacher is on a trajectory toward building an identity as a mathematics teacher and the other is not. future research could be directed at examining how this disposition can be identified so that teachers who possess this orientation can be recruited to work in urban environments. foote et al. mathematics teachers’ identity journal of urban mathematics education vol. 4, no. 2 94 references america’s choice®. 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(2005). mathematics teacher identity: a framework for understanding secondary school mathematics teachers/learning through practice. teacher development: an international journal of teachers’ professional development, 9, 315–345. http://www.nctaf.org/documents/nctaf_induction_paper_2005.pdf https://www.nycteachingfellows.org/purpose/impact.asp https://www.nystart.gov/publicweb/ http://www.highered.nysed.gov/tcert/pdf/draftteacherstandards2010.pdf journal of urban mathematics education  december 2008, vol. 1, no. 1, pp. 84–107  ©jume. http://education.gsu.edu/jume  pamela l. paek is a research associate at the charles a. dana center, university of texas  at  austin,  2901  north  ih­35,  suite  2.200,  austin,  tx  78722;  e­mail:  pame­  la.paek@mail.utexas.edu.  her  interests  include  developing  practitioner–researcher  partnerships,  secondary mathematics, the impact of policy and practice on teaching and learning, and issues of  equity and access in education.  practices worthy of attention:  a search for existence proofs of promising  practitioner work in secondary mathematics  pamela l. paek  university of texas at austin  the goal of the practices worthy of attention (pwoa) project was to surface in­  novative practices currently in use by urban schools and districts that show prom­  ise of improving students’ secondary mathematics performance. each school and  district explored has a different perspective and a unique set of practices in place  to improve secondary mathematics achievement. the goal of this project was not  always to discover innovations in how practitioners address similar issues, but  rather to document what practitioners are doing to strengthen secondary mathe­  matics education. thus, although the practice highlighted might be commonplace,  the specific structures and strategies being employed by the school or district to  implement it are worthy of attention. a cross­case analysis of the 22 practices re­  vealed  two main categories:  raising  student achievement and building  teacher  capacity.  keywords: secondary mathematics, student achievement, teacher capacity  in the last half of 2006, i led a national search for practices in urban schools  and districts that show promise—on the basis of early evidence and observation—  of  increasing student  learning  in  secondary  mathematics. i call  these “practices  worthy of attention” (pwoa), and my work on them had three overarching goals:  1.  to better understand existing initiatives,  innovations, and programs that are being used to  improve secondary mathematics teaching and learning around the country, and mark these for  further scientific inquiry.  2.  to identify common themes in these practices that can be used to strengthen student achieve­  ment in urban systems across the country.  3.  to provide research support to help the practitioners more rigorously evaluate how well their  practices are working, which in turn can help to strengthen their methods of operation. http://education.gsu.edu/jume mailto:pamela.paek@mail.utexas.edu paek  practices worthy  journal of urban mathematics education vol. 1, no. 1  85  theoretical framework and connection to the literature  recent federal and state education policies call for a substantial increase in  the breadth and depth of mathematical knowledge that students must acquire in  order to graduate from high school. for example, a growing number of states that  once  required  knowledge  only  of  middle­school­level  mathematics  for  high  school graduation have, over the past 5 to 7 years, begun to require that all stu­  dents demonstrate mastery of algebra i and geometry content (center on educa­  tion policy, 2006). to give students opportunities to take higher­level mathemat­  ics courses in high school, which will better prepare them for mathematics in their  postsecondary lives, many states and districts have policies encouraging students  to take algebra i in the 8th grade. these policies have had an effect: the national  assessment of educational progress (naep) shows that  in 2000, only 27% of  eighth­grade students nationwide took algebra i, whereas by 2005, 42% of eighth  graders nationwide had taken algebra i (mathews, 2007).  outside of policy requirements,  improving  student access to and achieve­  ment in mathematics is important because students’ performance in middle school  and  high  school  mathematics  correlates  with  their  overall  academic  success  in  high school and beyond. the national educational longitudinal study (nels)  indicated that students who took rigorous high school mathematics courses were  much more likely to go to college than those who did not take such courses (u.s.  department  of  education,  1997).  research  suggests  that  specific  mathematics  courses,  like  algebra  i,  serve  as  gatekeepers  to  more  advanced  mathematics  courses and can affect mathematics enrollment and achievement in high school,  which in turn affects enrollment in college and completion of a four­year degree  (adelman, 2006; ma, 2001). the nels study showed that 83% of students who  took algebra i and geometry enrolled in college within 2 years of graduating from  high school, whereas only 36% of those who did not take these courses enrolled  in college. therefore, understanding the factors that contribute to improved stu­  dent learning in algebra i and a successful transition to geometry is a critical first  step toward increasing the postsecondary opportunities available to students.  unfortunately, few school districts  in the nation have the capacity to help  their students meet these rigorous mathematics requirements. national­ and state­  level reports document a critical shortage in the supply of appropriately trained  and certified mathematics teachers as well as a high rate of attrition among those  teachers, especially in urban areas (national science board, 2006). many second­  ary mathematics teachers lack deep knowledge of the mathematics content they  are expected to teach (barth & haycock, 2004; massell, 1998). in fact, ingersoll  (1999) found that a third of all secondary school teachers of mathematics nation­  wide had neither a major nor a minor in mathematics. moreover, research shows  inconsistencies in instruction across classrooms within the same district and even paek  practices worthy  journal of urban mathematics education vol. 1, no. 1  86  within the same school. teachers interpret the same instructional ideas in various  ways (marzano, 2003; stigler & hiebert, 1998, 1999) and accordingly make in­  dependent decisions about whether to ignore, adapt, or adopt policymakers’ rec­  ommendations for instruction (spillane, reiser, & reimer, 2002).  in  urban  districts  faced  with  these  and  other  difficult  issues—including  heavy turnover among administrators, administrators who do not understand what  is needed to support a high level of mathematics learning, and low expectations  for student performance from both teachers and administrators—mathematics in­  struction has proven very difficult to improve (bamburg, 1994; beck­winchatz &  barge, 2003; tauber, 1997). as a result, all too often, students in urban school  districts are not given adequate opportunity to enroll and succeed in challenging  mathematics courses in their secondary years (national science board, 2006).  the pwoa project was  inspired by these challenges and  by the need for  education systems to invest resources wisely. thus began the work of identifying  practices in secondary mathematics education that might merit further attention,  greater investment, and wider dissemination.  defining practices worthy of attention  research on pwoa differs from other work describing “best practices” or  “promising practices” in that the pwoa work starts from where schools and dis­  tricts presently are, focusing on work and ideas currently in progress. starting by  investigating practices that have not yet been identified as “best” or “promising”  through specific national criteria, such as those of the what works clearinghouse  or the national center for educational achievement, means that there is often lit­  tle or no documentation of how a practice is being implemented and scarce evi­  dence  of  the  practice’s  impact  or  effectiveness.  therefore,  the  first  step  in  re­  searching a practice is spending time with the practitioners in each school or dis­  trict to discover the theory­of­action behind the practice and to document the im­  plementation of the practice and the evidence of its effectiveness so far. this step  not only provides a historical record of activities, but also honors the work, giving  practitioners  a  chance  to  see  their  ideas  and  efforts  documented  in  a way  that  shows a picture of the work to date. this step also provides a starting point for  researchers to continue to work with practitioners to better measure the effects of  the practices on secondary mathematics teaching and learning.  developing methods to accurately and comprehensively measure and assess  the impacts of these practices on mathematics teaching and learning helps to meet  a current need of urban districts and schools. ironically, just as policymakers and  district leaders are looking to raise the evidentiary standard for adopting a school  improvement practice, the size of district offices—including that of their research  and evaluation staff—is being greatly curtailed, or staff is being diverted to deal paek  practices worthy  journal of urban mathematics education vol. 1, no. 1  87  with  the  reporting  exigencies  related  to  no  child  left  behind  act  of  2001  (nclb). 1 thus, many urban districts do not have the staff and financial resources  to clearly determine what data are needed by each person in the system and how  such data can be used. most  important, these districts have not yet worked out  how to translate the knowledge gained from the data into effective decision mak­  ing at each level of the education system.  methods  initial selection of programs, schools, and districts  the  first  step  in  the  pwoa  project  was  to  interview  administrators  and  teachers at schools and districts across the united states that embody diverse edu­  cational systems but that primarily serve students classified as economically dis­  advantaged and/or as racial and ethnic minorities. i contacted networks of mathe­  matics leaders and teachers known to staff at our institution and partner institution  and drew on my knowledge of schools and districts to develop an initial pool of  administrators and teachers to interview regarding practices in their schools and  districts that were potentially worthy of further examination.  protocol for initial interviews  at the june 2006 urban mathematics leadership network (umln) 2  meet­  ing, i used a four­question protocol to interview mathematics administrators from  the 12 participating umln districts. the interview protocol included the follow­  ing definition of what constitutes a practice worthy of attention:  a practice worthy of attention (pwoa) is a practice being used in your district that shows  promise of improving mathematics education within your district and across districts. the  pwoa  i  seek  specifically  look  at  the  grade range  of  middle  school  through  college.  a  pwoa is an example of how you have solved problems or challenges your district faced,  ideally with tools that measure the effects of change.  the first question asked the administrators which practices they would no­  minate  for their district. the second asked what types of documentation of  the  practice existed (e.g., training protocols or documents describing school or district  1 no child left behind act of 2001, public law 107­110, 20 u.s.c., §390 et seq.  2  umln serves as a vehicle for rapid dissemination of advances and promising practices, and  enables state mathematics leaders and the leaders of large urban districts to work together to better  align their mathematics improvement efforts and thus raise student achievement. paek  practices worthy  journal of urban mathematics education vol. 1, no. 1  88  initiatives) and what evidence was used to show the effectiveness of the practice  (e.g., school/district evaluations of students and/or teachers, third­party evaluator  reports, improvement in test scores). the final two questions were logistical, con­  cerning scheduling a site visit and establishing a contact person at the school or  district.  follow­up phone interviews with umln district staff were conducted with  two main goals in mind: (1) to get more details about the nominated practice, in­  cluding documentation or evidence of effectiveness available to date, and (2) to  schedule a site visit. on the basis of these interviews, eight practices were chosen  for further investigation.  for  non­umln  schools  and  districts,  most  initial  interviews  were  con­  ducted by phone, although  in a handful of cases, i was able to learn about the  practices  by  attending  presentations  on  them  at  conferences.  the  protocol  for  these phone interviews was a combination of the two protocols already discussed.  ultimately, i gathered information on about 30 programs, schools, and dis­  tricts, and scheduled site visits with 22 of them.  the remaining eight were not  followed up on either because the practice did not fit the goals of the project or  because the site did not respond to requests for a visit.  site visits and profiles of the practices  i visited most of the 22 sites to develop a fuller picture of how the practices  were actually being implemented and evaluated. during most of these visits, i at­  tended a professional development workshop centered on the practice being stu­  died; this allowed me to get more detailed information about the practice by wit­  nessing how schools and districts were explaining and teaching it. the visits also  included time to talk further with the person interviewed on the phone and the op­  portunity to gather any materials related to the practice. i also had informal, face­  to­face conversations with other staff members to learn what they thought about  the practices. for a few sites, an actual visit was not feasible, but enough informa­  tion about the sites’ practices was available to write a profile, with feedback from  the district or program to ensure that the profile correctly reflected the practice.  practices that exemplify these categories are described next in two separate cross­  case analyses, with snapshots of each practice.  results  on the basis of the site visits, the interviews with teachers and campus and  district leaders, and the documentation of the practices, i concluded that the inno­  vations within the practices could be classified into one of two main categories: paek  practices worthy  journal of urban mathematics education vol. 1, no. 1  89  (1) approaches to raising student achievement and improving student learning in  mathematics, and (2) approaches to increasing teacher capacity.  raising student achievement through academic intensification  all  of  the schools,  districts,  and  programs  profiled  in  this  study  have  in­  creased their expectations for student achievement, but some of them focused par­  ticularly on academic intensification strategies to help students meet the higher  expectations. the types of practices that emerged in support of academic intensi­  fication  include:  building  summer  bridge  programs,  requiring  and  supporting  more rigorous mathematics courses, and providing intense and ongoing support  throughout the school day.  summer bridge programs  two of  the practices  deemed worthy  of  attention  involve  summer  bridge  programs, which help students transition from middle school to high school ma­  thematics:  the academic youth development (ayd) initiative and step up to  high  school  (a chicago  public  schools  program).  these programs  are  not re­  medial  programs;  rather,  they  focus  on  developing  problem­solving  skills  that  form a foundation for success in algebra i. both programs are based on the dem­  onstrated efficacy  of  social  interventions  on  student  engagement  and  academic  success. step up to high school, for example, models its format on the emerging  scholars  program,  a college­level  program  developed to  improve  minority  and  female participation in mathematics.  academic youth  development  (ayd)  is  an  algebra i  readiness  program  being implemented by many urban districts  in the united states (e.g., chicago,  atlanta, new york city) that focuses on helping students better understand con­  tent by presenting it from multiple perspectives and applying it in real­life situa­  tions. at the heart of ayd is a 3­week transitional summer school and yearlong  follow­up program. rather than focus on the behavior of all students, the initiative  focuses on the beliefs, attitudes, and behavior of a cadre of student allies upon  whom the algebra teachers can rely to model respectful engagement and academic  success and thus help shape the classroom culture during the regular school year.  teachers nominate for the program students who not only are at risk of fail­  ing a future algebra i course but also who have good attendance and show poten­  tial leadership skills. in addition to mathematics problem solving, ayd concen­  trates on teaching students persistence and giving them the power to be in charge  of their own learning. for instance, students who view intelligence as a factor that  can  be  improved  with  learning  and  habits  of  mind  are  more  likely  to  persist  through initial failure (dweck, 2002). ayd gives students information about the paek  practices worthy  journal of urban mathematics education vol. 1, no. 1  90  changing nature of intelligence and encourages them to see failure not as a sign  that they cannot learn, but as a signal to change strategy.  step up to high school, in the chicago public schools, is a 4­week literacy  and  mathematics  program  for  students  in  the  summer  before  their  ninth­grade  year.  step  up  targets  students  who  are  likely  to  be  overlooked  by  other  pro­  grams—their low test scores indicate that they are at risk for academic failure as  they transition into high school, but their scores are not quite low enough for them  to be placed automatically in other academic support programs.  in addition to building the academic skills in reading and mathematics that  are key to high school success, step up focuses on helping students build teach­  er–student relationships and student–student relationships around shared academ­  ic  interests.  step  up  includes  orientation  seminars  and  activities,  information  about high school resources, and discussions of study skills, such as organization  and time management. students attend step up at the high school they will attend  and are taught by teachers, who teach at that school in the regular academic year,  ideally by the teacher who will be their first­year algebra teacher. this arrange­  ment gives the students the opportunity to meet teachers and classmates before  high school begins and to learn to navigate through their new physical surround­  ings.  both ayd and step up to high school show promise for improving teach­  ers’ understanding of student learning processes and for supporting students’ ma­  thematical learning and academic engagement. pre­ and post­surveys in both pro­  grams show gains in students’ confidence about their ability to do well in chal­  lenging academic courses.  more rigorous course requirements  three sites profiled  in this study  set specific course completion goals  for  their  students  and  then  backward­mapped  the  curriculum  to  better  prepare stu­  dents on the strands and topics they would later be required to know. each site  also found ways to support students and help them do well in the more advanced  courses. el paso collaborative for academic excellence (epcae) has built and im­  plemented a cohesive k–16 mathematics program for all 12 of the school districts  it serves in the greater el paso, texas area. epcae leaders realized that if stu­  dents could successfully complete algebra ii in high school, they could usually  avoid  remedial  mathematics  courses  in  college  and  enter  college  algebra  fully  prepared.  large­scale collaborative effort: the 12 districts that epcae serves colla­  borate with the local community college, the local four­year university, and the  entire el  paso  community  in  an  effort  to  achieve coherence  in  their  curricula, paek  practices worthy  journal of urban mathematics education vol. 1, no. 1  91  promote success for students past high school graduation, and establish a common  vision for a k–16 effort.  curriculum alignment: epcae formed a k–16 mathematics alignment initi­  ative composed of mathematics educators—elementary, middle, and high school  teachers and college  and university  faculty—who spent 2 ½  years producing  a  curricular framework that aligned high school and first­year college mathematics.  this group then backward­mapped the curriculum to prepare students for success­  fully completing algebra ii before high school graduation. after the curriculum  frameworks were developed, epcae provided teachers with professional devel­  opment to use the frameworks as the foundation for algebra ii in high schools.  grant high school in portland, oregon set the goal of having all students  pass geometry by their sophomore year of high school. the school’s mathematics  teachers set this goal themselves when they became frustrated with what seemed  like two schools within one building—one in which students who were predomi­  nantly  racial  or  ethnic  minorities  took  the  pre­algebra  courses,  and  another  in  which predominantly white students  took the precalculus courses. the teachers  felt that this unequal access to higher­level mathematics courses would limit some  students’ postsecondary opportunities. four teachers developed an intensive ma­  thematics program, and the school started a freshman academies program to help  students transition successfully into high school.  intensive mathematics program: grant’s  intensive mathematics program is  for students who enter high school behind in mathematics. teachers  intensified  mathematics instruction by providing double periods of mathematics for 2 years,  in effect giving the students 3 years of mathematics—pre­algebra, algebra i, and  geometry—in just 2 years, beginning in their freshman year. one goal of the 2­  year  program  is  to  allow  students  to  have  the  same  mathematics  teacher  both  years. this arrangement has helped teachers create a culture of learning and sup­  port that students can benefit from in their two periods of mathematics and in their  first 2 years of high school.  norfolk public schools  in norfolk, virginia want to ensure that their stu­  dents have every opportunity not only to take geometry in high school, but also  algebra ii and other higher­level mathematics. school leaders believe that getting  students  through  algebra i  earlier—in  8th grade—creates  greater opportunities  for students to take and excel in the higher­level courses in high school. the dis­  trict developed the algebra for all project, which requires students to take and  pass an algebra i course and the state’s end­of­course algebra exam in 8th grade.  norfolk knew that the project could not only consist of changing enrollment pat­  terns, but also needed to involve an improvement in the quality of mathematics  instruction. to that end, the district is focusing on curriculum and extending the  time students spend working on mathematics each day. paek  practices worthy  journal of urban mathematics education vol. 1, no. 1  92  curriculum: norfolk focused on vertical articulation and coherence of ma­  thematics across grades. the district realized that a foundation of algebra content  was  needed  in  all  grades  preceding  algebra  i.  mathematics  content  staff  inte­  grated algebraic reasoning across all topics in the k–7 curriculum in a coherent  content  strand  involving  patterns,  functions,  and  algebra.  the  new  articulation  ensures a progression of concepts, so that when students reach algebra i, they are  prepared with basic algebraic ideas and concepts.  extended  instructional  time:  mathematics  is  taught  for  a  minimum  of  90  minutes per day at all grade levels. the district provides teachers with an instruc­  tional manual that shows how they can use those 90 minutes to fully engage stu­  dents in learning mathematics. teachers also help students learn mathematics in  “academic success sessions” during the school day or after school.  epcae, grant, and norfolk all show promise in helping students meet ri­  gorous mathematics course requirements. epcae has seen an increased number  of students enrolling and passing algebra ii as well as increased graduation rates.  at grant, the enrollment of black students in algebra ii has increased from 8.9%  to 17.9% since the first cohort completed the intensive 2­year course; 100% of  students in the 2­year course plan to enter college. in norfolk, the percentage of  middle school students enrolled in algebra i has increased, as has the percentage  of students passing the course and exam: from 41% to 69%.  embedded student support within the school day  schools and districts that engage in academic intensification must find ways  to support students who come to the mathematics classroom with diverse expe­  riences. two small schools, eastside college preparatory school and high tech  high, have found ways to embed such student support in the daily schedule as a  regular part of students’ schooling. this scheduling is especially important given  that most students do not arrive at eastside and high tech adequately prepared  for  high  school.  in  these schools,  a  low  student­to­teacher  ratio  helps  teachers  give students more individualized attention, and the school culture includes plan­  ning  for  college  as  a  regular  part  of  students’  schooling.  larger  schools,  like  evanston township high school, are challenged by  large  classrooms and  high  student­to­teacher ratios, so these schools must rely on strategies like tutorial pro­  grams and extra time for mathematics instruction.  eastside college preparatory school in east palo alto, california is an in­  dependent school serving students in grades 6–12 from populations that are his­  torically underrepresented  in  higher education. enrollment  is  just over 200 stu­  dents. eastside’s goal is to provide a strong, student­centered academic environ­  ment and requires that, at a minimum, students have completed precalculus before  graduating. students receive various forms of support that are embedded into the paek  practices worthy  journal of urban mathematics education vol. 1, no. 1  93  school day, including daily tutorials and individual advising, to meet these high  expectations.  tutorials: two 90­minute tutorials are  built  into the school day to ensure  that students are getting support to understand the core course content (english  and mathematics) and are completing their homework. the tutorial sessions come  immediately after the targeted course and are led by the same teacher, who tutors  around 20 students. the tutoring sessions ensure that students receive timely help  on concepts and ideas. the framework ensures consistency of instruction and al­  lows teachers extra time to work with students who are struggling and to provide  more intensive opportunities for students to engage with the academic content.  advisory  system:  students  meet  daily  with  an  advisor,  who  works  with  them specifically on their personal and academic challenges and issues. advisors  are teachers assigned to a group of 6–8 students with whom they work closely  over the 4 years of high school. advisors also provide students with resources for  extracurricular activities that can help support their academic interests and portfo­  lios for applying to college.  academic support: additional academic courses that focus on reasoning and  analytical skills as well as topics in college admission and transitioning to college  are required. these courses provide students with a strong foundation in the skills  and habits that are necessary for academic success in high school and beyond.  high tech high (hth) in san diego, california is a charter school that fo­  cuses  on  solutions  for  dealing  with  student  disengagement  and  low  academic  achievement. the school develops personalized, project­based learning environ­  ments and expects all students to graduate well prepared for college. the school’s  enrollment  is  just over 500 students. hth encourages student  learning through  project­based learning and close work with advisors and mentors.  project­based  learning:  hth  offers  hands­on  experiences  in  mathematics  through project­based learning. after mathematics teachers provide a lesson and  tasks for students to engage in, students break into small learning groups to work  on projects that require them to apply the mathematics concept to a hands­on ac­  tivity. because the classrooms are grouped by grade level and students come in  with differing levels of mathematical proficiency, classes are taught in ways that  cover the span of several mathematics courses; for example, algebra i, geometry,  and calculus are taught in the same class, and the teacher focuses on a mathemat­  ics strand and differentiates the difficulty in the project activity for students. stu­  dents work within and across groups to gain advice and input for their projects,  and the teachers check  in with each group to monitor the projects and provide  support and guidance as needed.  advising: the advisory program was designed to support students in their  academic preparation for college. each hth student  is assigned a staff advisor  who also acts as a liaison to the student’s family, so parents are aware of their paek  practices worthy  journal of urban mathematics education vol. 1, no. 1  94  child’s growth and challenges at hth. advisors work closely with students  to  help them plan for their futures, navigate the college admissions process, and ap­  ply for financial aid and scholarships.  internships: beginning in their junior year, students work as interns two af­  ternoons a week for at least one semester at local businesses, schools, nonprofit  organizations,  or  professional  associations.  each  student  works  on  a  specific  project overseen by a mentor who understands and supports hth’s design prin­  ciples and works individually with the student to cultivate a productive learning  experience  that  exemplifies  the  project­based  learning  in  school  in  an  actual  work­related setting.  evanston township high school in evanston, illinois has an enrollment of  over 3,100 students. the school is working on building student success in algebra  i and has taken steps to ensure that students receive daily, individual support in  mathematics.  intensive daily support: algebra i classes are structured to provide more in­  structional time for all students. students work in small groups to discuss an idea  and then share their findings with the whole class; students feel comfortable ask­  ing questions of each other and of the teachers when they do not understand a  concept. students in upper­level mathematics courses have been recruited to assist  in algebra i classes, helping students understand concepts and serving as teach­  ers’ aides. in addition, to make sure struggling students receive support, the chair  of the mathematics department meets individually with students who have failing  grades to discuss their performance and talk about what kind of help they need.  algebra i teachers also have 30 minutes each morning to work with struggling  students.  eastside,  high  tech,  and  evanston  township  all  have programs  in  place  that show promise for supporting students on a daily basis to ensure their long­  term success. in the two small schools that mainly serve economically disadvan­  taged,  first­generation  college­bound  students,  100%  of  students  graduate  high  school and enroll in four­year universities. in evanston, students are passing al­  gebra i at higher rates.  summary for raising student achievement  raising student achievement requires changes in the attitudes and practices  of  administrators,  teachers,  and  students.  in  summer  bridge programs,  students  learn about the value of academic effort and build peer and teacher relationships  that will support them throughout high school. success in these programs necessi­  tates firm belief on the part of teachers that their students really can succeed in  high school mathematics and that collegial student peer groups can be a strong  support for that success. requiring rigorous courses of all students demands both paek  practices worthy  journal of urban mathematics education vol. 1, no. 1  95  a change in how districts and schools think about student ability and much more  support for both students and teachers. intense, embedded daily support, for ex­  ample, constantly reiterates the idea that mathematics is important and that, with  hard work and a strong network of teacher and peer support, all students can take  and pass rigorous mathematics courses.  building teacher capacity  all  of  the schools,  districts,  and  programs  profiled  in  this  study  have  in­  creased their expectations for what teachers should do, but some of them have fo­  cused intense attention on improving teacher practices. the practices designed to  build  teacher  capacity  provide  opportunities  for  teachers  to  interact  with  other  teachers  in  focused and specific ways, share knowledge, and thus  improve and  expand their current practices. the practices designed to build capacity also in­  creased  individual support for  teachers and expanded their access to resources.  these practices require support from administrators if the traditional ways teach­  ers have interacted are to be overcome. as teachers are asked to support students  with various experiences and backgrounds, districts and schools are asked to sup­  port teachers the same way, instead of providing all teachers the same training and  expecting all of them to perform the same way. three main approaches to build­  ing capacity emerged: redefining mathematics teacher roles and responsibilities,  making instruction public, and having new, customizable tools for teaching.  redefining mathematics teacher roles and responsibilities  four  districts  focused on  broadening  the sphere  of  mathematics  teachers’  roles and responsibilities in two main ways: by improving the teaching of specific  subpopulations and by increasing teacher participation at the district level.  improving  teaching  for  specific  subpopulations.  in  new  york  city  and  denver  public  schools,  mathematics  teachers  work  closely  with  teachers  who  specialize in teaching students with special needs, learning how to maintain rigor­  ous content standards while supporting students  learning english or students  in  special  education.  the  practices  encourage  good  teaching  by  focusing  on  the  types of instructional tasks that teachers can use for differentiating instruction to  meet the diverse needs of students, encouraging the use of academic vocabulary,  and  providing  various  entry  points  for  students  to  learn  the  mathematical  con­  cepts. these practices also provide teachers with feedback on specific ways that  some students may struggle as a result of language acquisition issues or cognitive  impairment.  denver public schools developed a collaboration between mathematics and  special  education  teachers.  the district  believes  that  special  education  teachers paek  practices worthy  journal of urban mathematics education vol. 1, no. 1  96  often do not have expertise  in  mathematics and  thus have difficulty supporting  their students  in higher­level mathematics. mathematics teachers do not always  know how to accommodate special education students’ individualized education  plans without “dumbing down” the mathematics content. denver saw a need to  broaden  teachers’  roles  by  having  mathematics  and  special  education  teachers  work together to best support all of their students in secondary mathematics.  in denver’s program, teachers are matched in pairs (one special education  teacher and one  mathematics  teacher)  for  the academic  year. the whole group  meets about every 6 weeks. in each meeting, each pair of teachers writes a single  mathematics  lesson plan, working together  to build  in accessibility and accom­  modations to address the range of their students’ individual challenges and needs.  the goal is for teachers to maintain the integrity of the mathematics while also  following a process for planning accessibility strategies that address learning bar­  riers. to make their work concrete, the teachers each choose three students who  represent a range of mathematical abilities and write their lessons with those stu­  dents in mind. built into each meeting are opportunities for teachers to reflect on  their use of specific strategies and share their goals and cautions regarding acces­  sibility strategies. this type of sharing builds a supportive group that shares ideas  and actual practices in the field, giving the teachers a common set of goals to aim  for and cautions to keep in mind.  new  york  city  department  of  education  created  the  english  language  learners (ell) mathematics initiative to raise the academic achievement of ell  students through a strong network of district and school­based mathematics and  ell leaders. the initiative is designed to raise the quality of mathematics instruc­  tion while providing for the diverse needs of students with various language and  academic backgrounds.  at the core of the initiative is a professional development program for ma­  thematics teachers that emphasizes techniques specifically geared to teaching stu­  dents whose first language is not english. at the core of the program is the belief  that mathematics is not “language­neutral”—meaning that mathematics pedagogy  depends on the language of instruction—and therefore the professional develop­  ment  opportunities  focus  on  how teachers  must  teach  in  ways  that  incorporate  students’ native languages, english, and academic mathematics language.  teachers  are  trained  in  wested’s  quality  teaching  for  english  learners  (qtel), which  helps them develop a theoretical  foundation and corresponding  strategies for effectively teaching academic language to ell students. the tools  and processes taught in professional development modules focus on developing  adolescent students’ abilities to read, write, and discuss academic texts in english.  reflection activities for teachers provide opportunities to think about past lessons  and plan how to address specific challenges. teachers also analyze case studies  and  videos  that  show  a  range of  teaching  styles,  in  order  to  better  understand paek  practices worthy  journal of urban mathematics education vol. 1, no. 1  97  some obstacles to their own as well as their students’ understanding. additionally,  teachers are asked to develop resources and  lesson plans and to problem­solve  specific teaching and learning situations.  district roles and responsibilities. in lamoille south supervisory union and  portland public schools, mathematics teachers are taking on leadership roles and  working  with  district  leaders  to  learn  more  about  specific district  mathematics  needs;  this  in  turn  improves  their  own  practices.  in  the  partnership  for  high  achievement, district leaders and teachers work to communicate common goals  and sustain them with concrete steps for improving classroom practices.  lamoille south supervisory union in morrisville, vermont consists of three  school districts serving  students  in grades  k–12. lssu is creating a  local,  ba­  lanced assessment system in mathematics that is aligned with the k–12 curricu­  lum. to support that work, teachers’ responsibilities now include developing as­  sessments at the district  level. teachers receive training in assessment develop­  ment and assessment for learning, which helps them understand how assessment  can provide the information they need to improve their practices.  lssu incorporates the use of ongoing and embedded professional develop­  ment structures that broaden teachers’ knowledge and understanding of the devel­  opment, use, and analysis of assessment. lssu leaders involve teachers in writing  assessment items because they believe that, to affect instruction at the classroom  level, teachers need to understand what is expected at the district level. they also  believe that teachers need to be involved in the kinds of conversations that help  them reflect on their practice.  as they develop assessment  items, teachers talk about different types and  uses  of  assessments  (formative,  benchmark,  and  summative),  learning  how  to  make judgments about student learning depending on the type of student work or  data they  have  available.  in  addition,  given  that  teachers use  the  same  assess­  ments, they can collaborate to analyze the results and then plan interventions and  modifications together.  portland public schools in portland, oregon has developed a set of district­  level leadership opportunities for all interested mathematics teachers. the district  mathematics  specialists  believe  that  developing  local  leaders  at  each  school  as  agents of change is the most effective way to sustain a common set of mathemat­  ics goals across the district. they hope that this leadership development will in­  crease teacher capacity at each school and lead to better and more consistent ma­  thematics  teaching  so  that  students  have  equal  opportunities  for  mathematics  achievement.  leadership opportunities are organized within a large group of teachers and  district mathematics specialists. each year, the large group divides into subgroups  focused on different ways of approaching mathematics education improvement.  one year, the topics the subgroups focused on were determining the content for a paek  practices worthy  journal of urban mathematics education vol. 1, no. 1  98  new, third  year of high school  mathematics graduation requirement; supporting  the transition of students from eighth­grade to high school mathematics; and de­  veloping and piloting districtwide common formative assessments in grades 6–8.  the next year, the third­year math and transition to high school topics remained,  and two new topics were added, one focused on implementing the college prepa­  ratory mathematics program and the other on using technology in mathematics  classrooms.  the subgroups  and  topics  change shape  as  the responsibilities  and  needs of teachers change.  the  subgroups  generate  guidelines  for  interaction  to  support  individual  teacher voices and develop a clear set of steps to meet goals. teachers volunteer  to facilitate monthly meetings, and the district mathematics specialists help them  plan the agendas. in their teacher­leader roles, teachers feel they have the power  to make a difference beyond their own classrooms, and leading and participating  in these district­level groups is a way for them to be directly involved in district  improvement in student mathematics learning.  the  partnership  for  high  achievement  (pha)  is  a  program  designed  to  strengthen the capacity of leaders and teachers in texas school districts to imple­  ment a research­based instructional support model to continuously improve teach­  ing and  learning. the  model  integrates  leadership development  for department,  school, and district leaders with support for classroom teacher development.  pha’s strategy is to provide technical assistance and professional develop­  ment  to a district’s  teachers and  leaders  to support the district  in ensuring that  every student has access to the same curriculum. to implement this strategy, a  leadership advisor works with the district leadership team, and a mathematics ad­  visor works with designated teacher teams. the advisors teach district leaders and  teachers about the instructional support model and how to implement it, and pro­  vide supplementary resources based on the unique needs of the district. the advi­  sors  work  with  the  district  leadership  team  and  teacher  teams  throughout  the  school year to ensure that the elements of the instructional support model are ac­  complished.  in denver, new york, lamoille south, portland, and pha­partnered dis­  tricts, the broadened teacher roles and responsibilities promise to increase teacher  skill sets and renew investment in student learning. certainly, the teachers seem to  be embracing their new roles. in denver, reflective feedback collected from the  participating teachers indicates that they are learning more about content and im­  proving their teaching strategies. new york city teachers appear receptive to im­  proving their practice to accommodate ell students. in lssu, teachers are hav­  ing epiphanies about the role of assessment in learning and are eagerly engaging  with one another and their students. in portland public schools, 36% of secondary  mathematics teachers are involved in a mathematics leadership subgroup. in pha, paek  practices worthy  journal of urban mathematics education vol. 1, no. 1  99  participating districts’ mathematics and science scores have gone from below the  texas average to above the texas average.  making instruction public  deprivatizing instruction, or making instruction public, is a powerful means  for  changing  teacher  practice.  this  process  requires  teachers  to  open  up  their  classrooms, trusting that observers are not evaluating them but are providing val­  uable feedback to help them reflect on their practices. making instruction public  allows  teaching  and  learning  to  be  captured  in  multiple  ways  from  multiple  sources,  giving  teachers  regular  feedback  so  they  can  continually  work  on  im­  proving their teaching. three districts and one multi­district initiative have made  open classrooms a major part of their mathematics improvement plans.  bellevue school district in bellevue, washington has set the goal of “get­  ting rid of walls of classrooms” and building a culture of openness and sharing  among teachers and the district mathematics curriculum coaches. the curriculum  coaches observe classrooms, learn what teachers are doing successfully, share the  successful practices with all  mathematics  teachers, and help teachers with their  concerns and challenges. although some teachers were defensive at first, feeling  that observations were a threat to their autonomy, they soon saw the value in shar­  ing their successful practices, especially when they were working together toward  the same goals.  bellevue further encourages collaboration by sharing among teachers the re­  sults of common assessments, so that teachers can see how all students are per­  forming on the same types of tasks and discuss how their practices contributed to  their students’ performance. the district develops common assessments for every  unit at every grade level, and teachers are required to administer the assessments,  score students’ work, and post results on the district’s intranet. with assessment  results accessible to the entire professional community in bellevue, the hope is  that teachers will seek out and share best practices with each other in the ongoing  effort to improve work with students.  further, the operations and results of teacher practices are available to great­  er numbers of people, including parents, because the district requires all teachers  to have a classroom website that includes the course syllabus and/or grade­level  goals and expectations. the website also includes online access to grades.  columbus public schools in ohio has made classrooms public by instituting  a peer observation program for teachers. at each school, a teacher leader, trained  at the district level to support professional learning communities, conducts weekly  meetings to help other teachers work as a team to address challenges. most of the  time in these meetings is spent developing specific strategies for addressing stu­  dent needs, but the work also involves reviewing progress on school­specific ac­ paek  practices worthy  journal of urban mathematics education vol. 1, no. 1  100  tion plans, student testing results, and teacher­student survey results. these meet­  ings have helped encourage teachers to stop working in isolation and to open their  classrooms and their practices to observation.  teacher leaders have developed and refined a data collection tool they use  in observing classrooms and collecting information about instructional strategies.  the teacher leaders use the data they collect to promote discussions with teachers  about how to learn from these observation experiences; the culture surrounding  these discussions is collaborative, not evaluative.  principals observe classrooms to see if there is systematic use of the stan­  dards­based mathematics curriculum guides. most principals do classroom walk­  throughs daily, as required by the district. the principals have been trained to ask  reflective questions of teachers and have also learned how to focus on what they  should  be  seeing  in  mathematics  classrooms.  district­level  administrators  also  visit classrooms, and several mathematics curriculum specialists spend at least a  half­day per week visiting schools and monitoring the implementation of the ma­  thematics curriculum.  yes college preparatory school in houston, texas has embedded into the  teaching culture a teacher feedback and evaluation system that  includes regular  observations by coaches, mentors, peers, and supervisors. this system supports  teachers with goal setting and reflection, providing feedback to improve teacher  practices throughout the school year as part of their ongoing professional devel­  opment. at the beginning of the year, teachers set goals, using a summative rubric as  a guide. the rubric covers four domains: classroom management and culture, in­  structional  planning  and  delivery,  yes  responsibilities,  and  yes  values.  each  domain  has  multiple  indicators, so observers rate teachers on each  indicator to  develop a composite domain rating. this detailed rubric helps observers identify  the areas in which teachers need the most assistance and support, which enables  them to customize mentoring and coaching to improve teacher pedagogy.  throughout the year, teachers receive feedback from their peers, from su­  pervisors, and from students. at the end of the school year, the summative rubric,  along  with  a  teacher’s  course  material,  progress  on  professional  development  goals, self­reflections, self­evaluations, administrator evaluations, student perfor­  mance, and student feedback, is used to evaluate the teacher’s performance.  phoenix union high school district in phoenix, arizona uses professional  learning communities to create a culture that focuses on how to change the way  teachers engage with students. teachers  in phoenix union began to change the  culture of their practice by opening their doors to peer review and learning from  one another about best strategies for improving student learning in mathematics.  when teachers opened their doors to each other, no teacher worked in isola­  tion. teachers began to share what worked well and went to one another for help paek  practices worthy  journal of urban mathematics education vol. 1, no. 1  101  when they struggled with a concept or topic. they make all student work public  so they can analyze what students really know and what they are struggling with.  teachers began to change their thinking about classroom observers, no longer as­  suming they were evaluative and critical;  instead, teachers learned ways of  im­  proving their practice through observation of their peers. these changes resulted  in more consistent instruction and assessment strategies across the district.  the district also asks teachers to work in teams to provide meaningful les­  sons and assessments that are congruent with the curriculum. although methods  for building lessons and assessments are discussed in teacher preservice and in­  service workshops, the teams allow teachers to help each other better understand  the  development  process  as  they  look  at  specific  instructional  examples,  re­  sources, and strategies. by developing and working with common lessons and as­  sessments, teachers can learn from one another and develop more consistent me­  thods of delivering instruction.  silicon  valley  mathematics  initiative  (svmi)  in  the  san  francisco  bay  area believes that the key to improving student achievement is improving instruc­  tion through intensive, hands­on professional development for individual teachers.  to that end, the initiative has mathematics coaches frequently observe classrooms  and  discuss  their  observations  with  teachers  one  on  one.  this  practice  makes  teachers’  instruction  open  to  outside  feedback  while  providing  a  structure  for  teachers to learn how to improve their instruction.  the main job of the coaches is to assist the teachers they work with to focus  on student thinking and mathematical pedagogy. coaches visit the classrooms of  each of their teachers about 20 times per year. the general structure of each visit  includes a pre­conference, observation of a lesson, and a post­conference. coach­  es  encourage teachers  to reflect on the  lesson,  examining  student  work  as  evi­  dence, to help inform and adjust future instruction.  the  mathematics  coaches  tend  to  relate  to  their  teachers  in  one  of  three  ways—as collaborators, models, or leaders. in the collaborator role, coaches are a  resource to the teacher, providing materials, information, and encouragement, and  collaborating with the teacher to plan lessons. in this role, coaches do not give  direct feedback about the teacher’s pedagogy, but focus more on student work,  which makes the teacher feel less defensive about being evaluated or criticized. in  the model role, coaches model instruction of deep problem­solving tasks for stu­  dents. teachers can use this model lesson as a guide for developing their future  lesson plans. as a leader, the coach guides the teacher in nonevaluative ways. for  instance, the coach’s comments are grounded in what was just observed—what  the teacher understood about how well the lesson went and what students seemed  to learn. the coach then assists the teacher in figuring out how to address the con­  tent the students did not seem to understand well. paek  practices worthy  journal of urban mathematics education vol. 1, no. 1  102  the various strategies for making instruction public practiced in columbus,  at the yes school in houston, in the phoenix union district, and svmi schools  are  helping  teachers  better  understand  their  own  practices  and  improve  their  teaching. teachers in these districts have found that deprivatized instruction en­  courages collaboration and allows them to support each other. in bellevue, teach­  ers are much more comfortable now sharing their information with each other and  with parents. in columbus, teachers indicated that the weekly meetings were use­  ful for establishing collaboration and consistency of instruction, and they are now  accustomed to regular visitors in their classrooms. at yes, all teachers are meet­  ing a minimum standard for providing quality teaching to their students. in phoe­  nix union, teachers have an open­door policy that fosters consistent observation  and learning from one another. teachers  involved in svmi coaching are using  evidence of what students have learned rather than anecdotal information to gauge  students’ understanding.  new tools for teaching  an issue in training teachers  in the use of new tools and resources is that  professional development is usually the same for all teachers in a given school or  field. the success of such strategies and tools, however, differs significantly in  different cases, because teachers come into professional development workshops  with different knowledge, experiences, and pedagogical practices. to remedy this  problem, one program and three districts provide customizable trainings to assist  teachers appropriate new tools and strategies to improve their teaching practices.  agile mind is an online tool that supports and models sustainable teaching  in secondary mathematics courses (from middle school mathematics through ap  calculus).  curricula are  aligned  to  state standards  in  the states  in  which  agile  mind is used, the national council for teachers of mathematics (nctm) stan­  dards, and various mathematics textbooks so that teachers can use agile mind to  support the textbooks they are required to use.  instructional resources are available for teachers to use in planning and deli­  vering instruction and effective assessment. each course includes several topics,  and within each topic, an online instructional guidance system provides teachers  with specific resources for instruction planning, teaching, assessment, addressing  various  teaching  challenges,  and  alignment  to  state  standards  and  textbooks.  teachers can use all of these resources or select specific ones. within each online  resource, teachers have the option of adding their own notes, which helps them  customize their practice.  agile  mind  provides  instructional  guidance  for  all  aspects  of  the  lesson,  from opening questions that enable teachers to introduce key concepts and engage  students in discussion to framing questions that support teachers in helping stu­ paek  practices worthy  journal of urban mathematics education vol. 1, no. 1  103  dents apply the lesson to real life. further questions are suggested to help probe  students’ thinking and to uncover misconceptions. teaching tips offer strategies  for dealing with possible challenges students  might  face. assessments are built  into each topic, with different types of reports available so teachers can review  both what the entire class understands and what individual students understand.  teachers are offered a range of resources they can use in secondary mathematics  courses, giving them the flexibility to choose the resources best suited to their in­  structional goals.  anchorage school district in alaska has developed its own assessment re­  porting system, a comprehensive database system that follows students longitudi­  nally with all the data that was previously kept in their paper cumulative folders.  the purpose of this system is to give teachers access to data on their students at  any time. for instance, if a student transfers to another teacher or school within  anchorage, that student’s data are immediately transferred electronically into the  new classroom, so teachers have up­to­date access to all the student information  they need.  data are available for  individual student performance on district and state  assessments across several years. while teachers can view their own classroom  data,  school  administrators  can  view  an  entire  school  or  any  classroom  within  their  assigned school.  the system  allows  the district  to  customize  professional  development opportunities to the needs of  individual teachers and schools. dis­  trict­level mathematics curriculum specialists work with individual teachers and  schools that have lower than average performance in the district.  the assessment reporting system allows users to sort students’ proficiency  on  various  mathematics  assessments  by  demographic  information  like  race/ethnicity and gender according to  the entire assessment or selected mathe­  matical strands. the four proficiency levels are color coded to give teachers a vis­  ual snapshot of where students need the most help, allowing them to target specif­  ic students struggling in each strand. the format of all data output has been cus­  tomized based on teachers’ requests, and the reports continue to be revised in re­  sponse to teacher  feedback. because the system  is  homegrown, not an off­the­  shelf  product,  anchorage  has  the  flexibility  to  further  customize  the system  to  improve its usefulness as a tool to inform teacher practices.  the assessment reporting system also  features a grade­level expectation  item bank. teachers can pull  items from this bank that are linked to the grade­  level expectations they are focusing on and use those items to develop customized  mini­assessments. the data from these items can then be used as part of the in­  structional cycle for measuring and improving student learning on different ma­  thematics expectations.  boston public schools’ secondary mathematics coaches use asset­based in­  struction to develop teacher capacity. asset­based instruction encourages teachers paek  practices worthy  journal of urban mathematics education vol. 1, no. 1  104  to focus on students’ strengths rather than on their deficits. coaches model the  asset­based approach for teachers by emphasizing instructional experiences they  observe that enhance teachers’ understanding of and competence in teaching ma­  thematics. this approach builds on teachers’ strengths, helping them see how they  can then use those same techniques to engage their students. the asset­based ap­  proach allows teachers to customize their instruction and allows coaches to cus­  tomize their approaches to teacher professional development. because coaching is  at the individual teacher level, coaches can customize the training to emphasize  what they believe a teacher needs to work on.  after observing a teacher’s classroom, a coach talks with the teacher about  student­centered coaching and the strategies teachers can use to take advantage of  the known strengths of each student and the class as a whole. the coach usually  focuses on the interaction of the teacher with a particular student to exemplify the  techniques. the teacher and coach discuss the importance of both affective and  cognitive experiences in helping motivate students, again from the perspective of  building on students’ strengths. they also talk about how to improve ability be­  liefs. together, the teacher and coach also identify patterns of students’ strengths  by analyzing student work and assessments. the coach reinforces how to motivate  students with genuine positive support and encouragement as often as possible.  the teacher and coach also identify places in the curriculum where students are  currently successful and map out a lesson that guarantees at least one successful  experience for each student.  cleveland  municipal  school  district  is  using  a  program  called  keeping  learning on track (klt) in its 10 lowest­performing k–8 schools. klt is a for­  mative assessment program developed by educational testing service. klt fo­  cuses on using evidence of  learning to adjust and customize instruction as  it  is  taking place so that teachers can immediately address students’ learning needs.  because teachers’ instructional styles vary, klt provides a variety of ways  for teachers to measure student learning on the fly, giving teachers the flexibility  to  choose  the strategies  that  best  allow them  to make  instructional  adaptations  right  at  that moment.  these  types  of  formative  assessment  checks  can  provide  teachers  the  feedback  they  need  to  change  their  daily  practice,  and  that  small  change might result in large changes in teacher pedagogy, the classroom culture,  and student learning.  teachers using  klt  meet regularly to reinforce  and  build upon the tech­  niques, strategies, and ideas behind the program. teachers use these meetings to  discuss the implementation of assessment­for­learning techniques in their class­  rooms and to refine their understanding of klt techniques.  agile mind and the practices in use in anchorage, boston, and cleveland all  show promise for improving teacher practices. agile mind users tend to increase  the implementation of the resources each year they use it, and schools tend to ex­ paek  practices worthy  journal of urban mathematics education vol. 1, no. 1  105  pand the courses that can be supported by it. in anchorage, teachers report that  they appreciate the assessment reporting system and use it to analyze and under­  stand how their  instruction affects student performance. in boston, teachers ap­  preciate the  individual coaching and  modeling they receive and recognize  how  asset­based  instruction  changes  the  culture  of  their  classrooms.  in  cleveland,  teachers  report  that  they  regularly  use  assessment­for­learning  techniques;  the  schools using klt have seen substantially greater gains in student achievement  than have non­klt schools.  summary for building teacher capacity  building teacher capacity requires changes  in district and  school attitudes  about  how to  best  support  teachers  as  they  improve  their  teaching.  with  broa­  dened roles and responsibilities, teachers redefine how they think of teaching and  what they can contribute. they learn that they can gain the expertise to work suc­  cessfully with subpopulations of students in need of their help, be part of a devel­  opment team for building common assessments at the district level, or participate  as leaders in the district for promoting change in mathematics. when instruction  is public, teachers learn about the power of collaboration for improving their prac­  tice and lose the fear of having observers in the classroom. with structured obser­  vation  protocols  and  regular  opportunities  for  feedback,  teachers  forget  about  working in isolation and focus more on the ways they can work together to im­  prove  student  achievement.  finally,  with  new  tools  and  customized  support,  teachers can access the individual training and feedback they need to make good  practices part of their daily instruction.  discussion and next steps  the practices i have identified address challenges that virtually all american  school  districts  must  face.  in  too  many  cases,  however,  school  districts  create  their  solutions  to these challenges  from  scratch  and  in  isolation.  the practices  worthy of attention project is designed to offer a more effective approach to col­  laborative learning and to the dissemination of creative solutions to difficult edu­  cational problems. to successfully tackle the challenges  faced by  all educators  and  leaders  in  improving  mathematics  teaching  and  learning,  researchers  must  spend more time in schools and districts, observing and analyzing how the broad  approaches and big ideas are actually codified, implemented, and assessed within  and across districts. this project is a first step toward creating a nationwide group  of practitioners who can share specific strategies with and learn from one another,  which  will  serve  to  open  doors  across  districts  much  as  classrooms  have  been  opened within schools. by taking the time to observe and evaluate actual practic­ paek  practices worthy  journal of urban mathematics education vol. 1, no. 1  106  es, researchers can find out directly how research is interpreted and implemented  and therefore advise mathematics leaders and teachers in ways that directly affect  their work.  the next phase of this work is to partner researchers with schools and dis­  tricts to raise the standards of evidence by which the researchers measure the ef­  fectiveness of these practices. this partnership will allow for the fulfillment of a  key purpose of this work: not only to identify common themes in these practices  that can be used to strengthen teachers’ practices and student achievement in ur­  ban systems across the country, but also to determine the effects of districts’ initi­  atives for improving teacher practices and, in turn, the effects of those practices  on students’ secondary mathematics progress and achievement.  references  adelman, c. (2006). the toolbox revisited: paths to degree completion from high school through  college. washington, dc: u.s. department of education.  bamburg, j. (1994). raising expectations to improve student learning. oak brook, il: north cen­  tral regional educational laboratory.  barth, p., & haycock, k. (2004). a core curriculum for all students. in r. kazis, j. vargas, & n.  hoffman (eds.). double the numbers: increasing postsecondary credentials for underre­  presented youth (pp. 35–45). cambridge, ma: harvard education press.  beck­winchatz, b., & barge, j. (2003). a new graduate space science course for urban elementary  and middle school teachers at depaul university in chicago. the astronomy education re­  view, 1(2), 120–128.  center on education policy. (2006). state high school exit exams: a challenging year. washing­  ton, dc: center on education policy.  dweck, c. s. (2002). messages that motivate: how praise molds students’ beliefs, motivation, and  performance (in surprising ways). in j. aronson (ed.), improving academic achievement:  impact of psychological factors on education (pp. 37–59). san diego, ca: academic press.  ingersoll, r. m. (1999). the problem of underqualified teachers in american secondary schools.  educational researcher, 28(2), 26–37.  ma, x. (2001). a longitudinal assessment of antecedent course work in mathematics and subse­  quent mathematical attainment. journal of educational research, 94, 16–28.  marzano, r. j. (2003). what works in schools: translating research into action. alexandria, va:  association for supervision and curriculum development.  massell,  d. (1998). state  strategies  for building  local capacity: addressing  the needs of  stan­  dards­based reforms. philadelphia, pa: center for policy research in education, universi­  ty of pennsylvania.  mathews, j. (2007, march 12). adding eighth­graders to the equation: portion of students taking  algebra before high school increases. washington post. retrieved march 14, 2007, from  http://www.washingtonpost.com.  national  science  board.  (2006,  january).  america’s  pressing  challenge:  building  a  stronger  foundation. nsb 06­02. washington, dc: national science board.  spillane, j., reiser, b., & reimer, t. (2002). policy implementation and cognition: reframing and  refocusing implementation research. review of educational research, 72, 387–431.  stigler, j. w., & hiebert, j. (1998, winter). teaching is a cultural activity. american educator.  retrieved march 14, 2007, from http://www.washingtonpost.com paek  practices worthy  journal of urban mathematics education vol. 1, no. 1  107  http://www.aft.org/pubs­reports/american_educator/winter98/index.html.  stigler, j. w., & hiebert, j. (1999). the teaching gap: best ideas from the world’s teachers for  improving education in the classroom. new york: free press.  tauber, r. (1997). self­fulfilling prophecy: a practical guide to its use in education. westport,  ct: praeger.  u.s. department of education. (1997). mathematics equals opportunity. washington, dc: u.s.  department of education. retrieved march 24, 2007, from http://www.ed.gov/pubs/math. http://www.aft.org/pubs-reports/american_educator/winter98/index.html http://www.ed.gov/pubs/math journal of urban mathematics education  december 2008, vol. 1, no. 1, pp. 60–83  ©jume. http://education.gsu.edu/jume  jacqueline leonard is an associate professor of mathematics education in the college of  education at temple university, 1301 cecil b. moore avenue, ritter hall 434, philadelphia, pa  19122; e­mail: jleo@temple.edu. her research interests are equity and access issues as they pertain  to mathematics education and teaching for social justice and cultural relevance in the mathematics  classroom.  brian r. evans is an assistant professor of mathematics education in the school of educa­  tion  at  pace  university,  163  william  street,  11th  floor,  new  york,  ny  10038;  email: bevans@pace.edu. his research interests are social justice in urban mathematics education  and international mathematics education. he is also interested in alternative certification and pre­  service teacher preparation in mathematics.  math links: building learning  communities in urban settings  jacqueline leonard  temple university  brian r. evans  pace university  learning mathematics in urban settings is often routine and decontextualized ra­  ther than inquiry­ and culturally­based. changing prospective teachers’ attitudes  about pedagogy in order to change this pattern is often tenuous. the purpose of  this pilot  study  was  to provide opportunities  for  teacher  interns  enrolled  in  a  graduate certification program to interact with urban students in a community­  based program called math links. twelve interns completed 30 hours of field­  work at church­based sites. prior to  fieldwork,  the interns participated in a 3­  hour professional development and education session, in addition to their educa­  tion courses. three interns’ work with urban children and youth reveal that com­  munity­based experiences changed  their attitudes about practice and  their  ca­  pacity to teach urban children mathematics in culturally sensitive ways. one in­  depth case study of an asian teacher reveals not only changes in her attitudes and  beliefs about urban students but also changes in her pedagogy as she shifted from  teaching by telling to guided inquiry.  keywords: “at­risk” students, community­based programs, mathematics educa­  tion, teacher interns  for more than 2 decades, there has been an impetus of reform in mathemat­  ics education (martin, 2003, 2007). as mathematics teacher educators, we have  focused on reform­based pedagogy in our elementary and secondary mathematics  methods courses with the intent to inform preservice and beginning teachers  in  undergraduate and graduate teacher credential programs about the advantages and  challenges of using reform­based practices.  reform­based practices in mathematics classrooms can be viewed in one of  two ways: use of reform­based curriculum and/or use of reform­based pedagogy.  studies on reform­based curriculum show that teacher educators can successfully http://education.gsu.edu/jume mailto:jleo@temple.edu mailto:bevans@pace.edu leonard & evans  math links  journal of urban mathematics education vol.1, no.1  61  guide  preservice  teachers  in  developing  conceptual  knowledge  in  mathematics  (ebby, 2000; sherin, 2002; spielman & lloyd, 2004). in addition to the use of  reform­based  curriculum,  reform­based  pedagogy,  such  as  teaching  for  under­  standing (ball, hill,  & bass, 2005;  ma, 1999; sherin, 2002),  facilitating class­  room  discourse (cazden,  2001;  o’connor  &  michaels,  1993)  and  engaging  in  culturally based practices (brenner, 1998; leonard, 2008; lipka, hogan, webster,  yanez,  adams,  clark,  &  lacy,  2005)  are  common  elements  found  in  reform­  based classrooms. yet, effective reform­based teaching of mathematics requires  that  preservice  teachers  learn  by  actively  engaging  students  in  the  teaching­  learning  process  (ambrose,  2004;  ebby,  2000;  lowery,  2002;  sherin,  2002).  thus, it is important for preservice and beginning teachers to have opportunities  to apply the knowledge they gain from theory and research in education courses  to real settings where they can implement reform­based practices with children.  for the purpose of this article, reform­based teaching is characterized by in­  quiry and culturally sensitive approaches to teaching and learning. in an inquiry­  based approach, the roles of teachers and students are redefined. the teacher is no  longer the sole authority for building mathematical knowledge in the classroom  (nctm, 2000). instead students are encouraged to be proactive rather than pas­  sive, using their own knowledge and experience to justify solutions to mathemat­  ics problems. the principles and standards document (nctm, 2000) provides an  impetus  for  reform­based  teaching  in  mathematics.  the  document  falls  short,  however, when it comes to cultural pedagogy (leonard, 2008; leonard, in press;  martin, 2007). focusing only on content knowledge, without attending to peda­  gogy (ball, bass, & hill, 2005) or the students’ culture (ladson­billings, 1995;  nieto, 2002), does not lead to the development of high quality teachers (martin,  2007). because mathematics is not divorced from culture, teachers must also be  culturally competent in order to be prepared to work with diverse student popula­  tions (ladson­billings, 1995).  knowledge of diverse students’ learning styles and culture helps teachers,  especially those who are from different racial, ethnic, and/or social backgrounds,  to  develop  strong  teacher­student  relationships  with  culturally  diverse  students  (lipka et al., 2005; shade, kelly, & oberg, 1997; silverman, strawser, strohauer,  & manzano, 2001). culturally sensitive approaches should also link mathematics  to issues of social justice. teaching for social justice empowers historically mar­  ginalized  students  to use mathematics  as a  form of  liberation (gutstein, 2006).  diverse students are more likely to realize the importance of learning mathemat­  ics  if  it can be used to empower them to change their circumstances (gutstein,  2006; ladson­billings, 1994; leonard, in press).  yet, teaching preservice and beginning teachers reform­based strategies is  not a panacea. as methods instructors, we have found these teachers’ beliefs and  prior experience cause them to be resistant,  initially,  to reform­based practices. leonard & evans  math links  journal of urban mathematics education vol.1, no.1  62  among preservice teachers enrolled in our teacher credential program, a signifi­  cant number did not experience any kind of reform­based teaching. essays written  in our mathematics methods courses revealed that some preservice and beginning  teachers continue to be taught mathematics in traditional ways as osisioma and  moscovici (2008) had found in a similar study with science teachers. these pre­  service and beginning teachers were often taught to use rules and algorithms to  solve problems and were not allowed to question the teacher or their peers in the  learning context. excerpts of  three preservice teachers’ reflections of their pre­  vious experiences in k–12 mathematics classrooms are presented for considera­  tion:  i remember my previous math teachers back in middle school, and they did not incor­  porate any hands­on activities into their lessons. it was simply learning off what was on  the  board.  i  think i  would  have  been  less  intimidated  by  math  if  i  had  materials  and engaging activities to help me to learn the concepts. (female student, fall 2007)  for the most part, i don’t have a lot of clear memories of how exactly i learned math.  this is probably because the teachers i had rarely did anything extraordinary to support  their lessons. i do remember in second grade doing something called a mad minute,  which were 30 addition and/or subtraction problems that we had a minute to try and  complete. also in second grade, we could earn fake money for doing certain things in  class, and every other week or so we would have an auction where we could spend that  money on small prizes. other than that, i honestly don’t remember anything specific  from elementary school regarding learning math. (male student, fall 2007)  what i can remember from math classes when i was younger involves a lot of scrap  paper and many trips to the board. we would be given pages and pages of homework,  sometimes without the concept even grasped. i just remember asking the question why  a lot and never getting an answer. many math teachers are just concerned that you can  actually solve the problem, rather than why it is solved like that. (female student, fall  2007)  these excerpts reveal that despite almost 20 years since the publication of  the curriculum and evaluation standards for school mathematics (nctm, 1989)  the teaching­learning context  in k–12 mathematics classrooms has not dramati­  cally  changed  (martin,  2003).  mathematics  instruction  continues  to  be  discon­  nected from students’ culture and everyday experiences (silverman et al., 2001).  mathematics, as a content domain, continues to be viewed as a complex series of  algorithms  with  abstract  entities  that  have  nothing  to  do  with  the sociocultural  context of students’ lives.  these experiences are more pronounced among candidates in our graduate  certification program, who tend to be older adults planning to teach as a second  career or young adults with bachelor degrees in liberal arts who want to obtain a  teaching  credential.  these candidates  are  able  to  obtain  jobs  as  teacher  interns  while completing the credential program. in this study, a teacher intern is defined leonard & evans  math links  journal of urban mathematics education vol.1, no.1  63  as one who has a paid or unpaid field experience in an informal or formal school  setting. in some cases, teacher interns have full­time jobs as teachers while they  are taking courses at night. a recent study of science teacher interns revealed that  it is possible to change teacher beliefs among this population (osisioma & mos­  covici, 2008).  osisioma and moscovici (2008) examined nine science teacher interns’ be­  liefs about inquiry and reform­based methods of  instruction before, during, and  after taking two science methods courses. osisioma and moscovici collected data  from written reflections, lesson and unit plans, interviews, observations, class dis­  cussions, and peer­teaching in the methods courses. they found most participants  primarily used traditional methods of instruction and were more teacher­oriented.  at the end of the two methods courses, the researchers found the number of in­  terns who believed in the use of inquiry and student­centered instruction rose to  seven from initially only one. the authors concluded that beliefs about science  teaching and learning changed over the two semesters and recommended that this  area of research receive greater attention.  in order to address teaching mathematics from a cultural and social justice  perspective, we studied the impact of reform­based practices learned in two grad­  uate teacher education courses and the enactment of reform­based pedagogy  in  community­based settings. it is in such settings that prospective teachers’ percep­  tions of urban students of color might change. too often, perceptions of african  american and other underrepresented minority students are rooted in deficit theo­  ries that contend these students are “less than ideal learners and, therefore, in need  of  certain  kinds  of  teachers”  (martin,  2007,  p.  8).  more  often  that  not,  these  teachers are strong in discipline but weak in mathematics content knowledge and  cultural  sensitivity.  some  african  american  scholars  (gay,  2000;  ladson­  billings, 2006; leonard, 2008; martin, 2007; nasir, 2005) call for teachers to use  students’ cultural experiences as a springboard for learning mathematics. in this  article, we are particularly interested in how k–12 teacher interns interacted with  urban students who were enrolled in an after­school program or a specialized pro­  gram for “at­risk” high school students. the purpose of this article is to report on  the enactment of reform­based practices  in these non­traditional urban settings.  because students in such settings are often marginalized, a framework that con­  nects issues of social justice with education is needed.  theoretical framework  this pilot study is grounded in a framework that has social justice at its core.  the ability to view the world through the eyes of marginalized persons is critical  to developing culturally sensitive approaches to teaching (gay, 2000; nieto, 2002;  shade et al., 1997). paulo freire in pedagogy of the oppressed (1970/2000) de­ leonard & evans  math links  journal of urban mathematics education vol.1, no.1  64  scribed the importance of conscientização, which is the development of the skills  necessary  for  critical  consciousness  and  the  teaching  of  social  justice  (apple,  2003;  gutstein,  2003).  to  operationalize  conscientização,  however,  a  specific  type  of  emancipatory  pedagogy  is  needed.  culturally  responsive  teach­  ing/pedagogy  is  one  such  framework  (gay,  2000). 1  according  to  shade  et  al.  (1997), culturally responsive pedagogy builds bridges between the culture of the  school and the home. they contend that knowledge should be transmitted in three  areas: general skills needed for survival (reading, writing, and mathematics), cul­  tural  information  (art,  science,  and  history),  and  cultural  norms  (behaviors  and  mores). yet, the debate continues over whose knowledge is valued and taught as  official  (apple,  1995).  white­middle  class  values  and  examples  dominate  the  american  educational  system  while  the contributions  and  values  of  persons  of  color  are  often  neglected  (blanchett,  2006;  gutstein,  2006;  leonard,  2008).  freire’s  construct  of  conscientização  challenges  this  perspective  from  a  class  perspective while culturally responsive pedagogy grounds our work in racial and  social justice.  gay (2000) contends that culturally responsive pedagogy is crucial in moti­  vating urban students of color to learn. culturally responsive pedagogy derived  from multicultural education paradigm in the 1970s; it “simultaneously develops,  along with academic achievement, social consciousness and critique, cultural af­  firmation, competence and exchange; community building and personal connec­  tions; individual self­worth and abilities; and an ethic of caring” (p. 43). one of  the most consistent and powerful findings of research studies related to diverse  students’ academic achievement is the ethic of caring (gay, 2000). the ethic of  caring is demonstrated by teacher attitudes, expectations, and behaviors related to  children’s intelligence and academic success (gay, 2000). caring teachers believe  their students are competent and hold them in high esteem. students then live up  to teachers’ expectations and exhibit appropriate classroom behaviors (gay, 2000;  ladson­billings, 1994). cultural responsive pedagogy, then,  is an important as­  pect of the teaching­learning environment in urban school settings. thus, we use  the constructs of conscientização and culturally responsive pedagogy to ground  the math links study.  research questions  the research questions that emerged in the math links study were: (1) how  does  one  teacher­researcher’s  reform­based  practices  influence  teacher  interns’  1  gay (2000) uses  the concept culturally responsive  teaching; whereas, ladson­billings (1994)  uses the concept culturally relevant pedagogy. for a brief discussion of the similarities and differ­  ence of concepts that might be positioned under the umbrella of culturally responsive/specific pe­  dagogy see leonard (2008). leonard & evans  math links  journal of urban mathematics education vol.1, no.1  65  beliefs about culturally responsive pedagogy? (2) how does interaction with ur­  ban youth in a community­based internship influence the development of cultural­  ly  responsive  pedagogy among  teacher  interns?  to  answer  these  questions  we  conducted a year­long study with two cohorts of teacher interns to draw on data  collected and analyzed on participants enrolled in two types of education courses  in our graduate teacher certification program: elementary  mathematics  methods  (fall 2006) and effective teaching (spring 2007).  the math links study  the math links pilot study grew out of a similar study in science that stu­  died  changes  in  preservice  teachers’  science  instruction  when  they  engaged  in  community­based  internships (leonard, boakes, & moore,  in press). the math  links pilot study was designed to obtain process data about the supports and re­  sources needed to empower teacher interns to practice reform­based teaching in  k–12 diverse school settings. the purpose of the math links pilot was two­fold:  (1) to provide teacher interns with field­based experience to practice reform­based  mathematics instruction; (2) to provide teacher interns with critical understanding  of culturally responsive pedagogy. by exposing teacher interns to urban students  enrolled in informal school settings, we aimed to reduce stereotypes about urban  children and youth and to increase the capacity of prospective teachers to engage  in reform­based mathematics instruction and culturally responsive pedagogy.  study sample  a total of 12 preservice teachers (4 undergraduate and 8 graduate) were re­  cruited to participate in the study. six preservice teachers (1 undergraduate and 5  graduate) participated in fall 2006, and six preservice teachers (3 undergraduate  and 3 graduate) in spring 2007. some of our participants were secondary majors  and some were elementary majors. the variety of participants’ backgrounds adds  important caveats to our data analysis. however, the population of  interest was  preservice and beginning teachers enrolled in a graduate certification program. as  previously mentioned, the rationale for studying this population is studies on this  particular population are scarce and these teacher candidates have few if any field  experiences in education prior to student teaching or induction (osisioma & mos­  covici, 2008). the ages of the eight interns selected from the larger study ranged  from 25 to 39 years of age. five were white women, two were white men, and  one was a korean woman. leonard & evans  math links  journal of urban mathematics education vol.1, no.1  66  the teacher­researcher  one of the teacher­researchers of this study was also the instructor of the  courses in which the teacher interns were enrolled. one of the courses in which  the  interns  were enrolled was  an  elementary mathematics  methods  course (fall  2006), and the other course was a general pedagogical course on effective teach­  ing  (spring  2007).  the  teacher­researcher  will  be referred  to  as  bridget  (pseu­  donym)  for  the sake of anonymity.  although these were two different courses,  bridget’s philosophy of education was consistent in both courses. her strong be­  lief in equity and social justice influenced the texts and articles students read in  the courses. students  in the pedagogy course read texts that dealt with cultural  relevance and social justice on a general level (i.e. gloria ladson­billings’ aera  presidential  address  [ladson­billings,  2006],  the  dreamkeepers  [ladson­  billings, 1994] and diversity pedagogy [sheets, 2005]). students in the mathe­  matics methods course read culturally relevant and social justice articles that were  specific to mathematics education (i.e. gutstein, 2003; leonard, davis, & sidler,  2005; martin, 2003). the other teacher­researcher of this study, who did not have  a teaching role in this study, shares a similar philosophy of education with bridget  and also promotes equity and social justice in publications, presentations, and in  the classroom. this researcher teaches at a university in which the college of edu­  cation promotes teaching from a social justice perspective as its core mission.  these courses provided a springboard to discuss issues of equity and social  justice and to demonstrate pedagogical ways to infuse students’ culture into les­  son plans, particularly in mathematics. in both courses, teachers had to demon­  strate  teaching  effectiveness  by  presenting  a  micro­teaching  lesson  (short  20­  minute lesson focusing mainly on one concept as opposed to a full lesson plan) to  their peers. in the pedagogy course, students could present a lesson dealing with  any of the core content areas specific to their major field of study (english, ma­  thematics, science, social studies) or specialty areas (art, music, physical educa­  tion). for example, a student in the general pedagogy course read a book by maya  angelou to integrate art and literacy. in the mathematics methods course, students  focused on teaching a mathematics topic to students in grades prek–8. an exam­  ple of a lesson in the mathematics methods course consisted of using the faces of  actors, such as will smith and sandra bullock, to teach about the golden ratio  and symmetry. both of the above lessons made broad connections to american  culture. micro­teaching to peers, however, does not provide preservice teachers  with the field­experiences they need to teach. learning to teach involves practice  with real students. an important part of any teacher credential program is provid­  ing settings for prospective teachers to work with actual children (ambrose, 2004;  ebby, 2000). teacher  interns, who were also students  in bridget’s courses, had  the privilege of not only delivering the content but also practicing culturally res­  ponsive pedagogy in urban settings in philadelphia. leonard & evans  math links  journal of urban mathematics education vol.1, no.1  67  the community­based sites  the  teacher  interns  worked  at  two  african  american  churches  that  had  longstanding relationships in the communities they served. both churches are lo­  cated  in  urban  neighborhoods  in  north  philadelphia.  both  zion  and  haven  churches have served the north philadelphia community for more than 100 years.  in the last 5 years, zion collaborated with researchers at the university where the  math links study took place by supporting preservice teachers’ work with child­  ren in saturday science programs (leonard, moore, & spearman, 2007; leonard,  boakes,  &  moore,  in  press).  furthermore,  zion  has  served  as  a  site  for  after­  school  and  summer  enrichments  programs  for  early  childhood  and  elementary  students. in recent years, it became a site for an at­risk high school program sup­  ported by a grant from the city of philadelphia. haven, on the other hand, has not  been as involved with educational endeavors. the recent addition of a computer  lab and establishment of an after­school program has helped to thrust haven into  the community spotlight, however. programs for children and adults have been  developed.  because  of  their  educational  initiatives  and  community  efforts,  the  zion and haven sites were selected for the eight graduate student participants in  the math links study to obtain field­based experiences.  five of these interns (one man and four women) worked at the zion site dur­  ing the fall of 2006. three of these interns (two women and one man) worked at  the haven site during the spring of 2007. at­risk youth, 13 to 18 years of age,  were enrolled at the zion site. children, 6 to 12 years of age, were enrolled in an  after­school program at haven. thus, we were able to collect data on teacher in­  terns’  actions  with  elementary,  middle,  and  high  school  students.  it  should  be  noted,  however,  that  attendance at  the  two  sites  varied  because  both  programs  were relatively new and voluntary.  methods  we  used  qualitative  research  methods  to  collect  and  analyze  data  in  the  math links pilot study. because we report on two different cohorts of interns si­  multaneously, this study may be characterized as a study within a study. specifi­  cally, we use case study methodology to analyze and report our findings. case  studies are often used for in­depth examination of processes that emerge from a  small number of phenomena (bogdan & biklin, 1998). considerations were given  to ethnicity and gender to obtain a diverse sample for the case studies. three of  the teacher  interns were selected for further study (one white man, one white  woman, and one asian woman) because their backgrounds provide the research  community  with  rich  data  about  the  cultural  sensitivity  of  teachers  from  these  specific backgrounds. it is important to understand how these teachers enact cul­  turally responsive pedagogy in urban settings. whites, particularly white women, leonard & evans  math links  journal of urban mathematics education vol.1, no.1  68  continue to choose teaching as a career (martin, 2007; remillard, 2000). as a re­  sult, these teachers are more likely to work with urban students if african ameri­  can and other teachers of color continue to decline. data sources consisted of the  following  for  each  of  the  cases:  coursework,  informal  observations  at  project  sites,  logs,  and  interviews.  we  then  used  the  constant­comparative  method  to  compare and contrast the cases (glaser & strauss, 1967). each of the cases pro­  vided the researchers with rich data about the participants’ development of peda­  gogical content knowledge in mathematics and culturally responsive pedagogy.  the results of this study will be presented in two parts. to answer the first  research question about how the teacher­researchers’ reform­based practices in­  fluenced teacher interns’ beliefs about culturally responsive pedagogy, we analyze  the results of structured interviews obtained from the teacher  interns and class­  room vignettes. to answer the second research question about how interactions in  the community­based internship helped the interns to develop culturally respon­  sive pedagogy, we analyze three case studies and examine one of these cases in  depth. due to data source limitations, these participants qualify as a convenience  sample. while not appropriate practice for a quasi­experimental study, a conveni­  ence  sample  served  our  purposes  for  this  pilot  self­study  to  inform  future  re­  search.  procedures  prior to serving as an intern, participants completed a 3­hour professional  development  session  taught  by  a  mathematics  education  consultant  while  they  were  simultaneously  enrolled  in  one  of  bridget’s  courses  (as  previously  de­  scribed). the teacher  interns were trained to use guided inquiry  during profes­  sional development. windschitl (2003) characterizes guided inquiry by the level  of student involvement. a hallmark of guided inquiry is that students investigate a  prescribed  problem  using  their  own  methods.  while  teaching  the  education  courses, bridget modeled inquiry­based instruction. teacher interns also watched  episodes of kay toliver as she engaged students of color  in inquiry­based ma­  thematics instruction (foundations for the advancements in science and educa­  tion productions [fase], 1998). thus, teacher interns were exposed to examples  of inquiry­based instruction prior to working with students in the field.  in order to provide participants in this pilot study with field experience prior  to student teaching, we placed them in settings where they could obtain 30 hours  of fieldwork in informal education settings. the local churches sites were located  within a one­mile radius of the college. therefore, teacher  interns could easily  complete the required 30  hours over  the course of  the semester while simulta­  neously taking education courses. seven teacher interns, who were enrolled in the  graduate teacher credential program in the college (one of the original eight grad­ leonard & evans  math links  journal of urban mathematics education vol.1, no.1  69  uate teacher interns dropped out of the study due to a schedule conflict with her  job), were observed by the teacher­researcher and/or the graduate research assis­  tant as they worked with students in the community­based settings during the fall  2006 and spring 2007 semesters.  the interns were required to keep a log of notes to document their activities  with students each time they went to the site. these logs were analyzed by the re­  searchers to determine not only how the teacher interns’ pedagogy was changing  but also how their attitudes toward the students were changing. in other words, we  examined the logs for evidence of caring (gay, 2000) and actions that exemplified  behaviors that could be synonymous to having a culturally responsive or social  justice stance (i.e. advocating for students when rules or regulations are unjust or  unfair; teaching in a manner that informs students about the status quo and/or how  to challenge such systems) (gutstein, 2003; tate, 1995).  an interview protocol was also developed and administered to participants  after they completed their community­based field experiences. in particular, we  were interested in comments that reflected changes in practices or attitudes about  the student  population.  teacher  interns  were given a  four­digit  id  number  for  identification purposes. the structured interviews were read and coded to categor­  ize the teacher interns’ responses. the teacher interns’ responses were then ana­  lyzed  to  find  themes  and  patterns  among  their  experiences.  common  elements  informed  the  researchers  about  how  to  improve  the  field­based  aspect  of  the  project for future study.  limitations  one limitation in this pilot study is the sparse number of student participants  in the community­based field settings and the variant amounts of data collected  from the teacher interns’ logs. some teacher interns wrote a minimal amount in  their logs, while one in particular (a korean woman) kept copious notes and de­  tailed  descriptions  about  the  lessons  and  her  interactions  with  students.  thus,  these data sources are uneven. the interview protocol, however, was used to fill  in gaps in the data. thus, triangulation of data sources was used to increase the  validity of our findings.  a second limitation is the instructor of the general education and mathemat­  ics methods courses in which the participants were enrolled was also a partici­  pant­observer in the study. while teacher­research is common in qualitative stu­  dies in education, issues of power and researcher bias are threats to internal relia­  bility. in order to minimize these threats, a graduate research assistant was also a  participant­observer in each of the field­based settings, and a second mathematics  educator (the other teacher­researcher  in this study), who had previously taught  mathematics  methods courses at the same  institution and currently teaches  ma­ leonard & evans  math links  journal of urban mathematics education vol.1, no.1  70  thematics methods courses at a different  institution, corroborated the interpreta­  tion of data and the results. thus, checks and balances were put into place to mi­  nimize bias and increase the integrity of our findings.  results  structured interviews  six of the seven teacher interns who participated in the community­based in­  ternship during the fall or spring semesters of 2006–2007 participated in the struc­  tured interview. two reported on their experiences at haven and four reported on  their experiences at zion. as shown in appendix a, four categories emerged as a  result of qualitative analysis: (1) lessons learned from the program, (2) teacher  intern’s perceived strengths, (3) perspectives on the math links program, and (4)  perspectives on urban students.  analysis of intern responses in lessons learned from the program (category  1) reveal that three interns (2210, 0078, 1080) focused on classroom management  issues (i.e. organized lessons, firm and consistent discipline, classroom manage­  ment techniques) and three interns (3695, 9352, 0063) focused on care and/or re­  lationships (i.e. diverse needs of children, building relationships, mutual respect,  increased understanding of diverse students). teacher interns reported perceived  strengths (category 2) by describing their commitment (0078, 9352), experience  (3695), and lesson creativity (2210, 9352, 0063, 1080). perspectives on the math  links program (category 3) reveal the interns at haven (2210, 0078) did not be­  lieve they had the supervision and oversight they desired. one intern (9352) at  zion  mentioned  that  organization  and  communication  could  have  been  better.  however, teacher interns mentioned some of the benefits of the program, such as  the resources (0078), exposure to work with students (2210, 3695, 0352, 0063,  1080), and learning from peers (0063). one intern (1080) who was student teach­  ing at the same time that she participated in the study noted: “activities presented  to youth in the math links program were used the next day with students during  student  teaching.”  this  comment  highlights  the  importance  of  the  teaching­  learning process. one must actually engage in teaching in order to learn how to  teach  (ambrose,  2004;  ebby,  2000).  finally,  we  analyzed  the  comments  that  emerged in category 4: perspectives on urban students. three comments focused  on students’ behavior (0078, 0063) or opportunities (3695), but three commented  on  how  their  own  attitudes  and  perceptions  of  urban  students  changed  (2210,  9352, 1080). one intern at haven remarked: “children were intelligent, focused,  and dynamic.” one  intern  at zion  stated the program “challenged myths about  urban students: lazy, don’t want to learn; don’t care about education; don’t care  about work. students were hard working, wanted to learn, and wanted to under­  stand math.” leonard & evans  math links  journal of urban mathematics education vol.1, no.1  71  because  the college  did  not  provide  field­based  experiences  for  graduate  students prior to student teaching, the foregoing comments have important study  implications. overall, comments about urban students were consistently positive.  moreover, the community­based field experience allowed some of the teacher in­  terns to become students of students and challenged negative perceptions and ste­  reotypes about students of color (nieto, 2002). in order to learn more about their  interactions with students  in the community­based settings, we present the case  studies of three interns. pseudonyms are used for anonymity. these cases are pre­  sented as vignettes.  the vignettes  although a total of 12 teacher interns participated in community­based field  experiences in the 2006–2007 academic year, only eight were graduate students,  which is the focus of this article. the profiles of these eight  interns reveal two  were white men and six were women (5 white and 1 asian). of these eight par­  ticipants only six were also enrolled in one of bridget’s courses: 1 man; 5 women.  vignette 1: shawn. shawn was a 25­year­old european american man with  a sociology degree from the university of michigan. shawn was the only male  teacher intern who worked at zion. because of this, many of the predominantly  male students perceived him as a mentor. he was observed teaching an inquiry­  based lesson that  integrated mathematics and space science in the fall of 2006.  the  lesson  involved  having  students  calculate  the percentages  of  different  ele­  ments found in a sample of playdoh used to simulate moon rock. students learned  how to slice the rock samples like geologists and then estimated and extrapolated  the data to determine what type of rock sample they had by counting beads of dif­  ferent colors. the vignette taken from one of shawn’s reflection papers  is pre­  sented:  it seems as if my undergraduate education in sociology has laid the groundwork for a  deeper and more applicable understanding of social justice and equity, which i have  been able to build upon both in theory and in practice. ultimately, my understanding  of culture in mathematics education will be tested in the classroom, and that is why  my experience at zion this semester has been so valuable. while  the context that  children are raised in may not be the sole determining factor of their success, it un­  doubtedly will impact the rest of their lives. students who have limited access to re­  sources and effectual education will have limited opportunities to achieve success.  this reality is clearer after one day at zion than it could ever be in a journal article or  textbook. tutoring has become the ideal opportunity to apply what i am learning in  the classroom to situations i will face as an educator. i am discovering that educa­  tion, in particular my own, is a steady progression from abstract theory to more tang­  ible concepts, concepts that have practical implications for the classroom. it is envi­  sioning how to embrace the inquiry­based models of learning we are exposed to and  relate them to every teaching opportunity we are presented with. leonard & evans  math links  journal of urban mathematics education vol.1, no.1  72  shawn’s vignette reveals the math links experience was pivotal to his de­  velopment  of  critical  consciousness  and  his  understanding  of  equity  and  social  justice as it related to mathematics education. clearly, he exhibited culturally res­  ponsive pedagogy as he  learned to mesh theory  and practice. shawn, however,  hints at the complex nature of inquiry­based teaching. how do teachers relate the  pedagogies they learn in teacher education programs to students’ lives? how do  they include elements of student culture as a springboard for learning without wa­  tering down the curriculum or lowering expectations? these questions cannot be  answered by participating in 30 contact hours of field experience. nevertheless,  shawn had a better understanding of teaching diverse students after participating  in the math links study.  vignette 2: camille. camille was a 26­year­old european american woman  who had received an undergraduate degree in english from cornell university.  she had also lived and studied abroad in japan. she was an intern at the haven  site. during her observation in the spring of 2007, bridget (recall this was the in­  structor and participant­observer) noticed that camille had an  excellent rapport  with the 6 to 12 years old african american students. they were learning about  the story of sadako, origami, and how to make paper cranes, which they con­  nected together to make a long strand. camille also brought pictures of her travels  to japan so the students could see the lifestyle of the japanese people. impressed  by camille’s ability to retell sadako’s story, a japanese girl’s fight with leukemia  after being exposed to agent orange during world war ii, bridget loaned her the  storybook by coerr (1993). to help the african american students at haven un­  derstand the gravity of sadako’s plight, camille described an event that her stu­  dents could relate to. the vignette taken from one of her reflection papers on so­  cial justice is presented:  i would like to address what to me was one of the main strengths of this article [leo­  nard & hill, 2008]. the background material regarding analytical scaffolding and  social  scaffolding  is  extremely  helpful  and  profuse  as  is  its  later  exemplification  within the context of the [lesson]. the following discourse occurred during one of  my own sessions with six african american students. my boyfriend, c, and i gave a  joint presentation about  the blues (musically and historically) at  the haven after­  school program where i tutored once a week.  c [stated], “a contradiction is when you say one thing and then do another. the  united states contradicted itself when it took away african american rights. every­  body, how would that make you feel?” “sad,” one said. “unfair,” said another. “has  anyone ever heard of the blues?” i asked. another student said, “it’s when you are  sad.”  c and i found that when we quizzed the students later on, they remembered almost  word for word what was said. i had a very valuable first­hand experience with scaf­  folding and intend now, further confirmed by this article, to use it as often as possi­  ble in future lessons. leonard & evans  math links  journal of urban mathematics education vol.1, no.1  73  camille’s vignette reveals her ability to engage in culturally responsive pe­  dagogy. she and her boyfriend engaged students by using their cultural capital to  scaffold their learning. camille’s use of the blues as social scaffolding shows her  ability to move from theory to practice. the mathematics in this lesson was cultu­  rally relevant as she tried to link learning geometry and spatial skills to sadako’s  life with cultural  beliefs about her  illness. to make connections with sadako’s  story, camille used the blues, which is a part of african american culture. by  using the context of the blues, the students were able to understand the affective  nature of  the story, and they  were eager  to make paper cranes and  learn  more  about japanese culture. thus, camille’s lesson is an example of how to teach ma­  thematics concepts within a social justice context.  vignette 3: sun­lee. sun­lee was a 39­year­old korean woman attending  the college on a visa. she was a graduate of ewha woman’s university in seoul,  korea. she had an undergraduate degree in library science and was interested in  teaching english to esl students. therefore, she chose to dually enroll in the te­  sol and graduate certification program at the college. sun­lee taught and inte­  grated science and mathematics lessons to high school students in the fall of 2006.  she was instrumental in helping the students learn geometry, measurement, and  aerodynamics  by constructing kites (leonard, 2002). students  made tetrahedral  kites out of tissue paper, straws, and string (ceeo, 2001). they learned mathe­  matics vocabulary (tetrahedron, faces, vertices, edges, etc.), used rulers to meas­  ure accurately, and learned how lift, gravity, drag, and thrust worked together to  make a kite fly. sun­lee kept a meticulous journal of her experiences at zion. an  excerpt from her journal describes her work with the students during the kite ac­  tivity:  there were three more girls, but i did not know their names yet. since we had to  reattach two bridles, we had to measure two strings for the two small kites. i thought  that the students needed to find information from the text for themselves. when they  asked me how to do [it], i read the instruction with them while pointing out the part.  after reading, i asked them what it meant. for example, part of the instruction was  for  [a]  three­quarter  inch  of  space  between  the  loop  knot  and  the  straw.  rickie  showed me 3.4 inches by means of a ruler. therefore, i pointed to this part of the in­  struction, and we read together. rickie understood and made [a] three­quarter inch  space. carolyn said her ruler was not big enough to measure the  longer part of a  bridle. i asked her how she could measure the longer string. she thought about it and  said, “oh, moving like this.” she displayed the iteration on the ruler.  sun­lee  demonstrated structured  inquiry  in  the  foregoing  excerpt  (wind­  schitl, 2003). rather than teaching by telling, she led one of the students to figure  out the difference between 3.4 and ¾ inches and helped another student determine  how to measure multiple pieces of string using a ruler. by helping them to con­  struct their own knowledge, the students were more likely to remember the ma­ leonard & evans  math links  journal of urban mathematics education vol.1, no.1  74  thematics  concepts  they  learned.  this  vignette  hints  at  the  complexity  of  the  teaching­learning process. how much information should teachers tell students?  how much should students be responsible for learning? knowing “when to pro­  vide an explanation, when to model, when to ask the rather pointed questions…is  delicate and uncertain” (ball, 1993, p. 393).  the data presented above in vignette 3 as well as data presented in the in­  terview protocol show the unique characteristics and behaviors that sun­lee ex­  hibited during the study. moreover, the data she provided in her participant log  provided rich accounts of her work with students at the zion site. while she was  selected because of convenience,  the case of sun­lee provided the researchers  with a deep and informative account of her field experiences at zion.  the case of sun­lee  the case of sun­lee was quite unique and interesting. she was the only per­  son of color who participated in the math links study. furthermore, she made  tremendous strides in english fluency and literacy during the fall 2006 semester,  and  she became culturally  competent  as  a  result  of  her  experience  in  the pilot  study. her participant log consisted of 13 detailed accounts of teaching and learn­  ing mathematics within a cross­curricular context at the zion site from october  23, 2006 to december 13, 2006. appendix b summarizes the lessons, teacher ac­  tions, student actions, and teacher behaviors.  analysis of sun­lee’s case study  analyses of sun­lee’s journal entries, as shown in appendix b, reveal that  she progressed rather quickly  from using direct  instruction to inquiry­based  in­  struction over  the course of  the fall 2006 semester. on october 23rd, sun­lee  used direct  instruction to teach  andrew a part­whole  interpretation of  fractions  and subsequently the conversion of those fractions into percents:  i asked [andrew] whether or not he knew how to get percentage. he said, “no.” he  got the 41 white rocks, 13 red rocks, and 7 blue rocks. in order to explain [how] the  numbers could be transformed into numbers less than one, i helped him to draw a pie  chart. we sectored the pie into 61 pieces. in the comparison of the pie with pizza, i  explained 61 pieces as a whole number 1. and i told him that the concept could be  expressed 7/61, 13/61, and 41/61, which were less than the number one.  on october 24th, sun­lee still wrote about “showing” students how to do  things, but by october 25th, we have evidence that sun­lee began to use ques­  tioning techniques that allowed the students to develop their own understanding. leonard & evans  math links  journal of urban mathematics education vol.1, no.1  75  rather than teaching by telling, she was beginning to help students take responsi­  bility for their own learning:  i gave short instructions and wanted them to read the procedure again. i thought that  the students needed to find the information from the text for themselves. when they  asked me how to do it, i read the instructions with them…. after reading, i asked  them what it meant.  by november 1st, sun­lee attempted to engage students in discussion. yet,  she was hesitant to do so because students’ work was at different stages and be­  cause of her perceived limited english proficiency. however, a breakthrough oc­  curred on november 2nd when sun­lee had two students offer their own exam­  ples of newton’s third law of motion. by having the students experiment with a  rocket launch from a lesson derived from mission mathematics (hynes & hicks,  2005) before introducing newton’s third law, students were able to make con­  nections by collecting actual data and offering their own explanations of the law:  “for every action there is an opposite and equal reaction.” i gave an example with a  ruler and the edge of the desk. by hitting the ruler with weak force, the ruler dropped  down. however, the ruler flipped over and dropped down when i used strong force.  dante [used] a similar example. also, rickie [shared] his idea.  toward the end of the semester sun­lee engaged students in investigations  that  led  them  to  make  discoveries  about  other  theories  as  well.  mathematical  probability  was  connected  to  genetics  when  dante  experimented  with  punnett  squares using a coin to determine genetic outcomes. finally, it can be seen that  sun­lee  realized  that  inquiry,  although  time  consuming,  is  paramount to  good  instruction. at­risk students at zion were engaged at high levels when they were  given opportunities to investigate, discuss, and explain their reasoning. analyses  of sun­lee’s  journal show that she progressed over  the course of  the semester  from using direct instruction to reform­based, context specific instruction. most  importantly, she empowered at­risk students by helping them to be self­directed  and to take charge of their own learning, which is one of the tenets of culturally  relevant pedagogy (ladson­billings, 1995).  discussion  the results of this pilot study are promising. the major finding of the study  is that during the graduate certification program five of the seven interns who par­  ticipated in the pilot study, though older and more entrenched in their beliefs and  values, engaged in reform­based practices during the internship. evidence for this  claim is supported by the results of interview data and case studies. this finding leonard & evans  math links  journal of urban mathematics education vol.1, no.1  76  has important implications for the field if these changes can also occur among be­  ginning  teachers.  it  is  consistent  with  the  finding  by  osisioma  and  moscovici  (2008) in which they observed a shift in science teacher interns’ beliefs from tra­  ditional  methods  of  teaching  science  to  an  inquiry  and  reform­based  approach  about teaching and learning over the course of two semesters. data, such as that  published by the national association of educational progress (naep), continue  to show dismal performance in mathematics, particularly in grade eight (nces,  2007). the ability to think and reason is critical if one is to achieve above basic  and proficient levels in mathematics.  a  second  finding  is  five of  seven teacher  interns  who  participated  in  the  community­based internship changed their perceptions of the predominantly afri­  can american students participating in the math links program. they recognized  that “typical” stereotypes about these children simply were not true. the students  were eager and willing to learn  mathematics and were receptive to one­on­one  tutoring  and  whole  group  instruction.  furthermore,  these  interns  developed  an  ethic of care and exhibited teacher dispositions that martin (2007) characterized  as racial competence and commitment to anti­oppressive and anti­racist teaching.  nevertheless,  additional  studies  are  needed  to  learn  whether  the  perception  of  hard work and the ethic of care might be transported into the traditional school  setting.  a third finding is the importance of providing teacher interns at the graduate  level with field­based experiences prior to student teaching. because this popula­  tion generally enrolled in evening courses at the college to obtain a teaching cer­  tificate, field experience was not a part of the credential program. five of the six  teacher interns interviewed noted the value of the field experience. one specifical­  ly mentioned how she used materials and techniques learned in the math links  program during student teaching. the community­based internship provided these  interns  with  an  opportunity  to  learn  from  their  interactions  with  students.  this  finding concurs with the findings of our previous work with teacher interns (leo­  nard, boakes, & moore, in press) and with ebby’s (2000) work with preservice  mathematics  teachers.  prospective  teachers’  pedagogical  content  knowledge  in  mathematics was dependent upon the teaching­learning process (ebby, 2000; she­  rin, 2002). additional research, however, is needed to learn whether inquiry­based  practices learned during fieldwork can be sustained throughout induction.  finally, this pilot study has implications for researchers attempting to link  the teaching of mathematics to social justice. teaching preservice and beginning  teachers about social justice in a vacuum was not meaningful, as two interns who  participated  in the case studies attested. actually working with students whose  lives stood in stark contrast to their own privileged backgrounds was eye opening  for these two interns. they realized firsthand the powerful impact poverty has on  some urban students’ lives and the ramifications of the lack of educational oppor­ leonard & evans  math links  journal of urban mathematics education vol.1, no.1  77  tunity.  moreover,  the structured  interviews  and  case  studies  show how several  teacher interns were moved by the potential (realized and unrealized) of students  in the community­based programs. thus, learning to teach for social justice must  include critical work with appropriate student populations.  the results of the math links pilot study show that providing community­  based field experiences for teacher interns benefits both interns and students alike  as relationships and rapport are forged during the mentoring process. not only did  one of the interns continue for an additional semester, but site coordinators and  students  also  requested  additional  interns  the  next  semester.  given  limited  re­  search dollars, sustaining successful partnerships are challenging. however, the  math links study shows the educational possibilities when there is a nexus be­  tween research communities and civic responsibility.  acknowledgments  we acknowledge cara m. moore, a graduate student at temple university, for collecting data and  editing this  paper.  we  also  acknowledge  the  united  methodist  church,  eastern  pennsylvania  annual conference, for funding this pilot study. the views contained in this paper do not neces­  sarily reflect the positions or policies of the united methodist church.  references  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teacher intern’s perceived strengths  perspectives of math links program  perspectives of urban student  2210  haven  learned that highly organized lessons that  allow  some  flexibility  are  important;  dis­  cipline needs to be firm/consistent.  used  background  knowledge  in  art  and  history  to  blend  a  variety  of  media  and  context  to  lessons,  which  maintained  stu­  dent interest regardless of learning styles.  prior  teaching  experience  was  limited  so  extra  exposure  was  helpful.  program  al­  lowed me to work one­on­one with child­  ren. little structure and guidance provided,  which  caused  site  director  to  have  some  uncertainty  about  the  parameters  of  the  program.  children  were  intelligent,  focused,  and  dynamic. struck by how poverty and fami­  ly  situation  can  undermine  intelligent  students, causing them to miss school and  jeopardize their education. longed to exert  more  influence  on  parents  when  she  be­  comes a full­time teacher.  0078  haven  variety of resources available to teachers.  classroom management techniques.  commitment to teaching urban children.  provided  resources.  more  training  and  oversight needed.  broader view of behavior issues.  3695  zion  learned about the diverse  needs  of  child­  ren.  possessed  patience  and  caring  qualities;  had prior  experience  working  with pre­k  inner city children.  children  have  different  needs;  teacher  must  cater  to  the  needs  of  all  children;  provided  experiences  beyond  private  and  suburban settings.  program  provided  opportunities  for  child­  ren.  9352  zion  building  relationships  with  children  is  important. mutual respect between teacher  and  students  is  key  to  academic  success.  consistency is vital with students.  personal  creativity  came  out  during  the  internship. made connections with students  despite differences in age and appearance.  had prior experience teaching high school.  commitment  to  working  with  children  who have special needs.  wonderful  experience  that  exposed  the  intern to real­world setting and allowed her  to  build  confidence.  program  provided  opportunities to interact with students in an  informal way. organization and communi­  cation  could  have  been  better.  requested  lesson  templates  or  exact  lesson  plans  to  follow.  program helped intern  to  become a better  teacher by overcoming misconceptions and  insecurities about teaching youth. learned  how accepting students can be when given  special attention.  0063  zion  hands­on  lessons  can  serve  as  motivator.  increased understanding of the similarities/  differences  between  child  in  u.s.  and  korea.  constructing  hands­on  lessons;  ability  to  adapt what i learn to new situations.  providing  the  opportunity  to  observe  stu­  dents’ learning and to learn from peers and  mentors.  hands­on  activities  provided  learning opportunities for the students.  worried  about  student  motivation  and  attentiveness,  but  it  changed.  older  stu­  dents responded well to hands­on activities  and were highly engaged in learning.  1080  zion  learned  how  to  control  the  pace  of  the  lesson and make sure students are on task.  learned how to use appropriate classroom  management techniques.  discovered  ways  to  motivate  students;  one­on­one  interaction  led  to  direct  in­  volvement in one case.  program allowed intern to tutor students in  a small groups; experience allowed her to  develop  ideas  for  use  in  other  settings.  activities  presented  to  youth  in  the  math  links program were used the next day with  students during student teaching.  program  challenged  myths  about  urban  students:  lazy,  don’t  want  to  learn;  don’t  care  about  education;  can’t  do  the  work.  students  were  hardworking,  wanted  to  learn, wanted to understand math. leonard & evans  math links  journal of urban mathematics education vol.1, no.1  81  appendix b  journal analysis of one teacher intern’s pedagogy  date and lesson type  teacher actions  student actions  analysis of teacher behaviors  october 23, 2006  rock doctors  students  cross  cut  playdoh  repre­  sentation of moon rocks to deter­  mine what type of rock they had by  the  percentages  of  minerals  they  found in the playdoh.  i  asked  [andrew]  whether  or  not  he  knew  how  to  get percentage. he said, “no.” he got the 41 white  rocks,  13  red  rocks, and  7  blue  rocks.  in  order  to  explain  the  numbers  could  be  transformed  into  numbers less than one, i helped him to draw a pie  chart.  we  sectored  the  pie  into  61  pieces.  in  the  comparison  of  the  pie  with  pizza,  i  explained  61  pieces as a whole number 1. and i told him that the  concept could be expressed 7/61, 13/61, and 41/61,  which  were  less  than  the  number  one.  by  writing  down the numbers in the format, i modeled how to  compute the division 7/61.  andrew  computed  the  other  fractions.  anthony  brought his multiplication knowledge to divide the  fractions.  sun­lee worked one­on­one with andrew. she use  direct instruction to help him understand parts of a  whole and to teach how to calculate percentages.  october 24, 2006  tetrahedral kite  students  used  straws,  string,  and  tissue  paper  to  make  tetrahedral  kites. each student  made  one cell,  and  all  of  the  cells  were  put  to­  gether to make one large kite.  since joshua said, “i don’t know how to put these  straws,” i approached to help him. he was holding  his sixth straw, which needed to support the tetrahe­  dron.  i  showed  how  the  straw  could  uphold  the  tetrahedron  and  said  to  him  to  run  the  thread  through the straw and clip tightly.  the  runner  and  holder  took  their  positions  while  releasing  the  string  to  fly  it.  the  runner  and  the  holder  tried  several  times,  but  it  didn’t  work  out.  the kite kept falling down whenever the holder let  go, although we changed the position of the bridle.  sun­lee  worked  one­on­one  with  joshua.  she  supported  his  learning  by  helping  him  make  one  tetrahedral cell for the kite. leonard & evans  math links  journal of urban mathematics education vol.1, no.1  82  october 25, 2006  tetrahedral kite (cont.)  students redesigned the tetrahedral  kite.  we started taking the kite apart. while making the  knots,  i  looked  for  additional  help.  i  called  ca­  rolyn…. since  we  had  to  reattach  two  bridles,  we  had to measure two strings for the two small kites. i  gave short instructions and wanted them to read the  procedure again. i thought that the students needed  to find the information from the text for themselves.  when they asked me how to do it, i read the instruc­  tions with them…. after reading, i asked them what  it  meant.  since  we  had  two  different  kites,  i  sug­  gested two groups of students pull each kite….the  reason was their own experience of pulling the kite  will make them think in depth.  students  suggested  modifying  the kite:  place  wax  paper on the open cells, change position of bridle,  move to larger space to fly the kite. rickie said he  could make a new small kite.  andrew  and  joshua  flew  the  kites,  and  it  stayed  horizontal  while  the  students  were  running.  they  were running all over the place, and it did not go up  vertically.  in  this  lesson,  sun­lee  employs  some  inquiry­  based  practices.  she  allowed  the  students  to  have  some  autonomy  and  encouraged  them  to  find  out  information  for  themselves  and  to  think  in  depth  about how the kite flew.  november 1, 2006  alka rockets  students  used  fuji  film  canisters  and index  cards to  make a rocket.  after  making predictions, students  used  different  amounts  of  alka­  seltzer and  water into it  to  launch  the  rockets.  they  measured  the  height  each  time  the  rockets  were  launched.  i told the students to check whether they had all the  items [for] the procedure. also, i  show two rocket  pictures that i printed in  color  from the internet to  talk  about  the  force  and  direction  of…the  rockets.  students were asked to read the first procedure. by  referring  to  the  procedure,  i  intended  to  enable  students to practice applying necessary information  to their own work.  i asked what the function of the rocket fins and nose  cones were. the discussion could not develop well  partly because  each  student  was  working on a dif­  ferent  stage. the  other  reason  could  be  that  i  was  not  confident to lead the discussion  because i  was  concerned about my english proficiency. although i  understood  preparation  and  practice  in  real  class­  rooms  could  have reduced  my anxiety, the anxiety  in my mind still existed.  if  i  use  the  lesson  again,  i  will  have  the  students  read through the procedure first to grasp the whole  process.  also,  they  could  discuss  the  functions  of  the fins and cones more than they did this time.  the  students  completed  their  rockets  and  were  [asked] to predict the height. a table was given to  record  each  height.  each  student  tried  the  initial  variable (amount of water). the rockers went high  up from the scale of 2 to 8. while trying the other  variables (size of tablet). students figured out that  the less amount of water and more amount of alka­  seltzer went higher than the others. the lesson was  a  success.  marie  said  it  was  the  most  interesting  experiment. rickie said, “it was pretty cool.”  analysis  of  lesson  reveals  sun­lee  continues  to  utilize some aspects of inquiry. instead of teaching  by telling, she wanted students to find the informa­  tion and apply the knowledge learned to the task of  making the alka rocket. although she was ambi­  valent about her english, she tried to lead the stu­  dents in a discussion about how the  fins and nose  cones  would impact the rocket’s  flight.  while  she  was not able to engage students in such a discussion  at  this  juncture,  sun­lee  reflected  about  how  she  could  do a  better job the next time  she taught the  lesson. leonard & evans  math links  journal of urban mathematics education vol.1, no.1  83  november 2, 2006  alka rockets (cont.)  followed up  with  data analysis  of  the  results  from  previous  day’s  launch.  i asked…how many inches are in a foot? some said  6 inches because they knew that the temporary ruler  was…six inches. someone said 12 inches. i wrote 1  foot = 12 inches to help students transform inches to  feet. after the measurements, i told them the expe­  riment proves newton’s third law of motion: “for  every  action  there  is  an  opposite  and  equal  reac­  tion.” i gave an example with a ruler and the edge of  the desk. by  hitting the ruler  with  weak force, the  ruler dropped down. however, the ruler flipped over  and dropped down when i used strong force. dante  [used] a similar example. also, rickie [shared] his  idea.  students recalled their highest results and discussed  the influential factors for the highest rockets. eve­  rybody said that less water caused the rockets to go  higher. they  figured  out the real lengths  by  mea­  suring the heights and multiplying their results by  six  inches  because  the  temporary  ruler  was  seg­  mented  every  six inches. dante thought one  scale  of the temporary ruler was a foot. so he wrote his  rocket went, 6 ft., 8 ft., and 7 ft. but after the ex­  planation, he changed 6 to 36 inches and 8 to 48  inches.  sun­lee  probed  the  students  to  determine  their  background  knowledge  on  measurement.  they  knew  12  inches  was  one  foot  but  the  data  reveal  little  experience  with  measurement  tasks,  scaling,  and converting inches to feet. while sun­lee rein­  forced equations and laws to  help  students under­  stand the activity, she also demonstrated newton’s  law. this demonstration led two students to share  examples  and  ideas  about  newton’s  law,  that  is,  the students offered alternative explanations, which  is a hallmark of inquiry.  november 15, 2006  genetics and punnet squares  student used  probability and  pun­  nett squares to make predicts about  traits  while leaving the classroom, i thought it would be  better for us to go over the punnett square tomorrow  because i felt that [dante] was not sure yet.  in order to decide the components of genetic trait,  students  were  asked  to  flip  a  coin.  when  dante  flipped  a  nickel  twice,  he  got  a  pair  of  n  and  n  genes  for  each  flip. that  means his portrait  has a  round  nose,  which  carried  a  recessive  gene  of  a  pointy  nose. after  deciding all other traits includ­  ing the color of eyes and hair, he drew the portrait  with color[ed] pens. students were asked to apply  the  rule  to  real  life  problems.  dante  created  the  punnett square but got 2 squares wrong.  sun­lee  worked  one­on­one  with  dante  as  he  determined  genetic  traits  and  used  the  punnett  square.  realizing  his  struggle  with  the  punnett  square,  sun­lee  planned  to  review  concepts  with  him the next day.  december 1, 2006  measurement  students  learned  customary  and  metric units of length.  in  a  previous  lesson, alka  rockets  activity, dante  used  the  unit  of  length  (3  ft.,  8  in.)  to  record  the  measurement. dante uses the unit consistently [sic]  in the right form, but i wanted to know whether or  not  he  was  aware  of  inches,  square  inches,  and  square feet. when i asked him to explain…the rela­  tionship  between  a  meter  and  a  yard,  he  did  not  explain,  he  draw  [sic]  a  division  calculation. how  could  i  let  him  know  the  relationship  between  a  meter  and  a  yard?  step  by  step!  i  asked  dante  to  show  and  explain  his  thoughts  a  lot  because  i  learned that teachers  should ask  students, “explain  what you think.” however, this process takes time.  in a real class with many students, it would not be  easy to follow each student’s thought process.  dante recognized an inch and said [to] himself the  ruler was a foot when he saw the ruler. he meas­  ured a line of 3 inches on his own. when he meas­  ured 2 1/8, he asked [sun­lee] how to measure. (a  book was provided for a reference.) he measured a  yard  and  a  meter  of  the  table  again  by  using  the  paper ruler. [he measured] a meter length with the  paper inch ruler. he  got 39 5/16 inches. also, he  measured one yard with the paper centimeter rule.  he got 91 cm. then, he referred to the conversion  table and found that one meter is 100 centimeters.  now, he could understand one yard is 90% of a one  meter.  during methods class, sun­lee watched the movie  stand  and  deliver.  in  the  film,  jaime  escalante  helped his students learn calculus by teaching them  step  by  step.  sun­lee  borrowed  this  phrase  and  helped dante develop mathematical knowledge by  providing  hands­on  experiences  with  paper  inch  and  centimeter rulers. she also learned the impor­  tance of allowing students to explain their thinking  from  the  methods  course.  however,  she  was  also  aware of the challenges teachers face when they try  to use the method in regular classrooms. microsoft word final martin et al. vol 3 no 2.doc journal of urban mathematics education december 2010, vol. 3, no. 2, pp. 12–24 ©jume. http://education.gsu.edu/jume danny bernard martin is chair of curriculum and instruction and an associate professor of mathematics at the university of illinois at chicago, 1040 w. harrison street, chicago, il, 60607; email: dbmartin@uic.edu. his research has focused primarily on understanding the salience of race and identity in african american learners’ mathematical experiences, taking into account sociohistorical and structural forces, community forces, school forces, and individual agency. maisie l. gholson is a second-year doctoral student in mathematics education in curriculum studies of the college of education and graduate research assistant in the learning sciences research institute, at the university of illinois at chicago, 2075 sel, mc-250, 950 south halsted street, chicago, il, 60607; email: mghols2@uic.edu. her research interests include black students’ thinking and reasoning in first-year algebra, teacher use and implementation of standardsbased curricula with black students, and black students’ mathematics experiences throughout the african diaspora. jacqueline leonard is associate dean of teacher education and outreach and a professor of mathematics education in the school of education and human development at the university of colorado denver, 1380 lawrence street, suite 740, denver, co 80217, e-mail: jacqueline.leonard@ucdenver.edu. her research interests include access and opportunity in mathematics education and critical pedagogy, such as teaching for cultural relevance and social justice in mathematics classrooms. commentary mathematics as gatekeeper: power and privilege in the production of knowledge danny bernard martin university of illinois at chicago maisie l. gholson university of illinois at chicago jacqueline leonard university of colorado denver his research commentary is generated in response to two recent events, each occurring under the national council of teachers of mathematics (nctm) umbrella. first, in the march 2010 issue of journal for research in mathematics education (jrme), editor m. kathleen heid stated, “jrme publishes research in which mathematics is an essential component rather than being a backdrop for another area of inquiry. i encourage readers to continue to examine articles in jrme with the ‘where’s the math?’ question in mind” (p. 103). heid proceeded to describe seven studies,1 exemplars, which “make sense only in the context of mathematics” (p. 103). second, we refer to nctm hosting a research symposium titled keeping the mathematics in mathematics education research at the 2010 nctm research presession. the panelists for the symposium were deborah ball, michael battista, guershon harel, and patrick thompson (jere confrey was the discussant). the published symposium summary stated: this session focuses on the role of mathematics in mathematics education research. in particular, the session addresses a growing concern among many mathematics education scholars regarding the lack of attention to mathematics in much of the current work in mathematics education. (nctm, 2010, p. 60) 1 ely (2010); filloy, rojano, & solares (2010); ng & lee (2009); oehrtman (2009); speer & wagner (2009); stylianides, g. & stylianides, a. (2009); thanheiser (2009) t martin et al. commentary journal of urban mathematics education vol. 3, no. 2 13 the myth of neutrality: mathematics education is political we acknowledge that participation on a panel does not imply that all panelists hold the same views. however, because the session description referred to concerns among many unnamed mathematics educators, the statement could be interpreted to mean that the participants were among those who are concerned. during the session, each panelist presented her or his particular responses to the issue of “centering” mathematics content in mathematics education research. in the context of a philosophical argument, harel (2010), in his handout, raised the following questions about the role of mathematics in mathematics education research: • are the methodologies and theories of mathematics education research independent of the discipline of mathematics? • if no, how does mathematics factor in these methodologies and theories? • what are the impacts of the different perspectives on these two questions on the fulfillment of the ultimate goals of mathematics education research? he also stated, “these questions are neutral—they entail no political agenda” (p. 2). harel further noted: the body of literature on whole number concepts and operations, rational numbers and proportional reasoning, algebra, problem solving, proof, geometric and spatial thinking produced since the 70s and into the 90s has given mathematics education research the identity as a research domain, a domain that is distinct from other related domains, such as psychology, sociology, ethnography, etc. in contrast, many current studies, rigorous and important in their own right as they might be, are adscititious to mathematics and the special nature of the learning and teaching of mathematics. often, upon reading a report on such a study, one is left with the impression that the report would remain intact if each mention of “mathematics” in it is replaced by a corresponding mention of a different academic subject such as history, biology, or physics. there is a risk that, if this trend continues, research in mathematics education will likely lose its identity. (p. 4) we question the neutrality suggested by harel (2010). heid’s (2010) commentary and question, as well as the symposium summary and harel’s aforementioned statement, are not neutral (blair, 1998). they are political statements and represent particular stances and positions on the value and production of knowledge. they should be acknowledged, recognized, and deconstructed as such. in our view, these statements also represent very public displays of power and privilege. the implications for such exercises of power, under the auspices of an institutional and organizational entity such as nctm, are profound, as they have the potential to marginalize scholarship within particular areas of focus as well as to marginalize scholars who devote themselves to this work. young schol martin et al. commentary journal of urban mathematics education vol. 3, no. 2 14 ars and graduate students are particularly vulnerable if the subtext of these statements is on pursuing what is valued in the field, as decided by those in positions of power, versus choosing what they want to make their life’s work. clearly, these can be overlapping choices but for many scholars they are not, and real conflicts can arise between senior scholars and junior scholars, between faculty members and graduate students, and along many other lines where issues of power and identity emerge as relevant. what mathematics? for whom? and for what purposes? when these statements and stances are further mined for their political content, several questions emerge about knowledge production in the domain. for example, an ethnomathematical-inspired response (see, e.g., powell & frankenstein, 1997) would first require one to ask, to whose mathematics are heid (2010) and harel (2010) referring? is it the very same school mathematics that has been used to stratify students, affording privilege to some and limiting opportunities for others (dime, 2007; gutierrez, 2000, 2008; gutstein, 2003; leonard, 2008, 2009; martin, 2009a, 2009b, 2010; stinson, 2009; tate, 1995)? mathematics, as a subject domain, is not acultural, without context or purpose, including the political (leonard, 2008), yet many students perceive school mathematics to be a narrow set of rules and algorithms that have little or no meaning to their lives. is this the mathematics to which heid, harel, and, perhaps, the other panelists might be referring? mathematics can also be a tool for understanding the world and, in the case of marginalized students, it can aid in understanding the social forces that contribute to their marginalization (gutstein, 2006; martin & mcgee, 2009). is this the mathematics to which heid and the panel members refer? who decides what counts as mathematics education research? more generally, in the spirit of scholarly exchange, we ask who is empowered or entitled to decide what counts as mathematics education research?2 are some types of studies, areas of focus, and theoretical perspectives privileged over others? who is being silenced in the context of such exercises of power? who belongs to the list of “many mathematics educators” to which the symposium summary refers? why is there a “growing concern” among these scholars about particular areas of inquiry in mathematics education when the history of research in the field is characterized by shifts from behaviorism to cognitivism, to constructivism, to situated, to sociocultural analyses? which areas of study are now 2 we believe that “peer review” is an incomplete and insufficient response to this question given that the ideological parameters for what constitutes knowledge in the field are established and reinforced in many other contexts (e.g., advising of students, funding of grants). martin et al. commentary journal of urban mathematics education vol. 3, no. 2 15 the causes for such concern? cannot significant and insightful findings about mathematics learning result from studies where there is not a focus on specific mathematics content? we agree, without pause, with the basic premise that mathematics content is important. we do not wish to minimize its importance. nor do we seek to overstate it. in fact, we have taught, and continue to teach, mathematics content in contexts ranging from elementary schools to universities. yet, for many scholars, including ourselves, subsequent efforts to add needed complexity to the understanding of learners, their social realities, and the forces affecting these realities have led them (and us) to take social, sociopolitical, and critical turns in their (our) work, away from overly narrow concerns with mathematics content. these turns have made salient many issues not typically pursued in mathematics education research, including issues of identity, language, power, racialization, and socialization. are these the turns that have prompted recent replies within the nctm context? a historical review of jrme would show that the vast majority of articles published do indeed focus on (school) mathematics content. so, the extra scrutiny imposed by “where’s the math?” is unclear. moreover, it has been somewhat standard practice for jrme to confine issues of equity, for example, to “special” issues of the journal.3 in many ways this practice has helped to relegate these issues and the authors of such scholarship to the margins. the most recent equity effort by jrme is being published in an online context where the special issue designation remains intact. dealing with equity-focused scholarship in this way is all the more interesting considering that equity is the lead principle of the principles and standards for school mathematics (nctm, 2000), the signature document for nctm. the fact that equity is the first principle in this document would seem to imply that equity is nctm’s foremost guiding principle. if this is indeed the case, then a fundamental question that could (should?) be applied to all articles and reports being considered for jrme and other nctm publications is: to what degree does this article (or report) under consideration contribute to equitable mathematical experiences and outcomes? it is interesting to note that none of the seven studies cited in heid’s (2010) editorial explicitly attend to issues of equity. because this lack of attention is representative of a more general trend, it may be true that jrme is not regarded as a “go-to” journal for mathematics education scholars who employ research methods and take epistemological positions considered outside the mainstream. of course, alternative outlets do exist, but this should not minimize efforts and events that 3 see jrme 1984 volume 15, number 2: special issue – minorities in mathematics (edited by westina matthews) and jrme 1997 volume 28, number 6: special issue – equity, mathematics reform, and research: crossing boundaries in search of understanding (edited by william f. tate and beatriz s. d'ambrosio). martin et al. commentary journal of urban mathematics education vol. 3, no. 2 16 impact the kinds of scholarship and perspectives that appear in what some scholars regard as the flagship journal in the field. the changing faces of mathematics education examination of the excerpt from harel (2010) presented earlier also reveals a concern about losing the “identity” of mathematics education research. not only was that identity established based on studies in the areas he cited—whole number concepts and operations, rational numbers and proportional reasoning, algebra, problem solving, proof, geometric and spatial thinking—but also that identity can be linked to the researchers who carried out studies in those areas. in a very real sense, those researchers became, and perhaps remain, the faces of mathematics education. throughout the time period identified by harel, from the 1970s to the early 1990s, those faces were predominantly white and predominantly male and most studies in the areas he identified did not address issues of equity or attend to the “social” and “emotional” conditions that he noted as being important.4 we note that as new scholars have entered the field and turned their attention to equity-oriented and critical scholarship, they have increasingly drawn from theories and methods outside of mathematics education and raised questions that go far beyond issues of content. we also note that many of the scholars conducting this research are scholars of color, female, and critical white scholars, who, while appreciating and respecting traditional areas of focus and research approaches, have partially eschewed tradition. certainly, this research can make, and is making, positive contributions to the identity of mathematics education research. furthermore, we argue that the students on whom equity scholarship often focuses—african american, latina/o, native american, and poor students— serve as canaries in the mineshaft for the long history of content-focused scholarship in mathematics education; a history that is many times longer and more indicative of priorities in the field than any recent scholarship that might be implied in the symposium summary. data on mathematics achievement among these students show that, despite some small gains, they continue to be underserved by mathematics education despite a proliferation of theoretical perspectives and content-focused research paradigms focusing on cognition, curriculum development, and assessment (secada, 1992; tate, 1997). rather than generating concern about studies that do not give priority to mathematics content, it may be more informative to understand why studies that have continued to do so have offered so little in the way of progress for students who remain the most underserved. minimal progress for these students would 4 the research of ed silver and colleagues is one notable exception (see, e.g., silver, smith, & nelson, 1995). martin et al. commentary journal of urban mathematics education vol. 3, no. 2 17 seem to demand that we pursue all promising areas of inquiry informing us about how to help them experience mathematics in ways that allow them to change the conditions of their lives. it is important to document which approaches and practices are effective with these students (leonard, 2008). now is not the time for restricting the production of knowledge. as scholars who are deeply concerned with equity issues, not only for children but also within the domain itself—and clearly these should be concerns shared in the jrme and nctm contexts—we believe the stakes are simply too high to remain silent on such efforts. many scholars in mathematics education have written about and acknowledged the gate-keeping role that mathematics has served in limiting meaningful participation in schools and society. any move in the directions of (a) using mathematics as the critical filter in regulating the production of knowledge about mathematics learning and participation and (b) consequently including and excluding scholars and scholarly ideas because they fall outside of some preferred areas of focus, is, in our view, an unfortunate one. it is a move that merely appropriates and instantiates the most effective methods for creating hierarchies in our domain. it is also a move that appears to represent a “back-to-basics,” traditional approach to mathematics education research. this move is contradictory; it implicitly calls for a return to prescriptive, narrow approaches to the study of mathematics learning and behavior in an increasingly complex world (e.g., the lives and mathematical development of students are more complex than the strategies they do or do not demonstrate; the lives and practices of teachers are more complex than their level of content knowledge). is this move not reminiscent of the ideological and epistemological debates that characterized the math wars? relevant and insightful knowledge about mathematics learning and participation should be welcomed not discouraged. what’s the context? we agree that the seven examples cited in heid’s (2010) editorial do a fine job of attending to mathematics content and are informative in their own right; but, in our view, they represent a limited range of approaches for studying mathematics teaching and learning and children’s mathematical development. in that, mathematics teaching and learning and children’s mathematical development are intertwined with a number of complex micro-, meso-, and macro-level forces. understanding how and why children interact with mathematics content in the ways that they do as well as how and why they learn is not a question of mathematics content alone. consider a hypothetical study of children’s systematic errors in multi-digit subtraction problems involving whole numbers.5 although hypothetical, the study 5 an extended version of this narrative appears in martin (in press). martin et al. commentary journal of urban mathematics education vol. 3, no. 2 18 description is representative of many that have focused on children’s systematic errors in multi-digit subtraction problems involving whole numbers (see, e.g., verschaffel, greer, & decorte, 2007). in our hypothetical study, the (hypothetical) female researcher draws primarily on developmental and cognitive psychology and her prior work has sought to identify universals in children’s mathematical thinking. her recent work has turned to questions focusing on the role of culture. in addition to this new focus, she has decided to extend her work to urban settings, hoping that it can contribute to discussions of equity by highlighting key areas of intervention for urban elementary school children. highlighted in one portion of her study is a student identified only as omari. it is reported that omari demonstrated poor performance on a series of problems across clinical sessions such that even his pattern of errors differed significantly from known results presented in previous studies: many children develop only a concatenated single-digit conception of multi-digit numbers. she characterizes omari’s misconceptions as reflecting low cognitive ability. his case, in turn, is used as a data point in a larger argument about the at-risk status of poor, urban children and as evidence to support the claims that “most children from low-income backgrounds enter school with far less knowledge than peers from middle-income backgrounds” (national mathematics advisory panel, 2008, p. xviii) and that “although low-income children have pre-mathematical knowledge, they do lack important components of mathematical knowledge” (clements & sarama, 2007, p. 534). here, we offer one possible, hypothetical contextualization of omari’s mathematical behavior relative to some of the considerations that we believe would shed additional light on his mathematical behavior; considerations that do not typically make their way into many content-focused studies. we build the context by drawing on recent research on the political economy of urban schooling that paints a vivid picture of how issues like race, class, housing and school segregation, and school policy interact to affect thousands of children in public schools (lipman, 2004; neckerman, 2007). we draw especially from work focused on chicago; the nation’s most racially segregated large city and a city whose districts and schools are emblematic of urban public education (lipman; neckerman). our purpose in presenting a hypothetical contextualization of omari is to demonstrate that context provides for profoundly different understandings of his “mathematics identity” (martin, 2000, p. viii) that not only brings into question the findings of the hypothetical study but also much of the reported nonor under-contextualized findings found in existing mathematics education literature. we begin our contextualization by noting that omari is a black child. we note this aspect of his identity not to essentialize his being in the world but to suggest the inextricability of identity development—racial, mathematical, gender, and otherwise—and mathematics learning and development, not as a predictor of martin et al. commentary journal of urban mathematics education vol. 3, no. 2 19 behavior but as a factor that influences how students are socialized and enculturated into local mathematical practices (oppland, 2010). omari lives with his mother, father, and grandmother in a working-class neighborhood. his neighborhood consists primarily of black families. the history of the neighborhood reveals that it has existed as a cultural enclave, first for various european immigrant groups, then for black residents. as the black population in the neighborhood increased, city institutions have increasingly underserved the neighborhood (neckerman, 2007). recent investment in the surrounding area coincides with gentrification and the dislodging of long-time residents in favor of wealthier residents (lipman, 2004). omari attends the neighborhood school, but the school is under threat of reconstitution or closure despite growth trends in achievement. the school is located in a district where nearly half the students are identified as black and nearly 90% are identified as black or latina/o (lipman, 2004; neckerman, 2007). eight percent of the students are identified as white and half of those are in special education, where they receive additional educational services. the remaining white students in the city attend either private schools or one of the selective charter schools. district policy is driven by attempts to close what it identifies as a racial achievement gap between the nearly 90% black and latina/o student population and the 4% white student population (lipman; neckerman). community groups and leaders have protested against the growing number of charter schools and called for more school funding and school improvement plans that provide students in neighborhood, non-charter schools with equitable learning opportunities (lipman). these groups and leaders have also argued that racial achievement gap rhetoric throughout the district sends a damaging message to black children about their identities and contributes to a larger discourse in the city that pathologizes black communities and families. a number of new and inexperienced teachers have been hired recently at omari’s school (lipman, 2004; neckerman, 2007). many of these teachers know very little about the history of the community, and they struggle to engage the children they teach (neckerman). omari’s teacher reluctantly took her job after failing to be hired elsewhere and plans to leave when a better opportunity opens up, as many of the new teachers plan to do. omari’s teacher also struggles to teach mathematics and, as a result of her struggles, she has helped to proliferate some of the errors and misunderstandings that omari demonstrated in the findings of the (hypothetical) study. omari’s test scores from the previous year, however, show that he scored in the 90th percentile for mathematics and 85th percentile for reading on the state assessment test. this year, omari has been disciplined by his teacher many times, as have most of the african american boys in the classroom (ferguson, 2000; kunjufu, 2005). omari’s teacher is often upset with him for using non-standard methods in mathematics. omari has stated to his teacher that he martin et al. commentary journal of urban mathematics education vol. 3, no. 2 20 likes his methods better than the “school way” and, like many of his classmates, is often confused because the teacher makes mistakes in her explanations. mathematics is no longer omari’s favorite subject and he is resigning himself to doing “school math” in the way his teacher tells him. however, outside of school, omari prides himself on being able to help his mother and grandmother go shopping at the local grocery store. omari is particularly proud of his ability to correctly add all the numbers on the checkout receipt. omari’s grandmother decided it would be good practice for him and, in the evening after shopping, she reads the numbers back to omari so that he can add them cumulatively. initially, this activity was a struggle for omari but, with help from his grandmother, he began to develop efficient methods for carrying out the calculations. during the week in which omari participated in the study, his grandmother was seriously ill and he was very worried about her. context produces different knowledge although hypothetical, our contextualization of omari clearly demonstrates how context produces different knowledge, and why asking what’s the context? is important. recent equity-focused scholarship provides scientific evidence that forces us to consider the multiplicities of complexities of omari’s mathematics development and identity. this research, by a growing number of scholars, brings attention to issues of power, identity, language, and race (e.g., berry, 2008; dime, 2007; gutiérrez, 2000, 2008; jackson, 2009; johnson, 2009; leonard, 2008; 2009; malloy & jones, 1998; powell, 2002; spencer, 2009; stiff & harvey, 1988; stinson, 2009; tate, 1995; taylor, 2005; walker, 2006; weissglass, 2002) and has allowed us to begin altering the conversation on children like omari who, quite frequently, have been constructed in deficit-oriented ways. critical analyses of research (e.g., gutiérrez, 2008; martin, 2009a, 2009b, 2010; valero & zevenbergen, 2004) have shown that content-focused studies that ignore or simplify the larger social context have often helped to normalize these constructions by suggesting, for example, that poor and minority children enter school with only premathematical knowledge and lack the ability to mathematize their experiences, engage in abstraction and elaboration, and use mathematical ideas and symbols to create models of their everyday lives. left unanswered is whether researchers who report these findings understand, even partially, the everyday lives of these children (martin, 2009b). just 10 to 15 years ago, considerations such as those pointed out in omari’s story were either understudied or underconceptualized in children’s mathematical development. why return to a time when mathematics education research largely ignored such considerations (lubineski & bowen, 2000; secada & meyer, 1989)? martin et al. commentary journal of urban mathematics education vol. 3, no. 2 21 closing remarks it is important to note that invoking equity-oriented concerns does not represent our own attempt to privilege one kind of scholarship over other forms. nor is it an attempt to suggest that equity concerns are mutually exclusive with the personal commitments of heid (2010) and the members of the panel (nctm, 2010). our concerns rise well above the personal and are focused on knowledge production and regulation in the field, including contexts like jrme and the nctm research presession. misinterpreting and misconstruing our scholarly critique as personal would be disingenuous and misses the point. questions such as “where’s the math?” (heid, 2010) and expression of a “growing concern among many mathematics education scholars regarding the lack of attention to mathematics in much of the current work in mathematics education” (nctm, 2010, p. 60) represent political stances and are symbolic of larger power relations in the domain. they are not neutral and we find it necessary to ask whose interests are served by these political stances? what intellectual territory and spaces are being claimed or reclaimed by such concerns? we raise these and our earlier questions knowing that the enterprise of mathematics education is no different than other societal contexts characterized by power relations. mathematics education, as an enterprise, benefits from a variety of research perspectives and approaches. nevertheless, mathematics should not be the gatekeeper for the production of knowledge in the field. editor’s note: deborah ball, michael battista, jere confrey, guershon harel, and patrick thompson (the presession panelists) and kathleen heid (jrme editor) were provided an advance copy of this commentary and invited to write a response commentary; see jere confrey’s and michael battista’s responses, this issue. references berry, r. q., iii (2008). access to upper-level mathematics: the stories of african american middle school boys who are successful with school mathematics. journal for research in mathematics education, 39, 464–488 blair, m. 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(2002). inequity in mathematics education: questions for educators. the mathematics educator, 12(2), 34–39. journal of urban mathematics education december 2011, vol. 4, no. 2, pp. 96–130 ©jume. http://education.gsu.edu/jume michael meagher is an assistant professor in the department of secondary education in the school of education in brooklyn college, city university of new york, 2608 james hall, 2900 bedford ave., brooklyn, ny 11210; email: mmeagher@brooklyn.cuny.edu. his research interests are in the areas of mathematics teacher preparation in urban schools and advanced digital technologies in the teaching and learning of mathematics. andrew brantlinger is an assistant professor in the department of teaching, learning, policy, and leadership in the college of education – university of maryland, 2226 benjamin building, college park, md 20742; email: amb@umd.edu. his research interests are in the areas of the sociology of education, critical pedagogy, and mathematics teacher preparation. when am i going to learn to be a mathematics teacher? a case study of a novice new york city teaching fellow michael meagher brooklyn college city university of new york andrew brantlinger university of maryland college park in this article, the authors present a case study of a mathematics teaching fellow of the new york city teaching fellows program. the presentation focuses on the teaching fellow’s family and educational background, her beliefs as a novice teacher, preparation to teach mathematics, and first-year experience teaching middle school mathematics in a “high-needs” school in new york city. the authors contend that although the teaching fellow articulated reform-oriented instructional beliefs, she was unable to enact them in the classroom. this lack was due, in part, to the inadequacies in the induction support system that was promised to her. the authors situate the case study using results from a larger study of novice mathematics teaching fellows and analyze the case study from a perspective that supports reform-oriented approaches to mathematics teaching. keywords: alternative teacher education, mathematics education, reform mathematics teaching, urban education he new york city teaching fellows (nyctf) program was started in 2000 to address “the most severe teacher shortage in new york’s public school system in decades” (nyctf, 2010, p. 1) and to replace uncertified teachers with (transitionally) certified teachers in “high-needs” schools (goodnough, 2000a, 2000b, 2004). from 2004–2008, nyctf was the largest program in the united states providing an alternative route to teaching certification. currently, one in four mathematics teachers in new york city come through the teaching fellows program (nyctf, 2010) and, over the past decade, more than two-thirds of new middle and high school mathematics teachers entering the new york city public school system were teaching fellows (v. bernstein, personal communication, 2006; nyctf, 2010). t meagher & brantlinger urban mathematics teacher journal of urban mathematics education vol. 4, no. 2 97 in this article, we present a case study of kelly, 1 a first-year mathematics teaching fellow. we focus on kelly’s social and educational background, her beliefs as a novice teacher, preparation to teach mathematics, and first-year experience teaching middle school mathematics in a high-needs school in new york city. we situate kelly’s individual case study in a larger context by using results from a larger observational study of 8 novice mathematics teaching fellows (mtf) completing state required graduate coursework at four universities; survey data from 167 mtf who, like kelly, had taught for one year in schools; and other research that we and our colleagues have conducted relating to mtf (see brantlinger, cooley, & smith, 2009; donoghue, brantlinger, meagher, & cooley, 2008; foote, brantlinger, haydar, smith, & gonzalez, 2011). we analyze kelly’s case study from a perspective that supports reform-oriented approaches to teaching of mathematics (nctm, 2000). our perspective is one of looking for questioning, applying strategies, communicating, reasoning and reflecting, and tasks that engage students in higher-order thinking, novel problem solving, and communication of their developing ideas about mathematics. review of literature urban districts have had chronic difficulties recruiting and retaining teachers qualified to teach in such areas as mathematics, science, and special education (levin & quinn, 2003; liu, rosenstein, swan, & khalil, 2008). as a result, highneeds urban schools that serve lower-ses youth of color are more likely to be staffed by less qualified and less experienced teachers than schools that serve higher-ses student populations (lankford, loeb, & wyckoff, 2002; peske & haycock, 2006). given the link between teacher quality and student achievement, these raceand class-based gaps in human resources pose a serious problem for students in high-needs urban schools (peske & haycock, 2006; rivkin, hanushek, & kain, 2005; rockoff, 2004; sanders & horn, 1994). over the past decade, issues of teacher quality and teacher recruitment have received increased attention from policymakers, philanthropists, education researchers, the media, and the public (kramer, 2010; levy, 2000; peske & haycock, 2006; rotherham, 2008; u.s. department of education, 2010). the no child left behind act of 2001 (nclb) has put districts under considerable pressure to find “highly qualified” teachers in such subjects as mathematics, science, and special education. it must be noted, however, that the problem of recruiting and retaining qualified teachers for new york city schools pre-dates nclb by at least two decades. in 1990, some 14,000 uncertified or “temporary license” teachers worked in new york city schools, up from 7,000 in 1980 (goodnough, 2004). 1 a pseudonym, as are all names of people and places throughout. meagher & brantlinger urban mathematics teacher journal of urban mathematics education vol. 4, no. 2 98 in the spring of 2000, to help improve the quality and certification status of staff working in the more than 100 new york city public schools deemed “failing,” state officials paved the way for alternative routes to teaching (goodnough, 2000b). the nyctf program was a public relations coup for the new york city department of education (doe) because the newly minted “transitional” teaching license allowed education officials to count teaching fellows among the ranks of certified teachers after they fulfilled minimal preservice preparation program of 200 hours and passed state certification exams (goodnough, 2004). to allay concerns about what the media and even the doe referred to as “boot camp” preservice training (goodnough, 2000a), doe officials pointed out that the fellows were the “best and the brightest” who held prestigious educational credentials prior to admission to nyctf, and that many had real-world work experience as professionals, hence were fully qualified (see editorial by new york city schools chancellor levy, 2000). the district and nyctf literature (nyctf, 2010) also touted an intensive mentoring and induction program to support the fellows once they became teachers of record (new teacher center, 2006). alternative teacher certification alternative routes to teacher certification are the alternative to traditional, 4year undergraduate programs housed in university colleges (schools or departments) of education. early-entry alternative routes are those in which participants become certified teachers of record in a comparatively short timeframe, often after completing 200 combined hours of coursework and fieldwork. the theory behind early-entry programs is twofold: (a) that program participants bring particular skill sets, knowledge, and dispositions (e.g., content knowledge, real-world experience, professionalism, enthusiasm) that makes the full slate of traditional pre-employment coursework unnecessary, and (b) that effective teacher development takes place in the classroom with appropriate on-site induction and mentoring support (johnson & birkeland, 2008). in many early-entry alternative route programs, including nyctf, participants continue to take courses required for standardor full-teacher certification as they begin full-time teaching with transitional or temporary licenses. a number of studies have compared the characteristics of participants in early-entry alternative and traditional route programs who teach in similar school contexts and have the same number of years of experience. many of these comparative studies address the effectiveness with which alternative and traditional route teachers with similar experience and similar teaching placements effect student achievement or retention of classroom teachers (i.e., retention) (see, e.g., boyd, grossman, lankford, loeb, & wyckoff, 2006; darling-hammond, holtzman, gatlin, & heilig, 2005). taken as a whole, this research finds that traditionally and alternatively certified teachers with similar experience in similar teaching meagher & brantlinger urban mathematics teacher journal of urban mathematics education vol. 4, no. 2 99 contexts are, for all intents and purposes, similarly effective at raising student achievement. when it comes to the effectiveness of teachers from traditional and alternative pathways, there is more within-pathway variation than betweenpathway variation (boyd et al., 2006). at the same time, this research indicates that alternative route teachers, at least those in the nationally prominent nyctf and teach for america programs, have considerably higher rates of attrition than traditional teachers who teach in similar schools (boyd et al., 2006; darling-hammond et al., 2005; stein, 2002; veltri, 2008). there are several possible explanations for this. first, it could be that many participants in early entry alternative routes see teaching either as a resume builder or as a possible career they can “try out” (chin & young, 2007; veltri, 2010). second, it could also be because, as one study of new york city teachers prior to the nyctf program finds (darling-hammond, chung, & frelow, 2002), participants in early-entry alternative route programs report feeling, on average, less prepared to teach than participants in traditionally certified programs. third, it could be because the induction and mentoring components of early-entry alternative programs often fail to live up to their promise of helping new teachers learn to teach on the job (foote et al., 2011; humphrey, wechsler, & hough, 2008; veltri, 2008, 2010; zeichner & schulte, 2001). given the substantial amount of variation of teacher quality within pathways and substantial variation of program quality of both alternative and traditional route programs alike, some scholars make the argument that the field needs to move past simple alternative vs. traditional route comparisons and simplistic arguments about one route or the other being uniformly superior. humphrey and colleagues (2008) posit that, while early-entry alternative programs adequately prepare some types of candidates, they do not work for all types. based on a study of seven early-entry alternative route programs and their participants, these scholars argue that whether or not a particular route is effective depends on an interaction between the program, the participant, and the context of her or his initial teaching placement. reform-oriented ideals meet classroom realities research on novice teachers who have gone through traditional routes indicates that many hold idealistic and reform-oriented (e.g., student-centered, progressive, constructivist) views of teaching as preservice candidates, but adopt “traditional” teaching methods (e.g., lecturing, emphasizing student control and memorization) once they become teachers of record (costigan, 2004; flores, 2006; lortie, 1975). cohen (1988) argued that novice teachers do not maintain their reform-oriented perspectives because traditional teaching methods are more familiar and less demanding than reform-oriented methods. traditional methods also help novice teachers cope with discipline issues and navigate an unfamiliar meagher & brantlinger urban mathematics teacher journal of urban mathematics education vol. 4, no. 2 100 curriculum. some blame schools of education and their traditional approaches to teacher preparation (e.g., national council on teacher quality, 2006; walsh & jacobs, 2007) for this situation arguing that they over-emphasize re-form-oriented ideals, ideals that run counter to the culture of public schools and “counter to the main thrust of educational reform efforts in the u.s. in the early twenty-first century” (labaree, 2005, p. 277). yet, this phenomenon is not limited to traditional teacher preparations. a study by costigan (2004) indicates that a similar mismatch exists between how participants in early-entry alternative route programs imagine themselves teaching as preservice candidates versus how they teach as first-year teachers. based on a study of approximately three-dozen participants in the nyctf program, costigan concludes that there is a disjuncture between the classrooms alternate route teachers envision as preservice candidates and those they implement as novice teachers. as preservice candidates, these teaching fellows articulated high-minded ideals (e.g., “befriending” and “molding students”). however, once they begin teaching, their interview and journal narratives focus almost exclusively “on issues of daily survival, such as dealing with the troublesome children in their classes” (p. 133). costigan reports that the focus on management is fueled by the fellows’ desire to create what one called a “safe space” (p. 136) where learning can occur but also where they can realize the ideals they articulated as preservice candidates. goodnough’s (2004) observational study of one first-year teaching fellow in the nyctf program supports costigan’s (2004) findings. according to goodnough, the teaching fellow she studied was initially “motivated by idealism and naiveté” (p. 51) and expressed the desire to employ student-centered (e.g., whole language) instructional approaches. however, a mandated scripted curriculum and an in-school mentor compelled her to employ teacherand control-centered methods. initially resistant, this first-year teaching fellow eventually “accept[ed] the party line and knuckl[ed] under to a routine that instinct told her would not help the children or her in the long term” (p. 113). it is important to note here that the nyctf program partners with schools of education that provide teaching fellows with state required preservice and in-service graduate coursework. hence, it may very well be that university-based teacher educators strongly emphasize the same progressive and reform-oriented ideals in the minds of teaching fellows that they are accused of planting in the minds of traditionally certified teachers (walsh & jacobs, 2007). at the same time, national, standards-based reforms in mathematics adopted in the period from 1985–2000 recommend that teachers adopt non-traditional, reform-oriented teaching methods (cohen & hill, 1998). the national council of teachers of mathematics (nctm) (1989, 1991, 2000) asks mathematics teachers to emphasize student thinking and student-centered problem solving throughout meagher & brantlinger urban mathematics teacher journal of urban mathematics education vol. 4, no. 2 101 instruction. nevertheless, despite such professional opinions and evidence that support the effectiveness of reform mathematics strategies (stein & lane, 1996; schoenfeld, 2002), large-scale studies (e.g., third international mathematics and science study) suggest that reform instruction has yet to take hold among u.s. teachers—inclusive of urban mathematics teachers (newmann, lopez, & bryk, 1998; stigler & hiebert, 1999; weiss, pasley, smith, banilower, & heck, 2003). a number of scholars (e.g., kennedy, 1999; stigler & hiebert, 1999) conjecture that this situation is largely the result of the institutionalization of traditional teaching during preservice teachers’ own k–12 education—the apprenticeship of observation (lortie, 1975)—as well as experiences with cooperating fieldplacement teachers who too often use traditional methods. stigler and hiebert note further that, in contrast to teachers in some high-performing countries, japan in particular, u.s. teachers generally work in isolation and fail to have substantive exchanges about teaching and learning with colleagues. however, increased mentoring by informed and competent mentors and professional development in reform methods may be changing this scenario for novice teachers in the u.s. (new teacher center, 2006). methods here we present one of eight case studies detailing a fellow’s first-year teaching; it is from a large-scale, mixed methods case study project of mtf completed over a 2-year period. this case was selected for presentation because kelly, the young woman in the case, was representative of the large number of mtf who were recent college graduates: white, female, and middle class (donoghue et al., 2008). kelly was also chosen because she articulated her ideas clearly and in detail. study context metromath scholars at city university of new york (cuny) conducted the large-scale, case study project. metromath was a center for learning and teaching funded by the national science foundation from 2004–2009. as metromath scholars, we were among a dozen researchers who worked on the project and were responsible for collecting data in kelly’s classroom during the 2006–2007 school year. while our primary focus was on kelly’s instruction, we also collected data on kelly’s experiences as she completed her state-mandated master of the arts of teaching (mat) degree coursework at borough university (bu), one of four partnering universities that provided graduate courses for mtf during the 2006–2007 academic year. at that time, kelly was one of approximately 55 firstyear mtf taking graduate courses at bu; a slightly smaller number of second meagher & brantlinger urban mathematics teacher journal of urban mathematics education vol. 4, no. 2 102 year mtf were also taking graduate courses at bu. during this time, i, the primary author, worked as an assistant professor teaching methods and research courses to approximately 40 mtf at another nyctf partner university. while i was not kelly’s instructor, my regular contact with other mtf provided insights that informed data collection and analysis. school context and case study class in her first year, kelly taught at a large, non-selective, middle school in a new york city neighborhood predominantly populated by african american, caribbean, and hispanic families. the school is a “high-needs” school; new york city doe data indicates that approximately three-fourths of student families receive public assistance at the school (new york city doe, 2011). similar to many city middle schools, at the time of this study, the school was divided into academies to create a small school feel and give students a cohort model whereby they took most of their subjects together in academically differentiated groups of approximately 30. at the time that kelly began teaching, the school was undergoing a “restructuring year,” which meant that it had been failing to sufficiently raise student achievement for 2 consecutive years. the class that was the focal point of our study was, within the context of the school, a high-track class. kelly also taught two lower-track courses in her first year. the high-track class consisted of 17 black (i.e., caribbean and african american) girls, 4 black boys, 3 latinas, 4 latinos, 1 asian girl, and 1 asian boy. while advanced for the school, the students in this class tested at grade-level in mathematics on average. our decision to observe kelly’s high-track class rather than the lower-track classes should be viewed in the context of our larger set of case studies participants, who taught a mixture of advanced-, regular-, and remedial-track courses. our entire corpus of classroom observations reflected this diversity. to add depth, we observed kelly’s lower-track classes on two occasions in her first year and similar lowertrack classes in her second year. research questions in this article, we explore the following four research questions: 1. what is the nature of the preparedness of a typical first-year mathematics teaching fellow to teach in a high-needs middle school? 2. what are the instructional views and goals of a first-year mathematics teaching fellow? meagher & brantlinger urban mathematics teacher journal of urban mathematics education vol. 4, no. 2 103 3. what does teaching “look like” in the classroom of a first-year mathematics teaching fellow, and how consistent is this instruction with reform-oriented mathematics teaching? 4. what is the nature of the induction and support available to an alternatively certified mathematics teacher in nyctf, and how does it impact her professional development? reform-oriented research perspective our expectations for kelly’s teaching and our analysis of the observed teaching episodes are based on a reform approach to mathematics instruction (nctm, 2000). our perspective is one of looking for questioning, applying strategies, communicating, reasoning and reflecting, and tasks that engage students in higher-order thinking, novel problem solving, and communication of their developing ideas about mathematics (e.g., henningsen & stein, 1997; stein, grover, & henningsen, 1996; nctm, 1991, 2000). we believe this perspective should be at the center of mathematics teacher education programs and of early years mentorship of new teachers. the perspective we take is also appropriate because we demonstrate that kelly expressed an interest in learning and implementing reform-oriented methods of teaching middle school mathematics. it is further appropriate because students in high-poverty schools have typically been taught using traditional methods in teacher-centered, teacher-controlled classrooms (cwikla, 2007; haberman, 1991; lipman, 2004; newmann, lopez, & bryk, 1998; weiss et al., 2003) in spite of evidence that they would benefit from reformoriented approaches (boaler, 2002, 2006; boaler & staples, 2008; schoenfeld, 2002; stein & lane, 1996). data sources and analysis observational data. we collected classroom observational data on kelly in the form of field notes, videotapes, and audiotapes twice a month (on average) during her first year of teaching. the video and audio data were used as supplements to the field notes. interview data. after each classroom observation, during a postobservational interview, kelly was asked to reflect on the lesson that was just observed. we asked follow-up questions based on things that stood out to us in the lesson, things that we were confused about, and more general issues we were discussing with the larger group of metromath scholars. these post-observation interviews were recorded and transcribed. we also conducted in-depth, formal, question-and-answer interviews with kelly (and the seven other case study participants) at the beginning and end of the 2006–2007 school year. these interview meagher & brantlinger urban mathematics teacher journal of urban mathematics education vol. 4, no. 2 104 questions dealt with such aspects as the teaching fellows’ educational background, ongoing graduate coursework at bu (and three other institutions), and beliefs about teaching mathematics in new york city public schools. specifically, the interview was divided into six sections, questions that explored (a) decision to become a mathematics teacher, (b) preservice summer program at the partner university, (c) general goals for teaching and teacher identity, (d) teaching mathematics, (e) particular school setting that the mtf will be or would like to be teaching at fall semester, and (f) policy-related issues relevant to secondary and middle school mathematics. specific questions in these sections included: how were you taught mathematics in high school? based on your experiences in the nyctf preservice program, how prepared do you feel to enter the classroom? what strengths do you have that help or will help you succeed as a mathematics teacher? what do you believe to be the big ideas of (middle and secondary) school mathematics? describe the communities and neighborhood environment the school is situated? have you heard of the national council of teachers of mathematics or the standards-based reform movement in mathematics education? here, we collated portions of the interviews discussing the fellows’ beliefs about the nature of school mathematics and mathematics teaching and learning. survey data. we collected survey data from 167 in-service mtf at the four nyctf programs for mathematics in august of 2007 (approximately 70% of kelly’s mtf cohort who remained in the classroom after one year). the design of the surveys, informed in part by the observational component of the large-scale project, allowed us to examine the representativeness of our eight cases to the mtf who completed the surveys (e.g., their use of required textbooks, beliefs about students) and to compare these case participants’ ideas to the aggregate data of the entire cohort. the survey was a combination of open-ended questions (e.g., briefly explain what aspect or experience of the summer program you believe most helped you to prepare to teach math? what do you consider to be effective math teaching? what are some important similarities or differences between students in “high-needs” urban schools and students you went to school with?) and likert-like questions (e.g., use the following scale—never, briefly, occasionally, regularly, extensively—to rate how often students in low-track and high-track courses should engage in the following activities: e.g., hypothesis, theory, or generalizations; relearn basic skills (taught by past teacher); and use manipulatives and models). here, we examined only the survey questions related to the fellows’ views of mathematics and mathematics education. data coding. the coding scheme used to analyze field notes was produced in collaboration with the metromath scholars (i.e., research team) during our weekly research meetings. it emerged from an open-coding process (emerson, fretz, & shaw, 1995) whereby several members of the team coded early sets of meagher & brantlinger urban mathematics teacher journal of urban mathematics education vol. 4, no. 2 105 field notes from the eight case studies and, through discussion as well as repeated application to the new field notes being generated, agreed on a set of codes that could be applied to all the field notes of the project (e.g., professional development, classroom management, teacher math questions, and opportunity for meaning making). we used these broad codes to partition the data and to focus in on particular issues such as our research questions. we further coded within each of these categories (e.g., under classroom management we developed “sub-codes” such as positive and negative interactions, and classroom culture). we wrote memos and developed themes for each of the larger codes based on the within category coding and reading. reliability of the coding was established through the fact that the field notes from all eight case studies were being used by all the research team for various parts of the project and, thus the application of codes had multiple checks. in the case of kelly, we (the authors here) were the observing team for the classroom visits and reached consensus on the coding of the field notes. in this article, we used the codes to mark data (e.g., recorded classroom events, excerpts in interviews) relevant to the aforementioned research questions. the case of kelly in the remainder of the article, we describe kelly’s transition into mathematics teaching in a high-needs urban school in the context of her being an mathematics teaching fellow in the nyctf program. the description is divided into sections that address the following topics: (a) background information about kelly relevant to her current job as a mathematics teacher and her preparation to teach, (b) kelly’s views about mathematics teaching in urban schools and the 2-month preservice preparation she received, (c) a detailed description of an early lesson, (e) a mid-semester review of how kelly perceived her teaching, and the nature and effect of the induction support she was receiving, (f) a detailed description of lesson later in the year, and (g) a description of kelly’s reflections on her first year of teaching. these sections roughly conform chronologically to kelly’s first year of teaching. the first two sections are relevant to our first research question about kelly’s preparedness to teach, the second and fourth sections are relevant to our second research question about kelly’s instructional views and goals, the third and fifth sections are relevant to our third question on the nature kelly’s mathematics instruction, and the fourth section addresses our fourth research question on the nature and effect of the induction support kelly receives. kelly’s background and preparation to teach kelly was 24-years of age when she became a teacher of record. white and female, she was, in many ways, the prototypical mathematics teaching fellow. meagher & brantlinger urban mathematics teacher journal of urban mathematics education vol. 4, no. 2 106 indeed, more than 55% of kelly’s nyctf mathematics cohort members were between the ages of 21–25 when they began the program, more than 55% were female, and more than 50% were white (donoghue et al., 2008 2 ). and kelly, as did approximately 75% of her mathematics cohort, self-identified as middle or upper class (donoghue et al., 2008). kelly came to her position as an urban middle school teacher after having been away from exposure to any mathematics coursework for the 6 years she was in college and graduate school. in the interview conducted a week before she began teaching, kelly reported that in high school she had received high marks in all subjects, including mathematics. she also recalled that she was tracked into selective programs, including high-track mathematics courses, throughout her years of k–12 schooling. in the preservice interview she claimed: “i think i’m good with math but i was [also] in ap [i.e., advanced placement] english, history and science. so i’m a good student.” despite receiving the highest possible score on the ap calculus exam, she admitted that she was not interested in studying mathematics or science in college. statistics i was the only mathematics course she completed in college. again, kelly was typical of other fellows in her cohort in this regard. two thirds of kelly’s cohort indicated that they were in upper or honors tracks for mathematics in high school. and, while more than 90% took either ap calculus or calculus i in high school or college, the typical mathematics fellow only took a few mathematics courses in college and bit less than one third of her cohort had the equivalent of a mathematics minor or above (donoghue et al., 2008). kelly graduated from a large, prestigious eastern state university with a double major in international relations and religious studies. she spent her senior year at an american university in africa. she was interested in pursuing a career with the intelligence services but postponed pursuing this option in order to complete a master of arts degree in islamic studies. kelly said that she still retains a long-term goal to become a university professor. she became interested in teaching when it became apparent that the possibility of working in the intelligence community would involve commitments that she was not ready to make at that time. she researched alternative teacher certification programs and perceived the highest need area to be mathematics. feeling that her high school mathematics background was sufficient to teach middle school and with a desire to live in new york city, kelly interviewed with teach for america and nyctf. kelly reported that after her teach for america interview she “was completely turned off by” the other applicants and the interviewers and what she perceived to be a paternalistic attitude of the program, namely: “we’re gonna go into these poor places and we’re gonna be the savior” and 2 of the 92% of kelly’s cohort that reported their race on the preservice survey, 51% identified as white, 15% as east or south asian, 23% as black, and 11% as latino/a (both non-white and white). meagher & brantlinger urban mathematics teacher journal of urban mathematics education vol. 4, no. 2 107 “we’re gonna go in and save places that can’t pull it together themselves.” in her nyctf interview, she found “normal” interviewers and a more “laid back” approach and opted for that program. nyctf assigned kelly and another 55 mtf to a graduate program (mat) at bu. the remaining 240 (approximately) were assigned to one of three other nyctf partner universities for mathematics. kelly spoke positively of her experience in the 2-month nyctf preservice summer program at bu as did the majority of those mtf assigned to this university (brantlinger et al., 2009). she found the instructors, several of whom had worked as administrators in new york city schools, to be effective at providing general information and specific tools to survive teaching in new york city public schools. as she said, “it was a great experience. they’re so organized … everybody [i.e., her peers] had such creative, dramatic math lessons and then what’s amazing about bu, which is gonna get me through the year.” kelly and her bu cohort mates noted that the summer preservice coursework at that university focused extensively on “instructional design and delivery” and classroom management (brantlinger, et al., 2009). kelly reported, “the mantra at [bu] is ‘organization and discipline is gonna get you through your first year.’” program directors at bu confirmed this focus, claiming to emphasize the “pragmatic over the theoretical” in their preservice program. the preservice curricula at the other university programs were not so focused on classroom management and organization. however, it is important to note that, kelly and her bu cohort mates were more appreciative of their preservice preparation than mtf at the other university partners (brantlinger et al., 2009). as kelly put it: it’s nice like at [bu] that they prepare you for that. if you’re not being prepared for a hard to staff school or these kind of discipline issues-if you came from a college that didn’t even deal with that, then you’d be scared to death and completely unprepared. (august 2006) this quote speaks to an overarching theme of kelly’s discourse, namely, her perception that urban schools are riven by discipline issues and her first task in her teaching is to control the situation. in this sense, she was like the novice teaching fellows costigan (2004) interviewed. as we will see below, there is something of a contradiction of kelly’s considerable desire for control, as she also wanted to alleviate students’ fear of mathematics and use it as a liberating tool. kelly’s views of urban mathematics instruction prior to teaching in the preservice interview, we asked kelly about her initial thoughts about teaching mathematics in a high-needs new york city school. kelly described her readiness and gratitude: “i think that [bu] is amazing at getting us ready for that meagher & brantlinger urban mathematics teacher journal of urban mathematics education vol. 4, no. 2 108 first day of school.” while feeling prepared to begin the school year, kelly’s confidence seemed also to stem from a relationship she developed with a supervisor professor from bu. he had a professional relationship with kelly’s school and told her that he thought she would be successful. kelly reported that he had been proactive in putting her in contact with his mentees from previous years who worked at the school. he was also helpful in providing advice on how to survive in the first months of teaching. much of kelly’s calm in facing the coming school year appeared to be based on her sense that she would be able to call on this professor if she was having difficulty in her professional life. when asked about how her secondary teachers taught mathematics, kelly described traditional instruction and teacher-centered classrooms. she noted that teachers typically did one or two sample procedural problems on the board and then gave students similar problems to complete on their own. she observed that teachers only interacted with the students when called upon to do so. kelly felt that this system was possible because her schools were in a middle-class, midwestern suburb, hence were full of “highly motivated” students (for more on teaching fellows’ perspectives on teaching urban youth, see brantlinger, cooley, & brantlinger, 2010). kelly stated: i think the teachers pretty much thought, “i don’t have to do much and these kids will pick up on it” …they sat at an overhead, would do one or two examples, and put an entire list of problems on the board and tell us to start working. and we would do it but i think we were just highly motivated, good students that if ten problems were put on the board we’re gonna go at it. (august 2006) kelly was successful as she earned high grades and scored well on standardized tests. at the time, she felt that she was learning mathematics. however, poised to enter the classroom in the fall of 2006, she expressed different feelings about her secondary mathematics preparation. she now considered her former mathematics teachers to have used a deficient approach largely because they failed to link school mathematics to her own life or to real-world applications. as kelly elaborated: i never actually learned—which is what i’m struggling with now, is how to apply to real life. [my high school teachers] didn’t actually do any of those motivations to make that link. so when i’m sitting there trying to think, “hmmm, how does an imaginary number actually—how am i supposed to have—make them want to learn and tell them it’s important to learn imaginary numbers?” …and then our [bu] professors are telling us it’s not really about the formula, give them a calculator or both, or give them the formula. and i’m—wait a minute, that was all i learned, how to memorize the formula and plug it in. (august 2006) meagher & brantlinger urban mathematics teacher journal of urban mathematics education vol. 4, no. 2 109 kelly expressed a desire to address this situation by making the middle school curriculum she would teach more relevant to her future students’ lives. however, she admitted that she did not have clear ideas about how she could accomplish this in the days before she would begin as a teacher of record. as the above excerpt indicates, kelly judged her preservice program at bu as not particularly helpful in this regard as mathematics-specific teaching methods were not a focus (brantlinger et al., 2009). hence, it is not surprising that kelly did not know what is meant by “reform” or “student-centered” approaches to mathematics education. this lack of understanding was symptomatic of the larger lack of contextual and professional understanding exhibited by kelly and other preservice mtf we interviewed. when asked about nclb legislation, kelly reported that she was aware of it but unfamiliar with the details. to be fair, it may be that bu and the other universities cover these topics in more depth during in-service graduate coursework that follows the preservice program. kelly did not have much educational or practical mathematical background to draw on as she entered her first year. while kelly appreciated the management and lesson planning techniques she learned in the bu preservice program, she expressed concern about her lack of exposure to mathematics-specific teaching methods and mathematics content prior to becoming a teacher of record. when asked about the specifics of her teaching, she made general statements, and admitted that she had not developed a well-formed vision of mathematics instruction: i know how to get through my first week. i don’t really know how i’m gonna teach math beyond that … i don’t know how i’ll do it because it’s probably impossible but i would like to find some way to vary each class … i’m shooting for two or three days a week have an exciting motivation … i don’t understand how i can do instruction, independent work, group work … i bought a book, barnes & noble the other day of critical thinking problem solving. (august 2006) kelly was not unique. while the preservice mtf at bu felt generally well prepared to manage and organize their classrooms, like kelly, more than 45% reported on a survey item being only “somewhat prepared,” “under prepared,” and “not prepared at all” to “teach mathematics using a variety of instructional methods” in their first year (brantlinger et al., 2009). in a similar vein, kelly, the seven other case studies, and an additional 12 mtf in kelly’s cohort (at bu and other universities), were generally unable to articulate a complex, detailed, and coherent vision for teaching mathematics in interviews conducted prior to the 2006–2007 school year (donoghue et al., 2008). when asked in the preservice interview how they would “teach for understanding,” what they thought “effective mathematics teaching looked like,” or to describe what they envisioned as they “imagined themselves teaching,” only a few were able to articulate clear and detailed descriptions. many admitted that they meagher & brantlinger urban mathematics teacher journal of urban mathematics education vol. 4, no. 2 110 did not yet have a clear vision of what “effective” mathematics teaching looked like and, when pressed, fell back on the idea that the mathematics curriculum needed to be made more real-world relevant. one of kelly’s cohort mates at bu, also a metromath case study, reported the following in her preservice interview: the only thing that helped me out in the summer [program at bu] on how to teach math was to bring real-world situations—real-life situations—into the course. so i think that will be helpful because a lot of students when they want to understand something you have to relate to what they know. you can’t give a problem and say “when you play tennis” because most city kids don’t play tennis, so they won’t understand it. so that’s one thing i learned when teaching math, give problems that relate to them and their life so they can have a better understanding. (cristina, august 2006). in spite of being unable to articulate an alternative model of instruction, kelly and other mtf we interviewed verbally distanced themselves from the traditional model of mathematics instruction that many reported experiencing in their own schooling. instead, they articulated reform-oriented ideals (e.g., real-world motivations, focus on understanding, collaborative activities) for their instruction. admittedly, it might be difficult for the mtf to develop a detailed and coherent vision of an alternative to traditional mathematics instruction because of their streamlined preservice program, which comprised 140 hours of coursework and 60 hours of clinical fieldwork (brantlinger et al., 2009). as one participant enrolled in a different partner university reported: i taught 6th and 7th grade my first year and content wise was not a problem at all. as far as everything else that’s involved in teaching, i thought the program gave me enough to get through the first week and then after that [not much]—but at least i knew what to do on the first few days, and that’s important (james, august 2006) as mentioned, less than a third of the 2006 cohort of mtf took more than a few mathematics classes in college. a sizable number had only learned that they would be mathematics teachers a few weeks prior to beginning the nyctf program (donoghue et al., 2008). kelly’s primary stated goal was to gain control of her classroom. in the preservice interview, kelly reported: my goal is not—is to be organized but not to have—i know we’re supposed to have math goals but my biggest goal is to have control of the classroom. i am so afraid of having 12and 13-year-olds having more control over me than i have over them. (august 2006) kelly was not unique amongst her peers at bu and in her broader cohort in worrying about their ability to manage a classroom (brantlinger et al., 2009). this meagher & brantlinger urban mathematics teacher journal of urban mathematics education vol. 4, no. 2 111 concern for control is to be expected given the extensive exposure to classroom management techniques in her preservice preparation and costigan’s (2004) research on novice teaching fellows would suggest. kelly noted a clash between her ideas about classroom organization and management and the customs at her school. kelly said that the school she would teach at required her to have her students sitting in groups and that this was not her preference stating: what i got from my principal is that we have to have kids in groups from day one … and i think i would—in terms of discipline and getting control i would have liked … them to be sitting by themselves in rows for the first little while but that can’t happen. (august 2006) her school was not unique in this requirement as the district mandated “workshop” instructional model required all schools and teachers to include collaborative student work in their daily lessons (new york city doe, 2006; stein & coburn, 2007; traub, 2003). yet, kelly thought it would be easier to gain control of the class if the students were sitting in rows. she also had personal concerns about controlling the class because she did not feel that she was very “intimidating” and was afraid that students would see her as a “pushover.” however, she had what she perceived to be an advantage in that she has a moderate level of spanish, framing this as a control mechanism rather than an avenue for improved communication with students: so i don’t actually don’t want them to know that i don’t know very much [spanish]. like if they wanted to talk to me in spanish, i can try, but i quickly realized that i like that—having that little air of mystery that they don’t know how much i know so they are much more cautious to break into spanish because they think i don’t know it. (august 2006) despite the “me vs. them” issue of classroom control, kelly mostly expressed genuine compassion toward school-aged youth. she “loved” working with middle school age students during her many years as a swimming instructor. further, she stated that her main mathematical goal was to alleviate some of the math phobia she attributed to some students. she reported, “i just want to, somehow over the course of the year, alleviate any fear of math and make it interesting.” (august 2006). she wanted to give her future students reasons not to give up on mathematics. she also said that she wanted to find out about students’ lives and connect mathematics to what she finds out: “i want to have those overall concepts of what they’re learning in class can have that link to life” (august 2006). we see here an interesting juxtaposition of, on the one hand, a strong desire for control and close management and, on the other, a desire to allow students to engage and express themselves freely through mathematics. meagher & brantlinger urban mathematics teacher journal of urban mathematics education vol. 4, no. 2 112 a lesson early in the school year the lesson observation we discuss in detail here took place after kelly had been teaching for about five weeks. when the students arrived to class, a “do now” is written on the board. a do now is a posted problem or set of problems that students work on as they settle into their seats. it was a feature of the district mandated workshop instructional model (stein & coburn, 2007; traub, 2003) and most lessons we observed kelly and the other metromath case studies teach included a do now (donoghue et al., 2008). the do now for this lesson reads: solve/evaluate. 1. x – 4 = –9 2. –10 = 5y 3. 3a + 8 = 2 4. –12 = –8 + 4b the vignette that follows occurred about 20 minutes into the 90-minute period as kelly began to review the do now problems with her class. this review followed student “work time” on the do now, announcements, and a review of graded quiz questions. note that in the vignette, the evaluative remarks and commentary in italics were written in the first author’s field notes. the second author videotaped the lesson and added additional comments of his own after the field notes were completed. kelly: how did you guys solve these? let’s look at the ‘do now.’ many students are still writing the day’s behavioral objective. this is another instance of minor disorganization in not being clear about what students are supposed to be working on. kelly: how did you guys solve number one” [i.e., x – 4 = –9]? [student a], what did you do? student a: i back tracked backtracking was a procedure that kelly and other middle grades mathematics teachers introduced as an official part of the curriculum. kelly: which is? student a: i did –5. i put umm kelly: we’re on number one. how did you guys do that? student b: it’s –5. kelly: it is –5. how did you do that? we see here that kelly immediately affirms a correct answer, which is a typical feature of kelly’s instruction in all her classes and in all but one of the other seven case study teaching fellow’s mathematics instruction. student c: i added 4 and that makes it –5. meagher & brantlinger urban mathematics teacher journal of urban mathematics education vol. 4, no. 2 113 kelly: ok. your goal is to get the variables all on one side of the equation and get the values on the other side. so what you do to one side you have to do to the other side. that’s the key to the whole solving equations with variables on both sides. what you do to one side you have to do to the other. we see here that kelly takes ownership of the mathematics by expanding on the students’ response and putting it in the teacher’s words. again, this is typical for kelly and seven of eight of our case studies. kelly: so what did we do for number two? (–10 = 5y) [student d] what did you do? student d: i did ‘5 + 10.’ kelly: interesting. contrast her reaction here with the situation above where a correct answer is affirmed. here an incorrect response gets “interesting.” student d is subsequently ignored. kelly (to student e): what did you do? student e: it’s –2. kelly: –2. how did you get that? student e: 5 times –2 is –10. kelly: yeah. he just thought about it. 5 times what? that’s called ‘guess and check’ what he did. you could divide both sides by 5. again we see kelly take ownership of the mathematics discussion. kelly also shows the calculation on the overhead but writes it in over the “do now” questions so everything is starting to get a little cramped, although she does use different colored markers. kelly: how about number three [3a + 8 = 2] what did we do? [student f], what was the first thing you did? student f: it’s –2. kelly: –2? it’s hard to tell here but i think [student f] is solving by inspection or guess and check but kelly is looking for a first step. specifically she is looking for “subtract 8 from both sides.” student f: what is that? positive 8? kelly: positive 8 student f: negative 3 kelly: –3? who has a different idea, a different approach? yes? student g: subtract 8. kelly: subtract 8 would be the first thing. what do we get here? to get rid of the variables we do the exact opposite. so here ‘a’ is being multiplied by 3 so to get rid of it, to get a by itself you divide by 3. right? again we see kelly both privileging correct responses and taking over the explanation of the mathematics from the student. kelly also asks a “pseudo meagher & brantlinger urban mathematics teacher journal of urban mathematics education vol. 4, no. 2 114 question,” a typical feature of traditional instruction (cazden, 2001) and something we regularly observe in seven of the eight cases. kelly does not wait for students to respond to her question before responding herself. the above episode is representative of whole class discussions in kelly’s classroom at this early point in the school year. it also highlights a number of significant issues that surfaced in kelly’s teaching throughout her first year. in particular, despite the reform-oriented attitudes evident in her preservice interview, her whole class instruction is consistently teacher-centered. she poses mostly procedural problems that require students to rehearse previously taught methods. she controls both mathematical explanations and evaluations of responses, leaving little space for student thinking to surface or be addressed in a responsive manner. she privileges students’ correct answers by leaving them unchallenged but immediately challenges incorrect answers with comments such as “interesting” and “are you sure?” kelly silences student confusion in the apparent attempt to avoid unpredictable or uncomfortable instructional moments during whole class discussions or presentations (leinhardt 1989; skott, 2001; tanner & jones, 2000; westerman, 1991). a “mini-lesson,” a lecture or guided discussion portion of the required workshop model lesson format, followed the whole class discussion (new york city doe, 2006; stein & coburn, 2007; traub, 2003). in the “mini-lesson,” kelly leads students through three examples of solving equations. following the minilesson, kelly assigns a number of problems from the state-required textbook and, early in the year, students either worked on these individually or with a nearby student. rather than finding or creating tasks that might better build on students’ current understandings, kelly typically designs the “student work period” portion of the lesson around textbook problems. further, kelly often assigned textbook problems before she had worked through them herself and, as a result, the problems often did not relate very well to topics covered in the mini-lesson. there is a certain irony in her instruction because, at this early point in the school year, kelly already had fallen into a pattern of teaching that she had critiqued in her summer interview; namely, doing a couple of example problems and then asking the students to do a number of similar problems. again, kelly’s instruction remained teacher-centered in spite of her desire to teach differently and in spite of the district mandated workshop model that was designed to limit a teacher-centered instruction (new york city doe, 2006; stein & coburn, 2007; traub, 2003). in fairness to kelly, this traditional model of teaching is prevalent in u.s. schools (stigler & hiebert, 1999) and tends to override reform-oriented policies (cohen & hill, 1998; spillane, 2004). kelly used the assigned textbook as her primary curricular resource as do many novice teachers (kauffman, johnson, kardos, liu, & peske, 2002). kelly’s meagher & brantlinger urban mathematics teacher journal of urban mathematics education vol. 4, no. 2 115 demonstrations of new material rarely strayed from presentations of examples from the required textbook, hence were not particularly responsive to her students’ prior understandings or developing ideas. as the episode above indicates, this lack of responsiveness held for whole class discussions and also during lesson components that might allow her to pay more attention to student understanding. it also held for the “student work period” that kelly included as part of the workshop model. during this phase of the lesson, kelly frequently told individual or small groups of students explicitly or exactly how to solve the assigned textbook problems. at other times, she provided them with enough “clues” to allow them to progress efficiently through the assigned problems. from a student perspective, it seemed that finishing assigned work took precedence over understanding it. kelly’s students quickly learned to rely on her, rather than themselves or their classmates, to complete in-class assignments. it is important to note that kelly continued to express a desire to teach in a more reform-oriented manner at this early point in the school year. however, she continued to be at a loss as to how she would accomplish reform teaching. in the follow-up interview to the lesson above, kelly reported, “i still want to work on the students being more reliant on each other and less on me for answers [and] i am still struggling with making the material interesting.” kelly’s students were not blank slates as they likely had learned in prior years to rely on mathematics teachers for answers to difficult problems (stigler & hiebert, 1999). more positively, kelly seemed to have developed a positive relationship with many of her students. in a post-lesson reflection, kelly observed, “i am feeling much more confident as the weeks go by [and i am] much more relaxed with my position as the teacher, and giving the students more responsibility in their own learning environment” (post-lesson interview, september 2006). she reported having few substantive management issues with the focus class for this study. with this class, she was aware of “only a few management issues such as getting out of their seats.” kelly admitted that she continued to struggle with stressful and disruptive management issues in her two low-track classes in this early part of the school year. in our post-observation reflections in the early part of the school year, we both noted how comfortable kelly appeared in front of the focus class students. kelly seemed to be in a reasonable position to develop as a middle school mathematics teacher and she had every reason to look to her university classes, administrators, mentors, and colleagues at the school for support in improving basic aspects of her pedagogy. yet in a reflection on the above lesson, kelly wrote, “no one has helped me with much of anything in terms of math content.” it became apparent that, if reform-oriented instruction was the goal, kelly’s preservice training and in-service support around mathematics pedagogy was insufficient. meagher & brantlinger urban mathematics teacher journal of urban mathematics education vol. 4, no. 2 116 in summary, in the first part of the school year, kelly struggled to varying degrees with such aspects of her teaching as implementing cognitively-demanding and -appropriate mathematics tasks, listening to students, and facilitating classroom mathematics discourse. while these are concerns for most novice teachers (moyer & milewicz, 2002; tanner & jones, 2000), in some teacher education programs, such issues around mathematics instruction are addressed in student teaching or later in in-service professional development. these pedagogical ideas clearly were not being addressed for kelly. kelly’s teaching was again fairly typical of other mtf. for example, seven of the eight case studies we observed generally followed the workshop model and six of these seven implemented it in teacher-centered ways manner at the outset of the 2006–2007 school year. the eighth case study rejected the workshop model and adopted an even more teacher-centered lesson format, one that provided little space for collaborative student work. all of the case study mtf struggled with classroom management issues, albeit to varying degrees. as discussed at the end of the next section, kelly’s lacking mentoring and induction experiences were also fairly typical (foote et al., 2011). the middle part of the year kelly’s journey through the middle part of her first year is marked by her attempts to change the teacher-centered dynamic of her class. we also describe how some early direct support she received both from the school administration and the university representative to the school diminishes over time, limiting her capacity to implement the reformist ideals she aspires to. supports one of the major aspects of kelly’s first year, and a central story repeated by other novice mtf, revolves around ongoing professional development and support (foote et al., 2011). as do participants in other early-entry alternative route programs, participants in nyctf enter the classroom under the assumption that there will be a battery of supports in place to ensure optimal development. induction support is particularly important given the streamlined preservice preparation in such programs (humphrey, wechsler, & hough, 2008; johnson & birkeland, 2008). as an in-service teaching fellow, kelly could expect to receive professional development and support from five distinct sources: (a) her university classes, (b) a university supervisor, (c) assistant principals, (d) a mathematics coach appointed in a general consultant role in the school, and (e) a new york city doe mentor hired by the school. the assistant principal and coach are supports available to all teachers; however, the university supervisor and new york city doe mentor are unique to the teaching fellows program. meagher & brantlinger urban mathematics teacher journal of urban mathematics education vol. 4, no. 2 117 the university classes. during her first 2 years as an in-service teacher, kelly took two evening courses a semester at bu and a couple more over the summers. in her first semester following the preservice program, kelly took one class in special education, one class in literacy in the classroom, and one in web design. these classes were in addition to the six credit hours of preservice graduate coursework that kelly completed over the summer. kelly was looking to her graduate classes to help her develop not as a teacher in general but specifically as a mathematics teacher. kelly did not take a mathematics teaching methods course in her first year at bu and ended up frustrated with respect to the latter goal: i’m not taking any classes on how to teach math. so, how am i supposed to be a good math teacher? [bu instructors] purposefully talk about other subjects because we’re like, “well you guys know about mathematics so let’s talk about english, let’s talk about science. let’s do a debate on technology and science classroom, or special ed in the classroom” …you want us to be well rounded, we’re not even well rounded in our subject area yet. (may 2007) kelly’s critique of the in-service coursework at bu represented a shift as she had been extremely positive about bu’s preservice coursework and had felt well prepared for the beginning of the school year. in in-depth interviews, the other two case studies at bu were also quite critical of the bu program for similar reasons and the more general sense that much of the coursework was irrelevant to their teaching context. reflecting on his first year at bu, a different case study reported that he “wasn’t very satisfied with the program at [bu].” in contrast to kelly, he did take one mathematics methods course during his 2 years at bu and reported that this course, unlike most, was “very good.” he claimed that this course provided “great resources” and “good techniques” for “working with manipulatives, collaborative groups, how to write quality assessments, how to analyze student work, stuff like that” (may 2007). the university supervisor. university supervisors were to visit fellows on a monthly basis, and their formal observations impacted the decision to award full teacher certification or not. as mentioned above, kelly had a very positive initial relationship with her bu supervisor. he had taught at kelly’s school decades earlier and continued to maintain close contacts there. he also had taught in kelly’s summer program at bu. on the survey, kelly wrote the following: “he came in and helped with my classroom setup. also suggestions on how to deal with bad behavior.” however, the relationship lapsed as the school year progressed with kelly commenting in a post-observation interview that i’m not falling on my face, so he’s got other things to do. because he spends a lot more time with [teacher] than he ever did with me. he was here a lot in september. i really haven’t seen him since. he stops in to say hello, but he even, he’s sup meagher & brantlinger urban mathematics teacher journal of urban mathematics education vol. 4, no. 2 118 posed to see me once a month for borough university but the last three times he’s just stopped in for a lesson plans so he can write it out because he trusts me. (may 2007) as we discuss below, the university mentor’s failure to help kelly’s development as a mathematics teacher may have been as a result of the positive impression he had of her prior to her teaching. from our reform-oriented mathematics perspective, the problem is that his focus appeared to have been on control and survival and not on enhancing pedagogy and instructional competencies. hence, kelly was lacking on the support she need to develop as a reform mathematics teacher. although we can assume these mentors left kelly alone because she demonstrated basic survival skills, it might imply that other beginning teachers are not necessarily getting coaching in mathematics pedagogy and content. assistant principals. there were two assistant principals in the school who worked with new teachers, including kelly, on and off throughout the year. her somewhat cynical summary of the extent to which they helped was, “they saw me in september or october, they realized i was competent…and then they’ve never helped me at all” (may 2007). the relationship seemed to improve in the second half of the school year with the assistant principal for mathematics was coming into her class occasionally. mid-year kelly reported, “i asked [him], ‘can you stop in and see how things are going?’…so he’s stopped in a few times and gave me some pointers” (may 2007). she considered the specific advice he gave her about her mathematics instruction to be useful. as she related: he was telling me about how to develop from one idea to the next, connect, because i wasn’t connecting. …i was reviewing graphing and he said this is a good opportunity to do positive and negative numbers. he said that they’re not good at it, so if you are creating tables [of values] you should also try, find time to stick in a little revision on adding and subtracting integers. i was like, “oh, i never even thought about doing that. (may 2007) the assistant principal thought highly of kelly, and commented that she was “highly motivated, very effective, very dedicated. she’s excellent. she’s excellent. she has tremendous skills as a first year teacher” (march 2007). he continued: “what has impressed me is her knowledge. we have her teaching one of the more advanced eighth grade classes and teaching them high school. [i am impressed with] her knowledge in conveying the lesson, getting her lesson across to the kids” (march 2007). as noted in the previous section, our reform-tinted observations did not support as glowing an assessment, and kelly herself has expressed concern about her own instruction and preparation in this regard. the mathematics coach. the mathematics coach, appointed in a general consultant role in the school (e.g., to help align mathematics instruction with state standards), provides another level of support for novice and experienced mathe meagher & brantlinger urban mathematics teacher journal of urban mathematics education vol. 4, no. 2 119 matics teachers alike. while many coaches are assigned to schools by the new york city doe, other schools, such as kelly’s, hire their own coaches as part of decentralization efforts (o’day, bitter, & gomez, 2011). for the most part, kelly did not find the coach to be useful: “she was supposed to have come to my classroom once a week [but it never happened]” (may 2007). nevertheless, kelly found the coach was helpful on occasion, for example, with some suggestions, discussed below, for project work she could do with the students. doe mentors. by state law, the new york city school district was to provide the fellows with mentors. this mentorship was to be more extensive than the university-based mentorship. however, kelly had a doe mentor who, instead of visiting her bi-monthly as required, observed kelly teach one full lesson per month. on the in-service survey kelly wrote, “my mentor was great at helping me with personnel things-days off, death in the family, can i change schools, etc.” (may 2007). like many mentors, kelly’s carried around materials developed by the new teacher center (ntc, 2006). kelly wrote: “this massive book which we went through was useless. it was better to just talk about what was happening and my needs.” when we consider these five supports together, except for rare occasions, it appeared no one was systematically examining kelly’s teaching and helping her improve her general and mathematics specific instructional skills. because, in comparison to other new teachers in the school, her classroom management skills were seen as good, and she had a positive relationship with many of her students, she gained the reputation of being a successful teacher and administrators apparently felt their time would be better spent elsewhere. unfortunately, this inadequately applied support system did not allow kelly to fully develop as a mathematics teacher. with regard to a triumvirate of basic pedagogy: norms, tasks, and discourse (henningsen & stein, 1997), kelly’s training (together with her background and personality) was sufficient to help her achieve reasonable success with norms but left her struggling with tasks and discourse. kelly’s experiences with mentoring and induction were typical. as we report elsewhere (foote et al., 2011), the first-year mtf report typically receiving formal mentoring once or twice a month. while 3 in 10 teaching fellows report meeting with their mentor on a weekly basis, an equal amount report receiving no formal mentoring at all in the first year. like kelly, even those who received regular mentoring did not find it to inform their instructional practice. attempts to change the class dynamic after a few months of experience, when she had developed good working relationships with students, kelly attempted to change the dynamic of her class through the incorporation of more group work. she did so, in part, to conform to the culture of the school and the district-required workshop model, a model that meagher & brantlinger urban mathematics teacher journal of urban mathematics education vol. 4, no. 2 120 she expressed mixed feelings about. indeed, on the in-service survey, kelly “strongly agreed” with the statement “your school administration makes sure you teach with the workshop model.” kelly again was typical in this regard as the majority of kelly’s cohort we surveyed either “strongly agreed” or “agreed” with this statement. she also attempted to make instructional changes to facilitate more student-to-student communication and collaboration. in a mid-year interview, kelly reported her attention to group work was working to some extent: the amount of individual questions i have to answer [has gone down] as i have been stressing working with their groups. however, with that emphasis on group work, i am seeing a lot of work that looks identical and so i worry there is too much copying going on and not enough collaboration. classroom observations over this time period indicated that while kelly had seated her students together in pairs and asked them to rely on their partners and other nearby pairs, she had not changed the types of tasks she gave students nor the reward structures (as individual rather than collaborative performance was graded), so many continued to work individually. as was the case in lessons from earlier in the year, kelly gave her students tasks that generally consisted of a set of problems chosen in a rather ad hoc fashion from the textbook. later in the year, based on comments from her mathematics coach, kelly assigned group-oriented activities that required students to work together. for example, in one lesson she had groups collaborate on a poster that represented a set of data by using either a bar chart or a pie chart. she recalled: so, i asked the math coach what to do and she said choose things that [the students] know already like order of operations or reading graphs and that kind of stuff and make projects that you don’t have to teach them anything that they can just work on their own for the double period. and that you’re reinforcing skills that they already know (may 2007) evidently, kelly did not see collaborative student activities as spaces to advance mathematical ideas. she implied that mathematical ideas only develop when the teacher uses direct instruction. during the collaborative project-based lessons we observed, kelly spent more time commending students for their presentation style than for mathematical understanding. student mathematical errors often went uncommented on or uncorrected in group activities and presentations. such errors were not used to further student understandings. in interviews, kelly seemed generally aware of the limitations of these lessons. reflecting on what the students would have learned from one lesson, she stated: meagher & brantlinger urban mathematics teacher journal of urban mathematics education vol. 4, no. 2 121 not a whole lot. they, i mean the ones that did the percents got a quick review of that. you know what, not much. i mean they all knew how to do it, so i guess they got to look at how their class did the problems. (march 2007) while the shift toward more group work resulted from kelly’s desire to change the classroom dynamic, it seems also that she neither have the requisite understandings nor a complete commitment to this approach. she concluded that the students do not learn a lot when she is not directly “teaching them anything.” we see this as symptomatic of kelly’s view of herself as the holder of knowledge, which she has a duty to teach to the students. a lesson late in the school year in this section, we present a typical lesson that kelly taught towards the end of the school year. we again note that, by mid-year, we see kelly as a teacher who began the year with a good teaching presence in the classroom, a commitment to improving instruction, and a willingness to learn. it was also clear that, earlier in the year, she was falling into many of the common difficulties of novice teachers in terms of her relatively weak command of content knowledge for teaching, pedagogical skills such as questioning, and a tendency to take over explanations from students. she continued to rely to a large extent on the district-required textbook for lesson plans—although, she expresses ambivalence about the value of this textbook in interviews and on the survey. in part, because she has not received adequate support, kelly has not developed in terms of task selection, task implementation, and facilitation of discourse. the “do now” written on the board is: 1. 20 rolls cost $4.97 for brand a and 15 rolls cost $3.99 for brand b. which brand is cheaper and by how much? 2. represent as an expression: one side of a square is t. what is four times the perimeter minus 10? the focus of the lesson is unit conversion or proportional reasoning for comparison as the first do now problem indicates. kelly includes the second problem for the purposes of curriculum spiraling or review. after providing the students a few minutes to get settled and begin working on the do now, kelly calls on students to review the two problems. kelly searches for and repeats correct answers from students as she stands in front of the class. she does not write anything down to keep track of student responses. meagher & brantlinger urban mathematics teacher journal of urban mathematics education vol. 4, no. 2 122 before class, kelly wrote solutions to two unit conversion examples on a poster, which she next displays on the board. these are routine unit conversion problems (e.g., comparing 62 inches and 2 yards by converting yards to inches) but they involve quite a bit of formal notation and manipulation, and the work seems to be beyond the students’ current level of understanding. kelly reads out the problems and talks through her solutions without asking any questions. she then goes through a third example by reading it out from the book but without writing anything on the board. effective board work is not a skill that kelly has developed very well. she has developed the habit of requiring students to listen to her and follow along in a text without writing anything down. limited board writing places a considerable cognitive burden on the students and it is a burden that many appear unable to bear much of the time. after a few heads go down on desks, kelly next asks students to follow along in the textbook. kelly then sets the students the task of working on eight similar unit conversion problems from a test-prep booklet. some time later, a small group of students has called kelly over to discuss question number six: a pile of dimes weighs 1000 grams. if each dime weighs 2.3 grams, how much money is in the pile? [as above the evaluative comments in italics come from the researcher’s field notes.] student a: i got two thousand three hundred but it’s got to be [inaudible] kelly: a thousand is what? how many? in a? student a: gram. above is the first instance of many in this episode where kelly’s questioning technique is to say an almost complete sentence but substituting “what” for the last word and asking the students to fill in that word. she is” funneling” them towards the correct answer. kelly: in a gram. but you have: each dime weighs 2.3 so you want to take the total weight and do what? student a: times? kelly shakes her head “no.” student a: divide? the students have been doing a number of these problems and almost all of them involve the students either dividing or multiplying two quantities presented in the problem. [student a] knows that she must either multiply or divide so she picks one. she also knows that kelly will tell her if she is correct or not. does [student a] have a reason for saying “multiply” first? kelly (almost?) never asks students “why?” so we (and kelly) do not get insight into student thinking. meagher & brantlinger urban mathematics teacher journal of urban mathematics education vol. 4, no. 2 123 kelly: divide. [affirms] because you want to know how many dimes weighs a thousand grams. student a: so i would divide it? kelly: right now you’re… we see here a common kelly behavior, which is to take over the conversation. notice how much kelly speaks and how little the students speak. notice also that the students are never asked to explain anything but are brought along at each step by kelly. some laughter at tables in middle of the classroom. kelly turns and glares, a few students make eye contact with her. kelly: you can stay after if you want to play around. we don’t have time for this. this incident may point to some of the lack of quality in the discussions that kelly has with the students. it seems that she is sometimes anxious to get through discussions quickly and to move the students forward to the “student work period” because she is anxious about the behavior of other students in the class. kelly turns her attention back to the girls she’s been working with. kelly: ok, so you divide it. what do you get? student b (works on the calculator): divide what? kelly [to student a]: divide what? student a: you don’t do a thousand? kelly nods. it seems that student a really doesn’t know what’s going on here but through her conversation with kelly she can get through the problem and get the right answer. at no stage is student a really questioned about the problem or is any attempt made to make sense of the problem by analogy with smaller numbers or estimation of what might make sense as an answer. meanwhile student b has finished working on the calculator and shows the display to kelly. kelly points at calculator screen: that’s right. student a looks intently at the calculator and then leaves and goes to her own desk. student a is seemingly content at this point. it’s not clear that she understands the problem but she does have the answer. kelly: now, so the only question is what are your answers in? student a: money kelly: they’re in money but they’re in dollars right? and you’re in dimes so how many dimes are in a dollar. we see again here how kelly drives the entire episode forward. kelly is the one who points out that there is an issue still to be addressed and kelly is the one who points out the direction for addressing the issue. meagher & brantlinger urban mathematics teacher journal of urban mathematics education vol. 4, no. 2 124 student b: ten. kelly: ten. so you need to do what now. student c: divide? kelly: by? one can easily imagine that if student c had said “multiply?” then kelly would have shook her head and student c would have said, “divide?” student c: ten. student b still looks puzzled. it is not clear that student b is clear on what the answer is. one might argue that she comes out of this episode the best in that she is still confused! student a and student c have the answer but it is not at all clear that they have engaged with the problem in any meaningful way. this teaching episode was representative of many that took place during student work time in the last quarter of the school year. by this time, kelly has developed norms in her class such that she can have the students working in pairs and has the time to get involved in quite lengthy discussions with small groups of students with only minimal disruption from outside. we see, however, that the lesson was procedural in focus; kelly introduces an algorithm for solving unit conversion problems and does so without building on students’ prior understandings of, for example, money or real-world experiences with measurement. the tasks she sets her students also are highly “proceduralised” as she expects students to follow the approach that was outlined in the textbook. further, during student work time her discourse and question techniques serve to “funnel” (wood, 1998) students toward correct answers. typical of many u.s. mathematics teachers (stigler & hiebert, 1999), kelly does not want her students to experience a whole lot of frustration in learning new mathematical ideas. stigler and hiebert (1999), among others, have noted that many u.s. mathematics teachers may pose what might be more conceptually challenging mathematics, but are very quick to break the problem down into steps and procedures in order to limit the amount of time that students struggle intellectually with the mathematics. end of year interview an interview with kelly at the end of the school year showed kelly to be unhappy with many of her experiences in her first year of teaching. one of her greatest concerns was that the various supports, inasmuch as they were there for her, were providing help that was too general and not focusing enough on mathematics. in terms of her in-service graduate coursework at bu, she asked: “i’m not taking any classes on how to teach math. so, how am i supposed to be a good math teacher” [emphasis added]? she further complained that “the test is all the school cares about and they make that obvious.” she felt that she was progressing meagher & brantlinger urban mathematics teacher journal of urban mathematics education vol. 4, no. 2 125 as a teacher in general terms such as having control of the classroom but admitted that she struggled in her goals of finding real-world problems and helping alleviate students’ fear of mathematics. at the end of the school year, kelly resigned her position at her school and joined another school where she perceived the mathematics department to be more coherent and the administration to be more supportive of teachers’ development. our metromath survey study indicates that more than 15% of kelly’s mtf cohort did not return for their second year of teaching and another 20% changed schools (both voluntarily or otherwise) before they began their second year of teaching (donoghue et al., 2008). discussions and implications with two in three new mathematics teachers in new york city public schools presently coming through the teaching fellows program and with such early-entry alternative route programs increasingly common, particularly in large urban districts, there is a need for careful research of the experiences of such teachers in their early years of teaching. the research presented here tells the story of a typical experience of a new mathematics teaching fellow in a new york city public school. the story we tell is designed to give the reader insight into what the mathematics teaching of a typical recruit in an alternative certification program might look like. the story is of a teacher who we see as having the potential to be an effective middle grades teacher, but who was failed by the system of induction supports that were designed to help her reach that potential. we see in the evidence presented that despite a full set of university courses and at least three individuals who had supportive or mentorship roles, there was, in fact, almost no situation in which a mentor or other qualified individual was carefully examining kelly’s teaching and helping her develop pedagogical skills such as questioning, board work, and incorporation of students’ mathematical ideas. other skills left under developed were choosing tasks that engage students in higher-order thinking, novel problem solving, and communication of their developing ideas about mathematics. what is most striking about kelly’s case is that she is clearly conscious of the failure of the system, as she asks: “when am i going to learn to be a mathematics teacher?” many new york city public schools are difficult to staff and have difficulty retaining staff. to address this shortage, it is important that all aspects of training and support for incoming teachers are designed and implemented in such a way that allows for potential to be developed, for teachers to be supported, and students to have effective experiences in mathematics classes. the evidence of this case study suggests that such a system is not in place for some new teaching fel meagher & brantlinger urban mathematics teacher journal of urban mathematics education vol. 4, no. 2 126 lows in the new york city public school system resulting in students not benefitting from effective instruction. students in high-needs urban schools face a steady stream of inexperienced and under-qualified teachers (darling-hammond, 2004; peske & haycock, 2006). while they cannot address the former issue, nyctf and other high profile alternate route programs promise to address the latter issue by recruiting candidates who have prestigious educational credentials (e.g., having graduated from highly ranked universities, having scored in the top quartile on the sat exam). early-entry alternative route programs operate on the assumption that the “highly qualified” candidates they attract can be readied to teach in streamlined preservice programs and, with the proper induction support, learn to teach “on the job.” this case study, in particular when combined with our broader research on mtf, challenges such assumptions on several levels. first, nyctf program attracts a large number of recent college graduates and other candidates who, like kelly, lack both strong backgrounds in mathematics and prior work experience relevant to mathematics teaching (donoghue et al., 2008). second, our research indicates that mtf begin teaching without being well versed in mathematicsspecific teaching methods (brantlinger et al., 2009). third, more often than not, the promised mentoring and induction support for teaching fellows fails to materialize, leaving new mtf to fend for themselves (foote et al., 2011). research on nyctf and other early-entry alternative route programs similarly finds that most are unable to provide adequate mentoring and induction support for their candidates (humphrey et al., 2008; veltri, 2010). and finally, novice mtf rely on teacher-centered teaching scripts that contradict their own visions of effective instruction and, in this case, visions of effective instruction articulated in district policy documents (new york city doe, 2006; stein & coburn, 2007; traub, 2003). the results presented here are intended to contribute to the necessary understanding of the effect of alternative certification models on mathematics teaching in urban environments. with ever increasing numbers of mathematics teachers coming from such ranks this understanding is important for both teacher education programs and school districts as they adapt to the changing pathway to certification landscape. acknowledgments this material is based upon work partially supported by the national science foundation under grant no. esi-0333753. meagher & brantlinger urban mathematics teacher journal of urban mathematics education vol. 4, no. 2 127 references boaler, j. 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(2001). what we know and don’t know from peer-reviewed research about alternative teacher certification programs. journal of teacher education, 52, 266– 282. http://www.nytimes.com/2003/08/03/edlife/03edtraub.html journal of urban mathematics education december 2016, vol. 9, no. 2, pp. 153–184 ©jume. http://education.gsu.edu/jume emily joy yanisko is the lead clinical faculty for mathematics in the baltimore site of urban teachers at the johns hopkins school of education; email: emily.yanisko@urbanteachers.org. her research interests include urban mathematics teacher education and development, discourse in the mathematics classroom, and issues of equity in urban schools. negotiating perceptions of tracked students: novice teachers facilitating high-quality mathematics instruction emily joy yanisko urban teachers at johns hopkins university in this article, the author reports on a participant-observation case study that explored how alternatively certified, middle school teachers’ expectations of tracked students affect their ability to learn to teach in ways that promote students’ mathematical struggle and participation in productive mathematical discussions. two teachers—one teaching a “high-tracked” course and the other a “low-tracked” course—were participants. both teachers initially held perceptions of their students that limited their efficacy and self-efficacy with respect to providing high-quality mathematics instruction. however, through programand school-based mentoring, including participation in a modified reflective-teaching cycle, the teachers learned to learn from their teaching and modify their practice. both teachers began to allow their students opportunities to struggle with rigorous mathematics and participate in student-centered discussion. keywords: academic tracking, african american/black students, high-quality mathematics instruction, reflective teaching he national council of teachers of mathematics (nctm) states, “an excellent mathematics program requires that all students have access to a high-quality mathematics curriculum…[and] high expectations” (2014, p. 5). high-quality mathematics, according to the common core state standards for mathematical practice (ccssmp), provides students, among other things, opportunities to participate in mathematics instruction that asks them to “make sense of problems and persevere in solving them” (ccssmp, 2016, ¶2) and to “construct viable arguments and critique the reasoning of others” (¶4). in the interest of justice, all students should have access to rigorous, high-quality instruction. both the nctm and the ccssmp claim equity as a driving purpose for the importance of the implementation of standards of high-quality mathematics instruction. the development of both the nctm and the ccssmp was guided by the desire to make high-quality mathematics available to all students, regardless of race, ses, language, school placement, or course placement, so that all students become college and career ready and are internationally competitive. the language of all espoused throughout both the nctm and the ccssmp documents, however, may not be sufficient to t http://education.gsu.edu/jume mailto:emily.yanisko@urbanteachers.org yanisko perceptions of tracked students journal of urban mathematics education vol. 9, no. 2 154 ensure equity (apple, 1992), but rather suggests what may be considered a necessary step in the road to equity: equality of opportunity. martin (2015) contends that the documents perpetuate the goals and ideals of middleand upper-class white privilege and decenter the needs of those children and youth who he calls “the collective black” (p. 21). therefore, we must first think about how the mathematics being taught in schools to students of non-dominant backgrounds mirrors (or not) what the nctm and the ccssmp consider high-quality mathematics instruction. historically, the perceived student responsibility for learning mathematics focused on following rules presented by the teacher or the textbook, memorizing and applying those rules, and verifying correctness through an authority such as the teacher or the textbook (cobb & yackel, 1996; lampert, 1990; schoenfeld, 1992). this type of instructional practice can be formal and restricting and, for many students, can limit opportunities to develop their mathematical reasoning (brown, collins, & duguid, 1989; stein, grover, & henningsen, 1996). in addition to the strong language around problem solving and mathematical discussion of the ccssmp, the principles and standards for school mathematics (nctm, 2000) also states that students should be able to “make and investigate mathematical conjectures” and to “develop and evaluate mathematical arguments and proofs” (pp. 57–58). furthermore, this type of reasoning should be augmented through communication as students— organize and consolidate their mathematical thinking through communication; communicate their mathematical thinking coherently and clearly to peers, teachers and others; analyze and evaluate the mathematical thinking and strategies of others; [and] use the language of mathematics to express mathematical ideas precisely. (p. 60) if these understanding, communication, and reasoning goals and standards are to characterize norms of practice as evidenced in schools, there will need to be a fundamental adjustment in participation structures within the mathematics classroom (foreman & ansell, 2001; herrenkohl & guerra, 1998). through mathematical communication, students and teachers may work and reason together as they “do mathematics” in a way that augments the mathematical knowledge that students are expected to know (cobb & yackel, 1996). therefore, it is important to investigate how new teachers learn to teach in ways that encourage students’ perseverance in problem solving and participation in productive mathematical discussion with equitable practice and meaningful learning as the ultimate goal of these practices. in the study discussed here, i investigate two novice, alternatively certified, middle school teachers’ perceptions of their tracked students and how those perceptions affected their efficacy and self-efficacy with regard to infusing their classrooms with perseverance in problem solving, mathematical sense making, and productive discussions. two research questions guided the inquiry: yanisko perceptions of tracked students journal of urban mathematics education vol. 9, no. 2 155 1. how might alternatively certified, middle school teachers’ expectations of tracked students affect their ability to learn to teach in ways that promote students’ mathematical struggle and participation in productive mathematical discussions? 2. how might these teachers negotiate their perceptions of students and learn to teach in ways that promote students’ mathematical struggle and participation in productive mathematical discussions? review of relevant literature this study integrates and is informed by two primary bodies of literature, tracking and alternative certification. taken together, this literature shaped the rationale and purpose for this study. expectations in schooling, urban schooling, and tracking novice teachers from alternative certification programs are often used to fill vacancies in “hard-to-staff” districts or schools that are more times than not populated with underserved students (darling-hammond, 2000). too often, both novice and experienced teachers in these hard-to-staff schools have deficit perspectives of low-income students of color (delpit, 2012; habermann, 1991). these deficit perspectives may be more prevalent for teachers prepared in alternative certification programs than for those prepared in traditional teacher education programs (brantlinger & smith, 2013). furthermore, academic tracking may further compound deficit perspectives that many teachers hold of low-income students, particularly, low-income students of color. tracking is pervasive in education, and teachers of tracked students may have particular perceptions of these students’ ability, negatively altering their instruction based on these perceptions (futrell & gomez, 2008; oakes, 2005). not only are low-income students of color often placed in “low-track” courses, denying them equal access to high quality curricula, but also they are less likely to have certified, high-quality teachers (furtrell & gomez, 2008; oakes, 2005). this unequal access to both high-quality curricula and teachers is doubly detrimental in mathematics. access to high-quality mathematics instruction can provide a gateway to future academic achievement, and can be considered a new civil right (moses & cobb, 2001; oakes, 1990; smith, 1996; spielhagen, 2010). oakes and lipton (1996) contend that students placed in high-track courses have access to richer mathematics instruction as well as more content. access to cognitively demanding mathematics instruction and content provides students entrée to college preparatory high school courses and higher probability of engaging yanisko perceptions of tracked students journal of urban mathematics education vol. 9, no. 2 156 in mathematics related fields post-secondary school (oakes, 1990; oakes & lipton, 1996; smith, 1996; spielhagen, 2010). when analyzing the third international mathematics and science study (timss) data, stigler and hiebert (1997) found that education in the united states, in general, is often dominated by the teacher, following a pattern where teachers demonstrate a particular concept or skill, and then ask students to practice what they have “learned” by rote in an application phase. this prescription for instruction is reinforced over generations, regardless of reform efforts, because teachers often resort to providing their students the same instruction as they themselves experienced (lortie, 1975). furthermore, teachers in schools serving low-income students of color often are directed or feel the need to administer scripted curriculum that results in students participating in low-level tasks far more often than teachers placed in schools serving affluent, predominantly white populations (delpit, 2003). student expectations. teachers of low-income students of color often convey low expectations of the educational potential of their students (delpit, 2012; habermann, 1991; oakes, 1992), regardless of the way students are sorted into tracks, which can bear little relation to the students’ actual ability in mathematics (oakes, 2005). these low-expectations cause many teachers to engage their students in educational experiences that require little higher-order thinking, discussion, or sense making through problem solving. teachers of tracked courses often modify their instructional methods based on their perceptions of the educational potential of the students in those classes (braun, nielsen, & dykstra, 1975; eder, 1981; oakes, 1992; raudenbush, rowan, & cheong, 1993; rosenbaum, 1980; watanabe, 2008). reciprocally, tracking, as well as race and ses, can be deterministic with regard to the teachers’ and other stakeholders’ expectations for their students’ educational potential (oakes, 1992, 2005). student testing used to evaluate both schools and teachers is prevalent in schools that serve low-income students of color. the emphasis on testing, in turn, too often results in teaching methods that are likened to “a thin gruel of test preparation … [t]he drill-and-kill practices that guarantee students will not be ready for college, skilled employment, lifelong learning, or effective citizenship” (neill, 2012, p. 24). the tradition of administering low-level curriculum that is specifically focused on test-preparation may be pervasive in any track, regardless of the “objective” level of student ability (bol & berry, 2005; bol & nunnery, 2004; delpit, 2003). student placement in tracks due to expectations and resultant treatment due to both expectations and subsequent placement is a reciprocal relationship (braun et al., 1975). teachers believe that students are placed appropriately, whether due to race, ses, or perceived ability level (oakes, 2005), and therefore treat their students differently. teachers expect more from those whom they perceive as highly motivated or high achieving, and therefore ask questions that are more rigorous and yanisko perceptions of tracked students journal of urban mathematics education vol. 9, no. 2 157 provide higher-level educational experiences (braun et al., 1975; brophy & good, 1970; cooper, 1979; dusek, 1975; oakes, 1992; page, 1990; watanabe, 2008). track placements are often permanent, and those students in low tracks experience less rigorous instruction and lower-quality materials (rosenbaum, 1980). tracking in mathematics. tracking is pervasive in mathematics programs, and is used to sort students based on many different characteristics, including perceived ability, academic performance, test scores, or non-academic reasons (bol & berry, 2005; oakes & lipton, 1996). although tracking in mathematics has been defended as a mechanism to ameliorate the “achievement gap” (bol & berry, 2005), there is “no empirical evidence to justify unequal access to valued … mathematics curriculum, instruction, and teachers” (oakes, 1990, p. xi; emphasis in original). african american and latin@ students are over-represented in low-track courses and often have less qualified teachers, and are subjected to low expectations. however, low expectations of low-tracked students often begin before placement in tracks. there are many instances where african american and latin@ students are placed in remedial mathematics programs even when the measures of their academic ability are equal to or better than their white and asian peers (education trust, as referenced in love, 2002). these low expectations of students are reflected in instructional and assessment methods used in mathematics courses, which result in less emphasis on high-quality mathematics instruction (flores, 2007; irvine & york, 1993). tracking in mathematics has long-term effects on the trajectory of lowincome students of color. for example, access to algebra in eighth grade influences students’ mathematical knowledge, productive disposition, course taking patterns, and future mathematics achievement (smith, 1996). taking an algebra i course in middle school offers students experience with demanding mathematics programs and allows them access to the mathematics pipeline, as algebra i is often considered a “gatekeeping course” (moses & cobb, 2001; oakes, 1990; spielhagen, 2010). enrollment in algebra in middle school also has a positive effect on students’ access to a 4-year college (spielhagen, 2010). students of color, however, have less chance of being admitted to middle school algebra i courses (spielhagen, 2010). novice alternatively certified teachers and teacher education novice mathematics teachers seem to be fighting an uphill battle. they are faced with providing instruction that is consistent with the new common core state standards for mathematical practices (see csssmp, 2016). they are affected by the “apprenticeship of observation,” which dictates that when they encounter a situation in which they are unsure, they revert all too often to the instructional methods of their own k–12 experience (lortie, 1975). furthermore, if they are placed in schools that serve low-income students of color which are historically lowperforming, they may have expectations of their students that results in planning yanisko perceptions of tracked students journal of urban mathematics education vol. 9, no. 2 158 and enacting instructional routines which are low level in the intent to be testpreparatory (delpit, 2003; habermann, 1991). hiebert, morris, and glass (2003) suggest that it is unreasonable to expect that a teacher will learn everything they need to know to be an effective teacher during their teacher preparation program. this gap in knowledge is even more significant in the context of alternative certification, where the teacher education program is shorter in duration (darling-hammond, 2000). such gaps may affect their efficacy in providing high-quality mathematics education as well. hiebert and colleagues argue that in order for teachers to “learn to learn to teach” (p. 201), they must treat their teaching experience as an experiment, and be taught to learn from their own practice as a scientist would learn from an iteration of an experiment. with help, novice, alternatively certified teachers may be able to learn to learn to teach in ways that engage students in productive, high-level mathematics discussion regardless of their race, ses, or track placement. the literature reviewed highlights the importance of investigating how new teachers’ perceptions of tracked, low-income students of color affect their instruction, and how they may be supported to negotiate the challenges that the teachers’ notions of student ability pose in order to provide high-quality mathematics instruction. conceptual framework of mathematics instruction an in-depth understanding of mathematics includes not only the knowledge of rules and procedures but also the ability to engage in mathematical sense making, participate in productive mathematical discussions, and have productive dispositions toward mathematics (national research council [nrc], 2001). for the purposes of this article, mathematical sense making is defined as valuing and applying abstraction and using those tools to understand mathematical structures (schoenfeld, 1994). productive mathematical discussions are those student-centered discussions where ideas are communicated mathematically so the shared ideas can be publicly understood, critiqued, and guided toward a learning goal (smith & stein, 2011). and perseverance in problem solving is describe as when students make conjectures about different solution paths and monitor and evaluate their progress, making changes whenever and wherever necessary (ccssmp, 2016). mathematics classrooms that have students make sense of and persevere in mathematics problem solving through productive discussions and collaborations can be considered a more indirect form of instruction. some researchers have argued that when instructional schemes in mathematics classrooms focus on problem solving, communication in groups, and more indirect pedagogy (i.e., studentcentered and focused on student investigation and sense making), students from the dominant culture may be privileged (apple, 1992; delpit, 2006; lubienski, 2000). yanisko perceptions of tracked students journal of urban mathematics education vol. 9, no. 2 159 other researchers have found that classrooms organized around these principles can be equitable and beneficial to low-income students of color if support systems are in place, and careful attention is given to assigning competence and developing norms for problem solving (see, e.g., boaler & staples, 2008; gutiérrez, 2000; kitchen, depree, celedón-pattichis, & brinkerhoff, 2007). this conditional clarification is consistent with the recommendations of researchers who suggest that students from outside the dominant culture must be given explicit access to the “culture of power” that will allow them to be successful (bourdieu, passeron, & de saint martin, 1994; delpit, 2006). in theorizing about how to dismantle the culture of power in mathematics education as a shift toward equity, gutiérrez (2007) offers four dimensions to consider: access, achievement, identity, and power. access relates to the tangible resources that students have available to them to participate in mathematics, including teachers, technology, curriculum, and classroom environments that foster participation. achievement includes enrollment and participation in mathematics courses, standardized test scores, and persisting in the mathematics pipeline. identity addresses students seeing themselves in the mathematics curriculum, as well as seeing themselves and mathematics in the world. power refers to the presence or absence of student voice and decision-making abilities, ability to apply mathematics as a critical lens through which to see society, alternative forms of knowledge, and the representation of mathematics as a field that needs people. providing students with access to teachers who can provide a rigorous and affirming mathematical experience is beneficial to equity. incorporating tasks of high-cognitive demand and opportunities to participate through student-centered, productive discussion allows students access to achieving a type of mathematical proficiency that allows them the ability to apply their knowledge outside of the mathematics classroom (see nrc, 2001; smith & stein, 2011; stein, smith, henningsen, & silver, 2000). having student voices and decisions around mathematical problems with multiple possible solutions or solution strategies as a central feature of the mathematics classroom facilitates students’ power in the classroom. to facilitate student sense making and productive discussion, teachers may structure lessons around problematic or investigative tasks, set up these tasks to aid student access, and require students to complete the tasks collaboratively in small groups (boaler, 2002a, 2002b, 2006). teachers may assert and maintain expectations, and develop norms for students’ production of explanations of mathematical solutions strategies in smalland whole-group discussions, as well as require students’ active engagement with making sense of, responding to, and attempting to understand student explanations and justifications (wood, 1999; yackel, 2001; yackel & cobb, 1996). such instructional strategies mean that students must be able to answer questions such as “how” and “why,” and that teachers may ask probing, leading, or advancing questions to encourage students to develop these answers yanisko perceptions of tracked students journal of urban mathematics education vol. 9, no. 2 160 (bowers, cobb, & mcclain, 1999; mcclain & cobb, 2001; yackel, 2001; yackel & cobb, 1996). having students pose questions to others is another important component of sense making through discussion (borasi, 1992; ciardello, 1998; zack & graves, 2001). to promote this exchange between students, teachers may direct student questions to other students, asking students to revoice a students’ explanation or to pose a question if they cannot revoice. to structure discussions that promote sense making, teachers may also carefully choose solutions to be presented so that a discussion of reasonableness or correctness of those solutions may result (smith & stein, 2011). teachers may use instructional strategies of their own design, or use strategies that are presented to them during teacher preparation courses or through teacher-support systems, such as teacher mentoring. integrating these facets of equitable instruction is challenging work for any mathematics teacher; it is especially challenging for new teachers (achinstein & athanases, 2005). in particular, it is important to attend to the expectations that new teachers have of students. teachers’ expectations of students may affect teachers’ ability to provide students access to high-quality mathematics instruction (oakes, 2005). teacher’s expectations of students may be influenced by race, ses, and perceived ability level (delpit, 2012; habermann, 1991; oakes, 2005). in particularly, there is a dangerous and harmful essentialism present in the discourse around lowincome students of color and education: that most members of this collective subpopulation are inherently low performing, specifically so in mathematics (faulkner, stiff, marshall, nietfeld, & crossland, 2014; martin, 2012; stinson, 2006). methods design this study used participant observation within a case study (yin, 2009). yin defines case study as “an empirical inquiry that investigates a contemporary phenomenon in depth and within its real-life context, especially when the boundaries between the phenomenon and the context are not clearly evident” (p. 18) the methodology employed allowed me to describe in depth the instructional strategies considered and co-constructed by the novice teachers and me within the context of mentoring sessions and a teacher seminar (subsequently described). due to the nature of the study, and my relationship with the teachers, i learned much about them, both professionally and personally. the collected data provided firsthand notations and records describing their experiences and professional change evidenced through the talk and action of the teachers within the context of their mentoring and seminar support as well as their actions within their classrooms and schools. yanisko perceptions of tracked students journal of urban mathematics education vol. 9, no. 2 161 program context the context of this study was first-year teachers who were enrolled in an alternative certification program. alternatively certified teachers were chosen for two reasons. first, alternatively certified teachers are often less prepared to teach in ways that result in high levels of student mathematics achievement due to the truncated nature of their teacher preparation (brantlinger, cooley, & brantlinger, 2010). second, as previously noted, alternatively certified teachers may hold lower expectations for low-income students of color than traditionally certified teachers in the same context (brantlinger & smith, 2013; delpit, 2012; habermann, 1991). it is imperative, in the interests of equity, to investigate these teachers’ expectations of their tracked students, as well as how they may be taught to negotiate these expectations. the program for which i served as a mentor and in which the participants were enrolled was a collaborative effort affiliated with a large mid-atlantic university and a large local school district. locally, it was not unusual for schools in this district to be described as hard to staff; in that, many of the schools had a history of low student performance on the state’s high-stakes standardized assessments, had a high rate of teacher turnover, and had a predominant population of students who were low-income children of color. by design, this alternative certification program recruited prospective teachers who were committed to the community and/or the population of the student body. those who enrolled in and successfully completed the requirements of the program were certified as teachers of middle school mathematics or science and one other core middle school subject. while it encompassed some of the features of alternative resident-teacher certification programs, the program was designed so that prospective teachers were slowly introduced to teaching. during the summer, they were enrolled in a short field experience. when they entered teaching in the fall, they were partnered with a cooperating teacher for a month-long internship, and then subsequently assumed a half-time (rather than a full-time) teaching load while partnered with another novice teacher from the cohort. in this way, a pair of novice teachers filled a full-time teaching vacancy. through pairing, teachers were able to gain experience with the work of teaching and to observe other teachers during their non-teaching time. participants the participants in this study were two novice teachers enrolled in the alternative certification program described. during the year in which i collected data, there were 13 teachers enrolled in the program. eleven of those teachers’ primary certification was in mathematics. of those 11 teachers, nine agreed to participate in the study. after conducting preliminary observations of these nine teachers in their summer methods course, i selected two based on my perception of their initial pre yanisko perceptions of tracked students journal of urban mathematics education vol. 9, no. 2 162 dispositions for and conceptions of teaching and placement to solicit data from two contrasting cases (yin, 2009). the first, jack davis (all proper names are pseudonyms), was selected due to his unabashed commitments to student-centered instruction. the second, michelle miller, was selected because of my perception of her reticence to teaching in a conceptual, student-centered manner. these teachers were my mentees in the alternative certification program and participated in a teacher seminar while simultaneously participating in the requirements of the teacher preparation program. although both teachers are african american, it was not intentional; the alternative certification program attracted a high number of teachers of color as the program was focused on recruiting candidates that had an investment and commitment to the urban community in which they were going to teach. michelle and jack shared a vacancy in a k–8 public school academy. the students in the middle school program were tracked in all of their classes based on their reading scores. that is, the students’ scores on a reading assessment determined whether they were placed in the honors track for all of their classes or the comprehensive track (i.e., the low track) for all of their classes. michelle was assigned to teach the low-track courses; jack was assigned to teach the high-track courses. michelle miller. michelle miller is an african american woman. she had earned both a bachelor’s degree in physics and a master’s degree in mechanical engineering and previously worked as an automotive engineer before deciding to become a teacher. her experiences in school and in her subsequent profession were characterized by sexism. during her undergraduate education, all of her mathematics and science professors were men who rarely provided her with the substantive feedback necessary to achieve. as an engineer, she felt isolated and excluded from any employee bonding activities. she had no prior teaching experience before entering the program. the licensures that she subsequently earned through enrollment in the program were in middle grades mathematics and science. michelle’s initial one-month internship was in the same school where she was subsequently assigned as a paired, half-time teacher. this school was a public k–8 academy. michelle’s permanent placement assigned her to teach the low-tracked seventh-grade mathematics and science courses; she taught one section of each subject. jack davis. jack davis is an african american man. he entered the program immediately after receiving his undergraduate degree in economics. his k–12 experiences with mathematics were racialized; even though he was high-achieving he and his family still had to fight for his placement in the high-track mathematics courses. he also received his education in the same school district in which he was teaching. his previous instructional experience encompassed mentoring and providing tutoring for middle school students. the licensures that he earned through enrollment in the program were in middle grades mathematics and social studies. jack interned and was permanently placed in the same k–8 school as michelle. he yanisko perceptions of tracked students journal of urban mathematics education vol. 9, no. 2 163 taught one section of seventh-grade honors mathematics and one section of seventh-grade social studies. setting the school in which jack and michelle were placed enrolled nearly 700 elementary and middle school students in 2011. ninety-five percent of the students enrolled at the school were african american and 2% were hispanic or latin@. sixty-nine percent of the middle school students were considered low income, according to the data on students receiving free or reduced-price meals. fourteen percent of the middle school students were receiving special education services. both jack and michelle taught tracked mathematics courses, where the students were placed into all their courses based on a standardized reading inventory. although mathematics and reading scores may be correlated, this placement is significant because oakes (2005) suggests that often teachers tacitly accept the placement procedures as objective and appropriate regardless of the methodology and appropriateness of placement. this setting is also of particular importance because studies of the differential expectations and educational experiences of students in tracked courses are often studied in more diverse settings (e.g., oakes, 2005). there is a need to investigate the implications of tracking on the expectations of teachers in tracked settings where the majority of the population is low-income students of color, where expectations are already low, and the cognitive demand of the curriculum is often lowered across all tracks (delpit, 2012, 2013). researcher positionality as a researcher, i worked closely with the participants in this study. i was their program provided mentor and supported them during their first year of teaching. furthermore, i was an active participant in a seminar during their field placement. that is to say, during their initial year as a teacher, i influenced what instructional strategies were considered and potentially discussed as mechanisms for supporting and maintaining high cognitive demand during instruction (stein & smith, 1998). i am a white, middle-class woman. i had a middle-class upbringing, and attended majority-minority schools in my k–12 education. i benefitted from a type of tracking discussed here, specifically, a gifted program that was originally created as a way to desegregate schools by incentivizing white student attendance. i recall being offended as a young person by the comments my parents’ friends would make about their decision to send me to school with “those kids.” after undergraduate education in computer science and mathematics, i came into teaching through a grant-funded alternative certification program. this program was a partnership between the same university and school district that partnered together for jack and yanisko perceptions of tracked students journal of urban mathematics education vol. 9, no. 2 164 michelle’s teacher education. i taught high school for 8 years in that same district before attending the university as a full-time doctoral student to conduct this study. while i was teaching high school, i was also continuously enrolled in graduate coursework. the school in which i taught served a population of about 75% african and african american and 25% latin@ students. i once again noticed, and was offended by, the reactions and negative comments about my students that i received when i would tell people where i worked. i increasingly began to recognize how important it was for me to educate myself on critical perspectives in education, culturally relevant pedagogy, issues of inequity and racism in education, and the cultures and lived experiences of my students if i was going to effectively conduct my duties as an educator. i enrolled in as many courses at the university that addressed issues of equity, power, race, and class in education as i could, and i made a concerted effort to engage in conversations about equity and diversity with my colleagues of color, both in the school and at the university. when i enrolled as a fulltime student to complete my dissertation, my fellowship provided me the opportunity to work with an alternative certification program and mentor new teachers. i felt connected to this work because i had also been alternatively certified, i had taught in the same district, and the alternative certification program was attempting to bring in teachers who had a dedication or a connection to the district community. my experiences drove my dedication to mentor teachers to provide high-quality mathematics education for the low-income students of color whom my district served. mentoring features. the focus of the mentoring was on two interconnected features of instruction. first, i encouraged the teachers to find or develop tasks that provided students opportunities to perseverance with and make sense of mathematics to uncover relevant mathematics on their own through application of their prior knowledge to an accessible problem. second, i encouraged the teachers to provide students opportunity to have productive small-group and whole-class discussions (smith & stein, 2011) about the relevant mathematics and to develop strategies of effectively facilitating that discussion without taking ownership of the discussion from the students. because these teachers were novice teachers, many of the instructional strategies that i proposed arose either in discussion with them during individual mentoring sessions or group seminars. i served as a researcher collecting data and as mentor of these individuals operating with the intent of supporting each of them throughout their first year of teaching. this mentoring is important because several researchers argue that mentoring should be integrated into transformation of the teaching profession as well as an inquiry-based approach to incorporating all the features of education reform, including standards-based practices (feiman-nemser & parker, 1993; hargreaves & fullan, yanisko perceptions of tracked students journal of urban mathematics education vol. 9, no. 2 165 2000; stanulis & floden, 2009). research suggests that focusing on “balanced” approaches to instruction would improve the actual practice of teaching, rather than simply focusing on emotional support of new teachers (stanulis & floden, 2009). furthermore, there is a call for more research on how to support novices in their development of reform-based teaching by incorporating the standards into the base conception of induction mentoring practice (wang & odell, 2002). this mentoring treatment foregrounded two of the standards of mathematical practice (ccssm, 2016), “make sense of problems and persevere in solving them” (¶ 2), and “construct viable arguments and critique the reasoning of others” (¶ 4), while simultaneously incorporating content standards, mathematics teaching practices (nctm, 2014), other standards of mathematical practice (ccssm, 2016), and the nctm principals and standards for school mathematics (2000). during this mentoring process, i provided time and guidance during collaborative sessions for the teachers to reflect on videos of instruction as well as their own teaching by focusing on teacher moves that promoted or did not promote students’ sense making and productive discussion (smith & stein, 2011). during the planning sessions, michelle, jack, and i met as a group to either plan individual lessons or whole units. i shared information about tasks that, i believed, would promote student discovery of mathematics; the ordering of topics in a way that made mathematical sense; the different topics that would need to be incorporated; how to promote discussion; and what manipulatives would be appropriate for hands-on, concrete learning opportunities. during observations, i usually simply took notes, but there was one occasion that i co-taught with michelle while jack observed as a response to a request from michelle. during reflection sessions, i would focus on giving feedback on specific teacher moves or plans that, from my perspective, supported or inhibited student sense making and productive discussion. we would also collaboratively review student data and make decisions on next steps for student learning. the content of each mentoring cycle is summarized in table 1. the first cycle is disaggregated between jack (j) and michelle (m), as that planning session was conducted individually, while all the others were conducted in a group of three (jack, michelle, and me). differences in mentoring of michelle and jack. when specifically focusing mentoring resources on michelle, the work of mentoring became focused on lesson and unit planning. together, we worked on planning lessons that included tasks that were of a high level of cognitive demand, and included opportunities for student talk. when specifically focusing mentoring resources on jack, i attuned jack to the on-task discussions that the students had without his direct intervention, as well as mentored him to reduce the amount of scaffolding (e.g., leading questions) that he provided to small groups during their time working on problems that had high cognitive demand. also, because michelle and jack observed each other due to their half-time teaching arrangement, they were each able to participate in the other’s reflection conversation yanisko perceptions of tracked students journal of urban mathematics education vol. 9, no. 2 166 with me and support the positive strides that both students and they were making with respect to having students persevere with, make sense of, and discuss mathematics. table 1 mentoring cycle content cycle content 1 (j)  task with high cognitive demand (stein & smith, 1998)  individual work, small-group work, and whole-class discussion (lampert, 2003)  calling on students randomly during whole-class discussions  moving toward questions that elicit mathematical thinking rather than leading questions (driscoll, 1999)  using talk moves to engage more students in whole-class discussion (chapin, o’connor, & anderson, 2009) 1 (m)  task with high cognitive demand (stein & smith, 1998)  individual work, small-group work, and whole-class discussion (lampert, 2003)  (teacher suggested) student roles in groups (johnson & johnson, 1999)  feedback on student investigation implemented as direct instruction  feedback on initiation-response-evaluation model of student talk  using talk moves to give students agency in their learning (chapin, o’connor, & anderson, 2009)  allotting enough time for student sense making 2  co-planning open-ended task with high cognitive demand (stein & smith, 1998)  co-teaching and facilitating student whole-class discussion with michelle as jack observed  feedback on pacing of lesson; do not have to wait until all students have the right answer to have a summary discussion  feedback on focusing on mathematical goals  analyzing student work  (jack) feedback on effective use of talk moves (chapin, o’connor, & anderson, 2009) 3  teacher reported that administration wanted more use of hands-on activities and manipulatives  task with high cognitive demand (stein & smith, 1998)  unit planning – expressions and equations  guess-test-generalize (cme project, algebra i) and polya’s problem-solving techniques (polya, 1945)  guidance to avoid using keywords to decode word problems (nrc, 2001)  concrete-representational-abstract sequence of instruction (witzel, mercer, & miller, 2003) and the national library of virtual manipulatives  activating prior knowledge (hollingsworth & ybarra, 2009)  facilitating teachers doing the math 4  having students construct their own mathematics definitions based on examples  using multiple different representations of figures  unit planning – geometry  task with high cognitive demand (stein & smith, 1998)  heterogeneous grouping  small-group stations  using rubrics to hold students accountable for participation in small-group instruction  asking a random student to explain to the teacher to encourage students collaborating to develop an explanation (boaler & staples, 2008)  (michelle) feedback on slow pacing and lowering of cognitive demand yanisko perceptions of tracked students journal of urban mathematics education vol. 9, no. 2 167 5  unit planning – data analysis  tasks with high cognitive demand (smith & stein, 1998)  developing a project for project-based learning (krajcik & blumenfeld, 2006)  working through why constructions “work” based on properties of circles with the teachers 6  review of project for project-based learning (krajcik & blumenfeld, 2006)  (jack) review of the mathematics behind misleading graphs  (michelle) reflection on successful whole-class discussion on which data displays are appropriate for which situations 7  reflection on the project  discussion of the upcoming unit  discussion of upcoming testing and field trips  benefits of including tasks with high cognitive demand and small-group discussion  closing out coaching data collection and analysis i collected data from seven teacher-support reflection cycles. each support cycle took place over approximately two weeks. these sources allowed me access to teacher expectations of students as well as their instruction trajectory (see table 2). table 2 description of data sources data collected description collaboration: teacher seminar all mentees (four mentees and i) met to review teaching videos and their own teaching with an eye to student sense making and discussion. sessions were videotaped. planning: mentoring session jack, michelle, and i met bi-weekly to plan lessons and units. observations lessons were observed bi-weekly. there was one instance of coteaching with michelle while jack observed. reflection: mentoring session after the observations, the mentor and the participants met oneon-one to debrief the lesson. baseline baseline data was collected through observations of participants’ summer coursework, reading of their submissions in summer coursework, and interviews about how to teach with a task. follow-up interview participants were interviewed after the original treatment in order to reflect on questions on what transformed their teaching. within the teacher-support reflection cycle, all of the planning and reflection mentoring sessions were audio recorded and transcribed. each teacher seminar, as conducted between november 2011 and march 2012, was also video recorded and transcribed. although seven classroom observations of each participant were conducted, only three of these were sources for data collection. the teaching observations that served as sources for data collection were conducted in november 2011, during cycle 1 of the teacher-support reflection cycle; in january 2012, during cycle 4 of the teacher-support reflection cycle; and in march 2012, during cycle 7 of yanisko perceptions of tracked students journal of urban mathematics education vol. 9, no. 2 168 the teacher-support reflection cycle. i identified these particular classroom observations as sources for data collection to establish whether instructional change had transpired as documented by the beginning, middle, and end of the study. i took field notes during each of these three observations and made notations that indicated particular features of the lesson that were relevant to analysis pursuant to answering the research questions. these notations consisted of codes that i hypothesized before conducting the study. each teacher also carried an audio-recorder to document their verbal teacher moves as well as responses from students with whom they were interacting. these audio-recordings were transcribed. my observational data consisted of field notes and audio transcripts; however, my audio transcripts are limited to statements of teachers and the responses of students in interaction with their teachers. the codes for this corpus of data were centered on strategies and challenges. i looked for instances of strategies that the teachers were using to incorporate mathematical sense making and productive discussion, both those that the teacher used independently and those that we developed together. i also looked for the teacher using language that suggested they were struggling to implement high-cognitivedemand tasks that allowed students room for discussion, and what they expressed as their challenge. working with other colleagues in the doctoral program, we coded pieces of data independently to ensure the coding method was reliable. a subcode that arose during analysis in the “challenge” codes was student ability. teacher talk that referenced their notions of their students’ ability to participate in the mathematics arose regularly. findings during their interactions with me in the mentoring sessions, both michelle and jack expressed their expectations of their students’ ability due to their track and their experiences interacting with their students. they also, through mentoring and reflecting on their experiences with their students, began to develop the ability to provide high-quality mathematical experiences that allowed their students to engage in mathematical sense making through participation in productive mathematical discussion. michelle: low-tracked course initially, michelle had low expectations for her students in the low-tracked course. through mentoring and experience with her students, however, she began to teach in a way that provided opportunities for students to engage with highquality mathematics tasks and participate in mathematical sense making through productive discussion. yanisko perceptions of tracked students journal of urban mathematics education vol. 9, no. 2 169 michelle’s perception. as we discussed mathematical content in a manner that focused more on overarching mathematical ideas and topics rather than presentation of single, isolated skills, michelle’s perception of her students’ ability and the successes they would be able to experience when approaching mathematics through both concepts and skills often troubled her. in one mentoring session, i suggested that a teacher could teach equation writing and solving from a conceptual perspective based on generalization from arithmetic and working backwards to undo arithmetic procedures. i posited that then it would be unnecessary to teach solving onestep equations using addition, subtraction, multiplication, and division, and then two-step equations with combinations of those as individual elements. i suggested that initially thinking about an equation as a procedure with a result would allow students to think about what was done to the variable and then what would have to be undone to return back to the value of that variable. michelle responded by stating that she would separate one-step and two-step equations and then further break down one-step equations to focus on those that involved addition and subtraction as a single skill, and then subsequently address those one-step equations that involved multiplication and division: michelle: well, i don’t know, i probably would for [my class]. maybe i wouldn’t take 2 days for each [operation]…. i can combine addition, subtraction and then multiplication, division. jack: i can see that. michelle: i would put those together but i would do step-by-step. i wouldn’t combine them all at once and do several different operations with [my students]. so, okay, well i’m happy that it gives us a time to teach conceptually in step-by-step with the operations. okay. in this explanation, michelle immediately followed her statement of preference for isolating addition and subtraction one-step equations from one-step equations involving multiplication and division with the statement that she was happy to have the time to teach conceptually. this response suggests that although she wanted to use manipulatives to have the students visualize the individual operations, she still felt the need to isolate a single skill into a lesson. when i suggested that she could address solving equations as one concept, she resisted. she stated, “yeah, you probably could do it in your class” (michelle, mentoring session, december 9, 2012), referring to jack’s class. this highlights her concern about her students’ ability level, because the school labeled jack’s class as an honors class and labeled michelle’s class as comprehensive, ostensibly populated by students with a lower level of ability. michelle’s change in perception through experience. it may be that michelle was experiencing some dissonance between her perception of what she felt her students were able to accomplish successfully and what they actually were able to do. she had been seeing her students have successes in the classroom on problems that yanisko perceptions of tracked students journal of urban mathematics education vol. 9, no. 2 170 she felt might have been too difficult for her students to approach. she had stated previously: “sometimes you’re kind of leery in giving them stuff and then run off on their own. but sometimes they surprise you” (michelle, seminar, november, 21, 2012). the successes that she was seeing her students experience in the classroom from more discourse-based, conceptually focused instruction challenged her previous ideas regarding the limitations associated with perceptions of her students’ limited ability and allowed her to try more difficult mathematics problems with her students. not only did michelle begin to incorporate richer mathematics problems into her teaching but also she began to include more small-group work. she had students work together on real-world problems and allowed students to reason through the problems concretely before she asked questions that would help them think about the mathematical ideas behind them. she found that her students were successful in translating their thinking into mathematical terminology: so, for example, a lot of them, they started off with 30 and then $10 was for food, so they took out $10 out of the total 30 and i said, “what are you doing? what are you actually doing when you’re doing that?” and they get it! they say, “oh, i’m taking away. oh, what’s the mathematical term [for that]? oh, subtracting.” (michelle, mentoring session, december 30, 2011) michelle was happy to see that her students were “getting it” when they were approaching word problems, as michelle had initially considered word problems to be too difficult for her students. she saw her students have success working on a problem in their small groups without her intervention and without her leading the students through the whole process of problem solving. these instances of students working together to solve difficult mathematics problems allowed her to feel that she could take more risks in her mathematics classroom. she began to change her perception of scaffolding. originally, she thought of scaffolding as the practice of offering leading questions that directed students through the intended individual procedures and skills in the classroom. now she was scaffolding by asking guiding questions that facilitated students’ mathematical thinking while they investigated problems using concepts and ideas. michelle seemed to use what she knew or learned about student difficulties in a lesson as an insight for influencing her future instructional decisions, instead of abandoning her plans entirely. michelle taught a mathematics course and a science course to the same students. to engage her students in a conceptual discussion about different systems within the body, she had given her science class a homework assignment that required them to read about a certain system in the human body and then come prepared to discuss their knowledge with the class in small groups. however, she was disappointed that her students did not come to class prepared. she assigned blame to the students for not doing the homework assignment yanisko perceptions of tracked students journal of urban mathematics education vol. 9, no. 2 171 and therefore for sabotaging a lesson that would have focused on discussion. she noted, because of their lack of preparation, the students simply did not have anything to add to the discussion: i don’t know. i had this nice thing planned, and it’s one thing too when you’re trying to have group work and facilitate discussion, you have to have something to add. i had a homework assignment, they were supposed to investigate things that they identified that they were going to do. if half the class doesn’t do that, it’s like i wanted to rotate them so that they would [teach] each other and provide information for one another. it’s like, when they don’t do the task; it just shuts the discussion down. okay, you’ve got nothing to add. you know, they’ll just be sitting there. (michelle, seminar, february 1, 2012) michelle expressed frustration with her students for their inability to participate in the discussion. in this case, it seemed that the students were not able to participate in the discussion because of the individual, at-home nature of the preparatory assignment. michelle seemed to recognize this limitation, as she did not give up on having conceptual-themed discussions with this group of students. instead, she modified her instructional strategies to allow time in class for students to formulate their responses in their small groups before engaging in the discussion. subsequently, she tried again to organize a class based on the assumption that students would prepare for class in advance. but this time, michelle did so with her mathematics class. in a unit on data analysis, she provided her students with a list of scenarios and asked the students to decide which data display would be the most appropriate in the given scenario. she also provided them with a reference sheet of key terms, definitions of those terms, and their exemplar applications, to allow students access to a discussion where there could be multiple correct responses to a single question. instead of michelle’s initial scaffolding design, which was to assume control of the conversation and carefully guide the trajectory of student talk in the classroom, she positioned the students centrally in the discussion. she provided the students with a means of facilitating their discussion through their reference sheet and then allowed them to express their thinking to each other, only providing direction and comments when necessary to continue the flow of discussion. she began to place herself in a position of facilitator rather than director of classroom talk. after michelle made these different instructional adjustments, she saw that her students could successfully participate in a conceptual discussion: and we’re at the point [where] we’re talking about which data display to choose and why, depending on the circumstance. so i had two scenarios, and i asked each group to talk amongst themselves to determine which data display they’ll choose for [each of] the scenario[s]. and so, once they did that i opened it up to the floor. … and i called a group, a table, and they shared their response, and i said, “okay, who agreed or disagreed and want[s] to add to it?” so, that spawned a lot of discussion. “oh, look, i disagree because this, this, and this.” and they were able to use the terminology of why… and recommend something else. and so, that kind of went on back and forth for a good yanisko perceptions of tracked students journal of urban mathematics education vol. 9, no. 2 172 10 minutes, and it was a really good discussion. so i got to hear what they were thinking. so it was really good. positive. (michelle, seminar, march 7, 2012) although michelle had experienced earlier failures when expecting the students to prepare for and participate in whole-class discussions that focused on mathematical concepts and had multiple solutions, she had tried again and found success. she negotiated this challenge by providing not only class time to prepare in their small groups but also a reference sheet to remind students of the mathematical tools that they could use to respond to the tasks and to explain and justify their reasoning and solutions. this experience provided her with another example of student success in her classroom. summary of michelle. michelle taught 70-minute class periods throughout the year. in november, michelle incorporated 5 minutes of small-group work, some guided presentation, and individual work. in the january observation, she attempted to allow students to collaborate in small groups for 5 minutes, then co-opted their discussion when she felt the student were struggling. by march, she spent 34 minutes of her 70-minute class period facilitating students’ discussion in small groups, followed by student presentations and discussions of findings in the whole group. this transition can be attributed, in part, to the consistent mentoring that resulted in her having successful experiences with her students working on rigorous mathematics and having productive discussions. throughout the year, michelle saw more successes in terms of her students’ ability to access different mathematical problems in her classroom as she changed the types of scaffolds she provided. instead of leading students through systematic procedures that she considered difficult for her students, she began to facilitate their interaction with mathematical concepts and their constituent skills. where earlier in the year she would stop small-group time to demonstrate solutions to the exercises that the students were working on, later in the year she provided more indirect scaffolding and allowed the students to work together to solve problems. she discovered that her students were capable of working with other students on difficult mathematics with less teacher direction. she reflected on her surprise and happiness about her students’ successes: “they do it and then you’re shocked…. it went very, very well” (michelle, seminar, march 14, 2012). she seemed to continue allowing students more autonomy in their small groups as a result of her negotiation of this challenge, in addition to allowing her students more opportunities for working together. furthermore, instead of removing instances in which she thought her students would struggle, she made different instructional decisions to support her students’ efforts to working through not only mathematical skill but also the meaning of concepts. yanisko perceptions of tracked students journal of urban mathematics education vol. 9, no. 2 173 jack: high-tracked class although jack initially believed that his honors tracked students would easily participate in sense making through productive discussion, his initial experience in the classroom gave him a different perception. due to that experience, he became unsure and reticent about implementing instruction that promoted problem solving and discussion. jack’s perception. jack perceived that there were differences in his students’ and michelle’s students’ willingness to participate in student-centered instruction that included discussion. he stated: when i sit at my desk and watch [michelle] teach and i see her students doing…it’s easier for them to work off script…. i think my kids were technically supposed to be in honors classes, they like much more direct instruction and they don’t like to be asked to do something first. they’re a lot more resistant to it…. i think, like maybe, if you were in an honors class, you probably would be good at “doing school” and you’ll be good at “doing school” if you were good at just listening and taking notes. (jack, seminar, november 21, 2011). jack felt that his students were acclimated to instructional techniques that directed them as to what to do and what to recall. furthermore, he felt that his “honors” students were familiar with “doing school” in a particular way, a way that required them simply to sit quietly, listen, and take notes. he felt that michelle’s students might not have been as completely acculturated to direct instruction in the way that the honors students were, given that they were not considered to be “good at doing school.” therefore, he felt “being good at school” caused his students to be more resistant to participation in student-centered lessons and mathematical discussion, as compared to michelle’s students. jack also believed that his students’ familiarity with particular norms of schooling prevented them from productively collaborating; in that, instead of explaining and justifying solution strategies, asking clarifying questions, making sense of different solution strategies, and sharing the work, they were used to simply providing and sharing answers. he felt that his students were more comfortable with working individually rather than in small groups: i hope they drag [the students who do not understand] with them and not just leave them alone…. it’s hard to make them cooperate you know…. sometimes they just like to do it on their own. (jack, mentoring session, january 18, 2012) jack wanted his students to work together so that a student who had a greater understanding of a particular topic could assist the other students who were not as secure. however, the students’ comfort with working individually, rather than col yanisko perceptions of tracked students journal of urban mathematics education vol. 9, no. 2 174 laboratively, made it difficult for jack to get them to assist each other when addressing a mathematics problem in small groups. jack attributed his students’ resistance to student-centered mathematics and participation in discussion to the type of instruction with which his honors students were familiar. this resistance was not what he had expected as he entered into teaching believing that students who were labeled honors would find it easier, as compared to the “comprehensive” students, to participate in discussion and studentcentered instruction. his students’ resistance to collaborative problem solving and discussion, however, led him to believe that his students had repeatedly experienced instruction in their years of honors course placement which were organized around lecture, note taking, and independent seatwork. he reflected during a follow-up interview during the following school year: if anyone, i would expect them to be able to handle it more…like i said…it was harder for them to make the jump to doing something extra. i guess what they saw as something extra. as long as they could write it down, i think they thought that was sufficient. um, but i guess i had to spend time explaining like, “you learn more, you learn more by teaching.” (jack, follow-up interview, may 7, 2013) because jack believed that his students previously had solely been required to complete problems individually and to record their answers in written form, they were resistant to doing something they felt was unnecessary, or “extra.” therefore, jack had to reinforce his expectations consistently and to provide a rationale as to why students would benefit from discussing mathematics to entice his students into participating fully. jack’s change in perception through experience. initially, jack attempted to implement a procedure whereby those students who finished a particular problem first would serve as experts and assist other students in completing that problem. during an early seminar, jack explained the problem with this particular strategy: so, one thing i tried…in terms of getting students to appreciate each other, [or] what each other has to…say…when they work on something,…. whoever finishes first and gets it correct, they get to go around and explain it to the rest of the class and like check off the papers and stuff. but what i noticed was that…they weren’t explaining. (jack, seminar, november 7, 2011) jack wanted his students to discuss and explain their solution strategies to others in order for each of the students to develop an appreciation of other students’ thinking. the students who were serving as experts, however, were simply telling the others the answer and having them change their approaches to earn a check on their work. this telling was not the desired outcome; jack wanted students to learn to collaborate and explain their thinking to others. jack realized that the strategy of using student experts was not yielding the result of collaboration. yanisko perceptions of tracked students journal of urban mathematics education vol. 9, no. 2 175 in november, jack taught a lesson where students investigated patterns to discover how to calculate numbers raised to the zero power and negative exponents. to investigate negative exponents, jack had the students complete a table of positive exponents and extend the pattern backwards to a zero exponent. jack circulated around the classroom and encouraged students to explain their thinking about why the result of raising a number to a zero exponent would be 1. jack: can you explain why you think it may be 1? do think it may be 1? can you explain, [student name]? student 1: no. jack: so you just think it’ll be 1 but don’t know why. student 1: oh, i used the calculator (laughing). jack: (to another student) ok, you think you can [explain it]? student 2: yeah. jack: go ahead and explain it. student 2: i think it would be 1 because yeah i would say 3 [to the] 1. three [to the] 1s gonna equal 3 but as you have 9 over 9 that’s gonna be 1 whole. (jack, classroom observation, november 17, 2011) when one student said that she used the calculator to find her answer and could not explain her thinking further, another student in that group suggested that she could explain it and then proceeded to explain her thinking. jack persisted in pressing students to explain their thinking rather than just finding a solution. he asked for explanations from individual students as well as encouraged them to speak to each other, saying, “i need you to make sure that everybody at this table [understands]” (jack, classroom observation, november 17, 2011). jack was not only eliciting explanations from students so that they could relay their ideas to him, but also encouraging them to explain to each other when he was not present. later in january, jack began to incorporate center rotations with his students. in this particular lesson, half of the class was completing a review “scavenger hunt,” where folded papers were arranged around the classroom with a multiplechoice problem on the inside, and the answer to a different problem on the outside. students would move about the classroom searching for the answers to their problems, and, in this way, complete each of the problems. the other half of the class was split into two groups, one that was directed by jack to take notes on the definitions related to congruence, and the other was attempting to discover the surface area of a rectangular prism through the use of nets. most of jack’s focus was on the group to which he was providing direct instruction, so the students in the other group had to work collaboratively and mostly autonomously, as jack would check back with them only intermittently. jack previously challenged a small group of students to refine their formula by double-checking if the area of each face of the rectangular prism was the same. while jack worked with the other group, the following conversation transpired: yanisko perceptions of tracked students journal of urban mathematics education vol. 9, no. 2 176 student: mr. davis, we got it. jack: do [all the sides] go together? student: no, because [different side lengths of the different pieces]. jack: check each other, [student name] and [different student name] check each other’s. (jack, classroom observation, january 27, 2012) two things are salient about this conversation. it shows that jack was still, consistently, not only requiring students to work together, but also asking them to collaborate through validating their answers with other students. also, there is a shift in the language that the students were using. in the initial observation, the students used the work “i” when explaining their thinking. here, the student used the word “we.” jack’s consistent, high-level expectations were changing the norms of student discussion in the classroom. in march, jack reflected in seminar about a lesson in which he engaged students in error correction of a past assessment. by then, jack was aware of students engaging autonomously in their small groups in discussions that included both procedural and conceptual talk. i also affirmed, through observation, that his students were making sense of problems, procedurally and conceptually, in small groups, through discussion. jack was also aware of occasional incidents when students would give answers to each other and then would quickly redirect their conversation to explanation. he reflected: i think there was a mix [of procedural and conceptual conversation among students]. of course, there’s going to be some people just saying, “oh, the answer was ‘a’,” and so i did hear that. and so for those people i had to say, “but why is it ‘a’?” (jack, seminar, march 7, 2012) summary of jack. jack negotiated the challenge of his students’ receptiveness and ability to persevere, make sense of, and have productive discussions about mathematics. in november, jack’s students spent 50 minutes of his 70-minute class period working individually, while the rest of the time was spent participating in teacher-led talk. by the time i observed in january, jack had his whole class working in centers for the duration of the period, where they worked in small groups either autonomously, in student-centered investigations, or teacher-led notes. during the observation in march, jack spent 45 of 70 minutes in whole-class discussion where students presented their findings, challenged each other, and justified their arguments. although there had been a marked shift in both the occurrence and the manner through which students were discussing in the classroom, there still was an occasional deviation from that pattern. while jack was cognizant of these deviations, he persisted in enforcing his high expectations for these students by reminding them quickly to explain their thinking to each other. jack’s commitment to excellence in teaching and education motivated him to work diligently to change the yanisko perceptions of tracked students journal of urban mathematics education vol. 9, no. 2 177 culture of discussion in the class through teacher moves that reminded students of his persistent expectation of explanation and discussion. discussion for michelle and jack, the fact that their students were tracked had at least an initial effect on their instructional decision-making. research indicates that tracking has an influence on teachers’ perceptions of students, especially their assumptions about their lower-tracked students (oakes, 2005). in this case, tracking affected the perceptions that these teachers had with respect to their students both in the courses labeled comprehensive and in those labeled honors. michelle taught a course that was labeled as comprehensive, although this tracking was based on her students’ prior reading scores and not on any prior assessments of their mathematics proficiency. nevertheless, michelle often used her preconceived notions of her students’ low ability as an excuse for not engaging them in challenging, student-centered mathematics lessons that would require problem solving and provide the students with opportunities for sense-making discussions. in addition, michelle did not recognize the differing strengths or needs of her students. she spoke of her students as an aggregate, as a whole class only composed of students with low levels of ability. although she made attempts at incorporating opportunities for discussion in her class, her perception of her students’ ability caused her to fear student struggle, to provide direct instruction of procedures, and to focus much of her attention on rote memorization of vocabulary terms. nonetheless, although jack and michelle’s students took the exact same district-prescribed unit assessments, she never indicated that she had any evidence that her students were performing any differently that jack’s honors students were. her instructional decision making, particularly her decision to slow down her pacing and to include more direct instruction, seemed to be based solely on her preconceived notion of what the “limited” ability level of students placed in a lower-tracked course meant to her. however, she was able to negotiate this challenge and engage her students in productive mathematics learning in wholeand small-groups discussions that allowed students opportunities to make sense of mathematics. michelle used strategies to mitigate her students’ perceived struggle to encourage their participation in class discussions. she gave students both individual and small-group thinking time before requiring them to share their thinking to the class. she also provided reference sheets with mathematical terminology so that students could use the “correct” mathematical vocabulary in their discussions. she required them to verify individually calculated solutions with other small-group members. also, michelle began to appreciate using small-group work because it provided her with more contact time with individual students. during this time, she could correct any misconceptions held by individuals. she may not have changed yanisko perceptions of tracked students journal of urban mathematics education vol. 9, no. 2 178 her mind about her students’ initial ability, but she was able to see her students’ successes in mathematical sense making as the year progressed. jack, on the other hand, taught the class that was labeled honors; students were placed in this course by the same prior reading assessment that determined the placement of michelle’s students. although jack stated that he had automatically assumed that his honors students would be more capable of engaging in studentcentered mathematics investigations and discussions, he found his students resistant to this type of instruction. he attributed their resistance to the students’ assignment to high-tracked courses. he felt that because his students had been tracked into honors courses, a feature of scheduling that had been in place for more than one year, his students had become acclimated to learning in a particular manner: through listening, note taking during directed instruction, and completion of written solutions to mathematics problems reflecting their individual and independent efforts. he felt that his students’ prior experiences with tracked courses, and their prior teachers’ mode of instruction in these tracked courses, made his students appreciate and expect specific types of instruction over others. they were not only familiar with these instructional routines; they had learned how to be “good at school” in environments marked by these practices. jack felt that he had to struggle harder than michelle to teach students how to participate in student-centered learning environments where the expectation was for students to explain their thinking to others, and he attributed that struggle to the fact that his students were tracked into the honors section. jack, however, used his experience with building relationships with middle school students to establish a safe environment in his classroom. he built a rapport with his students that caused them to trust that he would not embarrass them publicly. furthermore, he established norms in his classroom whereby students would be respectful of others during smalland whole-group discussions and would not devalue the thinking of others. through these techniques, he began to develop a culture of participation in smalland whole-group discussion in his classrooms. his students became familiar with the practices of struggling with, and making sense of, mathematics collaboratively. implications tracking has long been targeted for discussion in the mathematics education community. many mathematics researchers have advocated for heterogeneous classrooms, claiming that they are better sites to support all students’ learning across and within all achievement levels, advancing both equity and mathematics achievement (e.g., boaler & staples, 2008; burris, hubert, & levin, 2006). tracking not only affects teachers’ preconceptions of students’ potential to achieve, but also it might affect students’ development of a productive disposition about mathematics (e.g., oakes, 2005). nevertheless, there has been much resistance to group yanisko perceptions of tracked students journal of urban mathematics education vol. 9, no. 2 179 ing students’ heterogeneously both in public discourse and in policy documents (e.g., loveless, 1998). education policy is not likely to eliminate tracking. thus, it is important to prepare teachers for the possibility of teaching in a tracked course. the study reported here highlights two points. first, teacher education courses must include an opportunity for both prospective and practicing teachers to think about and discuss tracking critically. teacher educators must provide counternarratives to the pre-conceptions that tracked former students bring to their classrooms as teachers. the teachers’ perceptions of tracked students influenced their desire and ability to engage their students in sense making and productive discussion. furthermore, it has been stated that teachers in alternative certification programs may be more likely to harbor low-expectations of tracked students than traditionally certified teachers (brantlinger & smith, 2013; delpit, 2012; habermann, 1991). therefore, it follows that critical discussion about tracking, the perceptions of students who are tracked, and how to actively work against tracking are arguably even more important in alternative certification programs than in traditional teacher education programs. it often is assumed that tracking is necessary to provide appropriate instruction for homogeneous groups of students. research has shown that detracking and heterogeneous instruction is powerful and equitable (boaler & staples, 2008; burris et al., 2006). moreover, oakes (2005) finds that the methodology of tracking is often tacitly accepted as appropriate, and therefore teachers believe that instruction with different levels of rigor is appropriate for their tracked population. as teacher educators, we need to problematize tracking and its implications. in addition, teachers need to be pushed to continue to try to engage their students in sense making through participation in productive discussion so they can gain the experience through experimentation that can further challenge their pre-conceptions of their students’ ability. through literature, hands-on experiences, and classroom discussions, we may be able to disrupt the misconceptions that new teachers might have about students who have been tracked. this disruption will help to address situations similar to that evidenced by michelle and jack, as their assumptions about their students’ abilities contributed to the challenges that they faced when developing student-centered instruction, instruction that included opportunities for students to discuss and make sense of mathematics collaboratively. furthermore, teacher educators need to prepare teachers to have discussions with their students that include counter-narratives to ability grouping. teachers must be equipped with ways to talk to students about their ability so that students placed in low-track courses are able to build the same types of productive dispositions that students in high-track mathematics courses might develop. additionally, high-tracked teachers need to be equipped with ways to challenge the norms of schooling that may be prevalent in courses that are labeled honors. often teacher educators focus solely on the low teacher perceptions of low-tracked students; how yanisko perceptions of tracked students journal of urban mathematics education vol. 9, no. 2 180 ever, student resistance to a “new way” of participating in mathematics courses also needs to be a focus, considering that schools that serve low-income students of color often heavily stress low-level direct instruction for test-preparation purposes even in those courses noted as honors (delpit, 2003). in this way, teacher educators can help to reduce the challenges that novice teachers face. teacher educators, however, cannot expect that teachers learn everything they need to know about teaching during their teacher preparation program (hiebert et al., 2003). therefore, it is important that teacher education programs prepare teachers to analyze their students’ abilities in ways that enable the teachers to recognize student strengths as well as weaknesses and to learn for themselves that students of any track are capable of being successful. additionally, it is important for teacher education programs to help novice teachers develop ways to de-track mathematics within their classrooms and at their schools. in this instance, jack and michelle were encouraged to group students heterogeneously in their classrooms. furthermore, because jack and michelle were planning together, the level and type of education that the students in the two different tracks were receiving were more similar than they were different. therefore, as jack and michelle learned how to implement lessons that included tasks with high levels of cognitive demand (stein & smith, 1998) and productive discussion (smith & stein, 2011), they were effectively de-tracking the seventh grade at their school. it should be encouraged in teacher preparation programs that teachers, no matter the level of the track of their placement, consistently hold students to high expectations, allow them opportunities to make sense of the mathematics, and allow them to work in heterogeneous groups (boaler & staples, 2008). concluding thoughts the participants were intentionally placed in hard-to-staff schools where the population was high-minority and low-income. in addition, students in these schools are tracked, which is pervasive in education. teachers who work in schools that serve a population that is high-minority and low-ses may have, or have to challenge, low expectations of the children in their classrooms (delpit, 2006; haberman, 1991). tracking may further the low expectations of this population of children. research indicates that tracking has an influence on teachers’ perceptions of students, especially their assumptions about their lower-tracked students (oakes, 2005). both the nctm and the ccssm, however, have goals for every student to experience an education that provides opportunity for mathematical success for each student at a high level. what this study elucidates is that tracking may negatively affect teachers’ perceptions on their ability to engage both low-tracked and high-tracked students in persevering in, making sense of, and productively discussing mathematics. yanisko perceptions of tracked students journal of urban mathematics education vol. 9, no. 2 181 references achinstein, b., & athanases, s. z. 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(2001). making mathematical meaning through dialogue: “once you think of it, the z minus three seems pretty weird.” educational studies in mathematics, 46(1/3), 229– 271. journal of urban mathematics education july 2011, vol. 4, no. 1, pp. 23–49 ©jume. http://education.gsu.edu/jume clarence (―la mont‖) terry, sr. is an assistant professor in the department of education at occidental college, 1600 campus road, education (f-1), los angeles, ca 90041; e-mail: lterry@oxy.edu. his research focuses on the impact of critical pedagogical interventions on the mathematics learning and identity development of male african americans across learning settings. mathematical counterstory and african american male students: urban mathematics education from a critical race theory perspective clarence l. terry, sr. occidental college in this article, the author argues that the persistent underachievement of many african american male students in urban school districts requires pedagogical interventions designed to re-engage and re-orient students to mathematics as a critical cultural activity. while counterstory, a critical race theory construct, has wide application across educational research, few researchers have explored its relevance for urban mathematics education. here, the author provides a grounded operationalization of counterstory by examining the pedagogy embedded in african american narrative on literacy. using data from a participatory action research project involving male african americans in south los angeles, the author then demonstrates how counterstory is embedded in the mathematical activity associated with graphical representation and trend analysis of data. through the theorized approach to counterstory presented, the author provides useful direction for education researchers interested in counterstory, and for mathematics educators attempting to broaden their pedagogical approaches to teaching and learning mathematics. keywords: african american male students, critical race theory, critical mathematics literacy, counterstory, mathematics education, urban education it’s a number game, but shit don’t add up somehow: like i got 16 to 32 bars to rock it but only 15% of profits ever see my pockets; like 69 billion in the last twenty years spent on national defense but folks still live in fear; like nearly half of america’s largest cities is one-quarter black, that’s why they gave ricky ross all the crack; 16 ounces to a pound, 20 more to a ki a five-minute sentence hearing and you’re no longer free… terry mathematical counterstory journal of urban mathematics education vol. 4, no. 1 24 *** ... crack mothers, crack babies and aids patients, youngbloods can’t spell but they can rock you in playstation. this new math is whippin motherfuckers’ ass you wanna know how to rhyme? you better learn how to add. it’s mathematics!  he numerically laced lyrics above, excerpted from hip-hop vanguardian mos def‘s song mathematics (smith & martin, 1999), capture an image of an urban american landscape replete with socioeconomic, political, health, and educational injustices for african american youth. in turn, the song constitutes an appeal for the development of a literacy that is commensurate with the expressed need to critically read and reflect on the urban american text, as well as on black life within that context. the artist–activist asserts that if african american youth are to effectively decode, process, and understand the meanings embedded in urban life—as well as to counteract the negative forces therein—they must develop mathematical literacy. the ―new math‖ is not the newest policy take on curriculum or instruction, or even contested national or state content standards for that matter; the new math requires one to critically mathematize everything from the goings-on of the neighborhood block to understanding global capital markets— and everything in between. indeed, for generations of african americans still coming up from slavery, ―reading the word and world‖ has very literally been the key to unlocking persistent racial, cultural, socioeconomic, political, and physical bonds (freire, 1970/2000; freire & macedo, 1987; perry, steele, & hilliard, 2003; wacquant, 2002; woodson, 1933/1969). mathematical literacy as such remains equally important, if not more so in the contemporary urban milieu. in this article, i discuss the role that the critical race theory (crt) notion of counterstory 1 can play in urban mathematics education. to provide context, i highlight problematic notions of equity that plague mathematics education as shaped by national policy documents, thereby arguing the need for a crt lens in research and practice in urban mathematics education. while ―counterstorytelling‖ is a crt methodology that has growing application across educational research, few researchers have explored its relevance for mathematics classrooms. to explore counterstory in mathematics education, i begin with an examination of  editors note: the lyrical excerpt was printed as written and performed to preserve the integrity of the art form. 1 i favor counterstory over the more common usage counter-story, also reflected in the shift away from counter-storytelling to counterstory-telling. the shift connotes an emphasis on the construction and telling of a particular counterstory, rather than a focus on the dominant narrative being countered. t terry mathematical counterstory journal of urban mathematics education vol. 4, no. 1 25 the pedagogical relationship between narrative and counterstory in an attempt to provide an operational definition of counterstory as a crt construct. while this operationalization will undoubtedly be of use to education researchers in general, it allows us, as mathematics educators, to identify counterstory in mathematical teaching and learning contexts in particular. as such, in the second half of the article, i apply this operationalization of counterstory to critical ethnographic data from a recent study involving high school-aged black male urban youth in south los angeles. focusing specifically on the conception and construction of mathematical counterstory (terry, 2010a), i propose this emergent mathematical activity as a potential pedagogical basis for developing productive curricular and instructional interventions in urban mathematics classrooms. national doctrines of equ(al)ity in mathematics education examining the relatively high level of ―illiteracy‖ in school mathematics, moses and cobb (2001) have fairly assessed the socioeconomic position of a number of african americans as serfdom. this positionality is particularly true considering the clear connections between prek–12 mathematics achievement, college access, and socioeconomic status. the persistent disparities in racially disaggregated mathematics achievement data over the past several decades have prompted efforts by the mathematics education community to chart new and equitable directions for the teaching and learning of mathematics on a national level (schoenfeld, 2002; secada, 1989; tate, 1997). the national council of teachers of mathematics (nctm) issued its standards documents (nctm, 1989, 2000) in language that recommended a corporate move toward ―equity‖ that was to be instantiated through ―high-quality instructional programs.‖ in short, nctm has largely been responsible for re-framing how we, as mathematics educators, think about our practice vis-à-vis disparate group achievement (schoenfeld, 2002). the critique is not new; this ―mathematics for all‖ framework, as several scholars have well argued (allexsaht-snider & hart, 2001; dime, 2007; martin, 2003), can be thought of as a liberal but ineffective attempt to redress claims about unequal group outcomes (i.e., the racial ―achievement gap‖) through complete inclusion. while it is clear that the national discourse on equity in mathematics education has shifted, the nctm‘s framing of inequity as disparate achievement ultimately frames equity qua equality. the implicit logic of this approach is that if inequity is equivalent to academic disparity, then equity has been achieved when there is a consistent parity between the achievement scores of the various racial and/or ethnic groups. though progress toward group parity is important, there are other types of inequity that mathematics educators must attend to. mathematics education research and practice predicated upon a ―group‖ unit of analysis largely leaves questions about the well being of individual students along various dimensions terry mathematical counterstory journal of urban mathematics education vol. 4, no. 1 26 unaddressed. further complicating the matter, the ―mathematics for all‖ framing is colorblind (bonilla-silva, 2006; delgado & stefancic, 2001; omi & winant, 1994). the ―for all‖ language prescribes an intervention that treats groups and individuals similarly without adequately addressing the differential realities of the racial and/or cultural experience of african americans over time (martin, 2003, 2009). if, as is the case, the construct of ―race‖ is implicated by the very fact that we, as a community of mathematics educators, can predict mathematical performance by race and/or ethnicity, what are we to make of solutions that ignore race altogether? that is to say, there is some sense in which we would naturally expect an attention to race, racialization, and racism to play a role in how we address a problem that presents itself along racial lines. as such, though the increased focus on equity represented in the standards documents is laudable to some extent, the documents (insofar as they promote policy and practice) are insufficient for driving an equitable approach to the teaching and learning of mathematics for african american students. for this reason, there is a much repurposing in mathematics education research needed in terms of attention paid to (a) race and racial identity as constructs of interest in studying the teaching and learning of mathematics and (b) the experiences of african american students as both racialized persons and cultural ―others‖ in the mathematics classroom. 2 these misleading doctrines of equity and parity hold premiere significance for african american male students in urban classrooms. key reports on the academic success of african american students in california reveal that black males are less likely to experience high school and post-secondary success as evidenced by data from the state high school exit exam, sat scores, high school graduation rates, post-secondary enrollment, as well as college degree earning (rosin & wilson, 2008); holzman (2006) reports similar trends across the united states, examining literacy and graduation rates, advanced placement (ap) coursework, assignment to special education services, and rates of discipline for black males in every state. if we consider that the u.s. census bureau reports that more than half of african americans (52%) live in the central city areas of the nation‘s major metropolitan regions, it becomes clear that our urban school districts play a key role in influencing these inequities (mckinnon, 2003). in fact, the 2009 national assessment of educational progress (naep) reported that students in 10 of the 18 urban districts participating in its trial urban district assessment performed below their counterparts in large u.s. cities (national center for education statistics, 2010). 3 2 see martin (2009) for an articulate and timely framing of these new directions. 3 among the large urban school districts mentioned are atlanta, baltimore, chicago, cleveland, detroit, the district of columbia, fresno, los angeles, milwaukee, and philadelphia; the mathematics achievement scores recorded and compared are those of fourthand eighth-graders in these districts and cities. terry mathematical counterstory journal of urban mathematics education vol. 4, no. 1 27 but even if an immediate and large-scale overhaul of the chronic political economic conditions that characterize urban communities and schools were granted (anyon, 2005; noguera, 2003a; 2003b, 2003c; oakes, 2005), those who measure ―gaps‖ could continue to expect the underperformance of male african americans in mathematics and other disciplines to persist on the whole. on one hand, the legacy of the education debt (ladson-billings, 2006) is not so easily dismissed; on the other, urban schooling as institutional conditioning continues to be a marginalizing for male african americans. as a result, scholars call for the development of research and teaching that reflects the unique pedagogical needs of black males (harper, 2010; howard, 2008; lynn, 2002; noguera, 2003b). in shaping a response to these needs, attention must be paid to the ―voice‖ of male african americans and their narratives about (mathematics) learning. in the next section, i discuss narrative as groundwork for understanding how the crt construct of counterstory contributes to african american education. counterstories as a crt construct: learning from enslaved africans although opposition to the education of enslaved africans was a characteristic position of the white governors of slavery in the american south (anderson, 1988; du bois 1903/1999; eaton, 1936), literacy was often a characteristic trait of individual leaders of various slave rebellions and revolts. kilson (1964) argues that the leaders of what are likely the most famous uprisings of enslaved africans in u.s. history—gabriel prosser (virginia–1800), telemaque ―denmark‖ vesey (south carolina–1821), and nat turner (virginia–1831)—were all literate men. anderson (1988) further points out that the movement among enslaved africans toward literacy was not a phenomenon unique to a few extraordinary individuals in the antebellum south: ―despite the dangers and difficulties, thousands of slaves learned to read and write. by 1860, about 5 percent of the slaves had learned to read‖ (p.16). ―their ideas,‖ anderson suggests, ―about the meaning and purpose of education were shaped partly by the social system of slavery under which they first encountered literacy…they viewed literacy and formal education as means to liberation and freedom‖ (p.17). thus, despite political opposition from the planter class, african americans reduced illiteracy from 95 percent in 1860 to 70 percent in 1880—and further to 30 percent by 1910 (anderson, 1988). african american stories (including research and scholarship) about lives of individual black folks, in the context of their respective struggles for literacy and literal freedom, provide a foundation for understanding the propagation of a de facto philosophy of education. narrative, as it were, is clearly interwoven throughout the philosophical underpinnings of african american education. perry (2003), for example, argues: terry mathematical counterstory journal of urban mathematics education vol. 4, no. 1 28 [african americans] developed a philosophy of education that was passed on in oral and written narratives. moreover, narratives were not only the vehicles for passing on this philosophy, but they also had a discursive function. they were central to identity formation of african americans as intellectually capable people. (p. 12) in other words, calling to mind what a good many of enslaved africans knew (i.e., how to read), and what they knew about their ―knowing‖ (i.e., that literacy was a tool for liberation), provide key insight into the pedagogical function of narrative in the african american community. my argument here is that the pedagogic force of slave narrative (and the subsequent philosophy of education enacted by their descendants) continues on in the crt construct of the counterstory. prior to the birth of crt, the notions of voice and narrative have been key in understanding the developing ontology of african americans. crt scholarship indeed relies heavily on the inherent pedagogies in the storytelling of african americans and others (delgado, 1989). while counterstories are key to pursuing racial justice in legal studies (bell, 1987, 1992; crenshaw, gotanda, peller, & thomas, 1995), they are also found in crt approaches to pursuing racial justice in education (decuir & dixson, 2004; dixson & rousseau, 2005, 2006; ladsonbillings, 1998; ladson-billings & tate, 1995; solórzano & yosso, 2001, 2002). solórzano and yosso (2002) argue that there are three general forms of counterstory-telling—those that are (a) our own stories or narratives told in first-person voice (see douglass, 1845/1997; knight, norton, bentley, & dixon, 2004; reese, 2006), (b) other people‘s stories or narratives told in third-person (see decuirgunby, 2006), and (c) composite stories or narratives constructed through the use of various forms of data (see bell, 1987, 1992; dixson, 2006; dubois, 1911/2004; rousseau & dixson, 2006; solórzano & yosso, 2002). solórzano and yosso (2001) posit that counterstories serve at least four theoretical, methodological, and pedagogical functions: 1. counterstories build community among the marginalized by personalizing educational theory and practice. 2. counterstories provide a context within which to challenge and transform the hegemonic wisdoms of those at society‘s center. 3. counterstories open new realities to marginalized peoples by helping them envision possible lives. 4. counterstories teach marginalized people to actualize those new possibilities through synthesizing elements of stories with current realities, thereby producing richer actual lives. delgado (1989) and solórzano and yosso (2001) provide excellent overviews of the ways in which counterstories are used, including moving examples. rather than rehash what has been previously documented about the methodological uses terry mathematical counterstory journal of urban mathematics education vol. 4, no. 1 29 of counterstory in educational and/or critical legal research and practice, here i present a different discussion that focuses on crt‘s pedagogical dimensions. 4 specifically, i offer an organizing conceptual frame for thinking about the pedagogical intentions and outcomes in counterstory-telling, as well as an operational definition of counterstory-telling that can be used when analyzing narrative. counterstory-telling as a dialectical synthesis counterstory-telling involves the production and reproduction of local (individual) narratives to counter the apparent and accepted wisdom of masteror meta-narratives (also called ―majoritarian‖ stories; see love [2004] and solórzano & yosso [2002]). in figure 1, i illustrate that in counterstory-telling a dialectical relationship exists between dominant narratives (expressions of white supremacist ideology) and african american narratives (reflective of a presuppositional belief in liberation as education, a tacit african american philosophy of education). that said, however, not all narration is intended to counter or challenge dominant narrative. figure 1. counterstory-telling as a synthesis of competing narratives. 4 rather than making any precise epistemological claims about the difference between method and pedagogy, i am arguing here that the aspects of teaching and learning that reside in the act of counterstory-telling are what make them methodologically useful to crt scholars—hence the helpfulness of shifting focus. terry mathematical counterstory journal of urban mathematics education vol. 4, no. 1 30 solórzano and yosso (2001) contend, ―a story becomes a counter-story when it begins to incorporate the five elements of critical race theory‖ (p. 39). conceptually, this distinguishes counter-story as a unique class of stories that actively engages the central tenets of crt in the act of storytelling. this engagement, however, does not fully capture the full force of these stories. to better appreciate the pedagogical moment bound in the act of counterstory-telling, as well as to provide researchers with a more applicable construct, here i provide an operationalization that can be used to further distinguish counterstory-telling from straightforward narration. as such, i claim that a narrative constitutes a counterstory when it satisfies three key criteria: (a) contains a kernel or representation of the dominant narrative such that it communicates a clear understanding of that dominant narrative and its implications to the communicant; (b) provides the communicant (in the form of a competing narrative that is grounded in a ―freedom reality‖) reasonable and sufficient grounds for contradicting the dominant narrative; and (c) allows the communicant to access the larger freedom reality toward which the competing narrative pushes. in this dialectical relationship, narratives that assert the ex-slave and his or her descendants as self-motivated persons who are organized around drives for literacy and education, for example, are antithetical to dominant narratives that position african americans on the margins of schooling as unintelligent and incapable underachievers who generally hold negative dispositions toward education and are relatively unmotivated to succeed. as illustrated in figure 1, the narrative and dominant narrative are in tension with one another because they are expressions of ideologies that are themselves in contention with one another (african american philosophy of education and white supremacist ideology). those tensions find synthesis in the pedagogical moment when the communicant, through hearing the counternarrative, understands the freedom reality. by freedom reality, i mean an acknowledgment of and striving toward freedom and self-determination as the natural state of the african. this acknowledgment stands in sharp contrast to the unnatural state of being african in america—a being ordered by and subjugated through racism (i.e., racial prejudice combined with institutional power). let us consider an example of how african american narrative functions as counterstory. frederick douglass published his narrative of the life of frederick douglass: an american slave in 1845. this text provides what is one of the classic depictions of awakening the slave experiences concerning bondage and liberation. douglass (1845/1997) recounts: [mrs. auld] assisted me in learning to spell words of three or four letters. just at this point of my progress, mr. auld found out what was going on, and at once forbade mrs. auld to instruct me further, telling her, among other things, that it was unlawful, as well as unsafe to teach a slave to read. to use his own words, further, he said, ―if you give a nigger an inch, he will take an ell. a nigger should know nothing but terry mathematical counterstory journal of urban mathematics education vol. 4, no. 1 31 to obey his master—to do as he is told to do. learning would spoil the best nigger in the world…if you teach that nigger (speaking of myself) how to read, there would be no keeping him at all‖…these words sank deep into my heart, stirred up sentiments within that lay slumbering, and called into existence an entirely new train of thought. it was a new and special revelation, explaining dark and mysterious things, with which my youthful understanding had struggled, but struggled in vain. i now understood what had been to me a most perplexing difficulty—to wit, the white man‘s power to enslave the black man…from that moment, i understood the pathway from slavery to freedom. (p.33) in this passage, we observe douglass‘s evolving understanding of the role illiteracy played in perpetuating the bondage of africans in america and the consciousness that emerged from witnessing the slave master‘s reaction to the prospect of his literacy. this experience is of particular interest because not only does douglass, herein, conceive of education as a means to a liberatory end but also his narrative quite clearly demonstrates the satisfaction of the key criteria by which we may establish its constitution as counterstory. the story identifies the dominant narrative. first, embedded in douglass‘s narration is an expression of the dominant narrative. it is communicated in the sharp rebuke given to mrs. auld by her husband. mr. auld expresses several key beliefs about douglass‘s illiteracy, which reflects a white supremacist ideology. specifically, mr. auld argues: a nigger should know nothing but to obey his master—to do as he is told to do. learning would spoil the best nigger in the world. this statement is a representation of dominant narrative as it asserts several key understandings—primarily, it asserts (a) frederick douglass‘s ontological status as nigger and slave; it also asserts (b) that knowing how to and that to obey one‘s master is necessary and sufficient knowledge for existence as nigger; and finally, (c) that douglass‘s existence as anything other than slave would be a spoiled or ruined version of his ultimate purpose (i.e., an undesirable corruption of his principal form). implicit in mr. auld‘s role as communicator of this dominant narrative is his positioning as possessor and distributor of knowledge—the one who ultimately knows. the story contradicts the dominant narrative. second, and fortunately for douglass, inherent in mr. auld‘s rebuke is the pretext for the competing narrative forming the basis for douglass‘s counterstory. auld claims: learning would spoil the best nigger in the world…if you teach that nigger (speaking of myself) how to read, there would be no keeping him at all. in this portion of auld‘s statement, we read several competing assertions: (a) the thoughts douglass has experienced as a result of his interactions with mrs. auld have to do with learning—which is fundamentally different in process and substance than possessing the knowledge required to be a nigger; (b) learning fundamentally alters one‘s perceived existence as nigger from the perspective of a slave master; and (c) that douglass‘s enslavement is in some important sense contingent upon his illiteracy. terry mathematical counterstory journal of urban mathematics education vol. 4, no. 1 32 the story frames access to freedom. third, and the final criterion, there is in this narrative a portal by which the communicant can access the larger freedom reality toward which the competing narrative points. douglass, in reflecting on the situation, relates the following: i now understood…the white man’s power to enslave the black man… from that moment, i understood the pathway from slavery to freedom. here, we are able to envision the ―promised land,‖ as it were, through the eyes of the subjugated african. we gain access to the broader reality where the african is free to self-determine, free to become—simply free. as douglass knows the pathway from slavery to freedom, we too (through his narrative) understand and know it. as such, the narrative presented by douglass constitutes a counterstory that he tells through oral presentation and through this written text. douglass engages in counterstory-telling as a critical pedagogical act—the relating of a narrative to an audience as a means to destroy internalized and actualized white supremacist ideologies. therein lies the dialectical synthesis: in relating his narrative, douglass assumes the role of critical pedagogue and liberator, moving ahead with the business of abolishing the racist institution of slavery. though the construct has been formulated and refined over the last 2 decades, counterstories have deep pedagogical roots. while it may seem anachronistic to characterize slave narrative as counterstory, a close reading of such narrative provides important insight into the purpose and intent of these kinds of stories. in the first half of this article, i have attempted to demonstrate the structural and pedagogical connections that help us understand why and how some narratives constitute counterstory. in the second half of the article, i intend to show what relevance this particular take on counterstory has for developing opportunities for engaging in critical cultural activity in urban mathematics classrooms. urban mathematics education and mathematical counterstory in the summer of 2008, i began work with seven high school-aged black male youth from a number of high schools across south los angeles on a participatory action research (par) project (mcintyre, 2000) exploring the incarceration and university attendance rates of 18–34 year old male african americans. the young men and i formed a research team that examined the narratives about incarceration and schooling embedded in quantitative data from california state prisons and state universities, and in qualitative data collected from black male interviewees across south los angeles. their research project became the subject of critical ethnographic analysis (terry, 2009) and of discussion of the role that participatory action research can play in the development of critical mathematics literacy (terry, 2010a). here, i explore data from this study to shed light on the ways in which counterstories can play out in mathematical contexts for urban stu terry mathematical counterstory journal of urban mathematics education vol. 4, no. 1 33 dents. in so doing, i hope to highlight the pedagogical force of these mathematical counterstories and the possibility they offer for shaping the mathematics experiences of male african americans in urban classrooms. envisioning possibility for mathematical counterstories the research team had a variety of compelling interests that drove us, as coresearchers, to ―look for mathematics‖ in interesting and engaging places. prior to our research project, we undertook a number of data analysis exercises as preparation; we ―mathematized‖ situations that struck us as having significant implications for the quality of life of african american men. for example, the team explored data regarding firearm-related homicides from across the united states. these data allowed us to ponder the impact that geographic location has on determining whether black men are subject to increased risk of gun murder. when we examined these data against the backdrop of the relative distribution of african american population (in major urban areas), the notion of black men as ―endangered‖ began to make some sense. a significant moment in our preparation came as we shifted our attention to discussing the murder rate at home in los angeles. while these young men understood firearm-related homicide as a regular occurrence in south los angeles communities (read: black-on-black crime), they also firmly believed that police officers frequently used firearms to ―justifiably‖ kill young black males. in fact, through televised community protest, the young men had become aware of the firearm-related deaths of three male african americans at the hands of the inglewood police department at the time of the project (bloomekatz, 2009). as such, the young men had become very interested in discussing how african americans are policed—particularly in south los angeles. the daily breeze printed an article titled ―big drop in l.a. killings‖ (watkins, 2008). in that article, the journalist reported the ―success‖ that the los angeles police department (lapd) was experiencing in lowering crime rates and homicide through increased police presence, a presence that was subsequently being funded by increased trash collection fees. the text quickly consumed our interest. in addition to the text of the article, the journalist provided a url to an online statistical crime report produced by the lapd that offered various graphical organizers in support of the narrative on lower crime rates (watkins, 2008). the first page showed that los angeles had fewer homicides in july of 2008 than in any other july ranging back to 1969 (presented in histograms) and that the 18 homicides in july of 2008 were the lowest of any month in any year since march 1970 (see figure 2). 5 our team confirmed the statistical reporting by examining 5 figures 2 and 3 reprinted with permission of the los angeles police department. tables originally appeared in the report state of the city: crime snapshot, retrieved from http://www.lapdonline.org/crime_snapshot. http://www.lapdonline.org/crime_snapshot terry mathematical counterstory journal of urban mathematics education vol. 4, no. 1 34 the bar graphs presented. the second page of the lapd statistical report had a number of graphs and tables focusing on various aspects of the homicide rate (gang homicides, shooting victims, homicide clearance rates, and unemployment). in the bottom left corner of the page, we saw a table titled ―historical data.‖ the table showed the numbers of police officers and city population for 1970 and 2008 (see figure 3). figure 2. july homicides in los angeles, 1969–2008. figure 3. historical data on police force and population in los angeles. terry mathematical counterstory journal of urban mathematics education vol. 4, no. 1 35 i saw here a chance to problematize the lapd‘s narrative using these data. i invited students to take these numbers into further consideration by calculating the person-to-cop ratios for these 2 years. in my mind, if students could show that the respective person-to-cop ratios were disproportional (i.e., that the 2008 person-to-cop ratio was less than the 1970 ratio), then we could use this as evidence to counter the story being constructed by the lapd brass. maybe the ratio of police officers to angelenos was not, in fact, responsible for lower crime rates—and urban angelenos were experiencing declines in crime for a heretofore undiscovered reason. perhaps the dropping rates were attributable to increased community organization and activism, to growth in local summer programs, to increased streetand community-level gang intervention efforts, or some other reasonable combination of factors. and, if that were the case, perhaps we were witnessing the construction of a problematic (dominant) narrative intended to support the increased militarization of south los angeles. the student researchers and i were surprised, however, by the results of our calculations (see figure 4): figure 4. in-class calculations of person-to-cop ratios. la mont terry (lt): it‘s the same [ratio]? kell: yeah, but…yeah, definitely. paris: they ‗turkey‘-ed it! they jiggled it all over. kell: no, [i swear] on everything! watch! redd got the same answer. redd: sure do! i was not prepared to see this result. neither were the students. paris, in fact, believed that his friends had finagled the numbers a bit—intentionally stretching their calculations to make the ratios equivalent. however, it was clear that kell terry mathematical counterstory journal of urban mathematics education vol. 4, no. 1 36 and redd had honestly arrived at the same ratio. after confirming the ratio with others, and to make sense of the result, we continued our conversation: lt: those [ratios] are really, really, really close—almost exactly the same. so, remember what the lapd chief said: what‘s the difference? why is there less crime? (quoting chief bratton in the article) c-o-p-s. cops! ok, you tell me. you look at those numbers. are there more cops? geronimo: yes. lt: yes, there are more cops, aren‘t there? redd: yeah. lt: but, for every one cop, does that cop have to police more or less people? deforest: more. geronimo: more. lt: is it really a difference though? redd: no. that‘s what i saw. geronimo: so what [the chief] said is false! lt: (to redd) so tell me what you saw. redd: because you see how they have them officers, it was like 433 [people] for every cop. and now that they have [more cops], it‘s still almost exactly the same [ratio]. lt: it‘s the same! redd: so they doing all that charging and extra taxes on trash pick-up for basically nothing. lt: (explaining) so they‘re saying, ―look we got more cops!‖ yeah, we got more cops, but guess what: there are more people too! just cuz you have ―more cops‖ doesn‘t mean they‘re gonna make more difference if you have more people to police, right? ...so what does that mean? geronimo: that mean, probably we changed…uh… lt: well, here‘s a question to ask—if there are the same number of people to cops, the same ratio of people-to-cops in these 2 years: is it really the number of cops that made a difference? what made the difference? the people-to-cop ratios turn out to be fairly equal (433:1 and 434:1) and, therefore, did not quite play the role in contradicting the lapd narrative that i suspected they might. however, in discovering the approximate equivalency of these ratios, the young men picked up on a hint of disguise in the lapd‘s narrative. in the passage above, geronimo claims the police chief has lied. redd concludes that the trash fee hikes are ―for basically nothing.‖ the students had been told a story about an under-policed city and a dwindling crime rate. it became clear to them, however, that police chief bratton, in cooperation with mayor villaraigosa, had authorized the use of these statistics as a pretext for rationalizing the lapd policy of ―over-policing‖ local low-income communities of color: east l.a., south central, and south los angeles more broadly. the implicit logic of these officials: if we can achieve parity with historic lows in homicide rates by achieving this (433:1) person-to-cop ratio, then hiring even more police officers should bring us to new historic lows in homicides. in so doing, the lapd narra terry mathematical counterstory journal of urban mathematics education vol. 4, no. 1 37 tive established the hike in trash fees as a crucial funding source for hiring these extra police while simultaneously providing the citizenry with a narrative that conveniently frames its compelling interest in accepting the higher municipal fee. it is clear that the police department did not need to create very sophisticated graphical representations in order to convince the public that their policies are effective and well-founded. as a team of critical mathematics researchers, however, we hoped to create a counternarrative that could contradict the stories offered by l.a. city officials. though we were not quite able to find the contradiction we had hoped for, it was clear that we had cast some level of doubt on whether the story city officials were telling was as ―clean-cut‖ and truthful as they would like the voting public to believe. this moment in our research represented an important point in our orientation to data; the high school-aged black males understood that mathematics is often used to tell stories, and some (often?) times in subjective, possibly disingenuous, ways. if that was the case, they now knew that they too could use mathematics to confirm and/or disconfirm the stories that people tell. therein was our initial experience of mathematical counterstory. mathematical counterstory in university and prison data as we, the research team, transitioned into the formal research project exploring the causes of the disproportionate incarceration of black men in state prisons and their underrepresentation in state universities, we collected and analyzed data accordingly (terry, 2010a). the california state prison and university data for 18–34 year old black men were available in disaggregated subgroups for the years 2000 through 2007. we were operating under the common assumption that more male african americans were in prison than in the california state university system. this assumption was consistently resonant in the interviews that we conducted with african american men across south los angeles (i.e., every one of the black men interviewed readily accepted this proposition and had wellarticulated explanations for why it was the case). and the assumption seemed mostly correct—with one small exception. when comparing the state prison and state university data, we found that this assumption held true in every case except one—the 18–24 year old subgroup. unlike the lapd narrative that revolved around the manipulation of personto-cop ratios, these data struck the team as standing in clear contradiction to the popular understanding regarding the incarceration of young black men. with the exception of the year 2000, 18–24 year old students outnumbered their prison counterparts in california state universities. there were mixed emotions about this finding. on one hand, the young men were disappointed that they had accepted an unchecked assumption about the homogeneity of these subgroups. clearly, the data were nuanced. on the other hand, within the youngest subgroup, an interesting (and contrasting) trend could be observed. as such, perhaps this ob terry mathematical counterstory journal of urban mathematics education vol. 4, no. 1 38 servation could serve as the basis of a new narrative about the improving academic achievement of african american men. in fact, according to the criteria for operationalizing counterstory-telling presented earlier in this article, these quantitative data did position the research team to construct a mathematical counterstory. let us examine each criterion individually. the story identifies the dominant narrative. to show the possibility for mathematical counterstory, we must clearly identify the relevant dominant narrative. in table 1, the disaggregated numbers of male african americans in california state universities and prisons are organized according to the 18–24, 25–29 and 30–34 year age groups. examining the net difference between the total numbers of black males in both systems reveals a severe disparity between those incarcerated and those attending university. setting aside important caveats about how the par team operationalized and measured these variables (terry, 2010a), it is clear that these data harbor what might commonly be understood to be the common sense understanding (read: dominant narrative) concerning black males and incarceration. table 1 net difference between black males in california state universities and prisons year california state universities california state prisons net dif. a b c total a b c total 2007 7845 1303 573 9721 5523 6642 6130 18295 -8574 2006 7262 1303 563 9128 5578 6814 6251 18643 -9515 2005 6791 1321 565 8677 5469 6595 6237 18301 -9624 2004 6365 1219 582 8166 5539 6461 6461 18461 -10295 2003 6056 1177 575 7808 5522 6488 6814 18824 -11016 2002 6158 1254 631 8043 5592 6515 7285 19392 -11349 2001 5967 1192 682 7841 5659 6493 7553 19705 -11864 2000 5606 1229 650 7485 5936 6843 8126 20905 -13420 note: a = 18–24 year-old age group; b = 25–29 year-old age group; c = 30–34 year-old age group. on average, for every male african american between the ages of 18–34 attending a california state university between 2000 and 2007, more than two are incarcerated in a california state prison. these data intimate the narrative that many, including the african american men interviewed by the research team, are familiar with. data like these are commonly used to forward deficit claims about the status of male african americans in u.s. society (reese, 2006). whether one interprets these data as evidence of the depravity and criminal degradation of the black male or, in turn, his criminalization within an over-policed society, there lies an implicit statement about prisons—they ―lock-up‖ black men. further, when compared to the relative numbers of those enrolled in corresponding univer terry mathematical counterstory journal of urban mathematics education vol. 4, no. 1 39 sities, these data seem to shepherd us toward an intuitive understanding that more black men are failing in their roles as citizens than not. herein lies the representation of the dominant narrative in the data. this is the aspect of the data upon which the counterstory turns. the story contradicts the dominant narrative. a second aspect of establishing counterstory is locating a critical pedagogical force with which to contradict the dominant narrative. here, we search for evidence of a competing narrative, and find it as we look specifically at the case of the 18–24 year old african american men (see figure 5). figure 5. more 18–24 year-old black men in state universities than prisons. in this bar graph, the par team discovered that, with the exception of the 2000 data, there were more 18–24 year old black men enrolled in california state universities than incarcerated in california state prisons. contrary to the trend observed in the broader sample that show twice as many black males incarcerated as attending university, here students outnumber prisoners. not only are there more students than prisoners in this subgroup from 2001 to 2007 but also there is a widening gap in these numbers due to both increases in the numbers of 18–24 year old black men enrolled in california state universities and an overall decrease in the respective number of prisoners. as such, while the dominant narrative might suggest that more african american males are failing than succeeding in their roles as citizens, a competing narrative might suggest that young black men are terry mathematical counterstory journal of urban mathematics education vol. 4, no. 1 40 actually reversing this trend. 6 rather than being forced to take a deficit perspective toward african american men in society, we may very well assert the agency of young black men in bucking these institutional trends. the story frames access to freedom. a final step to establishing a counterstory lies in being able to show how the data can lead one to a larger freedom reality; that is, a new understanding of one‘s position in society. university and prison, as institutions, have come to represent the very best and very worst of the african american socioeconomic experience insofar as they can be understood to be embodiments of success and failure. the data can be used to create space for a pedagogic moment of self-determination and self-realization for young black men. rather than allowing the overrepresentation of african american men in the prison system to be the defining mental image, the competing narrative shows 18–24 year old black men attending california state universities at greater rates than their counterparts are being incarcerated, thereby allowing african american men to envision and understand a different pathway and reality. the competing narrative suggests to young black men that they can be part of a growing trend of young african american men who are succeeding in california state universities—while also staying out of prisons. counterstory-telling as dialectical synthesis. while the key elements of counterstory are present in the data, the pedagogical force of the counterstory is limited without an actual ―telling‖ of the story. part of the difficulty in translating our inherent understandings of counterstory to mathematical contexts lies in knowing that mathematical data do not inherently constitute stories. the stories, in fact, are actualized through our interpretations of data and in the subsequent communication of those interpretations. in the present study, the high school-aged male black youth on the par team ―told‖ the mathematical counterstory within the context of the short film the team created. 7 the mathematical counterstory represented in this film is a blending of all three types of counterstories discussed in solórzano and yosso (2002); it incorporates the autobiographical experiences of the students, as well as qualitative and quantitative data collected during the par project. among other things, mathematical counterstory-telling allowed these students to act upon their world through the sharing of their data, analysis, and conclusions in a pedagogically-powerful way. though not explored here, the telling of the mathematics counterstory had a powerful impact on the mathematics, racial, social activist identities of the students (terry, 2009). this telling, 6 it is also important to note that, while black males ages 25 and older are enrolled in undergraduate and graduate programs, 18–24 is a more likely college-going age—thereby accounting for disproportionate population of prisoners in 25–34 year olds. further, this would also likely be the case for other racial and/or ethnic subgroups. 7 film available at the following url: http://scholar.oxy.edu/edu_faculty/1/. http://scholar.oxy.edu/edu_faculty/1/ terry mathematical counterstory journal of urban mathematics education vol. 4, no. 1 41 however, insofar as it is embodied within the text of the film, becomes an important opportunity for the critical examination and re-reading of the various narratives involved here by the students themselves, as well as the film‘s past, present, and future audiences (both local and national) (franke, spencer, & terry, 2009; terry, 2009; 2010b). discussion in this article, i have asked the reader to consider the value of importing the crt construct of counterstory into the mathematics education of urban african american students. young black men, in particular, rank among the highest casualties of our urban schooling efforts (garibaldi, 1992; howard, 2008; jackson & moore, iii, 2006; noguera, 2003b); this is no less true in their experience of mathematics classrooms (davis & martin, 2008; stinson, 2004, 2006; tate, 1997). in the seemingly unending hustle to invent interventions on behalf of these students (i.e., to close the achievement gap), teachers and mathematics educators rarely pause to ask questions about the wholesale impact of their efforts on nonacademic outcomes for african american students. as a result, i believe we, as a community of mathematics educators, miss key opportunities to re-orient disaffected and marginalized students to the utility of mathematics. mathematics counterstory provides mathematics educators with a unique opportunity to redress this missed opportunity. what lies at the heart of movements like ethnomathematics (anderson, 1990; brown, 2008; eglash, 1997; powell & frankenstein, 1997; powell & temple, 2001) and critical mathematics literacy (frankenstein, 1990; gutstein, 2006; gutstein & peterson, 2005) is a fundamental acknowledgment that mathematics is principally a cultural activity rooted in the lives and experiences of particular people groups. whether discussing the historic tasks of constructing egyptian and pre-columbian pyramidal structures, or contemporary efforts to model the biological growth of hiv and infection rates in the african american community, the mathematical content and process (nctm, 2000) emerge organically from our needs as people and communities—not just as synthetic requirements of our school work. the pedagogical significance of this acknowledgement is that it allows teachers and students of mathematics to reshape their perspectives about school mathematics in such a way that repositions mathematics as a tool to engage and act upon the world around them. my experiences as a mathematics educator lead me to believe that it is critical for young african american men in urban classrooms to re-experience mathematics in this way—that is, mathematics as a basic tool for acting upon the world—rather than as a collection of facts and skills that are taught strictly as test content for testing‘s sake (davis & martin, 2008; walker, 2009). terry mathematical counterstory journal of urban mathematics education vol. 4, no. 1 42 the ability to understand mathematics content and use mathematics processes in meaningful ways provides youth in urban classrooms with the inherent motivation to refine their existing set of mathematical tools, as well as to acquire more sophisticated ones through advanced curricula. the potential benefit of this approach is manifold: students see mathematics courses as means to developing enduring understanding and relevant skill—not simply to learning ―harder‖ mathematics; students develop persistence in their course-taking patterns as they subsequently reinterpret the purpose of mathematics class; and students are better positioned to make sense of the connections between coursework and content visà-vis the opportunity to apply mathematical knowledge in both traditional curricular and critical contexts. in the present study, the high school-aged black males encountered counterstory as a tool for exploring and making sense of contextualized data. because data analysis, graphical representation, and mathematical modeling in general are literacies emphasized in the nctm content and process standards, as well as throughout state k–12 mathematics standards, teachers can find multiple opportunities to establish points of contact between students‘ critical interests and curricular foci; this holds as true in the elementary grades as it does in advanced mathematics courses (gutstein, 2006; tan & min, 2003). while it is dangerous to assume that all male african americans experience the world in the exact same way vis-à-vis their multiple identities, mathematics educators should recognize that meaning and relevance provide important internal motivations to engage with mathematical content and process; this is particularly important in urban classrooms, where many of the intrinsic motivations that are assumed to be a part of schooling (such as maintaining grade-point average; earning one‘s diploma; and/or participation in school-based, co-curricular activity) fail to hold significance. teachers in general, however, are becoming increasingly aware of students‘ desires to critically interrogate their environments (duncan-andrade, 2005, 2007; morrell, 2004, 2008; rogers, morrell, & enyedy, 2007; tan, 2003). with that said, it is incumbent upon mathematics educators to create the pedagogical and curricular spaces for their students to engage in mathematics processes through critical lenses like that which the mathematics counterstory represents. as mathematics educators adopt increasingly critical pedagogical perspectives on mathematics content and processes, they open the door for lowperforming male african americans (as well as other marginalized groups) to self-identify as doers of mathematics. while this in no way suggests that black males fail to identify as doers of mathematics as a general rule (clark, johnson, & chazan, 2009; martin, 2000; stinson, 2009), critical takes on mathematics instruction do allow for a broader participation structure in the mathematics classroom (civil & planas, 2004; franke, kazemi, & battey, 2007), thereby creating vital support for mathematics identity development from sociocultural perspec terry mathematical counterstory journal of urban mathematics education vol. 4, no. 1 43 tives (nasir & hand, 2006). this broader participation is particularly true as male african americans develop more transformative forms of resistance through the content and process involved with constructing and telling mathematical counterstories (solórzano & delgado bernal, 2001). conclusion here, i have offered a pedagogically grounded operationalization of the crt construct of counterstory-telling in an attempt to argue its usefulness to urban mathematics education. enhancing the current conceptions of counterstories as useful manifestations of student voice from within the margins of mathematics schooling experiences (berry, thunder, & mcclain, 2011; stinson, 2009), i argue here for the potential for contentand process-based mathematical counterstories to provide an increased degree of relief in illustrating what role crt can play in urban mathematics education. the ethnographic data presented here suggest that there are powerful ways whereby mathematics educators can re-frame traditional math literacies in critical contexts in order to create broader opportunities for participation and inclusion of african american students in urban mathematics classrooms. this pedagogical operationalization of counterstory should be helpful to educators reflecting on how to develop more nuanced understandings of contentarea instruction. moreover, i also believe that educational researchers in general may find this operationalization useful in their work on counterstory insofar as it attempts to identify basic pedagogical elements of counterstory as viewed through an african american philosophy of education. furthermore, the ideas presented are implicit encouragement for researchers and practitioners in mathematics education to continue to thoughtfully consider the equity/equality distinction as it plays out in their work. in short, equality is an equity-driven carriage. in order to meet the deep and evolving pedagogical needs of low-performing black males in urban schools, we must be free to think outside of the box and beyond the equality rhetoric that hinders our policy, practice, and scholarship. the ever-present gaps in black male achievement are grounded in broader socioeconomic, political, and racial contexts that are firm (bell, 1992; ladson-billings, 2006). while this does not mean the abandonment of our commitments to improving test scores, we must not allow equalizing scores, as a goal, to interfere with the development of realistic positions that honor the differential realities of black males in our approaches. in this work, i allow the emic perspectives of the black male participants (as researchers and doers of mathematics) to re-write why and how mathematics is important. this attention to the individual is an important aspect of the scaffolding that will allow us to build toward improved academic achievement for black males in urban classrooms. terry mathematical counterstory journal of urban mathematics education vol. 4, no. 1 44 tate (2008) argues that a major focus in urban mathematics education scholarship must be to build and evaluate (and i would add, contest) theory. while i think there is a great deal of theoretical and practical value in the particular operationalization of counterstory presented in this article, it is clear that this particular pedagogical approach to the teaching and learning of mathematics would benefit from exploration in future studies. its present application is limited to a very small and particular group of low-performing african american male students in urban los angeles; a greater understanding of mathematical counterstory and its instructional possibilities would undoubtedly emerge from further application in a range of diverse settings—perhaps most important of all, schools. this study was conducted during traditional out-of-school months in a non-school space. as teachers and mathematics educators explore the notion of mathematical counterstory-telling in classrooms and other non-traditional academic settings like charter schools, themed-academies, and single-sex classrooms, the implications for curriculum, instruction, and assessment will be both contextualized with respect to the various structures of schooling and limitless in scope. there is much to be learned about the impact of this pedagogical approach on a number of key factors such as student identities, mathematics course-taking patterns, student understanding of the discipline as a whole, and, ultimately, academic performance. clearly, there is no essential black experience. not only are all black males not ―urban‖ but also not all black males experience urban space in the same ways. further, there really are no silver bullets in terms of intervening in the mathematics achievement of male african americans. with that said, if anything in this conversation is either universal or essential, it is the need of students in urban settings to be able to read and reflect upon their worlds through the lens of mathematics. if mathematics educators are willing and prepared to step away from traditional approaches to curriculum and instruction, we have reason to believe that those students who are regularly marginalized within our traditional approaches to teaching and learning mathematics might respond—both favorably and critically. acknowledgments this article is based upon work supported by the national science foundation under nsf grant no. 0119732. any opinions, findings, and conclusions or recommendations expressed in this article are those of the author and do not necessarily reflect the views of the national science foundation. this article is based on data from the author‘s doctoral dissertation. the author wishes to express his gratitude to occidental college graduate student lindsey fuller and undergraduate students asha canady and kamil lewis for their inspiring participation in conversations about the ideas in earlier versions of this article—as well as to the anonymous reviewers at jume for their helpful comments. terry mathematical counterstory journal of urban mathematics education vol. 4, no. 1 45 references allexsaht-snider, m., & hart, l. 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(original work published 1933) http://www.tcla.gseis.ucla.edu/reportcard/features/5-6/curriculum.html http://www.tcla.gseis.ucla.edu/reportcard/features/5-6/toyguns.html http://ed-osprey.gsu.edu/ojs/index.php/jume/article/view/19/2 journal of urban mathematics education  december 2008, vol. 1, no. 1, pp. 10–34  ©jume. http://education.gsu.edu/jume  julius davis is a doctoral candidate in mathematics education in the school of education  and urban studies at morgan state university, 1700 east coldspring lane, jenkins building 421,  baltimore, md 21217. his research focuses on understanding how issues of race and racism shape  the lived realities, schooling, and mathematics education of african american students.  danny  bernard  martin  is  chair  of  curriculum  and  instruction  and  an  associate  professor  of mathematics education and mathematics at  the university  of illinois at chicago,  college of education (mc 147), 1040 w. harrison, chicago, il 60607; e­mail:dbmartin@uic.edu.  his research has focused on understanding the salience of race and identity in african americans’  struggle  for  mathematics  literacy.  dr.  martin  is  author  of  the  book  mathematics  success  and  failure among african­american youth (lawrence erlbaum associates, 2000) and editor of the  forthcoming book mathematics teaching, learning, and liberation in the lives of black children  (routledge, 2009).  racism, assessment, and  instructional practices:  implications for mathematics teachers of  african american students  julius davis  morgan state university  danny bernard martin  university of illinois at chicago  couched within a larger critique of assessment practices and how they are used  to  stigmatize  african  american  children,  the  authors  examine  teachers’  instructional practices in response to demands of  increasing test scores. many  mathematics  teachers might be unaware of how  these  test­driven  instructional  practices can simultaneously reflect well­intentioned motivations and contribute  to the oppression of their african american students. the authors further argue  that  the focus of assessing african american children via comparison to white  children  reveals  underlying  institutionally­based  racist  assumptions  about  the  competencies of african american students. strategies are suggested for helping  teachers resist test­driven instructional practices while promoting excellence and  empowerment for african american students in mathematics.  keywords: african american students, assessment, instructional practice, racial  hierarchy, racism  although the phrase “teaching to the test” has been spoken in hallways and  teachers’  lounges  throughout  the  nation’s  public  schools  for  decades,  with  the  passage  of  the  no  child  left  behind  act  of  2001  (nclb), 1  the  phrase  has  become somewhat of a formalized instructional practice. the first author taught  and conducted research at a middle school  in the baltimore city public school  1 no child left behind act of 2001, public law 107­110, 20 u.s.c., §390 et seq. http://education.gsu.edu/jume mailto:dbmartin@uic.edu davis & martin  racism and standardized testing  journal of urban mathematics education vol.1, no.1  11  system that utilized teaching to the test as the dominant  instructional approach  with its african american students. our conceptualization of teaching to the test  is characterized by classroom practices that emphasize remediation, skills­based  instruction over critical  and conceptual­oriented thinking, decreased use of rich  curriculum  materials,  narrowed  teacher  flexibility  in  instructional  design  and  decision making, and the threat of sanctions for not meeting externally­generated  performance standards. reflecting  low­level  expectations  for african  american  children, these teaching­to­the­test approaches often require teachers to make use  of  remedial  mathematics  plans  and  strategies  that  focus  on  lower­level  mathematical content. while mastery of this lower­level content is necessary, it  often becomes the ceiling of the mathematics that students learn because it allows  students to meet minimum standards for what counts as success.  in baltimore, district and school administrators and teachers supported this  approach by developing and implementing a supplemental saturday mathematics  program  dedicated  to  preparing  african  american  students  for  the  state  administered  standardized  test.  the  administrative  staff  at  the  school  also  developed  and  implemented  an  additional  plan  devoted  to  increasing  african  american  students’  performance  on  the  test.  the  district  and  school  administrators selected students to participate in these remedial programs based  on  their  having  standardized  test  scores  that  were  at  basic  and  near­proficient  levels. students were required to participate in both the in­school and saturday  school mathematics program. nearly 25% of the student body at the researched  middle school was required to participate in these special programs. during the  school  day,  students  were  taken  out  of  their  elective  courses  twice  a  week  to  participate in the in­school mathematics program.  in the regular mathematics courses at the school, administrators instituted an  additional remedial mathematics plan that required teachers to spend the first 30  minutes of their 90­minute class period reviewing mathematical concepts taught  to students  in previous  mathematics courses. the remainder of  their class  time  was spent focusing on the state administered test in mathematics. students were  taught from textbooks that focused on this test. they were also inundated with  worksheets, board work, test­taking strategies, and other materials devoted to the  state administered standardized test in mathematics.  because our conceptualization of teaching to the test is based largely on the  first author’s observation in a single middle school, it is clearly not exhaustive of  the instructional practices found throughout baltimore. we believe, however, that  these practices are not isolated to the first author’s experiences. the practices that  were  observed  bear  a  striking  resemblance  to  those  documented  in  the  larger  literature  (see,  e.g.,  kozol,  1992;  lipman,  2004;  noguera,  2003;  oakes,  1990;  oakes, joseph, & muir, 2004) on school inequality and propelled us to use this  example  to  begin  a  conversation  among  mathematics  educators  about  such davis & martin  racism and standardized testing  journal of urban mathematics education vol.1, no.1  12  practices and approaches. the literature reveals that teachers with large numbers  of african american students reported more often that test scores were used to  evaluate  students’  progress,  select  textbooks,  provide  students  with  special  services,  and  make  curriculum  and  instructional  decisions  (madaus,  west,  harmon,  lomax,  &  viator,  1992;  strickland  &  ascher,  1992).  in  these  classrooms, teachers indicated that emphasis was placed on test content, teaching  test­taking skills, teaching topics known to be on the assessment, and preparing  students  for  the test more than a  month  before the test  (madaus, et al., 1992).  these practices force teachers to rush instruction and provide students with little  to no opportunity to learn more advanced­level mathematical concepts (madaus,  et  al.,  1992).  teachers  of  african  american  students  who  focused  mainly  on  preparing these students for tests in mathematics spend a significant amount of  time on rudimentary levels of mathematics (madaus, et al., 1992).  in her book, high stakes education, lipman (2004) provided evidence of  this approach to educating african american and latino/a children in the chicago  public school system (cps). she documented widespread remediation and test­  focused  instruction  in  the  schools  where  she  conducted  her  research.  lipman  stated:  cps leaders contend that the harshness of accountability is offset by new remedial  “supports”…including  after­school  remedial  classes,  mandatory  summer  “bridge”  classes  for  failing  students,  and  transition high  schools.  however,  these  remedial  programs  are  explicitly  aimed  at  the  [statewide  assessment  test]….the  impoverishment and redundancy of this basic skills education for students the district  has defined as “behind” can hardly be construed as an antidote for the inequities of  the  system,  particularly  as  african  american and  latino/as  are  disproportionately  assigned  to  this  type  of  schooling.  mandating  a rudimentary  curriculum  that  few  middle­class parents would choose for their own children publicly signals that low­  income children and children of color are deficient. (p. 47)  lipman’s  (2004)  analysis  is  especially  powerful  because  she  also  highlighted the voices of the teachers who carried out these practices on behalf of  the  district.  the  following  comments  come  from  interviews  that  lipman  conducted:  grover  teacher:  i’ve  been  at  this  school  for  five  years,  and  the  emphasis  on  standardized tests weighs more heavily than it ever has in my career. (p. 77)  westview teacher: we are test driven… everything is test driven. (p. 77)  eighth­grade teacher: with all the teaching strategies—teaching them how to take  tests. i have tested them to death to tell you the truth. (p. 78) davis & martin  racism and standardized testing  journal of urban mathematics education vol.1, no.1  13  we concur with  lipman (2004) when  she stated the following about this  narrow approach to educating african american children:  the  emphasis  on  analyzing  and  preparing  for  standardized  tests;  the  immense  pressure on administrators, teachers, and students to raise scores; the substitution of  test­preparation materials for the existing curriculum; practice in test­taking skills as  a  legitimate classroom activity—these constitute a meaning system that reinforces  the definition of education as the production of “objective,” measurable, and discrete  outcomes. (p. 80)  without an awareness that what was observed in a single middle school in  baltimore is also taking place in other  locations around the country, one might  easily  conclude  that  the  approach  to  teaching  mathematics  to  the  students  in  baltimore  was  appropriate.  there  would  be  no  linking  of  the  practices  in  one  context  to  similar  practices  in  another.  hence,  the  institutionalized  nature  of  african american students’ mis­education would be lost. one might argue that  these students needed support to help raise their level of achievement to that of  white  students  and  that  the district  and  school  officials  were  simply  providing  them  with  that  support.  many  administrators  and  teachers,  however,  might  be  unaware of how such practices can, on one hand, reflect well­intentioned goals  but  simultaneously  contribute  to  the  oppression  of  their  african  american  students.  reflecting  on  the  experiences  of  the  first  author,  both  of  us  agree  that  although this approach resulted in increased test scores for sixth­ and eighth­grade  students, these increases do not mitigate the oppression. we unequivocally oppose  such  a  narrow  instructional  approach  and  conceptualization  of  mathematics  education  for  african  american  children.  our  opposition  is  based  on  our  scholarly  analysis,  our  respective  teaching  experiences  in  diverse  african  american  contexts,  and  our  willingness  to  advocate,  as  african  american  scholars, on behalf of african american children.  yet, as lipman (2004) argued, it is insufficient to analyze such practices in  isolation of  the  larger  ideologies and political  movements  that undergird them.  utilizing a race­critical perspective (martin, 2009), we argue that such test­driven  instructional  practices,  particularly  within  hyper­segregated  african  american  schools, like those in baltimore and elsewhere, must be situated within a larger  system of assessment that has “scientifically” (a) supported the social construction  of  african  american  children  as  intellectually  inferior  and  (b)  facilitated  the  development of ranking systems that reify these negative social constructions.  although  a  full  deconstruction  of  this  assessment  system  is  beyond  the  scope of this article, we offer a partial deconstruction that links scientific racism,  race­based  ranking  systems,  and  instructional  practice  in  classrooms  predominated by african american children. in concert with this deconstruction, davis & martin  racism and standardized testing  journal of urban mathematics education vol.1, no.1  14  we suggest strategies for helping teachers resist narrow, test­focused, instructional  approaches while promoting excellence and empowerment for african american  children in mathematics.  assessment and scientific racism  historically, testing and assessment has been linked to larger eugenics and  white supremacy efforts  that have tried to prove,  through science,  that african  americans and other non­whites are intellectually and culturally inferior (gould,  1981;  herrnstein  &  murray,  1994;  jensen,  1969;  ladson­billings,  1999).  intelligence  testing  has  evolved  alongside  various  racist  beliefs  about  african  americans. according to ladson­billings (1999), “throughout u.s. history,  the  subordination of blacks has been built on ‘scientific’ theories (e.g., intelligence  testing), each of which depends on racial stereotypes about blacks that makes the  conditions appear appropriate” (p. 23). from the seventieth century to the first  half of the twentieth century, the scientific community participated in validating  the so­called inferiority of african americans when compared to whites and the  formation  of  a  racial  hierarchy  in  both  intelligence  and  culture  (gould,  1981;  herrnstein & murray, 1994; jensen, 1969; montagu, 1997; norman, 2000).  montagu  (1997)  argued  that  few  members  of  the  scientific  community  spoke  against  the  notion  of  a  hierarchy  of  races.  instead,  the  shared  beliefs,  values, and techniques exhibited by the scientific community formed the basis of  scientific racism (norman, 2000). scientific racism can be defined as the use of  scientific methods to support and validate racist beliefs about african americans  and other groups’ based on the existence and significance of racial categories that  form a hierarchy of races that support political and ideological positions of white  supremacy  (gould,  1981;  herrnstein  &  murray,  1994;  jensen,  1969;  montagu,  1997;  norman,  2000).  gould  (1981),  montagu  (1997),  and  norman  (2000)  asserted  that  the  science  establishment  invested  a  considerable  amount  of  resources  into  advancing  scientific  racism.  according  to  montagu,  “virtually  every  scientist  writing  during  the  nineteenth  century  was…caught  in  an  inexorable web of racist beliefs” (p. 32). similarly, norman argued, “despite an  impressive array of eminent scientific advocates, scientific racism had, from its  inception and even up to its modern­day manifestations, been nothing more than  the uncritical couching of popular racist beliefs in the idiom of science” (p. 3).  there are three faulty assertions guiding scientific examinations of race and  intelligence that conceal and  couch racist  beliefs about african  americans and  other groups (gardner, 1995; gould, 1981). first, there is widespread belief that  intelligence can be described by a single number. gould (1981) contended that  converting  abstract  concepts  such  as  intelligence  into  numerical  entities  is  a  fallacy.  gardner  (1995)  argued  that  the  belief  in  a  single,  standardized,  and davis & martin  racism and standardized testing  journal of urban mathematics education vol.1, no.1  15  inherent  human  intelligence  or  g  (general  intelligence)  ignores  the  concept  of  multiple  intelligences.  second,  these  faulty  assertions  fail  to  take  into  consideration jones’ (1995) arguments about the long­forgotten justifications of  slavery and segregation that rest on beliefs about african american intellectual  inferiority and the alleged intellectual superiority of whites; in that, there exists a  faulty  belief  that  intelligence can  be used to rank social groups  in some  linear  order. gould argued that such ranking requires a criterion that takes the form of  an  “objective  number”  to  assign  all  individuals  to  their  proper  status.  the  assumption is that “if ranks are displayed in hard numbers obtained by rigorous  and standardized procedures, then they must reflect reality, even if they confirm  what we wanted to believe from the start” (gould, 1981, p. 26). nonetheless, we  concur  with  gould,  who  also  argued:  “science must  be understood  as  a social  phenomenon, a gutsy, human enterprise, not the work of robots programmed to  collect  pure  information.…science,  since  people  must  do  it,  is  a  socially  embedded activity” (p. 21). third, the most conservative research on intelligence  suggests that it is genetically­based and immutable. this suggestion “invariably  [leads]  to  [the  conclusion]  that  oppressed  and  disadvantaged  groups—races,  classes, or sexes—are innately inferior and deserve their status” (gould, 1981, p.  25).  several  scholars  abandoned  the  word  intelligence  to  avoid  debates  and  endless  arguments  associated  with  intelligence  testing,  biological  determinism,  scientific rationalism, and scientific racism (gould, 1981; herrnstein & murray,  1994; jensen, 1969). herrnstein and murray (1994) suggested that scholars use  more neutral terms such as cognitive ability to subside criticism. essentially, the  discourse  about  intelligence  testing  was  minimized  throughout  the  1970s.  we  argue that contemporary, race­comparative analyses began to flourish on the heels  of  this changing discourse, however. since that time, a number of comparative  analyses of mathematics achievement have been conducted, typically supporting  and  serving  as  evidence  for  so­called  racial  achievement  gaps  (see,  e.g.,  lubienski,  2002;  strutchens  &  silver,  2000).  these analyses  have consistently  normalized white student performance and portrayed african american children  as lacking in mathematics skills and ability.  politics and purposes of standardized tests  apparently  neutral  assessments  are  not  objective  at  all,  but  rather  ‘objects  of  history’—created  to  fulfill  particular  social  functions,  which  have  shaped  the  assessments  in  particular  directions  that  are  not  readily  apparent.  the  seemingly  innocuous requirement for the results of a test to be reliable requires that the test  disperses individuals along a continuum so having the effect of placing a magnifying  glass over a very small part of human performance, and this is particularly marked in  mathematics. (williams, bartholomew, & reay, 2004, p. 58) davis & martin  racism and standardized testing  journal of urban mathematics education vol.1, no.1  16  the  history,  politics,  and  purposes  of  standardized  testing,  particularly  in  mathematics, are rooted in the research and discourse revolving around race and  intelligence that we outlined above. we contend that deconstructing this research  and  discourse  on  intelligence  testing  is  important  in  understanding  the  racist  underpinnings of contemporary standardized testing not only in mathematics, but  also in every discipline. as the quote by williams, bartholomew, and reay points  out, one of the purposes of assessment is to create hierarchies. we claim that the  use of a single form of assessment or a single statistic to describe mathematical  ability  is  limited in explanatory power. nonetheless, this happens for two main  reasons. the first—to convey certainty and absolute truth—stems from the fact  that numbers and statistics represent a special form of being “objective” (gould,  1981, 1995). they carry the weight of proof. the analysis of all types of data,  including statistics, however, involves interpretation that cannot be divorced from  social  and  political  contexts.  how data  is  chosen  and  used depends  on  who  is  doing the choosing and their purpose for conducting the analysis.  it is not uncommon for statistical reports to be presented in the absence of  important  qualitative  and  contextual  considerations.  even  when  inclusion  does  occur,  misunderstanding  of  these  contextual  forces  occurs  in  deference  to  supporting the validity of the statistics. for example, race, which usually appears  in achievement studies as an undertheorized independent variable, is said to cause  measured achievement differences among socially constructed racial groups. yet,  this faulty use of the concept of race usually reflects an inadequate understanding  of  racism  and  racialization  and  their  impact  on  educational  outcomes.  socioeconomic  status,  typically  described  by income,  is  also  said  to  determine  achievement outcomes  but  is  made  causal  without  a nuanced  understanding  of  wealth differentials (e.g., property ownership, investments, inheritance) within the  same  socioeconomic  (income)  strata  and  how  forces  like  racism  and  discrimination,  in  turn,  account  for  those  wealth  differences  (conley,  1999).  similarly, neighborhood effects are used to construct theories about opposition,  disengagement, and resistance to schooling that  leads to academic failure. yet,  these  analyses  fail  to  account  for  student  success  in  these  very  same  neighborhoods.  the second reason is that statistics allow students to be ranked and sorted  along  what  are  thought  to  be  racial  lines  (gould,  1981,  1995;  tate,  1993).  because  of  how  test  scores  are  used  in  race­comparative  analyses  (lubienski,  2002; strutchens & silver, 2000; u.s. department of education, 1997), african  american students are frequently constructed and represented as being inferior to  white and asian students  in mathematics (martin, 2007, 2009). in mathematics  education,  these rankings and  sortings  have  been used to produce what martin  (2009)  has  termed  the  racial  hierarchy  of  mathematical  ability.  this  racial  hierarchy  results  in  white  students  being  positioned  at  the  top  and  african davis & martin  racism and standardized testing  journal of urban mathematics education vol.1, no.1  17  american  students  at  the  bottom.  this  “ranking”  proves  to  be  particularly  interesting  because  asian  students,  collectively,  perform  better  than  white  students.  yet,  it  is  white  students  who  are  used  as  the  barometer  for  african  american  students’  performance.  for  example,  a  commonly  cited  research  finding  has suggested that african  american 12th graders perform at  the same  level  as  white 8th graders  (lubienski,  2002;  national  research  council,  1989;  thernstrom  &  thernstrom,  1999;  u.s.  department  of  education,  1997).  such  findings provide pseudoscientific support for racist assumptions (tate, 1993) that  suggest african american students are intellectually inferior to white students and  located at the lowest levels of a racial hierarchy.  a  belief  in  racial  hierarchies  undergirds  all  forms  of  intelligence  testing,  including  school­based  achievement  testing  (gardner,  1995;  gould,  1981;  herrnstein & murray, 1994), and is aligned with the same racist assumptions that  have allowed african americans to be exploited within the laws and practices of  the united states  more generally (ture & hamilton, 1992;  wilson, 1998).  we  suggest  that teachers who engage  in teaching to the test and other shortsighted  remediation must necessarily accept the existence of this hierarchy as evidenced  by their subsequent efforts to relocate african american students within it.  assessment in mathematics education: foregrounding race and racism  invoking a race­critical perspective, we claim along with others (see, e.g.,  hilliard, 2003; ladson­billings, 1999) that, beyond any knowledge that might be  gained  about  student  thinking  and  development,  the  larger  political  effect  of  standardized  testing,  particularly  in  the  area  of  mathematics  education,  is  to  maintain  white  supremacy  in  one  form  or  another  (e.g.,  u.s.  international  standing and competitiveness, normalization of white student behavior). 2 ladson­  billings (1999), for example, has  argued that  the school  curriculum suppresses  multiple voices and perspectives while simultaneously legitimizing the dominant,  white,  male, upper­class ways of knowing and  being  as the “standard” that all  students should be required to emulate (see also swartz, 1992). this dominance is  evidenced by the fact  that schools serving  african  american  students  typically  adopt curriculum from predominantly white school districts (davis, 2008; martin,  2007).  such  choices  are  often  not  done  in  response  to  the  authentic  needs  of  2 by this statement, we mean that the goals for testing are often framed in terms of improving the  “standing”  of  the  united  states  relative  to  other  countries  in  international  comparisons.  many  high­achieving asian countries are often discussed as threats to the standing of the united states.  the goals for african american children are often framed as increasing test scores for the purpose  of having outcomes match those of white children. davis & martin  racism and standardized testing  journal of urban mathematics education vol.1, no.1  18  african american learners, but suggest that what african american children need  is determined by what is best for white children.  we also claim that commonly used race­comparative analyses are one small  piece of the larger structural and institutional mechanisms that support this goal. it  is  this  larger  structural  effect,  above  and  beyond  the  efforts  and  intentions  of  individual white scholars and policy­makers, that continues to drive instructional  practices  for  teachers  of  all  children  but  especially  african  american  children  when  they  are  viewed  as  less  than  ideal  learners  and  mathematically  illiterate  (martin, 2007, 2009).  in  tate’s  (1993)  critical  race analysis  of  standardized  testing  practices  in  poor school districts serving large numbers of african american students, he used  the voluntary national mathematics assessment as a platform to discuss the racist  underpinnings  of  standardized  testing.  tate  argued  that  standardized  tests  are  “scientifically”  constructed  to  socially  reproduce  the  most  negative  aspects  of  african american students’ lived realities. he also argued that standardized tests  were  designed  to  prepare  poor  african  american  students  to  replicate  their  parents in the division of labor by providing them with instruction in mathematics  suitable  for  this  purpose.  tate  further  claimed  that  policies  governing  standardized test were designed to ensure that poor african american students did  not  receive  the  same  instruction  in  mathematics  as  middle­  and  upper­class  members  of  society.  he  believed  that  test  scores  are  not  intended  to  provide  feedback for the purposes of educational improvement in mathematics, but to rank  students and to determine their economic potential. in other words, standardized  test shape the lives of poor african american students in more significant ways  than middle­class or affluent students.  we  agree  with  tate’s  (1993)  analysis.  the  current  environment  of  high­  stakes  testing  engendered  by  nclb  has  caused  many  states  and  local  school  districts  to  shift  their  instructional  approaches  in  ways  where  satisfactory  outcomes  on  state  assessments—not  authentic  learning  and  development—  become the primary goal. these pressures have also positioned administrators and  teachers to appropriate much of the underlying ideology that characterizes african  american  children  as  mathematically  illiterate,  using  white  and  asian  student  performance as the standard.  the  current  environment  of  high­stakes  testing  is  not  only  just  a  contemporary  phenomenon,  but  also  one  that  has  historical  ties  to  intelligence  testing and the construction of racial hierarchies. nclb has repositioned state and  local policies and  instruction and standardized testing efforts  in public schools,  specifically, in mathematics, to carry out the construction of these hierarchies.  there are two aspects of nclb that shape our discussion of standardized  testing in mathematics education. first, one of the main goals of nclb is to close  the so­called racial achievement gap in reading and mathematics. martin (2009) davis & martin  racism and standardized testing  journal of urban mathematics education vol.1, no.1  19  argued that plans to move african americans and other marginalized groups from  their  perceived  positions  of  being  mathematically  illiterate  to  being  mathematically literate, an intellectual space supposedly occupied by white and  asian  students,  is  rooted  in  racist  beliefs  about  these students.  the underlying  assumption is that african american students’ performance in mathematics must  conform  to that of  white students  in  order  for  these  students  to  be  considered  mathematically  literate  (martin,  2009).  in  our  view,  the  performance  of  white  students as the benchmark for african american students sets an artificially low  standard for african american learners given that the collective averages of white  students on many large­scale mathematics assessments are less than the highest  levels of proficiency (secada, 1992; strutchens & silver, 2000; tate, 1997) and  ignores the needs of  african  american children as african american children.  connecting the discourse on african american students in mathematics education  to intelligence testing, the assumption is that “black inferiority is purely cultural  and  that  it  can  be  completely  eradicated  by  [mathematics]  education  to  a  caucasian standard” (gould, 1981, p. 32).  the accountability  measures  dictated by  nclb  require  states  to  publicly  identify  low­performing  schools.  this  practice  has  played  a  major  role  in  subjecting  african  american  students,  their  schools,  and  school  systems  to  inferior labels as a result of failing to meeting standardized testing goals (davis,  2008; lattimore, 2001, 2003, 2005a; sheppard, 2006). this practice was clearly  evident in the baltimore city school and district discussed in the introduction of  this article. currently, this african american school district is in its second year  of system improvement and the students are considered the lowest performers in  mathematics  in  the  state.  in  addition  to  these  labels,  the  failure  to  meet  standardized test goals places their schools in danger of losing federal dollars to  finance their education.  second,  nclb  has  explicitly  attempted  to  standardize  what  constitutes  highly  qualified  teachers  for  all  students. 3  the  policy  mandates  the  use  of  standardized  tests  to  quantify  what  constitutes  a  highly  qualified  mathematics  teacher. this policy treats the  instruction  african  american students receive  in  mathematics as a generic set of  teaching  competences that should work for all  students  (ladson­billings,  1999;  martin,  2007).  when  these  approaches  to  teaching  fail  to  produce  the  desired  results,  african  american  students  are  deemed  deficient—not  the  approaches  used  to  teach  these  students  (ladson­  billings, 1999; martin, 2007).  3 in this article, we do not give extensive attention to the forms of assessment used for teacher  certification. we do, however, claim that the same logic applies to these tests. the nclb policy  document defines highly qualified as a teacher who holds at least a bachelor’s degree and has  passed state­certification or licensing exams. davis & martin  racism and standardized testing  journal of urban mathematics education vol.1, no.1  20  later in this article, we will briefly revisit arguments made by martin (2007)  who problematized notions of highly qualified mathematics teachers by asking:  “who  should  teach  mathematics  to  african  american  students?”  we  use  the  characteristics described by martin as a catalyst to provide mathematics teachers  with  strategies  to  resist  contributing  to  the  oppression  of  african  american  students.  african american students’ experiences in  mathematics education  martin  (2007)  discussed  how  achievement  has  served  as  the  dominant  discursive  frame used  to talk  about  the competencies  of  african  americans  in  mathematics  within  the  context  of  mainstream  mathematics  education  research  and  policy.  he  demonstrated  how  a  framework  of  color­blind  racism,  in  turn,  supports  this  achievement­focused  discourse.  martin  challenged  mathematics  education researchers to construct an alternative discursive and assessment frame  focused  on  how  african  american  learners  experience  mathematics  education,  and  suggested  that  future  research  should  focus  on  mathematics  learning  and  participation  as  racialized  forms of experience,  not only  for  african  american  learners but also for all learners.  analysis  of  the  relevant  literature  reveals  two  important  insights  about  african american students’ experiences with standardized testing (berry, 2005;  corey & bower, 2005; lattimore, 2001, 2003, 2005a; lubienski, 2002; moody,  2003,  2004;  strutchens  &  silver,  2000;  u.s.  department  of  education,  1997).  first, school districts serving large numbers of african american students often  implement  remedial  strategies  to  comply  with  state  and  federal  regulations  surrounding standardized testing in mathematics (davis, 2008; lattimore, 2001,  2003, 2005a; tate, 1993). in support of this strategy, african american students  are inundated with practice materials that include worksheets and in­class practice  tests  devoted  to  state  assessments  (lattimore,  2001,  2003,  2005a).  the  mathematics instruction that these students are exposed to emphasizes repetition,  drill, right­answer thinking that often focuses on memorization and rote learning,  out­of­context  mathematical  computations,  and  test­taking  strategies  (davis,  2008;  ladson­billings,  1997;  lattimore,  2001,  2003,  2005a).  this  type  of  instruction  often  leaves  african  american  students  disengaged  and  viewing  mathematics as irrelevant and decontextualized from their everyday experiences  (corey & bower, 2005; davis, 2008; ladson­billings, 1997; lattimore, 2005b;  tate, 1995).  second,  standardized  tests  serve  as  a  “gatekeeper”  in  providing  african  american  students  access  to  higher­level  mathematics,  gifted  and  honors  programs,  and  future  aspirations  (berry,  2005;  davis,  2008;  lattimore,  2001, davis & martin  racism and standardized testing  journal of urban mathematics education vol.1, no.1  21  2003,  2005a;  moody,  2003,  2004;  oakes,  1990;  sheppard,  2006).  throughout  their schooling experiences, african american students are often denied access to  higher­level  mathematics and advanced programs  because of  their performance  on  standardized  test  (berry,  2005;  davis,  2008;  moody,  2003,  2004;  oakes,  1990), thereby leaving the majority of african american students in lower­level  mathematics courses (corey & bower, 2005; davis, 2008; lubienski, 2001, 2002;  moody, 2003, 2004; oakes, 1990; oakes, joseph, & muir, 2004). in high school,  state  administered  standardized  tests  have  also  been  found  to  serve  as  the  gatekeeper  to  african  american  students  receiving  a  high  school  diploma  (lattimore, 2001, 2003, 2005a). for example, since 2006, students in california  must pass an exit exam to graduate. students in maryland will have to pass the  state  administered  standardized  test  to  receive  their  high  school  diploma  beginning with the graduating class of 2009.  re­conceptualizing the assessment of  african american students in mathematics:  implications for teachers  there  is  very  little  consideration  given  to  the  argument  that  african  american  students  represent  a  distinct  cultural  group  (akbar,  1980;  ladson­  billings,  1994),  requiring  an  education  in  mathematics  that  reflects  their  lived  realities and collective conditions (martin, 2007; thompson, 2008). according to  ladson­billings (1999):  african american students are a part of almost every social strata and their social  context may affect what experiences they have and how they view the world, their  cultural knowledge, expressions, and understandings, which may be transmitted over  many  generations,  may  share  many  features  with  african  americans  across  socioeconomic and geographical boundaries. (p. 699)  we  argue  african  american  students  must  receive  an  education  in  mathematics  that  not only  prepares  them to  function  effectively  in  mainstream  society  but  also  builds  on  their  cultural  knowledge  base  and  value  systems  (ladson­billings,  1997).  this  argument  implies  framing  the purpose,  structure,  and  ideology  of  mathematics  education  for  african  american  learners  in  ways  that are responsive to their needs as african american learners (martin & mcgee,  in press).  in  reconceptualizing  and  reframing  mathematics  education  for  african  american learners, a growing number of african american scholars have begun  to  advance  liberatory  mathematics  education  agendas  for  african  american  students (martin &  mcgee,  in press;  moses  &  cobb, 2001; thompson, 2008).  the most notable example of this agenda is the algebra project (moses & cobb, davis & martin  racism and standardized testing  journal of urban mathematics education vol.1, no.1  22  2001) and its parallel youth development program, the young people’s project  (ypp).  through  the  algebra  project,  civil  rights  activist  and  mathematics  educator robert moses has led the charge to provide african american students  with a liberatory mathematics experiences via a curriculum anchored in culturally  relevant activities. moses argued the fight for mathematics literacy is a fight for  twenty­first  century  citizenship  and  that  african  american  youth  must  be  empowered to fight for their  liberation on their own terms. this empowerment  was clearly evident in baltimore where african american youth from the algebra  project  in  that  city  challenged  school  officials  for  not  providing  them with  an  adequate education (prince, 2006).  martin and mcgee (in press) argued, “any framing of the form, philosophy,  and  content  of  mathematics  education  for  african  americans  must  address  the  historical and contemporary social realities that they face.” they suggested that  history  “compels  us  to  frame  african  americans  mathematics  education  and  mathematics  literacy  in  the  same  way  that  education,  in  general,  was  framed  around their life conditions in the past, for the purposes of liberation.” in defining  liberation,  they  drew  on  the work of  watts,  williams,  and  jagers  (2003),  who  defined liberation as follows:  liberation  in  its  fullest  sense  requires  the  securing  of  full  human  rights  and  the  remaking  of  a  society  without  roles  of  oppressor  and  oppressed.…it  involves  challenging  gross  social  inequities  between  social  groups  and  creating  new  relationships that dispel oppressive social myths, values, and practices. the outcome  of this process contributes to the creation of a changed society with ways of being  that  support  the  economic,  cultural,  political,  psychological,  social,  and  spiritual  needs of individuals and groups. (pp. 187–188)  thompson (2008) similarly argued for a liberatory framing of mathematics  education for african american learners through what she calls nation building.  she  defined  nation  building  “as  the  conscious  and  focused  application  of  knowledge, skills, and abilities to the task of liberation” (p. 17). thompson further  argued  that  nation  building  “involves  the  development  of  behaviors,  values,  institutions, and physical structures that elucidate african history and culture” for  the purposes of ensuring “the future identity, existence, and independence of the  nation”  (p.  17).  she  believed  that  efforts  to  increase  african  americans  in  mathematics and science should be geared toward the liberation of african people  throughout the diaspora and to eradicate systems of racism (white supremacy).  these liberatory agendas typically stand little chance of being accepted in  mainstream  discussions  of  mathematics  education  because  “african  americans  seeking equal opportunity in education [specifically in mathematics] will only be  granted  when  the  opportunity  being  sought  converges  with  the  economic  self­  interest of whites” (tate, 1993, p. 17). bell (1980) has referred to this contingency  as interest convergence. davis & martin  racism and standardized testing  journal of urban mathematics education vol.1, no.1  23  critical  reflection  and  advocacy  by  mathematics  teachers  of  african  american students  we realize that teachers cannot expect to engage in liberatory instructional  practices with african american students and be rewarded by the same system  that  demands  that  they  contribute  to  the  negative  social  construction  of  these  students  (ture  &  hamilton,  1992).  yet,  we  would  appeal  to  what  is  morally  correct,  given  the  needs  and  social  realities  (ladson­billings,  1997)  of  these  students  and  frame  the  discussion  that  follows  as  both  a  challenge  and  an  invitation for teachers. we challenge teachers to engage in critical reflection on  their own practices and we invite them to consider the suggestions we make about  changes in these practices, where necessary.  martin  (2007)  suggested  that  teachers  should  (a)  develop  a  deep  understanding of the social realities experienced by african american students,  (b)  take  seriously  one’s  role  in  helping  to  shape  the  racial,  academic,  and  mathematics  identities  of  african  american  learners,  (c)  conceptualize  mathematics  not  just  as  a  school  subject  but  as  a  means  to  empower  african  american students, and (d) become agents of change who challenge research and  policy  perspectives  that  construct  african  american  children  as  less  than  ideal  learners  and  in  need  of  being  saved  or  rescued  from  their  blackness.  we  encourage  teachers  of  african  american  students  to  reframe  their  instructional  practices  by  taking  the  ideas  developed  by  martin  seriously.  in  our  view,  “teachers who are unable, or unwilling, to develop in these ways are not qualified  to teach african american students no matter how much mathematics they know”  (martin, 2007, p. 25).  while we do not offer prescriptive or formulaic approaches for how teachers  might utilize martin’s (2007) suggestions, we do point to some important initial  steps and underscore that many of these steps should occur simultaneously and  throughout teachers’ work with african american students. our conceptualization  of the instructional strategies and strategies of resistance offered in this section  are  in  many  respects  inextricably  linked  and  presented  in  concert  with  one  another. the strategies are intended to help teachers resist teaching to the test and  resist  contributing  to  the  negative  social  construction  of  african  american  students.  to  help  resist  these  efforts,  we  strongly  believe  that  teachers  must  continuously engage in critical reflection about their practice, their beliefs about  african american students, and their commitment to these students. individually  and  collectively,  teachers  must  engage  in  critical  reflection  on  how  they  conceptualize mathematics education for african american students. in so doing,  we believe that issues of race and racism must be at the forefront of discussions of  mathematics education for african american students. martin argued that there  are several documented cases where  failing to do so can stall  the progress and  design of meaningful mathematics education for these students. davis & martin  racism and standardized testing  journal of urban mathematics education vol.1, no.1  24  teachers have to realize that we are all socialized by institutions (e.g. media,  policies, laws, etc.) that support racist views and beliefs about african american  children. policies and ideologies associated with high­stakes testing, for example,  often position teachers in ways where critical reflection on their practices is de­  emphasized or derailed by progressive rhetoric. martin (in press) discussed how  he has used the  following three­question quiz  in professional development and  research contexts with teachers from various ethnic and racial backgrounds, years  of experience, and geographic locations: •  how many of you have heard of, and understand, what  is meant by the  racial achievement gap? •  how many of you have, or plan to, devote some aspect of your practice to  closing the racial achievement gap? •  how many of you believe in the brilliance of african american children?  after  noting that the vast  majority of  teachers answer affirmatively to all  three questions, martin (in press) goes on to point out how the second and third  questions  are  conceptually  and  practically  incompatible.  he  pointed  out  that  acceptance  of  the  racial  achievement  gap  rhetoric  necessarily  requires  that  teachers,  even  african  american  teachers,  accept  the  inferiority  of  african  american children, especially when closing the so­called racial achievement gap  is translated as raising african american children to the level of white children.  the  quiz  was  a  strategy  to  help  teachers’  resist  and  rethink  negative  social  constructions of african american students.  in addition to taking the quiz,  teachers  must ask themselves difficult and  uncomfortable  questions  about  african  american  students  and  their  conditions  that include, but are not limited to: do i believe african american students are  intellectually inferior? do i believe that issues of race and racism play a role in  shaping  the  lives,  schooling,  and  mathematics  education  of  african  american  students?  do  i  harbor  racist  beliefs  about  african  american  students?  do  i  believe that history has any bearing on african american students’ contemporary  lived realities, schooling, and mathematics education? questions of this nature are  reflective  strategies  of  resistance  that  must  be  considered,  thought  about,  and  answered truthfully. we pose these questions for all teachers but we particularly  direct  them to those white teachers whom the  literature has  identified as  being  particularly resistant to change in their negative or deficit beliefs about african  american children (sleeter, 1993).  we believe that teachers must decide whether they are willing to be agents  of social change for african american students. teachers should ask themselves  the  following  questions:  what  am  i  willing  to  sacrifice  for  african  american  students? am i willing to sacrifice or take the risk to provide african american davis & martin  racism and standardized testing  journal of urban mathematics education vol.1, no.1  25  students  with  a  liberatory  mathematics  education  in  the  face  of  policies  that  require me to do otherwise? am i willing to challenge policies that treat african  american students as less than ideal learners?  once teachers have explored these considerations, we believe that teachers  need  to  spend  time  seriously  thinking  about  how  they  envision  mathematics  education for african american students. in the process of conceptualizing what  mathematics education for african american students should look like, teachers  should ask themselves the following question: what do i want african american  students to be able to do as a result of their mathematics education? we suggest  that african american students should be able to use mathematics as a tool to (a)  reexamine  history  and  use  this  history  to  generate  critiques  and  better  understandings of their immediate life conditions and collective group conditions  in the world, and (b) gain access to areas in the larger opportunity structure where  mathematics  knowledge  has  often  been  used  to  keep  african  americans  out.  overall, the mathematics education african american students receive should be  designed  to  improve  their  life and  group  conditions  (martin,  2009;  thompson,  2008).  in  terms  of  the  goals  for  mathematics  learning,  teachers  might  consider  adopting  the  stance  that  effective  teaching  should  not  only  produce  growth  in  students’  mathematical  skills  but  also  connect  to  these  students’  lives,  experiences, and lead them to employ their mathematical knowledge in multiple  settings  and  develop  their  racial,  academic,  and  mathematics  identities.  for  teachers, this entails thinking of empowerment along three lines: mathematical,  social,  and  epistemological  (ernest,  2002;  martin  &  mcgee,  in  press). 4  this  stance does not mean that students cannot be shown how to carry out procedures  and learn to produce correct answers. however, if they do not see themselves as  legitimate doers of mathematics, then the acquisition of skills with little personal  identification on the part of students is not likely to sustain itself. in other words,  we argue that teachers of african american students should consciously attempt  to  integrate  these  students’  experiences,  home  and  community  lives  into  their  4 according to ernest (2002), mathematical empowerment concerns the gaining of power over the  language, skills, and practices of using and applying mathematics; that is, the gaining of power  over a relatively narrow domain, for example, that of school mathematics. social empowerment  through mathematics concerns the ability to use mathematics to better one’s life chances in study  and work and to participate more fully in society through critical mathematical citizenship. thus it  involves the gaining of power over a broader social domain, including the worlds of work, life and  social affairs. epistemological empowerment concerns the individual’s growth of confidence not  only in using mathematics, but also a personal sense of power over the creation and validation of  knowledge. this empowerment is a personal form: the development of personal identity so as to  become a more personally empowered person with growth of confidence and potentially enhanced  empowerment  in  both  the  mathematical  and  social  senses  (and  for  the  mathematics  teacher—  enhanced professional empowerment). davis & martin  racism and standardized testing  journal of urban mathematics education vol.1, no.1  26  conceptualization of  mathematics and teach them how to use mathematics as a  means to view and critique the world (lynn, 2001; tate, 1995).  several scholars’ work has provided insight into how mathematics teachers  have conceptualized mathematics along mathematical, social, and epistemological  lines  (ladson­billings,  1997;  lynn,  2001,  tate,  1995).  for  example,  ladson­  billings (1997) described a sixth­grade mathematics teacher of african american  students who went beyond the district curriculum by providing her students with  an  engaging,  rigorous,  and  challenging  education  in  algebra.  this  teacher’s  students  were  engaged  in  problem  solving  around  algebra,  pushed  to  think  at  higher  levels,  and  encouraged  and  reassured  by  their  teacher  that  they  were  capable doers of mathematics. in this class, a student with special needs benefited  from this teacher’s belief system and instruction in mathematics. at the end of the  school  year,  this  teacher  convinced  the  school  principal  to  remove  the  student  from  receiving  special  education  services  because  of  his  mathematical  performance in her class.  lynn  (2001)  captured  the  experience  of  a  middle  school  mathematics  teacher  who  reflected  seriously  on  issues  of  poverty  and  racism.  this  teacher  engaged students in a discourse about how the history of lynching and jim crow  racism  has  shaped  african  americans’  lives.  he  used  this  history  to  teach  his  students  the  importance  of  checking  their  work  and  knowing  their  math  facts.  this  teacher  connected  the  two  by  making  the  case  that  historically  african  americans  have  had  to  prove  that  injustices  actually  occurred  to  them  by  supporting their experiential claims with facts. in this lesson, the teacher situated  this  discourse  in  a  historical  analysis  of  african  american  experiences  with  racism  in society. the teacher provided his students with concrete examples of  how to use numerical data presented in the media to critically examine the ways  that numbers get utilized in an unjust society. for example, the teacher made the  case  that  a  media  report  describing  a  decrease  in  joblessness  does  not  always  translate  into increasing  jobs  for  the masses of  african  americans. essentially,  this  teacher  was  committed  to  raising  african  american  students’  social  consciousness about the uses and abuses of mathematics in society much like we  attempt to do in the article.  tate  (1995)  described  a  mathematics  teacher  who  engaged  her  african  american  students  in  real  problems  of  social  and  economic  importance  for  african  americans  and  their  community.  this  teacher  asked  students  to  pose  problems  they  felt  were  important  to  them  and  affected  their  community,  to  conduct research on one of the posed problems, and to develop strategies to solve  that  problem.  the  students  were  then  encouraged  to  execute  the  strategy  they  developed. the students posed a wide range of problems that included the aids  epidemic, drugs, ethics in medicine, and sickle cell anemia. in one class, students  posed problems about the excessive number of liquor stores in their community davis & martin  racism and standardized testing  journal of urban mathematics education vol.1, no.1  27  and “embarked on an effort to close and/or relocate 13 liquor stores within 1000  feet of their school” (p. 170). the students’ action resulted in “over 200 citations  to liquor store owners and two of the 13 stores closed down for major violations”  (p. 170).  we recommend that teachers spend time developing relationships with their  students  that  extend  beyond  the  mathematics  content  being  taught  in  their  classroom. teachers should not rely solely on secondary sources (e.g. principals,  other teachers, cumulative records, etc.) to define their outlook, views, and beliefs  about african american students. teachers can spend time learning about african  american  students’  home life, social realities, childhood experiences, and  likes  and dislikes. in this way, teachers can show that they are committed to african  american children and their families in ways that extend beyond just raising test  scores.  in  his  study  of  african  american  middle  school  students,  davis  (2008)  described an african american female mathematics teacher, mrs. rene taylor,  who got to know her students by spending time with them in and out of school  (e.g. lunch, after school, hallways, taking students to the movies, inviting students  to her home, etc.). mrs. taylor spent time listening and talking to students as they  spoke about their problems, interests, likes and dislikes. she engaged her students  in a discussion about herself that respected the boundaries of her position. mrs.  taylor’s  relationship  with  her  students  inevitably  resulted  in  developing  a  relationship with their parents. it should be noted that mrs. taylor also allowed  two students to move into her home, primarily because one student was homeless  and  the  other  student  had  problems  with  a  drug­addictive  parent.  we  are  not  suggesting that teachers should do everything that mrs. taylor did, but clearly her  actions demonstrate how teachers can develop relationships with students that are  genuine and meaningful. her actions also illustrated the level of commitment she  has to her students and what she was willing to do for them.  while getting to know their african american students, teachers should not  come  to  hasty  conclusions  or  generate  stereotypical  assumptions  about  their  abilities and  values. however,  teachers should  not  lose site of  the fact  that the  historical  legacy  of  racism  continues  to  shape  african  american  students’  contemporary  lived  realities  in  their  community,  home  life,  schooling,  and  mathematics education despite the absence of overt racist laws and social customs  (davis, 2008). research has shown that teachers who know or get to know their  african  american  students  provide  them  with  a  more  enriching  educational,  mathematical, and social experience (ladson­billings, 1994, 1997; lynn, 2001;  moody,  2003,  2004;  tate,  1995).  in  the  first  author’s  research  of  african  american middle and high school students, participants cited the impact that mrs.  taylor had on their educational, mathematical, and social experiences. davis & martin  racism and standardized testing  journal of urban mathematics education vol.1, no.1  28  in  the  midst  of  conceptualizing  mathematics  education  for  african  american students and getting to know these students, teachers should spend time  learning about and helping to positively shape african american students’ racial,  academic, and mathematics identities (see, e.g., martin, 2000; nasir, 2007; nasir,  jones,  &  mclaughlin,  2007).  martin  (2000)  argued  that  african  american  students’ racial, academic, and mathematics identities are linked and contribute to  these students larger sense of self. he characterizes mathematics identity as being  shaped  by  students’  beliefs  about  (a)  their  ability  to  perform  in  mathematical  contexts,  (b)  the  instrumental  importance  of  mathematical  knowledge,  (c)  the  constraints  and  opportunities  in  mathematical  contexts,  and  (d)  the  resulting  motivations and strategies used to obtain mathematics knowledge. nasir, jones,  and mclaughlin (2007) argued that african american students’ racial and ethnic  identities vary across individuals and that “the kind of racial  identities students  hold  has  implications  for  their  sense  of  themselves  as  students,  and  for  their  achievement” (p. 3). hence, these scholars’ work indicated that teachers play a  significant role in shaping these identities.  teachers  might  question  their  students  about  whether  they  believe  that  “being  african  american”  and  “being  a  doer  of  mathematics”  are  compatible  (martin,  2006).  ellington’s  (2006)  study  of  high  achieving  african  american  students found that  these students’ racial, academic, and  mathematics  identities  were  shaped  by  how  these  students  saw  themselves  in  the  larger  african  american  community.  if  african  american  students  do  not  perceive  “being  african american” and “being a doer of mathematics” as being compatible, then  rich  and  meaningful  discussions  that  affirm  students’  racial,  academic,  and  mathematics identities should become an ongoing part of teachers’ practice. this  practice would include explicitly addressing (through discussions or journals) and  shattering stereotypes about who can and cannot do mathematics and reducing the  “stereotype  threat”  (steele  &  aronson,  1995)  that  accompanies  practices  like  standardized  testing.  stereotype  threat  occurs  when a  negative stereotype (e.g.,  african american students are lacking in mathematics ability) becomes salient as  a  criterion  for  test  evaluation.  in  that,  students  become  concerned  about  confirming  the  stereotype  and  through  various  psychological  mechanisms;  the  concern  can  cause  one  to  perform  more  poorly  than  they  would  perform  in  a  neutral context.  we  are  not  suggesting  that  teachers  are  not  engaging  in  practices  that  contribute  to  the  development  of  their  african  american  students’  racial,  academic, and mathematics identities. however, for those teachers who might not  have  considered  this  aspect  of  mathematical  development,  we  strongly  believe  that  these  identities  are  important  constructs  for  teachers  to  understand  and  intentionally incorporate into their instructional practices with african american  students.  for  example,  as  a  mathematics  teacher,  in  a  high­stakes  testing davis & martin  racism and standardized testing  journal of urban mathematics education vol.1, no.1  29  environment at the high school level,  the first author engaged  in practices that  positively  shaped  his  african  american  students’  racial,  academic,  and  mathematics identities without formal knowledge of these identities. davis (2005)  developed  a  project  intended  to  expose  his  african  american  students  to  the  mathematical,  technological,  and  scientific  contributions  of  people  of  african  descent. 5  the project required students to do research, write a report, do an oral  presentation, create a display, and participate in a school­wide exhibit to expose  and  engage  their  school  community  in  a  discussion  about  the  person  they  researched and their contribution to these fields. the students’ oral presentation,  research  reports,  letters,  responses  to  the  project,  exhibit,  and  trip  ultimately  allowed  the  first  author  to  understand  how  the  project  assisted  in  shaping  his  students’ racial, academic, and mathematics identities.  with  respect  to  african  american  students’  being  characterized  as  low  performers, behavior problems, and disengaged in mathematics setting, teachers  should seriously consider alternative reasons for these students’ actions other than  the ones  commonly  cited (i.e.,  mathematically  incapable,  uninterested  in  being  doers of mathematics, etc.) by researchers, teachers, and administrators (akbar,  1980; berry, 2005; davis, 2008). akbar (1980) argued that boredom and cultural  disconnect  of  schools  are  the  primary  reasons  for  african  american  students’  behavior,  disengagement,  and  performance  issues  in  these  settings.  in  mathematics education, berry (2005) and davis (2008) found african american  students across achievement levels were bored and disengaged from mathematics  and other academic disciplines. berry and davis, similar to corey and bower’s  (2005) research,  made the case that  these students’  mathematics education was  disconnected from their culture.  in davis’s (2008) research, he found the african american middle school  students  who  he  studied  in  high­stakes  testing  environments  disengaged  from  mathematics because of the actions of some of their past and present mathematics  teachers. these students reported that their mathematics teachers often (a) would  not “teach” them; (b) were not able to help them learn mathematical topics and  concepts  in  which  they  were  experiencing  difficulty;  (c)  disrespected,  embarrassed, or humiliated them with respect to learning mathematics; (d) did not  provide  them  with  challenging  and  intellectually  stimulating  mathematics;  (e)  presented  them  with  mathematical  concepts  and  topics  that  they  had  already  learned;  (f)  provided  instruction  that  was  centered  around  worksheets,  rote  5  we use african descent in this context to connote that the racial and ethnic background of the  people did not just include african americans or africans, but included a wide range of black  people  from  around  the  world.  this  inclusion  was  done  to  help  expand  african  american  students’ conceptualization of what it means to be black or african american into a larger cultural  discourse that connects these students to a larger cultural history and heritage. davis & martin  racism and standardized testing  journal of urban mathematics education vol.1, no.1  30  memorization,  board  work,  and  test­taking  materials  and  strategies;  and  (g)  maintained classrooms that were in constant disarray. while these students were  disengaged from the learning process, they participated in activities (e.g., walking  around the classroom and school hallways, horse playing, etc.) that often led them  to be further marginalized by district and school rules and policies (e.g. federal,  state,  local  testing policies) and teacher  subjectivity when their  behaviors were  reflective of  their resistance to the realities of  their schooling and  mathematics  experience.  we strongly urge teachers to continuously engage in critical reflection about  african american students, their instructional practice, and conceptualization of  mathematics  by  asking  questions  such  as:  do  i  provide  african  american  students  with  lower­level  coursework  because  i  believe  they  are  incapable  of  doing higher­level coursework? do i perceive african american students as being  lazy  in  mathematics  because  of  their  racial  identities  or  because  they  do  not  engage in mathematics the way white students do? do i believe these students are  undeserving  of  a  mathematics  education  requiring  higher­level  thinking  and  coursework?  we  strongly  encourage  teachers  to consider  and  make  use of  our  questions, suggestions, and examples, where applicable.  conclusion  we started this article by framing our discussion about standardized testing  practices in a local school district and school serving large populations of african  american students where policy initiatives and administrators require teachers to  teach to the test in mathematics. our goal for this article was to present arguments  about  the racist  underpinnings  of  such  instructional  practices  and  how  federal,  state,  and  local  policies  institutionalize  racist  beliefs  about  african  american  students.  we  situated  our  analysis  of  these  instructional  practices  within  a  deconstruction of systems of assessment that seek to create racial hierarchies and  offer “scientific” support for african american intellectual inferiority.  based on our critical analysis, our request to mathematics teachers is simple.  mathematics  teachers  of  african  american  students  must  stop  engaging  in  teaching­to­the­test  and  other  narrow  instructional  practices  and  provide  these  students with a challenging and intellectually stimulating mathematics education  that  assists  these  students  in  improving  their  individual  and  collective  group  conditions. we are not dismissing the reality that teachers must operate under the  conditions  created  by  the  oppressive  forces  of  mandates  such  as  nclb.  nevertheless, out of genuine concern for african american students, this article is  an  instantiation  of  our  advocacy  for  these  students  to  receive  the  mathematics  education  they  deserve.  we  appeal  to  teachers’  moral  commitment  to  african davis & martin  racism and standardized testing  journal of urban mathematics education vol.1, no.1  31  american  students  by encouraging them to put these students’ well­being over  their fear of federal, state, and local sanctions.  we  urge  teachers  to  take  action  both  individually  and  collectively  at  the  district, school, and classroom level to provide african american students with a  liberatory education in mathematics. in so doing, we have provided teachers with  insight and examples of how that might be done. we urge teachers to be agents of  social change  in their own  school and classroom contexts, hopefully driven  by  beliefs that build on the following:  african  american  children  [must  be  prepared]  with  the  knowledge,  skills,  and  attitude needed to struggle successfully against oppression. these, more than test  scores, more than high grade point averages, are the critical features of education for  african americans. if students are to be equipped to struggle against racism they  need excellent skills from the basics of reading, writing, and math,  to understand  history, thinking critically, solving problems, and making decisions; they must go  beyond merely filling in test sheet bubbles with number 2 pencils. (ladson­billings,  1994, pp. 139–140)  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email: megan.staples@uconn.edu. her research focuses on teaching and learning interactions in secondary mathematics classrooms, with a particular interest in collaborative interactions and detracked classrooms. mary p. truxaw is an assistant professor in the department of curriculum and instruction in the neag school of education at the university of connecticut, 249 glenbrook road, unit 2033, storrs, ct 06269; email: mary.truxaw@uconn.edu. her research focuses on discourse in mathematics classrooms and teacher education, with particular interest in urban, linguistically diverse schools. the mathematics learning discourse project: fostering higher order thinking and academic language in urban mathematics classrooms megan e. staples university of connecticut mary p. truxaw university of connecticut in this article, the authors report results from a small-scale study of the mathematics learning discourse (mld) project that aimed to affect change in urban mathematics classrooms. the project focused on enhancing students’ understanding of mathematics through an emphasis on classroom discourse and higher order thinking. four teachers participated in a 3-day summer course and yearlong collaboration that was organized around three principles for supporting a learning discourse in their respective classrooms: appropriate and effective development of students’ academic language, student engagement in mathematical practices of justification and collective argumentation, and access for all students to rigorous mathematics. the authors discuss the research base for the mld program, its implementation, and its effect—as well as promise—by analyzing student scores on preand post-assessments both for mathematical performance and for the development of students’ proficiency with academic language and justification. keywords: academic language, collaboration, mathematics education, professional development, urban education n the current high-stakes testing climate, instruction in many urban public school settings is becoming increasingly controlled and, in some places, scripted, as basic skills are prioritized over higher levels of reasoning. this narrowing of urban students’ intellectual diet ultimately increases the education gap between these students and their more affluent counterparts (anyon, 1997; keiser, 2005). this education gap should not only be measured in mere test scores but also in the opportunities students have to learn to think and express themselves i staples & truxaw mld project journal of urban mathematics education vol. 3, no. 1 28 mathematically and reason in ways that will support their participation in a democracy and further their individual pursuits (goodlad, 1994; michelli, 2005). in this article, we report the results from the mathematics learning discourse (mld) project, a small-scale research and development project undertaken with four teachers in two schools in an urban school district in connecticut. the purpose of this project was to support teachers in fostering a mathematics learning discourse in their urban classrooms. specifically, teachers sought to create a teaching and learning environment that developed students’ academic language; promoted justification and argumentation (e.g., sense making); and provided all their students access to participation in cognitively challenging mathematical activities. such an approach runs counter to typical pedagogy in urban settings (leonard & evans, 2008; manouchehri, 2004). it was expected that this approach would enhance students’ engagement and mathematics learning (boaler & staples, 2008; brenner, 1998; hiebert, carpenter, fennema, fuson, wearne, murray, oliver, & human, 1997; national research council, 2001; silver & stein, 1996; stein, grove, & henningsen, 1996; wood, williams, & mcneal, 2006), which, in turn, would increase their proficiency at responding to open-ended prompts, akin to those on the state standardized tests that often require higher levels of reasoning. we first discuss the research base for the mld program and describe its implementation. we then evaluate the effect of the mld project, analyzing student scores on preand post-assessments both for mathematical performance and for the development of students’ proficiency with academic language and justification. we conclude with a discussion of the promise of the program’s model and future next steps. focusing on a mathematics learning discourse we made several deliberate choices in developing the mld program. one choice was to focus on student discourse. this focus is appropriate for two reasons. first, language is the predominant medium by which students learn and demonstrate their understandings. language mediates learning (vygotsky, 2002). verbal discourse in classrooms (supported by symbolic representations, visuals, hands-on materials, etc.) is used to introduce students to mathematical ideas and provide opportunities to make sense of these ideas. how a teacher organizes her or his instruction to provide students access to, and opportunities for, meaning making and concept development is crucial to student learning. acknowledging the centrality of language, we take the stance that it is through participation in practices such as justification and argumentation that students might expand their mathematical knowledge. second, the focus on student discourse is appropriate because it is not uncommon in urban settings to have students who are at various levels of language staples & truxaw mld project journal of urban mathematics education vol. 3, no. 1 29 proficiency including english language learners (ells). in the school district of focus, nearly half of the public school students spoke a language other than english at home (connecticut state department of education, 2008). as part of a needs assessment done prior to the project, teachers who were interviewed remarked that language issues affected their students’ performance on statemandated mathematics assessments. as noted in the literature, many ell students have mastered conversational english, but have little exposure to academic language (cummins, 2008). thus, they face the double challenge of mathematics and language as they work on open-ended prompts. the development of students’ academic language is a central function of schooling (schleppegrell, 2007; zwiers, 2008). consequently, specific instructional practices within these schools, and other urban schools with similar characteristics, should support ells and bilingual students (dalton & sison, 1995). the mld project: research basis and rationale the mathematics learning discourse (mld) project was undertaken during the 2007–2008 school year. in a 3-day summer workshop, a group of teachers from two urban public schools was introduced to the idea of a mathematics learning discourse. we presented this idea in terms of a model with three pillars (see figure 1)—a model developed through a review of relevant research literature. we explored each pillar with the teachers through a series of activities and discussions. given the research literature, it was expected that teachers who organize classrooms characterized by the three pillars might prove to be more effective with their students. mathematics learning discourse appropriate and effective development of students’ academic language student engagement in mathematical practices of justification and collective argumentation access for all students to rigorous mathematics on some level figure 1. three pillars of mathematics learning discourse. appropriate and effective development of students’ academic language. a key aspect of developing students’ academic language is to create a bridge from staples & truxaw mld project journal of urban mathematics education vol. 3, no. 1 30 everyday, informal language toward academic language and use of the mathematics register (halliday, 1978; pimm, 1987; zwiers, 2008). too often, attention to language in mathematics classrooms focuses almost solely on vocabulary. language-related instruction should move beyond simple vocabulary; it should include attention to how language is used to express mathematical ideas (functional linguistics) and the development of the mathematics register (i.e., language associated with the meanings of mathematics) (halliday, 1978; moschkovich, 2002; pimm, 1987; schleppegrell, 2007). this goal is pertinent for all students, but particularly so for students whose first language is not english (cummins, 2000; schleppegrell, 2007; valdés, bunch, snow, lee, & matos, 2005). these students may be socially fluent, yet may need strategic linguistic support for engaging cognitively challenging mathematical tasks (janzen, 2008), justification and higher order thinking. many classrooms, however, do not support such practices. teachers remain unaware of the language demands involved in learning mathematics, especially as they pertain to justification and higher order thinking (adler, 1999; moschkovich, 2002; pimm, 1987; valdés et al., 2005). the effectiveness of attending to language development has been demonstrated by many researchers, including those who developed and implemented the siop® model1 (echevarría, vogt, & short, 2007, 2010), which became one central feature of working on academic language with this group of teachers. we drew on siop® strategies and techniques because they have been proven effective for supporting the teaching of academic content to ells (echevarría, short, & powers, 2006). student engagement in mathematical practices of justification and collective argumentation. student participation in justification, meaning making, and argumentation have been implicated as critical components for supporting students in learning mathematics (see, e.g., hiebert et al., 1997; national research council, 2001; silver & stein, 1996; stein, et al., 1996; wood et al., 2006). there is also some evidence that participation in these practices is particularly effective in supporting the learning of lower attaining students and/or ells (boaler & staples, 2008; moschkovich, 2002). by justification and argumentation, we mean engaging students in the processes of sense making (hiebert et al., 1997) and having them offer claims, supported by evidence and warrants (toulmin, 1958) in order to support a result and convince others of the claim’s validity. access for all students to rigorous mathematics on some level. in addition, teachers need to ensure that all their students, who vary in their prior mathematical background, language ability, and other characteristics, have access 1 siop®, formerly known as the sheltered instruction observation protocol, is an instructional approach that offers teachers a framework for planning and implementing high quality instruction for english language learners. staples & truxaw mld project journal of urban mathematics education vol. 3, no. 1 31 to participate in the lesson’s mathematical activities (goodlad, 1994; michelli, 2005). inequitable access and participation to mathematics during class leads to inequitable learning opportunities and learning gains (cohen, lotan, & leechor, 1989; cohen & lotan, 1997; gee, 2003; martin, 2003). components of access for all were conceptualized to include developing productive classroom norms for discussion (yackel & cobb, 1996; wood, 1999) and groupwork (cohen, 1994a, 1994b), using manipulatives, and designing tasks that allow a range of students access to engaging the task. these tasks were multi-dimensional (cohen, 1994a; lotan, 2003) and cognitively demanding (stein et al., 1996). in addition, access for all attended to explicitly teaching what a good justification looks like and providing formative feedback (black & wiliam, 1998) to support student learning. the mld project: development and practices in the summer of 2007, a group of teachers from an urban school district participated in 3 days of summer professional development (pd) that focused on the three pillars. teachers participated in a range of activities and discussions related to the following: strategies to support language development, with special attention to ell students (echevarría, vogt, & short, 2007, 2010); analysis of cognitively challenging tasks (stein, smith, henningsen, & silver, 2000), including language demands related both to making sense of the task and offering justifications; and strategies to support engagement and access for all students (e.g., strategic groupwork, formative feedback, etc.). for example, we analyzed the cognitive demands of tasks by adapting an activity from stein et al.’s (2000) implementing standards-based mathematics instruction, and discussed various task features that could be modified to ramp up the cognitive demands (contrasted with those that made a task more complicated or harder to access). before this activity, we discussed what higher order thinking meant—a discussion we revisited throughout the summer sessions and academic year. focusing more on language, a colleague in bilingual education worked with the group to introduce them to the elements of the siop® model—for example, explicit inclusion of language objectives (along with content objectives) within mathematics lesson plans. additionally, she guided the group in analyzing the linguistic demands of open-ended prompts from our state assessments. we also introduced and modeled strategies such as math talk moves (chapin, o’connor, & anderson, 2003) and analyzed and discussed transcripts and videos from mathematics classrooms (boaler & humphreys, 2005) considering how the classroom discourse may promote conceptual understanding and higher order thinking. additionally, teachers participated in and reflected on cooperative problem solving activities that were strategically designed and implemented to enhance ac staples & truxaw mld project journal of urban mathematics education vol. 3, no. 1 32 cess for all in mathematics classrooms. these activities pressed for higher order thinking and included the use of “check points”—points in the task where the teacher is called over and all members of the group need to be prepared to explain the work and respond to questions. these check points provided an opportunity for formative assessment (as the teacher interacted with the group) as well as language development. the summer experience was followed by ongoing collaborative work across the course of an academic year involving the development, implementation, and debriefing of higher order thinking (hot) mathematics lessons. the hot lesson plans incorporated pedagogical strategies related to each of the three pillars (e.g., content and language objectives, verbal and written discourse that provided opportunities for higher order thinking and justification, and strategic support to allow all learners to engage in meaningful mathematics). the collaborative teams included teachers, university teacher educators/researchers, and preservice mathematics teachers who completed internships in the schools. the collaborative meetings took place weekly (planning one week and debriefing the next), and the hot lessons occurred approximately twice each month lasting about 1 hour each. the hot lesson plans were archived for public use (see the following website for archived lessons: http://www.crme.uconn.edu/lessons/). along with developing, implementing, and reflecting on hot mathematics lessons, the teachers were encouraged to infuse the three pillars of mathematics learning discourse in their everyday teaching practices. (for examples of practices associated with each of the pillars, see truxaw & staples, 2010.) the mld project: research to document the possible impact of the project and evaluate the potential of the underlying model, we address the following research questions: 1. what was the impact (if any) of the mathematics learning discourse project on student performance on open-ended math prompts? 2. what was the impact (if any) of the mathematics learning discourse project on student demonstrated proficiency with academic language and mathematical justification? in examining these questions, we focus primarily on student learning data. these research questions allow us to address a broader question of interest: does the mld model seem to hold promise as an approach to professional development in urban schools? in referencing the mld model, we intend to indicate both the conceptual underpinnings for the “content” of the work (the three pillars), and the program’s structural design (summer sessions with yearlong follow up in collaborative teams). we first report on findings related to the two research questions. staples & truxaw mld project journal of urban mathematics education vol. 3, no. 1 33 we take up the question of the model’s promise in the discussion and implications section. the project took place in five classrooms of four teachers in two urban schools (one k–8 school and one high school) in connecticut—one grade 4, one grade 5, and three grade 9 classes (one teacher had two sections of algebra and one had one section). the schools had partnered with the researchers’ university on other projects so there was already some level of rapport established. the grade 4 and 5 teachers had 18 and 29 years of teaching experience, respectively, and the grade 9 teachers had two and three years of teaching experience. the teachers volunteered to be involved with the project and received a small stipend for their participation. in the focus school district, 95% of students qualified for free or reducedpriced meals and 94% of the students were categorized as “minority” students. additionally, at the two schools more than half of the students spoke a language other than english at home (52% at the k–8 school and 71% at the high school) (connecticut state department of education, 2008). data sources. the principal source of evidence to address the research questions examined in this article was student preand post-assessment performance data on open-ended mathematics prompts from participating teachers’ classrooms as well as other classes within each school. for each of the grade levels, the prompt was a released or sample item from the state tests that required higher order thinking and/or a justification of the student’s response. the state assessments include two open-ended prompts for grades 3–8 and four open-ended prompts for grade 10. students were given up to 30 minutes to complete the pre and post-assessment prompts. all prompts were embedded in some context (as this is the priority of the state). we also administered a reflective survey after students completed the assessments. we asked students to restate the problem in their own words, circle confusing words and phrases, and identify other aspects of the problem that were confusing or difficult for them. to gauge whether changes in students’ performance in these classes were beyond what could be expected in a typical year, we collected data from the same single-item preand post-assessment from other classes at these schools. for grade 9, we collected data from eight other ninth-grade classes. for grades 4 and 5, we administered the same prompts in the project teachers’ classes the year prior to their involvement with the project. we chose to use prompts from the state assessment for several reasons; most notable of these was the tremendous pressure on teachers to improve student performance on such assessment. the teachers also identified these prompts as challenges for their students. for instance, in 2008, 40% of 8th graders across the state achieved “mastery” on this component of the state tests; in the urban school district of focus, only 12% of 8th graders achieved mastery. in addition, these staples & truxaw mld project journal of urban mathematics education vol. 3, no. 1 34 prompts are language intensive (generally both reading and writing are necessary) and require higher order thinking. thus, they aligned with the goals of the project. although the analysis for this article focuses on the student performance data, other data collected for the project include: materials and field notes from the summer pd; hot lesson plans; student work samples from hot lessons; audiorecordings and field notes of lesson planning and debriefing sessions as well as the implementation of hot lessons; and teacher interviews. data analysis. we analyzed the preand post-assessments from multiple perspectives. prompts were scored using the state rubrics for open-ended prompts; two trained scorers independently scored each prompt. if scores differed, another scorer scored the prompt. the scores were analyzed using descriptive statistics, as will be presented in the findings section. the analyses we conducted varied by grade level depending on the data available. for the ninth-grade classes (two teachers), we made two main comparisons. first, we considered all students who completed the prompt and compared the scores of students in mld classes with those in non-mld classes. second, we considered only students who were in mld classes all year and compared their results with those students who were in non-mld classes all year. this reduced our sample size, but may provide a more accurate picture of the possible effect of the project. for grades 4 and 5, we did not have a large comparison group. rather, we compared end-of-year scores on identical prompts for the mld teachers’ classes from the year prior to the project and mld teachers’ classes for the project year, allowing for group-level comparison between the two classes. to directly target student academic language and justification, we developed a rubric to score student work samples by applying research literature related to argumentation and justification (e.g., healy & hoyles, 2000; toulmin, 1958) and academic language/mathematics register (e.g., pimm, 1987; schleppegrell, 2007). the initial rubric described different levels of proficiency with respect to two categories: use of academic language and argument/justification. for argument/justification, we used as a working definition: the process of sense making (hiebert et al., 1997) to remove doubt about a claim using logical reasoning, including evidence of claims, warrants and evidence (toulmin, 1958). for academic language, we used the working definition: language appropriate to communicate the mathematics involved in the context of the problem/situation, including processes, properties, functions and relations (halliday, 1978; pimm, 1987). academic language entails use of vocabulary that is important for expressing ideas precisely and mathematically, as well as use of appropriate sentence structures, and so forth, that are needed to express mathematical ideas (e.g., generalizations, justifications, identification of a counterexample, etc.). we made some adjustments to the rubric as we applied it to student work samples. for example, we realized that the overlap of justification and academic staples & truxaw mld project journal of urban mathematics education vol. 3, no. 1 35 language was quite extensive on many prompts, as expressing a justification relied upon using language to express causal relationships and inferences. this overlap was particularly extensive with contextual problems that required little in terms of specific mathematical terminology; therefore, most of the academic language required to respond to the prompt was related to expressing the justification. we modified the rubric (see table 1) to indicate this overlap by showing a middle band that we could not attribute to either academic language or justification alone. the use of specific mathematical terminology (e.g., product, polygon) was attributed exclusively to the use of academic language category. the types of inferences and soundness of the reasoning students used was attributed exclusively to the argument/justification category. table 1 academic language and justification (alj) rubric score 3 2 1 0 use of academic language student uses appropriate mathematical terminology consistently (e.g., “equals” vs. “makes”) student uses appropriate mathematical terminology, as required by prompt. few uses of mathematical terminology are present, as required by prompt. student work reveals little or no command of academic language or use of mathematical terminology. use of academic language and argument/ justification student work reveals appropriate words and/or phrases to indicate logical connections and relationships; claim is expressed for all relevant cases. work reveals some appropriate indicators of logical connection and/or relationships; claim may or may not be expressed for all relevant cases. work demonstrates challenges with articulating logical connection and/or relationships; claim is not expressed for all relevant cases. work reveals little or no language that describes requisite logical connection or relationships; claim is not expressed, or not expressed for all relevant cases. argument/ justification claim holds for all relevant cases; argument demonstrates validity for all relevant cases; student offers a justification that includes a claim and explicitly identifies evidence, as well as the logical connection between the claim and evidence. student offers a justification that includes a claim and/or a warrant and/or evidence; connection between the claim and evidence is partially articulated; justification may or may not hold for all required cases. student offers a claim and may have work that supports the claim, but the student does not make the connection between these explicit. student does not produce work that includes a justification or claim, or student only offers a minimal response as a claim that could be seen as a guess. for each task, we operationalized the rubric based on the demands of the task and nature of required justification. refinements were made as we considered student work samples, and then rescoring was done using the refined criteria. based on the demands of the particular prompts we used, we opted to give a single score (rather than separate scores for each category). thus we scored holistically but accounted for all three categories of the rubric, as necessitated by the staples & truxaw mld project journal of urban mathematics education vol. 3, no. 1 36 task. twenty percent of prompts were double scored to ensure consistency across scorers, and one scorer scored the remainder. limitations. although the results we share are promising, we raise a caution to the reader that these results suggest value in this model and approach, but are not definitive. more work needs to be done. the results reported here are based primarily on a single prompt, albeit open-ended and requiring extensive thought and one that is drawn from the state assessment program and consequential for students’ performance on state tests. any one prompt may have unique features that may unknowingly impact the particular results. for example, solano-flores and trumbull (2003) found that, for ells, “each [assessment] item poses a different set of linguistic challenges” and that “ell performance varies considerably not only across items but also across languages” (p. 8). similar results may also hold for non-ell students who are developing their academic language and proficiency with justification. with respect to gauging students’ proficiency with academic language and with justification, it is important to note that content understandings are a confounding factor for any score related to justification or academic language (just as academic language is a confounding factor for students’ demonstrated proficiency with mathematical content on any question that requests a justification). without some level of content understandings and ability to read and comprehend, it is impossible for a student to demonstrate her or his level of proficiency with academic language or justification on these contextual prompts. thus, if a student leaves a prompt blank, it may be that the student does not understand the material or that the student could not access the problem (reading comprehension) and determine the mathematical work required by the prompt. this complexity is near impossible to sort, and is one reason why it is so challenging for teachers to focus on student growth in these areas, which we discuss later. findings in this section, we present evidence of student improvement in demonstrated proficiency regarding both content and academic language/justification, and subsequently argue for the value of this model for professional development to support students’ engagement in higher order thinking and the development of their academic language. student mathematical performance prompt score gains. students in participating classrooms in grades 4, 5, and 9 demonstrated improved performance on open-ended prompt scores. a score of 2 or 3 was considered in the “mastery” range (a term used by the state). across the staples & truxaw mld project journal of urban mathematics education vol. 3, no. 1 37 mld classes, the level of mastery increased overall. additionally, there were marked decreases in scores of 0 across all three grade levels. we review the results for each grade level and offer evidence that the improvement was greater than would be expected without the professional development program. grade 9 results – state scoring rubric the following prompt was administered in the ninth-grade classes, which included the three mld classes and eight other ninth-grade classes for purposes of comparison: for an original graphic design, lee charges a fixed fee of $50 plus $25 for each hour that he works. his main competitor charges a fixed fee of $40 plus $30 for each hour that he works on a design. lee’s competitor advertises that his rates are cheaper. is lee’s competitor correct? explain your reasoning. (the grid is provided in case you decide to use a graph as part of your explanation.) remember to show your work. (2003 released item, connecticut state department of education, 2009) this prompt requires students to understand the fee structure for lee and his competitor and to determine a way to assess the validity of the competitor’s claim. a full analysis reveals that lee is cheaper for any job that takes longer than 2 hours; his competitor is cheaper for a job that takes less than 2 hours; and they charge the same amount at 2 hours. students can solve this problem in a wide variety of ways, including graphing, using equations, generating specific points, and analyzing the relative rates of change. we observed all strategies being used. table 2 reports the results of the fall administration (pre-assessment). these data show that the students across the ninth-grade were not faring well on this prompt and that students in the ninth-grade mld classes were generally doing more poorly. eighty-eight percent of the students scored a 0; only 4% were considered at mastery level. table 2 ninth-grade student scores on open-ended prompt, fall (pre-assessment) number of students score percent mastery fall 0s 1s 2s 3s mld classes 49 43 (88%) 4 (8%) 0 (0%) 2 (4%) 4% non-mld classes 131 99 (76%) 14 (11%) 10 (8%) 8 (6%) 14% all ninth-grade classes 180 142 (79%) 18 (10%) 10 (6%) 10 (6%) 12% staples & truxaw mld project journal of urban mathematics education vol. 3, no. 1 38 table 3 reports the post-assessment scores showing the improvement of the mld classes and the relative steadiness of the results from the other ninth-grade classes. whereas the number of 0s in the mld classes decreased dramatically from 88% to 47% and the percent mastery increased from 4% to 31%, the distribution of scores of other ninth-grade students remained approximately the same between fall and spring. note that the attrition/retention rates for the mld classes are similar to those of the larger sample of ninth-grade students. table 3 ninth-grade student scores on open-ended prompt, spring (post-assessment) number of students score percent mastery spring 0s 1s 2s 3s mld classes 38 18 (47%) 8 (21%) 5 (13%) 7 (18%) 31% non-mld classes 100 71 (71%) 14 (14%) 10 (10%) 5 (5%) 15% all ninth-grade classes 138 89 (64%) 22 (16%) 15 (11%) 12 (9%) 20% we also considered the scores of the subset of students who completed the prompt in both the spring and the fall (30 students in the mld classes and 76 students in non-mld classes, for a total of 106 students). the data showed the same trends held when considering only that subset of students, although the overall results of this subset of students were slightly better, perhaps not surprisingly. overall, these are positive results for the ninth-grade mld classes. as we conducted our scoring, we looked for evidence of growth of academic language and student justification that might be captured by using the state rubric. we found two indicators that students in the mld classes were outperforming those in the non-mld classes with respect to justification and academic language. first, students in the mld classes were more likely to maintain a connection between the mathematics they were doing and the context of the prompt, and to directly relate their mathematical work to making an argument about the competitor’s claim. for example, in the spring, there were similar percentages of students in the mld and non-mld classes using algebra, 33% and 35%, respectively. this method is relatively sophisticated, requiring students to write two functions modeling the two different price schemes. students can then generate a table of values, or can solve the system of equations (to find where the costs are equal). to successfully answer the prompt question, students must make a connection between the solution of the system and its meaning in context of the problem. many non-mld students offered no (or an erroneous) interpretation of the meaning of staples & truxaw mld project journal of urban mathematics education vol. 3, no. 1 39 the results of their solution method. they simply wrote x = 2 (the correct result obtained when solving the system of equations). in the mld classes, more students connected their mathematical work with the context and offered an interpretation of their results. of students using algebra, 46% of the mld students received a score of 2 or 3, compared to 24% of those in the non-mld classes, a score that could only be obtained with some interpretation of the results in the context of the problem. the second indicator was whether a student attempted a written response. a blank response is often a sign of being overwhelmed or not being able to comprehend the word problem. thus, a shift from a blank paper (or the response “i can’t do this”) to some writing likely indicates increased comprehension of the problem. we conducted a simple count on the number of students who left the problem blank (no writing and no calculations). the percentage of students leaving the problem blank decreased for both mld and non-mld classes, but considerably more for the mld classes. for the non-mld classes, 21% left the problem blank initially and 18% left the problem blank at the end of the year. for the mld classes, the initial percentage was 29%, and this was reduced to 10% at the end of the year. for the 30 students in the mld classes who completed both the spring and fall prompt, 37% (11 of 30) offered some written explanation in the fall and 70% (21 of 30) offered some written explanation in the spring, nearly double the fall numbers. although it is difficult to determine whether we should identify this result as evidence of increased proficiency with academic language, we take it as a potential positive indicator of language use. grade 9 results – academic language and justification scores to explore changes in students’ proficiency with academic language and justification in a more targeted way, we applied the academic language and justification (alj) rubric (table 1) to the graphic design prompt. we scored the prompts of the 30 mld students for whom we had both preand post-scores. one important challenge that emerged related to tracking students’ academic language was the nature of the graphic design problem itself. this particular prompt required academic language to structure a justification appropriately (stating claims, warrant and evidence and linking those) and to use language of comparison (which strongly overlaps with everyday language). there was little need, however, for students to use mathematical vocabulary (an aspect of academic language), perhaps in part because the prompt is couched in a “real-world” context. thus, our scoring was based primarily on two of the three components—the one that indicates the overlap of justification and academic language, and the justification category. for example, a student could write, “i think lee’s competitor is wrong. for 5 hours, lee’s competitor charges more. see my table below.” staples & truxaw mld project journal of urban mathematics education vol. 3, no. 1 40 mathematically, this is a complete justification. however, such a justification does not require much use of mathematical academic vocabulary. as a second example, a student might write, “my table shows that lee is more expensive for 1 hour. they charge the same price for 2 hours. lee is cheaper for more than 2 hours. so lee’s competitor is not correct.” we would like to highlight one difference with the state rubric that we found particularly interesting. the prompt asks whether the competitor’s claim is correct. from a mathematical standpoint, as well as legal, the student needs to find only one counterexample to lee’s competitor’s claim that his (the competitor’s) rates are cheaper to prove the statement false. a few students in the spring indeed found one counterexample (e.g., at 3 hours) and claimed lee’s competitor was incorrect. our interpretation of the state scoring rubric, however, indicates that the assessors would award 1 out of 3 points for this response. the point would be awarded for showing appropriate calculations for one data point. although this argument structure (specifically, showing one counterexample to prove a statement false) is mathematically sound, and potentially reflects a sophisticated understanding of the problem and role of a counterexample, the state required students to offer a full analysis of the situation (including identifying the “breakeven” point) to receive a score of 3. the problem, however, does not explicitly indicate this requirement. thus, we felt that some students who had a sophisticated mathematical understanding may have scored low on the state prompt. the justification rubric, however, counts this strategy as a valid approach to justifying one’s response that the competitor is not correct. table 4 reports sample student responses at each of the scoring levels, along with a commentary that indicates some features noted for determining the score. table 4 scoring examples for the graphic design academic language and justification rubric (grade 9) score examples commentary 0 a. left blank. b. shows only an equation for each cost structure. c. “lee’s competitor is right”; no other work is shown. these responses provide no indication that the student formulated an argument or could express it. 1 “the competitor is right.” shows computations of 40+30 = 70, labeled competitor. shows computations of 50+25 = 75, labeled lee. offers claim and evidence. evidence is not linked to claim; warrant is implicit. no use of academic language such as “because.” staples & truxaw mld project journal of urban mathematics education vol. 3, no. 1 41 2 “his rates would be even, it depends on each hour. so each one is not cheaper neither the highest amount. they both are the same amount counting each hour.” $50 + $25 + $25 = $100 $40 + $30 + $30 = $100 claim is implicit (the competitor is not correct); includes a warrant (the rates can be even) and offers evidence (showing computations) linking them by saying “they both are the same amount.” includes some indication that who is cheaper varies by the hours worked (“it depends”). 3 “no lee’s competitor is not correct because if it takes him three hours or more the price is more expensive than lee’s.” work shows table for hours and dollars for lee and competitor, with all values labeled. offers a claim, a warrant (there are jobs for which the competitor is more), and points to the evidence (“if it takes him 3 hours or more” and includes a table). logical connectors used (because) and evidence explicitly linked. the subordinate clause (if it takes…) specifies the domain for which lee’s competitor costs more (3 hours or more). table 5 reports the students’ score results using the alj rubric for the 30 students in the mld classes who completed both fall and spring prompts. in this group of students, we see a clear trend towards higher levels of demonstrated proficiency with academic language and justification. in the spring, students were more successful in constructing a valid and complete argument and in using appropriate language to present their argument and response to the question. as noted, content understandings are required to demonstrate proficiency in these areas; therefore, some of the increase in these scores is likely a product of improved content understandings. similarly, changes in scores could reflect an increase in reading comprehension and understanding particular language such as “fixed fee,” which surveys revealed many students did not understand. we cannot identify all the factors that contributed to this change. we certainly expect an influence by the work the mld teachers were doing with their students. it could also be, however, that students’ exposure to other learning opportunities, such as an outstanding english course, had a considerable effect. table 5 ninth-grade student academic language and justification scores on open-ended prompt (preand post-assessment, n = 30) score 0s 1s 2s 3s mld classes pre-assessment 17 (57%) 8 (27%) 3 (10%) 2 (7%) mld classes post-assessment 9 (30%) 6 (20%) 5 (17%) 10 (33%) to see if we might more specifically identify and describe the nature of changes between students’ fall and spring prompts, we also examined each pair of student responses side by side. for the most part, we found that we were unable to staples & truxaw mld project journal of urban mathematics education vol. 3, no. 1 42 make definitive judgments about whether differences noted in a student’s responses between the fall and the spring offered specific evidence of a change in their proficiency with the practice of justifying, their use of academic language, or their mathematical understandings. these areas are naturally interdependent and our data were not sufficient to allow us to parse these components for all students. there were some instances, however, where, due to the particular nature of the responses, we felt that we could make a claim about one of these three areas. we offer some examples and analyze the nature of the changes we observed. in the following example, we argue that we can see growth in a student’s use of academic language in producing a justification. below is the written component of the student’s fall and spring responses: fall: his rate is cheaper cause if you add the fee and 1 hour pay his charge of money is larger. spring: so lee’s competitor is not cheaper. it cheaper for the first hour. the second they are even. but the next hours lee is cheaper as you see on my graph…. (student 11103) in terms of language, we see that the first response does not explicitly name lee or the competitor. each is referenced once by the term “his.” thus, there is an increase in the specificity of references. the first response also does not offer a claim that directly addresses the question “is lee’s competitor correct?” whereas the second response does in the first sentence. mathematically, we see growth as well. in the fall, the student only considered what happened at 1 hour, despite seeming to understand (by the written response and her work) that there was an hourly fee. in the spring, the student analyzed the situation, fully considering all possible cases for the number of hours. in the second example, we see a difference in a student’s mathematics, which provides more evidence of the student’s academic language use. we are not certain, however, if there is evidence of growth in both of these areas, or just mathematics (which then provided us the opportunity to see more academic language). one interesting feature of the following example is that both the fall and spring responses have the same structure, namely, the student makes a claim, uses the linking word because, and then provides evidence. thus, we do not see a difference in how the student produces a justification. fall: no lee’s competitor is not correct because lee’s competitor is cheaper. (the student’s calculation demonstrates computations for a 1-hour job for each.) spring: lee’s competitor is not correct because when i did a line graph lee’s competitor line increased more than lee’s. (the student has a graph for lee and his competitor, demonstrating costs for hours 1 through 5.) (student 11109) staples & truxaw mld project journal of urban mathematics education vol. 3, no. 1 43 note that, mathematically, the first response is incorrect. the evidence does not support the claim. furthermore, the student only explored a 1-hour job. in the spring, however, the student produced a set of values and a graph, referencing the graph as well as a particular feature of the graph (the line increased more) as evidence to support his claim. through his academic language, we could “hear” him interpret his graph using mathematical language. grade 4 results – state scoring rubric to gauge impact on student performance in grade 4, we report on scores from identical fourth-grade prompts that were administered to the mld teacher’s non-mld fourth-grade class prior to the mld project (spring 2007) and to the mld fourth-grade class near the end of the project year (spring 2008). this comparison allows for a group-level comparison between the two classes. the prompt is shown in figure 2. hot dog buns you estimate that you’ll need 40 buns for a class picnic. hot dog buns are sold in packages of 8 and packages of 12.  the package of 8 costs $1.00  the package of 12 costs $1.20 a. show three different ways you could buy packages to get at least 40 buns. b. which packages would you buy if you wanted to spend the least money? show or explain how you arrived at your answer. c. which packages would you buy if you wanted exactly 40 buns? (sample item, connecticut state department of education, 2009) figure 2. fourth-grade prompt. this prompt required that students work through multiple tasks and constraints, including mathematical, contextual, and linguistic challenges. the language challenges included some unfamiliarity with “everyday” words and phrases (e.g., package and purchase), as well as other words that are germane to the mathematical work they are expected to engage. for example, phrases such as “at least” and “the least” might seem familiar, but may be misinterpreted or not understood, leading to very different mathematical work. in terms of mathematical content, students need to demonstrate facility with estimating, display fluency with addition and multiplication of whole numbers and money amounts expressed as decimals, and be able to compare and evaluate different solutions. staples & truxaw mld project journal of urban mathematics education vol. 3, no. 1 44 to achieve a mastery score on the prompt (score of 2 or 3), according to the state rubric, students needed to provide appropriate answers for at least 2 of the 3 parts to the prompt. for part a, they could show pictures, numbers, words, or diagrams to show the different ways to make at least 40 buns. for part b, it was reasonable to make comparisons from responses to part a. although the least cost overall is $4.40 (2 packs of 12 buns + 2 packs of 8 buns (2 x $1.20) + (2 x $1.00) = $4.40), given the grade level, it was considered acceptable if students compared from the three combinations made in part a, selecting the least of the three, or if they articulated a reasonable cost per item explanation. however, it was not acceptable if students simply stated that the packs of 8 buns cost less because $1.00 is less than $1.20 (the cost of 12 buns). for part c, the students needed to show a combination equal to 40 buns exactly (e.g., 5 packages of 8 buns = 40 buns). this prompt was selected because it was typical of those used in the state assessment; it required interpretation of everyday and academic language, and, although it did not strictly require them, it provided opportunities for written explanation and justification. table 6 reports the results of the administration of the prompt using the state scoring rubric for the classes of our participating fourth-grade teacher in the year prior to the project (spring 2007) and for the project year (spring 2008). although the two classes of students differed, the prompts were identical, both classes were heterogeneously grouped, and both classes were taught by the same teacher. for the mld class, scores were considered only for students who were in the class the full year. the spring 2008 (mld) class demonstrated greater mastery (43%) than the same teacher’s class prior to mld involvement (22%)—nearly double the rate of mastery of the students from the previous year. further, the mld class showed a lower percentage of scores of 0 (14% mld vs. 56% pre-project). the results suggest an impact of the mld project on student performance on open-ended prompts. table 6 fourth-grade student scores on open-ended prompt, spring (post-assessment) number of students score percent mastery 0s 1s 2s 3s pre-mld class (spring 2007) 18 10 (56%) 4 (22%) 3 (17%) 1 (6%) 22% mld class (spring 2008) 21 3 (14%) 9 (43%) 5 (24%) 4 (19%) 43% note: rounded to closest whole %; therefore, may not sum to exactly 100%. staples & truxaw mld project journal of urban mathematics education vol. 3, no. 1 45 grade 4 results – academic language and justification scores to explore growth in grade 4 students’ proficiency with academic language and justification, we adapted the general alj rubric for use with this grade level and for the specific prompt. the spring hot dog bun prompt had three parts (a, b, c). because part b (“which packages would you buy if you wanted to spend the least money? show or explain how you arrived at your answer.”) included the greatest opportunity for academic language and justification, it was the main focus of the alj scoring. indicators for the alj rubric were identified that related to this part of the prompt in particular, but that also considered the other two sections. for example, to justify part b, students typically needed to refer to work done in part a (“show three different ways you could buy packages to get at least 40 buns.”) and make an argument for which of their three ways was the least expensive. table 7 shows sample responses at each of the scoring levels, along with some commentary. table 7 scoring examples for academic language and justification rubric hot dog bun prompt (grade 4) score examples commentary 0 “i would buy the low price ones. this was my answer because i don’t want to waste my money.” little command of academic language; claim does not have a logical connection to question being asked; no relevant evidence or warrants. 1 “i would take the package of 8 because it cost $1.00 and $1.00 is less than $1.20.” includes a claim (minimal use of academic language), a “because” statement, but does not include logical connection to the context of the problem. 2 student shows computation of prices for two different combinations of packages. show prices $4.40 and $5.00 and says, “easy compare” and shows the claim, “2 of the 8 dog packs and 2 of the twelve” (as the least expensive). uses language—e.g., packs, compare, twelve. (in parts a and c, used “packages of.”) claim is shown (“2 of the 8 dog packs and 2 of the twelve”), though not clearly labeled as “least expensive.” evidence (student work) and warrants (“easy compare”) are shown. the reader needs to infer logical connections. 3 student refers to own work from part a, organized as 1, 2, and 3, presenting the claim, “my third idea” followed by “because it is $4.40 and my 2 and 1 idea are a higher price.” the student then lists the three prices, and circles the lowest of the three. demonstrates ability to use academic and appropriate contextual language to represent responses. claim is explicit (“my third idea” [is least expensive]), provides a warrant “because it is $4.40 and my 2 and 1 idea are a higher price” (compares with prices for other two cases), and evidence (refers to work showing prices for each case). table 8 reports the alj scores for the fourth-grade classes of our participating fourth-grade teacher in the year prior to the project (spring 2007) and in the project year (spring 2008). the mld class (those who participated the full year) staples & truxaw mld project journal of urban mathematics education vol. 3, no. 1 46 demonstrated higher percentages of mastery, scores of 2 or 3 (28%), as compared with the class before mld involvement (6%). further, the mld class demonstrated lower percentages of scores of 0 (14% mld; 39% pre-mld). these results suggest that, overall, the mld students demonstrated greater proficiency with academic language and justification than students who did not participate in the project—on the same prompt, at the same time of year, with the same teacher. interestingly, there was not a clear relationship between the state scoring and alj scoring. for example, of the fourth-grade students who achieved mastery (scores of 2 or 3) according to the state rubric, only 60% scored a 2 or 3 using the alj rubric—suggesting that the state rubric does not attend directly to academic language and justification, even on open-ended prompts. overall, approximately 23% of the students scored lower on the alj rubric than the state rubric, approximately 18% scored higher on the alj rubric than the state rubric. these results suggest that rubrics such as ours might be necessary to unpack students’ proficiency with academic language and justification. table 8 fourth-grade student scores for academic language and justification on open-ended prompt number of students score 0s 1s 2s 3s pre-mld class (spring 2007) 18 7 (39%) 10 (55%) 1 (6%) 0 (0%) mld class (spring 2008) 21 3 (14%) 12 (57%) 3 (14%) 3 (14%) note: rounded to closest whole %; therefore, may not sum to exactly 100%. grade 4 results –student perceptions of prompt challenges as noted in the data sources section, students were asked to reflect in writing about the prompts. in grade 4, student reflections revealed some awareness of contextual and linguistic challenges. for example, responses related to what was confusing included: “i was get confused when they said if you wanted to spend the least amount of money,” and, “another thing that was confusing to me is that the question was telling me stuff i couldn’t understand.” of course, not all students were able to recognize and/or communicate what they did or did not know. for example, many fourth-grade students who did not achieve mastery indicated that they did not find anything confusing about the problem (e.g., “nothing was difficult for me.”). others circled whole questions or the entire page. these are indicators that parsing out what was confusing seemed challenging for many of these students. this result is of concern given that students’ facility with unpack staples & truxaw mld project journal of urban mathematics education vol. 3, no. 1 47 ing academic and contextual language in problems can influence their ability to make sense of what is required of them mathematically. this concern may be particularly pertinent for ells who are socially fluent but have not yet achieved academic fluency, as ells’ social fluency may mask their need for linguistic support in academic contexts such as written prompts (cummins, 2000). grade 5 results – state scoring rubric to gauge impact on student performance in grade 5, we report on scores from identical fifth-grade prompts that were administered to the mld teacher’s non-mld fifth-grade class prior to the mld project (spring 2007) and to the mld fifth-grade class near the end of the project year (spring 2008), allowing for a group-level comparison between the two classes. the prompt is shown in figure 3. this prompt was selected because it had similar demands to those described for the fourth-grade prompt; that is, it included the need to consider and work through multiple tasks and constraints that included mathematical, contextual, and linguistic challenges. in order to achieve a mastery score on the prompt, according to the state rubric, students needed to demonstrate reasonable estimates for the number of burgers and rolls based on the information given (approximately 85 hamburgers and 65 rolls), along with determining how many of each item is “enough” to meet the estimates (should be equal to or greater than estimates—within a reasonable range). additionally, students needed to calculate the cost of the packages and total cost accurately based on the number of each type of package purchased. a limitation of the prompt in terms of the alj scoring was that, although it says to “show how you arrived at your answer,” academic language was not required. consequently, we were unable to apply the alj rubric to analyze the fifth-grade prompts. we report here only the scores from the state rubrics and some themes drawn from the students’ written reflections. table 9 reports the results of the administration of the prompt using the state rubric for the classes of our participating fifth-grade teacher in the year before the project (spring 2007) and for the project year (spring 2008). although the two classes of students differed, the prompts were identical and were taught by the same teacher. for the mld class, scores were considered only for students who were in the class the full year. the spring 2007 class (pre-mld) was ability grouped (high ability); the spring 2008 group (mld) was heterogeneously grouped. this “ability” grouping would suggest the likelihood that the pre-mld class’ scores would surpass those of the mld class. the mastery level of the mld class (43%) that was not ability grouped was higher than the same teacher’s class the prior year (33%) that was identified as a high ability group. further, the mld class showed a lower percentage of scores of 0 (13%) than the pre-mld class (29%). staples & truxaw mld project journal of urban mathematics education vol. 3, no. 1 48 hamburger rolls you are going to have a fourth of july picnic for friends and family. you estimate that:  25 people will have 2 hamburgers and 2 rolls each;  15 people will have 1 hamburger and 1 roll each;  20 people will have 1 hamburger and no roll each. hamburgers and rolls are sold two ways each: hamburgers rolls 8 hamburgers for $1.75 6 rolls for $0.75 12 hamburgers for $2.15 18 rolls for $1.80 use this information to order enough hamburgers and rolls for the people coming to your picnic. show how many packages of each size of hamburgers and rolls you will buy. compute the final cost of all the items. show how you arrived at your answer. items number of packages cost 8 hamburgers/$1.75 12 hamburgers/$2.15 6 rolls/ $0.75 18 rolls/$1.80 total cost: ____________________ (sample item, connecticut state department of education, 2009) figure 3. fifth-grade prompt. table 9 fifth-grade student scores on open-ended prompt, spring (post-assessment) grade 5 results – student perceptions of prompt challenges the students’ written reflections on the prompts revealed similar trends to those reported in grade 4. an interesting, though perhaps not surprising finding, number of students score percent mastery 0s 1s 2s 3s pre-mld class (spring 2007) 18 6 (29%) 8 (38%) 2 (10%) 5 (24%) 33% mld class (spring 2008) 23 3 (13%) 10 (43%) 6 (26%) 4 (17%) 43% staples & truxaw mld project journal of urban mathematics education vol. 3, no. 1 49 was that students who had difficulty restating the question in their own words (e.g., “i don’t know” or “it was about hamburgers and rolls”) tended to score below the mastery level; those who could restate the problem (e.g., “i think the main idea was to get the amount of food they need but don’t need to be perfect” or “to find out how many packages of each size of hamburgers and rolls that you will need to buy”) tended to perform well overall. again, we see connections between language and mathematical performance. being able to make sense of the written text and unpack the purpose of the problem was a challenge for many students and critical to performance. discussion and implications promise of the model this small-scale implementation and study of the mathematics learning discourse project offers promising results. overall, we take these results to indicate that this model for promoting a mathematics learning discourse, grounded in the three pillars, is a productive one. students in mld classes demonstrated improvement in their proficiency responding to open-ended prompts beyond what was typical for their teacher or school. analyses that focused specifically on academic language and justification also indicated improvement. although we cannot claim that the mld project was solely responsible for the observed differences, or that the model can be successful in all environments, urban or otherwise, at any scale, it is reasonable to assert that the teachers’ participation in the program likely had an impact on student performance and that further exploration of the model’s effectiveness is warranted. in this discussion, we reflect on some of the components of the program that seemed critical given that these were teachers working in an urban setting, as well as some of the contextual factors that seemed to support this successful case. as we reflect on the model and its value, we are drawn to its emphasis on promoting the development of academic language, which, when coupled with justification, seems to address an important gap—especially for mathematics instruction in urban schools with linguistically diverse students. language is a critical part of mathematics teaching, learning, and assessing. the results from the project, as well as informal conversations with the teachers, indicate that this focus was productive in bringing to the fore ideas related to language that extended beyond the learning of vocabulary words and low-level drilling of computation, aspects perhaps over-emphasized out of a lack of know-how for engaging students in other kinds of more cognitively and linguistically challenging mathematical activities. as noted earlier, before the project, we had the opportunity to conduct a needs assessment with approximately 10 teachers at one of the participating staples & truxaw mld project journal of urban mathematics education vol. 3, no. 1 50 schools. teachers focused their discussion of the language challenges on students not knowing enough vocabulary words. this “deficiency” was important to them because they felt that their students often missed an open-ended prompt on the state assessment because they did not know specific vocabulary like museum or scuba diver. in addition to highlighting the critical nature of this work, it indicates how significant a shift it may be for teachers of mathematics to make as they begin to see the full scope of the role of language in teaching and learning mathematics. the work of language and mathematics extends well beyond vocabulary—whether everyday or mathematical terminology—and includes engaging students in expressing mathematical ideas and core mathematical practices such as justification. the model’s focus on higher order thinking has also seemed to play a critical role. rather than focusing narrowly on more routine skills, as is often the case in urban schools (anyon, 1997; keiser, 2005), and which was certainly emphasized in the schools we worked with as well, this project called attention to higher order thinking that may expand students’ opportunities to learn to think mathematically. principals and curricular directors prioritize as they try to respond to the pressures of high-stakes tests. we have heard numerous times that, in some schools, students’ scores on the open-ended response items (a relatively small portion of the test in elementary grades) are so low, that teachers are told to not spend (i.e., “waste”) time trying to address these areas. the “cost” to produce a measurable change is too high. the mld project’s design allowed the teachers to not only focus regularly on higher order thinking but also to have resources and a community with which to work in order to build their capacity to design and implement these lessons. there were contextual factors that seemed to support the success of the mld project. first, prior partnerships between the schools and university likely facilitated aspects of the project. this familiar relationship also gave us the opportunity to place three teacher education interns at these schools. the interns provided additional classroom support in the mld classrooms, for example, organizing manipulatives, collaborating on the lesson design, or helping with responding to student work. in the context of the teachers’ incredibly full work lives, these contributions helped keep the core idea of the mld project on their radar. second, though receiving a small stipend, these teachers volunteered for the project. thus, they had some prior level of commitment for engaging in this kind of work. finally, the fact that the state-mandated assessments included open-ended prompts that required language and higher order thinking provided an “in” for the project in urban schools where “adequate yearly progress” is central to the thinking of administrators; not all states include similar items on their mandated assessments. staples & truxaw mld project journal of urban mathematics education vol. 3, no. 1 51 future steps overall, this project’s implementation has led us to conclude that the model for promoting a mathematics learning discourse, grounded in the three pillars, provides beginnings from which to build. although the mld project was a successful case, there is still much room for improvement as the majority of students still were not at mastery level. there are many challenges still to be addressed as well. for example, although not a main finding of this project, we did identify that some students struggled to articulate what they did and did not understand when reading an open-ended prompt (and subsequently deciding what mathematics to do). as part of the mld project, we did very little with teachers regarding reading strategies and how students make sense of open-ended prompts. rather, we focused on building background with students before engaging them in such a prompt and student production of mathematical language (verbal and written) and arguments. such reading and comprehension activities seem important to attend to in moving forward as a key piece of students’ success is interpreting from a word problem what mathematics one is to do. for example, following work done by ratner and epstein (2009), we could incorporate activities where a teacher asks a student to follow a think-aloud protocol as the student reads through an open-ended response, pausing to comment on what she knows, is questioning, or thinking about. this kind of activity could reveal to the teacher more about how students approach and make sense of these prompts, and how they then infer the nature of mathematical work they need to do. such information could then guide the teacher in planning targeted instructional activities. another challenge is that we need to better understand the key components of teacher learning with respect to this model and the project’s design so that we can scale up the intervention. in this first year, we were able to work closely with this group of teachers and we had additional school-based support with teacher education interns. such a model is likely too human-resource intensive to replicate on a larger scale. based on follow-up work, we suggest that it may be possible to scale up the work through a combination of professional development and grade-level team collaboration (e.g., using a modified lesson study protocol) that support teachers’ sustained work on these issues with a lower level of additional personnel support (staples & truxaw, in press). the effort must be one of overall capacity building for a building-based community. in conducting this work, we have become acutely aware that, as a community of education researchers, we need to develop better frameworks and methods for understanding and capturing growth in academic language in mathematics and proficiency with justification. if teachers are expected to actively promote the development of student proficiency in these areas, they need to have concrete ways to describe and understand proficiency at any moment, as well as ways to track staples & truxaw mld project journal of urban mathematics education vol. 3, no. 1 52 growth over time. this issue is also one for researchers, as we need to find ways to do the same kind of documentation and description of growth. the alj rubric developed for this project is perhaps a productive first step. once adapted to a particular task, we found it useful for focusing our attention beyond students’ content understandings. as we adapted the rubric to a task, we found it made us carefully examine the actual demands of the task and more fully deliberate what would be considered a complete justification and the range of ways students might approach a justification. we also had to look at the opportunities there were to express mathematical ideas and use academic language and compare that to what was required of the prompt. although we did not directly engage teachers in using these rubrics for scoring, we have worked with teachers in pd sessions on identifying claims, warrants and evidence, and whether students articulate the links among these. these efforts seem like a productive route to continue to pursue. future work should explore how to generate teacherand/or student-friendly versions of such rubrics. thinking about change over time, however, may require different tools. we found that the set of tasks that teachers developed and implemented over the course of the year varied greatly in their demands and opportunities to demonstrate proficiency with language and justification. consequently, it was quite challenging to understand whether variation over time was because of the task demands and opportunities or because of changes in students’ proficiencies. the variation in the demands of the tasks seemed to make comparisons challenging. for example, the competitor’s claim in the graphic design prompt can be shown false with one counterexample. this argument structure is very different from other standard justification tasks, for example, where students might be asked to generalize and offer a method for computing the perimeter of the nth figure in a given pattern. this latter type of task requires students to be able to express a claim about all relevant cases (a generalization) and offer a justification that likely requires coordination between a visual model and claim. these justifications make for very different mathematical work. to our knowledge, there are no frameworks that look at mathematics academic language development and/or the development of proficiency with justification. given the centrality of these two proficiencies to students’ success in mathematics (consequential for both understanding and ability to demonstrate understanding), the development of such a framework seems quite worthwhile. to address this issue, one route to consider is the development of tasks, or sets of tasks, that would vary in their demands, but which, when taken together, paint a more complete picture than is possible with any single prompt at one point in time. these tasks potentially could allow us to better identify whether an observed change is most related to a growth in mathematical understanding, language proficiency, or proficiency with justification. these more developed measures and staples & truxaw mld project journal of urban mathematics education vol. 3, no. 1 53 frameworks are also potentially valuable tools to support teachers as they work to improve students’ proficiencies by providing a clearer sense of the “target” and a sense of different levels of proficiency. at this point, we seem to have an underdeveloped sense of what developing proficiency might look like over time. thus, generating such tools might also require basic research into these areas as well. given the growing consensus that mastery of academic language is subject specific and critical to students’ education and future success (moje, dillon, o’brien, 2000; shanahan & shanahan, 2008), and given the influence of state assessments on teachers’ classroom practice (and administrator directives), we also think it would be constructive for the state assessment system to explicitly value academic language and practices such as justification. as noted in our data, we found that the state rubrics do not explicitly attend to the students’ use of academic language or expression of a justification. yet, at the same time, the openended prompts on the state assessments demand these on some level: the prompts are couched in a “real-world” scenario where students must read and interpret verbal and other information in order to make sense of the situation, determine the mathematics they must do to address the question, and make some kind of claim (supported by evidence) about the correct result. there is almost an irony here. an effort has been made to not judge students explicitly along these dimensions (academic language and justification), perhaps trying to not disadvantage students unduly for language competencies, yet the prompt inherently demands great attention to language—academic (explain how you know) and otherwise (vocabulary terms)—in order for the student to begin to engage in the mathematics purposefully. changes to the system could take the form of a rubric similar to the alj rubric or supporting materials that offer more explicit documentation of the demands placed on students for reading, interpreting, and responding to these openended prompts. it is important to stress that this call is not only about decoding or reading comprehension. inferring from a contextual problem what mathematics one must work on to address the problem is 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(2008). building academic language: essential practices for content classrooms. san francisco, ca: jossey-bass. microsoft word final waddell vol 3 no 2.doc journal of urban mathematics education december 2010, vol. 3, no. 2, pp. 116–154 ©jume. http://education.gsu.edu/jume lanette r. waddell is an assistant professor of the practice in the teaching and learning – mathematics education department in peabody college at vanderbilt university, 230 appleton drive, 261 wyatt, nashville, tn, 37203; email: lanette.r.waddell@vanderbilt.edu. her work and research interests include implementation and evaluation of professional development models in teaching mathematics with understanding for k–8 preand in-service teachers; using representations, tools, and technology to enhance mathematics teaching and learning; and the effects of culturally relevant pedagogies on teaching practices in mathematics. how do we learn? african american elementary students learning reform mathematics in urban classrooms lanette r. waddell vanderbilt university in this article, the author uses qualitative methodology to investigate how african american elementary students in an urban school engaged with a national council of teachers of mathematics standards-oriented mathematics curriculum and how their engagement converged with or diverged from the offered patterns of teaching practices in classrooms. the findings suggest that student practices converged with teaching practices that reflected the african american cultural dimension of social/affective interactions such as focused collaboration and active participation and diverged when students enacted practices that reflected expressive creativity and nonverbal interactions as with dramatic expression and improvisation. rather than looking at the divergent behaviors as social problems or behaviors needing remediation or punishment, considering what can be learned from these behaviors could enhance the mathematical identity and academic achievement of african american students. keywords: african american children, elementary mathematics, student learning, reform mathematics, urban education n the united states, the mathematical under-achievement of many african americans has been a source of concern for numerous reasons. studies have shown that mathematics achievement is affected more by the school environment (e.g., curriculum, teacher qualifications and expectations, materials) than by the home environment (lee, 1998; roscigno, 1998); that mathematics is critical in advancing to higher education due to its filtering effect (schoenfeld, 2002); and that mathematics is a civil rights issue in that, if children are not mathematically literate, they will be relegated to second-class economic status (moses & cobb, 2001). these studies demonstrate the importance mathematics holds in the arena of academic advancement and success. researchers such as gloria ladsonbillings (1998), william tate (1995), na’ilah nasir (2002), danny bernard martin (2000) and others have studied african american students and mathematics i waddell reform elementary mathematics journal of urban mathematics education vol. 3, no. 2 117 teaching and learning in an effort to understand many african american students’ persistent low levels of achievement on standardized assessments and in schools (the college board, 1999; langland & emeno, 2003; national center for educational statistics, 2001). many solutions have been suggested in the effort to provide better educational opportunities for african american and low-income students: higher curriculum standards, testing regiments and accountability measurements, supplemental education programs, improved teacher–student relationships, and more qualified teachers (see, e.g., the college board, 1999; ferguson, 2001; national center for educational statistics, 2001; ogbu, 2003). these solutions focus closely on improving schools, curricula, and the work of teachers to create better and more specific opportunities for african american students to excel. what is missing are studies that look closely at how students respond and interact with mathematics in light of these efforts by teachers and other school personnel. in this article, i describe a study about how a group of african american elementary students in an urban school negotiated and learned mathematics in national council of teachers of mathematics (nctm) standards-oriented classrooms. by studying their patterns of interaction and engagement, i consider how their actions connected to and supported theories of african american cultural dimensions. four questions guided the study: 1. what kinds of individual mathematical interactions occur when african american students engage in standards-oriented activities, tasks, and events? 2. what patterns of practice emerge across these african american students in their mathematical interactions? 3. in what ways do student practices converge with or diverge from the classroom practices and teacher anticipated norms? 4. to what extent do these student patterns of practice connect with research on african american cultural dimensions? what is standards-oriented teaching? the nctm (2000) standards consist of two main parts: content standards and process standards. they are “inextricably linked” (p. 4); the foundation of the reform vision is built upon the fusion of these parts. the content standards are what nctm has selected as the most important mathematical topics for each grade level. they are comprised of related ideas, concepts, skills, and procedures that form the foundation for understanding and lasting learning as defined by nctm. the process standards describe the five teaching and learning processes that nctm promotes as necessary elements in the development of an investigatory mathematics classroom: problem solving, communication, connections, reasoning and proof, and representation. these processes are crucial as the waddell reform elementary mathematics journal of urban mathematics education vol. 3, no. 2 118 framework of a classroom that strives to develop students who are mathematical inquirers and critical thinkers. the process standards support what is called mathematizing, “developing mathematical understandings from initial instructional activities that focus on material objects, actions, and events, through the process of coming to see concrete situations in mathematical terms” (lampert & cobb, 2003, p. 240). a standards-oriented mathematics classroom is supported by a social constructivist view of learning. ernest (1996) described social constructivism in terms of the interconnectedness between the individual and the social. he stated, “human subjects are formed through their interactions with each other as well as by their individual processes” (p. 342). in his work on social interactionism, voigt (1996) also explained that, in learning, the focus is on the interaction between the individual, or subject, sense making and the social processes in which the individual participates. students bring their prior knowledge and cultural understandings to bear on mathematical activities or tasks. they then negotiate meanings within these tasks through individual cognitive dissonance, through their work with others, or through teacher feedback (explicit or implicit). this negotiation of meaning occurs mainly through discourse, which can be rigidly directed by teacher actions or more fluidly built through numerous interactions between students, teachers, and tools. as students have more and more opportunities to interact with others around particular meanings and concepts, “taken-as-shared” (p. 33) meanings develop. only through the interaction of the participants in the development of routines and obligations can the stabilization of meanings and the creation of mathematical themes occur (p. 41). wood’s (1994) concept of mutual orienting also describes the actions and interactions of classroom community members. she explains, “patterns of interaction are seen as emerging from the individual’s interpretation of another’s actions and from the mutual orienting that occurs between the teacher and students…they build up negotiated expectations and obligations over the course of the school year” (p. 151). cobb (2000) called this negotiation of classroom actions and interactions “social norms”; these norms are established through the interactions between the teacher and the students and amongst students themselves. from the view of these researchers, the vision of standards-oriented mathematics learning can be described as the negotiated mathematical meanings that develop through participation with provided tasks and tools; meanings that become shared by the group or community through interactions leading to the establishment of classroom norms. to enhance this process of negotiating mathematical meanings within an activity or task, teachers provide students with opportunities to interact with other students and tools. students should have opportunities, when working on mathematical tasks, to reflect on their own processes and the processes of others; this opportunity provides time to reorganize waddell reform elementary mathematics journal of urban mathematics education vol. 3, no. 2 119 information and to be taught to engage in the critique of and inquiry into mathematical concepts. these opportunities to reflect, critique, and inquire also provide the teacher with more information about the nature of students’ prior knowledge, backgrounds, worldviews, and interests, thereby allowing more connections to the students’ lives and understandings. research on the implementation of nctm standard-oriented curricula in the current political landscape, it is important to consider if standardsoriented practices for teaching mathematics will help engage all students in learning mathematics deeply and with understanding and increase levels of achievement in course grades and testing for all students. numerous studies have analyzed and considered the effects nctm standards-oriented curricula have had on student test score achievement and how african american students have fared in these classrooms and schools. one such study that focused on overall test score achievement was conducted by the alternatives for rebuilding curriculum (arc) center (2002), a part of the consortium for mathematics and its application in massachusetts. they conducted a study that focused on the three national science foundation (nsf, 2002) funded standards-oriented curricula to establish what effect these types of curricula had on student test score achievement. the three curricula— investigations in data, number, and space; everyday mathematics; and math trailblazers—were implemented in three states: massachusetts, illinois, and washington.1 schools and grades chosen to participate in the study were selected by their usage of the nsf curricula and the length of implementation (at least 2 years) determined through telephone surveys and standardized test score record availability. it is important to note, however, that this data collection process did not provide details into particular teaching practices, nor did it examine the meaning or level of curricula implementation beyond the requirement that the curricula be in place for 2 years. nevertheless, it yielded 742 classrooms and more than 100,000 students from grades 3 to 5 in the three states. each school was then matched to a comparison school based on reading test scores, income measures, and percentage of white students in the school. overall, they found standardsoriented schools outperformed their matched counterparts on all test measures and mathematics strands, with measurement and computation showing the largest 1 everyday mathematics is a comprehensive pre-kindergarten through sixth-grade mathematics curriculum developed by the university of chicago school mathematics project, published by wright group mcgraw-hill. math trailblazers is a full mathematics curriculum for grades k–5 that was developed by the teaching integrated math and science (tims) project at the university of illinois at chicago, published by kendall/hunt publishing company. a complete description of the investigation curriculum is described later in the methods section of this article. waddell reform elementary mathematics journal of urban mathematics education vol. 3, no. 2 120 gains. when the results for all three states were disaggregated by race/ethnicity and ses, all reform schools again outperformed the comparison schools. when examining data on african american students as a racial group, students in the standards-oriented classrooms performed approximately 4 percentile points better on all subtests and total scaled score than comparison students. what was disconcerting, however, was that african american students, as a group, scored far below all other race/ethnic categories of students regardless of the type of curriculum implemented in the school. a study by briars and resnick (2000) of the pittsburgh, pennsylvania school district analyzed achievement test scores and curriculum implementation levels to determine how well students achieved when the school district adopted the standards-oriented curriculum everyday mathematics. they grouped schools by two levels of teacher curriculum implementation, strong and weak, and matched these schools by demographics (e.g., free and reduced-priced lunch, family structure, mobility rate, and percentage of african american students). for their study, they defined strong implementing schools as having a majority of teachers who used all of the components of the curriculum and provided students with opportunities to engage in mathematics according to the nctm process and content standards. weak implementing schools were schools that had only one or two teachers using the curriculum as prescribed by the curriculum developers. they found that students in the strong implementing schools outperformed those in weak implementing schools on three strands of the standardized test used to measure achievement: computation skills, conceptual understanding, and problem solving. the scores of african american students in the strong implementing schools also rose, and, although the test score gap between them and white students narrowed, it was still large except on computational skills where the african american students outscored the white students (see also mccormick, 2005; secada, 1992). in a similar study, riordan and noyce (2001) also found a positive correlation between the length of time (in years) that a teacher used the everyday mathematics curriculum and the achievement of the students, though as in the arc study, they did not have a detailed way of analyzing the level of implementation other than the length of time the curriculum was in use. their study, using regression analysis on test score data, compared three groups: early implementers (more than 4 years of implementation), late implementers (less than 4 years), and non-implementer schools. the data were collected from the statewide massachusetts standardized test scores from 1999. they found that the student scores of the early implementer schools were the highest, followed by the late implementer schools. the non-implementer schools had the lowest average scores, though just slightly lower than late implementing schools. thus, the longer a school has been using the standards-oriented curriculum, the higher the students scored on the waddell reform elementary mathematics journal of urban mathematics education vol. 3, no. 2 121 standardized test. an important note for this study, however, was the improved achievement of the african american students. the african american students in early implementing schools had average scores 9 points higher than african american students in the non-implementing schools. this point difference between test schools was larger than all other racial groups except hispanics. however, in all schools, the overall scores of the african american students were lower than those of other student groups. overall, these studies suggest that standards-oriented curricula have a positive impact on the achievement of african american students after at least 2 years of implementation. however, the test score achievement gap continues. this points to a need to consider the implementation of curricula and teaching practices found in standards-oriented classrooms to have a more detailed view of the impact standards-based teaching practices have on students. interactions with standardsoriented teaching practices, teacher beliefs and knowledge, response of the students to these practices; each of these can create markedly different patterns of practice in classrooms generally regarded as standards-oriented. being able to document the practices, established norms, and cultures in standards-oriented classrooms can assist in creating a sharper picture of implementation and what standards-oriented classrooms might “look like.” african american cultural dimensions and learning for several decades, social scientists have studied and described how (some) african americans interact with and interpret the world and how this interpretation affects learning (see, e.g., boykin, 1983, 2001; boykin & bailey, 2000; jones, 2003; ladson-billings, 1994; shade, 1982, 1992). it is important to note, however, that this growing body of research does not intend to essentialize african americans into some monolithic group of sameness but rather attempts to highlight shared beliefs, values, and customs (i.e., culture) that might be uniquely shared historically among black americans. for instance, boykin (1983) identified nine cultural dimensions that might be considered unique among african americans: spirituality, harmony, movement, verve, affect, communalism, expressive individualism, orality, and social time. he offered these dimensions as “motifs, patterns of behavior, and predilections” (p. 348) that distinguish african americans from other groups of people in our country. more recently, boykin and bailey (2000) researched three of the african american learning dimensions that stemmed from boykin’s earlier work on dimensions of african american cultural themes: communalism, an acculturation toward social and family relations; movement, rhythmic and expressive orientation toward life; and verve, heightened appreciation toward physical stimulation. their study, conducted with 163 lowincome african american students in grades 2 through 5 in a large urban city, waddell reform elementary mathematics journal of urban mathematics education vol. 3, no. 2 122 was designed to understand how these three cultural dimensions (might) affect learning preferences and orientations. they administered six surveys to see how cultural practices related to movement, communalism, and verve in the students’ homes correlated to the same areas in the students’ learning preferences in school. their findings suggested that students most strongly oriented toward movement factors both at home and in school, which were expressed by items related to how music is incorporated into and improved everyday well-being. the communalism theme was shown through student preferences focused on a sense of common duty and support within the family (and extended family) on the home scale and a similar preference for working within a group and valuing friends on the school scale. verve, while having a higher correlation between home and individual learning preferences than communalism, showed average scaled scores that were at or below the likert mid-point range for those same items in school, which included student preferences in active oriented activities in the classroom and when playing. though boykin and bailey’s (2000) work did not focus specifically on the mathematics learning of students, it could be said that most u.s. mathematics classrooms have social and participation norms that are at odds with these three themes of african american cultural dimensions. vervistic and movement opportunities, such as activities that require high levels of energy and participation, do not regularly show up in classrooms (other than physical education). berry (2002), in his study of successful african american middle school boys, found that boys who displayed vervistic behaviors were kept out of higher-level mathematics classrooms, as the teachers and administrators did not consider the students’ academic abilities, only their social ones. according to boykin (1983), this mismatch in cultural styles is a challenge that needs to be addressed as we move toward improving the academic achievement of african american students. similarly, shade (1992) studied an expanded view of cognitive style— perceptual, intellectual, and social domains—to examine the possibility of a unique african american learning style. she defined cognitive style as “a culturally induced way in which individuals organize and comprehend their world” (p. 256). in her research, 178 grade 9 students were stratified by achievement levels and sex, and consisted of 92 african americans and 86 european americans with similar socioeconomic backgrounds. the students were administered three tasks that aligned with three areas of cognitive style under consideration. her results found a significant difference between the african american and european american students on the perceptual processing task (p<0.0001), which aligned with the domains of field sensitive/field dependent, in which african american students showed preference for field sensitive tasks. this finding suggested that african american students focused on ideas holistically, in context, and in relation to the environment rather than in a field independent view—using analytic waddell reform elementary mathematics journal of urban mathematics education vol. 3, no. 2 123 and systemic approaches to situations found in many mathematics classrooms (stiff & harvey, 1988). on the social interaction style instrument, a small, but statistically significant difference (p<0.025) was found between african american and european american students in the area of orientation toward the world around us, where african american students tended to be more adaptive and spontaneous to occurrences in the world, as opposed to making plans and attempting to organize the outside world. these findings led shade to suggest that african americans demonstrated a preference for considering the world in a more spontaneous, flexible, and less structured way than european americans. in other research on african american cultural orientation, jones (2003) developed trios, a theory that represents the attitudes, beliefs, and values of african american culture that arose out of african americans’ need “for selfprotection and self-enhancement in a universal context of racism” (p. 239). trios is defined by five main components: time, rhythm, improvisation, orality, and spirituality. through these components, jones noted a collaborative, rather than oppositional, duality of individualism and collectiveness in african american culture: a sense of independence, “be yourself at all times” through improvisation and orality; along with interdependence, “harmony in my group” through a sense of collectiveness with other african americans. this duality means that there is a sense of a collective group identity that supports a need for individuality, rather than conformity. jones’s (2003) research on trios, which in many respects mirrored boykin’s (1983) work, was conducted with 1415 respondents of different races and ethnicities ranging in age from 14 to 62 from across the country, with most respondents being college-aged students. they responded to a 77-item survey in which they were asked to respond to statements concerning different aspects of the trios dimensions. in his study, he found that four dimensions of trios were captured well in the statements on the survey: spirituality, belief in a higher power that influences all living things; improvisation, goal directed individualistic and creative problem solving in a distinctive style; orality, preference to face-toface communication and personal expression; and time (also called present orientation), personal perspectives are in the present and time is derived from tasks and not prescribed by them. in an analysis of racial and gender differences in orientation to the different dimensions of trios, the study showed that african americans tended to exhibit more dimensions of trios than non-african americans. in particular, spirituality and time provided the strongest african american correlations and greatest black–white differences in his study. although this study focused on adolescent and adult respondents, the findings can also be relevant to younger african americans. willis’ (1989) review of the research on african american learning styles allowed her to categorize the research on cultural dimensions into four areas. in waddell reform elementary mathematics journal of urban mathematics education vol. 3, no. 2 124 this study, those categories were used to frame an overarching organization to the cultural themes and orientations found in the work of boykin (1983), shade (1992) and jones (2003) (see table 1): social/affective response, african americans are people-oriented, emphasize the affective domain, interaction and learning in social groups is important; harmonious, interdependence and communal aspects of people and environments are respected, knowledge is to be relevant and thought about holistically; expressive creativity, african americans tend to prefer environments that offer adaptive, variable, and novel situations, and stylistic simultaneous stimulation is preferred with verve and oral expression; and nonverbal response, use of intonation and body language are important ways of communicating, movement and rhythm are vital. here, i added a fifth category, spirituality, to capture the belief that a greater power than humans influences all things and actions, as reflective of the work by both boykin (1983) and jones (2003). when considered in school settings, these cultural dimensions act as filters for the students’ understanding of and participation in classroom interactions. table 1 categories of african american cultural dimensions (adapted from willis, 1989) african american cultural dimension categories related cultural dimensions as described by researchers boykin (1983) jones (2003) shade (1982) social/affective – affective response to stimuli, values social interactions above object interactions communalism affect orality social time time orality social cognition harmonious – interdependence, holistic approaches to the world, purposeful uses of information harmony communalism rhythm field sensitive worldview expressive creativity – creative, adaptive, spontaneous, vervistic, dramatic and enthusiastic oral expression, multiple stimuli preferred orality verve expressive individualism improvisation orality stimulus variety conceptual tempo nonverbal – use of intonation in oral language, body language, movement and rhythmic expression movement verve affect harmony rhythm stimulus variety field sensitive spirituality – belief that a greater power than humans is at work and influences all things and actions spirituality spirituality conceptual tempo in more recent work, boykin and jones (2004) discussed how african american cultural dimensions and themes can be used to analyze the interactions between the learner and the learning environment provided by teachers and schools. using the previous research on african american cultural and trios dimensions, they developed a psychosocial integrity approach to teaching that can waddell reform elementary mathematics journal of urban mathematics education vol. 3, no. 2 125 enhance opportunity for learning and achievement not only for african american students but also for all students. the notion of psychosocial integrity considers that people experience life through: complexity, experiences are understood in multiple facets, variations, and depths; coherence, making sense of life through our frame of reference; and texture, experiences can be examined through many points of view. boykin and jones believe that schooling practices that draw on psychosocial integrity ideas as assets of the students could lead to enhanced outcomes for all students. such an approach incorporates five practices that should be the focus of school activities: promoting meaning making through multiple modalities; teaching thinking and learning strategies while fostering critical thinking processes; building a learning community; utilizing cultural resources of students, families, and their communities; and providing a supportive yet demanding learning environment. this approach is supported by the work on social constructivism; individual sense making is considered and built upon in the process of developing and negotiating community norms and practices in the classroom (ernest, 1996). psychosocial integrity also closely connects with the research on culturally relevant pedagogy as both approaches draw heavily on understanding and incorporating students’ lives into classroom practices (ladson-billings, 1994). table 2 shows how psychosocial integrity practices relate to research on african american cultural and learning dimensions as well as culturally relevant pedagogy. promoting meaning making through multiple modalities is guided by cultural dimensions that encourage students to draw on their knowledge and the knowledge of others to make sense of tasks. students draw on ideas and beliefs of the world around them and on feelings and thoughts within themselves to understand the world. the dimensions of harmony, social/affective responses, and spirituality are reflected in this theme. by teaching thinking and learning strategies, teachers are providing students with opportunities to focus deeply on tasks and topics in many forms and in many ways and use whatever tools are necessary to accomplish those tasks. allowing movement and physical and oral expression when working, along with paying close attention to task understanding and completion, supports the dimensions of harmony, nonverbal response and expressive creativity. to have the collaboration skills necessary to engage with others on a topic and use others as learning resources, teachers build learning communities in their classrooms. working communally, speaking with others while working, and interpreting others’ ideas encourage students to turn to others as learning partners. understanding that students come from a neighborhood or community that supports particular cultural and social habits and these habits have a large impact on their lives in school is an important reason for considering and using students’ cultural resources in teaching. using individual differences as a teaching tool and making clear connections to students’ lives outside of school can provide students waddell reform elementary mathematics journal of urban mathematics education vol. 3, no. 2 126 with a sense of belonging. the african american cultural dimensions of harmony, expressive creativity, and nonverbal response support these ideas. finally, creating a supportive but demanding environment can be used to develop a sense of harmony among the students as they rally together in support of success for all (ware, 2006). learning about each student and their individual levels of understanding can help teachers focus on student improvement and create attainable goals and expectations. these psychosocial integrity practices could allow school personnel and teachers to develop pedagogical practices for african american students that draw on their often-preferred modalities and cultural and psychological schemas. table 2 psychosocial integrity practices and related african american cultural dimensions psychosocial integrity teaching practices psychosocial integrity practices defined supporting cultural dimension categories (willis, 1989) culturally relevant pedagogy (ladson-billings, 1994) promoting meaning making making connections among topics, to students experiences, and prior and common knowledge harmonious social/affective spirituality drawing on prior knowledge, cultural themes, individual differences teaching thinking and learning strategies providing tools for deep, constructive, active engagement with academic content; multiple stimuli and a variety of activities are used in teaching harmonious nonverbal expressive creativity considering knowledge as something constructed and critiqued building a learning community learning is interdependent, collaboration is fostered through personal interactions social/affective harmonious development of learning community and collective empowerment through collaboration using cultural resources drawing on students’ every day practices, cultural habits, and worldviews in every day practices harmonious expressive creativity nonverbal community connections outside of the school walls creating a supportive and demanding learning environment maintaining high expectations, focus on effort and improvement social/affective academic skill development and achievement is paramount standards-oriented practices in mathematics most strongly correlate to the psychosocial integrity practices of promoting meaning making, teaching thinking and learning strategies, and building a learning community. these practices offer opportunities for teachers to develop mathematical understandings in their students, draw on students’ prior mathematical knowledge, help students use a variety of tools and modalities to communicate mathematically, and support student collaboration and interdependence. through the explicit development of mathematical ideas in discussions and in tasks, and supported by continuous classroom discussions on expected behaviors and norms, standards-oriented practices en waddell reform elementary mathematics journal of urban mathematics education vol. 3, no. 2 127 courage the development of a supportive and demanding learning environment as well. understanding how to incorporate these psychosocial practices into the everyday work of schooling will require long-term investment into teachers’ professional development. african american student identity, engagement, and agency in mathematics nasir (2002) and nasir and hand (2008) studied how student identity and engagement might create opportunities for students to learn. studying african american students in mathematics-based activities that occurred outside of school as well as in-school mathematics activities raised the question of how these students could be successful with mathematics in out-of-school activities but unsuccessful with mathematics in school, and the impact of student identity in each these settings. nasir framed identity as “being constructed by individuals as they actively participate in cultural activities.…[it] both shapes and is shaped by the social context” (p. 219). the concern was how the learning communities or settings influenced the development of identity. in particular, nasir and hand focused on practice-linked identities, “identities that people come to take on, construct, and embrace that are linked to participation in particular social and cultural practices” (p. 147). drawing on wenger’s (1998) idea of how identities are formed within a community of practice through engagement, alignment, and imagination, nasir analyzed how students developed identities and goals when playing dominoes and basketball. she focused on the practice-linked goals the students created for themselves that allowed them to participate in the games. embedded in these were mathematical goals, such as calculating shot percentages or making number combinations, that allowed the students to reach their practicelinked or learner’s goal. she noticed that students would develop goals that permitted them to be a part of the play and part of the player community and this, in turn, helped them identify with the game. this desire to be a part of the community of dominoes or basketball players motivated the students to create personal goals that meshed with the goals and rules of the game. in her view, learning was intertwined with student goals and their developing identity within the community of practice, in this case the community of dominoes or basketball. students engaged and developed goals that allow them to participate in the community, which in turn helped them imagine and create new identities relative to the community. this engagement led to a desire to learn more about the activity/game and generate and align new goals to continue participation in that community. in this case, the ability of the students to imagine themselves as part of the basketball or dominoes community provided a motivation to engage with the community, when given the opportunity. in another study that focused on african american students and their waddell reform elementary mathematics journal of urban mathematics education vol. 3, no. 2 128 mathematics learning, martin (2000) studied the mathematical socialization of african american middle school students through a framework that allowed him to analyze how community, school, and individual themes and beliefs affected the success of african american students in school mathematics. students in his study were interviewed and observed over a 1-year period, along with interviews of their guardians and teachers. he found that successful students were able to overcome a great number of negative forces, such as poor past mathematical experience or pressure to underperform, through student agency, which, for these students, was defined as “actively constructing meanings for mathematics learning and mathematics knowledge and acting on those meanings accordingly” (p. 170). the negative forces came from different areas of the students’ lives—some from family, some from school, most strongly from peers—leading martin to suggest that the students were motivated to succeed “by an inner drive and selfdetermination to succeed” (p. 183). he was less certain about how these students developed a strong sense of identity and motivation toward success in school because the students had widely varied lives and family backgrounds. martin wrote, “although community and school forces do have the potential to affect their mathematics socializations and identities, these forces are not deterministic” (p. 185). how these students developed and used their identity and agency to manage negative forces and make beneficial academic choices determined their level of success in mathematics. why might students decide to engage with particular communities of practice? in these examples, students chose to participate in the community of domino or basketball players; they invoked their agency to make choices about what they needed to learn and what actions they needed to take in order to participate fully. basketball and dominoes, for these students, were part of the social and historical culture of african americans; these students saw other african americans engage in these activities and that allowed them to imagine that they could participate in these activities as well. african american cultural dimensions such as improvisation could occur when making moves during these activities, the communalism among players that emerged when participating in these activities, or the verve with which these games can be played made these activities attractive to young african americans. standards-oriented mathematics has the potential to provide a framework in which the identity and engagement needs of african american learners could be met. teachers can create mathematical tasks and lessons that consider students’ prior knowledge and ways of knowing, which can encourage students to develop personal goals in conjunction with the tasks. the whole class/community can share ideas, which can spark student participation in mathematics lessons. with the increase in participation, students begin to see mathematics as something they can engage in and, in turn, can increase their identification and alignment with waddell reform elementary mathematics journal of urban mathematics education vol. 3, no. 2 129 mathematics. developing such a mathematics community can provide a space for students to interact with mathematical ideas through many modalities and preferred learning styles. in doing so, opportunities can emerge that entice students to engage with mathematics and align their practices and interactions with the mutually established meanings, routines, and obligations of the classroom. methods the work in this study was part of a larger ethnographic study, entitled mathematics plus (mathplus), which examined the relationship between teacher learning and student performance in two urban elementary schools. the study reported here focuses on student performance and interaction within one of the schools. this school enrolled approximately 275 students from a range of ethnic and social class backgrounds during the data collection period. sixty-five percent of the students were eligible for free or reduced-priced lunch. the school drew many of its students from the surrounding neighborhood, which was predominately african american, and included many low-income families along with some families from middle class backgrounds. as designated by the school district, the school was a desegregation school and, as such, drew a small number of white students who were admitted from outside the immediate neighborhood. from this small school of 11 classrooms, nine classroom teachers are represented; the teachers’ classroom experience ranged from 6 years to over 30 years by the end of the study. because this study focused on the learning and interactions of african american students over a period of at least three years, seven african american students from these classrooms had sufficient data to be included (see table 3; names of all students and teachers are pseudonyms). table 3 students with years and grades during study student name years in study grades kiana ali 1999–2003 1st through 4th candace brown 1999–2003 1st through 4th maya connor 1999–2002 2nd through 4th jordan jones 2000–2003 1st through 3rd samuel quon 1999–2003 1st through 4th felix robinson 1999–2002 2nd through 4th royce rush 1999–2003 1st through 4th the investigations curriculum the school in the study adopted the mathematics program, investigations in number, data, and space (terc, 2004), and this program was used throughout waddell reform elementary mathematics journal of urban mathematics education vol. 3, no. 2 130 the 4 years of the study. investigations, a standards-oriented curriculum, focuses on developing conceptual understanding and mathematical thinking through visual models, strategy development, inquiry, collaboration, and communication. the goals of the curriculum, as stated by the developers, are to offer students meaningful mathematical problems, to emphasize depth in mathematical thinking rather than superficial exposure to a series of fragmented topics, to communicate mathematics content and pedagogy to teachers, and to substantially expand the pool of mathematically literate students. during lessons, students spend the majority of the mathematics period working on mathematical tasks, activities, or games that are designed to promote mathematical communication, questioning, and problem-solving skills. teachers attended monthly professional development sessions throughout the study, facilitated by the principal investigator, to support their use of the program and discuss issues related to implementing a standardsoriented curriculum. data sources and analysis researchers visited the study classrooms informally once per week and took field notes on the classroom activities. once a month, more formal observations were conducted in which classroom actions were audiotaped and transcribed. during each audiotaped observation, one researcher would be in the classroom observing the students in the room during the lesson, while another would observe the teachers. the teachers and students were also interviewed three times per year, in the fall, winter, and spring. the teacher interviews were audiotaped and transcribed, while the student interviews were videotaped and transcribed. data analyzed for this study included teacher and student observations and the clinical student interviews. to discuss how the african american children of this study interact with standards-oriented teaching practices, there was a need to define what those practices were and whether the teachers in this study enacted practices that could be regarded as standards oriented. the teachers’ practices were drawn from either a 1 or 2-year period of observations, depending on the years the students in the study were members of each teacher’s class. in order for a practice to be considered a pattern for an individual teacher, the teacher must have engaged in that practice in more than half the observations for the years being analyzed. and at least six teachers had to enact a particular practice for it to be included as a pattern of practice for this study. in all, there were nine standards-oriented practices enacted by the teachers that became common experiences for the students throughout the study: whole group discussion, encouraging active listening, questioning and probing, small group work, teacher support, sharing and explaining ideas, activating prior knowledge, modeling strategies and thinking, and modeling means of waddell reform elementary mathematics journal of urban mathematics education vol. 3, no. 2 131 communication (see table 4). the patterns that emerged from the coding process for each teacher were analyzed to determine how they might fit with and reflect the themes reflected in the theories of african american cultural dimensions and psychosocial integrity themes as described earlier (see tables 1 and 2). a compilation of all the teachers was then created to determine what learning opportunities were common across all the teachers, and thus, all the students (see table 5). table 4 nine patterns of practice common among teachers teacher practice description of practice leading whole group discussions whole group discussion about mathematics topic of the day encouraging active listening encouraging students to listen to other students and focus on topic at hand during discussion questioning and probing student thinking teacher questioning used to solicit student ideas or information; probe student thinking creating opportunities for small group/partner work small group work offers opportunities to actively engage with task or activity facilitating student independent work the teacher supports student learning during independent or small group mathematical tasks encouraging the sharing and explaining of student ideas allowing opportunities for sharing and explaining ideas and solutions activating prior math knowledge teachers consider and use prior mathematical knowledge as a mathematical tool modeling thinking and solution strategies modeling games and activities to emphasize strategies and thinking modeling multiple means of communication students are encouraged to use a variety of means and media to communicate in the classroom (e.g., models, drawings, graphs, materials, manipulatives) the analysis of the research data continued with the individual coding of the student observations. each transcribed observation was broken into events, defined as a teacher or student initiated segment of a lesson in which interactions between teachers, students, and their ideas are focused around the same mathematical goal. each event was described using a number of categories: whole group participation, preparing for independent work, doing independent work, small group/partner interactions, choice and use of mathematics tools, communicating mathematical ideas, mathematical errors, mathematical understandings, and other. these categories reflected the different interactions and classroom practices that are expected in standards-oriented classrooms (nctm, 2000). for the student data, codes were used that captured how students interpreted and responded to mathematical tasks, what they did in the face of confusion or lack of knowledge, waddell reform elementary mathematics journal of urban mathematics education vol. 3, no. 2 132 how they engaged with their work, and how they engaged with peers, teachers, and the classroom environments or routines. each category had between four and eight codes to describe the students’ interactions. these codes were drawn in part from the research on african american cultural dimensions and provided multiple ways to interpret what was happening for the students beyond a more typical interpretation, such as on-task or off-task. after the coding of all students was completed, counts of each code were made to provide initial information on patterns in students’ interactions. these counts were tabulated for each student by grade and then again over the entire study. patterns were determined by the percentage of times a particular practice was enacted by each student and then whether the practice was enacted over the number of years the student was in the study. the individual student practices were compared; those practices that were consistently found among at least five students were considered patterns (see appendix a). some patterns occurred among smaller student subgroups (e.g., just the girls) and this was noted and discussed as well. table 5 psychosocial integrity practices and african american cultural dimensions related to teacher patterns of practices psychosocial integrity themes themes defined supporting cultural dimension categories (willis, 1987) teachers’ patterns of practice promoting meaning making making connections among topics, to students experiences, and prior and common knowledge harmonious social/affective spirituality sharing and explaining ideas, leading whole group discussions, activating prior mathematics knowledge teaching thinking and learning strategies providing tools for deep, constructive, active engagement with academic content; multiple stimuli and a variety of activities are used in teaching harmonious nonverbal expressive creativity leading whole group discussion, encouraging active listening, questioning and probing, facilitating student independent work, modeling multiple means of communication, modeling thinking and solutions strategies, activating prior math knowledge building a learning community learning is interdependent, collaboration is fostered through personal interactions social/affective harmonious creating opportunities for small group/partner work, leading whole group discussions, facilitating student independent work using cultural resources drawing on students’ every day practices, cultural habits, and worldviews as a tool in everyday practices harmonious expressive creativity nonverbal modeling multiple means of communication creating a supportive and demanding learning environment maintaining high expectations, focus on effort and improvement social/affective facilitating student independent work, modeling multiple means of communication waddell reform elementary mathematics journal of urban mathematics education vol. 3, no. 2 133 how have we chosen to learn? coherence between student and teacher patterns of practice candace, felix, jordan, kiana, maya, royce, and samuel—these seven african american students each had a story to tell. each student’s pattern of practice was connected to three areas: which classroom opportunities would allow or encourage the student pattern to emerge, which african american cultural dimension(s) were reflected in the practice, and which of the enacted teacher practices supported or allowed the engagement of a particular student practice. once the student patterns of practice were connected to the teacher patterns of practice, the coherence, or lack thereof, was defined to be convergent, episodic convergence, supported divergence, or divergent. for a practice to be considered convergent, the students would have consistently engaged with that practice in about 80% of the events noted for that practice. that is to say, the classroom practices provided space and opportunity for the students to respond to mathematical tasks as the teachers intended. episodic convergence practices occurred when students, for the most part, engaged with a particular practice in a way that was anticipated by the teacher and classroom norms, but occasionally interacted with activities and tasks in ways that are not expected or anticipated. these practices occurred in more than 50% but less than 80% of the noted events for that teacher practice. supported divergence practices were practices enacted by the students that did not cohere with a pattern of practice of the classrooms in this study. these patterns, however, were often unnoticed, overlooked or ignored by teachers, possibly leading to the continued use of the pattern. divergent practices were patterns of practice enacted by the students that were not supported by teacher practices or classroom norms. even in the face of redirection or reprimand, students continued to engage with these divergent practices throughout their years in the study (see appendix a). focused collaboration the students most often displayed direct and focused collaboration on the assigned tasks or activities when working in small groups with their peers. they were mathematically engaged during this time, with their collaborative focus becoming stronger in grades 3 and 4. teachers consistently provided small group independent work time for students. work on learning styles by dunn et al. (1990) found that african american students preferred to work with others more often than other ethnic groups (e.g., mexican american, greek american, chinese american), supporting the finding that the students look for opportunities to interact with each other when working. this finding is also supported by the work of nasir (2002). she found that, when african american children were engaged waddell reform elementary mathematics journal of urban mathematics education vol. 3, no. 2 134 with others during activities, they would work toward learning more about that activity in order to align their practices with the activity’s expectations. the students in this study made the effort to be a part of the classroom mathematics community and worked toward aligning their practices with the expectations of the teacher, thus establishing a norm about the importance of working together with others when learning mathematics. considering african american cultural dimensions, the strong collaborative focus during independent time allowed the students to enjoy social/affective interactions with others while learning. in this example from grade 3, candace was using the traditional regrouping algorithm to subtract. the students were given place value mats and base ten blocks (see figure 1) to help them solve the problems. she was then assigned to work with kiana, and together they worked through solving the problem 95 minus 56. figure 1. illustration of base ten blocks. observation: they carried a supply of base ten blocks with them and a place value mat. kiana instructed candace: “show me 95.” candace built 95 without any problem. kiana asked her if she could subtract six from five and candace said she needed more ones. kiana indicated the tens section and said, “take one away from here and add ten more.” candace did that correctly. kiana said, “make sure you have 95 still.” she then took candace through this same problem on the paper. “you would have to turn that 5 into a 10. so what do you have up here? you have a 15.” candace crossed out the five in 95 and writes 15. kiana continued her coaching, “so the nine… if you are taking one away from nine what do you have?” candace didn’t seem to follow. “what comes before nine?” kiana hinted. candace said, “you cross that out and make an eight.” then they worked the subtraction: “15 minus 6.” candace counted on her fingers and kiana did too in order to confirm candace’s answer of nine: “8 minus 5.” they got the answer 39. we see here the collaborative nature between these two students; they were focused on the task and worked through the problem without interruption. although to some, this interaction may appear to be more directed by kiana in that she is telling candace what to do and is not focused completely on understanding the mathematics. the interaction, however, demonstrates the efforts of one student to tens ones waddell reform elementary mathematics journal of urban mathematics education vol. 3, no. 2 135 support the work of another, with kiana acting as the more knowledgeable other. in many “traditional” classrooms, this type of student interaction is often discouraged. kiana took on the role of coach and supported candace’s effort to make sense of the subtraction by asking candace questions and allow her to participate in the problem solving—moving the blocks and writing all the steps. active participation students actively participated in whole group lessons, by listening, watching, and engaging in mathematics discussions without engaging in other activities unrelated to the mathematics at hand. they followed the lesson or activity, and many students eagerly supply responses and ideas. by creating almost daily opportunities for the students to come together for whole group discussions and using questioning and probing techniques to encourage participation, the teachers provided a space in which the students interacted with mathematical ideas. in the following fourth-grade observation, maya is actively engaged with the lesson, even though her engagement pushed against the norm of quiet listening and turn taking. observation: teacher laura began working on the poster board at the front of the rug and wrote: “how many hundreds” and below that question wrote 200+400. she asked the students to show with their fingers how many hundreds there were. maya held up six fingers. teacher laura then wrote 201+403, and again asked how many hundreds. maya whispered “604,” and then raised her hand. maya said, “it would be six hundred and something, these are easy.” teacher laura wrote the next one 199+404, and then asks what they should do. maya whispers, “it’s 603”. teacher laura then asked, “what if we had 199 pennies and something cost $2.00….” maya commented softly, “if they gave her $2.00 then she can give one penny back.” teacher laura asked, “how much is 404, how many hundreds?” maya said softly, “can i blurt out the answer? i want to so badly.” the class decided there are four hundreds in 404. as the class prepared to move on to the next activity, maya raises her hand and says, “teacher laura, can somebody say the answer now?” teacher laura seemed confused. “can i say the answer?” teacher laura said, “we said the answer.” maya responded, “that’s not the answer, the answer is 603.” this observation illustrates that within most of the study classrooms, there were limits on how students were allowed or encouraged to interact during whole group lessons. students were expected to listen closely to others, respond to the given question or prompt, and stay focused and attentive to what was happening in the discussion. when students called out a response or idea without acknowledgment from the teacher, chose to attend to other things during the lesson (active inattention), or did not pay attention to what was going on (passive inattention), they were most often reprimanded or redirected toward the expected behavior. here, maya maintained a steady stream of comments while continuing to focus on the waddell reform elementary mathematics journal of urban mathematics education vol. 3, no. 2 136 lesson, even though she was not interacting in the discussion as teacher laura had intended. looking back to african american cultural dimensions, calling out by students or engaging in other related activities during whole group time could be construed as a student’s need for orality or movement expressiveness (boykin & cunningham, 2001; foster, 1992). physical tool use the next convergent pattern of practice that focused on the students’ interactions with mathematical tasks and activities was the practice of using physical tools to solve problems and accomplish tasks. physical tools were the concrete mathematics materials for modeling and counting (blocks, cubes, etc.), drawings, pictures, or written algorithms, but not mental tools or models. the students were adept at using physical tools in many different ways to support their problem solving. the teachers supported tool use as they supplied the students with access to tools, modeled tool use often, and encouraged students to share how they used tools in their work. thus, heavy use of tools was expected and the data showed that the percentage of physical tool use in both the classroom and interview events ranged from a high of 95% of the time by felix to a low of 51% by samuel. as an example, when solving problems, felix was adept in using tools and models to support his efforts. in grade 4, he was able to use snap cubes to solve the problem 4 times 8: interviewer: what is 4 times 8? felix: (counting the cubes by 4, taking sticks of 4 cubes from the long stick, lining up the short sticks on the desk): these are 4s and then add them up. add 4 eight times. and then…4, 8…(counting the cubes of the third stick one by one) 12, (counting the cubes of the fourth stick) 16, (counting the cubes of the fifth stick) 20, (counting the cubes of the sixth stick) 24, (counting the seventh) 28, (counting the eighth) 32. interviewer: great. so just how you explained to me. can you explain to me one more time? felix: i put all these 4s and put them into one group (piling up the sticks, holding them together, separating the sticks and placing them one by one on the desk, murmuring). then you add them together (putting the sticks back together). you’ll come up to 32. he also used physical tool modeling for addition and subtraction problems, as well for solving story problems for all 3 years in the study. having access to many tools connects to the african american cultural dimension of nonverbal interactions and expressive creativity; tools allowed students physical activity while working with mathematics, drawing on the ideas of waddell reform elementary mathematics journal of urban mathematics education vol. 3, no. 2 137 rhythm/patterns and movement as integral parts of the african american life experience. tool use, as it was enacted in these classrooms, also allowed personal interactions with others in that students generally shared tools and worked together on mathematical tasks requiring tools. although using physical tools was well supported by the teachers in this study, the reliance on tools was expected to lessen, as the students got older. by the end of third grade and all of fourth grade, students were expected to rely on mental models to support solving problems more than was expected of them in the earlier grades, particularly with basic facts and to support work on more complex problems. however, only two of the students, samuel and maya, used mental models efficiently and effectively. the other five students had some basic strategies for using mental models, but only for simpler problems. they instead continued to rely on physical tools to support their problem solving efforts. fuson (2003) found that some students using a reform curriculum continued to use lower-level counting strategies to solve computation problems without moving on to higher-level strategies up into the fourth grade. similarly, siegler’s (2003) review of literature on individual differences in students’ mathematics cognition found that students with mathematics difficulties had limited strategies for retrieving correct answers from memory and would rely on elementary counting strategies. he contended some of these mathematics difficulties arose from the children’s “limited exposure to numbers before entering school” and “from poor families with little formal education” (p. 295). although i do not agree with siegler’s deficit family view, he did suggest that greater practice and instruction in how to execute strategies as well as addressing limited background knowledge and conceptual understanding would allow students with mathematics difficulties to learn with reasonably high levels of proficiency. addressing the issue of bridging concrete and physical representations to more abstract mathematical symbols and language is an important topic and provides for a possible extension of this research study. direct communication students in the study made attempts to clearly and directly respond to questions and prompts by the teacher during whole group discussions and, most times, their responses stayed on the topic under discussion. the teachers encouraged strong communication through the practices of encouraging active listening and modeling clear mathematical communication; the teachers’ practices also supported students’ communication efforts by allowing the students to share their ideas during whole group discussions and independent work time, which also created opportunities for peer modeling. considering african american cultural dimensions that would support these practices, social affective opportunities entail waddell reform elementary mathematics journal of urban mathematics education vol. 3, no. 2 138 encouraging and valuing interaction among classroom members. by engaging in whole group discussions orally and through the use of tools, students were given the opportunity to engage in personal interactions with other classroom members and to develop a sense of community in the classroom. lampert and cobb (2003) discussed using communication to learn as a goal in standard-oriented classrooms; as students are provided opportunities to participate in mathematical tasks, teachers model ways of communicating and make explicit the acceptable forms of classroom communication. as students have additional opportunities to express themselves mathematically, they become more adept in using mathematical language to express their ideas. in this study, the girls were more vocal than the boys; they averaged over 104 events where they engaged orally with mathematics topics, while the boys averaged only 81 events in comparison. this finding could be related to the fact that the girls used more dramatic expression (discussed in the next section) than the boys and consistently used oral means for expressing their vibrancy when interacting with mathematics. direct communication is considered an episodic convergence practice because, although the students did communicate mathematically in the way the teachers expected during whole group lessons most of the time, the number of times that they had difficulty being clear and concise when explaining their thinking and solutions during interviews and small group settings was quite large (33% of all communication events). the students often used imprecise language when attempting to explain a solution or idea—using language that was vague; talking in circles; or starting to talk, stopping, then starting again, but staying on or close to the topic under discussion. this imprecise form of communication was evident in all years of the data analyzed. in the classroom, students were often questioned and guided through imprecise oral explanations by the teacher, which often times interrupted the flow of the speech. in her review of literature on sociolinguistics of african american children, foster (1992) also found that african american students were interrupted more often by teachers when speaking imprecisely than were white students who spoke in a more factual, linear, lecture style. she noted: teachers failed to comprehend or appreciate the stories being narrated. frequently interrupting students with inappropriate questions or attempting to redirect the narrative to focus on a particular but often insignificant aspect of the story, the teachers questioned the african american students’ intellectual competence and emotional stability (p. 305). learning to query, rather than curtail, imprecise speech would honor the students’ cultural funds of knowledge and allow teachers to consider whether a student is using speech to coordinate their thinking or if the student needs support through the thinking and communicating process. waddell reform elementary mathematics journal of urban mathematics education vol. 3, no. 2 139 understanding how students’ cultural norms impact the communication of their thinking and how to support that communication is important in standardsoriented classrooms, due to the emphasis placed on the use of communication to enhance and solidify learning. without considering the frameworks students use in their interactions, teachers can get caught up in funneling the student toward a more desired response or action rather than working toward understanding what in the student’s frame of reference produced the particular outcome or solution (wood, 1994). moschkovich (2002), in her study on the communication practices of bilingual students in mathematics classroom, concurred: in particular, this perspective can affect how teachers assess a student’s competence in communicating mathematically. for example, if we focus on a student’s failure to use a technical term, we might miss how a student constructs meaning for mathematical terms or uses multiple resources, such as gestures, objects, or everyday experiences. we might also miss how the student uses important aspects of competent mathematical communication that are beyond a vocabulary list. (p. 193) in this study, although the teachers may have had difficulty understanding the verbal reasoning of students who were attempting to solve a problem, the students were either able to solve the problem or were working toward solving the problem. in many cases, it appeared as though the students used their oral speech as a thinking tool; speaking their thoughts out loud as a way to solidify their solution. if students were using their speech as a way to organize their thinking, interruptions to that speech could impede the thinking process and make problem solving and explaining difficult for the students. this type of speech was evident in an example of felix as he tried to explain how to mentally add 10 to 128: interviewer: what number is 10 more than 128? felix (murmuring and looking down): i’m not sure. interviewer: well, if you started up from 128, and you counted up 10 numbers, what would you get? felix: i’ve counted 10 numbers, but i made a mistake…so stop counting…kind of get to 38. interviewer: okay. what number is 11 more than 128? felix (looking ahead and touching his chin with his left hand): the second one is 138, and the third one is 139. felix attempted to solve the problem of adding 10 to 128, but did not readily have a solution. when trying to explain, he sounded as if he was talking to himself about what he was doing in his head, attempting to understand what was being asked of him and how he could go about solving the problem. the interviewer, waddell reform elementary mathematics journal of urban mathematics education vol. 3, no. 2 140 not appearing to understand what felix was doing to solve the problem, continued on to the next problem. after a few moments, felix was able to put his explanation together and give correct solutions to both problems. having an opportunity to verbalize his thinking and mentally grapple with the wording and ideas in the problem appeared to be needed communication for felix to solve the problem. divergent and supported divergence student patterns of practice in wood’s (1994) research on patterns of interactions, she stated, “teachers have unwittingly undermined their own goals by failing to realize that the consequences of their interaction (and inaction) are often quite different from their intentions” (p. 149). in supported divergence patterns of practice, the teacher would encourage particular behaviors or beliefs either because they were unaware of the students’ engagement in that practice or belief or they were unaware of how their actions helped create or continue particular patterns or beliefs. divergent and supported divergence patterns of practice occurred far less frequently than convergent patterns, but appeared, in many ways, to be common practices of interaction and communication for the students. activating personal knowledge when solving problems, students used their personal knowledge as a tool. students made connections to ideas they held about the world or to the experiences they had in their daily lives. these connections not only occur when solving word problems that were designed to draw on student background knowledge but also when students used personal knowledge to solve problems about patterns or to develop solution strategies for computation problems. in most instances, however, the teachers did not solicit personal connections in problems; the students initiated these connections. nor did teachers explore the personal solutions in an attempt to connect them back to the original problem. particularly in the interviews, students, when asked to explain, made connections to their understanding of the world and used that understanding for making sense of the problem. in the following interview from grade 4, royce was working on problems related to theoretical versus experimental probability. in answering, she considers what is important in trusting someone to tell you the truth rather than considering the mathematics of the situation: interviewer (after discussing problems of drawing red and yellow cubes from a bag): suppose someone in your class said, “it’s been yellow five times in a row so i’m due for a red one now,” or, “it doesn’t matter that i got all those yellows. it’s still an equal chance of red or yellow.” what would you say to that person? waddell reform elementary mathematics journal of urban mathematics education vol. 3, no. 2 141 royce: disagree. it depends what classmate says it. interviewer: would you agree if brent (a “high level” mathematics student) said it? royce: no, i wouldn’t agree with him. interviewer: would you agree if…who would you agree with then? royce: i would agree with jeri because she’s my best friend. she’s my best friend in the classroom. here, by considering what is important when believing what someone tells you, royce does not attend to the mathematical concepts of the probability problem. she privileged her personal knowledge over her mathematical knowledge. this action does not mean, however, that she does not possess the mathematics knowledge to solve the problem. exploring her reasoning and then connecting her response back to the problem would allow her response to be valued, considered, and redirected. additionally, by allowing her to present her understanding of the task, a teacher can expose misunderstandings or roadblocks to strengthening mathematical ideas. this practice of drawing on personal knowledge when learning resonates with the african american cultural dimension of harmonious interactions: providing students with the opportunities to draw on their personal and previously held knowledge to support their mathematical development through the purposeful use of information. interestingly, drawing on students’ background knowledge and making connections to the world at large are also practices supported by standards-oriented teaching. unfortunately, the teachers in this study, as a group, did not consistently make personal connections an explicit part of their teaching practices. it is important to note that although the teachers attended ongoing professional development designed to support the implementation of the curricula, the content of those sessions was not available for analysis to know whether teachers discussed issues of using personal versus school knowledge as a learning tool. dramatic expression another divergent pattern of practice enacted by the students was their use of dramatic expression when working on or talking about mathematics problems. all seven students demonstrated the use of dramatic intonation and movement during interviews and in small group work to demonstrate ideas or punctuate speech. one student, maya, displayed dramatics in all areas of the classroom setting as well as in small groups and interviews. foster (1992) also noted that african american students, when sharing their writing, “resembled performed narratives, with stylistic features—gestures, dialogue, sound effects, asides, repetitions, shifts in verb tenses for emphasis—similar to those in a dramatic stage performance” (p. 304), which supports the findings of this study. in the following waddell reform elementary mathematics journal of urban mathematics education vol. 3, no. 2 142 observation, royce displayed her dramatic side during a grade 4 interview task: observation: royce (reading the problem aloud): jeanine has 40 cookies. she wants to share them evenly among 8 friends. how many cookies will each friend get? (sighing loudly) ok, this is what i hate. (she pulls out blocks and separates them into 8 piles.) royce (with an exasperated voice): evenly. each of them will get 5 cookies. i put these together, each of them equals 10 and ten 4’s equal 40. and then i broke them up by 5s, and 4 plus 4 equals 8. that’s how i got it (she looks at the interviewer with a dramatic pause). any questions? this pattern of practice is considered divergent because the teachers in the study reprimanded students who used dramatic or excessive movement while working or speaking. particularly during whole group sessions, students who moved around or talked exuberantly were redirected to sit down, sit still, quiet down, and so forth, because the behavior diverged from the expectation of being a quiet classroom participant and not disturbing other students from learning. thus, most of the events where students displayed dramatic expression were found outside of the view of the teacher in a setting where disciplinary action would not be taken. when engaged in dramatic and expressive speech and movement outside of the teacher’s view, they were less likely to be reprimanded and thus were able to engage in this behavior. as an example, jordan used dramatic action when interacting with mathematics tasks during interviews. he used verbal expressiveness— shouting “timber” when he dumped out a container of cubes—as well as movement expression—pretending to be deep in thought while quickly pacing back and forth when working on a difficult problem—throughout his interviews. he also moved around quite a bit during the end of his interviews; he stood up, lay on the floor, and walked around the interview area while working on the tasks. these dramatic examples and constant movements, however, did not deter jordan from solving the problems he was assigned to do; he still worked on the tasks and focused his attention on them. mathematical communication is an important part of standards-oriented teaching practices. the emphasis is on providing opportunities to verbally communicate and encourage students to develop a vocabulary that will allow them to engage in reasoning, justifying, conjecturing, and explaining; however, other ways in which students might communicate are not addressed in detail in standards-oriented practices. consider the african american dimension of orality, which is described as using oral communication to express feelings and emotions with movement and verve. expressing oneself in dramatic and vervistic ways departs from traditional norms of classroom interaction—raising your hand and waiting for your turn to speak, letting one person speak at a time, not embellishing your speech or dramatizing parts of your ideas—as the students in this study have waddell reform elementary mathematics journal of urban mathematics education vol. 3, no. 2 143 done. boykin and cunningham’s (2001) research on movement expressiveness of elementary african american children showed that the children had better retention of information when learning in an environment that was movement oriented as well as learning with materials that incorporated high movement themes. improvisation another practice that students enacted outside of the attention of the teacher was improvising. as defined, improvising occurred when students followed an activity or task in an unexpected or un-prescribed manner, but still engaged with the task in a mathematical way. studies on african american elementary students provided other examples of students creating alternative routes though activities as a way to complete them or make them personal (dyson, 1999; gadsden, 2001). in classrooms, improvising most often occurred when students had a limited understanding of the required activity. teachers modeled activities and held discussions about what was expected in an activity prior to sending the students off to work on the activity with their partner or group. some of the students may not have fully understood all the requirements for the activity but knew enough to create an alternative version of what was expected of them. in grade 1, kiana was playing a card game with royce in which the players had to choose, from an arrangement of 20 cards, two cards that summed up to 10. observation: once kiana finished arranging the cards, she said to royce, “go, what makes 10?” royce chose a 0 and a 10 and started to record it. the two had a brief argument over who should record, and finally agreed that the person who drew the cards should record the combination. royce then wrote 10+0=11 on her paper. when it was her turn, kiana dug through the pile of unused cards to find a seven. she replaced a six in the original arrangement with this seven card and said, “7 and 2.” she counted the pictures on the cards and realized that it is only nine. she tried 1, 5, and 3, in that order, all coupled with the seven card. each time she chose a new second card, she counted the pictures again, always starting with the seven card and counting each picture by ones. eventually, she found that 7+3=10, but she wrote on her paper, “5+5=10.” next it was royce’s turn. she also tried to dig through the pile of unused cards to find a specific card, but kiana got very angry and accused her of cheating. kiana not only took on the role of activity director but also improvised the rules to suit her needs of finding a combination of 10. throughout grades 1 and 2, kiana would often change the rules of an activity to match her mathematical understanding, but did not like others to do the same. as discussed by forman (2003), research on emergent goals in mathematics learning indicates that “children, especially while working in group settings, will establish their own priorities for problem solving” (p. 339) as they learn what it waddell reform elementary mathematics journal of urban mathematics education vol. 3, no. 2 144 means to work collaboratively on mathematical tasks. dyson’s (1999) work with african american children in an early literacy study also found students using the limited knowledge they held about a task in order to complete it, creating their own version of the final product. she focused on what students transferred from other tasks—the mechanics of tasks completion such as materials to be used, types of solution displays, and how to “talk” about your work—that helped them fill in missing information as the students then improvised and completed the current task. her view was that a negotiation needed to occur between the experiences students draw on and the experiences the teacher wished to provide in order to develop a common meaning of the proposed task. another form of improvisation occurred during the clinical interviews. students would be given a task to do by an interviewer and if they knew how to solve the problem, they might solve it in an unexpected way or if they were unsure of how to solve the problem, they might change the problem to a form a problem they could solve. what made the acts of improvisation during interviews intriguing was that although the interviewer was a proxy for the teacher, the students appeared to be less driven to hide their improvisation during interview tasks. an example of improvising or answering in an unexpected way was when jordan was asked to circle three numbers from 1 through 9 that totaled up to 15. interviewer: can you circle three numbers on this page that equal 15? jordan (uses marker, pauses and thinks for 5 to 6 seconds, then circles): i can’t circle four? interviewer: just three. (jordan circles three numbers, 9, 3, 2) how do you know they equal 15? jordan: doesn’t. interviewer: it doesn’t? jordan: uh-uh. interviewer: what does it equal? jordan: 14. here, jordan appeared to understand the task—he asked if he could circle 4 numbers instead of 3—but still completed the task by creating a sum of 14 rather than 15. these acts of improvising, although infrequent in total number, raised the question: why would a student engage with a task in an un-prescribed manner? is it a natural occurrence for young children? although forman (2003), drawing on the work of saxe and colleagues, points out that young children will modify or improvise an activity in an effort to simplify it, there was no mention of why students would improvise when it was clear they were mathematically able to accomplish the task as given. if considered from the african american cultural waddell reform elementary mathematics journal of urban mathematics education vol. 3, no. 2 145 dimension of expressive creativity, this practice could be construed as an attempt by the students to express their individualism, particularly in the interview setting where they were less likely to be sanctioned by the interviewer for deviating from the expected path. by creating an alternate, yet mathematically sound response to a problem, students gain control over their work and present themselves as proficient and creative problem solvers, which allows a student to “save face” and not be negatively labeled or seen in a stereotypical light (erikson & shultz, 1992; perry, steele, & hilliard, 2003). considering divergent student practices considering divergent practices forces a closer look at the frameworks students draw upon when interacting with classroom activity. for many african american students, african american cultural learning dimensions provide students with a framework that they may use to make sense of classroom activities and tasks, which at times is at odds with the expectations of the teachers and/or school. as teachers attempt to establish particular norms of behavior and interaction for students, the disconnect between students’ cultural understandings and the expected responses can, in some cases, make it seem as though the student is being willfully disobedient or is less competent than those students who are adhering to the norms. what is important to consider, however, is that students have an implicit knowledge of the meta-discursive rules of school. sfard (2000) suggested that the ideas of mutual orienting, patterns of interaction, and negotiation of obligations all fall under the idea of meta-discursive rules, which regulate the discourse in mathematics classrooms (and other classrooms) and the ways in which classroom community members speak and interact with each other without overt acknowledgment. in essence, students knew that their dramatic expressiveness and improvisation would be censured by the teachers, but still engaged in these practices. what would cause children at such a young age to engage in behaviors that could possibly get them into trouble? as these students worked on mathematics activities, they attempted to engage with mathematics in a way that made sense to them, which allowed them to participate in the activities and show that they are mathematically “sound.” in other words, they engaged in mathematical activities as a way to align their practices with the classroom goal of successful mathematics participation, as might occur with the african american cultural dimension of improvising, or to divert attention away from the fact that they might not be successful, as might occur with the dimension of dramatic expression. this idea is similar to the findings of nasir (2002) with african american children who aligned their practices to those of domino and basketball players as a way to engage with that community. in her study, even when the children did not know how to participate in parts of the waddell reform elementary mathematics journal of urban mathematics education vol. 3, no. 2 146 domino or basketball activity, they would find a way around their obstacle so they could be seen as successful participants. in this study, the students may have been attempting to create a mathematics identity that would be sufficient to allow them to engage and align with the mathematics practices of the classroom. nasir (2002) also described how many african american children, when sufficiently motivated in activities outside of school, engaged, imagined, and then aligned themselves as part of a community where they felt they could belong. she claimed that this process has not occurred in schools for african american students. this study suggests that, in elementary school, students do engage with mathematics, and attempt to align themselves with the practices in a way that makes sense to them. what is missing appears to be the level of imagination. did the students see themselves as being in the community of strong mathematicians? only one student in the study consistently stated that he believed himself to be a good mathematics student. the other six students, when asked whom they knew was good at mathematics, often named white boys in their classrooms. the students gave very similar reasons for why these boys were considered good mathematics students: they could answer the teacher’s questions quickly, they studied and practiced at home and over the summer (or so the students thought), and had mathematics tricks that help them get the correct answer. considering the racial demographics of carter school, the fact that african american students outnumbered european american students in every classroom at a ratio of at least 4 to 1, and the presence of some very bright african american students in each classroom (consider maya and samuel of this study), it was a surprising finding that an african american student was mentioned only once. the students’ choices, however, reflect the ideas that society holds as a whole regarding who is a good mathematician (burton, 1994). it would appear that the students held a view of a good mathematician that they did not reflect and thus could not imagine for themselves. this lack of imagination could play an important role in the increasing disidentification noted in many african american students as they progressed through mathematics in school (erikson & shultz, 1992; martin, 2000; nasir, 2002; osbourne, 1997). nasir (2002) stated: the importance of imagination in this process offers evidence that becoming is more than just what one does as a participant. it also includes the meanings one makes of that participation. children’s ability to imagine (and the affordances for such imagination in practices) their own learning trajectories and their place in relation to others is critical to the development of new goals and the access to new identities. (p. 241) although the teachers in this study did work toward breaking down the traditional view of mathematics through their enactment of standard-oriented mathematics practices, it is clear that the current societal view of what it means to be waddell reform elementary mathematics journal of urban mathematics education vol. 3, no. 2 147 be successful in mathematics is still a barrier for increasing the successful participation of many african american students. by expanding the definition of who is considered a strong mathematics student to include the ability to explain solutions or create multiple solutions to a problem, more students may begin to imagine themselves as good mathematics students, capable of succeeding in mathematics. this expansion appears to be a difficult task, as the teachers in this study developed classrooms that reflected a less traditional view of what it means to do mathematics but the students still continued to adhere to the beliefs that reflected a narrow view of mathematics success. although the divergent practices of the students do relate to some of the dimensions of african american culture, they could also be explained through research that explored how lower socioeconomic children interact with the world. lareau (2003), in her book unequal childhoods discussed what she called “the accomplishment of natural growth” (p. 238) to describe the development of children in working-class and poor (i.e., low-ses) families. she found that these children were raised with similar parenting practices that led to certain ways of interacting with the world. one characteristic she found was that working-class and poor children tended to spend time creating and recreating their own games and activities for play with other children, unlike middle-class children who were more often engaged with organized, adult-led activities. these independent playing opportunities could lead the children to be more comfortable with being inventive when engaging in games, similar to improvising that was noted in this study. moreover, lareau found that low-income children were more responsible for their lives outside of the home; this led the children to become more selfreliant and believe in their ability to take care of themselves without adult help. being self-reliant encouraged the children to rely on their own personal knowledge as a tool for problem solving and decision-making. these findings from lareau’s study support the idea that ses can play a role in how students interact in classrooms and, in this case, the behaviors of low-income african american children. however, other studies also show that race also plays a part in student behavior and achievement in school (lee, 1998; roscigno, 1998; wilson, 1998). considering the work on african american cultural dimensions and how these dimensions take into consideration the historical evidence that african americans have generally been part of the lower economic station in american society, some coherence between the practices and habits of low-ses families and african american families could be expected. additionally, ses as well as race tend to explain only some of the differences in achievement and attainment for african americans in school (wilson, 1998) and, although more affluent african americans score higher on standardized tests and have higher grade point averages than less affluent african americans, they still score lower than students of other races (nettles, millet, & ready, 2003; steele, 1992; vars & bowen, 1998). although waddell reform elementary mathematics journal of urban mathematics education vol. 3, no. 2 148 considering ses when discussing the relevance of african american cultural dimensions in student behavior and achievement confounds the issues, it appears that both factors play a role in these students’ possibilities of achieving, and as such any practices that might support and enhance their success would be warranted. implications and conclusion in my analysis of the data, there appeared to be alignment with the practices the students enacted, african american cultural dimensions, and the research supporting those dimensions. the students enjoyed social interactions, talking and sharing, and connecting with others. they also used physical tools often and relied on them to help solve problems. dramatic expression and improvising relate to the idea of expressive creativity; students wanted to be individual and unique in their work and using dramatics or creating their own rules allowed for this individualization and uniqueness. more comparative and in-depth research that focuses closely on the interactions and practices of african american students in standards-oriented classrooms is needed to better understand the influence of african american cultural dimensions on their learning and achievement and how to use those dimensions to better support african american students’ learning and achievement. rather than looking at the divergent behaviors as social problems or behaviors needing remediation or punishment, teachers and schools could look at what can be learned from these behaviors that would enhance the academic achievement of students. as an example, the students desire to use physical tools and oral communication to support their thinking could lead teachers to make more efforts in bridging students’ learning of abstract mathematics concepts. by lengthening the time that students have to use tools to solve problems, and making more explicit connections between how the tool use bridges to symbolic representations, teachers will be drawing on students’ strengths while moving them toward deeper levels of mathematical understanding. other areas of divergent behavior might be more difficult to incorporate into classrooms, such as dramatic expressions, consistent oral communication, or improvisation. however, if teachers understood and used these behaviors as tools to support students’ learning in mathematics, african american students may have more success in learning mathematics and come to perceive themselves as mathematicians. studies that explore the surrounding activities that accompany divergent behaviors can provide clues to understanding how african american students think about the mathematics they are learning while engaged in what may, to some teachers, seem to be unrelated activities or actions. most of the students in this study did not appear able to imagine themselves waddell reform elementary mathematics journal of urban mathematics education vol. 3, no. 2 149 as good mathematics students. we are still left with the question: how can african american students see themselves as part of the mathematical community? breaking down the traditional view of what it means to do mathematics and who can succeed in it would be a step toward providing african american students with a vision of mathematics success that includes themselves. teachers need to be encouraged and supported to develop and enact explicit practices that work to directly break down the cultural barriers and traditions about who can be successful in mathematics. more in-depth studies of teachers’ practices over time in standards-oriented classrooms could help consider the effects of using cultural connections and resources when teaching african american students mathematics. candace, felix, jordan, kiana, maya, royce, and samuel represent the variety, complexity, and hope for african american students in our nation’s schools. although the vision and philosophy of standards-oriented practices in mathematics appears to be sound pedagogy, there is still much to learn about its implementation and effects on student achievement and attainment in mathematics. in the same way, african american cultural dimensions may not answer all questions about learning styles and preferences for learning by african american students, but by considering the ideas and research that support these dimensions, we are presented with opportunities to reflect on how teaching practices and school cultures affects the achievement of all students. acknowledgments the research reported in this article was funded in part by the national science foundation (grant no. rec-9875739). the views expressed are the author’s and are not necessarily shared by the grantors. references alternatives for rebuilding curriculum center. 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(1994). patterns of interaction and the culture of mathematics classrooms. in s. lerman (ed.), cultural perspectives on the mathematics classroom (pp. 149–168). dordrecht, the netherlands: kluwer. waddell reform elementary mathematics journal of urban mathematics education vol. 3, no. 2 153 appendix a enacted student patterns of practice compared to teacher practices and african american learning and cultural themes enacted student patterns, interactions, beliefs opportunity or space within classroom that allow or encourage pattern african american cultural dimension that supports pattern enacted teacher pattern practice that allow or encourage pattern student–teacher pattern coherence focused collaboration – students work on task at hand with little tangential or unrelated activities when working in small groups but occasionally needed teacher assistance in tasks small group work, teacher engaging as a support during independent or small group mathematical tasks, modeling of tasks for students prior to independent work, developing classroom norms of collaborative habits social affective harmonious small group/partner work teacher support modeling strategies and thinking convergent – students occasionally needed teacher support in understanding the task or the mathematics of the task when working independently active participation – students listen, watch, and engage in mathematics discussions without a noticeable number of other activities unrelated to mathematics discussion at hand whole group discussion about math topic of the day, developing norms around participation behaviors, modeling participation behaviors, lively energetic discussions social affective whole group discussions active listening questioning and probing convergent – students can actively participate but within set norms and limits of classroom physical tool use – students understood and used blocks, cubes, paper, rulers, calculators, shapes, number cards, number lines, pictures, graphs, drawings, and so on to solve problems and accomplish tasks small group work to collaborate on tool use, modeling the use of tools, access to a variety of tools and ideas nonverbal expressive creativity sharing and explaining ideas small group work whole group discussion means of communication modeling strategies and thinking convergent – students used tools effectively and regularly; sometimes would rely on concrete tools instead of moving toward more abstract tools direct oral communication – student speaks on the topic/task at hand without moving to a tangential or unrelated area when talking about mathematical ideas, tasks, or activities establishing norms during whole group discussion of listening to other classroom members, allowing opportunities to share ideas, modeling ways of responding and speaking social affective harmonious sharing and explaining ideas whole group discussions encourage active listening means of communication modeling strategies and thinking episodic convergence – listening and clear communication are important elements in classrooms; however, students sometimes used mathematical language that was limited or vague, did not always talk in a linear fashion, had false starts, retraced ideas; talked off-topic during speech activating personal knowledge – students used their personal background knowledge to understand and solve problems whole group discussions about tasks and activities; small group work, encouraging personal solutions to problems; sharing solutions in whole group setting harmonious sharing and explaining solutions small group/partner work whole group discussion means of communication supported divergence – students used their prior knowledge and ideas often but sometimes would privilege their ideas over the ideas presented in the problem or task; teachers neither actively engage these connections nor did they sanction them dramatic expression – students displayed movement while speaking; got up often while working; used body language and hand movements while working; used intonation, excitement, expression when speaking whole group discussion, small group work, using energetic activities, music and rhyme in activities; allowing movement throughout classroom often nonverbal expressive creativity small group/partner work whole group discussion divergent – although students engage in whole group discussion almost daily, they are not encouraged to be expressive or movement oriented in class; episodes of dramatic expression occur most during interviews cont. on next page waddell reform elementary mathematics journal of urban mathematics education vol. 3, no. 2 154 appendix a cont. improvisation – students worked on a task or activity by making new rules or changing the rules of prescribed activity, followed activities in unexpected manner while maintaining mathematical integrity of activity, created mathematically unexpected solution paths small group work time, student developed solutions; sharing of student solutions; allowing individualism in work habits and products expressive creativity harmonious sharing and explaining ideas small group/partner work whole group discussions divergent – classroom expectations were to follow norms and rules, improvising falls outside this realm; students tended to improvise out of the sight of teacher, although improvising was sometimes an attempt to continue working on an activity or tasks in the face of difficulties self-reliance – students believed in relying on oneself to understand and do mathematics independent and individual work time and space spirituality expressive creativity modeling strategies and thinking divergent – classrooms practices supported collaboration and interdependence, but students maintained this belief in spite of the classroom practices boys as mathematicians – students often professed the belief that boys, and most often white boys, were good at mathematics using male students examples during whole group discussions; using male students as partners for struggling students n/a n/a divergent – standardsoriented mathematics posits all students can be strong and capable mathematicians.